Hartree-Fock Energy Expression

A more complete treatment of density matrices (without applying second quantization) can be found in the classical papers by Lowdin (1955) and McWeeny (1960). 

By means of the first- and second-order density matrices introduced above it is a trivial task to derive the expression of the electronic energy in the Hartree-Fock theory. The goal is simply to evaluate the expectation value of the Hamiltonian H, which in the second quantized formalism is given by Eq. (4.40)  

Expressing the second-order density matrix in terms of the first-order one via Eq. (7.20) one obtains  

To bring this result into a more familiar form, we may interchange the indices v and X in the first term, and apply the interchanges v a than a A, in the second 

A pictorial interpretation of the Hartree-Fock energy is obtained by writing this latter expression in terms of molecular orbitals. Then, cf. Eq. (7.15), we have  

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The starting point for our discussion is the Dirac-Hartree-Fock energy expression in Eq. (8.121) which we may write as... 

For the next example, let us rewrite the Hartree-Fock energy expression (cf. Sect. 7.3) in terms of spatial orbitals. The problem for the one-electron part has been solved above. Let us be concerned with the two-electron part E ... 

Although it appears simple in the context of this chapter on SCF, the idea to consider blends of the HF and density functional theories was revolutionary. As Figure 1 shows, it originated with Slater in 1951 who proposed that the Hartree-Fock energy expression (21) be simplified by replacing F30 with D30 to yield... 

The extrapolation to the complete basis set energy limit is based upon the MoUer-Plesset expansion E= + E + E + E + E +. .. as described earlier in this appendix. Recall that E + E is the Hartree-Fock energy. We will denote E and all higher terms as E , resulting in this expression for E ... 

Density functional methods (DFMs) have also been used to estimate NLO properties,43-53 although this approach has not yet been applied to organometallic complexes. Hartree-Fock theory expresses the energy in... 

The portion of the surface for the HeH+-H2 system calculated by Benson and McLaughlin,104 and referred to above, lies within the range of H-H separations found by Kutzelnigg et a/.105 to be adequately described at the Hartree-Fock limit. Their Hartree-Fock SCF results should, therefore, be accurate enough to warrant their use in classical trajectory calculations. Such calculations have indeed been performed by McLaughlin and Thompson.111 Spline interpolation functions were used to express the tabular near-Hartree-Fock energies in analytical form. The trajectory calculations for the reaction... 

If one works out this expression one obtains equations that are identical to equations (316) and (317). These equations were first derived by Talman and Shadwick [45]. Since in our procedure we optimized the energy of a Slater determinant wavefunction under the constraint that the orbitals in the Slater determinant come from a local potential, the method is also known as the optimized potential method (OPM). We have therefore obtained the result that the OPM and the expansion to order e2 are equivalent procedures. The OPM has many similarities to the Hartree-Fock approach. Within the Hartree-Fock approximation one minimizes the energy of a Slater determinant wavefunction under the constraint that the orbitals are orthonormal. One then obtains one-particle equations for the orbitals that contain a nonlocal potential. Within the OPM, on the other hand, one adds the additional requirement that the orbitals must satisfy single-particle equations with a local potential. Due to this constraint the OPM total energy ) will in general be higher than the Hartree-Fock energy Fhf, i.e., Ex > E. We refer to Refs. [46,47] for an application of the OPM method for molecules. 

For the lithium ground state, Clementi and Roetti used a basis set consisting of two Is STOs (with different orbital exponents) and four 2s STOs (with different orbital exponents). The lithium Is and 2s Hartree-Fock orbitals were each expressed as a linear combination of all six of these basis functions. The Hartree-Fock energy is —202.3 eV, as compared with the true energy —203.5 eV. 

To express the Hartree-Fock energy in terms of integrals over the basis functions x, we first solve (13.154) for 2, 2/ 2Jij - K j) and substitute the result into (13.155) to get... 

Let us generalize the above results to obtain an expression involving spatial integrals for the Hartree-Fock energy of an iV-electron system containing an even number of electrons. The analogue of the minimal basis H2 Hartree-Fock wave function,... 

Introducing, however, a basis set expansion as of Eqs. (10.2) and (10.3) in the relativistic case for the molecular spinors ipi, this leads to a complicated expression for the Dirac-Hartree-Fock energy of Eq. (10.21),... 

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