Rogers Computational Chemistry Using the PC

In Computer Projects 8-1 and 8-2, we used the STO and CBS basis sets stored as part of the data base of GAUSSIAN. The general basis case (keyword gen) in GAUSSIAN permits us to bypass the stored basis sets (there is no stored STO-1G basis set) and make our own basis functions. To run GAUSSIAN under the general basis input to determine the SCF output for the ground state of the hydrogen atom using a single Gaussian trial function, the input file is

# gen

hatom gen

0 2 h

1 0

S 1 1.00 0.282942 1.0

****

Input File 8-1. The General Basis Input for an STO-1G Calculation of the Ground State Energy of the Hydrogen Atom.

The first line # gen (route section) tells the system that we want to define our own function. The lines 2, 3, and 4 are a blank line, program label (for human readers), and a blank line. The next line that is read by the system is 0 2, specifying that the ground state of H has a 0 charge and is a spin doublet (one unpaired electron). The next line, h, specifies hydrogen, followed by a blank.

The remainder of the input file gives the basis set. The line, 1 0, specifies the atom center 1 (the only atom in this case) and is terminated by 0. The next line contains a shell type, S for the 1s orbital, tells the system that there is 1 primitive Gaussian, and gives the scale factor as 1.0 (unscaled). The next line gives g ¼ 0:282942 for the Gaussian function and a contraction coefficient. This is the value of g, the Gaussian exponential parameter that we found in Computer Project 6-1, Part B. [The precise value for g comes from the closed solution for this problem 8=9p (McWeeny, 1979).] There is only one function, so the contraction coefficient is 1.0. The line of asterisks tells the system that the input is complete.

When we run this program, we get a good deal more information than we are ready for at this point, but one thing is obvious: the energy, found in the last block of output,

HF ¼ 0:4244132

This result agrees with Computer Project 6-1, but it is not very good, 0.4244 hartrees, as compared to the exact solution of 0.5000, a 15.1% error. What went wrong?

The Gaussian, with r2 in the exponent, drops off faster than the true 1s orbital, which has r in the exponent. The Gaussian is too ‘‘thin’’ at larger distances r from the nucleus (Fig. 8-2).

Figure 8-3 Approximation to the 1s Orbital of Hydrogen by 2 Gaussians. The upper curve is the sum of the lower two curves.

We find that there are two Gaussian primitives and one unpaired electron from the output

which agrees with the picture of the STO-2G basis set that we are trying to build. Of course, we want to know what the parameters are for the two Gaussians. The keyword GFinput inserted after # sto-2g in the route section of the input file produces an output file with the added information

Basis set in the form of general basis input: 1 0

S2 1.00

****

Output File 8-1. Parameters for the STO-2G Basis Set.

The parameterized STO-2G basis function is

which is the function graphed in Fig. 8-2. The smaller exponent contributes to the ‘‘tail’’ of the composite function by causing it to drop off less rapidly with r, and the

approached is called the Hartree–Fock limit, of which more will be said later. The calculation carried out up to this point is called a Hartree–Fock calculation, which is why the energy is labeled HF in the output file.

Semiempirical Methods

If we are willing to use empirical observations in place of the integrals in the secular matrix, we can avoid calculating some or all of the matrix elements. For example, as seen in the EHT method, the negative of the spectroscopic atomic ionization energy makes a good substitute for the calculated Coulomb energy on the logic that the amount of energy necessary to drive a valence electron away from its core is equal and opposite to the amount of energy that held it there in the first place. (This is not strictly true because of rearrangement of the core electrons during the ionization process.) Filling in the matrix elements by fitting them to spectroscopic or other experimental data leads to a semiempirical calculation of the eigenvalues and eigenfunctions. Such methods are not fully empirical, even though they use empirical information, because they are rooted in quantum theory as expressed through the variational principle.

Semiempirical methods, of which there are quite a few, differ in the proportion of calculations from first principles and the reliance on empirical substitutions. Different methods of parameterization also lead to different semiempirical methods. Huckel and extended Huckel calculations are among the simplest of the semiempirical methods. In the next two sections, we shall treat a semiempirical method, the self consistent field method, developed by Pariser and Parr (1953) and by Pople (1953), which usually goes under the name of the PPP method.

