A Computer Study of Molecular Electronic Structure

Transcription

1 A Computer Study of Molecular Electronic Structure The following exercises are designed to give you a brief introduction to some of the types of information that are now readily accessible from electronic structure calculations. The writeup, which should summarize the results obtained and address the questions raised in each of the exercises, must be each student's own independent work. Once you have read the Molecular Electronic Structure Calculations handout, you will be ready to follow these exercises. The computer you will use for this exercise, a PowerMac G4, is located in my lab on the lower floor of Burke, room 010a. Enter through the white door that opens onto the hall and has a Chem 81 lab sign on it; this door is the first on your left as you walk past the restrooms on that floor. Exercises For each of these exercises, you will follow the procedure described in the Molecular Electronic Structure Calculations handout. Items and questions you need to consider in your writeups are indicated by a symbol. 1. The first exercise involves an exploration of variational calculations of the ground state electronic state of H 2. The goal is to see how the calculated equilibrium bond length (R e ) and the calculated equilibrium dissociation energy (D e ) change as the variational wavefunction is improved. (i) Build an H 2 molecule, use the Calculations... menu entry to select the Hartree-Fock STO-3G basis set of atomic functions, Submit the calculation, and name your H 2 files something memorable. Once the calculation has completed, Display the Properties floating window. Note and record the energy. It will be in atomic units (hartrees), and note that 1 hartree = ev. Measure the calculated value of the equilibrium bond length (R e ), and record this value. Display the Output file, and copy it to your Microsoft Word collection document, as described in the companion handout. We will return to this output later on. (ii) Repeat the calculation of part (i) using a) the 3-21G* basis; b) the 6-31G* basis; c) the G** basis. For each basis change, you need only redisplay the Calculations... dialog, switch basis sets, and Submit the revised calculation. You do not need to save the Output file for these other basis set calculations. 1

2 For your writeup, calculate a value for D e, the bond dissociation energy, at each level of calculation. Recall that the total energy of two 1s H atoms infinitely far apart is exactly 1 hartree. Your writeup should include the following table summarizing these calculations: Calculated values of R e and D e for H 2 Basis Total Energy/hartrees R e /Å D e /ev STO-3G 3-21G* 6-31G* G** 1. Do your results support the prediction of the Variational Theorem? 2. How well do your calculated R e values agree with the experimental value, Å? 3. How well do your calculated D e values agree with the experimental value, ev? (See page 516 in the text.) MacSpartan cannot perform calculations at so-called higher levels of theory, such as second, third, or fourth-order perturbation theory, but other software we have in the department can. Tabulated below are results from several of these higher-order calculations, all of which used R e = Å. Compare these results to yours, calculate D e values for them, and compare them to experiment. Do you see an improved agreement between theory and experiment as the level of theory increases? Perturbation theory Total energy/hartrees none second order third order fourth order Your Output file from the STO-3G basis has some interesting information we can use to compare to our lecture discussion of H 2. As you look through this file, you will find early on a listing of the steps the program took to reach the minimum energy. Look for the lines that resemble the following: Max. Max. Neg. Cycle Energy Grad. Dist. Eigen 1-1.xxxxxxx xxxxxxx xxxxxxx xxxxxxx These lines show, iteration by iteration, how the energy changed (I ve x ed out the real numbers!), how rapidly it changed ( Grad. means gradient ), how far the atoms moved ( Dist. ), and a final blank column that would have 2

3 entries only if the iteration went too far and had a so-called negative eigenvalue, one that represented an overstep in energy below the true minimum. Later in the file, you will find lines like these: Cartesian Coordinates (Angstroms) Atom X Y Z H H xxxxxxx 2 H H xxxxxxx that show you the final Cartesian coordinates of the atoms. (Again, I ve x ed out the real values.) The molecule is aligned along the z axis with the origin at the bond mid-point for H 2. The difference between these two z values should agree with your measured bond length from part (i). Finally, you will find the table of MO energies and the AO coefficients: Closed-Shell Molecular Orbital Coefficients MO: 1 2 Eigenvalues: -0.xxxxx 0.xxxxx (ev): -xx.xxxxx xx.xxxxx Sg+ Su+ 1 H1 S x.xxxxx x.xxxxx 2 H2 S x.xxxxx -x.xxxxx The MOs are numbered 1 and 2, in increasing energy, and the notation Sg+ and Su+ stand for, as I hope you guessed, σ g + and σ u +, the symmetry labels we have used all along (with the + label one we have used only for certain molecular states). The energy eigenvalues are shown in hartree and ev units, and the final two lines show the AO coefficients. Our basis set used only two AOs: 1s Slater Type Orbitals on each atom. (The atoms are called H1 and H2 here; we called them A and B in class.) You will find the two σ g coefficients are equal and the two σ u coefficients have equal magnitude but opposite sign. Our theory, using real H 1s orbitals, lead to an expression for these coefficients in terms of the overlap integral S. (See page 508 in the text.) Calculate our coefficients for the internuclear distance your STO-3G calculation predicted and compare them to the output data file values. You should find that they are close, but not exactly equal due to the use of Slater Gaussian orbitals by MacSpartan. 2. In this exercise, you will examine the bonding in our prototypical ionic heteronuclear, LiF. Close your H 2 file, open a new file, build LiF, set up a calculation using the STO-3G basis, submit, and name your file. 3

