Microsoft Word - An Introduction to Basis Sets.doc

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1 An Introduction to Basis Sets: The "Holy Grail" of computational chemistry is the calculation of the molecular orbitals (MOs) for a given molecule. IF we can calculate the MOs for a molecule, THEN we can know lots of things about the molecule, including its: energy electron density electrostatic potential transition state (if any) frequency However, calculating MOs is not that easy. The computer, however, does this for us. But, you, the chemist, must tell the computer some information first. Input files for all of the major computational chemistry packages contain these three parts: 1. Geometry which includes o Bond lengths o Bond Angles o Dihedrals 2. Kind of Calculations: o Single point energy o Frequency o Transition state o Electron density o Electrostatic potential 3. Starting set of mathematics and approximations o Calculation method (molecular mechanics, semiempirical, or ab initio o Type(s) of approximation (Hartree-Fock, Moller-Plesset, etc.) o Basis Set Approximation The graphic below "captures" the essence of what is the responsibility of the chemist and what is the responsibility of the computer:

2 In this lab we will be looking at the third ingredient, basis sets. A molecular-orbital theory calculation is a mathematical expression of an electron in a molecule. Although there are many types of molecular-orbital functions, in this lab we will only look at the Slater Type Orbitals (STOs) and the Gaussian Type Orbitals (GTOs). Although there is not a major difference in these two methods when calculating small molecules, major discrepencies arise for larger molecules of 30 or more atoms. STOs require more calculating, which takes tremendous amounts of time, however their calculations have been found to be more accurate than GTOs. On the other hand, GTOs, although less accurate, are much faster to calculate than STOs. This forced scientists to compromise time or accuracy. Eventually, scientists realized that by adding several GTOs, they were able to mimic the STOs accuracy. In fact, as the number of GTOs used increased, the better they were able to model the STO equation. When using GTOs to model STOs, the new equations are given a new name. They are identified as STO-kG equations where k is a constant that represents the number of GTOs used. For instance, two common equations are the STO-3G and the STO-6G in which 3 and 6 GTOs are used respectively.

3 Objective for the "Comparing STOs and GTOs" Lab Activity: In this lab you will be required to create a spreadsheet that will allow you to compare a set of STOs and GTOs. This will be done for a hydrogen atom which has only one electron. You will make the assumption that the Slater Type Orbital answers are the most accurate and then attempt to model your Gaussian Type Orbitals to fit the STOs. Objectives for the "Comparing STOs and GTOs" Lab Activity: 1. To gain an understanding of the relationship between the size of the molecule and the CPU time. 2. Define a formula that calculates the number of basis functions required for molecules with an (x) number of carbons and a (y) number of hydrogens.

4 Background Reading for Basis Sets: Key Points The objective of computational chemistry is to be able to mathematically express the properties of molecules. One of the most important properties is that of the molecular orbitals. Scientist use something called a basis set to approximate these orbitals. There are two general categories of basis sets: Minimal Basis Sets A basis set that describes only the most basic aspects of the orbitals Extended Basis Sets A basis set that describes the orbitals in great detail In the most general sense, a basis set is a table of numbers which mathematically estimate where the electrons can be found. An American physicist named J. C. Slater developed algorithms by fitting linear least-squares to data which could easily be calculated. Suddenly, basis sets became much easier to express. Slater's equation is known as the Slater Type Orbital or (STO). The equation is given below. The general expression for a basis function is given as: (-alpha * r) Basis Function = N * e where: N = Normalization constant alpha = Orbital exponent r = Radius in angstroms This expression given as a Slater Type Orbital equation is:

5 At this point, the STO equation looks like a great solution to finding the basis set of a molecule. Although the STO equation is a wonderful approximation for the molecular orbitals, the problem lies in the calculations. You see, calculating the STO of a molecule requires enormous amounts of work. This uses way too much time, even for super computers. Instead, a scientist by the name of S. F. Boys developed a method of using a combination of Gaussian Type Orbitals in order to express the STO equation. The following is the Gaussian Type Orbital (GTO) equation. Notice that the difference between the STO and GTO is in the "r." The GTO squares the "r" so that the product of the gaussian "primitives" (original gaussian equations) is another gaussian. By doing this, we have an equation we can work with and so the equation is much easier. However, the price we pay is loss of accuracy. To compensate for this loss, we find that the more gaussian equations we combine, the more accurate our equation. In Basis Set Lab, you will be looking at a STO-3G basis set: The "STO" tells you that you are attempting to represent a STO equation for your molecule. The 3G tells you that in order to do this you are combining 3 gaussian primitives. In this case, STO-NG, the number "N" before the "G" is the number of gaussian primitives that are used to simulate the STO equation. Remember, the bigger your N, the more accurate your results. All basis set equations in the form STO-NG are considered to be "minimal" basis sets. (Remember our definition of minimal.) The "extended" basis sets, then, are the ones that consider the higher orbitals of the molecule. The way to think about extended basis sets is to think of cleaning the kitchen floor. First you must sweep it to get all the big pieces of dirt. Then you go back and mop it to get the little bits of grime. Then you go over again and wax it to make sure everything is cleaned off and the floor is shiny. That is what you are doing to the molecules. Using the minimal basis set is like sweeping the floor. You are only describing the big properties. Using the extended basis sets is like mopping and waxing. You are fine tuning your description. One way to mop up the little grime in your molecule descriptions is to use a method called "Split-Valence Basis Sets." The theory behind this method is that it would be extremely tedious to attempt calculating the expression for every single atomic orbital. Instead, by combining an

6 expression for an orbital larger and smaller than the one we are looking at, we can approximate the orbital we are interested in. This method entails combining two or more STO's in order to describe an orbital. The STO's differ only by the value of their exponent (zeta) and can be manipulated by a constant "d" so that you can size the orbital to what you are looking for. Now we need to wax the molecule. This is done by taking into account the Polarization effect. The polarization effect is the resulting effects when two atoms are brought close together. Taking into account the polarization effect means accounting for the d and f-shells by adding STOs of higher orbital angular momentum. The asterisk is the symbol used when taking into account polarization. One asterisk symbolizes accounting for the d-shell and two asterisks symbolizes the f-shell. If an asterisk is in parentheses symbolizes that polarization functions are added only to second-row elements. Standard polarization basis sets are 3-21G* and 6-31G*. It is important that you understand how to read each basis set. For example, let us consider the 6-311G* basis set. The 6 represents the 6 gaussian primitives used to calculate the s-shell, the 3 represents the number of GTOs for one of the sp-shells and each 1 represents the number of GTOs for the other two sp-shells. The * represents the consideration of the d-shell. Other standard basis sets are STO-3G, 3-21G, 3-21G*, 4-31G, 6-31G, 6-31G*, 6-311G, and 6-311G*. When dealing with anions, the use of diffuse functions is recommended. Indeed, this is the case when the electron density of a molecule is far removed from its nuclear center. Diffuse functions are also useful for systems in an excited state, for systems with low ionization potential, and systems with some significant negative charge attached. The presence of diffuse functions is symbolized by the addition of a plus sign, +, to the basis set designator: 6-31G+ or 6-31+G. Again, a second +, such as G, implies diffuse functions added to hydrogens. The use of doubly-diffuse basis sets is especially useful if you are working with hydrides. When choosing a basis set, choose the best basis set for your time limits. Remember, the better the basis set, the more time it will take to calculate! Now, let's look at a typical basis set printout. Here is a dump for H and C, showing data for the STO-2G basis set. You can see that for the hydrogen, 2 gaussian primitives are used to construct

7 the s orbital. The first value of is the orbital exponent, and the is the contraction coefficient. With carbon, you see the addition value, so the first is the orbital exponent, the second is the s-part of the sp-hybrid, and the third part is the p-part of the sp-hybrid. STO-2G BASIS="STO-2G" H 0 S **** C 0 S SP *************** Here is a dump of the 6-31G* for H and C. For H, there is only one valence electron, and it is represented by two orbitals, one constructed of 3 primitives and the other with 1 primitive. With C, the s-orbital is a core orbital, and is represented by 6 gaussian primitives. The sp-orbital, on the other hand, is a valence orbital, and is represented by two orbitals, one with 3 gaussians and the other with 1. Since this is a "*", or polarized basis set, notice that there is some "d" character to the carbon atom. 6-31G* BASIS="6-31G*" H 0 S S ****

8 C 0 S SP SP D **** The relationship between basis sets and accuracy is represented in the diagram below. Our ultimate goal is to calculate an answer to the Schroëdinger's Equation (right bottom corner). However, we are still a long way from being able to complete this calculation. Right now we are in the top left corner of the chart. In that first box, we are treating each electron independently of the others. As you move across to the right, you find calculations that account for the interactions of electrons. As you move down the column you find more complex and more accurate basis set calculations. You will only be expected to understand the shaded regions.

9 There are other trade-offs for using each type of basis set. The more complex basis sets are more accurate but, they use up a great deal of computing time. Whenever you run a computational chemistry calculation you will be using time on a computer. Normally, the computer that will be running the calculation will be one that is shared with many other people doing other calculations. Thus, it is important that you act responsibly when choosing which basis set to use. You should pick one that is efficient for your use. This means that you should consider how much time it will take to run the molecule and use the basis set that will run the fastest without compromising your desired level of accuracy.

10 "Comparing STOs and GTOs" Lab Procedure Procedure: For this lab, you will need access to a spreadsheet. Part I: Below is the equation for a Gaussian Type Orbital where alpha ( radius from the nucleus. ) is and r is the Data Entry: 1. At the top of the spreadsheet, label the first cell "alpha." In the next cell define the constant. 2. Leaving a few rows blank, identify one cell "Radius." This is where we will give the equation various values of r. 3. Beneath Radius, you are going to create a list of values from -3 to +3. Divide the values into 0.01 increments. In order to do this, you are going to use a "Fill" comand. First, type a -3 in the first cell beneath the "Radius" label. Next, highlight the -3 and all the cells down to at least the 600th line. In the menu bar, look under Edit to find the "Fill" command. You want to choose the "Series" option. This will then prompt you for the increment by which you want your column to increase. The default is set to 1, but you will type Click on OK. If your values do not reach +3, then repeat these steps until you reach this value. 4. Next to the Radius column, title the next cell "Gaussian" or "GTO." 5. Beneath this label you will need to enter you GTO function as given above. Remeber to identify alpha and the radius by their cell names and that pi=pi( ), e x =EXP(x). Hints: When referring to a cell, make sure you use, $, before the column or row if it remains constant.

11 The first value should be Use a "Fill" command, (this time use "Down" instead of "Series") in order to get values for all the radii. Graphing: 1. Highlight the lists "Radius" and "GTO" including labels. 2. In the toolbar, click on the button with a wand and bar graph. 3. Select the area on the spreadsheet where you want the graph to appear. 4. Follow the steps to create a line graph. Hints: In Excel, make sure you choose the XY scatter plot and the second type of line graph.

12 Part II: Now we will compare this approximation to the STO. Below is the STO equation. The "squiggle" is a Greek character standing for an orbital coefficient, similar to alpha in gaussian curves. For hydrogen, the "squiggle" (better known as "xi", is equal to Under alpha, type "xi." In the next cell, type the value. 2. Now, label the cell next to your GTO heading "Slater" or "STO." In the cell beneath the heading, enter the Slater equation. Fill down your column. 3. With the "Radius," "GTO," and "STO" columns highlighted, create a new graph to compare the GTO and STO calculations. Notice the major discrepency for small radii values. Also, notice how the approximations become similar for larger radii. Part III: Now we would like to use several gaussians to model the hydrogen orbital. To get all of the numerical constants required, we will have to use a basis set as a reference. These are stored in a large database, from which you can "order" a specific basis set. 1. Go to the Gaussian Basis Set Order Format Your browser should look like

13 this: 2. Order your constants: o Select the STO-3G basis set. By selecting the STO-3G we are requesting to receive constants for 3 gaussians. o Next type an h for hydrogen in the "Elements" blank. o Don't put in an !! o Click Submit. The result should be two columns of three numbers each. This is a basis set for hydrogen. The left list is alpha values, and the right list is coefficients which will be used to sum up the three gaussians. 3. Copy these constants so you can enter them into your spreadsheet. 4. At the top of your spreadsheet, next to the other constants, label each constant and define the value in the next cell. 5. Next to the STO column, put three new columns, labelled "g1," "g2," and "g3". 6. Under each new label, enter the gaussian equation using the values of alpha from the basis set and the radius as needed. 7. Fill down the column, and repeat the procedure for the other two columns. 8. Change the first number under Radius to be 0. When you do this, the whole list should change to go from 0 to 6 instead of -3 to 3. Also, the values in your gaussian lists should change, since we are computing them for different radii now. Check your values against those of your classmates to make sure you are all getting correct answers. ***Since we looked at the equations on both sides of the nucleus before, you should have observed that it is exactly the same on both

14 sides. This is important! We are only interested in the positive side of the graph now, since we know that both sides will be exactly the same (and since negative and positive do not have any true physical meaning here!).*** 9. Compare the answers from each new gaussian equation to those of the slater equation. Notice that no single one of these three gaussians is an approximation on its own; they must be combined. The way of combining them is called "Linear Combination of Atomic Orbitals" (LCAO). The other list of numbers in the basis set are coefficients for adding up the three gaussians. They are used in the following equation. 1. Create another column next to the three gaussians, labelled "LCAO". 2. Put the equation, referring to the appropriate constants, in the first cell. 3. Fill the column. 4. Create a graph of all three new gaussians and the LCAO. 5. To make this contrast clearer, we are going to examine several gaussian approximations and compare them to the STO. 6. Label three new columns for the two equations needed for a STO-2G model, and the STO-2G model itself. 7. Do the same for a STO-6G 8. Now go back to the order form for basis sets and find the basis sets, both alpha and d values, for STO-2G and STO-6G. 9. Put these values into your spreadsheet as before. 10. Set up each column with the neccessary equations. 11. In two new columns, use the LCAO equation for both of the 2G and 6G approximations. 12. Graph the 1 gaussian (the first one created), 2 G, 3 G, 6G, and STO all on a single graph. **DO NOT put all of the gaussians for each approximation on there, ONLY THE SUMMATIONS.**

15 Basis Sets, Functions, and CPU Time Lab Procedure: Procedure: Group Part: In order to conduct this lab, you will need a spreadsheet and a Waltz Interface. The molecules we are using in this lab are alkanes. They have the formula C n H 2n+2. Alkanes are made up of n carbons linked together in a line with hydrogens jutting off every carbon. All of the bonds in alkanes are single bonds. The entire class will be collecting data to use for analysis. Split up the molecules so that each group will run a geometry optimization for two molecules. The more molecules the class runs, the better your results. When dividing up the molecules, skip every third alkane in the sequence. (ex. CH 4, C 2 H 6, C 4 H 10 ) You should set your spreadsheet up as follows: You will use Waltz to run each molecule THREE times: once with an STO-3G, once with a 3-21G, and once with a 6-21G. Once your run has completed, you will need to extract two pieces of information from the output file, 1. The number of basis functions used. 2. The CPU time. (Remember that the CPU time is different from the Wall time. The CPU time is the time it took to complete the run on the computer. The Wall time is the time it took for you to get your answer back after you clicked the execute button.)

16 Class Part: 3. Once the class has collected all the data, compile it all into one spreadsheet. 4. Graph the number of basis functions versus the number of carbons in each molecule. Do this for each of the 3-21G and 6-21G basis sets. You should have two graphs. What do you observe about the shape of each graph? 5. Try to write two equations (one for each basis set) which give the number of basis functions (bf) for a certain number of carbons (c) plus a certain number of hydrogens (h). (Clue: Use systems of equations) y bf = (a*c) + (b*h) What are the coefficients (a) and (b) in this equation (both are integers)? 6. Let's test your equation. Calculate the number of basis sets needed to calculate the following molecules. 7. Now graph the CPU time, in seconds, versus the number of basis functions calculated. What relationship do you see? Describe what happens to the CPU time required as the molecule gets extremely large. 8. Try to predict what the CPU time will be for the molecules which haven't been calculated yet. Each group should take one molecule that had been skipped earlier and one that is larger than any that have already been tried. How accurate do you think your predictions are? Do you think that the prediction for a molecule between two that we calculated is any more or less accurate than the prediction for a molecule not in between two that we calculated? If so, why? Discuss the difference between these predictions with your group. Record your predictions. 9. Now, each group should run the calculations for their two molecules. Record the data and share the results with the class.

17 Questions for "Comparing STOs and GTOs" Lab Activity: Part I: 1. What do you notice about the graph on either side of the nucleus? 2. Are the two sides similar? 3. If so, what does that show? Part II: 1. What do you notice about the shape of the three gaussians? 2. How similar are they? 3. How similar is each one to the simple gaussian we first created? 4. How do they relate to the LCAO? Discuss these questions and compare results with you lab parter and classmates? Part III: Remembering that the STO is the most accurate approximation of the waveform: 1. How accurate is each of the gaussian approximations? 2. Which approximation is the most accurate? 3. Which one takes the most time to compute? 4. How does accuracy relate to number of gaussians used? 5. Which is more important, the time required or the accuracy? 6. How should we decide which approximation to use for a specific project? Questions for Basis Sets, Functions, and CPU Time Lab Activity: Class Part: 1. How accurate were your predictions about the number of basis functions? Were all of the predictions of basis functions equally accurate or inaccurate? 2. How accurate were your predictions about CPU time? Were all of the predictions of CPU time equally accurate or inaccurate? 3. Did the patterns of accuracy/inaccuracy fit what you expected? How so? Why or Why not? 4. Which predictions of CPU time were most accurate? Why? 5. Which predictions of CPU time were least accurate? Why? 6. How could you have made these predictions more accurate without doing calculations on a molecule for which n is larger that what we have already done? 7. Predicting values for data points which are inside a set of data that you already have is called interpolation. Predicting values for data points which are outside the data you have is called extrapolation. From your experiment, which one do you think gives more accurate predictions in general? Discuss why there is a difference in accuracy between the two types of predictions.

18 Background Reading for Basis Sets: As a quick review, remember that scientists are mainly interested in the properties of molecules. One characteristic of a molecule that explains a great deal about these properties is their molecular orbitals. Recall the following diagram regarding what the scientist must know in order to calculate the molecular orbitals. One of the three major decisions for the scientist is which basis set to use. There are two general categories of basis sets: Minimal basis sets a basis set that describes only the most basic aspects of the orbitals Extended basis sets a basis set with a much more detailed description Basis sets were first developed by J.C. Slater. Slater fit linear least-squares to data that could be easily calculated. The general expression for a basis function is given as: (-alpha * r) Basis Function = N * e where: N = normalization constant alpha = orbital exponent r = radius in angstroms This expression given as a Slater Type Orbital (STO) equation is:

19 Now it is important to remember that STO is a very tedious calculation. S.F. Boys came up with an alternative when he developed the Gaussian Type Orbital (GTO) equation: Notice that the difference between the STO and GTO is in the "r." The GTO squares the "r" so that the product of the gaussian "primitives" (original gaussian equations) is another gaussian. By doing this, we have an equation we can work with and so the equation is much easier. However, the price we pay is loss of accuracy. To compensate for this loss, we find that the more gaussian equations we combine, the more accurate our equation. All basis set equations in the form STO-NG (where N represents the number of GTOs combined to approximate the STO) are considered to be "minimal" basis sets. (Remember our definition of minimal.) The "extended" basis sets, then, are the ones that consider the higher orbitals of the molecule and account for size and shape of molecular charge distributions. There are several types of extended basis sets: Double-Zeta, Triple-Zeta, Quadruple-Zeta Split-Valence Polarized Sets Diffuse Sets Double-Zeta, Triple-Zeta, Quadruple-Zeta Previously with the minimal basis sets, we approximated all orbitals to be of the same shape. However, we know this is not true. The double-zeta basis set is important to us because it allows us to treat each orbital separately when we conduct the Hartree-Fock calculation. This gives us a more accurate representation of each orbital. In order to do this, each atomic orbital is expressed as the sum of two Slater-type orbitals (STOs). The two equations are the same except for the value of (zeta). The zeta value accounts for how diffuse (large) the orbital is. The two STOs are then added in some proportion. The constant 'd' determines how much each STO will count towards the final orbital. Thus, the size of the atomic orbital can range anywhere between the

20 value of either of the two STOs. For example, let's look at the following example of a 2s orbital: In this case, each STO represents a different sized orbital because the zetas are different. The 'd' accounts for the percentage of the second STO to add in. The linear combination then gives us the atomic orbital. Since each of the two equations are the same, the symmetry remains constant. The triple and quadruple-zeta basis sets work the same way, except use three and four Slater equations instead of two. The typical trade-off applies here as well, better accuracy...more time/work. Split-Valence Often it takes too much effort to calculate a double-zeta for every orbital. Instead, many scientists simplify matters by calculating a double-zeta only for the valence orbital. Since the inner-shell electrons aren't as vital to the calculation, they are described with a single Slater Orbital. This method is called a split-valence basis set. A few examples of common split-valence basis sets are 3-21G, 4-31G, and 6-31G. An example is given below. It is strongly encouraged that you take part of the class time to walk through this example with your class. It will be a tremendous help to the students understanding of this subject. Here we are using a 3-21G basis set to calculate a carbon atom. This means we are summing 3 gaussians for the inner shell orbital, two gaussians for the first STO of the valence orbital and 1 gaussian for the second STO. This is the output file from the gaussian Basis Set Order Form for carbon given a 3-21G basis set.

21 Here is another common method of displaying data. Notice the numbers are labeled so it is easy to match this data with the corresponding data in the output file. Once you have retrieved a basis set output file, you can use these numbers to calculate your equations. For a carbon, you will need three equations: 1s orbital, 2s orbital, and 2p orbital. This equation combines the 3 GTO orbitals that define the 1s orbital. This equation combines the 2 GTO orbitals that make up the first STO of the double-zeta, plus the 1 GTO that represents the second STO for the 2s orbital.

22 This equation combines the 2 GTO orbitals that make up the first STO of the double-zeta, plus 1 GTO that represents the second STO for the 2p orbital. Now, using these three equations, we can calculate the LCAO for the carbon atom. Polarized Sets In the previous basis sets we have looked at, we treated atomic orbitals as existing only as 's', 'p', 'd', 'f' etc. Although those basis sets are good approximations, a better approximation is to acknowledge and account for the fact that sometimes orbitals share qualities of 's' and 'p' orbitals or 'p' and 'd', etc. and not necessarily have characteristics of only one or the other. As atoms are brought close together, their charge distribution causes a polarization effect (the positive charge is drawn to one side while the negative charge is drawn to the other) which distorts the shape of the atomic orbitals. In this case, 's' orbitals begin to have a little of the 'p' flavor and 'p' orbitals begin to have a little of the 'd' flavor. One asterisk (*) at the end of a basis set denotes that polarization has been taken into account in the 'p' orbitals. Notice in the graphics below the difference between the representation of the 'p' orbital for the 6-31G and the 6-31G* basis sets. The polarized basis set represents the orbital as more than just 'p', by adding a little 'd'. Original 'p' orbital

23 Modified 'p' orbital Two asterisks (**) means that polarization has taken into account the 's' orbitals in addition to the 'p' orbitals. Below is another illustration of the difference of the two methods. Original 's' orbital Modified 's' orbital

24 Diffuse Sets In chemistry, we are mainly concerned with the valence electrons which interact with other molecules. However, many of the basis sets we have talked about previously concentrate on the main energy located in the inner shell electrons. This is the main area under the wave function curve. In the graphic below, this area is that to the left of the red dotted line. Normally the tail (the area to the right of the dotted line), is not really a factor in calculations. However, when an atom is in an anion or in an excited state, the loosely bond electrons, which are responsible for the energy in the tail of the wave function, become much more important. To compensate for this area, computational scientists use diffuse functions. These basis sets utilize very small exponents to clarify the properties of the tail. Diffuse basis sets are represented by the '+' signs. One '+' means that we are accounting for the 'p' orbitals, while '++' signals that we are looking at both 'p' and 's' orbitals, (much like the asterisks in the polarization basis sets). The tradeoff/relationship between basis sets and accuracy is represented in the diagram below. Our ultimate goal is to calculate an answer to the Schroëdinger's Equation (right bottom corner). However, we are still a long way from being able to complete this calculation. Right now we are

25 in the top left corner of the chart. In that first box, we are treating each electron independently of the others. As you move across to the right, you find calculations that account for the interactions of electrons. As you move down the column you find more complex and more accurate basis set calculations. Students will only be expected to understand the shaded regions. There are other trade-offs for using each type of basis set. The more complex basis sets are more accurate but, they use up a great deal of computing time. Whenever you run a computational chemistry calculation you will be using time on a computer. Normally, the computer that will be running the calculation will be one that is shared with many other people doing other calculations. Thus, it is important that you act responsibly when choosing which basis set to use. You should pick one that is efficient for your use. This means that you should consider how much time it will take to run the molecule and use the basis set that will run the fastest without compromising your desired level of accuracy.

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27 An Introduction to Basis Sets: The "Holy Grail" of computational chemistry is the calculation of the molecular orbitals (MOs) for a given molecule. IF we can calculate the MOs for a molecule, THEN we can know lots of things about the molecule, including its: energy electron density electrostatic potential transition state (if any) frequency However, calculating MOs is not that easy. The computer, however, does this for us. But, you, the chemist, must tell the computer some information first. Input files for all of the major computational chemistry packages contain these three parts: 1. Geometry which includes o Bond lengths o Bond Angles o Dihedrals 2. Kind of Calculations: o Single point energy o Frequency o Transition state o Electron density o Electrostatic potential 3. Starting set of mathematics and approximations o Calculation method (molecular mechanics, semiempirical, or ab initio o Type(s) of approximation (Hartree-Fock, Moller-Plesset, etc.) o Basis Set Approximation The graphic below "captures" the essence of what is the responsibility of the chemist and what is the responsibility of the computer:

28 In this lab we will be looking at the third ingredient, basis sets. A molecular-orbital theory calculation is a mathematical expression of an electron in a molecule. Although there are many types of molecular-orbital functions, in this lab we will only look at the Slater Type Orbitals (STOs) and the Gaussian Type Orbitals (GTOs). Although there is not a major difference in these two methods when calculating small molecules, major discrepencies arise for larger molecules of 30 or more atoms. STOs require more calculating, which takes tremendous amounts of time, however their calculations have been found to be more accurate than GTOs. On the other hand, GTOs, although less accurate, are much faster to calculate than STOs. This forced scientists to compromise time or accuracy. Eventually, scientists realized that by adding several GTOs, they were able to mimic the STOs accuracy. In fact, as the number of GTOs used increased, the better they were able to model the STO equation. When using GTOs to model STOs, the new equations are given a new name. They are identified as STO-kG equations where k is a constant that represents the number of GTOs used. For instance, two common equations are the STO-3G and the STO-6G in which 3 and 6 GTOs are used respectively.

29 Objective for the "Comparing STOs and GTOs" Lab Activity: In this lab you will be required to create a spreadsheet that will allow you to compare a set of STOs and GTOs. This will be done for a hydrogen atom which has only one electron. You will make the assumption that the Slater Type Orbital answers are the most accurate and then attempt to model your Gaussian Type Orbitals to fit the STOs. Objectives for the "Comparing STOs and GTOs" Lab Activity: 1. To gain an understanding of the relationship between the size of the molecule and the CPU time. 2. Define a formula that calculates the number of basis functions required for molecules with an (x) number of carbons and a (y) number of hydrogens.

30 Background Reading for Basis Sets: Key Points The objective of computational chemistry is to be able to mathematically express the properties of molecules. One of the most important properties is that of the molecular orbitals. Scientist use something called a basis set to approximate these orbitals. There are two general categories of basis sets: Minimal Basis Sets A basis set that describes only the most basic aspects of the orbitals Extended Basis Sets A basis set that describes the orbitals in great detail In the most general sense, a basis set is a table of numbers which mathematically estimate where the electrons can be found. An American physicist named J. C. Slater developed algorithms by fitting linear least-squares to data which could easily be calculated. Suddenly, basis sets became much easier to express. Slater's equation is known as the Slater Type Orbital or (STO). The equation is given below. The general expression for a basis function is given as: (-alpha * r) Basis Function = N * e where: N = Normalization constant alpha = Orbital exponent r = Radius in angstroms This expression given as a Slater Type Orbital equation is:

31 At this point, the STO equation looks like a great solution to finding the basis set of a molecule. Although the STO equation is a wonderful approximation for the molecular orbitals, the problem lies in the calculations. You see, calculating the STO of a molecule requires enormous amounts of work. This uses way too much time, even for super computers. Instead, a scientist by the name of S. F. Boys developed a method of using a combination of Gaussian Type Orbitals in order to express the STO equation. The following is the Gaussian Type Orbital (GTO) equation. Notice that the difference between the STO and GTO is in the "r." The GTO squares the "r" so that the product of the gaussian "primitives" (original gaussian equations) is another gaussian. By doing this, we have an equation we can work with and so the equation is much easier. However, the price we pay is loss of accuracy. To compensate for this loss, we find that the more gaussian equations we combine, the more accurate our equation. In Basis Set Lab, you will be looking at a STO-3G basis set: The "STO" tells you that you are attempting to represent a STO equation for your molecule. The 3G tells you that in order to do this you are combining 3 gaussian primitives. In this case, STO-NG, the number "N" before the "G" is the number of gaussian primitives that are used to simulate the STO equation. Remember, the bigger your N, the more accurate your results. All basis set equations in the form STO-NG are considered to be "minimal" basis sets. (Remember our definition of minimal.) The "extended" basis sets, then, are the ones that consider the higher orbitals of the molecule. The way to think about extended basis sets is to think of cleaning the kitchen floor. First you must sweep it to get all the big pieces of dirt. Then you go back and mop it to get the little bits of grime. Then you go over again and wax it to make sure everything is cleaned off and the floor is shiny. That is what you are doing to the molecules. Using the minimal basis set is like sweeping the floor. You are only describing the big properties. Using the extended basis sets is like mopping and waxing. You are fine tuning your description. One way to mop up the little grime in your molecule descriptions is to use a method called "Split-Valence Basis Sets." The theory behind this method is that it would be extremely tedious to attempt calculating the expression for every single atomic orbital. Instead, by combining an

32 expression for an orbital larger and smaller than the one we are looking at, we can approximate the orbital we are interested in. This method entails combining two or more STO's in order to describe an orbital. The STO's differ only by the value of their exponent (zeta) and can be manipulated by a constant "d" so that you can size the orbital to what you are looking for. Now we need to wax the molecule. This is done by taking into account the Polarization effect. The polarization effect is the resulting effects when two atoms are brought close together. Taking into account the polarization effect means accounting for the d and f-shells by adding STOs of higher orbital angular momentum. The asterisk is the symbol used when taking into account polarization. One asterisk symbolizes accounting for the d-shell and two asterisks symbolizes the f-shell. If an asterisk is in parentheses symbolizes that polarization functions are added only to second-row elements. Standard polarization basis sets are 3-21G* and 6-31G*. It is important that you understand how to read each basis set. For example, let us consider the 6-311G* basis set. The 6 represents the 6 gaussian primitives used to calculate the s-shell, the 3 represents the number of GTOs for one of the sp-shells and each 1 represents the number of GTOs for the other two sp-shells. The * represents the consideration of the d-shell. Other standard basis sets are STO-3G, 3-21G, 3-21G*, 4-31G, 6-31G, 6-31G*, 6-311G, and 6-311G*. When dealing with anions, the use of diffuse functions is recommended. Indeed, this is the case when the electron density of a molecule is far removed from its nuclear center. Diffuse functions are also useful for systems in an excited state, for systems with low ionization potential, and systems with some significant negative charge attached. The presence of diffuse functions is symbolized by the addition of a plus sign, +, to the basis set designator: 6-31G+ or 6-31+G. Again, a second +, such as G, implies diffuse functions added to hydrogens. The use of doubly-diffuse basis sets is especially useful if you are working with hydrides. When choosing a basis set, choose the best basis set for your time limits. Remember, the better the basis set, the more time it will take to calculate! Now, let's look at a typical basis set printout. Here is a dump for H and C, showing data for the STO-2G basis set. You can see that for the hydrogen, 2 gaussian primitives are used to construct

33 the s orbital. The first value of is the orbital exponent, and the is the contraction coefficient. With carbon, you see the addition value, so the first is the orbital exponent, the second is the s-part of the sp-hybrid, and the third part is the p-part of the sp-hybrid. STO-2G BASIS="STO-2G" H 0 S **** C 0 S SP *************** Here is a dump of the 6-31G* for H and C. For H, there is only one valence electron, and it is represented by two orbitals, one constructed of 3 primitives and the other with 1 primitive. With C, the s-orbital is a core orbital, and is represented by 6 gaussian primitives. The sp-orbital, on the other hand, is a valence orbital, and is represented by two orbitals, one with 3 gaussians and the other with 1. Since this is a "*", or polarized basis set, notice that there is some "d" character to the carbon atom. 6-31G* BASIS="6-31G*" H 0 S S ****

34 C 0 S SP SP D **** The relationship between basis sets and accuracy is represented in the diagram below. Our ultimate goal is to calculate an answer to the Schroëdinger's Equation (right bottom corner). However, we are still a long way from being able to complete this calculation. Right now we are in the top left corner of the chart. In that first box, we are treating each electron independently of the others. As you move across to the right, you find calculations that account for the interactions of electrons. As you move down the column you find more complex and more accurate basis set calculations. You will only be expected to understand the shaded regions.

35 There are other trade-offs for using each type of basis set. The more complex basis sets are more accurate but, they use up a great deal of computing time. Whenever you run a computational chemistry calculation you will be using time on a computer. Normally, the computer that will be running the calculation will be one that is shared with many other people doing other calculations. Thus, it is important that you act responsibly when choosing which basis set to use. You should pick one that is efficient for your use. This means that you should consider how much time it will take to run the molecule and use the basis set that will run the fastest without compromising your desired level of accuracy.

36 "Comparing STOs and GTOs" Lab Procedure Procedure: For this lab, you will need access to a spreadsheet. Part I: Below is the equation for a Gaussian Type Orbital where alpha ( radius from the nucleus. ) is and r is the Data Entry: 1. At the top of the spreadsheet, label the first cell "alpha." In the next cell define the constant. 2. Leaving a few rows blank, identify one cell "Radius." This is where we will give the equation various values of r. 3. Beneath Radius, you are going to create a list of values from -3 to +3. Divide the values into 0.01 increments. In order to do this, you are going to use a "Fill" comand. First, type a -3 in the first cell beneath the "Radius" label. Next, highlight the -3 and all the cells down to at least the 600th line. In the menu bar, look under Edit to find the "Fill" command. You want to choose the "Series" option. This will then prompt you for the increment by which you want your column to increase. The default is set to 1, but you will type Click on OK. If your values do not reach +3, then repeat these steps until you reach this value. 4. Next to the Radius column, title the next cell "Gaussian" or "GTO." 5. Beneath this label you will need to enter you GTO function as given above. Remeber to identify alpha and the radius by their cell names and that pi=pi( ), e x =EXP(x). Hints: When referring to a cell, make sure you use, $, before the column or row if it remains constant.

37 The first value should be Use a "Fill" command, (this time use "Down" instead of "Series") in order to get values for all the radii. Graphing: 1. Highlight the lists "Radius" and "GTO" including labels. 2. In the toolbar, click on the button with a wand and bar graph. 3. Select the area on the spreadsheet where you want the graph to appear. 4. Follow the steps to create a line graph. Hints: In Excel, make sure you choose the XY scatter plot and the second type of line graph.

38 Part II: Now we will compare this approximation to the STO. Below is the STO equation. The "squiggle" is a Greek character standing for an orbital coefficient, similar to alpha in gaussian curves. For hydrogen, the "squiggle" (better known as "xi", is equal to Under alpha, type "xi." In the next cell, type the value. 2. Now, label the cell next to your GTO heading "Slater" or "STO." In the cell beneath the heading, enter the Slater equation. Fill down your column. 3. With the "Radius," "GTO," and "STO" columns highlighted, create a new graph to compare the GTO and STO calculations. Notice the major discrepency for small radii values. Also, notice how the approximations become similar for larger radii. Part III: Now we would like to use several gaussians to model the hydrogen orbital. To get all of the numerical constants required, we will have to use a basis set as a reference. These are stored in a large database, from which you can "order" a specific basis set. 1. Go to the Gaussian Basis Set Order Format Your browser should look like

39 this: 2. Order your constants: o Select the STO-3G basis set. By selecting the STO-3G we are requesting to receive constants for 3 gaussians. o Next type an h for hydrogen in the "Elements" blank. o Don't put in an !! o Click Submit. The result should be two columns of three numbers each. This is a basis set for hydrogen. The left list is alpha values, and the right list is coefficients which will be used to sum up the three gaussians. 3. Copy these constants so you can enter them into your spreadsheet. 4. At the top of your spreadsheet, next to the other constants, label each constant and define the value in the next cell. 5. Next to the STO column, put three new columns, labelled "g1," "g2," and "g3". 6. Under each new label, enter the gaussian equation using the values of alpha from the basis set and the radius as needed. 7. Fill down the column, and repeat the procedure for the other two columns. 8. Change the first number under Radius to be 0. When you do this, the whole list should change to go from 0 to 6 instead of -3 to 3. Also, the values in your gaussian lists should change, since we are computing them for different radii now. Check your values against those of your classmates to make sure you are all getting correct answers. ***Since we looked at the equations on both sides of the nucleus before, you should have observed that it is exactly the same on both

40 sides. This is important! We are only interested in the positive side of the graph now, since we know that both sides will be exactly the same (and since negative and positive do not have any true physical meaning here!).*** 9. Compare the answers from each new gaussian equation to those of the slater equation. Notice that no single one of these three gaussians is an approximation on its own; they must be combined. The way of combining them is called "Linear Combination of Atomic Orbitals" (LCAO). The other list of numbers in the basis set are coefficients for adding up the three gaussians. They are used in the following equation. 1. Create another column next to the three gaussians, labelled "LCAO". 2. Put the equation, referring to the appropriate constants, in the first cell. 3. Fill the column. 4. Create a graph of all three new gaussians and the LCAO. 5. To make this contrast clearer, we are going to examine several gaussian approximations and compare them to the STO. 6. Label three new columns for the two equations needed for a STO-2G model, and the STO-2G model itself. 7. Do the same for a STO-6G 8. Now go back to the order form for basis sets and find the basis sets, both alpha and d values, for STO-2G and STO-6G. 9. Put these values into your spreadsheet as before. 10. Set up each column with the neccessary equations. 11. In two new columns, use the LCAO equation for both of the 2G and 6G approximations. 12. Graph the 1 gaussian (the first one created), 2 G, 3 G, 6G, and STO all on a single graph. **DO NOT put all of the gaussians for each approximation on there, ONLY THE SUMMATIONS.**

41 Basis Sets, Functions, and CPU Time Lab Procedure: Procedure: Group Part: In order to conduct this lab, you will need a spreadsheet and a Waltz Interface. The molecules we are using in this lab are alkanes. They have the formula C n H 2n+2. Alkanes are made up of n carbons linked together in a line with hydrogens jutting off every carbon. All of the bonds in alkanes are single bonds. The entire class will be collecting data to use for analysis. Split up the molecules so that each group will run a geometry optimization for two molecules. The more molecules the class runs, the better your results. When dividing up the molecules, skip every third alkane in the sequence. (ex. CH 4, C 2 H 6, C 4 H 10 ) You should set your spreadsheet up as follows: You will use Waltz to run each molecule THREE times: once with an STO-3G, once with a 3-21G, and once with a 6-21G. Once your run has completed, you will need to extract two pieces of information from the output file, 1. The number of basis functions used. 2. The CPU time. (Remember that the CPU time is different from the Wall time. The CPU time is the time it took to complete the run on the computer. The Wall time is the time it took for you to get your answer back after you clicked the execute button.)