PPP Self-Consistent Field Calculations

In the Huckel method, we assumed an initial constant p electron density q of one electron per carbon about all carbon atoms in a conjugated p electron system. We also took the electron exchange integrals between atoms to be one arbitrary unit of energy, according to whether the atoms are connected (b ¼ 1) or not connected (b ¼ 0). The eigenvectors (coefficients) generated in diagonalizing the secular determinant, however, yield electron densities and bond orders that are in contradiction to the original assumptions. In particular, if a bond order between atoms p and q is large and that between r and s is small, then the resonance integral b is not the same for these atom pairs, but bpq > brs.

It seems reasonable that, by taking into account the information we have generated in a set of calculated eigenvalues and eigenvectors, we can repeat the calculation and get a new and better set of eigenvalues and eigenvectors. If this works once, it should work many times. There may be convergence to a result that, though not exact, is self-consistent and is a better description of the molecule than the single matrix diagonalization of the Huckel method. This is the essence of the PPP-SCF method.

We have the makings of an iterative computer method. Start by assuming values for the matrix elements and calculate electron densities (charge densities and bond orders). Modify the matrix elements according to the results of the electron density calculations, rediagonalize using the new matrix elements to get new densities, and so on. When the results of one iteration are not different from those of the last by more than some specified small amount, the results are self-consistent.

Both onand off-diagonal elements are modified, but for simplicity, we shall reset the diagonal elements to zero after each iteration. In this way, orbital energies will be found that are above and below an arbitrary zero energy, stressing the analogy between the PPP-SCF method and the Huckel method. This is an acceptable procedure for hydrocarbons with alternating double and single bonds, called alternant hydrocarbons.

The PPP-SCF Method

In PPP-SCF calculations, we make the Born–Oppenheimer, s-p separation, and single-electron approximations just as we did in Huckel theory (see section on approximate solutions in Chapter 6) but we take into account mutual electrostatic repulsion of p electrons, which was not done in Huckel theory. We write the modified Schroedinger equation in a form similar to Eq. 6.2.6

^

to emphasize that F is an operator similar to the one-electron Hamiltonian operator. The linear combination

analogous to the Huckel matrix. The F matrix can be written succinctly as

01

if the S matrix is taken to be I (overlap integrals are approximated as zero or one). The corresponding matrix equation

leads to E, the diagonal matrix of eigenvalues, and A, the matrix of eigenvector coefficients.

In the Huckel theory of simple hydrocarbons, one assumes that the electron density on a carbon atom and the order of bonds connected to it (which is an electron density between atoms) are uninfluenced by electron densities and bond orders elsewhere in the molecule. In PPP-SCF theory, exchange and electrostatic repulsion among electrons are specifically built into the method by including exchange and electrostatic terms in the elements of the F matrix. A simple example is the 1,3 element of the matrix for the allyl anion, which is zero in the Huckel method but is 1.44 eV due to electron repulsion between the 1 and 3 carbon atoms in one implementation of the PPP-SCF method.

The elements of the F matrix depend on either the charge densities q or the bond orders p, which in turn depend on the elements of the F matrix. This circular dependence means that we must start with some initial F matrix, calculate eigenvectors, use the eigenvectors to calculate q and p, which lead to new elements in the F matrix, calculate new eigenvectors leading to a new F matrix, and so on, until repeated iteration brings about no change in the results. The job now is to fill in the elements of the F matrix.

The diagonal matrix element Frr is broken up into three parts

X

where Urr is the localized one-electron Hamiltonian relating to the interaction of electron i with the core at carbon atom r. The term 12qrgrr is the potential energy of repulsion between electron i and the charge due to all other electrons that can occupy the same orbital. The factor 12 appears because, to occupy the same orbital with i, electrons must have opposite spin, that is, they are one-half the total. The sum includes repulsions at all other atoms, t 6¼r. In Huckel theory, we were free to pick an arbitrary zero of energy a, and in PPP-SCF theory we can do the same thing.

The reference point Urr is set equal to zero. This leaves only one term and the sum

P

qtgrt on the right of Eq. (8-43), wherewith to obtain the matrix element Frr.

t6¼r

Let us illustrate the meaning of Frr by the example of carbon atom 1 in the linear, three-carbon allyl anion C3H6 . There are two carbon atoms other than C1, one adjacent and the other nonadjacent. Equation (8-44) has three terms, one for each carbon atom

There are similar on-diagonal terms for C2 and C3 in the allyl anion. Expect to see these matrix elements again.

The off-diagonal elements in the F matrix Frt are defined for neighboring atoms, which are not necessarily adjacent. There are no rr interactions for neighbor atoms.