4 When the calculation is completed, save the output file to your Word collection file and measure the bond length. Record this length. Open the Properties window and record the dipole moment. How does the bond length compare to the experimental value, Å? Is the disagreement in the same direction as for your H 2 STO-3G calculation? How does the calculated dipole moment compare to the experimental value, D? (Properties that involve the full electronic structure of a molecule, such as the dipole moment, require very accurately calculated wavefunctions, so spectacular agreement is not to be expected here.) Now repeat the calculation using the G** basis, the best we have available. No need to save the output, but measure R e and record the dipole moment again. Are they improved over the lower-level STO-3G values? Look at the MO table in your STO-3G output. Do the coefficients recorded there for the lowest three MOs support our discussion of ionic bonding in LiF? What is the nature of the HOMO and LUMO for this molecule, i.e., where is each orbital localized? 3. In this exercise you will examine the occupied and vacant molecular orbitals for the ground electronic states of C 2 and F 2. You will perform the same calculations and observations on each of these diatomics; so, do one, then save its file and repeat for the other. Our goal is to look at the ordering of the orbital energies to see if calculations agree with the predictions of Table 14.3 on page 524 of the text. Build a structure, select the 3-21G* basis, submit the calculation, and name your file. Save the output files to your Word collection file for later analysis. The question to ask is the relative energy ordering of the MOs as shown in your output versus the experimental ordering in the text. Are they the same? Next, for each diatomic in turn, set up calculations for the following wavefunction images using the Surfaces and Add Surface dialogs boxes: the HOMO, the LUMO, the HOMO-1, the HOMO-2, and the LUMO+1. (Note that for the HOMO-x and LUMO+x choices, you enter the number x directly in the Add Surface dialog box.) Submit this calculation, then when it is complete, use the Surface dialog box to display one at a time. You may want to orient them in a way that displays their symmetry to best advantage. Note (or copy the images to your Word file for later study) the symmetry and nature of each. What is the energy and the designation of the HOMO? What is the energy and the designation of the LUMO? 4

5 What is the energy of the 3σ g MO, and what is the spatial nature of this MO? This MO can be described approximately as a linear combination of two atomic functions, one on atom A and one on atom B. What two functions would you use to produce the 3σ g MO for each of these diatomics? Compare the spatial extent of the 3σ g MOs for C 2 and F 2. Rationalize any differences that you find. 4. In this exercise you will examine the nature of the HOMO and the LUMO for heteronuclear CO, an important ligand in organometallic chemistry. Build a structure for CO, select the 6-311G* basis, submit and name your file. Measure the calculated value of R e, and compare it with the experimental value, Å. Save the output file for later study. Select wavefunction surface images for the HOMO, the LUMO, and the π- bonding MOs. (Look at the output file to decide which MOs are the π- bonding MOs. There will be two of them at the same energy, remember!) What is the nature of the HOMO? Is the probability of finding an electron in the HOMO at the oxygen end of the CO bond > or < 50%? Explain your answer. Is the probability of finding an electron in the π-bonding MO at the oxygen end of the CO bond > or < 50%? Explain your answer. Is the probability of finding an electron in the LUMO at the oxygen end of the CO bond > or < 50%? Explain your answer. From an inspection of the forms of the HOMO and the LUMO, would you expect the CO ligand to be: a σ donor or a σ acceptor? a π donor or a π acceptor? Explain your answer. What is the HOMO-LUMO energy difference, in ev, as predicted by your calculation? How does this energy compare to the energy of the first excited electronic state of CO, 8.07 ev? 5