42 Class Part: 3. Once the class has collected all the data, compile it all into one spreadsheet. 4. Graph the number of basis functions versus the number of carbons in each molecule. Do this for each of the 3-21G and 6-21G basis sets. You should have two graphs. What do you observe about the shape of each graph? 5. Try to write two equations (one for each basis set) which give the number of basis functions (bf) for a certain number of carbons (c) plus a certain number of hydrogens (h). (Clue: Use systems of equations) y bf = (a*c) + (b*h) What are the coefficients (a) and (b) in this equation (both are integers)? 6. Let's test your equation. Calculate the number of basis sets needed to calculate the following molecules. 7. Now graph the CPU time, in seconds, versus the number of basis functions calculated. What relationship do you see? Describe what happens to the CPU time required as the molecule gets extremely large. 8. Try to predict what the CPU time will be for the molecules which haven't been calculated yet. Each group should take one molecule that had been skipped earlier and one that is larger than any that have already been tried. How accurate do you think your predictions are? Do you think that the prediction for a molecule between two that we calculated is any more or less accurate than the prediction for a molecule not in between two that we calculated? If so, why? Discuss the difference between these predictions with your group. Record your predictions. 9. Now, each group should run the calculations for their two molecules. Record the data and share the results with the class.

43 Questions for "Comparing STOs and GTOs" Lab Activity: Part I: 1. What do you notice about the graph on either side of the nucleus? 2. Are the two sides similar? 3. If so, what does that show? Part II: 1. What do you notice about the shape of the three gaussians? 2. How similar are they? 3. How similar is each one to the simple gaussian we first created? 4. How do they relate to the LCAO? Discuss these questions and compare results with you lab parter and classmates? Part III: Remembering that the STO is the most accurate approximation of the waveform: 1. How accurate is each of the gaussian approximations? 2. Which approximation is the most accurate? 3. Which one takes the most time to compute? 4. How does accuracy relate to number of gaussians used? 5. Which is more important, the time required or the accuracy? 6. How should we decide which approximation to use for a specific project? Questions for Basis Sets, Functions, and CPU Time Lab Activity: Class Part: 1. How accurate were your predictions about the number of basis functions? Were all of the predictions of basis functions equally accurate or inaccurate? 2. How accurate were your predictions about CPU time? Were all of the predictions of CPU time equally accurate or inaccurate? 3. Did the patterns of accuracy/inaccuracy fit what you expected? How so? Why or Why not? 4. Which predictions of CPU time were most accurate? Why? 5. Which predictions of CPU time were least accurate? Why? 6. How could you have made these predictions more accurate without doing calculations on a molecule for which n is larger that what we have already done? 7. Predicting values for data points which are inside a set of data that you already have is called interpolation. Predicting values for data points which are outside the data you have is called extrapolation. From your experiment, which one do you think gives more accurate predictions in general? Discuss why there is a difference in accuracy between the two types of predictions.

44 Background Reading for Basis Sets: As a quick review, remember that scientists are mainly interested in the properties of molecules. One characteristic of a molecule that explains a great deal about these properties is their molecular orbitals. Recall the following diagram regarding what the scientist must know in order to calculate the molecular orbitals. One of the three major decisions for the scientist is which basis set to use. There are two general categories of basis sets: Minimal basis sets a basis set that describes only the most basic aspects of the orbitals Extended basis sets a basis set with a much more detailed description Basis sets were first developed by J.C. Slater. Slater fit linear least-squares to data that could be easily calculated. The general expression for a basis function is given as: (-alpha * r) Basis Function = N * e where: N = normalization constant alpha = orbital exponent r = radius in angstroms This expression given as a Slater Type Orbital (STO) equation is:

45 Now it is important to remember that STO is a very tedious calculation. S.F. Boys came up with an alternative when he developed the Gaussian Type Orbital (GTO) equation: Notice that the difference between the STO and GTO is in the "r." The GTO squares the "r" so that the product of the gaussian "primitives" (original gaussian equations) is another gaussian. By doing this, we have an equation we can work with and so the equation is much easier. However, the price we pay is loss of accuracy. To compensate for this loss, we find that the more gaussian equations we combine, the more accurate our equation. All basis set equations in the form STO-NG (where N represents the number of GTOs combined to approximate the STO) are considered to be "minimal" basis sets. (Remember our definition of minimal.) The "extended" basis sets, then, are the ones that consider the higher orbitals of the molecule and account for size and shape of molecular charge distributions. There are several types of extended basis sets: Double-Zeta, Triple-Zeta, Quadruple-Zeta Split-Valence Polarized Sets Diffuse Sets Double-Zeta, Triple-Zeta, Quadruple-Zeta Previously with the minimal basis sets, we approximated all orbitals to be of the same shape. However, we know this is not true. The double-zeta basis set is important to us because it allows us to treat each orbital separately when we conduct the Hartree-Fock calculation. This gives us a more accurate representation of each orbital. In order to do this, each atomic orbital is expressed as the sum of two Slater-type orbitals (STOs). The two equations are the same except for the value of (zeta). The zeta value accounts for how diffuse (large) the orbital is. The two STOs are then added in some proportion. The constant 'd' determines how much each STO will count towards the final orbital. Thus, the size of the atomic orbital can range anywhere between the

46 value of either of the two STOs. For example, let's look at the following example of a 2s orbital: In this case, each STO represents a different sized orbital because the zetas are different. The 'd' accounts for the percentage of the second STO to add in. The linear combination then gives us the atomic orbital. Since each of the two equations are the same, the symmetry remains constant. The triple and quadruple-zeta basis sets work the same way, except use three and four Slater equations instead of two. The typical trade-off applies here as well, better accuracy...more time/work. Split-Valence Often it takes too much effort to calculate a double-zeta for every orbital. Instead, many scientists simplify matters by calculating a double-zeta only for the valence orbital. Since the inner-shell electrons aren't as vital to the calculation, they are described with a single Slater Orbital. This method is called a split-valence basis set. A few examples of common split-valence basis sets are 3-21G, 4-31G, and 6-31G. An example is given below. It is strongly encouraged that you take part of the class time to walk through this example with your class. It will be a tremendous help to the students understanding of this subject. Here we are using a 3-21G basis set to calculate a carbon atom. This means we are summing 3 gaussians for the inner shell orbital, two gaussians for the first STO of the valence orbital and 1 gaussian for the second STO. This is the output file from the gaussian Basis Set Order Form for carbon given a 3-21G basis set.

47 Here is another common method of displaying data. Notice the numbers are labeled so it is easy to match this data with the corresponding data in the output file. Once you have retrieved a basis set output file, you can use these numbers to calculate your equations. For a carbon, you will need three equations: 1s orbital, 2s orbital, and 2p orbital. This equation combines the 3 GTO orbitals that define the 1s orbital. This equation combines the 2 GTO orbitals that make up the first STO of the double-zeta, plus the 1 GTO that represents the second STO for the 2s orbital.

48 This equation combines the 2 GTO orbitals that make up the first STO of the double-zeta, plus 1 GTO that represents the second STO for the 2p orbital. Now, using these three equations, we can calculate the LCAO for the carbon atom. Polarized Sets In the previous basis sets we have looked at, we treated atomic orbitals as existing only as 's', 'p', 'd', 'f' etc. Although those basis sets are good approximations, a better approximation is to acknowledge and account for the fact that sometimes orbitals share qualities of 's' and 'p' orbitals or 'p' and 'd', etc. and not necessarily have characteristics of only one or the other. As atoms are brought close together, their charge distribution causes a polarization effect (the positive charge is drawn to one side while the negative charge is drawn to the other) which distorts the shape of the atomic orbitals. In this case, 's' orbitals begin to have a little of the 'p' flavor and 'p' orbitals begin to have a little of the 'd' flavor. One asterisk (*) at the end of a basis set denotes that polarization has been taken into account in the 'p' orbitals. Notice in the graphics below the difference between the representation of the 'p' orbital for the 6-31G and the 6-31G* basis sets. The polarized basis set represents the orbital as more than just 'p', by adding a little 'd'. Original 'p' orbital

49 Modified 'p' orbital Two asterisks (**) means that polarization has taken into account the 's' orbitals in addition to the 'p' orbitals. Below is another illustration of the difference of the two methods. Original 's' orbital Modified 's' orbital

50 Diffuse Sets In chemistry, we are mainly concerned with the valence electrons which interact with other molecules. However, many of the basis sets we have talked about previously concentrate on the main energy located in the inner shell electrons. This is the main area under the wave function curve. In the graphic below, this area is that to the left of the red dotted line. Normally the tail (the area to the right of the dotted line), is not really a factor in calculations. However, when an atom is in an anion or in an excited state, the loosely bond electrons, which are responsible for the energy in the tail of the wave function, become much more important. To compensate for this area, computational scientists use diffuse functions. These basis sets utilize very small exponents to clarify the properties of the tail. Diffuse basis sets are represented by the '+' signs. One '+' means that we are accounting for the 'p' orbitals, while '++' signals that we are looking at both 'p' and 's' orbitals, (much like the asterisks in the polarization basis sets). The tradeoff/relationship between basis sets and accuracy is represented in the diagram below. Our ultimate goal is to calculate an answer to the Schroëdinger's Equation (right bottom corner). However, we are still a long way from being able to complete this calculation. Right now we are

51 in the top left corner of the chart. In that first box, we are treating each electron independently of the others. As you move across to the right, you find calculations that account for the interactions of electrons. As you move down the column you find more complex and more accurate basis set calculations. Students will only be expected to understand the shaded regions. There are other trade-offs for using each type of basis set. The more complex basis sets are more accurate but, they use up a great deal of computing time. Whenever you run a computational chemistry calculation you will be using time on a computer. Normally, the computer that will be running the calculation will be one that is shared with many other people doing other calculations. Thus, it is important that you act responsibly when choosing which basis set to use. You should pick one that is efficient for your use. This means that you should consider how much time it will take to run the molecule and use the basis set that will run the fastest without compromising your desired level of accuracy.

52

53 Notes on the Density Matrix Theory Xihua Chen Department of Chemistry, New York University, New York, NY 10003, USA Summer, 2005 Contents 1 Density Operators Electron density The th-order density matrix Density operator Ensemble density operator Reduced density matrix The second-order density matrix The first-order density matrix representability problem Spinless Density Matrices Second-order spinless density matrix First-order spinless density matrix Energy with spinless density matrices Exchange-correlation hole

54 C C + C P PO C P Some basics of elementary quantum mechanics, in particular, the concepts of electron density and density matrix that are fundamental to the Density Functional Theory (DFT) methods are summarized. 1 Density Operators 1.1 Electron density Let s consider a molecular system which is composed of atomic nuclei and electrons. The geometrical structure of the molecule is defined by the positions of nuclei which are positively charged with. Within the Born- Oppenheimer approximation and non-relativistically, the stationary state of the molecule is completely specified by an antisymmetric electronic wave function (or state function),! " # $%, which belongs to a state space consisting of all possible states of the system and depends on the coordinates of nuclei, & ' () ( (as parameters according to the Born-Oppenheimer approximation), and the full coordinates of electrons, * +,-. / " +0 $ is solved from the Schrödinger equation of electrons, 7! " 3 8:9<;,=>;,?@5 (1.1) in which the electronic Hamiltonian is 7 BA +ED/F GIHKJ +ML +ED/F CNO " N + A C +ED/F The total energy of the system with the fixed nuclei is 91RES,R%:9<;,=>;,? LUT%V!V B9<;,=>;,? L 1D/F C W 1D/F 8 " +Q (1.2) X / W & W (1.3) According to the probability interpretation of wave function, we call 2Y5 Z2J the probability distribution function, 2Y *2J [2Y! " $ Z2J :MF\ ] J_^`^`^ <6a*5MF\ ] JX^`^`^ (1.4) and note that 2Y5 Z2Jb " 2Y! " $ Z2Jcb 2 " (1.5)

55 t t t is the probability of finding the system with electrons at a spatial region between " and " L b " with spin coordinates of $. We define electron density as the number of electrons per unit volume. For a given state of an -electron system, d! " F]8 (e 2f Z2Jgb $hf b Ji^j^`^ b (1.6) " and the Note is a one-electron function of only three spatial varibles, integration over all space results in the total number of electrons, e d! " F] b " Fk (e 2Y5 *2Jgb MF b Ji^`^`^ b (1.7) 1.2 The l th-order density matrix Mathematically, we can define an abstract Hibert space which consists of all - electron wave functions (vector functions). Then the ket of 5 is the coordinate representation of, 2f 5 3mX)noMFp JI^`^`^ 2f m (1.8) and the conjugate bra of 5 is n, *2 )n0 2MFp Jq^j^`^ m (1.9) The probability distribution function of Eq. (1.4), 2f Z2J )n, Z2f ]mxrsf\ ] JX^`^`^ < a MF\ 3 JX^`^`^ is a special case of the more general quantity, %u ] 8)n05%u Z2Y5 ]mxrhuf ]%uj ^`^`^ %u < a MF 3 J ^j^`^ (1.10) when u v, that is, uf BsF ] uj B J ^j^`^ u v. The quantity t u ] defined in Eq. (1.10) depends on two set of electron coordinates, * u and *, each settings of which determines a definite value. We thus can consider t u ] as the definition of the element of a so called density matrix or the th order density matrix. Therefore the probability distribution function 2Y *2J gives an diagonal element of the density matrix. From the definition, Eq. (1.13), w is positive semidefinite and Hermitian, 5 ] yx{z} t 5 u ] 8 t a 5 3 u (1.11) 3

56 t t tw t t 1.3 Density operator The density matrix element of Eq. (1.10) can be written as 5%u ] X~5%uF ]%uj ^`^`^ %u < a sf ] J_^^`^ n uf uj ^`^`^ u 2Y mn0 2MF Jq^`^^ m n uf uj ^`^`^ u 2w 2MF Jq^`^^ m (1.12) Here we have defined a projection operator, the density operator by, [2f m!n0 2 (1.13) We see the density matrix element, Eq. (1.10), is just the coordinate representation of the density operator. 1. The normalization of the wave function is equivalent to the unitization of the trace of the density matrix, n, *2Y ]mx e 5 36a 5 b e 5 u ]6a 5 s2!ƒ D b e nohuf huj ^`^^ %u 2Y mn0 2MFp JI^`^`^ mm2!ƒf D * b e no uf uj ^`^^ u 2ow 2MFp JI^`^`^ mm2!ƒf D * b " w (1.14) 2. Recall that the expectation values of the operator w which are Hermitian linear for an oberservable, in terms of wave function (which is assumed to be normalized) is n w ]mx n, 5 6*2 ˆ Z2Y w 3m n0 Z2f m5 Šn0 5 *2 w Z2f 5 3m (1.15) Now we would like to express the expectation values in terms of density 4

57 tw t t tw t t t t t t t operator, n w ]m_ n0 Z2 w *2Y 5 ]m e b no uf uj ^`^^ u 2Y m 5 w!n0 2MF Ji^`^^ m e b no uf uj ^`^^ u 2Y m 5 w!n0 2MF Ji^`^^ mz2!ƒ D e b nohuf huj ^`^^ %u 2 ˆ w Z2f mn, 2MFp Ji^`^`^ mœ2!ƒf D * e b no uf uj ^`^^ u 2 ˆ w w 2MF Ji^`^^ mz2!ƒ D " 5 w jw X{ " w 5 w 3 Now we have shown that Eq. (1.15) is just the coordinate representation of n mxb w " w w _v "!w w (1.16) is unique and contains all information. A quantum state that can be descibed by a wave function is said to be a pure state. Otherwise, it is called a mixed state in a sense that it can access a pool of pure states. The necessary and suffcient condition of a pure state is that the density matrix is idempotent, that is, ^ w [2Y mn0 2Y mn0 2jŽw (1.17) 3. Given an -electron state function 2Y m, w 4. The concepts of density matrix and density operator are applicable to timedependent problem. Here the wave functions are solutions to the timedependent Schrödinger equation, 4 1 And the density operator is defined as in Eq. (1.13), The commutor 7 pw t 2Y m_ 7 2f m (1.18) [2Y mn0 2 8 is found to be, 7 `w k 4@ 1 5 tw (1.19)

58 C w 7 t t t t 7 t since tw 1 2Y 1 mn0 2Y 2Y 1 m] šn0 2 L 2Y m n0 1 2f 4@ 2Y mn0 2 L 2f mn, 2Q A 4 4@ 7 w Aw w 8 Obviously, for a stationary state, and w commute, 7 w 8 z} (1.20) sotw has the same eigenfunctions of Ensemble density operator 7 A mixed quantum state (with a Hamiltonian ) can t be described by vector functions 7 of the state space or, in other words, by a linear superposition of eigenstates of. For example, in a many-electron system, it is impossible to write a one-electron Hamiltonian that doesn t depend on the degrees of freedom of other electrons. In this case, a single wave description of an individual electron doesn t exist. It is necessary to use a proper density operator to describe the electron s quantum state. Although a mixed 7 quantum state can t be described by a linear superposition of pure eigenstates of, it can be described by a probability distribution over all accessible pure states. Assume the set (ensemble) of all accessible pure states of a mixed state is +. We shall define an ensemble density operator, œ C +ž +}w t +/ C +š +32Y +om!n0 +]2 (1.21) in which +'x{zÿ + is the probability of the system to be found at the4 th statey+ and +š +hš sum over all accessible pure states. (1.22) 6

59 œw t œ C N N N tw N tw œ œ N N t N N t w 1. The ensemble density matrix operator defined by Eq. (1.21) is the ensemble generalization of the pure state density operator by Eq. (1.13). When the system is a pure state, the former reduces to the latter. As the pure state density operator w is normalized by Eq. (1.14), the ensemble density operator is also normalized, " œw 8B " C +pw t +58 C +* " <w t +5X C +%( (1.23) The ensemble generalization of the expectation values for an observable, given in Eq. (1.16) for a pure state, is, n m_ w C +ž + n0 +]2 I2Y +5m w C +ž + + " qw w t +5X{ " œw 8B w " w œw (1.24) 3. Unlike the density operator of a pure state which is idempotent as shown in Eq. 1.17, the ensemble density operator is generally non-idempotent, that is, œ C t +/ C t + J6 C œ (1.25) ^ w +š +pw t + ^ +ž +pw +š + J w +ž t +_ w 4. Like the pure state density operator, the idea and definition of ensemble density operator, Eq. (1.21), is also applicable to time-dependent problem. Similar to Eq. (1.19), the commutor 7 w can be found to be, 7 w 7 CN C`N 7 `w C N 4 {4 C`N {4 œ w (1.26) 7 < 1 1 œ For a stationary state, and w commute, 7 w :zÿ (1.27) œ 7 has the same eigenfunctions of. so w + J w 7

60 A t 1.5 Reduced density matrix Recall the previous definition of the th order density matrix, Eq. (1.10), t u ] X n05 u Z2Y5 ]m_: uf ] uj ^^`^ u <6aMF 3 J ^`^`^ We can define the reduced density matrix of order electron coordniates, by integrating out the rest tœ u ] 8 tœ 5 uf uj ^^`^ u ]MF0 J ^`^j^ e b \ F ^`^`^ b uf uj > u \ FY ]sf J Y \ FY (1.28) to which the binomial coefficient g enters because electrons are indistinguishable. When the system Hamiltonian involves only p-electron terms, then using the th order reduced matrix is more convenient than the full th order density matrix. Since the electronic Hamiltonian given in Eq. (1.2) only have one-electron and two-electron terms, the most important reduced density matrices are of the first- and second-order. operator as shown in Eq.(1.12), tj 5 u ] is the coordinate representation 1. Just as the density matrix is the coordinate representation of the density of the reduced density operatortj w which acts on the -electron Hilbert state function space. Liketw 2. The, w tœ 5 u ] t is positive semidefinite and Hermitian, tœ 5 ] _Šno uf uj ^j^`^ u 2ow tœ 2MFp Ji^`^`^ m (1.29) 6x{zŸ tœ u ] 8 t a 5 ] u (1.30) th-order density matrix normalizes to its binomial coefficient, " tœ 8 e ]6a b e b MF ^`^`^ b tœ u ] Z2!ƒ> D 0 fªfªfª «ƒ D e b t 5 u ] *2 «ƒ D Œ 0 fªfªfª!ƒ D 8 (1.31)

61 t t t œ t G w t t 3. Similar to the mixed state ensemble density operator is given by Eq. (1.21), the th-order mixed state density matrix can be generalized from the reduced pure state density matrix of Eq. (1.28), œ %u 3 X e b \ F ^^`^ b ^ uf uj Y u \ FY ]MF J > «Fc> (1.32) œ C +ž +/w tœ + (1.33) which corresponds to a reduced mixed state ensemble density operator, 1.6 The second-order density matrix By the definition of Eq. (1.28), the second-order density matrix (also known as the pair density matrix) is J uf uj ]MF0 J 8 AvZ e b ^j^`^ b 5 uf uj!> ]MF0 J Y G Ā Z e b ^`^`^ b uf ] uj ] ^`^`^ <6a*5MF\ ] J 3 * ^^`^ (1.34) 1. The second-order density matrix t J 5 uf uj ]MF0 J is the coordinate representation of the reduced density operator w which acts on the two-electron J Hilbert state function space. Liketw J %uf huj ]MF J 8 no%uf %uj 2w t 2MFp J m (1.35),t is positive semidefinite and Hermitian, J J 5MF J 3MF J x{zÿ t J %uf huj ]MF J _ t Ja MF@ J ]%uf %uj (1.36) Besides, in the electronic problem, the wave function is antisymmetric and so is the electron density, J uf uj 3MF J XŠA t J uj uf ]MF@ J XA t J 5 uf uj 3 J MF]8 t J 5 uj uf 3 J MF] (1.37) 9

62 œ w tw t + t G + œw + t w t + œ 2. According to the normalization of the reduced density matrix, Eq. (1.31), the second-order density matrix is normalized to, " t J %uf %uj ]MF J ]8 G AvZ G (1.38) 3. The second-oeder reduced mixed state ensemble density operator is obtained from those of the accessible pure states, J C + +hw J +, (1.39) and the second-order mixed state density matrix can be generalized from the reduced pure state density matrix, J 5%uF %uj 3MF J X AvZ e b ^`^^ b %uf %uj *Y ]sf J Y \ (1.40) 4. The density operator is a projection operator. The eigenfunctions of the second-order reduced density operator w belong to the two-particle Hilbert J vector space and are called natural geminals and their eigenvalues are called occupation numbers, e b MF b J J uf uj ]MF0 J < g+0mf@ J X{±Œ+ g+@5 uf uj (1.41) The natural geminals ²+ MF0 J are antisymmetric and the occupation numbers are non-negative because the operator is positive semidefinite, that is *±Œ+ xbz}. 5. Becasue an Hermitian operator can always be decomposed into its eigenfunctions, I2f g+mx:³c+]2f g+ m µ i2f g+ mn0 <+]2 ³c+2f g+ mn0 g+32 µ w w C 2Y <+5mn, g+]2 C ³+32Y g+5mn0 <+32 in which we have used the closure relation, we can express the second-order reduced density operator as J C ±Œ+]2Y <+5m!n0 g+32 (1.42) J 5 uf uj 3MF@ J X C ±Œ+Œ g+ uf uj < /a + 5MF@ J (1.43) 10

63 G t t w t C t + N t We note ± + is proportional to the probability of the two-electron state h+ is occupied., Eq.(1.15), n w ]mx n, 5 *2 ˆ w Z2Y 3m n0 Z2f m5 Šn0 5 *2 w Z2f 5 3m is the coordinate representation of Eq. (1.16), n mxb w " w w _v "!w w 6. We have shown the expectation values of the operator w Analogously, for any two-electron operators which are usually local, NO (1.44) J + the expectation value can be written as n w J mx " w J w e b MF b J J MFc ] J t 1.7 The first-order density matrix J 5%+0 ] By the definition of Eq. (1.28), the first-order density matrix is J 5 uf uj ]sf J ƒ D ƒ¹ D ¹ (1.45) t F«uF 3MF3Xe b JX^`^`^ b uf J Y 3MF J Y e b JX^`^`^ b uf ] J ^`^^ <6a*5MF ] J_^`^`^ (1.46) e Av b J J 5%uF %uj ]sf@ J Z2 ƒ¹ D ¹ (1.47) 1. The first-order density matrixt F«uF 3MF3 is the coordinate representation of the reduced density operatort w F which acts on the one-electron Hilbert state function space. t F«uF ]MF3X)no uf 2ow tœ 2MF3m (1.48) Liketw,t is positive semidefinite and Hermitian, J t F«5MF ]MF]x{z} t F«uF ]MF3X t Fa 5MF ] uf (1.49) 11

64 C C w + w œ + 2. According to the normalization of the reduced density matrix, Eq. (1.31), the first-order density matrix is normalized to the number of electrons, " t F«uF ]MF33_ e b MF t F«5 uf ]MF Z2 ƒ D (1.50) 3. The first-oeder reduced mixed state ensemble density operator is obtained from those of the accessible pure states, œ FX C +š +hw t F +, (1.51) and the first-order mixed state density matrix can be generalized from the reduced pure state density matrix, œ F!huF ]MF3X e b JX^`^`^ b %uf J > ]sf@ J Y e b JX^`^`^ b +š + t + (e b JX^`^`^ b +š + + uf J Y ]MF\Y ]6a + uf J Y 3MFY (1.52) t F belong to the one-electron Hilbert vector space and are called natural orbitals and their eigenvalues are called occupation numbers, e b MF t F«5 uf ]MF < g+@5mf3xbº²+ g+ uf (1.53) 4. The eigenfunctions of the first-order reduced density operator w The occupation numbers are non-negative because the operator is positive semidefinite, that is Zºh+'x{z}. 5. In terms of eigenfunctions, we can also express the first-order reduced density operator as t Fk C º/+2f g+ mn0 <+]2 (1.54) t F«huF ]MF3X C º²+ g+ %uf < + a sf] (1.55) We note º/+ is proportional to the probability of the two-electron state h+ is occupied. 12

65 t w C w w t C C t C t C P 6. Similar to Eq. (1.16) for the expectation value of an -electron operator, for a general one-electron operator, the expectation value is n mxb w " w w _v "!w w Fk +QD/F F\%+ 3 u+ (1.56) n w F@m v " w F w 8 e b MF b uf F«sF ] uf t F\5 uf ]MF (1.57) F«5MF ] uf X w F«uF 3»}MFMA¼ uf (1.58) If the one-electron operator is local, that is, then F8 +QD/F F«5%+ (1.59) n w F m_ " w FŒw e b MFh F!5MF3 t F«5 uf ]MF ƒ D (1.60) and its expectation value can be written, aimilar to Eq. (1.45), as 1.8 l -representability problem At this point, consider the electronic Hamiltonian in question, Eq. (1.2), 7 {A +ED/F G H +ML½ J! " F3MA +ED/F which includes only one- and two-electron operators, and use Eqs. (1.16), (1.45), and (1.60), we can write the Hamiltonian expectation value (the electronic energy) 1D/F X " +Q 13

66 t t C t G G t C C + C N t + t as 9<;,=>;,?s " 7 w X " A C +ED/F G H +ML J +QD/F ½! " + L CNcO N +QD/F " + " EA +QD/F G HKJ +ML ½! " +3]jw 8 L " CNO +ED/F " + jw 8 e b MF% A G H JF L ½! " F ] t F«uF 3MF3 ƒ D L e b MF b J jw t X " F J J MF@ J ]MF@ J (1.61) This equation shows that the electronic energy is a functional of the second- J. However it remains a question to find out the necessary and sufficient conditions for t to correspond to certain antisymmetric wave function, that is, to satisfy Eq. J (1.34), order density matrix 9g;,=>;,? 9<;,=>;,? t J 5%uF %uj ]sf J 8 AvZ e b ^`^`^ b huf ]%uj ^`^`^ %u < a MF 3 J ^`^j^ This is the so-called -representability problem. Obviously, once this problem is solved, the variational method is ready to be used to minimize the energy with respect tot J without having to solve the¾ -dimensional wave function5. 2 Spinless Density Matrices " +, $1+]24y. It is often desirable to inte- We have defined in the previous section the density matrices with the complete electronic coordinates, * +. / grate out the spin coordinates because most quantum operators (for example, the Hamiltonian of Eq. (1.2)) involve only the spatial coordinates. 2.1 Second-order spinless density matrix Integrate out the spin coordinates in the second-order density matrix, Eq. (1.34), J 5%uF %uj ]sf J 8 Av* e b ^`^`^ b %uf ]huj ] ^`^^ < a MF\ 3 JX^`^`^ 14

67 d d " G d " G " " t " " " " " " " " " " " " we have the second-order spinless density matrix J! " uf " u J ` " F J X e b $hf b $ J J! " uf $hf " J u $ J `" F@$hFj J $ J AvZ e b $hf b $ J b ^`^`^ b ^! " uf $hfc `" J u $ J 3 ^`^ ^ 3 < a! " F ]$hf\ ` J $ J 3 ^`^^ ] (2.1) 1. Considering the two electrons involved in the second-order spinless density matrix can haveà orá spins, the density matrix can be decomposed, J! " uf " J u `" FŒ J X d`â*â J Â*Â! " uf " J u `" FŒ J L d`âã J ÂÃ! " uf " J u `" d F J àJ ÃÂ! " uf " J u `" F J L d Ã!à J Ã!Ã! " uf " J u `" F J (2.2) 2. The diagonal elements of the second-order spinless density matrix are J! " F\ ` J 8 d J! " F J `" F J Ā Z e b $hf b $ J b ^`^`^ b 2Y! " F\ $hf ` J $ J ] ^`^ ^ ] Z2J (2.3) 3. For spin-free two-electron operators which are usually local, Ä J! " Fj J ], the expectation value can be written as, compared to Eq. (1.45), n]ä J m e b " F b J EÄ J! " Fj J d J! " uf " J u `" Fj J ÆŃ D ÆÅ Å Æ ƒ¹ D Æ Å¹ (2.4) 2.2 First-order spinless density matrix Integrate out the spin coordinates in the first-order density matrix, Eq. (1.46), t F«uF 3MF3X e b J_^^`^ b 5 uf ] JX^`^`^ <6a*5MF\ ] JX^`^`^ we have the first-order spinless density matrix d F!! " uf `" F3X e b $hf t F«! " uf $ uf `" F $hf] e b $hf b J_^`^`^ b! " uf ]$hf\ ] JX^`^ ^ ] <6aZ! " Fc $hf\ ] J ^`^^ ] (2.5) 15

68 G t " " J J d d " " 1. Considering one electron can have À or Á spins, the first-order spinless density matrix can be decomposed, d F«! " uf `" F38 d Â* F! " uf `" F3 L d Ã!à F! " uf `" F3 (2.6) The diagonal elements, that is, the electron density is then d! " X d Â* F! " Fc ` " F3 L d Ã!à F! " F ` " F3X d Â! " L d Ã! " (2.7) We thus define a spin density which reflects the spin polarization when d Â! " Ç d Ã! ", d È! " X d Â* F! " F\ ` " F3MA d Ã!à F! " Fc ` " F]8 d Â! " MA d Ã! " (2.8) 2. The diagonal elements of the first-order spinless density matrix are d F!! " F3X d F«! " F\ ` " F38 e b $hf b J_^`^^ b 2Y! " F ] J ^`^^ ] *2J (2.9) which is the electron density defined in Eq. (1.6). Integration over all space results in the total number of electrons, e d! " F] b " Fk (e 2Y *2J²b MF b Ji^`^^ b (2.10) 3. As the first- and second-order density matrices are related by t F«huF ]MF3X G Av e b J J %uf huj ]MF@ J Z2 ƒ¹ D ¹ (2.11) the first- and second-order spinless density matrices are related by d F«! " uf `" F3X For the diagonal elements, d F«! " F]8 e AU b G e AU b J! " uf J `" Fj J \ (2.12) J! " F\ ` " J (2.13) 16

69 w C w C C C P " J d " J t " d " " 4. If the one-electron operator is local, that is, then F«5MF ] uf X w F«uF 3»}MFMA¼ uf (2.14) F8 +QD/F F«5%+ (2.15) " F is, similar to The expectation value of a local one-electron operator, Ä F«! Eq. (1.60), n]ä#f m_ e b 2.3 Energy with spinless density matrices The electronic Hamiltonian in question, Eq. (1.2), 7 {A +ED/F " Fh EÄ#F«! " F] d F«! " uf `" F] ÆŃ D ÆÅ (2.16) G H +ML½ J! " F3MA +ED/F is spinless. Use Eqs. (2.4), and (2.16), we can rewrite the electronic energy, Eq. (1.61), in terms of spatial coordinates, 9<;,=>;,?s e b MF% A GIH JF L ½! " F ] t F«uF 3MF3 ƒ D L e b MF b J " F J J MF@ J ]MF@ J e b " F% A GIH JF L½! " F3] d F!! " uf `" F3 ÆŃ D ÆÅ L e b " F b " F J J! " F\ ` J e b " F% QA G HKJÅ Æ d F!! " u ` " ÆŃD ÆÅ L e b " ½! " d! " L e b " F b " F J J! " F ` J ˆ d F LUÉ1V ;! d LÊÉ ; ; d J (2.17) 1. The first term in Eq. (2.17) is the electronic kinetic energy and involves a " u ` ". 1D/F one-electron density function of six coordinates,d! 2. The second term in Eq. (2.17) is the external potential energy (usually the electron-nuclear potential energy) and involves a one-electron density function of only three coordinates,d! ". 3. The third term in Eq. (2.17) is the electron-electron potential energy and involves a two-electron density function of six coordinates,d! " F\ ` 17 X " +Q J.