In the case of adjacent atoms, a bond exists characterized by a bond energy bSCF analogous to the b of Huckel theory but modified by an electron exchange term

^

that is, the value of F is made more negative (bonding) by the electron density prt between atoms r and t times the parameter grt. The matrix elements Frt are unlike b in the Huckel treatment in that they change during iteration.

The parameters grr or grt are empirical estimates of how effective repulsion is between an electron in orbital i and the charge clouds on ‘‘its own’’ carbon atom r or the neighboring carbon atoms t in the molecule. For more distant carbon atoms t, grt is smaller, as expected for a smaller orbital interaction. Different recipes for obtaining empirical grt values have been used (Pilar, 1990). They give similar values. By one scheme, grr is taken to be the ionization energy of the carbon atom. More generally, a physical model of interacting negatively-charged spheres is used to calculate repulsive energies 12 prtgrt and the results are fitted to conform with experimental measurements.

Pariser and Parr adjusted the necessary parameters to the empirical singlet and triplet excitation energies in benzene to obtain

where the subscript 12 indicates nearest neighbors and 13 and 14 are the next most distant carbon atoms, etc. Fitting bSCF to the HOMO-LUMO transitions in benzene in a manner similar to Computer Project 6-2 yields

Having filled in all the elements of the F matrix, we use an iterative diagonalization procedure to obtain the eigenvalues by the Jacobi method (Chapter 6) or its equivalent. Initially, the requisite electron densities are not known. They must be given arbitrary values at the start, usually taken from a Huckel calculation. Electron densities are improved as the iterations proceed. Note that the entire diagonalization is carried out many times in a typical problem, and that many iterative matrix multiplications are carried out in each diagonalization. Jensen (1999) refers to an iterative procedure that contains an iterative procedure within it as a ‘‘macroitera-

tion.’’ The term is descriptive and we shall use it from time to time.

Like Frt ¼ bSCF 12 prtgrt, the zero point, which we may denote aSCF, changes during iteration. Because it is an arbitrary reference point to begin with, we can redefine it as zero after each iteration, ending up with a set of energy levels that qualitatively resembles the set of Huckel energy levels. As in Huckel theory for

alternant hydrocarbons (Smith, 1996), orbital energies are symmetrically distributed above and below a (defined) zero, although the calculated values of the energies are not the same. Energy distribution about aSCF is not symmetrical for molecules other than alternant hydrocarbons.

Ethylene

The simplest application is to ethylene. There are only two Frr elements and they are identical, so, completing the analogy with Huckel theory, let us define their energies aSCF. The SCF matrix is

We can calculate F12 ¼ F21 for the first diagonalization as F12 ¼ bSCF 12 p12g12, where bSCF ¼ 2:37 eV from Eq. (8-47) and g12 is the repulsion integral for

electrons on atoms 1 and 2, adjacent carbon atoms, which are the only kind in ethylene. Equation set (8-46) gives 7.19 eV for g12. The initial bond order (electron density between atoms) from Huckel theory is 1.00, hence

F12 ¼ F21 ¼ 2:37 121:00ð7:19Þ

¼ 5:96 eV

The form of the SCF matrix is the same as the Huckel matrix; hence, we substitute

0 5:96

ð8-49Þ

5:96 0

which is diagonalized as the Huckel matrix was to yield

The solution comes out to be very similar to the Huckel solution for ethylene except that the two energy levels, specified as, are 5.96 eV above and 5.96 eV below the reference level (Fig. 8-5).

On the basis of this calculation, one would expect to find a p ! p spectroscopic transition at

11:92 eVð8065Þ ¼ 9:61 104cm 1

where 8065 is the conversion factor from eV to cm 1. In fact, in semiquantitative agreement with the calculated p ! p energy separation, ethylene does have a strong absorption band at 6:21 104cm 1 in the vacuum ultraviolet.

In the case of ethylene, we have reached self-consistency in one iteration, that is, the output of the calculation is the same as the input F matrix. In general this will not be true.

Exercise 8-5

Extend the PPP-SCF calculation from ethylene to the allyl anion, C3H6 .

0.50 0.00 0.50

Solution 8-5

Find the eigenvectors (eigenfunctions), charge densities q, and bond orders p of C3H6 by the Huckel method. This provides a starting input matrix.

To form the first SCF input matrix from the HMO calculation, fill the charge densities and bond orders into the matrix