HF-LCAO

Hiickel s model was not originally presented in terms of the HF model, but I want you to think in HF-LCAO terms for the remainder of the chapter. So, imagine a simple rr-electron molecule such as ethene (Figure 7.1). 

Where might these one-electron wavefunctions come from I explained the basic ideas of HF and HF-LCAO theory in Chapter 6 we could find the molecular orbitals as linear combinations of appropriate atomic orbitals by solving the HF eigenvalue problem... 

For the minute, imagine an HF-LCAO treatment of just the jr-electrons in ethene where each carbon atom contributes just one electron and one atomic orbital of the correct symmetry to the conjugated system. Without any particular justification except chemical intuition, we make the following assumptions. 

Not only that, the elements of the HF-LCAO matrix are taken to be constants that depend only on the nature of atoms and atom pairs as follows. 

The Extended Hiickel model treats all valence electrons within the spirit of the TT-electron model. Each molecular orbital is written as an LCAO expansion of the valence orbitals, which can be thought of as being Slater-type orbitals (to look ahead to Chapter 9). Slater-type orbitals are very similar to hydrogenic ones except that they do not have radial nodes. Once again we can understand the model best by considering the HF-LCAO equations... 

You probably noted that the original papers were couched in terms of HF-LCAO theory. From Chapter 6, the defining equation for a Hamiltonian matrix element (in the usual doubly occupied molecular orbital, closed-shell case) is... 

The calculation usually proceeds along the traditional lines of HF-LCAO theory. We make an initial estimate of the electron density matrix P, calculate a revised h and iterate until the electron density and the HP matrix are self-consistent. 

In HF-LCAO theory, the electronic ground state of pyridine is We... 

Table 8.2 Pariser-Parr-Pople HF-LCAO treatment of pyridine...

A CNDO all-valence-electron HF-LCAO Hamiltonian matrix has elements... 

In a fascinating variation on the usual theme of comparing with experiment, the atomic values were chosen by comparison with matrix elements of accurate HF-LCAO Hamiltonians. 

Not only that, there are usually a lot of them. An HF-LCAO calculation with n basis functions requires the calculation and manipulation of about such integrals. 

Clementi and Raimondi refined these results by performing atomic HF-LCAO calculations, treating the orbital exponents as variational parameters. A selection of their results for H through Ne is given in Table 9.3. 

The next step on the road to quality is to expand the size of the atomic orbital basis set, and I hinted in Chapters 3 and 4 how we might go about this. To start with, we double the number of basis functions and then optimize their exponents by systematically repeating atomic HF-LCAO calculation. This takes account of the so-called inner and outer regions of the wavefunction, and Clementi puts it nicely. 

Atoms are special, because of their high symmetry. How do we proceed to molecules The orbital model dominates chemistry, and at the heart of the orbital model is the HF-LCAO procedure. The main problem is integral evaluation. Even in simple HF-LCAO calculations we have to evaluate a large number of integrals in order to construct the HF Hamiltonian matrix, especially the notorious two-electron integrals... 

Of interest is the SCF=Direct option. There are three ways of dealing with two-electron integrals over the basis functions in ab initio HF-LCAO calculations. The Conventional way is to calculate them once and store them on a... 

Then comes the HF-LCAO calculation (Figure 10.12). The procedure starts with an INDO run (Chapter 8) for the initial estimate of the electron density Notice once again the internal use of molecular symmetry. In early packages such as POLYATOM, the use of molecular symmetry was essential for fast execution but had to be explicitly included by the user. 

Here, occ means occupied and virt means virtual. In the restricted Hartree-Fock model, each orbital can be occupied by at most one a spin and one (i spin electron. That is the meaning of the (redundant) Alpha in the output. In the unrestricted Hartree-Fock model, the a spin electrons have a different spatial part to the spin electrons and the output consists of the HF-LCAO coefficients for both the a spin and the spin electrons. 

There is very little point in trying to obtain information from the 73 x 47 z= 3431 numbers that constitute the HF-LCAO coefficients for the occupied orbitals. Mulliken population indices are given next, together with Mulliken atomic charges (Figure 10.14). 

In Chapter 4, we considered a simple LCAO treatment of dihydrogen and calculated the potential energy curve reproduced in Figure 11.2. The LCAOs we deduced correspond to HF-LCAO MOs for a minimal basis set. 

Figure 11.2 Dihydrogen potential energy curve HF-LCAO model...

The reason usually advanced is that whilst the occupied orbitals are determined variationally within the HF-LCAO procedure, the virtual orbitals are not. Consequently, the virtual orbitals give a very poor description of excited states. 

The HF-LCAO calculation follows the usual lines (Figure 11.10) and the frozen core approximation is invoked by default for the CISD calculation. CISD is iterative, and eventually we arrive at the improved ground-state energy and normalization coefficient (as given by equation 11.7) — Figure 11.11. 

Table 11.1 shows an interesting point about CISD. The energy of the dineon pair at the arbitrarily large separation of 5000 pm is exactly twice the energy of two free atoms at the HF-LCAO level of theory, but this is not the case at the CISD level of theory. We say that HF theory scales correctly, whilst CISD does not. 

The first-order energy involves only the perturbation operator and the unperturbed wavefunction. In an HF-LCAO treatment, the integrals would be over the LCAOs, and this implies a four-index transformation to integrals over the basis functions. 

The idea is simple. We take the zero-order problem as the HF (or HF-LCAO) one, where each electron moves in an average field due to the nuclei and the... 

As a simple example, let s return to the dineon problem discussed above. Here are the salient points from a Gaussian run at 300 pm. Figure 11.12 shows the standard HF-LCAO calculation. 

If the neon-neon interaction were a pure dispersion one, then the HF-LCAO calculation would give a fully repulsive curve. The fact that the HF-LCAO calculation gives a shallow minimum implies an element of covalency. 

In Chapter 6, I discussed the open-shell HF-LCAO model. 1 considered the simple case where we had ti doubly occupied orbitals and 2 orbitals all singly occupied by parallel spin electrons. The ground-state wavefunction was a single Slater determinant. I explained that it was possible to derive an expression for the electronic energy... 

It is also a common experience that traditional Cl calculations converge very poorly, because the virtual orbitals produced from an HF (or HF-LCAO) calculation are not determined by the variation principle and turn out to be very poor for representations of excited states. 

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