70 G " " d J " J " J " J " J " " d " " " J " " " " " " " " J " J " " " J d " " " " " " J " 2.4 Exchange-correlation hole The electron-electron interaction term in Eq. (2.17) É ;,;«d J e b " F b " F J J! " F\ ` J (2.18) has included the quantum effects because if the electron interaction were classical, then However,d É ; ;? G e b " F b J! " F ` J F d! " F3 d! J. " F J d! " F] d! " J \ (2.19) " F ` J, to account for the nonclas- " F\ ` J (2.20) Let s define a pair correction function,ë! sical effects and write J! " F ` Recall Eq. (2.13), we have J _ G d! " F] d! " J! Q L Ë! d F!! " F3X e Ā b J! " F\ ` J e Ā b d! " F] d! J! Q L Ë! " F ` J Ā d! " F] e b d! J L Av d! " F3 e b d! " J 3Ë! " F\ ` " J Ā d! " F] L Av d! " F e b d! J ]Ë! " F\ ` J µ e b d! J ]Ë! " F\ ` J 8ŠA~ (2.21) d! J ]Ë! " F ` J contains the charge of one electron. Ë1Í?c! " F\ ` J 8 d! J ]Ë! " Fc ` J e b J Ë1Í?\! " F ` J XŠAÎ (2.22) That is, the integralì b We should define the exchange-correlation hole as 18

71 d " J " J " J d " " J " " " " " " " J " Or, in complete coordinates J 5MF\ ] J X G d MF] d J «E L Ë'5MF ] J ËŸÍ?k d J 3Ë MF\ ] J \ e b J ËŸÍ?\sF ] J 8ŠA~ (2.23) Now the electron-electron repulsion energy term can be written as É ;,;8 e b " F b " F J J! " F\ ` J G e b " F b " F d! " F] d! J J! Q L Ë! " F\ ` J G e b " F b " F d! " F] d! J J L G e b " F b " F d! " F] d! " J J ]Ë! " F\ ` " J Ï8 d L G e b " F b " F d! " F]3Ë1Í?c! " F ` J J (2.24) (a) The first term is the classical electrostatic energy which is a functional of the three-variable electron density. This term also includes the unphysical self-interaction. Even when there is only one-electron in the system, it won t be zero. (b) The second term can be seen as the interaction between the electron density and the charge distribution of the exchange-correlation hole. We can split the exchange-correlation hole into a Fermi hole and a Coulomb hole, ËŸÍ?! " F\ ` J 8rË1ÐÑ ; DÆ0Ò Ð ¹ +! " F ` J L Ë ÐÓ S,Ô\= S Ð ¹ ÒMÕ! " F ` J (2.25) The Fermi hole arises because of the Pauli principle and applies only to electrons of the same spin, while Coulomb hole applies to all electrons with either spin. Acknowledgment Notes are taken mainly from the books by C.Cohen-Tannoudji, B. Diu and F. Laloë [3], by Robert G. Parr and Weitao Yang [4], by A. Szabo and N. S. Ostlund [5] and by Wolfram Koch and Max C. Holthausen [6]. An early review (1960) by R. McWeeny [7] is very enlightening. 19

72 References [1] P. A. M. Dirac, The Principles of Quantum Chemistry, 3rd Ed. (Oxford University Press, New York, 1947). [2] K. Husimi, Proc. Phys. Math. Soc. Japan 22, 264 (1940). The reduced density matrix was introduced. [3] C.Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Chemistry Vol. 1 (Oxford University Press, New York, and Clarendon Press, Oxfrod, 1989). [4] Robert G. Parr and Weitao Yang, Density Functional Theory of Atoms and Molecules, Chapter 2 (Oxford University Press, New York, and Clarendon Press, Oxfrod, 1989). [5] Attila Szabo and Neil S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory. 2nd Ed., Chapters 1 and 2 (Dover Publications Inc., Mineola(NY), 1989). [6] Wolfram KOch and Max C. Holthausen, A Chemist s Guide to Density Functional Theory. 2nd Ed., Chapter 2 (Wiley-VCH, Verlag GmbH, Weinheim, Federal Republic of Germany, 2001). [7] R. McWeeny. Some Recent Advances in the Density Matrix Theory. Rev. Mod. Phys. 32, 335 (1960). 20

73 ( Introduction to Gaussian 03 Package Xihua Chen Chemistry Department, New York University January, 2006

74 A variety of capabilities : (a) 1. Molecular structure : structural measurement optimization 2. Electronic structure : electronic energy molecular orbitals electron affinities ionization potentials atomic charges multipole moments electron densities electrostatic potentials polarizabilities and hyperpolarizabilities By Andrew Weiser 3. Chemical reaction : bond energies reaction pathways transition states; reaction energetics potential surface 4. Thermochemistry analysis : thermal energies thermal enthalpies 2

75 A variety of capabilities : (b) 5. Vibrational frequencies normal frequency analysis IR intensities Raman intensities vibrational circular dichroism(vcd) intensities anharmonic vibrational analysis vibrational-rotational coupling By Jack Haas 6. Chiroptics : Electronic circular dichroism(ecd) intensities Optical rotation 7. NMR : NMR shielding magnetic susceptibilities spin-spin coupling g-tensor; hyperfine NMR 8. More. molecular dynamics (ADMP, BOMD) 3

76 An arsenal of executables grouped into overlays (a) Overlay 0 (initialization): Link L0 : Initializes program and controls overlaying Overlay 1 (initialization, optimization, miscellaneous): Link L1 : Processes route section, builds list of links to execute, and initializes scratch files L101 : Reads title and molecule specification L102 : Nonderivative Fletcher-Powell (FP) optimization L103 : Berny optimizations, Synchronous Transit-Guided Quasi-Newton (STQN) search of transition states L105 : Murtaugh-Sargent optimization (obsolete) L106 : Numerical differentiation of forces/dipoles to obtain polarizabilities / hyperpolarizabilities 4

77 An arsenal of executables grouped into overlays (b) Overlay 1 (continued) : L107 : Linear-synchronous-transit (LST) search of transition states L108 : Potential energy surface scan L109 : Newton-Raphson optimization L110 : Double numerical differentiation of energies to produce frequencies L111 : Double numerical differentiation of energies to compute polarizabilities & hyperpolarizabilities L113 : Eigenvalue-following (EF) optimization using analytic gradients L114 : Eigenvalue-following (EF) optimization using numeric gradients 5

78 An arsenal of executables grouped into overlays (c) Overlay 1 (continued) : L115 : Follows reaction path using the intrinsic reaction coordinate (IRC) L116 : Numerical self-consistent reaction field (SCRF) L117 : Post-SCF SCRF L118 : Trajectory calculations (dynamics) L120 : Controls ONIOM calculations L121 : Atom Centered Density Matrix Propagation (ADMP) molecular dynamics calculations L122 : Counterpoise calculations for Basis Set Superposition Error-correction (BSSE) 6

79 An arsenal of executables grouped into overlays (d) Overlay 2 (molecular coordinates): L202 : Reorients coordinates, calculates symmetry, and checks variables Overlay 3 (integrals): L301 : Generates basis set information L302 : Calculates overlap, kinetic, and potential integrals L303 : Calculates multipole integrals L308 : Computes dipole velocity and Rx Del integrals L310 : Computes spdf 2-electron integrals in a primitive fashion L311 : Computes sp 2-electron integrals L314 : Computes spdf 2-electron integrals 7

80 An arsenal of executables grouped into overlays (e) Overlay 3 (continued) : L316 : Prints 2-electron integrals L319 : Computes 1-electron integrals for approximate spin orbital coupling Overlay 4(initial guess): L401 : Forms the initial MO guess L402 : Performs semi-empirical and molecular mechanics calculations L405 : Initializes an Complete Active Space Multiconfiguration SCF (MCSCF) calculation 8

81 An arsenal of executables grouped into overlays (f) Overlay 5 (Self-consistent field -- SCF): L502 : Iteratively solves the SCF equations L503 : Iteratively solves the SCF equations using direct minimization L506 : Performs a restricted open-shell HF (ROHF) or perfect-pairing General Valence Bond (GVB-PP) calculation L508 : Quadratically convergent SCF program L510 : MC-SCF Overlay 6 (analysis of electronic properties): L601 : Population and related analyses (including multipole moments) 9

82 An arsenal of executables grouped into overlays (g) Overlay 6 (continued): L602 : 1-electron properties (potential, field, and field gradient) L604 : Evaluates MOs or density over a grid of points L607 : Performs (Natural Bond Orbital) NBO analyses L608 : Non-iterative DFT energies L609 : Atoms in Molecules properties Overlay 7 (derivatives): L701 : 1-electron integral first or second derivatives L702 : 2-electron integral first or second derivatives (sp) L703 : 2-electron integral first or second derivatives (spdf) L716 : Processes information for optimizations and frequencies 10

83 An arsenal of executables grouped into overlays (h) Overlay 8 (transformation): L801 : Initializes transformation of 2-electron integrals L802 : Performs integral transformation L804 : Integral transformation L811 : Transforms integral derivatives & computes their contributions to MP2 2nd derivatives Overlay 9 (post-hf): L901 : Anti-symmetrizes 2-electron integrals L902 : Determines the stability of the HF wavefunction L903 : Old in-core MP2 L905 : Complex MP2 L906 : Semi-direct MP2 11

84 An arsenal of executables grouped into overlays (i) Overlay 9 (continued) : L908 : Outer Valence Green s Function (OVGF) closed shell L909 : OVGF (open shell) L913 : Calculates post-scf energies and gradient terms L914 : Excited states: CI-Singles; Random-phase approximation(rpa, a time-dependent HF) and Zindo; SCF stability L915 : Computes fifth order quantities (for MP5, QCISD(TQ) and BD(TQ)) L916 : Old MP4 and coupled cluster (CCSD) L918 : Reoptimizes the wavefunction 12

85 An arsenal of executables grouped into overlays (j) Overlay 10 (Coupled Perturbed Hartree-Fock --CPHF): L1002 : Iteratively solves CPHF equations; properties (e.g.nmr) L1003 : Iteratively solves the CP-MCSCF equations L1014 : Computes analytic CI-Singles second derivatives Overlay 11 (Fock matrix derivatives): L1101 : Computes 1-electron integral derivatives L1102 : Computes dipole derivative integrals L1110 : 2-electron integral derivative contribution to Fx L1111 : 2 particle density matrix (PDM) & post-scf derivatives L1112 : MP2 second derivatives Overlay 99 (Termination): Link L9999 : Finalizes calculation and output 13

86 Gaussian job Route: a sequence of links (a) A simple example: Compute the electronic energy of formaldehyde (CH 2 O) using Hartree-Fock method with a minimum basis set STO-3G Standard route: in input file, we specify in a line # HF/STO-3G Non-standard route : Link 1 reads HF/STO-3G and generates the following route in output file # HF/STO-3G /38=1/1; 2/17=6,18=5,40=1/2; 3/6=3,11=9,16=1,25=1,30=1/1,2,3; 4//1; 5/5=2,32=1,38=5/2; 6/7=2,8=2,9=2,10=2,28=1/1; 99/5=1,9=1/99; To call Link 101 To call Link 202 To call Links 301, 302, 303 To call Link 401 To call Link 502 To call Link 601 To call Link 9999 Non-standard route: Overlay / Opt=val, Opt=Val, / Link, Link, 14

87 Gaussian job Route: a sequence of links (b) # HF/STO-3G Input file CH2O.gjf L1 L101 Link 1 : Reads and parses route section. Builds links. Link 101 : Reads in title and molecule specification Link 202 : calculates molecular symmetry L202 Link 301 : Generates basis set information L301 L302 L303 Link 302 : Overlap, kinetic, and potential integrals Link 303 : Calculates multipole integrals L401 Link 401 : Forms the initial MO guess L502 Link 502 : Iteratively solves the SCF equations L601 Link 601 : Assigns orbital and wavefunction symmetries; Performs Mulliken population analysis L9999 Link 9999 : Finalizes calculation and output Output file: CH2O.out 15

88 Example of CH 2 O with HF/STO-3G 3G: : Input file Input file 1 : molecule in Z-matrix # HF/STO-3G Example: aldehyde 0 1 C1 O2 1 A H3 1 B 2 C H4 1 B 2 C 3 D A = 1.28 B = 1.10 C = D = Input file 2 : molecule in Z-matrix # HF/STO-3G Example: aldehyde 0 1 C1 O H H Input file 3 : molecule in Cartesian # HF/STO-3G Example: aldehyde 0 1 C O H H

89 More about Gaussian Input file: Section layout Input file 1 : molecule in Z-matrix # HF/STO-3G Example: aldehyde 0 1 C1 O2 1 A H3 1 B 2 C H4 1 B 2 C 3 D A = 1.28 B = 1.10 C = D = % Resources management (link 0 commands) # Route card [blank line] Title section [blank line] Molecular coordinates [blank line] Internal coordinates variables [blank line] Other specification [blank line] 17

90 More About Gaussian Input file: Link 0 Commands %Mem=N Sets the amount of dynamic memory used to N words (8N bytes). The default is 6MW. N may be followed by a units designation: KB, MB, GB, KW, MW or GW %NProcShared=N Requests that the job use up to N processors %Chk=file %RWF=file Locates and names the checkpoint file Locates and names a single, unified Read-Write file %RWF=loc1,size1,loc2,size2, Locates and names split Read-Write file %Int=spec Locates and names two-electron integral file(s) %D2E=spec Locates and names two-electron integral derivative file(s) %KJob LN [M] Tells the program to stop the run after the M th occurrence of link N %save (%nosave) Causes Link 0 (not) to save scratch files at the end of the run %subst LN dir Tells Link 0 to the executable for a link from alternate directory 18

91 Example of CH 2 O: : Output file (a) : Link 1 Copyright (c) 1988,1990,1992,1993,1995,1998,2003, Gaussian, Inc. All Rights Reserved.. Cite this work as: Gaussian 03, Revision B.03, M. J. Frisch, G. W. Trucks, H. B. Schlegel, (many many other), and J. A. Pople, Gaussian, Inc., Pittsburgh PA, ********************************************* Gaussian 03: x86-win32-g03revb.03 4-May Jan-2006 ********************************************* %chk=nosave # HF/STO-3G /38=1/1; 2/17=6,18=5,40=1/2; 3/6=3,11=9,16=1,25=1,30=1/1,2,3; 4//1; 5/5=2,32=1,38=5/2; 6/7=2,8=2,9=2,10=2,28=1/1; 99/5=1,9=1/99; Input file 2 : molecule in Z-matrix # HF/STO-3G Example: aldehyde 0 1 C1 O H H Link 1 : Reads and parses route section. Builds links. 19

92 Example of CH 2 O: : Output file (b) : Link test aldehyde Symbolic Z-matrix: Charge = 0 Multiplicity = 1 C1 O H H Link 101 : Reads in title and molecule specification The following is not printed unless requested Isotopes and Nuclear Properties: Atom IAtWgt= AtmWgt= IAtSpn= AtZEff= AtQMom= AtGFac=

93 Example of CH 2 O: : Output file (c) : Link 202 Input orientation: Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z Distance matrix (angstroms): C O H H Stoichiometry CH2O Framework group C2V[C2(CO),SGV(H2)] Deg. of freedom 3 Full point group C2V NOp 4 Largest Abelian subgroup C2V NOp 4 Largest concise Abelian subgroup C2 NOp 2 Standard orientation: Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z Rotational constants (GHZ): Link 202 : calculates molecular symmetry 21

94 Terminology reminder 22

95 Example of CH 2 O: : Output file (d) Link 301 : Generates basis set information. Link 302 : Overlap, kinetic, and potential integrals Link 303 : Calculates multipole integrals Standard basis: STO-3G (5D, 7F) There are 7 symmetry adapted basis functions of A1 symmetry. There are 0 symmetry adapted basis functions of A2 symmetry. There are 2 symmetry adapted basis functions of B1 symmetry. There are 3 symmetry adapted basis functions of B2 symmetry. Integral buffers will be words long. Raffenetti 1 integral format. Two-electron integral symmetry is turned on. 12 basis functions, 36 primitive gaussians, 12 cartesian basis functions 8 alpha electrons 8 beta electrons nuclear repulsion energy Hartrees. NAtoms= 4 NActive= 4 NUniq= 3 SFac= 2.05D+00 NAtFMM= 60 Big=F One-electron integrals computed using PRISM. NBasis= 12 RedAO= T NBF= NBsUse= D-06 NBFU=

96 Example of CH 2 O: : Output file (e) : Link 401 Link 401 : Forms the initial MO guess. Harris functional with IExCor= 205 diagonalized for initial guess. ExpMin= 1.69D-01 ExpMax= 1.31D+02 ExpMxC= 1.31D+02 IAcc=1 IRadAn= 1 AccDes= 1.00D-06 HarFok: IExCor= 205 AccDes= 1.00D-06 IRadAn= 1 IDoV=1 ScaDFX= Initial guess orbital symmetries: Occupied (A1) (A1) (A1) (A1) (B2) (A1) (B1) (B2) Virtual (B1) (A1) (B2) (A1) The electronic state of the initial guess is 1-A1. 24

97 Example of CH 2 O: : Output file (f) : Link 502 Link 502 : Iteratively solves the SCF equations. Warning! Cutoffs for single-point calculations used. Requested convergence on RMS density matrix=1.00d-04 within 128 cycles. Requested convergence on MAX density matrix=1.00d-02. Requested convergence on energy=5.00d-05. No special actions if energy rises. Keep R1 integrals in memory in canonical form, NReq= SCF Done: E(RHF) = A.U. after 6 cycles Convg = D-04 -V/T = S**2 =

98 Example of CH 2 O: : Output file (g) : Link 601 Link 601 : Assigns orbital and wavefunction symmetries; Performs Mulliken population analysis. ********************************************************************** Population analysis using the SCF density. ********************************************************************** Orbital symmetries: Occupied (A1) (A1) (A1) (A1) (B2) (A1) (B1) (B2) Virtual (B1) (A1) (B2) (A1) The electronic state is 1-A1. Alpha occ. eigenvalues Alpha occ. eigenvalues Alpha virt. eigenvalues Condensed to atoms (all electrons): C O H H

99 Mulliken atomic charges: 1 1 C O H H Sum of Mulliken charges= Atomic charges with hydrogens summed into heavy atoms: 1 1 C O H H Sum of Mulliken charges= Electronic spatial extent (au): <R**2>= Charge= electrons Dipole moment (field-independent basis, Debye): X= Y= Z= Tot= Quadrupole moment (field-independent basis, Debye-Ang): XX= YY= ZZ= XY= XZ= YZ= Traceless Quadrupole moment (field-independent basis, Debye-Ang): XX= YY= ZZ= XY= XZ= YZ= Octapole moment (field-independent basis, Debye-Ang**2): XXX= YYY= ZZZ= XYY= XXY= XXZ= XZZ= YZZ= YYZ= XYZ= Hexadecapole moment (field-independent basis, Debye-Ang**3): XXXX= YYYY= ZZZZ= XXXY= XXXZ= YYYX= YYYZ= ZZZX= ZZZY= XXYY= XXZZ= YYZZ= XXYZ= YYXZ= ZZXY= N-N= D+01 E-N= D+02 KE= D+02 Symmetry A1 KE= D+02 Symmetry A2 KE= D+00 Symmetry B1 KE= D+00 Symmetry B2 KE= D+00 Link 601 : Assigns orbital and wavefunction symmetries; Performs Mulliken population analysis. 27

100 Example of CH 2 O: : Output file (h) : Link 9999 Link 9999 : Finalizes calculation and output. 1 1 UNPC-UNK SP RHF STO-3G C1H2O1 PCUSER 16-Jan # HF/STO-3G t est aldehyde 0,1 C O,1,1.28 H,1,1.1,2,121. H,1,1.1,2,121.,3,180.,0 V ersion=x86-win32-g03revb.03 State=1-A1 HF= RMSD=4.290e-005 Dipole=0.,0., PG=C02V You don't have to suffer to be a poet. Adolescence is enough suffering for anyone. -- John Ciardi Job cpu time: 0 days 0 hours 1 minutes 17.0 seconds. File lengths (MBytes): RWF= 11 Int= 0 D2E= 0 Chk= 8 Scr= 1 Normal termination of Gaussian 03 at Mon Jan 16 16:13:

101 Back to check route : Internal Options (IOP) Non-standard route: Overlay / Opt=val, Opt=Val, / Link, Link, # HF/STO-3G /38=1/1; 2/17=6,18=5,40=1/2; 3/6=3,11=9,16=1,25=1,30=1/1,2,3; 4//1; 5/5=2,32=1,38=5/2; 6/7=2,8=2,9=2,10=2,28=1/1; 99/5=1,9=1/99; Example: 3/6=3,11=9,16=1,25=1,30=1/1,2,3; Means: run through links 301, 302, and 303, with options 6 set equal to 3, option 11 set to 9, etc. in each of thelinks (in overlay 3). Check the options and their values at Consider options that control methods, basis set and integrals IOp(3/6) Number of Gaussian Functions. Here IOp(3/6=3) for STO-3G IOp(3/11) Control of 2-electron integral storage format IOp(3/16) Pseudopotential option (=1: Yes, read if general basis. ) IOp(3/25) Number of last 2-electron integral link (=1 : Direct SCF. ) IOp(3/30) Control of 2-electron integral symmetry (=1: symmetry turned on) 29

102 Option and values for basis sets (in overlay 3) For example, compare: # HF/STO-3G 3/6=3,11=9,16=1,25=1,30=1/1,2,3; # HF/6-31+G(d,p) 3/5=1,6=6,7=111,11=9,16=1,25=1,30=1/1,2,3; Explanation: STO-3G IOp(3/6) Number of Gaussian Functions. Here IOp(3/6=3) for STO-3G 6-31+G** IOp(3/5) Number of Gaussian Functions. Here IOp(3/6=6) for 6-31+G(d,p) IOp(3/6) Type of basis set. Here IOp(3/5=1) for Extended 4-31G,5-31G,6-31G IOp(3/7=111) Diffuse and polarization functions. Here IOp(3/7=111) for d-functions on heavy atoms (1) PLUS a set of sp functions on heavy atoms (10) PLUS p-functions on H atoms (100) 30

103 An array of Basis sets are available Basis Set Applies To Polarization Functions Diffuse Functions STO 3G H--Xe * 3-21G H--Xe * or ** G H--Cl (d) 4-31G H-Ne (d) or (d,p) 6-31G H--Kr (3df,3pd) G H--Kr (3df,3pd) ++ D95 H--Cl except Na, Mg (3df,3pd) ++ D95V H--Ne (d) or (d,p) ++ SHC H--Cl * CEP-4G H--Rn * CEP-31G H--Rn * CEP-121G H--Rn * LanL2MB H--Ba, La--Bi LanL2DZ H, Li--Ba, La--Bi SDD all but Fr and Ra cc pv[dtq5]z Dcc-pV[DT]Z H-He, B-Ne, Al-Ar included indefinition added via AUG prefix cc-pv6z H, B-Ne included indefinition added via AUG prefix SV H--Kr SVP H--Kr included indefinition TZV H--Kr (3df,3pd) MidiX H, C, N, O, F, P, S, Cl included indefinition EPR-II,III H, B, C, N, O, F included indefinition You can also specify particular basis functions, using keyword Gen. 31

104 Option and values for DFT methods (in overlay 3) For example, compare: # HF/6-31+G(d,p) 3/5=1,6=6,7=111,11=9,16=1,25=1,30=1/1,2,3; # B3LYP/6-31+G(d,p) 3/5=1,6=6,7=111,11=2,16=1,25=1,30=1,74=-5/1,2,3; Explanation: IOp(3/74) Type of exchange and correlation potentials. IOP(3/74=-5) means Becke s three-parameter hybrid exchange functional with Lee-Yang-Parr correlation functional. IOP(3/74=100) or by default is Hartree-Fock 32

105 DFT methods: IOP(3/74) -34 BMK. -33 X3LYP -32 t-hcth hybrid -31 t-hcth -30 OmPW3PBE. -29 OmPW1PBE. -28 OmPW1LYP. -27 OmPW1PW O3LYP. -23 HCTH HCTH B B HCTH B B1B BA3PBE. -15 BA1PBE. -14 PBE3PBE. -13 PBE1PBE -12 mpw3pbe. -11 mpw1pbe. -10 mpw1lyp. -9 LG1LYP. -8 B1LYP. -7 mpw91pw Becke3 with Perdew 91 correlation. -5 Becke3 using VWN/LYP for correlation. -4 Becke 3 with Perdew 86 correlation. -3 Becke "Half and Half" with LYP/VWN correlation. -2 Becke "Half and Half": 0.5 HF LSD -1 Do only coulomb part; skip exchange-correlation. 00 Default, same as No correlation. 01 Vosko-Wilk-Nusair method 5 correlation. 02 Lee-Yang-Parr correlation. 03 Perdew 81 correlation. 04 Perdew 81 + Perdew 86 correlation. 05 VWN 80 (LSD) correlation 06 VWN 80 (LSD) + Perdew 86 correlation 07 OS1 correlation 08 PW91 09 PBE 10 VSXC 11 Bc96 33

106 DFT methods: IOP(3/74) more 18 VWN5+P86 19 LYP+VWN5 for scaling 20 KCIS 100 Hartree-Fock exchange. 200 Hartree-Fock-Slater exchange (Alpha = 2/3). 300 X-alpha exchange (alpha= 0.7) 400 Becke 1988 exchange. 500 LG exchange. 600 PW91 exchange 700 Gill 96 exchange 800 PW86 exchange 900 mpw exchange 1000 PBE exchange 1100 BA exchange 1200 VSXC exchange 1400 B98 (JCP 108,9624(1998) eq.2c ) exchange 1500 HCTH (JCP 109,6264 (1998) exchange 1600 B97-1 (CPL 316,160(2000)) exchange 1700 B97-2 (JCP 115,9233(2001)) exchange 1800 HCTH147 exchange 1900 HCTH407 exchange 2000 OPTX exchange 2100 OPTX exchange as in O3LYP 2800 Old mpw exchange (local scaling in non-local ter 3/74= Ex+Corr 3/74=100 is Hartree-Fock 3/74=200 is Hartree-Fock-Slater, 3/74=205 is Local Spin Density, 3/74=402 is BLYP. 3/74=-5 is B3LYP etc. 34

107 Option & values for post-hf methods For example, compare: # HF/6-31+G(d,p) 1/38=1/1; 2/17=6,18=5,40=1/2; 3/5=1,6=6,7=111,11=9,16=1,25=1,30=1/1,2,3; 4//1; 5/5=2,32=1,38=5/2; 6/7=2,8=2,9=2,10=2,28=1/1; 99/5=1,9=1/99; # MP2/6-31+G(d,p) Link 801 : Performs integral transformation. Link 906 : Semi-direct MP2. IOP(9/16) : Control of Semi-/ Direct MP2 IOP(9/16=3) : Try to minimize integral evaluations, using fully direct methods if possible, otherwise spilling to disk. 1/38=1/1; In HF, IOP(5/32=1) means 2/17=6,18=5,40=1/2; sleazy (low-quality) SCF, 3/5=1,6=6,7=111,11=9,16=1,25=1,30=1/1,2,3; using loose integral cutoffs, 4//1; convergence on either energy 5/5=2,38=5/2; or density. 8/10=2/1; 9/16=-3/6; In MP2 IOP(5/32) by default is 6/7=2,8=2,9=2,10=2/1; not to use sleazy SCF. 99/5=1,9=1/99 35

108 Input file CH2O.gjf L1 L101 MP2 Route Link 1 : Reads and parses route section. Builds links. Link 101 : Reads in title and molecule specification L202 Link 202 : calculates molecular symmetry L301 L302 L303 Link 301 : Generates basis set information Link 302 : Overlap, kinetic, and potential integrals L401 Link 303 : Calculates multipole integrals L502 Link 401 : Forms the initial MO guess L601 Link 502 : Iteratively solves the SCF equations L801 Link 601 : Assigns orbital and wavefunction symmetries; Performs Mulliken population analysis L906 Link 801 : Performs integral transformation. L9999 Output file: CH2O.out Link 906 : Semi-direct MP2. Link 9999 : Finalizes calculation and output 36

109 Output electronic energy : a comparison # HF/6-31+G(d,p) After Link 502 (Iteratively solves the SCF equations.) SCF Done: E(RHF) = A.U. after 5 cycles Convg = D-04 -V/T = S**2 = # B3LYP/6-31+G(d,p) After Link 502 SCF Done: E(RB+HF-LYP) = A.U. after 5 cycles Convg = D-04 -V/T = S**2 = # MP2/6-31+G(d,p) After Link 502 SCF Done: E(RHF) = A.U. after 11 cycles Convg = D-08 -V/T = S**2 = After Link 906 (semi-direct MP2) E2 = D+00 EUMP2 = D+03 37

110 An array of methods are available Some New, for example, 1. Periodic Boundary Conditions (PBC) calculations for periodic systems 2. Atom-Centered Density Matrix Propagation (ADMP) molecular dynamics. 38

111 Methods / basis set A theoretical model chemistry is defined by the two main approximations that are made in order to make solution of the time-independent chrodinger equation tractable. These are the level of correlation treatment and the extent of completeness of the set of basis functions which are used to represent the molecular orbitals. (Martin Head-Gordon, J. Phys. Chem. 1996, 100, ) 39

112 Scaling features of computational methods Method Formal CPU Formal Memory Formal Disk Actual CPU Actual Disk Conv. SCF N 4 N 2 N 4 N 3.5 N 3.5 Incore SCF N 4 N 4 - N 4 N 2 Direct SCF N 4 N 2 - N 2.3 N 2 Conv. MP2 ON 4 N 2 N 4 ON 4 N 4 Dir MP2 SP ON 4 OVN - O 2 N 3 N 2 SD MP2 SP ON 4 N 2 VN 2 O 2 N 3 VN 2 Conv. MP2 ON 4 N 2 N 4 ON 4 N 4 Force Dir MP2 ON 4 N 3 - O 2 N 3 N 2 Force SD MP2 ON 4 N 2 N 3 O 2 N 3 N 3 Force MP3, CISD, O 2 N 4 N 2 N 4 O 2 N 4 N 4 QCISD MP4, QCISD(T) O 3 V 4 N 2 N 4 O 3 V 4 N 4 O: Number of occupied orbitals; V: Number of virtual orbitals; N: Number of basis functions (Table by Carlos P.Sosa and Patton L. Fast) 40

113 Tell G 03 what to do: keywords in # route card Input file 1 : single-point # B3LYP/6-31G Example: aldehyde 0 1 C1 O2 1 A H3 1 B 2 C H4 1 B 2 C 3 D A = 1.28 B = 1.10 C = D = Input file 2 : optimization # B3LYP/6-31G OPT Example: aldehyde 0 1 C1 O2 1 A H3 1 B 2 C H4 1 B 2 C 3 D A = 1.28 B = 1.10 C = D = Options of OPT, for example: # B3LYP/6-31G OPT(Tight, Z-matrix) Opt=tight: tightens the cutoffs on forces and step size that are used to determine convergence 41

114 Structural optimization: Option & values For example : # B3LYP/6-31G OPT(Tight, Z-matrix) 1/7=10,10=7,14=-1,18=40,26=3,38=1/1,3; 2/17=6,18=5,29=3,40=1/2; 3/5=1,6=6,11=2,16=1,25=1,30=1,74=-5/1,2,3; 4//1; 5/5=2,38=5/2; (missing 5/32=1) 6/7=2,8=2,9=2,10=2,28=1/1; 7/29=1/1,2,3,16; 1/10=7,14=-1,18=40/3(1); 99//99; 2/29=3/2; 3/5=1,6=6,11=2,16=1,25=1,30=1,74=-5/1,2,3; 4/5=5,16=3/1; 5/5=2,38=5/2; 7//1,2,3,16; 1/14=-1,18=40/3(-5); 2/29=3/2; 6/7=2,8=2,9=2,10=2,19=2,28=1/1; 99/9=1/99; 42

115 Input file Opt.gjf L1 L101, L103 L202 Structural optimization: Route with loop L202 L301, L302, L303 L301, L302, L303 L401 L401 L502 Loop! L502 L601 L701, L702, L703, L716 L103, converged? No L701, L702, L703, L716 L103, converged? No Yes. L202 L601 Yes. L9999 Output file: Opt.out L

116 Structural optimization: links and options Link 103 : Performs optimization using Berny algorithm Link 701 : 1-electron integral first or second derivatives Link 702 : 2-electron integral first or second derivatives (sp) Link 703 : 2-electron integral first or second derivatives (spdf) Link 716 : Processes information for optimizations and frequencies Example of option values for # B3LYP/6-31G OPT(Tight, Z-matrix) IOP(1/7) : Convergence of the 1 st derivative (forces). IOP(1/7=10) : 10^(-10), that is, Tight IOP(1/26) : Accuracy of function being optimized. IOP(1/26=3) : Fine grid accuracy for DFT (Energy 1.d-7, gradient 1.d-6) IOP(1/10) : Input of initial Hessian. IOP(1/10=7) : Use semiempirical force constants. IOP(2/29) : Update of coordinates from current Z-matrix. IOP(2/29=3): YES. IOP(7/29) : Mode of Use of L 716. IOP(7/29=1) Normal + Generate estimated initial force constants. 44

117 CH 2 O optimization: : Output (a) : Link 103, 1 st Link 103 : Performs optimization using Berny algorithm. GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad Berny optimization. Initialization pass ! Initial Parameters!! (Angstroms and Degrees)! ! Name Value Derivative information (Atomic Units)! ! A 1.28 estimate D2E/DX2!! B 1.1 estimate D2E/DX2!! C estimate D2E/DX2!! D estimate D2E/DX2! Trust Radius=3.00D-01 FncErr=1.00D-07 GrdErr=1.00D-06 Number of steps in this run= 20 maximum allowed number of steps= 100. GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad 45

118 CH 2 O optimization: : Output (b) : Link 103, 2 nd Link 103 : Performs optimization using Berny algorithm. GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad Berny optimization. Search for a local minimum. Step number 1 out of a maximum of 20 All quantities printed in internal units (Hartrees-Bohrs-Radians) Second derivative matrix not updated -- first step. The second derivative matrix: A B C D A B C D Eigenvalues RFO step: Lambda= D-03. Linear search not attempted -- first point. Variable Old X -DE/DX Delta X Delta X Delta X New X (Linear) (Quad) (Total) A B C D Item Value Threshold Converged? Maximum Force NO RMS Force NO Maximum Displacement NO RMS Displacement NO Predicted change in Energy= D-03 GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad 46

119 CH 2 O optimization: : Output (c) : Link 103, last Link 103 : Performs optimization using Berny algorithm. GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad Berny optimization. Search for a local minimum. Step number 5 out of a maximum of 20 All quantities printed in internal units (Hartrees-Bohrs-Radians) Update second derivatives using D2CorN and points Trust test= 9.73D-01 RLast= 7.12D-05 DXMaxT set to 3.00D-01 The second derivative matrix: A B C D A B C D Eigenvalues RFO step: Lambda= D+00. Quartic linear search produced a step of Variable Old X -DE/DX Delta X Delta X Delta X New X (Linear) (Quad) (Total) A B C D

120 CH 2 O optimization: : Output (d) : Link 103, last Link 103 : Performs optimization using Berny algorithm. Item Value Threshold Converged? Maximum Force YES RMS Force YES Maximum Displacement YES RMS Displacement YES Predicted change in Energy= D-12 Optimization completed. -- Stationary point found ! Optimized Parameters!! (Angstroms and Degrees)! ! Name Value Derivative information (Atomic Units)! ! A DE/DX = 0.0!! B DE/DX = 0.0!! C DE/DX = 0.0!! D DE/DX = 0.0! GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad Largest change from initial coordinates is atom Angstoms. 48

121 CH 2 O optimization: : Output (e) : Link 202, first Link 202 : Reorients coordinates, calculates symmetry Z-MATRIX (ANGSTROMS AND DEGREES) CD Cent Atom N1 Length/X N2 Alpha/Y N3 Beta/Z J C 2 2 O ( 1) 3 3 H ( 2) ( 4) 4 4 H ( 3) ( 5) ( 6) Z-Matrix orientation: Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z Distance matrix (angstroms): C O H H

122 CH 2 O optimization: : Output (f) : Link 202, first Link 202 : Reorients coordinates, calculates symmetry Interatomic angles: O2-C1-H3=121. O2-C1-H4=121. H3-C1-H4=118. Stoichiometry CH2O Framework group C2V[C2(CO),SGV(H2)] Deg. of freedom 3 Full point group C2V NOp 4 Largest Abelian subgroup C2V NOp 4 Largest concise Abelian subgroup C2 NOp 2 Standard orientation: Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z Rotational constants (GHZ):

123 CH 2 O optimization: : Output (g) : Link 202, last Link 202 : Reorients coordinates, calculates symmetry Z-MATRIX (ANGSTROMS AND DEGREES) CD Cent Atom N1 Length/X N2 Alpha/Y N3 Beta/Z J C 2 2 O ( 1) 3 3 H ( 2) ( 4) 4 4 H ( 3) ( 5) ( 6) Z-Matrix orientation: Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z Distance matrix (angstroms): C O H H

124 CH 2 O optimization: : Output (h) : Link 202, last Link 202 : Reorients coordinates, calculates symmetry Interatomic angles: O2-C1-H3= O2-C1-H4= H3-C1-H4= Stoichiometry CH2O Framework group C2V[C2(CO),SGV(H2)] Deg. of freedom 3 Full point group C2V NOp 4 Largest Abelian subgroup C2V NOp 4 Largest concise Abelian subgroup C2 NOp 2 Standard orientation: Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z Rotational constants (GHZ):

125 Tell G 03 what to do: keywords in # route card Input file 1 : single-point # B3LYP/6-31G Example: aldehyde 0 1 C1 O2 1 A H3 1 B 2 C H4 1 B 2 C 3 D A = 1.28 B = 1.10 C = D = Input file 2 : frequency # B3LYP/6-31G Freq Example: aldehyde 0 1 C1 O2 1 A H3 1 B 2 C H4 1 B 2 C 3 D A = 1.28 B = 1.10 C = D = Example of options of Freq: # B3LYP/6-31G Freq(Anharmonic, VibRot, Raman) 53

126 Gaussian 03 keywords ADMP AM1 Amber Archive B3LYP BD BOMD CASSCF CBS Keywords CBSExtrapolate CCD Charge ChkBasis CID CIS CNDO Complex Constants Counterpoise CPHF Density DensityFit Density Functional Methods Dreiding ExtendedHuckel External ExtraBasis Frozen Core Options Field FMM Force Frequency G* Keywords Gen Geom GFInput GFPrint Guess GVB Hartree-Fock Huckel INDO Integral IOp IRC IRCMax LSDA MaxDisk MINDO3 MM MNDO MP* Keywords Name NMR ONIOM Opt Output OVGF PBC PM3 Polar Population Pressure Prop Pseudo Punch 54

127 Gaussian 03 keywords (continued) QCISD ReArchive SAC-CI Scale Scan SCF SCRF SP Sparse Stable Symmetry TD Temperature Test TestMO TrackIO Transformation UFF Units Volume W1U Zindo Link 0 Commands Non-Standard Routes Obsolete Keywords Program Development Keywords ( 55

128 Gaussian 并行计算的简单实现 技能要求三年以上计算机使用经验, 了解操作系统的概念, 熟悉 Linux 或手头上有本 Linux 入门级的书 会上网查资料和在 BBS 上问问题 会使用 Gaussian 如果熟悉一些网络协议和基本原理更好 一 硬件 : 1:P42.8G/1G ram/120g HD 2:Ce2.0G/768M ram/80g HD 二 操作系统 : WinXP pro, Redhat Linux(Fedora Core 1, 从网上下载的光盘镜像 (.iso 文件 )) 三 软件 : VMware 天体验版 Linda 节点免费版 Gaussian98 并行版 ( 已编译好 ) 软件说明 : VMware 是一款口碑不错的虚拟机软件, 这个软件运行在一个操作系统上, 可以模拟出一个硬件环境, 然后将另外一套操作系统安装到这个模拟的环境中 Linda 是 Gaussian 并行计算所需要的一个在计算机节点间进行通讯, 传输计算数据和控制指令的软件 我得到的 Gaussian 是已经编译好的版本, 不知道是在什么环境下编译的, 但是可以在 Fedora Core 1 下运行 以上软件的详细介绍见各自的官方网站 四 安装步骤 1. 安装 VMware 如果你想直接在一台机器上安装 Linux, 则可以跳过 1,2 部分 通过一下这个网址按要求注册后可以获得 30 天体验版的 VMware5.0: 注册后 vmware 网站会给你注册的 发一封信, 其中有安装需要的序列号 同时会给出下载链接 2. 在 vmware 中创建 Linux 虚拟机菜单 File->New->Virtual Machine Virtual machine configuration 选 Custom Virtual machine format 选 New - workstation5 Guest operation system 就是你要安装的操作系统的类型, 选 Linux Memory 的设置要尽可能大些 一般可以指定自己机器一半的内存 Network connection 我选择的是 bridged, 这样你安装的操作系统就和你的网络上的机器具有相同的网络环境, 网络设置上也和你自己用的机器类似, 就是 IP 地址不同 注意最后设置 Disk file 时, 一定要自己指定 Disk file 的路径到一个剩余空间最大的驱动器, 在本例中,disk file 将达到 2G 多的大小 其他选项默认就可以了

129 3. 安装 Linux 注 : 这一部分及以下部分需要 Linux 基础知识才能看懂 双击 Devices 那一栏中的 CD-ROM,Connection 中选 Use ISO image,browse 到 Linux 第一张光盘的镜像文件 然后 Start this virtual machine. Linux 的安装不再详述, 这方面网上的资料很多, 只需注意如果你硬盘空间足够大, 选择安装所有模块就可以了, 这样省事 如果空间有限, 注意安装几个和 C 有关的编译器, 以及 rlogin,rsh,rexec,telnet 这些服务一定要安装 如果对 linux 不是很熟练, 建议把所有的管理工具和系统设置工具也装上 4. 安装 Gaussian 我得到的是编译好的版本, 所以无需再次编译, 直接解包解压缩就可以了 我把解包后的文件夹放到了 /usr 目录下 目录结构大概如下 : /usr/g98 /usr/g98/bsd /usr/g98/linda-exe 安装 Linda 下载的 Linda 是 rpm 文件, 在图形界面中双击就自动安装了, 默认安装到 /usr/sca 目录下, 我忘了字符模式下的安装方法 6. 设置 Linux 的 rlogin 这是很重要的一部分, 最终的目标就是在控制台中输入 rlogin noden 就可以直接登录到主机名为 noden 的计算机上 达到这个目的有以下几个要求 1) 所有节点机的用户名和密码必须相同, 而且用户名不能为 root 2) 所有节点机的 gaussian 和 linda 必须装在完全相同的目录下 3)/etc/hosts 中包括所有节点机的 IP 和主机名, 格式如下 : IP 主机名例如 : node node node )/etc/hosts.equiv 中包括所有节点机的主机名, 格式如下 : 主机名例如 : node1 node2 node3...

130 5)/home/ 你的用户名 /.rhosts 中的内容与 /etc/hosts.equiv 中的相同 6)/etc/securetty 这个文件中加入以下几行 : rlogin rsh rexec 这时候在这台机器上输入 : rlogin 这台机器的主机名就应当可以不提示输入密码登录 7. 设置 Linda 只需要一个文件, 在 /home/ 你的用户名 /.tsnet.config 中输入所有的节点的主机名, 和 /etc/hosts.equiv 中的内容一样 8. 设置计算节点在 /home/ 你的用户名 / 下建立一个文件, 名字任意, 内容为你在计算中将要用到的节点. 格式与 /etc/hosts.equiv 中的相同 在本例中这个文件为.tsnet.nodes 如下箭头所指处 9. 设置运行环境在 /home/ 你的用户名 /.bashrc 文件的最后加入以下内容 : export PATH=${PATH}:"./" export g98root=/usr export PATH=${PATH}:"${g98root}/sca/linda7.1/intel-linux2.4-ws3/bin" export LINDA_PATH="${g98root}/sca/linda7.1/intel-linux2.4-ws3/bin". ${g98root}/g98/bsd/g98.profile 在 /home/ 你的用户名 /.bash_profile 文件的最后加入以下内容 : export GAUSS_LFLAGS="-nodefile /home/ 你的用户名 /.tsnet.nodes" <-- export g98root=/usr export PATH=${PATH}:"./" export PATH=${PATH}:"${g98root}/g98/linda7.1/intel-linux2.4-ws3/bin" export LINDA_PATH="${g03root}/g03/linda7.1/intel-linux2.4-ws3/bin". ${g98root}/g98/bsd/g98.profile 注意以上两个内容应用到你的系统中时需要修改相应的路径 五 复制系统完成以后工作后将 Linux 操作系统关闭, 退出 VMware, 然后将虚拟机文件的整个文件夹以及扩展名为 vmdk 的 Disk file( 一般为 4 个 : 机器名.vmdk, 机器名 -s001.vmdk, 机器名 -s002.vmdk, 机器名 -s003.vmdk) 拷贝到另外一台也安装了 VMware 的机器上, 注意修改虚拟机文件 redhat.vmx 中 diskfile 的路径, 以及机器名.vmdk 中其他三个文件的路径. 复制完成后打开 vmware, 选择 File->Open,browse 到复制的机器名.vmdk, 打开 将网卡删除后再添加一块新的网卡 ( 要不然会因为 mac 地址和原来的相同而发生冲突 ) 启动系统, 利用系统设置工具修改 IP, 主机名 依次操作, 然后将所有的节点机都打开, 看能否相互无密码 rlogin

131 在机器上单独安装 Linux 可以参照修改 六 运行 Gaussian 建立一个 Gaussian 输入文件, 尽量复杂一些, 比如对一个几十个原子的分子进行结构优化, 这样好有时间察看运行状态 注意在输入文件中指定使用的 CPU 数目 : %NprocLinda= 你做的节点数目 ( 这个版本的 Linda 最多支持 4 个 ) 在其中任何一台计算机上, 打开控制台, 输入 : g98l < 输入文件.gjf 不指定 out 文件是为了即时察看运行状态 在本地机器进入迭代求解自洽场后 (l502.exel), 在其他的机器上察看进程 : ps -a 如果有 l502.exel, 那么就是成功了 参考文献 & 致谢 : [1] yinpenggang, 建立 PC-Cluster for QC, [2] Clarence K. Din and James C. Ianni,Gaussian03-on-Coffee Newbie Guide, Department of Chemistry,University of Pennsylvania,June 2005 [3] 水木社区 LinuxApp 版 [4] Elizerbeth, UniPod, Stanford.edu [5] 量子化学网, [6] VMware,VMware Inc. [7] sysop, 简易 cluster 架设, 动力论坛,

132 Gaussian 计算中分子总能量各项的意义 在 HFSCF 方法计算中, 体系采取固定核近似后, 体系的哈密顿算符分为电子哈密顿和核哈密顿两部分, 电子哈密顿分为三部分, 电子动能, 电子间斥能, 及核电排斥, 通过单电子近似, 得到单电子的 HF 方程, 自洽场的结果得到了电子的能量. 核哈密顿分为两部分, 核动能及排斥, 由于核固定动能为零, 只剩下一项, 因此体系的能量就是电子能量加上核斥能,, 这就是我们计算得到的分子总能量, 因为分子在 OK 时仍在平衡位置振动, 要加上零点校正. 最后结果即例子中的 : Sum of electronic and zero-point Energies= 在分子的热力学计算中, 包括了零点能的输出, 零点能是对分子的电子能量的矫正, 计算了在 0K 温度下的分子振动能量. 但我们计算的热力学数据并不总是在 OK 时, 为了计算在较高温度下的能量, 内能也要考虑到总能量中, 它包括平动能, 转动能, 和振动能. 注意在计算内能的时候, 已经考虑了零点能. 下面是一个计算实例 Temperature Kelvin. Pressure Atm. Zero-point correction= Thermal corection to Energy= Thermal correction to Enthalpy= Thermal correction to Gibbs Free Energy= Sum of electronic and zero-point Energies= Sum of electronic and thermal Energies= Sum of electronic and thermal Enthalpies= Sum of electronic and thermal Free Energies= 后面四行的四个能量分别为 E0, E, H, G. 计算为 E0=E(elec) + ZPE E=E0 + E(vib) + E(rol) + E(transl) H=E + RT G=H - TS 上例是在 K 和 1.0Atm 下计算的热力学状态函数,EO 即我们的分子体系总能量, 而在求 K 下的内能 E 时, 要加上平动能, 转动能, 和振动能, 由统计热力学知, 这三项处于激发能级, 注意这里的 ZPE 实际上就是对应 E(vib) 在 OK 时的能量, 但现在并非 0K, 有关能量随温度变化公式可参考统计热力著作, 最后得到的 E 就是平常所指的内能 U, 这正是统计热力学中对内能的定义. 对 H,G 状态函数, 不再叙述. 由于本版无法上传公式, 还请见谅, 有关内容请参看相关著作 Gaussian 中关于基组设置的简单介绍基组有两种, 一种是全电子基组, 另一种是价电子基组 价电子基组对内层的电子用包含了相对论效应的赝势 ( 缩写 PP, 也称为模型势, 有效核势, 缩写 ECP) 进行近似 以铟原子为例, 可以输入下面的命令来观察铟原子 3-21G 基组的形式 : # 3-21G GFPrint In 0,2 In 输出结果为 : 3-21G (6D, 7F)

133 S N=1 层,s 轨道 E E E E E E+00 SP N=2 层,s 和 p 轨道 E E E E E E E E E+00 SP N=3 层,s 和 p 轨道 E E E E E E E E E+00 D N=3 层,d 轨道 E E E E E E+00 SP N=4 层,s 和 p 轨道 E E E E E E E E E+00 D N=4 层,d 轨道 E E E E E E+00 以上都是内层电子轨道, 每个轨道用三个高斯型函数拟合, 也就是 3-21G 中的 3 SP N=5 层,s 和 p 轨道 E E E E E E+00 这是价电子轨道, 每个轨道用二个高斯型函数拟合, 也就是 3-21G 中的 2 SP N=6 层,s 和 p 轨道 E E E+01 这是空轨道, 每个轨道用一个高斯型函数拟合, 也就是 3-21G 中的 1 可见 3-21G 是全电子基组 再看看铟原子 LanL2DZ 基组的结果 : LANL2DZ (5D, 7F) S E E E E+01 S E E+01 P E E E E+01

134 P E E+01 这些都是价电子轨道 在 ECP 上加上了双 zeta(dz) 基组 所以 LanL2DZ 是价电子基组 在 GAUSSIAN 中,CEP 也是很重要的价电子基组, 并且 98 版中对元素的适用范围扩大到了 86 号元素 Rn 需要注意的是, 超过了 Ar 元素后, 每一种元素的 CEP-4G,CEP-31G, CEP-121G 是等价的 ( 和加了极化 * 的 CEP 还是有区别的 ) SDD 适用于除了 Fr 和 Ra 以外的所有元素, 但是结果的准确性并不太好 另外, 在下面的两个网址, 有几乎整个周期表元素的赝势下载, 不过我想不出那些短寿命的超铀元素有什么用 : 需要注意的是有些方法不需要输入基组, 像各种半经验方法,G1, G2 等等

135 Gaussian 中 IOP 的使用简介 在 Gaussian 的高级输入文件中, 经常可以看到如下格式的 IOP 指令 : Iop(n1/n2=n3) 说到底,Iop 就是 Gaussian 程序内部的一些高级指令, 从更加底层设置程序运行的各种参数 gaussian 是个傻瓜型软件, 所以很多比较深入的细节都要通过 IOP 来设置 上述指令中,n1 表示 overlay,n2 表示指令序号,n3 表示 n1/n2 指令的值 具体的 IOP 的 overlay 和指令号及所对应值的含义可以在 Gaussian 网上的 IOP 大全上找到 具体网址为 : 比如有 IOP(2/15=3) 的指令, 则我们找到 Overlay2 的第 15 个指令序号, 点击进入后可以看到如下的信息 IOp(2/15) SYMMETRY CONTROL. -1 Turns on symmetry; same as 0 for molecules but turns on assignment of space group ops. for PBC. 0 Leave symmetry in whatever state it is presently in. 1 Unconditionally turn symmetry off. Note that symm is still called, and will determine the framework group. However, the molecule is not oriented. 2 Bring the molecule to a symmetry orientation, but then disable further use of symmetry. 3 Don't even call symm. 4 Call Symm once with loose cutoffs, symmetrize the resulting coordinates, then confirm symmetry with tight cutoffs. 5 Recover the previous symmetry operations from the rwf, and confirm that the new structure has the same symmetry. 6 Same as 5, but get symmetry info from the chk. 00 Default (10) 10 Do re-orientation for PBC. 20 Suppress re-orientation for PBC. 100 Turn on symmetry operations for PBC. 由于我们将其值设为 3, 则表明计算中不要调用对称性判定的程序

136 另外常用的 IOP 的设置有 Iop(5/33=3,3/33=1) 用来输出 Fock 矩阵和 overlap 矩阵

137 Gaussian 中分子的几何构型 分子的几何构型 ************************************ 分子的几何构型 (Molecular Geometry) ************************************ 分子的平衡构型 (molecular equilibrium geometry) 是分子电子能量和核间排斥能量最小时分子的核排列 分子势能一个含有 N 个原子核的非线性分子的几何构型可以用 3N-6 个独立的核坐标决定, 分子的电子能量,U(q1,q2,,q3N-6) 是这些坐标的函数 U = Ee +VNN 注意到 3 个平移和 3 个转动自由度 ( 线性分子的转动自由度为 2) 对 U 是没有贡献的, 因 此对一个双原子分子,U 的表达式中仅仅保护一个变量, 即两个核之间的距离,U? 对一个多原子分子,U 是每两个原子核之间距离的函数, 是分子势能面 (potential energy surface, PES) 的一部分 对某一特定的分子核排列下 U 的计算被成为单点 (single-point) 计算, 因为这一计算仅仅涉及到分子 PES 上的一个点 一个大分子可能在其 PES 上有多个极小点, 对应于不同的平衡构象和鞍点 分子构象 (molecular conformation) 可以通过指定围绕单键的二面角的指得到 在能量极小点处的分子构象称为构型 (conformer) 几何构型优化从初始几何构型出发寻找 U 的极小值的过程称几何构型优化 (geometry optimization) 或者能量极小化 (energy minimization) 极小化的算法同时计算 U 和 U 梯度 在一个局部最小点,U 的 3N-6 个偏微分都是 0 PES 上 U = 0 的点称为稳定点 (statio nary point) 或者判据点 (critical point), 它可以是极小点, 极大点或者鞍点 除了 U 之外, 一些最小化方法使用到 U 的二阶偏微分, 从而生成 Hessian 矩阵, 又称为 力常数 (force constant) 矩阵, 因为 d^2u/qi^2 = fi 为力常数 如果一个稳定点是电子能量面上的一个极小点, 其力常数矩阵的所有特征值都是正值 然而, 若一个稳定点是过渡态 (transition state, TS), 其中一个特征值是负值 Newton-Rapson Newton-Rapson 方法是一种非常有效的寻找多变量函数的局部极小点的算法, 它将函数用 Taylor 展开到二次项, 包括函数的一次和二次微分, 并以此作为函数的近似 Quasi-Newton-Rapson 计算自洽场 (self consistent field, SCF) 能量的二阶微分是非常耗时的, 因此在优化时经常使用一种修正的方法, 即 quasi-newton( 或 quasi-newton-rapson) 方法 这种方法在每一步优化中通过计算梯度对 Hessian 值进行初始估算

138 优化方法为了优化几何构型, 要先对平衡构型做一个估算, 通常使用键长和键角的经验值 此外, 我们还要选择好适当的方法和基组, 然后就可以在所估算的平衡构型附近进行极小点的搜索了 软件对电子 Schrodinger 方程进行求解并得到 U 及其在初始构型处的梯度 通过这些数值对 Hessian 矩阵进行估算, 并调整 3N-6 个核坐标以得到在初始构型附近但能量更低的新的分子构型 对新构型的 U 和 U 进行计算以继续改进分子坐标使分子构型更接近于极小点 SCF 计算 对新的分子构型不断重复, 直到 U 和 0 之间的差足够小能满足对极小点的判据 (critia) 一般而言, 需要进行 3N-6 至两倍的 SCF 循环次数以找到一个极小点 (Gaussian 默认的循环次数为两倍要优化的变量, 有时候对 OPT=Tight/Vtight 需要增大循环次数 ) [Schafer, J. Mol. Struct., 100, 51 (1983)] 中间体局域最小点代表反应物, 产物或者一个中间体 (Intermediate), 该中间体在多步反应中是既是一个反应的产物, 同时又是另一个反应的反应物 因为反应中间体通常很快被反应掉或者其寿命很短而不能被光谱仪器检测到, 中间体结构和能量的计算是计算化学中一个重要的应用 反应表面为了得到一个完整的反应表面 U, 假设我们需要在解电子 Schrodinger 方程时在反应面上对 3N-6 个变量各取约 10 个点, 这样一共要进行 103N-6 个计算 实际上并非如此, 我们利用反应面别的重要性质来得到局部最小点和过渡态 对于利用 U 表面特征的分析算法可以参照 Szabo & Ostlund, Modern Quantum Chemistry, pp TS 计算算法为寻找过渡态, 在 Gaussian 程序中有几种方法可供选择 TS 在 Gaussian 中的 OPT=(TS,CalcFC) 算法, 需要输入一个估计的过渡态结构, 该过渡态是反应物和产物之间的一个中间体 CalcFC 指定对初始结构进行 Hessian 数值计算 ( 用 Newton-Rapson 方法 ), 而不是默认地对其二阶微分进行估算 ( 用 quasi-newton-rapson 方法 ) 这一步骤增加了计算时间, 但增大了 TS 寻找地成功率 QST2 另一个 Gaussian TS 搜索算法, 由 OPT=QST2 指定, 让程序自己通过反应物和产物的结构计算一个 TS 的初始结构 然后 Gaussian 用这一初始 TS 作为 TS 搜索的起始点 QST3 第三种 Gaussian TS 搜索方法, 用 OPT=QST3 指定, 在反应物, 产物和用户给定的初始 TS 基础上进行搜索 判断一个鞍点和 TS 构型, 我们需要进行频率计算 如果我们想知道一个鞍点是否是 TS 构型, 并要研究反应机理, 需要从鞍点出发, 沿 U 表面的反应路径分别得到反应物和产物

139 对一个 TS, 其虚频所对应的简振模式下的原子位移方向分别指象反应物和产物 当计算出 TS 构型和振动频率以及能垒高度后, 就可以通过过渡态理论 (Transition State Theory) 对反应速率进行计算了 一阶鞍点如果一个稳定点, 即电子能量面上的一个极小点, 是一个过渡态 TS, 其 Hessian 矩阵有且仅有一个负特征值 这个一阶鞍点条件意味着 TS 的能量在一个方向 ( 负特征值 ) 取得最大值, 这一方向即反应坐标方向 ; 而在其它方向上能量都是最小值 ( 正特征值 ) TS 振动模式一个 TS 有 3N-7( 线性分子为 3N-6) 个标准振动, 比正常的分子少一个 负的力常数的物理意义在于, 与 Hooke 定律所定义的力不同, 与 TS 虚频 v# 相关的力沿反应坐标, 指向相反的, 朝着产物的方向 频率 v#, 沿着反应坐标, 是一个虚数

140 用 Gview 设定体系的对称性 1 输入一个分子构型 [ 以 Ni(N5)2 为例, 点击 edit-point group, 如下图 2 出现如下图的窗口, 选中下图中对话框的左上角的方框, 对体系施加体系对称性 ;

141 3 体系当前的对称性是 C1, 选用更松的判定标准, 如下图 ; 4 改变判定标准后, 可以发现体系可能的对称性有好几个, 选定想要的对称性 ; 5 点击 symmetries, 然后 OK 就可以了

142 感谢董昊的帮助!

143 思路与 Mulliken 布居分析一样, 但在以下几个方面有所不同 (1) 计算原子上的电荷时所用的原子轨道不同 : Mulliken 分析用的是构建分子轨道的原子轨道, 因而对基组有依赖性 ; 而 NBO 分析用的是 NAO [natural atomic orbital] 原子轨道 NAO 是在对角化密度矩阵中一中心的块矩阵得到的 [ 原话 :diagonalization of one-center blocks of the density matrix]; Since the NAOs form an orthonormal set, completely spann ing the space of (generally nonorthogonal ) basis orbitals, the natural populations are inher ently positive, and sum correctly to the total number of electrons. 重叠区电荷的分割是通过权重而不是均分的方式分给两个原子. (2) 给出了分子中各原子的 NHO (Natural Hybrid Orbitals), 易于为普通的化学工作者掌握和解决问题. 构建 NHO (Natural Hybrid Orbitals) 的步骤如下 : [1]find the density matrix P in a basis set of natural atomic orbitals and diagonalize eath atomic subblock PAA to find the lone-pair hybrid on that center. NBO 程序默认布居数超过 1.999e 的轨道是 core orbital, 超过 1.90e 的轨道是 lone pair orbital. [2]for each pair of atoms A L, form the two-center density matrix P (AL). NBO 程序默认只有布居数超过 1.90e 的轨道才被考虑, 如果电子对的数目不够, NBO 程序会继续搜索三中心的 block, 直到体系的总电子数被满足 [3] symmetrically orthogonalize the hybrids found in step 1 and 2 to find the final natural hybrids orbitals. (3) 可以定量的给出轨道之间的相互作用能.

144 Thermochemistry in Gaussian Joseph W. Ochterski, Ph.D. c 2000, Gaussian, Inc. June 2, 2000 Abstract The purpose of this paper is to explain how various thermochemical values are computed in Gaussian. The paper documents what equations are used to calculate the quantities, but doesn t explain them in great detail, so a basic understanding of statistical mechanics concepts, such as partition functions, is assumed. Gaussian thermochemistry output is explained, and a couple of examples, including calculating the enthalpy and Gibbs free energy for a reaction, the heat of formation of a molecule and absolute rates of reaction are worked out. Contents 1 Introduction 2 2 Sources of components for thermodynamic quantities Contributions from translation Contributions from electronic motion Contributions from rotational motion Contributions from vibrational motion Thermochemistry output from Gaussian Output from a frequency calculation Output from compound model chemistries Worked-out Examples Enthalpies and Free Energies of Reaction Rates of Reaction Enthalpies and Free Energies of Formation Summary 17 1

145 1 Introduction The equations used for computing thermochemical data in Gaussian are equivalent to those given in standard texts on thermodynamics. Much of what is discussed below is covered in detail in Molecular Thermodynamics by McQuarrie and Simon (1999). I ve crossreferenced several of the equations in this paper with the same equations in the book, to make it easier to determine what assumptions were made in deriving each equation. These cross-references have the form [McQuarrie, 7-6, Eq. 7.27] which refers to equation 7.27 in section 7-6. One of the most important approximations to be aware of throughout this analysis is that all the equations assume non-interacting particles and therefore apply only to an ideal gas. This limitation will introduce some error, depending on the extent that any system being studied is non-ideal. Further, for the electronic contributions, it is assumed that the first and higher excited states are entirely inaccessible. This approximation is generally not troublesome, but can introduce some error for systems with low lying electronic excited states. The examples in this paper are typically carried out at the HF/STO-3G level of theory. The intent is to provide illustrative examples, rather than research grade results. The first section of the paper is this introduction. The next section of the paper, I give the equations used to calculate the contributions from translational motion, electronic motion, rotational motion and vibrational motion. Then I describe a sample output in the third section, to show how each section relates to the equations. The fourth section consists of several worked out examples, where I calculate the heat of reaction and Gibbs free energy of reaction for a simple bimolecular reaction, and absoloute reaction rates for another. Finally, an appendix gives a list of the all symbols used, their meanings and values for constants I ve used. 2 Sources of components for thermodynamic quantities In each of the next four subsections of this paper, I will give the equations used to calculate the contributions to entropy, energy, and heat capacity resulting from translational, electronic, rotational and vibrational motion. The starting point in each case is the partition function q(v, T ) for the corresponding component of the total partition function. In this section, I ll give an overview of how entropy, energy, and heat capacity are calculated from the partition function. The partition function from any component can be used to determine the entropy contribution S from that component, using the relation [McQuarrie, 7-6, Eq. 7.27]: ( ) ( ) q(v, T ) ln q S = Nk B + Nk B ln + Nk B T N T The form used in Gaussian is a special case. First, molar values are given, so we can divide by n = N/N A, and substitute N A k B = R. We can also move the first term into the 2 V

146 logarithm (as e), which leaves (with N = 1): ( ) ln q S = R + R ln (q(v, T )) + RT T ( ) ln q = R ln (q(v, T )e) + RT T V ( ( ) ) ln q = R ln(q t q e q r q v e) + T T The internal thermal energy E can also be obtained from the partition function [Mc- Quarrie, 3-8, Eq. 3.41]: ( ) ln q E = Nk B T 2, (2) T V and ultimately, the energy can be used to obtain the heat capacity [McQuarrie, 3.4, Eq. 3.25]: ( ) E C V = (3) T N,V These three equations will be used to derive the final expressions used to calculate the different components of the thermodynamic quantities printed out by Gaussian. 2.1 Contributions from translation The equation given in McQuarrie and other texts for the translational partition function is [McQuarrie, 4-1, Eq. 4.6]: ( ) 3/2 2πmkB T q t = V. The partial derivative of q t with respect to T is: ( ) ln qt = 3 T 2T h 2 which will be used to calculate both the internal energy E t and the third term in Equation 1. The second term in Equation 1 is a little trickier, since we don t know V. However, for an ideal gas, P V = NRT = ( ) n NA N A k B T, and V = k BT. Therefore, P ( ) 3/2 2πmkB T k B T q t = h 2 P. which is what is used to calculate q t in Gaussian. Note that we didn t have to make this substitution to derive the third term, since the partial derivative has V held constant. The translational partition function is used to calculate the translational entropy (which includes the factor of e which comes from Stirling s approximation): S t = ( ( )) 3 R ln(q t e) + T 2T = R(ln q t /2). 3 V V V (1)

147 The contribution to the internal thermal energy due to translation is: ( ) ln q E t = N A k B T 2 T ( ) 3 = RT 2 2T = 3 2 RT V Finally, the constant volume heat capacity is given by: C t = E t T = 3 2 R. 2.2 Contributions from electronic motion The usual electronic partition function is [McQuarrie, 4-2, Eq. 4.8]: q e = ω 0 e ɛ 0/k B T + ω 1 e ɛ 1/k B T + ω 2 e ɛ 2/k B T + where ω is the degeneracy of the energy level, ɛ n is the energy of the n-th level. Gaussian assumes that the first electronic excitation energy is much greater than k B T. Therefore, the first and higher excited states are assumed to be inaccessible at any temperature. Further, the energy of the ground state is set to zero. These assumptions simplify the electronic partition function to: q e = ω 0, which is simply the electronic spin multiplicity of the molecule. The entropy due to electronic motion is: S e = R ( ln q e + T = R (ln q e + 0). ( ) ln qe Since there are no temperature dependent terms in the partition function, the electronic heat capacity and the internal thermal energy due to electronic motion are both zero. 2.3 Contributions from rotational motion The discussion for molecular rotation can be divided into several cases: single atoms, linear polyatomic molecules, and general non-linear polyatomic molecules. I ll cover each in order. For a single atom, q r = 1. Since q r does not depend on temperature, the contribution of rotation to the internal thermal energy, its contribution to the heat capacity and its contribution to the entropy are all identically zero. 4 T V )

148 For a linear molecule, the rotational partition function is [McQuarrie, 4-6, Eq. 4.38]: q r = 1 σ r ( T Θ r ) where Θ r = h 2 /8π 2 Ik B. I is the moment of inertia. The rotational contribution to the entropy is S r = ( ( ) ) ln qr R ln q r + T T V = R(ln q r + 1). The contribution of rotation to the internal thermal energy is ( ) ln E r = RT 2 qr T V ( ) 1 = RT 2 T = RT and the contribution to the heat capacity is C r = ( ) Er T = R. For the general case for a nonlinear polyatomic molecule, the rotational partition function is [McQuarrie, 4-8, Eq. 4.56]: q r = π1/2 σ r ( T 3/2 ) (Θ r,x Θ r,y Θ r,z ) 1/2 ) Now we have ( ) ln q = 3, so the entropy for this partition function is T V 2T ( ( ) ) ln qr S r = R ln q r + T T = R(ln q r ). Finally, the contribution to the internal thermal energy is ( ) ln E r = RT 2 qr T V ( ) 3 = RT 2 2T = 3 2 RT 5 V V

149 and the contribution to the heat capacity is C r = ( ) Er T V = 3 2 R. The average contribution to the internal thermal energy from each rotational degree of freedom is RT/2, while it s contribution to C r is R/ Contributions from vibrational motion The contributions to the partition function, entropy, internal energy and constant volume heat capacity from vibrational motions are composed of a sum (or product) of the contributions from each vibrational mode, K. Only the real modes are considered; modes with imaginary frequencies (i.e. those flagged with a minus sign in the output) are ignored. Each of the 3n atoms 6 (or 3n atoms 5 for linear molecules) modes has a characteristic vibrational temperature, Θ v,k = hν K /k B. There are two ways to calculate the partition function, depending on where you choose the zero of energy to be: either the bottom of the internuclear potential energy well, or the first vibrational level. Which you choose depends on whether or not the contributions arising from zero-point energy will be coputed separately or not. If they are computed separately, then you should use the bottom of the well as your reference point, otherwise the first vibrational energy level is the appropriate choice. If you choose the zero reference point to be the bottom of the well (BOT), then the contribution to the partition function from a given vibrational mode is [McQuarrie, 4-4, Eq. 4.24]: e Θ v,k/2t q v,k = 1 e Θ v,k/t and the overall vibrational partition function is[mcquarrie, 4-7, Eq. 4.46]: q v = K e Θ v,k/2t 1 e Θ v,k/t On the other hand, if you choose the first vibrational energy level to be the zero of energy (V=0), then the partition function for each vibrational level is q v,k = and the overall vibrational partition function is: q v = K 1 1 e Θ v,k/t 1 1 e Θ v,k/t Gaussian uses the bottom of the well as the zero of energy (BOT) to determine the other thermodynamic quantities, but also prints out the V=0 partition function. Ultimately, the 6

150 only difference between the two references is the additional factor of Θ v,k /2, (which is the zero point vibrational energy) in the equation for the internal energy E v. In the expressions for heat capacity and entropy, this factor disappears since you differentiate with respect to temperature (T ). The total entropy contribution from the vibrational partition function is: ( ( ) ) ln q S v = R ln(q v ) + T T V ( ( Θ v,k = R ln(q v ) + T K 2T + (Θ v,k /T 2 )e Θ )) v,k/t 2 K 1 e Θ v,k/t ( ( ) ( Θv,K = R K 2T + ln(1 e Θ v,k/t Θ v,k ) + T K 2T + 2 K ( ( = R ln(1 e Θv,K/T (Θ v,k /T )e Θ )) v,k/t ) + K K 1 e Θ v,k/t = R ( ) Θv,K /T K e Θ v,k/t 1 ln(1 e Θ v,k/t ) (Θ v,k /T 2 )e Θ v,k/t 1 e Θ v,k/t To get from the fourth line to the fifth line in the equation above, you have to multiply by e Θ v,k /T e Θ v,k /T. The contribution to internal thermal energy resulting from molecular vibration is E v = R K ( ) 1 Θ v,k e Θ v,k/t 1 Finally, the contribution to constant volume heat capacity is )) C v = R K e Θ v,k/t ( Θv,K /T ) 2 e Θ v,k/t 1 Low frequency modes (defined below) are included in the computations described above. Some of these modes may be internal rotations, and so may need to be treated separately, depending on the temperatures and barriers involved. In order to make it easier to correct for these modes, their contributions are printed out separately, so that they may be subtracted out. A low frequency mode in Gaussian is defined as one for which more than five percent of an assembly of molecules are likely to exist in excited vibrational states at room temperature. In other units, this corresponds to about 625 cm 1, Hz, or a vibrational temperature of 900 K. It is possible to use Gaussian to automatically perform some of this analysis for you, via the Freq=HindRot keyword. This part of the code is still undergoing some improvements, so I will not go into it in detail. See P. Y. Ayala and H. B. Schlegel, J. Chem. Phys (1998) and references therein for more detail about the hindered rotor analysis in Gaussian and methods for correcting the partition functions due to these effects. 7

151 3 Thermochemistry output from Gaussian This section describes most of the Gaussian thermochemistry output, and how it relates to the equations I ve given above. 3.1 Output from a frequency calculation In this section, I intentionally used a non-optimized structure, to show more output. For production runs, it is very important to use structures for which the first derivatives are zero in other words, for minima, transition states and higher order saddle points. It is occasionally possible to use structures where one of the modes has non-zero first derivatives, such as along an IRC. For more information about why it is important to be at a stationary point on the potential energy surface, see my white paper on Vibrational Analysis in Gaussian. Much of the output is self-explanatory. I ll only comment on some of the output which may not be immediately clear. Some of the output is also described in Exploring Chemistry with Electronic Structure Methods, Second Edition by James B. Foresman and Æleen Frisch Thermochemistry Temperature Kelvin. Pressure Atm. Atom 1 has atomic number 6 and mass Atom 2 has atomic number 6 and mass Atom 3 has atomic number 1 and mass Atom 4 has atomic number 1 and mass Atom 5 has atomic number 1 and mass Atom 6 has atomic number 1 and mass Atom 7 has atomic number 1 and mass Atom 8 has atomic number 1 and mass Molecular mass: amu. The next section gives some characteristics of the molecule based on the moments of inertia, including the rotational temperature and constants. The zero-point energy is calculated using only the non-imaginary frequencies. Principal axes and moments of inertia in atomic units: EIGENVALUES X Y Z THIS MOLECULE IS AN ASYMMETRIC TOP. ROTATIONAL SYMMETRY NUMBER 1. ROTATIONAL TEMPERATURES (KELVIN)

152 ROTATIONAL CONSTANTS (GHZ) Zero-point vibrational energy (Joules/Mol) (Kcal/Mol) If you see the following warning, it can be a sign that one of two things is happening. First, it often shows up if your structure is not a minimum with respect to all non-imaginary modes. You should go back and re-optimize your structure, since all the thermochemistry based on this structure is likely to be wrong. Second, it may indicate that there are internal rotations in your system. You should correct for errors caused by this situation. WARNING-- EXPLICIT CONSIDERATION OF 1 DEGREES OF FREEDOM AS VIBRATIONS MAY CAUSE SIGNIFICANT ERROR Then the vibrational temperatures and zero-point energy (ZPE): VIBRATIONAL TEMPERATURES: (KELVIN) Zero-point correction= (Hartree/Particle) Each of the next few lines warrants some explanation. All of them include the zeropoint energy. The first line gives the correction to the internal thermal energy, E tot = E t + E r + E v + E e. Thermal correction to Energy= The next two lines, respectively, are H corr = E tot + k B T and where S tot = S t + S r + S v + S e. G corr = H corr T S tot, Thermal correction to Enthalpy= Thermal correction to Gibbs Free Energy= The Gibbs free energy includes P V = NRT, so when it s applied to calculating G for a reaction, NRT P V is already included. This means that G will be computed correctly when the number of moles of gas changes during the course of a reaction. The next four lines are estimates of the total energy of the molecule, after various corrections are applied. Since I ve already used E to represent internal thermal energy, I ll use E 0 for the total electronic energy. Sum of electronic and zero-point energies = E 0 + E ZPE 9

153 Sum of electronic and thermal energies = E 0 + E tot Sum of electronic and thermal enthalpies = E 0 + H corr Sum of electronic and thermal free energies = E 0 + G corr Sum of electronic and zero-point Energies= Sum of electronic and thermal Energies= Sum of electronic and thermal Enthalpies= Sum of electronic and thermal Free Energies= The next section is a table listing the individual contributions to the internal thermal energy (E tot ), constant volume heat capacity (C tot ) and entropy (S tot ). For every low frequency mode, there will be a line similar to the last one in this table (labeled VIBRATION 1). That line gives the contribution of that particular mode to E tot, C tot and S tot. This allows you to subtract out these values if you believe they are a source of error. E (Thermal) CV S KCAL/MOL CAL/MOL-KELVIN CAL/MOL-KELVIN TOTAL ELECTRONIC TRANSLATIONAL ROTATIONAL VIBRATIONAL VIBRATION Finally, there is a table listing the individual contributions to the partition function. The lines labeled BOT are for the vibrational partition function computed with the zero of energy being the bottom of the well, while those labeled with (V=0) are computed with the zero of energy being the first vibrational level. Again, special lines are printed out for the low frequency modes. Q LOG10(Q) LN(Q) TOTAL BOT D TOTAL V= D VIB (BOT) D VIB (BOT) D VIB (V=0) D VIB (V=0) D ELECTRONIC D TRANSLATIONAL D ROTATIONAL D

154 3.2 Output from compound model chemistries This section explains what the various thermochemical quantities in the summary out of a compound model chemistry, such as CBS-QB3 or G2, means. I ll use a CBS-QB3 calculation on water, but the discussion is directly applicable to all the other compound models available in Gaussian. The two lines of interest from the output look like: CBS-QB3 (0 K)= CBS-QB3 Energy= CBS-QB3 Enthalpy= CBS-QB3 Free Energy= Here are the meanings of each of those quantities. CBS-QB3 (0 K): This is the total electronic energy as defined by the compound model, including the zero-point energy scaled by the factor defined in the model. CBS-QB3 Energy: This is the total electronic energy plus the internal thermal energy, which comes directly from the thermochemistry output in the frequency part of the output. The scaled zero-point energy is included. CBS-QB3 Enthalpy: This is the total electronic energy plus H corr, as it is described above with the unscaled zero-point energy removed. This sum is appropriate for calculating enthalpies of reaction, as described below. CBS-QB3 Free Energy: This is the total electronic energy plus G corr, as it is described above, again with the unscaled zero-point energy removed. This sum is appropriate for calculating Gibbs free energies of reaction, also as described below. 4 Worked-out Examples In this section I will show how to use these results to generate various thermochemical information. I ve run calculations for each of the reactants and products in the reaction where ethyl radical abstracts a hydrogen atom from molecular hydrogen: C 2 H 5 + H 2 C 2 H 6 + H as well as for the transition state (all at 1.0 atmospheres and K). The thermochemistry output from Gaussian is summarized in Table 1. Once you have the data for all the relevant species, you can calculate the quantities you are interested in. Unless otherwise specified, all enthalpies are at K. I ll use 298K K to shorten the equations. 11

155 C 2 H 5 H 2 C 2 H 6 H C E E ZPE E tot H corr G corr E 0 + E ZPE E 0 + E tot E 0 + H corr E 0 + G corr Table 1: Calculated thermochemistry values from Gaussian for the reaction C 2 H 5 + H 2 C 2 H 6 + H. All values are in Hartrees. 4.1 Enthalpies and Free Energies of Reaction The usual way to calculate enthalpies of reaction is to calculate heats of formation, and take the appropriate sums and difference. r H (298K) = f Hprod(298K) f Hreact(298K). products reactants However, since Gaussian provides the sum of electronic and thermal enthalpies, there is a short cut: namely, to simply take the difference of the sums of these values for the reactants and the products. This works since the number of atoms of each element is the same on both sides of the reaction, therefore all the atomic information cancels out, and you need only the molecular data. For example, using the information in Table 1, the enthalpy of reaction can be calculated simply by r H (298K) = (E 0 + H corr ) products (E 0 + H corr ) reactants = (( ) ( )) = = 8.08 kcal/mol The same short cut can be used to calculate Gibbs free energies of reaction: r H (298K) = (E 0 + G corr ) products (E 0 + G corr ) reactants = (( ) ( )) = = kcal/mol 4.2 Rates of Reaction In this section I ll show how to compute rates of reaction using the output from Gaussian. I ll be using results derived from transition state theory in section 28-8 of Physical Chemistry, 12

156 Compound E 0 E 0 + G corr HF HD Cl HCl DCl F FHCl FDCl Table 2: Total HF/STO-3G electronic energies and sum of electronic and thermal free energies for atoms molecules and transition state complex of the reaction FH + Cl F + HCl and the deuterium substituted analog. A Molecular Approach by D. A. McQuarrie and J. D. Simon. The key equation (number 28.72, in that text) for calculating reaction rates is k(t ) = k BT hc e G /RT I ll use c = 1 for the concentration. For simple reactions, the rest is simply plugging in the numbers. First, of course, we need to get the numbers. I ve run HF/STO-3G frequency calculation for reaction FH + Cl F + HCl and also for the reaction with deuterium substituted for hydrogen. The results are summarized in Table 2. I put the total electronic energies of each compound into the table as well to illustrate the point that the final geometry and electronic energy are independent of the masses of the atoms. Indeed, the cartesian force constants themselves are independent of the masses. Only the vibrational analysis, and quantities derived from it, are mass dependent. The first step in calculating the rates of these reactions is to compute the free energy of activation, G ((H) is for the hydrogen reaction, (D) is for deuterium reaction): G (H) = ( ) = Hartrees = = kcal/mol G (D) = ( ) = Hartrees = = kcal/mol Then we can calculate the reaction rates. The values for the constants are listed in the appendix. I ve taken c = 1. k(298, H) = k BT hc e G /RT 13

157 = ( ) (298.15) exp (1) = e = 1.38s 1 k(298, D) = ( ) exp = e = s 1 So we see that the deuterium reaction is indeed slower, as we would expect. Again, these calculations were carried out at the HF/STO-3G level, for illustration purposes, not for research grade results. More complex reactions will need more sophisticated analyses, perhaps including careful determination of the effects of low frequency modes on the transition state, and tunneling effects. 4.3 Enthalpies and Free Energies of Formation Calculating enthalpies of formation is a straight-forward, albeit somewhat tedious task, which can be split into a couple of steps. The first step is to calculate the enthalpies of formation ( f H (0K)) of the species involved in the reaction. The second step is to calculate the enthalpies of formation of the species at 298K. Calculating the Gibbs free energy of reaction is similar, except we have to add in the entropy term: f G (298) = f H (298K) T (S (M, 298K) S (X, 298K)) To calculate these quantities, we need a few component pieces first. In the descriptions below, I will use M to stand for the molecule, and X to represent each element which makes up M, and x will be the number of atoms of X in M. Atomization energy of the molecule, D 0 (M): These are readily calculated from the total energies of the molecule (E 0 (M)), the zeropoint energy of the molecule (E ZPE (M)) and the constituent atoms: D0 (M) = atoms xe 0 (X) E 0 (M) E ZPE (M) Heats of formation of the atoms at 0K, f H (X, 0K): I have tabulated recommended values for the heats of formation of the first and second row atomic elements at 0K in Table 3. There are (at least) two schools of thought with respect to the which atomic heat of formation data is most appropriate. Some authors prefer to use purely experimental data for the heats of formation (Curtiss, et. al., J. 14

158 Chem. Phys. 106, 1063 (1997)). Others (Ochterski, et. al., J. Am. Chem. Soc. 117, (1995)), prefer to use a combination of experiment and computation to obtain more accurate values. The elements of concern are boron, beryllium and silicon. I have arranged Table 3 so that either may be used; the experimental results are on the top, while the computational results for the three elements with large experimental uncertainty are listed below that. Enthalpy corrections of the atomic elements, H X(298K) H X(0K) : The enthalpy corrections of the first and second row atomic elements are also included in Table 3. These are used to convert the atomic heats of formation at 0K to those at K, and are given for the elements in their standard states. Since they do not depend on the accuracy of the heat of formation of the atom, they are the same for both the calculated and experimental data. This is generally not the same as the output from Gaussian for a calculation on an isolated gas-phase atom. These values are referenced to the standard states of the elements. For example, the value for hydrogen atom is 1.01 kcal/mol. This is (HH 2 (298K) HH 2 (0K))/2, and not HH(298K) HH(0K), which is what Gaussian calculates. Enthalpy correction for the molecule, H M(298K) H M(0K) : For a molecule, this is H corr E ZPE (M), where H corr is the value printed out in the line labeled Thermal correction to Enthalpy in Gaussian output. Remember, though, that the output is in Hartrees/particle, and needs to be converted to kcal/mol. The conversion factor is 1 Hartree = kcal/mol. Entropy for the atoms, S X(298K) : These are summarized in Table 3. The values were taken from the JANAF tables (M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald, and A. N. Syverud, J. Phys. Ref. Data 14 Suppl. No. 1 (1985)). Again, these values are referenced to the standard states of the elements, and will disagree with values Gaussian calculates for the gas-phase isolated atoms. Entropy for the molecule, S M(298K) : Gaussian calculates G = H T S, and prints it out on the line labeled Thermal correction to Gibbs Free Energy. The entropy can be teased out of this using S = (H G)/T. Putting all these pieces together, we can finally take the steps necessary to calculate f H (298K) and f G (298K): 1. Calculate f H (M, 0K) for each molecule: f H (M, 0K) = x f H (X, 0K) D 0 (M) atoms = ( ) x f H (X, 0K) xe 0 (X) E 0 (M) atoms atoms 15

159 Experimental a Element f H (0K) H (298K) H (0K) S (298K) H 51.63± ±0.004 Li 37.69± ±0.006 Be 76.48± ±0.005 B ± ±0.008 C ± ±0.21 N ± ±0.01 O 58.99± ±0.005 F 18.47± ±0.001 Na 25.69± ±0.006 Mg 34.87± ± d Al 78.23± ±0.01 Si ± ±0.008 P 75.42± ±0.01 S 65.66± ±0.008 Cl 28.59± ±0.001 Calculated b Element f H (0K) H (298K) H (0K) S (298K) Be 75.8 ± ±0.005 B ± ±0.008 Si ± ±0.008 Table 3: Experimental and calculated enthalpies of formation of elements (kcal mol 1 ) and entropies of atoms (cal mol 1 K 1 ). a Experimental enthalpy values taken from J. Chem. Phys. 106, 1063 (1997). b Calculated enthalpy values taken from J. Am. Chem. Soc. 117, (1995). H (298K) H (0K) is the same for both calculated and experimental, and is taken for elements in their standard states. c Entropy values taken from JANAF Thermochemical Tables: M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald, and A. N. Syverud, J. Phys. Ref. Data 14 Suppl. No. 1 (1985). d No error bars are given for Mg in JANAF. 2. Calculate f H (M, 298K) for each molecule: f H (M, 298K) = f H (M, 0K) + (HM(298K) HM(0K)) x(hx(298k) HX(0K)) atoms 3. Calculate f G (M, 298K) for each molecule: f G (M, 298K) = r H (298K) (S (M, 298K) S (X, 298K)) Here is a worked out example, where I ve calculated f H (298K) for the reactants and products in the reaction I described above. 16

160 First, I ll calculate f H (0K) for one of the species: f H (C 2 H 6, 0K) = (2 ( ) + 6 ( ) ( )) = = 9.56 The next step is to calculate the f H (298K): f H (C 2 H 6, 298K) = ( ) ( ) = = 5.70 To calculate the Gibbs free energy of formation, we have (the factors of 1000 are to convert kcal to or from cal) : f G (C 2 H 6, 298K) = ( ( )/ ( )/1000 = ( ) = Summary The essential message is this: the basic equations used to calculate thermochemical quantities in Gaussian are based on those used in standard texts. Since the vibrational partition function depends on the frequencies, you must use a structure that is either a minimum or a saddle point. For electronic contributions to the partition function, it is assumed that the first and all higher states are inaccessible at the temperature the calculation is done at. The data generated by Gaussian can be used to calculate heats and free energies of reactions as well as absolute rate information. 17

161 Appendix Symbols C e C r = contribution to the heat capacity due to electronic motion = contribution to the heat capacity due to rotational motion C tot = total heat capacity (C t + C r + C v + C e ) C t C v E e E r = contribution to the heat capacity due to translation = contribution to the heat capacity due to vibrational motion = internal energy due to electronic motion = internal energy due to rotational motion E tot = total internal energy (E t + E r + E v + E e ) E t E v G corr H corr = internal energy due to translation = internal energy due to vibrational motion = correction to the Gibbs free energy due to internal energy = correction to the enthalpy due to internal energy I = moment of inertia K = index of vibrational modes N = number of moles N A = Avogadro s number P = pressure (default is 1 atmosphere) R = gas constant = J/(mol K) = kcal/(mol K) S e S r = entropy due to electronic motion = entropy due to rotational motion S tot = total entropy (S t + S r + S v + S e ) S t S v = entropy due to translation = entropy due to vibrational motion T = temperature (default is ) V = volume Θ r, Θ r,(xyz) = characteristic temperature for rotation (in the x, y or z plane) Θ v,k k B ɛ n ω n = characteristic temperature for vibration K = Boltzmann constant = J/K = the energy of the n-th energy level = the degeneracy of the n-th energy level 18

162 σ r = symmetry number for rotation h = Planck s constant = J s m = mass of the molecule n = number of particles (always 1) q e q r q t q v E ZPE E 0 = electronic partition function = rotational partition function = translational partition function = vibrational partition function = zero-point energy of the molecule = the total electronic energy, the MP2 energy for example H, S, G = standard enthalpy, entropy and Gibbs free energy each compound is in its standard state at the given temperature G = Gibbs free energy of activation c = concentration (taken to be 1) k(t ) = reaction rate at temperature T 19

163 对于复杂的大分子的计算, 目前由于受到计算机条件的限制, 从头算还有困难, 故往往采用各种近似方法. 在半经验方法的计算中, 从电子结构的一些试验资料估计最难以计算的一些积分. 当使用模型法计算分子电子结构时, 不再从原始的完正 Hamilton 量出发, 而是从最简单的模型 Hamilton 量出发. 这种 Hamiltom 量只是粗略地考虑了分子中相互作用, 通常包含一些待定的参数. 半经验法引入的化简极大地简化了必须的计算工作量, 并且可能计算一些更复杂分子的电子结构. 这种计算所得到的资料带有定性的和半定量的特性. 实际上, 如果该方法用于一些它力所能及的问题. 计算结果的精确度通常足以说明所研究分子的性质, 肯定或否定某中物理化学假定. 尽管在半经验方法中依据试验值对一些计算所进行的参数化补偿了计算方案的不足. 但是, 却不能苛求半经验方法面面俱到, 使分子的各种电子性质的计算都有同样好的结果. 因此, 通常保证分子的某些电子结构性质好的计算方案对于另外一些电子结构性质有可能导出不适当的结果. 于是, 对于每种半经验方法, 可以因各类具体参数化方案的不同而变的多样化. 在相当大程度上, 每类参数化都是局限于分子的一些性质或一定种类分子的计算. 因此, 半经验方法不是以描述分子的全部特性为基本内容, 而是着眼于比较同系物的某些性质. 当足以正确地引入参数时, 可以得到复杂化合物电子结构的定性或定量的资料, 同系物分子的某些特性的变化规律, 以及建立他们同试验观测的物理与化学性质的联系, 显然, 这些问题都是现代化学关注的中心.

164 磁化学 研究分子磁性与化学结构关系的物理化学分支学科 物质在磁场中显示出抗磁性 顺磁性和铁磁性, 它们产生于组成物质的原子或分子中的电子轨道磁矩 电子自旋磁矩和核自旋磁矩在空间的取向, 以及各种磁矩间的相互作用 从分子的磁性及其变化规律出发, 研究分子的价键性质 分子的电子组态 分子的结构 分子的空间构型 分子运动的力学性质 分子间相互作用 化学平衡及化学动力学等化学问题, 统称磁化学 磁化学的研究方法很多, 常用的研究手段有磁天平 电子顺磁共振 核磁共振 铁磁共振 磁圆二色和磁旋光法等 其中核磁共振技术发展异常迅速, 已成为鉴定有机化合物结构的重要手段 采用超导磁铁的脉冲傅里叶变换核磁共振谱仪的出现, 使仪器灵敏度及分辨率提高很多, 应用范围从化学扩展到生物学及医学 核磁共振成像技术在医学上已经用于各种疾病的检查, 是非常有发展前途的一种医学诊断方法 电子磁矩电子是发现较早的一种基本粒子, 存在于原子核外 各种化学元素便是根据该元素原子的原子核中的质子数目, 也就是该元素原子在非电离的正常状态下的原子核外的电子数目决定的 原子中的电子磁性是由电子的自旋产生的自旋磁矩和电子环绕原子核作轨道运动产生的轨道磁矩 对于不处于原子中的自由电子说来, 就只有自旋磁矩, 是电子具有的内禀磁性, 常简称电子磁矩 一般电子学只考虑运动电

165 子的电荷所产生的电流, 但是在上个世纪 (20 世纪 ) 末, 由于现代磁学和高新技术的发展, 诞生了磁学与电子学交叉的称为磁电子学 又称自旋电子学的新的交叉磁学或称边缘磁学 这样在磁电子学中电子电流和电子磁矩 ( 自旋 ) 都得到研究和应用 电子磁矩研究的一项很重要又很有意义的成果是对电子磁矩的精密测量和理论计算 这表现在 20 世纪中期的 30 年研究中, 对应用于电子磁矩与电子角动量关系的电子 g 因数的反常因数 ( 简称 g 反常因数 ) α 的精密测量和理论计算上 按早期的理论研究,g 因素 g=2, 即 g 反常因数 α=0, 但是在长期的越来越精密的实验研究中却表明, α 并不等于 0, 如表 1 中所示, 在 1948~1978 的 30 年实验研究中,α 的实验测量值从 3 位有效数字增加到 10 位有效数字 同时更值得注意的是, 对 g 反常因数 α 的理论计算, 在考虑了多种对电子磁矩的影响因素后, 得到的理论计算值也达到 10 位有效数字和很高的精度 ( 很低的不确定度 ) 还值得注意的是,g 反常因数 α 的实验测量值和理论计算值在 10 位有效数字中竟有 8 位有效数字相同, 这些都从表 1 中可以清楚地看出 总的说来, 关于电子 ( 自旋 ) 磁矩的实验测量和理论计算达到这样高的有效位数, 而实验测量值与理论计算值达到这样高的符合程度, 在磁学和其他自然科学中都是非常罕见的 表 1 电子 g 反常因数 α=0.5*(g-2) 的实验测量值和理论计算 年代 α 的实验测量值

166 α 的测量不确定度 ± ± ± ± ±

167 ± ± a 的理论计算值 原子磁矩 magnetic moment of atom 原子内部各种磁矩总和的有效部分 原子核具有磁矩, 但核磁矩很小, 通常可忽略, 原子磁矩则为电子轨道磁矩与自旋磁矩的总和的有效部分 一般地原子磁矩 μj 与原子的总角动量 PJ 有简单的关系, 大小为 μj=g(e/2m)pj, 方向相反, 式中 e/m 是电子的荷质比,g 称为朗德 g 因子, 它可以根据原子中的耦合类型计算出来, 是表征原子磁性质的量 原子磁矩在塞曼效应中起重要作用 分子固有磁矩一个分子中的电子的轨道运动产生的轨道磁矩和电子自旋产生的自旋磁矩的总和就构成分子的分子磁矩, 或者分子固有磁矩

168 我们把原子通过共用电子对结合的化学键成为共价键 (covalent bond) 路易斯 (G.N.Lewis) 曾经提出原子共用电子对成键的概念, 也就是俗称的 八隅律 ( 高中阶段也只是停留于此 ) 然而, 我们知道很多现实情况都无法用八隅率解释, 包括 :PCl5,SCl6 分子 更重要的是, 八隅率从来没有本质上说明共价键的成因 : 为什么带负电荷的两个分子不会排斥反而是互相配对? 随着近代的量子力学 (quantum mechanics) 的建立, 近代形成了两种现代共价键理论, 即是 : 现代价键理 (valence bond theory) 简称 VB( 又叫作电子配对法 ) 以及分子轨道理论 (molecular orbital theory) 简称 MO 价键理论强调了电子对键和成键电子的离域, 有了明确的键的概念 也成功的给出了一些键的性质以及分子结构的直观图像 但是在解释 H2+ 氢分子离子的单电子键的存在以及氧分子等有顺磁性或者大 键的某些分子结构时感到困难 而分子轨道理论可以完美的进行解释, 这里我就主要阐述 MO 法的相关理论 洪特 (Hund) 和密里肯 (R.S Mulliken) 等人提出了新的化学键理论, 即是分子轨道理论 这是人们利用量子力学处理氢分子离子而发展起来的 ( 一 ) 氢分子离子的成键理论氢分子离子 (H2+) 是由两个核以及一个电子组成的最简单分子, 虽然不稳定, 但是确实存在 如何从理论上说明氢分子离子的形成呢? 分子轨道理论把氢分子离子作为一个整体处理, 认为电子是在两个氢核 a 和 b 组成的势场当中运动 电子运动的轨道既不局限在氢核 a 的周围, 也不会局限于氢核 b 的周围, 而是遍及氢核 a 和 b 这种遍及分子所有核的周围的电子轨道, 成为 分子轨道 如何形成这样的分子轨道呢? 我们必须通过波函数来描述原子当中的运动状态, 而波函数是薛定谔方程的解 因为得到精确的薛定谔方程的解很困难, 因此我们才取了近似方法, 假设分子轨道是各个原子轨道的组成 仍然以氢分子离子为例 : 当这个单电子出现了一个氢原子核 a 附近时候, 分子轨道 Ψ 很近似于一个院子轨道 Ψa 同样, 这个电子出现在另外一个氢原子 b 附近时候, 分子轨道 Ψ 也很像原子轨道 Ψb 不过这个只是两种极端情况, 合理的应该是两种极端情况的组合即是 Ψa 与 Ψb 的组合 分子轨道理论假定了分子轨道是所属原子轨道的线性组合 (linear combination of atomic orbital, 简称 LCAO), 即是相加相减而得得 例如氢分子离子当中就有 : ΨI=Ψa+Ψb ΨII=Ψa-Ψb 其中 Ψa 和 Ψb 分别是氢原子 a 以及氢原子 b 的 1s 原子轨道 它们的相加相减分别可以得到 ΨI 以及 ΨII 相加可以看出处在相同相位的两个电子波组合时候波峰叠加, 这样可以使得波增强 如果两个波函数相减, 等于加上一个负的波函

169 数, 因此相减可以看成是有相反相位的两个电子波组合时的波峰叠加, 这样似的波消弱或者抵消 故在 ΨI 当中, 波函数数值和电子出现的几率密度在两个核当中明显增大, 相应的能量比氢原子 1s 轨道能量低, 进入这个轨道的电子将促进这个两个原子的结合因此成为 成键轨道 (bonding orbital), 用 δ1s 表示 ΨII 当中的波函数值在两核中间为零, 电子在两核出现的几率很小, 甚至为 0, 其电子能量高于氢原子的 1s 轨道, 进入这个轨道的电子将促使这个两个原子分离, 所以称为反键轨道 (antibonding orbital) 用 δ1s* 表示 ΨI 以及 ΨII 原子轨道能量之差基本相同, 符号相反 如果两个原子轨道的能量不一样高, 则它们所组成的分子轨道中, 能量低的成键轨道比原来能量低的原子轨道的能量还低 ; 能量高的反键轨道比原来能量高的原子轨道能量还高 ( 二 ) 分子轨道理论要点 从氢分子离子的形成可以得出分子轨道理论是着眼于整个分子, 把分子作为一个整体来考虑 分子当中的电子是在多个原子核以及电子的综合势场中运动 因此没有明确的 键 的概念 现在将分子轨道理论扼要的介绍如下 : 1 分子当中的电子在遍及整个分子范围内运动, 每一个电子的运动状态都可以用一个分子波函数 ( 或者称为分子轨道 )Ψ 来描述 Ψ ^2 表示了电子在空间各处出现的几率密度 2 分子轨道可以通过相应的原子轨道线性组合而成 有几个原子轨道相组合, 就形成几个分子轨道 在组合产生的分子轨道中, 能量地域原子轨道的称为成键轨道 ; 高于原子轨道的称为反键轨道 3 原子轨道在组成分子轨道时候, 必须满足下面三条原则才能有效的组成分子轨道 (1) 对称性匹配原子 : 两个原子轨道的对称性匹配时候她们才能够组成分子轨道 那么什么样子的原子轨道才是对称性匹配呢? 可将两个院子轨道的角度分布图进行两种对称性操作, 即旋转核反映操作, 旋转势绕键轴 ( 以 x 轴为键轴 ) 旋转 180 度, 反映是包含键轴的某一个平面 (xy 或者 xz) 进行反映, 即是照镜子 若操作以后她们的空间位置, 形状以及波瓣符号均没有发生改变称为旋转或者反应操作对称, 若有改变称为反对称 s 和 px 原子轨道轨道对于旋转以及反应两个操作均为对成 ;px 以及 pz 原子轨道对于旋转以及反应两个操作均是反对成, 所以她们都是属于对称性匹配, 可以组成分子轨道, 同理我们还可以得到 py 与 py,pz 与 pz 原子轨道也是对称性匹配 (2) 能量近似原则 : 当参与组成分子轨道的原子轨道之间能量相差不是太大时候, 不能有效的组成分子轨道 原子轨道之间的能量相差越小, 组成的分子轨道成键能力越强, 称为 能量近似原子 (3) 最大重叠原子 : 原子轨道发生重叠时, 在可能的范围内重叠程度越大, 形成的成键轨道能量下降就越多, 成键效果就越强 4 当形成了分子时, 原来处于分子的各个原子轨道上的电子将按照泡利不相容原理, 能量最低原理, 洪特规则这三个原则进入分子轨道 这点和电子填充原子轨道规则完全相同 泡利不相容原理指在原子中不能容纳运动状态完全相同的电子 又称泡利原子 不相容原理 1925 年由奥地利物理学家 W. 泡利提出 一个原子中不可能

170 有电子层 电子亚层 电子云伸展方向和自旋方向完全相同的两个电子 如氦原子的两个电子, 都在第一层 (K 层 ), 电子云形状是球形对称 只有一种完全相同伸展的方向, 自旋方向必然相反 每一轨道中只能客纳自旋相反的两个电子, 每个电子层中可能容纳轨道数是 n2 个 每层最多容纳电子数是 2n2 另外 : 核外电子排布遵循泡利不相容原理 能量最低原理和洪特规则. 能量最低原理就是在不违背泡利不相容原理的前提下, 核外电子总是尽先占有能量最低的轨道, 只有当能量最低的轨道占满后, 电子才依次进入能量较高的轨道, 也就是尽可能使体系能量最低. 洪特规则是在等价轨道 ( 相同电子层 电子亚层上的各个轨道 ) 上排布的电子将尽可能分占不同的轨道, 且自旋方向相同. 后来量子力学证明, 电子这样排布可使能量最低, 所以洪特规则可以包括在能量最低原理中, 作为能量最低原理的一个补充. ( 三 ) 分子轨道的类型在价键理论当中共价键可以分为 σ 和 键 在分子轨道当中我们如何区分 δ 和 键呢? 在氢分子离子形成过程当中我们看到了由两个 1s 轨道形成了一个成键的 σ1s 轨道 ( 形状像橄榄 ) 和另一个反键 σ1s*( 形状像两个鸡蛋 ) 凡是分子轨道对成周形成圆柱形对成的叫做 σ 轨道 在成键 σ 轨道上的电子称为 成键 σ 电子, 她们使得分子稳定化 ; 在反键 σ* 轨道上的电子称为 反键 σ 电子, 她们使得分子有解离的倾向 由成键 δ 电子构成的共价键称为 σ 键 同样, 我们可以用参加组合的原子轨道图形, 按照一定的重叠方式定性的绘出其他的分子轨道 比如沿着 x 轴考紧则两个 px 轨道将 头碰头 的组成两个 σ 型分子轨道, 如果时 py 和 py,pz 和 pz 就是 肩并肩 的组合称为另一种形状的分子轨道, 称为 轨道 它们有一个通过键轴与纸面垂直的对称平面, 好像两个长型的冬瓜, 分别置于界面的上下 成键 轨道上的电子叫做成键 电子, 她们使得分子稳定 图下部反键 2p* 轨道, 她们能量较高, 好像四个鸡蛋分别置于节面上下 反键 轨道上的电子叫做 反键 电子, 她们有使得分子解离的倾向 由成键 电子构成的共价键称为 键 由两个 p 原子轨道形成的 键称为 p-p 键 处此之外,p 轨道还可以和对称性的 d 轨道形成 p-d 键, 例如 px-dxz 相同对称性的 d 轨道之间还能形成 d-d 键, 例如 dzx-dzx 我们可以看出, 无论是 δ 型轨道还是 轨道, 成键轨道中的都是电子云在两核之间的密度比较大, 因此有助于两个原子的组合 在反键轨道中, 电子云原理两核中间区域偏向于两核的外测, 从而使得两个原子的分离

171 优化第一步 : 确定分子构型, 可以根据对分子的了解通过 GVIEW 和 CHEM3D 等软件来构建, 但更多是通过实验数据来构建 ( 如根据晶体软件获得高斯直角坐标输入文件, 软件可在大话西游上下载, 用 GVIEW 可生成 Z- 矩阵高斯输入文件 ), 需要注意的是分子的原子的序号是由输入原子的顺序或构建原子的顺序决定来实现的, 所以为实现对称性输入, 一定要保证第一个输入的原子是对称中心, 这样可以提高运算速度 我算的分子比较大, 一直未曾尝试过, 希望作过这方面工作的朋友能补全它 以下是从本论坛, 大话西游及宏剑公司上下载的帖子 将键长相近的, 如 B B B 键角相近的, 如 A A A 二面角相近的如 D D 都改为一致, 听说这样可以减少变量, 提高计算效率, 是吗? 在第一步和在以后取某些键长键角相等, 感觉是一样的 只是在第一步就设为相等, 除非有实验上的证据, 不然就是纯粹的凭经验了 在前面计算的基础上, 如果你比较信赖前面的计算, 那么设为相等, 倒还有些依据 但是, 设为相等, 总是冒些风险的 对于没有对称性的体系, 应该是没有绝对的相等的 或许可以这么试试 : 先 PM3, 再 B3LYP/6-31G.( 其中的某些键长键角设为相等 ), 再 B3LYP/6-31G( 放开人为设定的那些键长键角相等的约束 ) 比如键长, 键角, 还有是否成键的问题,Gview 看起来就是不精确, 不过基本上没问题, 要是限制它们也许就有很大的问题, 能量上一般会有差异, 有时还比较大如果要减少优化参数, 不是仅仅将相似的参数改为一致, 而是要根据对称性, 采用相同的参数 例如对苯分子分子指定部分如下 : C C 1 B1 C 2 B2 1 A1 C 3 B3 2 A2 1 D1

172 C 4 B4 3 A3 2 D2 C 1 B5 2 A4 3 D3 H 1 B6 2 A5 3 D4 H 2 B7 1 A6 6 D5 H 3 B8 2 A7 1 D6 H 4 B9 3 A8 2 D7 H 5 B10 4 A9 3 D8 H 6 B11 1 A10 2 D9 B B B B B B B B B

173 B B A A A A A A A A A A D D D D D D D D D

174 参数很多, 但是通过对称性原则, 并且采用亚原子可以将参数减少为 : X X 1 B0 C 1 B1 2 A1 C 1 B1 2 A1 3 D1 C 1 B1 2 A1 4 D1 C 1 B1 2 A1 5 D1 C 1 B1 2 A1 6 D1 C 1 B1 2 A1 7 D1 H 1 B2 2 A1 8 D1 H 1 B2 2 A1 3 D1 H 1 B2 2 A1 4 D1 H 1 B2 2 A1 5 D1 H 1 B2 2 A1 6 D1 H 1 B2 2 A1 7 D1 B0 1.0 B1 1.2 B2 2.2 A D1 60.0

175 对于这两个工作, 所用的时间为 57s 和 36s, 对称性为 C01 和 D6H, 明显后者要远远优于前者 好, 我真的很想知道这句话 你手工把坐标写好输入, 输入为 C2 对称性 是什么意思? 老是见到版面上有人说对输入施加对称性限制, 就是不明白是怎么回事 您能详细解释一下吗? 有例子就更好了 麻烦了谢谢! 比如, 你要输入 CO2, 那么你这么输入 C O O 写得很精确, 那就是对称性输入了 这样 g98 可自动判定为 D h 如果你输入的差不多, 比如 : C O O 精度不够, 那么 g98 就只能判定为 C h 了 再给你举个例子, 比如甲烷的 Td 点群的输入坐标, 考虑对称性, 把 C 放在直角坐标的原点, 四个氢放 在四个角的顶点 C H H H H 这样出来的结果点群判断就会是正确的, 当然这里的 1 我是随便取的一个值, 只是来说明这个问题而矣 谢谢二位指教, 也就是说第一个输入的原子应该是对称中心, 其它输入的原子应保持数值上的对应性 那 对于用 Gview 或 3D 生成的分子构型, 也可以根据这个原则将处于对称位置的原子的坐标改为一致了? 如果 分子很大 ( 如我这次算的 C14H14N4S20) 有三个环, 整体性的对称输入就有些麻烦, 是不是可以先生成分 子构型, 然后利用对称性对它的笛卡儿坐标或矩阵坐标进行修改, 将相近的坐标的数据改为一致? " 第一个输入的原子应该是对称中心, 其它输入的原子应保持数值上的对应性 " 没错, 做到这点就很方便了," 对于用 Gview 或 3D 生成的分子构型, 也可以根据这个原则将处于对称位置的原子的坐标改为一致 " 那还不如用手动输入, 大体系很难办了, 如果体系太大, 不用保持严格的对称性吧. 有机分子, 原子多了以后构型就没有什么对称性了 但对于有些系统, 比如 C60, 你当然得按照它的实际的对称性来输入 如果你希望你的计算按照某种对称性来做, 那就一定要手工输入严格的满足对称性的坐标. 用软件生成的坐标, 不是严格满足对称性, 即使

176 用 symm=loose, 有时也不一定能判断出来 各位大虾是用什么软件建模的呀, 我是用 SGI 里 sybyl 软件进行构像搜索后取最低能量构像导入高斯程序的 ; 有的是用 chem3d 直接建模, 这样子有没有什么问题啊, 应该怎么做, 谢谢 通常也就是这些方法, 建议还是认真的核实分子结构的是否合理, 然后再试着增大迭代圈数, 降低收敛判据. 用什么软件建模并不是问题, 对于我们量化计算, 如 vliant 所说, 核实分子的合理构型才是最重要的, 包括对个原子间的成键分析 就算不同的软件, 也不可能保证给出的构型合理, 因此我们一般都提倡手动输入坐标, 就是为了加强对构型的理解, 只有在这个基础上使用软件建模才能尽可能保证不出错 一个合理的计算结果不能寄希望于软件 我们都知道常用的分子坐标输入有 : 内坐标直角坐标混合坐标分子构型输入方法的不同会对计算过程产生影响么? 或者说, 对于哪些体系, 用哪种方法更好优化? 混合坐标 怎么讲? 我通常用内坐标, 它能很好的控制对称性, 且时间要短 混合坐标 是直角坐标和内坐标的混合使用, 高斯手册上有讲这部分内容 有看过讲哪种体系使用哪种方法更好优化, 好像这都是凭经验的结果 在 免费资源 论坛中有本 计算化学, 其中第八章 (67-72 页 ) 好像提到你所关心的问题, 你可以下来看一下, 或许有所帮助 如果你使用缺省的优化方案且初始几何构型完全相同, 那么, 不论你使用何种坐标输入方式, 优化结果应该是一样的 简言之, 优化结果只取决于优化方案和初始几何构型, 与坐标录入方式无甚关联 于对称性限制的优化来说, 显然以内坐标输入, 且选用 opt=z-matrix 可以减少变量, 使得优化效率提高 免费资源中 gauss 入门 有介绍 您冗余坐标是指有过剩的内坐标 冗余内坐标算法可以用于结构优化 优化第二步 : 对于大分子, 一般要先预优化 ( 如用 PM3 或 AM1), 有时是用部分优化, 然后才用能达到实验精度方法和基组进行优化, 就我所了解大多数情况下,6-31G(D) 基组就差不多 不好意思, 我是新手, 刚刚接触 guassian, 最近要算两个较大有机分子中原子的静电荷急相关能量和几何性质, 分子中已经超过了 80 个原子 ( 包括氢 ), 用 chemdraw 画出图形, 在 chem3d 中生成的 input 文件, 可是, 软件报错 2070~~ 好象说是连接 (last link) 错误, 怎么回事, 哪里出错了, 各位指点一下 ~ 怎么, 先用 pm3 预优化就可以了??

177 我的意思是, 初试构象对计算量有这么大的影响么?? 我以前也遇到过相似情况, 同一个分子, 相同的计算条件, 用 gview 画出的模型进行计算报 2070 的错, 用 chemdraw 画的模型,chem3d 生成的 input 文件进行计算就 OK, 完成可以计算完成 我还以为你画分子式的时候出什么问题了呢 ~~ 只想优化 H 原子, 其他重原子不做优化 怎样进行分子坐标输入, 下面的坐标输入方法对吗? 就是仅 H 原子的参数设置成 Vairable. %chk=sample #p HF/6-31G* opt Title line 0 1 cu cu cu p p p p p p s s n n n H 1 B14 2 A13 7 D12 H 1 B15 2 A14 7 D13 H 1 B16 2 A15 7 D14 H 1 B17 2 A16 7 D15 H 2 B18 1 A17 3 D16 H 2 B19 1 A18 3 D17 H 2 B20 1 A19 3 D18 H 2 B21 1 A20 3 D19 H 3 B22 1 A21 2 D20 H 3 B23 1 A22 2 D21 H 3 B24 1 A23 2 D22 H 3 B25 1 A24 2 D23 H 1 B26 2 A25 7 D24 H 2 B27 1 A26 9 D25

178 B B B B B B B B B B B B B B A A A A A A A A A A A A A A D D D D D D D D D D D D D D 你如果要这么做的话, 必须加 opt=z-matrix, 这样的话它就只优化后面的 H, 前面的直角坐标就不会优化了 你

179 只用了 opt 的话, 那就是全优化了 你也可以通过 molredunt 来实现部分优化 固定原子的方法有许多, 最简单的是这样的 : #rhf/sto-3g opt nosymm test the two C atoms are frozen 0, 我不知道你的那种方法行不行, 但我这个肯定可以 前面我都贴出来了 :) 这可是我的心血噢 ========= opt=z-matrix 的部分优化 %Chk=reac-hf3 #p RHF/STO-3G opt(z-matrix,maxcyc=500) freq=noraman scf(maxcyc=200) reactant fopt 1 1 N H 0 x2 y2 z2 H 0 x3 y3 z3 H 0 x4 y4 z4 O H 0 x6 y6 z6 H 0 x7 y7 z7 H 0 x8 y8 z8 x2= y2= z2= x3= y3= z3= x4= y4= z4= x6=

180 y6= z6= x7= y7= z7= x8= y8= z8= *********************************************** opt=modredundant 的部分优化 分子坐标部分可以为具体的数值, 而不用为参数形式 %Chk=reac-hf3 #p RHF/STO-3G opt(modredundant,maxcyc=500) freq=noraman scf(maxcyc=200) reactant fopt 1 1 N H 0 H 0 H 0 O H 0 H 0 H F 感谢各位大侠的帮助! 看了之后有个不明白的地方, 还请大侠不吝赐教! 在 opt=modredundant 的部分优化时最后一行的 1 5 F 是什么意思? 就是将第一和第五号原子的距离固定问一个菜鸟的问题 :N 后面的 -1 和 H 后面的 0 是什么意思? 是像手册上所说的把坐标增加所需要的值吗? 见笑了!! -1 指的是固定,0 指的是优化, 前面 helpme 贴的输入方法也是这样的 部分优化输入格式之二

181 opt=z-matrix 的部分优化 %Chk=reac-hf3 #p RHF/STO-3G opt(z-matrix,maxcyc=500) freq=noraman scf(maxcyc=200) reactant fopt 1 1 N H 0 x2 y2 z2 H 0 x3 y3 z3 H 0 x4 y4 z4 O H 0 x6 y6 z6 H 0 x7 y7 z7 H 0 x8 y8 z8 x2= y2= z2= x3= y3= z3= x4= y4= z4= x6= y6= z6= x7= y7= z7= x8= y8= z8= *********************************************** opt=modredundant 的部分优化 分子坐标部分可以为具体的数值, 而不用为参数形式 %Chk=reac-hf3 #p RHF/STO-3G opt(modredundant,maxcyc=500) freq=noraman scf(maxcyc=200)

182 reactant fopt 1 1 N H 0 H 0 H 0 O H 0 H 0 H F 冻结坐标的目的是什么了? 我虽然会做但不太清楚 Gaussian 为什么会有这个功能. 还有最近做了一个体系. 有一个键如果不冻结的话. 就优化不出那个反应中间体. 那个键会断. 我这样冻结键优化出来的结构计算频率也无虚频. 它能看作为一个 minima 吗? 这个结构可以作为我反应机理的中间体吗? 我用溶剂化优化结构可以在不冻结坐标的情况得到一个和这个结构很接近的结构. 冻结坐标有时是有用的, 举几个例子 : 1, 如优化溶质分子和 nh2o 的 supermolecule, 周围的水分子, 可能结构变化不大, 这时就可以固定水的键长或键角, 减少工作量 2, 如用 cluster 模拟大块固体, 这时也许就要按照晶格参数固定原子键长和键角 3, 又如不固定 NH3-H-H2O 中 N-O 键长是得不到质子传递势垒 4, 还有在优化稳定构型或过渡态得时候也有时用到部分优化得到接近得结果, 再放开优化, 这样逐步得到你想要得结果 冻结坐标对一些大的体系, 超分子体系好象是很有用的. 好象做生物分子的反应时氨基酸残基必须先固定. 但我这个体系如果不固定, 只要一放开, 结构就跑了会优化到其它构型上去. 就算 stepsize 设得很小也没有用. 优化第三步 : 优化结果分析, 一般要进行频率计算, 无虚频, 表示是极小值 ; 有虚频, 表示是过渡态 如果出现虚频, 可根据虚频的简正模式进行调整以获得极小值, 我还没有碰到这种情况, 希望各位朋友能给些详细的资料 如前信所说, 键长键角有差别是很正常的, 因为气相分子和晶体内分子的结构不可能完

183 全吻合, 只可能近似相同 另外, 对不同元素的限制主要看基组是否能计算该类元素, 例如你所采用的方法中用了 6-31G(D) 基组, 该基组支持 H-Kr 的元素, 即可以计算 Br 原子但是不能够计算 I 原子 如果你所用的基组不支持你所计算的某个原子可以采用扩展基组来进行计算, 或者降低基组的精度 我用 Gaussian 优化一个分子结构, 由于目前实验上没有办法得到纯净的样品来实验验证, 那么我怎么判断优化得到的结构是正确的? 一些主要的键长和键角的实验值或者别人计算的值是否和与你计算的分子类似? 这个判断不好做 所谓正确不正确, 如果计算出来的频率没有虚频, 就表示你找到了一个局域稳定点 要确定是不是全局稳定点很难, 只有通过别的方式比如 HOMO-LUMO 能级差等信息看你算出来态是不是稳定 也觉得高级的方法和大基组的结果更可信, 谢谢各位高手的指点!! 再问一个问题, 如果出现虚频如何修改得到分子稳定的结构用 GV 看一下, 根椐产生虚频的振动模式调整结构 消去虚频有什么技巧? 我有一个计算结果, 虚频有三个, 都只有 -30~-20 左右, 如果结构改动过大, 一不小心, 虚频就会更大 头痛 ing~~~ 那你就把结构朝振动的方向改小一点试一下 确实据我了解, 消除虚频的主要方法是改变构型 其次在计算上还可以尝试 :nosymm; 加大循环次数 ; 提高收敛度 ;iop(1/8=1) 等 其实我自己也是有这个问题, 对于势能面很平, 较小的虚频很难消除 首先, 1 在优化时采用 Scf(tight) 的选项, 增加收敛的标准 再去计算频率 如果还有虚频, 参见下一步 2. 对称性的影响, 很多情况下的虚频是由于分子本身的对称性造成的 这样, 在优化时, 如果必要, 要将对称性降低, 还有, 输入文件有时是用内坐标 建议如果有虚频的话, 将内坐标改成直角坐标优化 3. 如果上述方法还有虚频, 看一下虚频, 找到强度较大的, 将在频率中产生的原子的振动坐标加到相应的输入文件中 这样, 重新计算 直到虚频没有 4 实际上, 如果分子柔性较大, 很难找到最低点, 这是电子结构计算的问题, 这种情况下, 需要动力学的东西, 用构象搜寻的办法解决 如 : 模拟退火, 最陡下降法, 淬火法等 将得到的能量最低的构象做一般的电子结构计算, 这样, 应当没有问题 不要讲你还没有得到最稳定的结构, 那么, 是你的分子有问题, 要么计算错了, 要么就是游离出现代计算的范畴

184 zixia 上大虾的回复 : 将内坐标改成直角坐标优化 这一方法没有多大作用, 我试过, 还是一样的结果 另外如果是根据振动频率的模式来调整分子结构的话, 还要注意调整完后的分子结构 对应的能量是不是比未调整之前有所降低, 如果没有, 说明还是不行 alwens 兄所讲的第三点, 经过我的猜测和与 alwens 的探讨, 终于知道应该怎么做 : 分子结构输入用直角坐标, 将这些产生虚频的振动坐标直接与原始坐标相加 alwens 兄进一步提到 : 这个方法有时候不太好 不一定将坐标全部加上 有时候可以选择加 1/2 等 主要的目的是去掉由分子对称所形成的虚频 象上面的那个计算, 完全可以再利用 scf(tight) 的选项消掉 1-2 个虚频 强度很小 又多学了一招, 自我感觉很幸福中 ^_^ 怎么把内坐标改成直角坐标啊 看你的输出文件 Full point group CS NOp 2 Largest Abelian subgroup CS NOp 2 Largest concise Abelian subgroup CS NOp 2 Standard orientation: Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z 这一部分就是, 如果没有, 看 Input orientation: 也行 在 opt 的选项中加入 tight 或 saddle=0, 若是计算过度态, 应是 saddle=1 但是, 也不一定能消除虚频!! 最好, 将结构重新做一下, 注意分子的构型, 如 CH3 的三个 H 与相邻基团是交叉, 还是重叠!! 请教虚频 请问如何知道虚频的结构是什么, 是不是频率计算负值都可以认为是虚频, 那我用 gaussianview 看有三个负值, 接下来我想

185 做过渡态, 用 IRC, 那我应该怎么弄呢? 谢谢指导 你的输出文件中这部分就是看是否有虚频的 A" A' A" Frequencies Red. masses Frc consts IR Inten Raman Activ Depolar Atom AN X Y Z X Y Z X Y Z Frequencies 那一行数值为负, 即有虚频 有三个负值, 说明找的不是过渡态 但是如果另外两个虚频很小, 而另外一个虚 频很大, 而且它的振动模式对应于你所计算的反应模式 ( 比如反就前后某一键的拉长, 缩短等 ), 在计算所用的方法充许校 正范围内, 也还可以认为是找到过渡态 但是一般要求能够把另外两个虚频去掉 方法是要据振动模式, 调整结构, 再优化 般认为, 过渡态仅有一个虚频, 中间体没有虚频!! 建议仔细阅读 Exploring Chemistry with Electronic Structure Methos 和 G98W 程序使用入门, 后者稍微简单一点, 厦大的免费资源里有!! 有一个虚频是过渡态的必要条件 ; 有几个虚频肯定不行了, 有时候有一个虚频也不一定就找准了, 做 IRC 成功就可靠了 另 : 复合物也会有小的虚频, 只要不大于 100, 也可认为是一极小值 没错, 有些体系可能存在 several or many local minima, 那么找到的具有一个虚频的过渡态可能连接两个 loccal minima, 而非连接反应物和产物, 因此必须做 IRC 来判断. 关于虚频的物理意义 自恨理论功底浅薄, 所以特地在此向诸位高手求教 : 在寻找过渡态计算中 opt=ts, 需要计算频率, 如果有一个虚频就说明优化所得的构型是个过渡态 有时在做部分限制优化 opt=minimal 时, 也能得到一个虚频 ( 一般发生在 frozen 部分的化学键上 ) 我想问一下, 虚频的数值大小本身具有什么物理意义 后一种情况是否也可以认定为过渡态 我的理解 : 虚频就是在这个振动方向上力常数是负的过渡态之所以有一个虚频, 是因为它是一个鞍点, 鞍点上有一个负的梯度的方向分子沿这个方向振动时将转化为反应物或产物 就象从山上掉下来, 受到的力可以

186 认为是负的 但并不是有一个虚频就是过渡态, 还要在这个虚频的振动方向上分别指向反应物和产物 有一个虚频并不一定是你想要的过渡态, 你要看虚频的振动方向是不是对应着反应的方向 从概念上说, 虚频就是化学反应方向上的简正坐标的二次偏导小于零 另外, 根据我的经验, 这个虚频的负值应该大一些, 如果这个值太小, 那也不足信 不知对不对, 呵呵! 优化输出信息 : 这说明优化失败了, 没有收敛, 你可以尝试 : 1 降低收敛标准( 算一般的有机分子不要紧, 如果是过渡金属, 那就不要了 ) 2 增大循环次数 3 换基组和算法 4 改变初始猜测 是有机分子阿, 呵呵, 用的是 AM1, 还是太大吗 hsi 这种不收敛的情况可能是你最初用软件建模的时候构型不合理, 仔细考虑一下你的分子的构型是否合理, 是否还存在其它的构型, 单纯地降低收敛标准意义不大, 不过你可以先尝试一下增大循环次数, 最多 scf(maxcyc=300), 再不收敛的话就不是循环的问题, 是构型不对 S 能不能说说,π 轨道和 σ 轨道的组成上有什么差别? 这方面我比较欠缺 ure. You can directly analyze the components of the molecule orbitals from the population in the *.log file. 我在版面上说过了, 当初的讨论还真是热烈 :) 你可以翻看一下 我这里简述几句 : π 轨道主要是由 npx,npy,nd-1,nd+1 组成 σ 轨道主要是由 ns,npz( 键轴方向 ),nd0 还有,nd+2,nd-2 主要是组成 δ 轨道 opt 是可以 restart, 前面的中间文件是少不了的

187 1, %chk=... #.. opt Stopped 2, %chk=... #.. opt(restart) guess=read geom=allcheck 对弈 freq 的 restart, 是要采用数值计算方法才可以 restart ( 以下我省略 chk 和方法的输入, 自己加上 ) 1, freq(numerical) 2, freq(restart,numerical) guess=read OPT=restart 不能进行,why 因为死机 OPT 被中断, 我采用 OPT=restart 继续进行, 但是出现以下情况 : Restoring state from the checkpoint file "Mn7O14V11O2-PCOPT1-2.chk". FileIO operation on non-existent file. FileIO: IOper= 2 IFilNo(1)= -997 Len= IPos= 0 Q= dumping /fiocom/, unit = 1 NFiles = 16 SizExt = WInBlk = 512 defal = T LstWrd = FType=2 FMxFil=10000 Number Base End End Wr Pntr Rd Pntr Number Base End End Wr Pntr Rd Pntr

188 dumping /fiocom/, unit = 2 NFiles = 5 SizExt = 0 WInBlk = 512 defal = F LstWrd = FType=2 FMxFil=10000 Number Base End End Wr Pntr Rd Pntr dumping /fiocom/, unit = 3 NFiles = 1 SizExt = WInBlk = 512 defal = T LstWrd = FType=2 FMxFil=10000 Number 0 Base End End Wr Pntr Rd Pntr Error termination in NtrErr: NtrErr Called from FileIO. 为什么我的 file IO 会不存在? 是什么参数设漏了吗? 请各位大虾支持, 谢谢! 在你前一次的计算中, 第一单点的计算未完成, 所以,checkpoint 文档中尚无优化过程的信息, 无法 restart. 过渡态优化过程中遇到一个这样的问题我在过渡态优化过程中用这样的方法 # hf/6-31g(d) opt=(ts,calcfc) freq 当运行到 L999 时 Link died 请问可能是什么原因? 是不是我的过渡态输入有问题? 另外, 寻找过渡态有哪些技巧, 请指教! 你的计算在设定的步数里面没有找到过渡态 这有两个可能 : 一是初始输入的构型不好 二是循环次数不够 ( 可以 maxcycle= 大一点的数 ) 具体还是等高手来指点吧

189 对于决定收敛结构的标准, 做了一些很小但很重要的变动 当力比截断值小两个数量级时 ( 即, 极限值的 1/100), 结构即被认为是收敛的, 即便是位移比截断值还大 这个变动有利于大分子的优化, 因为它们的最小值周围可能有非常平缓的势能曲面 (G98manual p106) 所以你的 opt 有 2 个 yes 频率计算完成后还有一次优化, 可能这次就优化到 3 个了! 做频率分析的时候只需要用 Freq 就可以, 不需要加 Raman 选项, 对吗? freq=noraman 不作拉曼强度计算, 可以提高 10%-20% 的计算速度. 算一个分子的电离能, 里面有 I,CL 在优化起正离子的时候老是没有没有结果体系的能量趋势不固定, 有时大有时小, MAXCYCLE=80 后出现 2 个 YES, 然后 DIE LINK 999 但是 COPY OUT 最后的构型再去优化后 结构 4 个全是 NO 现在还在算, 但是还是 NO 怎么办? 要 OPT=CALCALL 吗? 有什么用? 另外, 还有一个题外的问题这样写对吗? OPT(MAXCYCLE=80,STEPSIZE=10) 可以吗? 增大优化次数,optcyc=N; N 给大一些 opt=(maxcycle=n) 也一样 前面的老帖子已经讲到这些, 好象在免费资源里的 Gaussian98 tips. 一部分引用如下 : First, whenever you encounter SCF convergence failures you need to look at the progress of the SCF by using #P in place of # which will print out the status at each iteration. It is useful to know if the SCF is converging slowly, oscillating, taking large jumps etc. Slow convergence can be a sign that a higher order converger like SCF=QC would be helpful. Oscillations can be a sign that a small HOMO/LUMO gap is causing a flip in the occupied orbitals and SCF=VShift=n is useful. Large jumps can indicate that a poor initial guess or a change in the electronic state from the initial guess has occured and normal convergence will eventually

190 achieved but after a larger number of cycles and a better initial guess would help. Second, whenever possible it is helpful to start with a smaller basis set, i.e. STO-3G, LANL1MB, etc for single points. This is cheaper and so you can often generate a better initial guess which can be used with GUESS=READ and the larger basis set and without the more expensive options. In your case the SCF converges initially quite smoothly and then bounces around without making progress. This makes SCF=QC a likely approach to clean up this problem, at least at the initial point. You might want to look at adding IOP(5/13=1) which allows the SCF to print out the MO's even in the event that convergence was not reached. This should only be used with single points, i.e. remove OPT. As to optimization constraints. To freeze atomic coordinates you should turn on NoSymm. This method of freezing structures causes problems for redundant internals which are being looked into but NoSymm fixes the most serious ones. 下面是血红素的化学合成的模型, 由于分子有点大, 又加上有个 Fe, 所以优化非常困难, 请求斑竹支高招 : (1) 这是一个高度对称的分子, 如何利用加虚原子, 提高计算速度? (2) 优化 Fe, 我采用了 LANL2DZ, 请问有没有更好的基组? 怎么手工加基组? (3) 由于变量太多, 您认为我通过怎样的办法, 减少变量来提高计算速度? 非常感谢!!! 做出文章了, 我一定挂你的大名, 呵呵 S O 1 B1 Fe 2 B2 1 A1 N 3 B3 2 A2 1 D1 N 3 B4 2 A3 1 D2 N 3 B5 2 A4 1 D3 N 3 B6 2 A5 1 D4 C 4 B7 3 A6 2 D5 C 8 B8 4 A7 3 D6 C 9 B9 8 A8 4 D7 C 4 B10 3 A9 2 D8 C 5 B11 3 A10 2 D9 C 12 B12 5 A11 3 D10 C 13 B13 12 A12 5 D11 C 5 B14 3 A13 2 D12 C 6 B15 3 A14 2 D13 C 16 B16 6 A15 3 D14 C 17 B17 16 A16 6 D15

191 C 6 B18 3 A17 2 D16 C 7 B19 3 A18 2 D17 C 20 B20 7 A19 3 D18 C 21 B21 20 A20 7 D19 C 7 B22 3 A21 2 D20 C 23 B23 7 A22 3 D21 H 24 B24 23 A23 7 D22 C 11 B25 4 A24 3 D23 H 26 B26 11 A25 4 D24 C 16 B27 6 A26 3 D25 H 28 B28 16 A27 6 D26 C 20 B29 7 A28 3 D27 H 30 B30 20 A29 7 D28 H 1 B31 3 A30 2 D29 H 9 B32 8 A31 4 D30 H 10 B33 9 A32 8 D31 H 13 B34 12 A33 5 D32 H 14 B35 13 A34 12 D33 H 17 B36 16 A35 6 D34 H 18 B37 17 A36 16 D35 H 21 B38 20 A37 7 D36 H 22 B39 21 A38 20 D37 B B B B B B B B B B B B B B B B B B B B B

192 B B B B B B B B B B B B B B B B B B A A A A A A A A A A A A A A A A A A A A A A A A A A

193 A A A A A A A A A A A A D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D

194 D D D D D 我也在做大分子的计算, 希望能交流交流 我的电子邮件 我的经验是先做 PM3 优化, 然后再用精度高的方法和基组 你的前三个原子, 不是一条直线吗? 如果是直线, 那对称性就可以是 C4v 了 如果不是直线, 那至少要做成 cs 对称性吧? 把前三个分子所在的平面作为对称面 你现在的对称性是没有 虽然看上去很对称 Fe 用赝势基组不知道会怎么样, 你可以看看别人的文章 做优化的时候, 可以固定一些外层的原子, 只让和中间的原子相连的动 大约有 3M 另外,Fe O S 分子是在一条直线上么? 如果不是, 你可以将吡啶环上的参数先减少了, 然后再加 Fe O S, 肯定能够减少不少参数, 但是不知道由没有实际意义了 1) 我谈一下我做过的体系加虚原子的经验, 希望对你有帮助 : 一般情况下, 虚原子加在分子的对称中心, 或键轴 Cn 轴上, 如要加第二个, 可放在垂直于 Cn 轴的方面上 (2) 你可以到基组库去寻找基组, 可能要比单单的 Lanl2Dz 效果好 (3) 你的体系高度对称, 肯定有不少的 CH 键长键角是一样的 ( 或者差别很小 ), 一律设为一个参数, 让它同步优化 PS: 你的输入文件的参数也太多了点 我看了一下你的结构, 你说是高度对称? 我想问你一下这是你手建的吗? 从你的坐标看好象没有对称性 32 号 H 怎么位置这么怪! 我认为你的结果最高对称性是 C(4V), 最好手建一下为好 ( 如不考虑外场作用 ) 这种体系的计算现在非常热门也非常有难度, 上次我们听的以色列教授介绍的也是这方面的内容, 这种体系以我们的水平可能很难处理好 你可能要找到做这方面的专家指导才会有很好的效果, 否则很难分析清楚计算出的结果 上次以色列教授沙松过来我们这边就是介绍的有关 P450 的计算, 虽然我没有听懂什么, 不过以我自己计算含金属的小分子经验来看, 关于这种体系很难处理的部分就在于如果控制中心金属原子铁和周围配体的配位情况, 以及了解中心金属原子的电子占据, 如果你仅是靠软件产生体系的输入坐标, 可能计算很难达到你想要的结果 这些坐标的产生可以要靠自己的经验来步步设置, 所以就像你所说的太细节的东西是很难说清楚的, 因为他们在计算过程中拥有比你更多的经验 只能给你这些参考性建议了, 没有什么实质性的帮助

195 我的输入文件是 : %chk=zux3 #p hf/lanl2dz opt symm=loose scf=(maxcycle=200) test zux 分子的点群是 c2, 但是优化第二轮是就出错了, Omega: Change in point group or standard orientation. Error termination via Lnk1e in d:\g98w\l202.exe.

196 如果我不用关键词 symm=loose, 优化时分子点群就变成了 c1 请高手指教, 这是何原因呢? 若我想保持分子仍是 c2, 改如何处理? 3x! 换内坐标吧 opt=z-matrix 你用内坐标可能会好一点吧? 而且也能保证你的对称性的 [em12][em12] Angle between quadratic step and forces= degrees. Linear search not attempted -- first point. Iteration 1 RMS(Cart)= RMS(Int)= Iteration 2 RMS(Cart)= RMS(Int)= Variable Old X -DE/DX Delta X Delta X Delta X New X (Linear) (Quad) (Total) R R R R R R R R A A A A A A A A A A A A D D D D D D D D D D Item value Threshold Converged? Maximum Force YES

197 RMS Force YES Maximum Displacement YES RMS Displacement YES Optimization completed. -- Stationary point found. 详细解释一下好吗, 这一段几乎全看不懂 Iteration 1 RMS(Cart)= RMS(Int)= Iteration 2 RMS(Cart)= RMS(Int)= 这应该是按笛卡尔坐标和内坐标计算的均方误差, 比较的标准不清楚! Variable 列是输入的变量 ;Old X 列是上一次的变量值 ;-DE/DX 是能量对坐标的一阶导数, 是判断驻点的依据 ;Delta X 是与上次相比变量的变化 ;New X 是新的变量数值! SCF Done: E(RHF) = A.U. after 12 cycles Convg = D-08 -V/T = S**2 = NROrb= 34 NOA= 8 NOB= 8 NVA= 26 NVB= 26 最后一行是什么东东? 总轨道数 34,Alpha 占居轨道 8 个,Beta 占居轨道 8 个,Alpha 虚轨道 26 个,Beta 虚轨道 26 个! NROrb=number of orb NOA= numberof alpha occupied orb NVA=number of virtual alpha orb 结构优化的时候有没有可能出现这种情况 : 即构型在两个或几个结构循环变化, 总也不能达到终点可能出现么? 如果这样的话该如何处理? 非线形搜索? 多谢! 构型在两个或几个结构循环变化, 总也不能达到终点, 可能出现么? 这个问题是你实际遇到的, 还是凭空想出来的? 几个结构在势能面上对应着局部最小, 自洽场的结果是会到达一个局部最小值, 而不会出现在几个个结构循环变化 我觉得你想说的是收敛振荡的问题吧? 他指的可能是结构优化振荡, 我曾经用过 iop(1/8) 里的一些选项, 对控制结构优化有一定帮助 我现在遇到的问题是这样的, 优化了很多圈, 但每次的能量总是在很小的范围内起伏只能在 Force 上达到两个 YES, 而 Displacement 上总是 NO, 我觉得就象是在几个结构中振荡一样, 这是所

198 谓的优化振荡么? iop(1/8) 里有设定优化步长的功能 : 0 DXMAXT = 0.1 BOHR OR RADIAN (L103, Estm or UnitFC). = 0.3 Bohr or Radian (L103, Read or CalcFC). = 0.2 Bohr or Radian (L105). = 0.3 Bohr or Radian (L113, L114). N DXMAXT = 0.01 * N 哪里有关于 iop(1/8) 的详细内容呢?? 在免费资源版有 IOP 的 pdf 文件谢 supi! 使用了 iop(1/8=1) 结构终于优化完了, 但这说明什么呢? 体系的能量对于结构的微小变化很敏感? 还是说势能面比较平坦? 版上的今天刚贴的 Guassian tips 这篇贴子中最后有提到收敛遇到的一种情况就是 "taking large jumps". 对于优化默认采用的步长为 0.3Bohr 对你优化的结构来说太大, 所以你看到结果能量总是在很小的范围内起伏. 因此你使用 IOP(1/8=1) 设置步长为 0.01Bohr, 而达到收敛的最终结果, 说明这个结构的势能面是起伏不平的, 对结构的变化非常敏感. 不过, 这种情况也像是 oscillating, 建议用 SCF(Vshift=300) 试一下, 看能不能收敛到同一结果? 试了好几次, 总觉得使用二次收敛方法 scf=qc, 对收敛没有什么太大的效果, 反而不如以前. 而且对于 oscillating 采用 SCF(Vshift=N) 来说, 由于 DFT 已经使用 N=200, 即使增大 N 的值对收敛的结果也无明显效果, 比不上减小优化步长. 但是对于减少步长来说, 机时消耗太大, 虽然在能量上比前者来得低, 但是仍可能达不到收敛的预期效果, 因此这些都说明了一个问题 :The initial guess isn't very good. 关于如果处理这些问题, 计算化学 上给出了一些更好的建议 看了关于学术腐败的讨论, 很惭愧, 学习高斯已经有好几个月, 还停留在 Opt 阶段, 而且还有不少问题, 先问两个吧, 请莫见笑 1. 我发现构型优化时,Opt(tight) 和只有 Opt 时, 机时差别若干倍, 但结果好像差的不是很多, 为什么? 2. 对于大分子, 可以先用小基组优化, 而后再用更高级基组优化, 具体写输入文件时, 小基组时的分子说明部分用初始分子构型的数据, 而更高基组优化时的分子说明部分应该用小基组的计算结果, 怎么写呢? opt=tight 采用了更为严格的收敛标准, 使用选项 Opt=Tight 的优化计算比使用默认截止点的优化计算要多算几步 所用的时间当然也大得多了 要注意我们使用的能量的原子单位是个很大的数值 1au= kcal/mol, 也许用两种方法算出来的能量只差了 0 05 个单位, 可换算一下就有 30 多的 kcal/mol! 2 你可以先用小基组算, 然后用前面的 chk 文件来作大基组的计算 在控制行加入 geom=check,guess=read, 就不用分子构型输入了 结构优化不收敛, 一般有什么 OPT=ReadFc, opt=calcfc, opt=calcall, 或者设置 maxcycle,opt=restart 等等, 但有没有人知道那种最为有效? 具体方法怎样? 当然有示例是最好的了! 呵呵 :) 谈谈我的想法 : 我感觉结构不收敛可能与你的初始猜测有关, 也许先固定某些键长键角再优化, 事后再放开优化会比较容易

199 收敛. 采用低级算法先计算只是其中的方法之一, 你为何一直要用这种方法? 有什么依据? 如果你顺利解决了你的计算, 那就谈谈你的经验如何? 我觉得采用 OPT=CALCFC 并不一定得到好的结果, 有时候收不收敛实际和我们处理的化学问题相关, 而不仅仅是用计算上的技巧进行解决. 比如两个原子间电子占据没有 sigma 成键, 但是存在强的 sigma 非键作用, 那么它们之间的斥力很强, 计算就很难收敛形成稳定的体系. 因此在处理收敛的问题时, 我们应该多分析不收敛的问题实质, 是电子的什么运动状态引起的, 而非看到不收敛就想利用一些 key 来解决, 我自己也试过一些 key, 包括 opt=calcfc,allcfc,nodiis,qc 等等 ( 包括一些强制性的手段 ), 并不是很有效, 因为它们都是从技术的角度来解决问题. 如果一个体系能形成稳定的体系, 那么它不依靠这些 key 也是能解决的. 反而如果一体系不收敛只是利用了这些技术就收敛了, 那么这种结果如何可信? 前面的贴子中, 有些技巧诸如调整原子间的距离和键角, 这些手段才是比较有效的, 因为它直接影响了原子间的电子运动状态, 即波函数, 我们有时候初始猜测不正确就是因为程序给出的电子的占据状态错误而直接影响了原子间的成键, 这时就需要对电子占据的轨道进行调整,Guess=alter, 这对于过渡金属的计算来说特别要注意, 因为 G98 总不能给出正确的初始猜测. 谢谢诸位意见, 收益非浅 我起初一直认为我的初始模型是正确的, 但后来我试过 4 个模型, 才发现初始结构非常重要 开始三个是 MnO2 理想的晶体结构, 通过改变键长希望结构优化收敛, 但是失败 我又重新定义采用 MnO2 的真实晶胞参数, 现在结果有些眉目, 能收敛只是没有正常结束, 可能是我吸附分子的坐标出了点问题, 出现 Error on Z-matrix card number 23 angle Alpha is outside the valid range of 0 to 180. 所以我深切体会到楼上师兄的话 : 调整原子间的距离和键角, 这些手段才是比较有效的 我现在才发觉,G98 真是一个费内存, 费时间的东西, 很多计算要的是经验, 你计算的垃圾多了, 你才知道你需要什么! 这个过程真够悲惨

200 中国科学 SCIENCE IN CHINA 2000 Vol.30 No.5 P 一种选择从头算基函数的有效方法 张瑞勤步宇翔李述汤黄建华韩克利何国钟 摘要通过分析从头计算中不同基函数的作用, 提出一种经济有效的基函数选取方案. 依照该方案, 基函数的选取应考虑体系中不同原子的性质及实际的化学环境. 描述一般体系时可根据该原子在元素周期表中的位置从左向右依次增加基函数用量. 对荷负电原子, 则使用更多的基函数以及适当的极化函数或弥散函数. 对荷正电原子, 基函数的用量可适当减少. 对正常价态的饱和共价键合原子不需加极化或弥散函数, 对氢键 弱相互作用体系 官能团及零价或低价金属原子等敏感体系则需加极化或弥散函数. 据此, 可在适中基函数和能承受的计算量下得到具有相当可靠性的计算结果. 对一系列体系计算分析充分证明, 该方案是非常实用和有效的. 此方案可适用于 Hartree-Fock, Mφller-Plesset 和密度泛函等计算中, 并对化学 材料科学和生命科学研究广泛的大体系计算具有重要的实际应用. 关键词从头计算基函数布居分析极化函数弥散函数 量子化学从头计算是求解多粒子 多电子体系的量子理论全电子计 [1~ 算方法, 已成为从理论上探讨物质分子体系结构与性质的重要工具 4]. 对中小分子体系, 利用涉及电子相关的高水平算法和较完备基组, 在分子结构 能量及光谱性质等多方面已获得了与实验非常一致的结果, 并准确预言了一些未知及亚稳态分子的有关性质和规律. 这标志着人们完全从理论上探讨物质的各类性质已成为可能. 然而, 随着学科的交叉与渗透以及所涉及的体系的复杂化, 在诸如材料 团簇 溶剂化和生物等领域应用从头算和密度泛函法会导致计算量与占用计算机储存空间急剧增加. 传统上, 对一研究体系具体使用的基组的大小一般靠经验确定, 缺乏清晰可循的原则, 几乎每一位计算化学工作者都需要进行大量的试算, 以获取必要的经验. 为保证结果精确可靠通常使用了较大的基组, 而导致计算量过大, 费用昂贵. 因此寻找一种使计算量适中而结果又具相当可靠性的基函数选择新方案是很有实际意义的. 在传统的基组选取中, 同周期的不同原子皆用相同数量的波函数描写 [5]. 在处理同一体系中的同一原子时, 不管其化学性质及荷电量如何, 都选取同样数量的基函数. 这显然是不合理的. 准确有效的描述应考虑实际分子中各原子性质和化学环境的差异. 本文将对此类问题进行深入分析, 借助大量的例证来求证一种经济实用的基函数选择方案.

201 1 基函数系与较经济有效基函数的选择 1.1 基函数系量子化学的从头计算通常是选取分子中所有原子的原子轨道构成基函数空间 [1~3]. 此类基函数主要由角度部分和径向部分组成. 角度部分用球谐函数表示, 而径向部分函数通常采用 STO 和 GTO 两大系列. 鉴于 GTO 及 STO 的优缺点, 人们提出了 STO-GTO 系列, 即用 GTO 的线性组合来近似 STO. 根据组成分子时各原子内外层电子对成键的不同贡献, 通常采用混合 z 基. 它是目前从头算中最常用的基组体系. 考虑到分子中电子云较原子中的电子云存在着较大差别及轨道的变形极化, 人们又引进了极化函数来修正 [2]. 使用极化函数虽然增加了基函数的数量和类型, 但使得描述的分子中电子运动图像更加真实. 另一类修正分子轨道的辅助基函数是弥散函数 [6,7], 它对描述电子富集区松散电子云较为有用. 显然它对于一些电子离核相对较远的富电子体系的描述尤其重要, 如 : 带有孤对电子, 或半径较大的负离子等体系. 1.2 较经济有效的基函数的选择在广泛考虑电子相关效应的基础上, 利用较完备的基组是精确探讨一具体体系的前提. 基组越完善, 所得结果可靠性越高, 与实验值越接近. 然而这对大体系的计算将会变得非常困难. 根据基函数集合中各类函数的性质可知它们所起的作用是不同的. 即使是同类函数, 在不同体系中所起的作用也不尽相同. 而对同一体系中不同原子也应区别对待. 这为针对不同的研究体系采用经济有效基组创造了可能性. 据此, 描述每个原子的波函数用量和类型应根据该原子在化合物中价层电子实际占有数和行为来确定. 电子数越少, 使用的基函数应越少. 因此, 描述原子的基函数可根据该原子在元素周期表中的位置从左向右依次增加, 而且如果是带负电荷, 则基函数应多一些. 相反, 如果是带正电荷, 则使用的基函数可适当减少. 另一方面, 极化和弥散函数是描述轨道变形和弥散行为的修正函数. 成键原子间因电荷交换, 造成电荷分布不均匀. 失去电子的原子其电子云分布收缩. 这种收缩的电子云不难被描述, 有时不需极化函数. 但对得电子的原子, 其电子云易变形且膨胀, 有时还过于弥散, 使用极化和弥散函数是必要的. 因此根据分子体系的实际成键特征, 可适当选用有效的极化和弥散函数. 虽然同一原子在不同的分子体系中环境千差万别, 但其环境在同类分子体系中在某种程度上是相近的. 本文从几类分子体系入手, 对一般氢键 醇酮羧酸 烷烯烃 氧化物及金属化合物等体系, 在 HF(Hartree-Fock), MP2(Mφller-Plesset) 和 B3LYP/DFT 水平上来考察不同的基组以及极化和弥散函数对它们的几何构型的影响. 所有的计算工作均是用 G94W 程序包完成的 [8]. 2 结果与讨论 本文具体对 (BH 3 ) 2, (NH 3 ) 2, (H 2 O) 2, (HF) 2, CH 3 OH, CH 3 CH 2 OH, CH 3 COCH 3,

202 CH 3 COOH, CH 3 CONH 2, CH 2 =CH-CH=O, CH 4,CH 3 CH 3, CH 3 CH 2 CH 3, CH 2 =CH 2, CH 2 =CH-CH=CH 2, (HCOOH) 2 H +, (SiO) 2, (SiO) 3, Fe + OH 2, Al(OH) 3, Al(CN) 3 和 NaCl 等几类特殊体系在不同计算水平上利用不同基函数进行了几何全优化. 仅将部分体系构型绘于图 1 中, 而一些相关的几何参数则列于表 1~4 中. 表 5 为部分构型的净电荷分布. 可以看到, 许多体系中已有相当大的净电荷转移. 对其中许多荷负电的原子的描述远不能用依据中性原子或微小电荷转移体系确定的基组描述. 下面通过考查不同的基函数对这些体系的几何构形的影响来论证基函数的选取方案. 对一些氢化物体系, H 原子一般显正电性. 由于氢原子周围电子较少, 受核的束缚力较强, 电子云分布较紧凑且为 s 型, 故一般用 s 型函数就可精确描述. 有关不同基函数对氢的作用在本文不同的例举中都有体现. 一般来说, 对含氢且并不复杂的体系, 不管是 HF, MP2 还是 B3LYP 方法, H 的 p 极化函数对它们的结构影响很小, 键长差别大多数只有约 10-4 nm, 键角差别也很小 (<1.0o). 对有机化合物也是如此. 对此在以往的文献中已有很多例证 [9,10], 与本文的分析是一致的. 这也是传统基组规定中将氢和其他元素分开考虑的原因. 2.1 烷烯烃体系烷烃体系是成键度比较高的一类体系, 且具有四面体共价键骨架. 由于这类体系无多余的孤立电子, 共价键电子云密集地分布于键合原子的连线上, 分子轨道易于描述. 因此较小的基组便可得到较准确的结果. 本文在不同自洽场方法 (HF, B3LYP/DFT 和 MP2) 和不同基组水平上优化了一系列烷烯烃体系的几何参数. 表 1 CH 3 CH 2 OH(C s ) 分子键长 键角的 B3LYP/6-31++G* 结果以及与其他基组结果的相对偏差 C C C O O H COH CCO G** G G G* G** G G O/6-31G*/C,H/6-31G C/6-31G*/O,H/6-31G C2,O3/6-31G*/C1,H/6-31G C2,O3/6-31G*/C1,H/3-21G 表 2 (NH 3 ) 2 (C s ), (H 2 O) 2 (C s ) 和 (HF) 2 (D h ) 二聚体的部分 H A 键长和有关键角的 6-31G** 结果以及与其他基组结果的相对偏差

203 (NH 3 ) 2 (H 2 O) 2 (HF) 2 方法基组 N1-H3 N5-H O H O4-H5 542 F H HF MP2 B3LY P 6-31G* * G G 6-31G* 6-31G* * 6-31G 6-31G* 6-31G* * (DFT ) 6-31G 6-31G* 表 3 (HCOOH) 2 H + (C s ) 分子的 6-31G* 键长 键角值以及与其他基组结果的相对偏差 ( 其中 6-31 # : O/6-31G*/C,H/6-31G) 方法基组 H1-O2 O2-C3 C3-O4 O4-H5 O4-O6 O6-C7 C7-O8 C3-H1 0 HF 6-31G * G G # MP2 6-31G *

204 6-31G G # B3LY P 6-31G * (DFT ) 6-31G G # H 10 HF 6-31G * G G # MP2 6-31G * G G # B3LY P 6-31G * (DFT ) 6-31G G # 表 4 M(AB) 3 分子的键长键角的 B3LYP/6-31++G** 结果以及与其他基组结果的偏差 基组 Al(OH) 3 (C 3h ) Al(CN) 3 (D 3h ) Fe + -OH 2 (C 2v )

205 6-31++G** 3-21G 6-31G 6-31G* 6-31G** 6-31+G G 6-31+G* A/6-31G*/M,B/6-31 G A/6-31G*/M,B/3-21 G M/6-31G*/A,B/6-31 G M/6-31+G/A,B/6-31 G A/6-31+G/M,B/6-31 G M,A/6-31+G/B/6-31 G M = Al A = O B = H M = Al A = C B = N Al O O H AlOH M = Fe A = O B = H Al C C N Fe O O H HOH 表 5 在 HF/6-31G* 水平上诸分子中各原子的净电荷分布 ( 原子编号与图 1 相一致, 且忽略了等价原子 ) 分子净电荷 ( 原子单位 ) (BH 3 ) 2 B1: 0.13 H3: H7:

206 (H 2 O) 2 O1: H2: 0.36 H3: 0.33 O4: (NH 3 ) 2 N1: H2: 0.05 H3: 0.28 H4: 0.04 (HCOOH) 2 H + H1: 0.51 (HF) 2 H1: 0.54 Fe1: Fe + OH Al1: Al(OH) O8: H9: 0.53 Al1: Al(CN) C2: 0.06 C1: CH 3 COCH C2: 0.49 C1: CH 3 CH 2 OH C2: 0.01 O2: C3: 0.59 H10: 0.24 F2: H3: 0.53 O2: H3: 0.52 O2: H5: 0.46 N5: C3: O4: H11: 0.31 F4: O4: O3: H4: 0.44 H4: 0.35 H5: 0.35 N5: H6: 0.26 H5: 0.60 O6: H5: 0.17 H6: 0.21 H5: 0.14 H7: 0.20 对烷烃, 几套基组的结果都非常一致. 特别是对 C H 键, 极化与弥散函数对结果影响甚微. 因为 H 原子用的是球对称的轨道, 与 C 形成的键轨道较原来原子轨道变形性小. 可以推测对烷烃同系物的 C H 键长和 HCH 键角来说, 不用极化和弥散函数就可得到很好的结果. 对 C C 键, 加极化函数可稍微减短, 更接近精确值. 而加弥散函数几乎无影响. 实际上此类体系中 C 略显负电性, H 略显正电性, 如丙烷在 MP2(full)/ G** 水平上仅为 C C H 若以基团为单位, 则为 (CH 3 ) (CH 2 ) (CH 3 ) , 而偶极矩也仅为 Debye. 整个体系中电荷交换量很小, 轨道极化程度不明显. 在计算中仅需用中等大小基组 ( 如 6-31G) 就可得较准确的结果. H8: 0.28 C7: 0.61 H8: 0.17

207 图 1 部分体系的分子结构 (a) CH 3 CH 2 OH; (b) (HF) 2 ; (c)(nh 3 ) 2 ; (d) (H 2 O) 2 ; (e) (HCOOH) 2 H+; (f) Al(OH) 3 ; (g) Al(CN) 3 ; (h) Fe + OH 2 同样对烯烃, 3 种计算方法在所用的几种不同基组水平上的结果均表明了很好的一致性. 对一般烯烃极化和弥散函数对 C H 键及 HCH, CCC, HCC 诸夹角影响较小. 但对 C C 键长, 如同烷烃中的基组效应, 极化函数有一定的影响, 弥散函数效应可忽略. 虽然在烯烃中有较 s 键弱的 p 键相互作用, 但所用的 p 轨道并不弥散, 未受较强的电荷中心诱导极化, 并且平面内骨架 s 键间夹角的变化并不影响该 s 键键连原子间的 p 键作用, 因此结合所得数据可知, 在一般的有关 C== C 体系的计算中无需考虑弥散效应. 但极化效应仅对 C C 键有较小的影响. 另外对各类烷基取代苯做了有关的计算, 也发现了同样的规律性. 2.2 醇酮酸胺类及其他含氧氮等杂原子体系在生物大分子或材料分子领域中, 一类很重要的单体就是含羟基 羧基或胺基等活泼基的化合物, 其官能团决定了该物质分子很多重要性质. 前已论述, 对 C== C 双键体系, 无需增加极化函数就能给出准确的几何参数. 但由于一个 O 或 == NH 替代了其中的一个 C== 基团, 导致了双键的极化. 并且在 O, N 的端位还有多余的孤电子对, 使得 N, O 周围电子较富集, 双键 C 上的电子较缺乏. 这为是否增加极化函数或弥散函数提供了依据. 本文利用 B3LYP 法在各种基组水平上优化了大量有关醇 酮 醛 酸 胺及其他含杂原子 N, O, S 和 P 的有机分子体系的几何结构. 仅将 CH 3 CH 2 OH 的结果列在表 1 中. 对 CH 3 CH 2 OH 来说, 弥散函数对构型影响较小. 但若对 C, O 增加极化函数, 结果变化较大, 更靠近 G** 结

208 果. 若仅对 O 增加极化函数, 结果与精确基组值也非常接近. 然而, 若仅对 C 增加极化函数, C C 和 C O 键长虽比较准确, 但对 O H, C H 键及夹角 COH 和 OCH 仅有轻微改进. 因此分析表明对 CH 3 CH 2 OH 仅需对 O 增加极化函数就可得到与 C, O, H 同时增加极化和弥散函数非常一致的结果. 显然前者大大减少了基函数的数目. 这与 CH 3 CH 2 OH 中的电荷分布是密切相关的. 在 B3LYP/6-31++G** 水平上, 电荷分布为 H C C H O H 显然由于 C, O 带负电荷, 电子密集, 需要考虑极化效应. 但与 C 相连的 H 上呈现较大正电荷. 若以 CH 3, CH 2 基团为单位, 则电荷分布又为 (CH 3 ) (CH 2 ) 0.2 O H 这表明 CH 3 和 CH 2 都有类似烷烃基团性质, 用一般基组即可, 仅需 O 加极化函数就可得到与精确值较为一致的结果. 同样对 CH 3 COCH 3 和 CH 3 COOH, 其电荷分布为 (H C ) 2 C 0.28 O 和 H C C 0.40 O O -0.5 H 0.36, 以基团为单位则为 (CH 3 ) 0.06 C 0.28 O (CH 3 ) 0.06 及 (CH 3 ) 0.07 C 0.40 O (O H 0.36 ), 也仅需对 O 进行加极化函数就可得较准确的结果. 对一些其他氧化物体系也是如此. 本文优化的 (SiO) 3, (SiO) 2 的键长和键角也表明不管是哪一种算法, 仅 O 加极化或弥散函数就可给出非常接近 Si, O 都加极化函数所得构型. 2.3 质子化甲酸二聚体及弱氢键体系氢键及其弱相互作用是理论化学计算中的一类难点问题. 一般来说, 形成氢键和弱相互作用的两原子间距较长, 轨道间重叠程度较小. 因此在计算中必须尽可能进行精确标度. 表 2 给出了上述氢化物二聚体的 H- 键以及与 H- 键相关的键连基团的键长键角受基组影响的情况. 对 (HF) 2 来说, 荷微量正电荷的 H 与荷负电的 F 相互静电作用后有少量电荷 (0.1e) 进一步由 F( 一分子片 ) 向 H( 另一分子片 ) 转移, 造成 F 电子云极化, 来加强 H- 键的形成. 此时 H F 键较长 (MP2/6-31G**: nm), H,F 间作用较弱, 对环境因素比较敏感. 且 H F 间电子云较未形成 H- 键前有所变形, 故极化函数对此将有较大影响. 若仅对 F 增加极化函数, 结果有相当大的改进, 但仅增加弥散函数改进甚微. 若进一步仅对 H- 键键连的 F 增加极化函数, 结果与精确值也吻合较好. 同理, 对 (NH 3 ) 2 中的 N H 及 (H 2 O) 2 中的 O H 键也是如此. 仅需增加 N, O 的极化函数就获得合理的 N H 和 O H 氢键长. 同时增加极化函数还敏感地影响形成 H- 键的重原子的有关键角. 在 (NH 3 ) 2 中, H N 键与其他 N H 键的键角改变了约 7, 而 H N 键中 H 的 H N 键与其他 N H 键夹角也改变了 4. 在 (H 2 O) 2 中更为明显, 夹角变化了 15. 对有质子 H + 参与的 H- 键体系, 极化函数影响更大. 表 3 列出了几个不同的基组下质子化甲酸二聚体中键长和键角相对 6-31G** 结果的差值. 总的来说, 6-31G 基组的结果与 6-31G** 结果的差别要比仅 O 加 d 极化函数后所得结果的差别大得多. 对基组最敏感的键长是将两个甲酸键合在一起的氢键, 即 O4-H5. 此键长对有无 d 极化函数差别较大 (HF: nm; MP2: nm). 实际上, 在此体系中, O 上带有较多的负电荷, 而 C, H 上带正电荷, 因此 O 的 d 极化函数较为重要. 这种影响还从体系的键角随基组的变化更明显地表现出来. 在 6-31G 计算中, 键角最大的变化 ( O4O6C7) 在 HF, MP2 和 B3LYP 计算中分别达 , 和 当荷负电的 O 加 d 极化函数时, 这种差别减小到

209 , 和 , 其他键角的差别也具同样的规律性. 若仅对 O 使用 d 极化函数就可得到与 O, C 和 H 全部都加极化函数非常相近的结果. 这样就减少了近三分之一的基函数. 作者曾采用这种方法处理质子化甲酸聚合物, 能够优化高达六聚物 (HCOOH) n H + (n = 6) 的几何构型, 进而获得与实验符合很好的蒸发能并解释了二聚物蒸发现象 [11]. 作者等人进一步将该基组用于含 H 2 O 的质子化甲酸聚合物, 也得到了满意的结果 [12]. 2.4 金属化合物体系含金属的化合物也是一类很重要的化合物. 金属元素的独特性质以及与各功能团间的相互作用控制着生物分子乃至材料的一些重要功能性, 同时在催化反应机理方面也起着至关重要的作用. 根据金属元素在周期表中的位置以及其不同的电子构型, 在所形成的化合物中表现了不同的性质. 因此, 应根据其金属性以及所形成的化合物的性质在有关的理论计算中对金属原子区别考虑. 一般对第 Ⅰ, Ⅱ 主族的金属元素来说, 因其高能级的 s 电子极易失去而具强的给电子能力, 在所形成化合物中, 一般以正离子的形式存在, 且较稳定. 因此在一般的计算中较小基组而不加极化和弥散函数就可给出较准确的结果. 如在 B3LYP/Cl/6-31G*/Na/6-31G/ 水平上所优化的 NaCl 键长 ( nm) 就与 Na,Cl 均用 6-31G* 所得结果 ( nm) 非常一致. 但对富电子体系中的 Na, 如 Na-NH 3, 其极化函数对构型影响较大, 应同时对 Na, N 进行极化函数修正. 对于 p 区的金属元素来说, 情况又有所不同. 因 p 区元素的 d 轨道是空的或全充满的, 起关键作用的是价 p 轨道. 但因主量子数和价电子数的不同, 它们表现出了不同的成键性质. 表 4 也同时列出了 Al(OH) 3 和 Al(CN) 3 在 B3LYP 及各种基组水平上的几何优化结果. 对 Al(OH) 3, 由于 Al 显正电性, 核对外层电子的吸引力较强, 使得其电子结构更加紧凑. 而 O 原子上电子较富集. 因此可以断定, 仅需对 O 原子进行极化和弥散函数修正就可得到较精确的结果. 分析表 4 可知, 对 Al O 键, 总的来说对弥散函数修正不大敏感 ; 但若仅在 O 原子上考虑极化和弥散函数修正, 结果会更接近 6-31+G* 结果. 键角 AlOH 对极化函数较敏感, 特别是 O 的极化函数. 若对 O 原子考虑极化修正, 即使 Al 和 H 都用较小的基组 ( 如 3-21G) 也能得到较好的结果 ( 将 AlOH 夹角修正了约 20 ). 这是显然的, 因为决定 AlOH 大小的是富电子的 O 原子的电子构型. 同样, 对 Al(CN) 3 也是如此. 对 Al 加极化函数对 C N 键影响较小, 但可使 Al C 键长减小. 若对 C, N 同时加极化函数, 则 C N 键长比较接近较大基组的结果, 而 Al C 键稍微变长. 然而, 若讨论类似的 AlR 3 (R 为烷基 ), 则需主要对 Al 进行极化修正. 对副族元素来说, 由于其 d 轨道是价轨道, 参与成键, 且具有多变的自旋态, 使得原子或相应的离子半径较大, d 轨道伸张弥散, 易变形. 由于它们在相关化合物中易表现为功能活性中心, 因此在计算中应谨慎考虑基组效应. 表 4 给出了在 12 种基组水平上的 Fe + -OH 2 离子的有关计算结果. 由于 Fe + 具有较大的正电荷, 因此与 H 2 O 键合后, 导致 0.244e 电荷 ( 在 MP2/ G** 水平上 ) 从 H 2 O 分子向 Fe + 转移并主要分布于 Fe 和 O 间, 从而加强了 Fe O 键的强度. 这种电荷转移导致了 H 2 O 分子中的分子轨道变形极化, 因此需要增加极化和弥散函数来描述. 从表 4 可以看出, 增

210 加弥散函数对 O H 键长 HOH 键角影响较小, 但增加极化函数可使键角减小 3 多. 但这种减小夹角的贡献主要来自 O 的极化函数. 仅需考虑 O 原子轨道的极化修正就可得到配体 H 2 O 的正确构型. 但对 Fe O 键来说, 情况就有所不同. 因为 Fe O 键密切相关于 Fe 和 O 原子轨道及电荷分布. 比较表 4 可知, 在一般 6-31G 基组水平上的结果 ( nm) 与较大基组 ( G**) 结果 ( nm) 偏差较大. 但仅考虑 O 的极化函数修正结果就有很大的改进 ( nm). 若仅考虑 Fe 的极化函数修正, 结果反而与精确结果相差更大 ( 约 nm). 从弥散函数的影响也可得到同样的规律. 这表明在此类体系的计算中, 对 O 考虑极化和弥散修正就可得到较精确的结果. 类似地对 FeOH 2, Fe 2+ OH 2 和 Fe 3+ OH 2 体系也进行了计算, 结果表明 FeOH 2 需要增加 Fe, O 的极化函数, 而 Fe 2+ OH 2 和 Fe 3+ OH 2 却仅需考虑 O 的极化函数修正. 2.5 CPU 机时比较以上讨论了基组对几类体系几何构型的影响. 可以看出, 仅对部分特殊环境中的原子加极化或弥散函数就可获得比较精确的结果. 这为计算生物大分子或材料分子在保证有效精度的前提下尽可能减少基组总数, 提高计算效率及可能性提供了依据. 由于基组的减小, 减缓了计算机内存的限制, 更重要的是可以节约有效的 CPU 时间. 表 6 给出了质子化的甲酸二聚体 (HCOOH) 2 H + 利用微机 (Pentium-II/ 双 350 MHz 处理器 ) 在不同的计算方法和基组下单点计算所用的 CPU 时间与基组大小的关系. 从中可以看出, 计算机的 CPU 时间与基组大小是密切相关的. 据前面分析, 仅在 O 原子上增加极化函数 (O/6-31G*/C, H/6-31G) 就可获得与所有原子都增加极化函数非常一致的结果, 然而函数个数却减少了 35 个, 计算机 CPU 时间缩短了一半, 非常接近用 6-31G 基组所用的时间. 推而广之, 对计算较大的分子体系, 节约的计算机内存和 CPU 时间将会是非常可观的. 基组 表 6 (HCOOH) 2 H + 单点计算的 CPU 时间比较 基函数的数 微机 CPU 时间 目 HF B3LYP MP2 6-31G G* G** G G G** O/6-31G*/C,H/6-31G O4,O6/6-31G*/others/6-31G O,H5/6-31G*/C,H/6-31G

211 3 结论 通过上面理论分析和例证可以得出以下几点结论 : (1) 一般来讲, 对计算中所涉及的原子的基函数 极化函数或弥散函数可按该原子在元素周期表中的位置从左向右以及荷负电的程度依次增加. 对以非极性或弱极性共价键键合的原子, 中小基组即可, 一般不用极化函数. 对于强极性共价键或离子键键连原子, 荷正电的可用一般基组, 不用极化函数, 荷负电的应用较大基组, 并加极化或弥散函数. (2) 需要特别指出的, 对生物分子或材料中常见的化合物类型, 如由醇类 羧酸 水等极性溶剂形成的大团簇以及材料和生命科学中含有杂原子的有机聚集体, 其 N, O, P, S 等电负性大的原子一般要用较大的基组, 且加极化函数. 对一般 C, H 原子中小基组即可, 不需加极化函数修正. 对所涉及的烷烯烃基团, 可统一用中小基组, 且不加极化函数. 重点官能团可重点加大基组. (3) 对体系中仍保持金属特性的金属原子或呈现负价和低的正价态, 该原子一般需加极化和弥散函数 ; 若呈现高正价态 (>+1), 一般不需极化和弥散函数修正. (4) 对于以分子间氢键或其他分子间弱相互作用相连接的分子, 各小分子单元按上述原则外, 对形成 H- 键或弱相互作用的活性原子, 一般需要考虑极化和弥散函数修正. 张瑞勤 ( 香港城市大学物理与材料科学系, 香港 ) 步宇翔 ( 香港城市大学物理与材料科学系, 香港 ) 李述汤 ( 香港城市大学物理与材料科学系, 香港 ) 黄建华 ( 中国科学院大连化学物理研究所分子反应动力学国家重点实验室, 大连 ) 韩克利 ( 中国科学院大连化学物理研究所分子反应动力学国家重点实验室, 大连 ) 何国钟 ( 中国科学院大连化学物理研究所分子反应动力学国家重点实验室, 大连 ) 参考文献 1, 唐敖庆, 杨忠志, 李前树. 量子化学. 北京 : 科学出版社, , 徐光宪, 黎乐民, 王德民. 量子化学 : 基本原理和从头算法 ( 中册 ). 北京 : 科学出版社, , 廖沐真, 吴国是, 刘洪霖. 量子化学从头计算方法, 北京 : 清华大学出版社, ,Labanowski J K, Andzelm J W. Density Functional Methods in Chemistry. NewYork: Springer-Verlag, ,Hehre W J, Ditchfield R, Pople J A. Self-consistent molecular orbital methods. Ⅻ. Further extensions of Gaussian-type basis sets for use in molecular orbital studies of organic molecules.

212 J Chem Phys,1972, 56: ,Duke A J. A Hartree-Fock study of the methyl anion and its inversion potential surface: Use of an augmented basis set for this species. Chem Phys Lett, 1973, 21: 275 7,Chandrasekhar J, Andrade J G, Schleyer Pv R. Efficient and accurate calculation of anion proton affinities. J Am Chem Soc, 1981, 103: ,Frisch M J, Trucks G W, Head-Fordon M, et al. GAUSSIAN94W (RevisionA). Pittsburgh: Gaussian Inc, ,Boggs J E, Cordell F R. Accurate ab initio gradient calculation of the structures and conformations of some boric and fluoroboric acids. Basis-set effects on angles around oxygen. J Mol Struct Theochem, 1981, 76: ,Pang F, Pulay P, Boggs J E. The structure of some nitrogen heteroaromatics. J Mol Struct Theochem, 1982, 88: 79 11,Zhang R Q, Lifshitz C. Ab initio calculations of hydrogen-bonded carboxylic acid cluster systems: dimer evaporations. J Phys Chem,1996, 100: ,Aviyente V, Zhang R Q, Varnali T, et al. Structures and energetics of proton-bound formic acid-water clusters. Int J Mass Spectrom Ion Proc, 1997, 161: 123 收稿 : 修稿 :

213 量子化学计算的一般性规则的验证 一 原理 1. Hartree-Fock 方法的精确性 在从头计算法 (ab initio) 的三个近似下 ( 非相对论近似 绝热近似和轨道近似 ) 分子或原子的全电子能量包括以下几个 : (1) ET: 电子的动能 ( 恒正 ); (2) EN: 核与核的势能, 对于计算单点能, 这一项能量总是不变的 ; (3) EV: 核与电子的势能 ; (4) EJ: 电子与电子的库仑势能 ; (5) EX: 电子与电子的交换势能 ; (6) EC: 电子与电子的相关能, 单电子近似时不考虑这项能量 所以电子的总能量可以写成 : E = ET + EN + EV + EJ + EX + EC 只要基函数足够完备,Hartree-Fock 方法就可以精确计算出前 5 项能量, 但是对于最后一项 EC 却无能为力 尽管 EC 在数值上相比前 5 项小得多, 但对于计算反应热等物理性质, 它的误差却是不能忽视的 2. 密度泛函方法的精确性 密度泛函方法中最基本的公式是 Kohn-Sham 方程 : E - EN = ET(ρ) + EV(ρ) + EJ(ρ) + EXC(ρ) 虽然该方程是严格成立的, 但是处理起来有很多困难, 首先是 ET(ρ), 它无法精确求得, 然后是 EXC(ρ), 它除了包括电子之间的交换势能和相关能以外, 还包括 ET(ρ) 中的误差 EJ(ρ) 中由电子自相互作用而产生的误差, 等等 其中, 最严重的问题是电子自相互作用, 目前还没有很好的解决方案 EXC(ρ) 计算由一系列经验公式给出, 所以它是一种半经验的方法 二 验证 1. 氢原子方法 / 基组 STO-3G 6-31G G** HF B3LYP QCISD 讨论 : (1) QCISD 的结果和 HF 的完全一致, 原因是 QCISD 在 HF 的基础上计算组态相关 (CI), 而 CI 对于氢原子是没有意义的 (2) B3LYP 方法总是比 HF 方法能量低 0.002Hartree 左右, 这是电子自相互作用矫正的结果, 自相互作用矫正通常会使能量偏低, 这在只含 H 的体系中非常显著 (3) HF/ G** 的结果更接近于实验值 (1H 原子的 Rydberg 常数 ), 仅仅是绝热近似误差和基组误差互相抵消的原因

214 (4) 用 HF/AUG-cc-pV5Z 计算的结果是 , 说明基组数量足够大时可以无限 接近于理论值, 绝热近似下的理论值为 Hartree( 即 Rydberg) 2. 氢分子离子方法基组键长 (A) 核排斥能轨道能总能量 HF STO-3G G G** B3LYP STO-3G G G** QCISD STO-3G G G** 讨论 : (1) 由于仍旧是单电子体系, 所以 HF 和 QCISD 的计算结果仍旧一致 ; (2) 单电子体系的 HF 方法严格遵循 " 核排斥能 + 轨道能 = 总能量 "; (3) DFT 的自相互作用会使轨道能偏高, 需要矫正, 所以不符合 HF 的能量公式 ; (4) DFT 的不适合做开壳层结构的优化, 由于自相互作用, 结构偏于松散 ; (5) 用 HF/AUG-cc-pV5Z 计算的键长为 A( 实验值为 1.06 A), 轨道能为 Hartree, 总能量为 Hartree, 换算成解离能为 2.79eV( 未考虑振动零点能 )( 实验值为 2.97eV) 3. 氦原子方法 STO-3G 6-31G G** 基组轨道能总能量轨道能总能量轨道能总能量 HF B3LYP QCISD 讨论 : (1) 轨道能的 2 倍不等于总能量, 因为轨道能不包括轨道中另一个电子对它的作用 ; (2) HF 和 QCISD 的轨道能完全一致, 因为 QCISD 首先要用 HF 方法计算 SCF 能量, 轨道能是从 SCF 能量中获得的 ; (3) QCISD 的总能量比 HF 低, 因为它考虑了电子的相关能 ; (4) B3LYP 的能量比 HF 更接近于 QCISD, 它也考虑了电子的相关能 ; (5) 用 QCISD/cc-pV5Z( 氦原则不能加弥散基组 AUG-) 计算的轨道能为 Hartre e ( ev, 实验值为 ev, 即第一电离能 ), 总能量为 Hartree( ev, 实验值为 78.98eV, 即第一电离能和第二电离能之和 ) 4. 氢分子方法基组键长 (A) 轨道能总能量

215 HF STO-3G G G** B3LYP STO-3G G G** QCISD STO-3G G G** 讨论 : (1) 由于是闭壳层,B3LYP 方法比 HF 更接近于 QCISD; (2) 用 QCISD/AUG-cc-pVQZ 计算的键长为 A( 实验值为 0.74 A), 轨道能为 Hartree, 总能量为 Hartree, 换算成解离能是 4.73eV( 未考虑振动零点能 )( 实验值是 4.75eV), 振动频率为 cm-1( 实验值为 cm-1) 三 结论 通过上述计算, 可以得到以下一些一般性的结论, 为方法和基组的选择提供参考 (1) 所有的从头计算都是以三个近似为前提的, 因此再好的方法和再大的基组也不可能得到完全精确的结果, 例如 H 原子绝热近似的误差就为 0.05%; (2) HF 的最大缺点就是它没有考虑相关能, 即便使用再大的基组也无法克服这个困难 ; (3) 密度泛函的优点是考虑了相关能, 缺点是矫正电子自相互作用的困难 ; 密度泛函计算的相关能是不完全的, 这会导致能量偏高, 而矫正电子自相互作用会使能量偏低, 这两个效应相互抵消, 这是某些密度泛函 ( 如 B3LYP 等 ) 计算精度比较高的原因之一 ; (4) B3LYP 不适宜计算开壳层体系, 因为矫正未成对电子的自相互作用会使体系的能量偏低很多 ; 在计算轨道时结果正好相反, 由于电子自相互作用无法矫正的原因, 轨道能量会偏高很多 ; (6) 计算组态相关的方法 ( 如 MP2 MP4 CCSD QCISD 等 ) 可以得到很好的相关能, 大幅度提高计算精度, 但是轨道能量由 SCF 能量得出, 结果与 HF 的一致, 所以仅需要计算轨道能量时, 没有必要用计算组态相关的方法

216 量子化学中的基组是在量子化学中用于描述体系波函数的若干具有一定性质的函数, 基组是量子化学从头计算的基础, 在量子化学中有着非常重要的意义 基组的概念最早脱胎于原子轨道, 随着量子化学的发展, 现在量子化学中基组的概念已经大大扩展, 不局限于原子轨道的原始概念了 在量子化学计算中, 根据体系的不同, 需要选择不同的基组, 构成基组的函数越多, 基组便越大, 对计算的限制就越小, 计算的精度也越高, 同时计算量也会随基组的增大而剧增 目录 1 斯莱特型基组 2 高斯型基组 3 压缩高斯型基组 3.1 最小基组 3.2 劈裂价键基组 3.3 极化基组 3.4 弥散基组 3.5 高角动量基组 4 参见 斯莱特型基组斯莱特型基组就是原子轨道基组, 基组由体系中各个原子中的原子轨道波函数组成, 其形式为 : 斯莱特型基组是最原始的基组, 函数形式有明确的物理意义, 但是这一类型的函数, 数学性质并不好, 在计算多中心双电子积分时, 计算量很大, 因而随着量子化学理论的发展, 斯莱特型基组很快就被淘汰了 高斯型基组高斯型基组用高斯函数替代了原来的斯莱特函数, 其形式如下 : 高斯型函数在计算中有较好的性质, 可以将三中心和四中心的双电子积分轻易转化为二中心的双电子积分, 因而可以在相当程度上简化计算, 但是高斯型函数

217 与斯莱特型函数在处的行为差异较大, 直接使用高斯型函数构成基组会使得量子化学计算的精度下降 压缩高斯型基组压缩高斯基组是用压缩高斯型函数构成的量子化学基组 为了弥补高斯型函数与处行为的巨大差异, 量子化学家使用多个高斯型函数进行线性组合, 以组合获得的新函数作为基函数参与量子化学计算, 这样获得的基组一方面可以较好地模拟原子轨道波函数的形态, 另一方面可以利用高斯型函数在数学上的良好性质, 简化计算 压缩高斯型基组是目前应用最多的基组, 根据研究体系的不同性质, 量子化学家会选择不同形式的的压缩高斯型基组进行计算 最小基组最小基组又叫 STO-3G 基组,STO 是斯莱特型原子轨道的缩写,3G 表示每个斯莱特型原子轨道是由三个高斯型函数线性组合获得 STO-3G 基组是规模最小的压缩高斯型基组 STO-3G 基组用三个高斯型函数的线性组合来描述一个原子轨道, 对原子轨道列出 HF 方程进行自洽场计算, 以获得高斯型函数的指数和组合系数 STO-3G 基组规模小, 计算精度相对差, 但是计算量最小, 适合较大分子体系的计算 劈裂价键基组根据量子化学理论, 基组规模越大, 量化计算的精度就越高, 当基组规模趋于无限大时, 量化计算的结果也就逼近真实值, 为了提高量子化学计算精度, 需要加大基组的规模, 即增加基组中基函数的数量, 增大基组规模的一个方法是劈裂原子轨道, 即使用多于一个基函数来表示一个原子轨道 劈裂价键基组就是应用上述方法构造的较大型基组, 所谓劈裂价键就是将价层电子的原子轨道用两个或以上基函数来表示 常见的劈裂价键基组有 3-21G 4-21G 4-31G 6-31G 6-311G 等, 在这些表示中前一个数字用来表示构成内层电子原子轨道的高斯型函数数目, - 以后的数字表示构成价层电子原子轨道的高斯型函数数目 如 6-31G 所代表的基组, 每个内层电子轨道是由 6 个高斯型函数线性组合而成, 每个价层电子轨道则会被劈裂成两个基函数, 分别由 3 个和 1 个高斯型函数线性组合而成 劈裂价键基组能够比 STO-3G 基组更好地描述体系波函数, 同时计算量也比最小基组有显著的上升需要根据研究的体系不同而选择相应的基组进行计算 极化基组劈裂价键基组对于电子云的变型等性质不能较好地描述, 为了解决这一问题, 方便强共轭体系的计算, 量子化学家在劈裂价键基组的基础上引入新的函数, 构

218 成了极化基组 所谓极化基组就是在劈裂价键基组的基础上添加更高能级原子轨道所对应的基函数, 如在第一周期的氢原子上添加 p 轨道波函数, 在第二周期的 C 原子上添加 d 轨道波函数, 在过渡金属原子上添加 f 轨道波函数等等 这些新引入的基函数虽然经过计算没有电子分布, 但是实际上会对内层电子构成影响, 因而考虑了极化基函数的极化基组能够比劈裂价键基组更好地描述体系 极化基组的表示方法基本沿用劈裂价键基组, 所不同的是需要在劈裂价键基组符号的后面添加 * 号以示区别, 如 6-31G** 就是在 6-31G 基组基础上扩大而形成的极化基组, 两个 * 符号表示基组中不仅对重原子添加了极化基函数, 而且对氢等轻原子也添加了极化基函数 弥散基组弥散基组是对劈裂价键基组的另一种扩大 在高斯函数中, 变量 α 对函数形态有极大的作用, 当 α 的取值很大时, 函数图像会向原点附近聚集, 而当 α 取值很小的时候, 函数的图像会向着远离原点的方向弥散, 这种 α 很小的高斯函数被称为弥散函数 所谓弥散基组就是在劈裂价键基组的基础上添加了弥散函数的基组, 这样的基组可以用于非键相互作用体系的计算 高角动量基组高角动量基组是对极化基组的进一步扩大, 它在极化基组的基础上进一步添加高能级原子轨道所对应的基函数, 这一基组通常用于在电子相关方法中描述电子间相互作用

219 如何阅读高斯的计算输出文件 高斯输出文件的关键是看 Link, 每个 Link 对应不同的部分, 以单点计算为例 : 单点计算的流程 ( 见附件 ) 输入行在 # 后加入 P, 不然的话就只会显示 Link 1 和 0, 输出文件一般很长 对于单点计算, 只 有 300 多行 第一部分 : 基本运行信息输入输出文件名, 初始命令 进入 Link 1, 显示进程号 首先是版权说明, 然后是作者, 利用 guassian 计算所发的文章, 参考文献上必须列的 进入 Link 101 读入输入文件, 将控制命令转化为程序能够看得懂的 IOP 占位段的选项和应用设置 Leave Link 101 进入 Link 202: 判断体系对称性, 并决定实际计算中对称性应该如何利用判断体系点群例如 Stoichiometry CH2O Framework group C2V[C2(CO),SGV(H2)] Deg. of freedom 3 Full point group C2V NOp 4 Largest Abelian subgroup C2V NOp 4 Largest concise Abelian subgroup C2 NOp 2 再将输入的分子坐标转换为标准内坐标, 就是 Standard orientation: 这是程序默认, 充分利用分子的对称性来达到方便计算的目的 就是将体系的质量中心放在坐标轴的主轴或者原点上 可以用 nosym 来禁止这个操作 Leave Link 202 进入 Link 301 产生基组信息基组函数对称性

220 计算出核排斥能 Leave Link 301 进入 Link 302,303, 这部分是计算积分的具体算法, 一般不会列出详细过程, 一句话带过 leave Link302,303 进入 Link 401 在实际计算之前, 必须有初始猜测, 这部分的功能就是产生初始猜测, 可以从 ch k 文件读入, 也可以程序自动产生 这部分可以用 guess 关键字来指定 我这个体系, 程序默认是 : Projected INDO Guess( 请仔细看手册, 不同的体系 Guess 是不一样的 ) Leave Link 401 进入 Link 502 ( 主角登场!) 自恰迭代求解 SCF 方程设定收敛标准 : 除了对包含比 Ar 重的原子的分子全电子 ( 非 ECP) 计算外, 积分只计算到 10-6 的精度 λ SCF 计算的能量和电子密度都收敛到 10-4, 或只有能量收敛到 10-5 为止, 无论哪一个先达到 λ 给出每一步迭代的信息 (?)( 这只有在程序输入中加入 #p 才会显示 ) 如果成功计算完毕 SCF Done, 接下来就是打印计算出来的能量, 自旋, 维力值和收敛判据了. Leave Link502( 一般计算经常会在这一步挂掉, 哈哈 ) 如果你的单点计算只写了这些, 输出结束, 如果你加入了 population 的内容, 那么, 基于 scf 计算的结果, 布局分析开始 进入 Link601 Link 601 利用优化后的波函数, 进行 Mulliken population, 得到轨道对称性电子态各个轨道的本征值 然后是一些电荷, 自旋密度的分析 leav Link601 最后是整个计算结果的一个总结, 各小节之间用 \ 分开

221 单点计算看明白之后看几何优化的就比较清楚了优化计算的第一步必须和以后各步骤分开处理, 因为有几个操作只需做一次 例如读入最初的 Z- 矩阵和产生初始分子轨道 必需有一个对分子几何构型做优化计算的循环, 用优化程序 ( 本例中使用 Berny 优化模块 L 103) 决定是否需要做另一个优化, 以及是否获得了最优的分子结构 λ 如果分子结构收敛, 则程序计算能量梯度, 确认已得到最优的分子结构后, 退出 λ 只在第一个和最后一个分子结构进行布居数分析并打印轨道, 不要在不感兴趣的中间结构进行这些操作 λ 第一个点的处理分成两部分, 计算路径各由一系列的积分 猜测 SCF 以及积分求导的计算组成 第一部分包括模块 L101( 读入分子初始结构 ) 和模块 L103( 进行本身的初始化 ), 并有选项指示模块 L401 产生最初的猜测波函数 第二部分计算在优化过程中使用由模块 L 103 产生的分子结构, 并且有选项指示模块 L401 取回前一结构的波函数, 做为下一步的最初猜测波函数 第八行中指示向前跳行, 表示如果模块 L103 正常结束 ( 没有任何特殊的操作 ), 下一行 ( 模块 L9999) 则跳过不执行 通常在第二次激活模块 L103 时, 会检查最初的能量梯度, 然后选出新的结构 下一个要执行的是模块 L202, 用来处理 Z- 矩阵, 接着是第二部分其它的能量和梯度计算, 它们构成主要的优化计算循环部分 如果模块 L103 第二次激活时发现几何结构已经收敛, 计算将会终止并产生禁止向前跳行的信号, 然后执行下一行的模块 L9999, 完成整个计算 第 行构成主要的优化计算循环 该循环计算积分 波函 以及优化计算中第二个点与后面各点的梯度 它以模块 L103 结束 如果几何结构还未收敛, 模块 L103 选择新的结构并正常结束计算, 这会造成执行次序由第 15 行向后跳到第 10 行, 进行新的计算循环 如果

222 模块 L103 发现几何结构已经收敛, 将退出并禁止跳行, 执行结束行第 行 结束前使用模块 L601 在最后的分子结构产生多极积分, 将打印出多极矩 分子轨道和布居 数分析, 如果需要的话 最后模块 L9999 产生存档项目并结束计算任务

223 收敛问题的调整 如果 SCF 计算收敛失败, 你首先会采取哪些技巧呢? 这里是我们强烈推荐的首选方法 1. 在 Guess 关键字中使用 Core,Huckel 或 Mix 选项, 试验不同的初始猜测 2. 对开壳层体系, 尝试收敛到同一分子的闭壳层离子, 接下来用作开壳层计算的初始猜测 添加电子可以给出更合理的虚轨 道, 但是作为普遍的经验规则, 阳离子比阴离子更容易收敛 选项 Guess=Read 定义初始猜测从 Gaussian 计算生成的 checkpoint 文件中读取 3. 另一个初始猜测方法是首先用小基组进行计算, 由前一个波函 得到用于大基组计算的初始猜测 (Guess=Read 自动进行 ) 4. 尝试能级移动 (SCF=Vshift) 5. 如果 接近 SCF 但未达到, 收敛标准就会放松或者忽略收敛标准 这通常用于不是在初始猜测而是在平衡结构收 敛的几何优化 SCF=Sleazy 放松收敛标准,Conver 选项给出更多的控制 6. 一些程序通过减小积分 精度加速 SCF 对于使用弥散函数, 长程作用或者低能量激发态的体系, 必须使用高积分精度 : SCF=NoVarAcc 7. 尝试改变结构 首先略微减小键长, 接下来略微增加键长, 接下来再对结构作一 点改变 8. 考虑使用不同的基组 9. 考虑使用不同理论级别的计算 这并不总是实用的, 但除 此之外, 增加迭代数量总是使得计算时间和使用更高理论级别差不多 10. 关闭 DIIS 外推 (SCF=NoDIIS) 同时进行更多的迭代 ( SCF=(MaxCycle=N) ) 11. 更多的 SCF 迭代 ( SCF(MaxCycle=N), 其中 N 是迭代数 ) 这很少有帮助, 但值得一试 12. 使用强制的收敛方法 SCF=QC 通常最佳, 但在极少数情况下 SCF=DM 更快 不要忘记给计算额外增加一千个左右的迭代 应当测试这个方法获得的波函, 保证它最小, 并且正 好不是稳定点 ( 使用 Stable 关键字 ) 13. 试着改用 DIIS 之外其它方法 (SCF=SD 或 SCF=SSD) 精度和 CPU 时间比较 执行方法的列表可以从下面找到 : M. Head-Gordon, J. Phys. Chem. 100, (1996). J. B. Foresman, Æ. Frisch, "Exploring Chemistry with Electronic Structure Methods" Gaussian, Pittsburgh (1996). W. J. Hehre, "Practical Strategies for Electronic Structure Calculations" Wavefunction, Irvine (1995). W. J. Hehre, L. Radom, P. v.r. Schleyer, J. A. Pople, "Ab Initio Molecular Orbital Theory" John Wiley & Sons, New York (1986). T. Clark, "A Handbook of Computational Chemistry" John Wiley & Sons, New York (1985). J. G. Csizmadia, R. Daudel, "Computational Theoretical Organic Chemistry" Reidel (1981). Journal of Molecular Structure 杂志每年有一期包含一张表格, 涉及上一年发表的所有理论计算 类似的信息还 有 : "Quantum Chemistry Literature Data Base Bibliography of Ab Initio Calculations for " K. Ohno, K. Morokuma, Eds., Elsevier, Amsterdam (1982). 理论 著作的评论有 : Reviews in Computational Chemistry Annual Review of Physical Chemistry Advances in Quantum Chemistry Advances in Chemical Physics 理论著作评 论有时也可以在 Chemical Reviews 和 Chemical Society Reviews 找到 练习找到 Gaussian 实例文 件 Gaussian 带有一套测试任务 它们也可以作为输入的例子 Windows 版 :\G98W\tests\tests.idx 测 试任务目录列表 \G98W\tests\gjf 测试输入文件目录 Unix/Linux 版 : /g98/tests/tests.idx 测 试任务目录列表 /g98/tests/com 测试输入文件目录 /usr/local/apps/g98/tests/cray1 测试输出 文件目录手动输入基组 关键字 gen 用于代替基组说明, 允许用户手动输入基组或对不同原子使用不同的 基组 基组说明必须包含在几何结构说明之后的输入文件中 手动输入说明的格式为 : center1 center basis set ***** center1 center basis set ***** 其中 center 是元素缩写, 例如 C 表示基组用于所有的碳原 子 或者是一个数字, 例如 3 表示基组应用到结构定义部分第三个原子 基组可以是已经存在的基组名, 例如 6-31G, 也可以是以下格式的 Gaussian 指数和系数列表 symmetry num_primitives scale exponent1 coefficient1 exponent2 coefficient2 : : repeat for next contraction 其中表示对称性的 symmetry 可以是 S, P, D, SP, 等 对于 SP 壳层, 接下来的行有一个指数和两个因子 如果比例因子设定为 1.0, 程序将考虑归一 化 关键字 5d 和 6d 可以用来定义是否使用 5 个实 d 函数或 6 个笛卡尔 d 函数 6d 等于 5d 加上一个附加 的 s 函数 5 个实 d 函数有时被称作纯 d 函数 测试任务 6, 21,22, 55, 87, 121, 122, 180, 186,187, 194, 234, 235, 305, 325, 356, 357 给出了这样的例子 手动输入 ECP Gaussian 程序中包含许多核势基组, 有 CEP-4G, CEP-31G,CEP-121G,LanL2MB, LanL2DZ, LP-31G, SDD 它们可以像其它基组一样被指定 核势可以 手动输入, 使用关键字 pseudo=read 因为必须输入价电子基组, 所以需要与 gen 关键字结合使用 输入包含

224 后面接一空行的几何结构定义, 接着是 gen 输入, 之后是 ecp 输入 Ecp 输入格式是 : element 0 ECP_name max_angular_momentum num_electrons_replaced comment number_of_primitives power_of_r exponent coefficient : : comment number_of_primitives power_of_r exponent coefficient : : 见测试任务 11, 56, 57, 180, 356, 357, 433 ZINDO Gaussian 98 的 ZINDO 关键字定义使用 ZINDO-1 半经验哈密顿量的计算 ZINDO 用于预测电子光谱, 特别是包含过渡金属的化合物 在 Gaussian 98 中,ZINDO 不能进行几何优化 参见测试任务 437 CIS 参见测试任务 , 153, , 165, 228, 229, 279, 280, 319, 320, 359 TD TD 在 Gaussian 以前版本中等价于 RPA 关键字 RPA 关键字在 Gaussian 98 中仍可用, 但已经不在手册中列出 见测试任务 399, 400, 438 CASSCF CASSCF 要求两个参量, 像 CASSCF(2,2), 分别为相关电子数和产生激发的轨道电子数 默认情况下, 使用最高占据轨道和最低空轨道 然而, 虚轨道的顺序并不总是非常稳定, 所以这并不总是最好的相关轨道的选择 活性空间可以达到 12 个轨道 一旦超过, 用户应当考虑 G2,CI,CC, 和 CBS 计算 CASSCF 的 Nroot 选项不同于 CIS 或 TD, 其中的 1 代表基态,2 代表第一激发态, 等等 关键字 StateAverage 和 Spin 通过包含态平均和自旋轨道耦合, 给出更准确的结果 然而, 这些都是更困难的计算 参见测试任务 43, , 164, , 300, 301, 304, 355, , 390 (Zork 译 ) Gaussian98 使用经验谈 经过一段时间对 GAUSSIAN 的使用, 发现不仅有助于加深对量子化学的了解, 而且可以巩固光谱学方面的理论知识 至少在研究光谱学理论方面,GAUSSIAN 是确实一个相当不错的东西 在使用中, 也遇到很多问题, 而且在 GAUSSIAN 的帮助文件中没有涉及 这些问题有的是靠自己解决的, 有的是和别人探讨的, 还有的是从网上或者文献中查到的 受本人研究方向所限, 文中涉及的大多数都是处理分子激发态方面的问题 如果在分子光谱学 ( 包括红外 拉曼 可见 - 紫外 ) 方面的知识有欠缺的话, 建议先看一看哈里斯和伯特卢西写的 对称性与光谱学 ( 胡玉才, 戴寰译, 高等教育出版社,1988), 这是一本浅显易懂的速成书籍 本人水平有限, 对有些问题的理解可能有误, 欢迎批评指正 ****************** 目录 ****************** ( 一 ) 一些杂七杂八的小问题 1-1. 内存与速度问题 1-2.GAUSSIAN 输入的一个注意事项 1-3.GAUSSIAN 的缺陷 ( 二 ) 关于基组的设置 2-1. 基组 2-2. 基组的选择 ( 三 ) 用 GAUSSIAN 处理激发态 3-1. 输入文件中的电子态多重度 3-2.GAUSSIAN 中研究激发态的方法 3-3. 含时方法 (TD) 输入方式 3-4.CIS,TDHF,TDDFT 方法激发能比较 3-5.GAUSSIAN 处理激发态的 BUG 3-6. 画激发态的势能曲线 3-7. 高对称性的分子使用 CIS 方法 ( 四 ) 求分子的光谱常数

225 4-1. 求离解能 4-2. 求电离能 4-3. 求激发态的光谱学参数 Te,ωe,Re 4-4. 如何获得光谱项的对称性 [ 附 ]Gaussian 98 的补充 1. 谐振强度 f 2. 用 TDDFT 做几何结构优化 3. 自定义赝势 ( 模型势, 有效核势 ) ( 一 ) 一些杂七杂八的小问题 1-1. 内存与速度问题 G98W 默认的内存是 128 兆, 我以前一直用 64 兆, 计算机是 P233, 所以奇慢无比 我后来发现, 在计算之前把所有的杀毒程序都关闭 ( 我的计算机上装了三个 ), 可以节省时间 20% 后来又加了一条 64 兆内存, 时间就减少到原来的一半 如果把计算机换成 P300, 用 128 兆内存, 只需要从前五分之一的时间 而且大的内存支持更大的基组 所以计算机配置可马虎不得 1-2.GAUSSIAN 输入的一个注意事项输入的分子坐标 ( 包括键长和键角 ) 必须是实型, 不能是整型, 否则会报错 例如, 键长 2.0 埃, 可以写成 r=2. 或者 r=2.0, 不能写成 r=2 1-3.GAUSSIAN 的缺陷对于含有重原子的分子来说, 光谱项的自旋 轨道角动量已经失去了意义, 这个时候的光谱项应该用自旋 - 轨道耦合总角动量的各个分量 Ω 表示 例如, 一氯化铊分子的 3Π 电子态, 在实验上观察到的是两个分得很开的 Ω 分量 :A0+ 态和 B1 态 但是 GAUSSIAN 仅能处理含有从 H 原子到 Cl 原子的分子 ( 因为不是重原子, 所以没有必要 ) 目前处理这类问题, 一般是使用免费的 MOLFDIR,GAMESS(USA),MOLPRO, NWChem,DIRAC,COLUMBUS( 其中的 SO-MRCIS 模块 ) 或商业的 GAMESS(UK),ADF 等程序 ( 二 ) 关于基组的设置 2-1. 基组基组有两种, 一种是全电子基组, 另一种是价电子基组 价电子基组对内层的电子用包含了相对论效应的赝势 ( 缩写 PP, 也称为模型势, 有效核势, 缩写 ECP) 进行近似 以铟原子为例, 可以输入下面的命令来观察铟原子 3-21G 基组的形式 : # 3-21G GFPrint In 0,2 In 输出结果为 : 3-21G (6D, 7F) S N=1 层,s 轨道 E E E E E E+00 SP N=2 层,s 和 p 轨道

226 E E E E E E E E E+00 SP N=3 层,s 和 p 轨道 E E E E E E E E E+00 D N=3 层,d 轨道 E E E E E E+00 SP N=4 层,s 和 p 轨道 E E E E E E E E E+00 D N=4 层,d 轨道 E E E E E E+00 以上都是内层电子轨道, 每个轨道用三个高斯型函数拟合, 也就是 3-21G 中的 3 SP N=5 层,s 和 p 轨道 E E E E E E+00 这是价电子轨道, 每个轨道用二个高斯型函数拟合, 也就是 3-21G 中的 2 SP N=6 层,s 和 p 轨道 E E E+01 这是空轨道, 每个轨道用一个高斯型函数拟合, 也就是 3-21G 中的 1 可见 3-21G 是全电子基组 再看看铟原子 LanL2DZ 基组的结果 : LANL2DZ (5D, 7F) S E E E E+01 S E E+01 P E E E E+01 P

227 E E+01 这些都是价电子轨道 在 ECP 上加上了双 zeta(dz) 基组 所以 LanL2DZ 是价电子基组 在 GAUSSIAN 中,CEP 也是很重要的价电子基组, 并且 98 版中对元素的适用范围扩大到了 86 号元素 Rn 需要注意的是, 超过了 Ar 元素后, 每一种元素的 CEP-4G,CEP-31G,CEP-121G 是等价的 ( 但和加了极化 * 的 CEP 还是有区别的 ) SDD 适用于除了 Fr 和 Ra 以外的所有元素, 但是结果的准确性并不太好 需要注意的是有些方法不需要输入基组, 像各种半经验方法,G1,G2 等等 2-2. 基组的选择 中国科学 B30(5),2000,419 页的文章给出了一种选择基组的方案, 认为基函数的选择应该根据体系中不同原子的性质及实际的化学环境 : 1 描述一般体系时可根据该原子在元素周期表的位置从左到右依次增加基函数用量 2 对核负电原子, 使用更多的基函数以及适当的极化函数或弥散函数 对核正电原子, 基函数用量可适当减少 3 对正常饱和共价键原子不需要增加极化或弥散函数, 对氢键 弱相互作用体系 官能团 零价或低价态金属原子等敏感体系, 要增加极化或弥散函数 这样就可以用适中的基组和可承受的计算量的到相当可靠的计算结果 方案适用于 HF,MPn,DFT 等计算中 在 GAUSSIAN 中有关的关键字是 ExtraBasis 和 Gen 二者都是在 routesection 指定基组的基础上, 再加上附加的基函数 不同的是 Gen 可以自己定义基组 实际上可并不那么简单 下面是一个使用 ExtraBasis 的例子 : # HF/3-21G ExtraBasis AgCl molecule specification Cl G(d,p) **** 这里对分子中所有的原子都使用 3-21G, 但是对 Cl 原子使用更大的基组 6-311G(d,p) 如果计算过程中报错, 一般是由于下面两点原因造成的 : 1 并不是所有的基组都可以用( 哪些基组适用于哪些原子需要去参考帮助文件 ), 即使是分别适用于不同原子的各个基组, 放在一起也不一定就行, 可能有匹配或收敛的问题, 也有可能是 BUG 所以需要反复多试 2 基组设置的顺序问题 还是看上面的例子, 我认为 route section 一行中设置的基组, 应当适用于该分子中所有的原子 ( 一般来说是小一些的, 虽然在计算中并不对所有出现的原子都使用这个基组 ), 而在分子坐标下面的基组只要对所定义的原子适用就可以了 ( 一般来说是大一些的基组 ) 比如, 如果把上面的输入变成 : # HF/6-311G(d,p) ExtraBasis AgCl molecule specification

228 Ag G **** 虽然都是对 Ag 原子用 3-21G 基组, 对 Cl 原子用 6-311G(d,p) 基组, 但是可能会报错 当然实际情况并不总是如此, 换个分子, 基组哪个在前哪个在后可能并没有关系, 但是注意基组顺序确实可以防止出错 这可能是一个 BUG ( 三 ) 用 GAUSSIAN 处理激发态 3-1. 输入文件中的电子态多重度输入文件中的多重度, 指的是具有这种多重度的最低能量的电子态, 不一定就是基态, 不过一般输入文件都用基态 激发态的多重度是在输入文件的 route section 行控制的, 如 :singlets,triplets 虽然在 GAUSSIAN 使用帮助中没有明说, 但是从给出的例子中可以体会出来 另外, 在 HyperChem 的帮助文件中有类似的说明, 虽然是针对 HyperChem 的, 但我想对 GAUSSIAN 一样适用 3-2.GAUSSIAN 中研究激发态的方法 GAUSSIAN 可以用 CIS,CASSCF 和 ZINDO 方法计算激发态 GAUSSIAN 中的 CIS 实际是对分子激发态的零级组态相关处理 有些从头计算程序 ( 如 GAMESS) 包含计算激发态的一级组态相关 (FOCI) 和二级组态相关 (SOCI), 我不清楚是不是和 CIS 相对应 从结果来看,FOCI 和 SOCI 比 CIS 略有改善 ZINDO 方法适用范围窄, 只能用于周期表前 48 种原子构成的分子, 这里不谈 98 版还新增加了含时 (Time Depend) 方法处理激发态 ( 包括含时 Hatree-Fock(TDHF) 和含时密度泛函 (TDDFT)) 我们一般计算的激发态称为 Low-lying Excited States, 也就是由原子价电子构成的分子电子激发态 更高的电子态是 Rydberg 态 从头算法和其他半经验方法对空轨道, 特别是较高空轨道的计算结果不好, 因而得到的 Rydberg 态的结果缺乏明确的物理意义 3-3. 含时方法 (TD) 输入方式帮助文件对 TDDFT 这么重要的方法说得真是太简单了, 让人去参考 CIS 的输入方法, 连个例子都没有 CIS 的输入方法是 : # CIS(Triplets,NStates=10)/3-21G Test 那就用 B3LYP 泛函试试看吧 : # TDB3LYP(Triplets,NStates=10)/3-21G Test 但是不行 其实正确的输入方法是 : # B3LYP/3-21G TD(Triplets,NStates=10)Test 这个问题浪费了我很长时间 为了表示它的重要, 单独分为一节 3-4.CIS,TDHF,TDDFT 方法激发能比较一般来说,CIS 和 TDHF 方法结果的准确度是比较差的,TDDFT 的结果是最好的 而各种 TDDFT 相比, B3LYP,B3P86,MPW1PW91 是最好的 ( 依具体的分子和使用的基组而不同 ) 可以参考文献: J. Chem. Phys., 111(7), 1999, 2889 Chem. Phys. Lett., 297(1-2), 1998, 60 J. Chem. Phys., 109(23), 1998, 这些文章都是计算 C,H,O 组成的有机物在基态平衡位置的垂直激发能 我对含有重元素 In 的分子激发态进行计算, 发现如果选好基组, 很多 TDDFT 的结果和实验值的差距甚至小于 500 个波数, 而 CIS 的结果则可能要差 5000 多个波数 所以现在使用 TDDFT 方法的比较多 另外还有人用 TDDFT 方法计算 DNA 这样的大分子 3-5.GAUSSIAN 处理激发态的 BUG 1 用 94 版进行 CIS 计算,50-50 选项不支持 LanL2DZ 和 3-21G* 基组 98 版则不会出现这个问题 这个问题可能还跟计算的分子有关 2 同时计算单重激发态和三重激发态( 也就是使用 这个选项 ), 有时候会出现致命错误 : 某些电子态在某些位置的能量有不规则起伏, 而单独计算单重态或三重态 ( 使用 Singlets 或者 Triplets 选项 ) 则不会出

229 现这个问题 我在计算中曾经遇见过两次 下面这个例子是从网上看到的, 我想这位德国的大哥也遇见了同样的问题 他对同一个任务执行了三种计算 : 1) #Bp86/6-31G* td=(nstates=5,50-50) scf=direct density 2) #Bp86/6-31G* td=(triplets,nstates=1) scf=direct density --link1-- #Bp86/6-31G* td=(nstates=10) scf=direct guess=read geom=check --link1-- #Bp86/6-31G* td=(triplets,nstates=10) scf=direct guess=read geom=check 3) #Bp86/6-31G* td=(nstates=10) scf=direct #Bp86/6-31G* td=(triplets,nstates=10) scf=direct density 第一种计算产生 10 个态 (5S 和 5T), 后两种产生 20 个 关键字 Density 对结果没有影响 结果是 : 1) 2) 3) Triplet-B1U Triplet-B3G Triplet-AG Triplet-B2U Triplet-B3G Singlet-B3G Singlet-B1U > > Singlet-B2U > > Triplet-AG > > Singlet-B3G > -- > 通过比较发现, 对前六个态, 三种处理的结果相同, 但是后面的态, 使用了 的方法就和另外两种方法的结果不一样了 我个人认为, 尽量不要使用 这个选项 推荐单重态 三重态分开计算, 虽谈多花一倍时间, 但是结果保险 3-6. 画激发态的势能曲线最笨的方法莫过于算完一个点后, 更改输入文件的坐标, 计算下一个点了 这样费时费力 在优化基态几何构型的时候, 可以使用 Scan 关键字进行扫描, 实际上 Scan 对激发态一样适用 这样开上一晚上计算机, 到早上所有激发态的势能曲线数据就全有了 例如计算从 R=0.6 埃到 1.2 埃氢分子的 10 个单重激发态 : # RCIS(NStates=10)/6-311G Scan H2 Excited States 0,1 H1 H2 H1 r r 需要注意三点 : 1 结果给的是垂直激发能, 必须换算成相对于基态平衡位置的能量 2 扫面范围不要离基态平衡位置太远, 一般从 0.75Re 到 2Re 就可以了, 太远了会报错 3 某些理论和某些基组的搭配可能不支持 Scan, 这时候就只好手工控制了 最后是数据处理, 这一步是最枯燥的了 数据处理需要用到 Excel Oringin Matlab 这些软件用不着钻研很深, 知道几个最基本的命令就足够用了 3-7. 高对称性的分子使用 CIS 方法默认的 CIS 计算求出最低的三个激发态 以它为例,GAUSSIAN 在初始的猜测中选择电子态数目两倍的向

230 量 ( 也就是 6 个 ) 进行迭代, 直到有三个收敛 对于大多数分子体系来说, 这种默认的向量数目足够了 但是对于高对称性的分子, 则需要特殊的方案, 否则不会得到比较高的收敛态 方法是增加 NStates 的数量, 以增加初始向量的数量 推荐 NStates 值为最大阿贝尔点群操作的数量, 也就是输出文件在 Standardorientation 之前的 Symmetry 部分 ( 在输入文件中使用 # 控制打印输出就可以得到,#T 不行 ) 这是一个计算苯分子的例子 : Stoichiometry C6H6 Framework group D6H[3C2'(HC,CH)] Deg. of freedom 2 Full point group D2H NOp 8 Largest concise Abelian subgroup D2 NOp 4 Standard orientation 因此在 route section 中使用 CIS(NStates=8) ( 四 ) 求分子的光谱常数 4-1. 求离解能首先必须注意, 原子化能和离解能不是同一个概念 帮助文件和 Exploring Chemistry with Electronic Structure Methods 一书中的解释并不确切 计算原子化能和离解能是不同的两个过程 以三原子分子 GaCl2 为例, 计算离解能的过程是 : GaCl2(X2A1)---->Cl(2P)+GaCl(X1Σ+) 过程中的能量变化称为离解能 很明显, 离解能还可以细分为第一步离解能 第二步离解能..., 上面的是计算第一步离解能, 计算第二步离解能的过程是 : GaCl(X1Σ+)---->Cl(2P)+Ga(2P) 而计算原子化能的过程是 : GaCl2(X2A1)---->Cl(2P)+2Ga(2P) 可见还是有区别的 不过对于双原子分子来说, 这两个概念是相同的 第二个需要注意的是, 离解能一般用符号 D0 或者 De 表示 ( 前者不包括零点能, 后者包括 ), 但是在分子光谱中, 还有一组 D0 和 De 为避免混淆, 这里需要作特别说明 第一组 D0 和 De, 也就是离解能, 数值比较大, 数量级一般在 0.1 到 1eV 这个参数多出现在真空- 紫外 振动光谱中 有一个公式 : ωe^2 ωeχe = (^ 是指数符号 ) 4De 其中的 De 就是离解能 另一组 D0 和 De 称为离心畸变常数, 数值很小, 作为转动光谱中的微扰项 如 : h^3 De = (π^6)(μ^3)(c^3)(re^6)(ωe^2) 其中的 De 就是离心畸变常数 GAUSSIAN 中有个求 PH2 分子的原子化能的例子, 可以作为求离解能的参考 也就是文件 e7-01 结果是原子化能 = E(P)(Hartree)+2E(H)(Hartree)-E(PH2)(Hartree)-ZPE(Hartree) = ( )+( )*2-( )-( )(Hartree) 其中的零点能 ZPE 经过了矫正 = Kcal/mol 和实验值 Kcal/mol 的差距相当小了 4-2. 求电离能

231 电离能缩写为 I.P. 在 GAUSSIAN 中有两种求 I.P. 的方法 法一 : 见文件 e7-03, 这个例子求 PH2 分子的电离能 结果为 : E ZPE PH (ZPE 经过矫正 ) PH ( 这是上面求原子化能的结果 ) I.P. = (E(PH2+)+ZPE(PH2+))-(E(PH2)+ZPE(PH2)) = ( )-( )(Hartree) = (Hartree) = 9.95(eV) 法二 :OVGF 方法 帮助文件里有详细的使用说明 4-3. 求激发态的光谱学参数 Te,ωe,Re 有两种方法 1 CIS 可以用结构优化的方法 Starl 的文章里说得很详细了, 这里不再细讲 仅仅需要补充的是, 如何强制保留设置后的波函 : 在使用 Guess=Alter 设置波函之后, 程序总会回到设置前的波函 这个时候需要使用 SCF 关键字中的 Symm 选项, 强制程序保留设置以后的波函 使用方式是在 route section 部分添加 SCF=Symm 2 势能曲线扫描法 TDDFT 方法不能计算激发态的波函, 所以不能用来研究激发态的平衡构型和分子属性 上面的结构优化方法用在 TDDFT 中就不适合了 但是可以用势能曲线扫描的方法 但是, 多原子分子能量是势能曲面, 控制多个输入坐标变量, 扫描整个曲面是困难的, 所以这个方法用于双原子分子 如何扫描激发态的势能曲线见前面的 3-6 节 有了势能曲线以后, 就可以找到势能最低点 Re( 为了精确, 这一区域的点要取密一些 ),Re 位置的能量相对于基态平衡位置能量的差值就是电子态高度 Te 势阱的深度就是离解能 De 这些都简单, 关键是求振动参数 ωe 求 ωe 也有两种方法 一种是用 Morse 势函数拟合的方法 Morse 势函数形式为 : V(R) = Te+De{1-EXP[-α(R-Re)]}^2 ( 对基态 Te 是 0) Te De Re 是已知的, 拟合出了 α 后, 用公式 α = ωe ((2(π^2)c*μ)/(h*De))^0.5 ( 高斯单位制 ) 就可以得到 ωe 这种方法太麻烦, 我们不这样做 实际只要把扫描出来的 Morse 势函数对 R 求二阶导数, 在 Re 处的导数值 f2 和 ωe 的关系有 : 4(π^2)(ωe^2)c*μ f2 = h*(10^18) 其中 c h μ 是国际单位制,ωe 的单位是波数 cm^-1 另外在扫描的势函数中,De 的单位是波数 cm^-1, R 和 Re 的单位是埃 否则上面的 f2 需要乘上个系数 求 f2 用 Origin 软件就可以做 注意要取 Re 位置的 f2 有了 f2, 就可以算出 ωe, 省去了计算 De 和拟合 α 的步骤, 避免引入更大的误差 4-4. 如何获得光谱项的对称性如果用 GAUSSIAN 计算比较轻的分子, 一般都能给出正确的激发态光谱项对称性, 但是对含有重原子的分子, 可能得不到对称性 这个时候就需要根据轨道跃迁的的信息获得对称性 下面是用 CIS 方法 LanL2DZ 基组计算 InCl 分子的例子, 输出文件给出 InCl 的电子轨道信息 : Occupied (SG) (SG) (PI) (PI) (SG) Virtual (PI) (PI) (SG) (PI) (PI) (SG) (PI) (PI) (SG) (SG) (SG) 这里结果只给出前 4 个单重激发态 : CIS wavefunction symmetry could not be determined.

232 Excited State 1: Singlet-?Sym ev nm f= > > This state for optimization and/or second-order correction. Copying the Cisingles density for this state as the 1-particle RhoCI density. CIS wavefunction symmetry could not be determined. Excited State 2: Singlet-?Sym ev nm f= > > Excited State 3: Singlet-SG ev nm f= > > > CIS wavefunction symmetry could not be determined. Excited State 4: Singlet-?Sym ev nm f= > > 由于 In 原子是重原子, 所以大多数电子态并没有给出对称性 ( 第三个激发态除外 ) 我们发现第一激发态是从第五个轨道跃迁到第六 七个轨道, 而第六 七两个轨道是简并的 (PI DELT 轨道都是二重简并的 ) 也就是说第一激发态的电子组态是 : InCl: ( 内层电子 )(1σ)2 (2σ)2 (1π)4 (3σ)1 (2π)1 这个组态耦合出来的单重态光谱项有一种 ( 当然还有一个三重的 ):1Π, 所以就是 1Π 态 再往下看, 第二激发态和第一激发态的电子组态 能量完全相同, 说明这是一个简并态, 而实际上除了 Σ 态以外的态都是二度简并的 所以第一激发态和第二激发态实际都是 InCl 的第一激发态 1Π 态 第三个激发态给出对称性了 如果没给, 方法和上面的一样, 先找到它的电子组态, 它是由两个组态叠加而成的 : InCl: ( 内层电子 )(1σ)2 (2σ)2 (1π)3 (3σ)2 (2π)1 InCl: ( 内层电子 )(1σ)2 (2σ)2 (1π)4 (3σ)1 (4σ)1 第一个组态可以耦合出的单重态光谱项有 :1Δ,1Σ+, 1Σ- 因为这不是一个简并态, 所以只能是 1Σ+ 和 1Σ- 其中之一 这就需要进一步知道 + - 对称性 方法是看跃迁系数 : 3 -> > 由于正负号相同, 所以是 1Σ+ 态 那么第四激发态必然是 1Σ- 了 另外还可以看第二个组态耦合出来的光谱项, 只有 1Σ+, 因此第三个激发态必然是 1Σ+ 两个结果一致 以上是自旋限制的计算 非限制计算 ( 每个轨道再分成自旋 Alpha 和自旋 Beta) 分析类似 *************************************************************************** Gaussian 98 的补充 1. 谐振强度 f 谐振强度 f 出现在计算激发态的结果中, 如 :

233 Excited State 3: Singlet-?Sym ev nm f= > 在 Gaussian 98 早期版本中,TD 方法计算激发态的谐振强度 f 是错误的 ( 其它计算激发态的方法没有这个问题 ) 从 A7 版开始做了修正 这点要注意 谐振强度能干什么呢? 可以预测电子激发态跃迁的寿命 用公式 t=1.499/(f*e^2) 这里的 E 是电子态之间的能量间隔, 单位是波数,f 是电子态谐振强度,t 是电子态寿命, 单位是秒 默认的下能级是基态 如果计算各激发态之间的 f, 在 Route Section 中加入关键字 AllTransitionDensities, 就得到各电子态之间的偶极矩 m(x),m(y),m(z) 求解 f 用公式 : 2 f= 3*E*[m(x)^2+m(y)^2+m(z)^2] 因为 Gaussian 给出的 m(x),m(y),m(z) 是 au 单位, 所以激发态之间的能量间隔 E 要换算成 au 单位, 才能代入上面的公式 2. 用 TDDFT 做几何结构优化一般认为, 因为 TDDFT 不能计算激发态的波函, 所以不能用来研究激发态的平衡构型和分子属性 所以用 Gaussian 98 提供的 TDDFt 方法做研究激发态的研究工作, 只能得到垂直激发能, 对于计算平衡键长 Re 键角和振动频率 ωe 无能为力 不过, 最近的报道说,TDDFT 做激发态几何优化在理论上证明可行 已经有消息说,Gaussian 公司准备在新版本的 Gaussian 中加入 TDDFT 几何优化了 可以参考文献 :Van Caillie C, Amos R. D., Geometric derivatives of density functional theory excitation energies using gradient-correct functionals, Chem. Phys. Lett., 317(1-2), 2000, p 和 Hsu, Hirata, Head-Gorden, J. Phys. Chem., A105, 2001, 关于线性多烯烃的 TDDFT 激发能和谐振强度 3. 自定义赝势 ( 也称为模型势, 有效核势 ) Gaussian 给出的 Pseudo 示例如下 : # Becke3LYP/Gen Pseudo=Read Opt Test HF/6-31G(*) Opt of Cr(CO)6 0 1 Cr molecule specification continues... C O G(d) **** Cr 0 LANL2DZ **** Cr 0 LANL2DZ 这里,Cr 原子的基组 LanL2DZ 输入了两次 这两个 LanL2DZ 是不同的 第一个描述的是价电子, 或者核势之外的电子 ; 第二个描述内层电子, 也就是核势部分 如果自定义 ECP 基组, 也这么输入 当然还有其他输入方法 这是在 CIS 计算中自定义 In 原子的基组,In 的 ECP 使用程序内部定义的 SDD 赝势, Cl 原子使用 Route Section 中的 LanL2DZ # CIS(NStates=3,50-50)/LanL2DZ ExtraBasis pseudo=read InCl Excited States 0,1 In Cl In r r=2.401 IN 0 S S E P P E **** In 0 SDD 如果自定义 ECP 的话, 可以把上面的 SDD 换成 : IN 0 IN-ECP 4 46 G POTENTIAL S-G POTENTIAL P-G POTENTIAL D-G POTENTIAL F-G POTENTIAL 价电子基组和核势顺序不要颠倒 下载更多的 ECP 基组, 可以到 : 区分核势和价电子基组的方法是 : 核势第一行有一个被核势代替的内壳层电子个数, 而且就是固定的那么几个偶数 像上面的例子, 核势代替 In 的 46 个内壳层电子 如何得到 Gaussian 中 ECP 基组的赝势呢? 只要在 Route Section 行添加 IOp(3/18=1) 就行了 IOp 的功能是极其强大的, 可以参考 Gaussian 公司 web 页 这只是其中之一 例如要输出 Cl 原子完整的 ECP 基组 CEP-121G, 就必须用到 GFPrint 和 IOp(3/18=1) 关键字, 像下面这样 : # HF/CEP-121G GFPrint IOp(3/18=1) 其中 GFPrint 输出 CEP-121G 的价电子基组部分 当然, 如果观察 cc-pv[dt]z,6-31g 等非 ECP 基组, 就不用加 IOp(3/18=1) 了, 仅用 GFPrint 就行 这是输出结果中 Cl 原子的 CEP-121G 基组部分, 第一部分是 GFPrint 结果, 第二部分是 IOp(3/18=1) 的结果 : **************************************************************************** ATOMIC

234 ORBITAL * GAUSSIAN FUNCTIONS **************************************************************************** FUNCTION SHELL SCALE * NUMBER TYPE FACTOR * EXPONENT S-COEF P-COEF D-COEF F-COEF **************************************************************************** 1-4 SP D D D D D SP D D D D D D D D D D SP D D D D D+00 **************************************************************************** ============================================================================ PSEUDOPOTENTIAL PARAMETERS ============================================================================ CENTER ATOMIC VALENCE ANGULAR POWER NUMBER NUMBER ELECTRONS MOMENTUM OF R EXPONENT COEFFICIENT ============================================================================ D and up S - D P - D =========================================================================== 以上内容大部分来自 讨论区 里面内容除少部分是讨论使用 GAMESS 及其它程序外, 大多是关于 Gaussian 的, 而且 Gaussian 公司的几个人也经常参加讨论 建议大家有机会去看看 首先你要 untar g03.tgz, 然后就是下面了 你可以修改./cshrc 里面的文件, 我给你 copy 下来 vi /home/local/.cshrc setenv G03ROOT /home/g03/ setenv GAUSS_SCRDIR /tmp setenv LD_LIBRARY_PATH /home/g03/ setenv PATH /home/g03/:$path 如果修改这个不行的话, 你可以用 env 查看 cshrc 的运行内容 再看, 实在没有错误还是不运行, 你可以关掉再开一次 如果要所有用户都可以使用 Gaussian 的话应该要建一个用户和组比如建一个 gaussian 用户, 一个 gaussian 组把 /home/g03 及其下的目录和文件都为 gaussian 用户所有,gaussian 组有读和执行权限把要运行 Gaussian 的用户都加入的 gaussian 组中其它设置同用户方式

235 现代价键理论基础 分子是由原子组合而成的 是保持物质基本化学性质的最小微粒, 并且又是参与化学反应的基本单元, 分子的性质除取决于分子的化学组成外, 还取决于分子的结构 分子的结构通常包括两方面内容 : 一是分子中直接相邻的原子间的强相互作用即化学键 (chemical bond), 化学键的成键能量约为几十到几百千焦每摩 ; 二是分子中的原子在空间的排列, 即空间构型 (geometry configuration) 此外, 在相邻的分子间还存在一种较弱的相互作用, 其作用能约比化学键小一 二个数量级 物质的性质决定于分子的性质及分子间的作用力, 而分子的性质又是由分子的内部结构决定的, 因此研究分子中的化学键及分子间的作用力对于了解物质的性质和变化规律具有重要意义 化学键按成键时电子运动状态的不同, 可分为离子键 共价键 ( 包括配位键 ) 和金属键三种基本类型 在这三种类型化学键中, 以共价键相结合的化合物占已知化合物的 90% 以上, 本章将在原子结构的基础上着重讨论形成化学键的有关理论和对分子构型的初步认识, 同时对分子间的作用力作适当介绍 第一节现代价键理论 现代价键理论的基础现代价键理论 (valence bond theory, 简称 VB 法, 又称为电子配对法 ) 量子力学对氢分子系统的处理表明, 氢分子的形成是两个氢原 1s 轨道重叠的结果 氢分子的形成曲线示意图只有两个氢原子的单电子自旋方向相反时, 两个 1s 轨道才会有效重叠, 形成共价键 氢原子间形成的稳定共价键, 是氢分子的基态 共价键的本质是电性的, 但因这种结合力是两核间的电子云密集区对两核的吸引力, 成键的这对电子是围绕两个原子核运动的, 出现在两核间的概率较大, 而且不是正 负离子间的库仑引力, 所以它不同于一般的静电作用 现代价键理论的要点 : 1. 两个原子接近时, 只有自旋方向相反的单电子可以相互配对 ( 两原子轨道重

236 叠 ), 使电子云密集于两核间, 系统能量降低, 形成稳定的共价键 2. 自旋方向相反的单电子配对形成共价键后, 就不能再和其他原子中的单电子配对 所以, 每个原子所能形成共价键的数目取决于该原子中的单电子数目 这就是共价键的饱和性 3. 成键时, 两原子轨道重叠愈多, 两核间电子云愈密集, 形成的共价键愈牢固, 这称为原子轨道最大重叠原理 因此共价键具有方向性 共价键特点 : 共价键具有饱和性和方向性如何理解共价键的饱和性和方向性? 例如 : 在形成 HCl 分子时,H 原子的 1s 轨道与 Cl 原子的 3px 轨道是沿着 x 轴方向靠近, 以实现它们之间的最大程度重叠, 形成稳定的共价键 [ 图 9-1] 其他方向, 因原子轨道没有重叠和很少重叠, 故不能成键 理论解释 : 共价键的形成将尽可能沿着原子轨道最大程度重叠的方向进行 原子轨道中, 除 s 轨道呈球形对称外,p d 等轨道都有一定的空间取向, 它们在成键时只有沿一定的方向靠近达到最大程度的重叠, 才能形成稳定的共价键, 这就是共价键的方向性 根据要点 1, 只有自旋方向相反的单电子可以相互配对 ( 两原子轨道重叠 ), 形成稳定的共价键, 因此共价键具有饱和性 共价键的类型 : 根据原子轨道最大重叠原理, 成键时轨道之间可有两种不同的重叠方式, 从而形成两种类型的共价键 σ 键和 π 键 σ 键 -- 以 头碰头 方式进行重叠, 轨道的重叠部分沿键轴呈圆柱形对称分布, 原子轨道间以重叠方式形成的共价键, π 键 -- 原子轨道中两个互相平行的轨道如 py 或 pz 以 肩并肩 方式进行重叠, 轨道的重叠部分垂直于键轴并呈镜面反对称分布 ( 原子轨道在镜面两边波的瓣符号相反 ), 而形成的共价键 成键分析 : 对于含有单的 s 电子或单的 p 电子的原子, 它们可以通过 s-s s-px px-px py-py pz-pz 等轨道重叠形成共价键 为了达到原子轨道最大程度重叠, 其中 s-s s-px 和 px-px 轨道沿着键轴即成键两原子核间的连线, 如图 9-4(a) 所示, 形成的共价键, 这种共价键为 σ 键 σ 键示意图, ( 图 9-1 σ 键成键方式 ), 当 py-py pz-pz 以 肩并肩 方式进行重叠时, 如图 9-4(b) 所示, 也可形成共价键, 这种共价键为 π 键

237 π 键成键方式 ( 图 9-2 π 键成键方式 ), 此时 xy 平面为对称镜面 实例分析 : 分析 N2 分子中化学键成分 N 原子的电子组态为 1s22s22px12py12pz1, 其中 3 个单电子分别占据 3 个互相垂直的 p 轨道 当两个 N 原子结合成 N2 分子时, 各以 1 个 px 轨道沿键轴以 头碰头 方式重叠形成 1 个 σ 键后, 余下的 2 个 2py 和 2 个 2pz 轨道只能以 肩并肩 方式进行重叠, 形成 2 个 π 键 所以,N2 分子中有 1 个 σ 键和 2 个 π 键, 其分子结构式可用 N N 表示 由于 σ 键的轨道重叠程度比 π 键的轨道重叠程度大, 因而 σ 键比 π 键牢固 π 键较易断开, 化学活泼性强, 一般它是与 σ 键共存于具有双键或叁键的分子中 σ 键是构成分子的骨架, 可单独存在于两原子间, 以共价键结合的两原子间只可能有 1 个 σ 键 共价单键一般是 σ 键, 双键中有 1 个 σ 键和 1 个 π 键, 叁键中有 1 个 σ 键和 2 个 π 键 因在主量子数相同的原子轨道中,p 轨道沿键轴方向的重叠程度较 s 轨道的大, 所以一般地说,p-p 重叠形成的 σ 键 ( 可记为 σp-p) 比 s-s 重叠形成的 σ 键 ( 可记为 σs-s) 牢固 正常共价键和配位共价键根据成键原子提供电子形成共用电子对方式的不同, 共价键可分为正常共价键和配位共价键 正常共价键 -- 如果共价键是由成键两原子各提供 1 个电子配对成键的, 称为正常共价键, 如 H2 O2 HCl 等分子中的共价键 配位共价键 -- 如果共价键的形成是由成键两原子中的一个原子单独提供电子对进入另一个原子的空轨道共用而成键, 这种共价键称为配位共价键 (coordinate covalent bond), 简称配位键 (coordination bond) 通常为区别于正常共价键, 配位键用 表示, 箭头从提供电子对的原子指向接受电子对的原子 实例分析 : 分析 CO 分子中化学键成分在 CO 分子中,O 原子除了以 2 个单的 2p 电子与 C 原子的 2 个单的 2p 电子形成 1 个 σ 键和 1 个 π 键外, 还单独提供一孤对电子 ( lone pair electron) 进入 C 原子的 1 个 2p 空轨道共用, 形成 1 个配位键, 这可表示为 ( CO 图 ) 配位键必须同时具备两个条件 : 一个成键原子的价电子层有孤对电子 ; 另一个成键原子的价电子层有空轨道 配位键的形成方式虽和正常共价键不同, 但形成以后, 两者是没有区别的 键参数 能表征化学键性质的物理量称为键参数 (bond parameter) 共价键的键参数主要有键能 键长 键角及键的极性