Total energy in the Hartree-Fock method

In Appendix M, p. 986, we derived the following expressions for the mean value of the Hamiltonian using the normalized determinant (without a constant additive term for the nuclear repulsion energy V n, SMO means summation over the spinorbitals i = I.N in the RHF method, the MO summation limit means summation over the orbitals N/2) 

Electronic Motion in the Mean Field Atoms and Molecules 

If double occupancy is assumed (i.e. the flexibility of the variational wave function is restricted) we may transform this expression in the following way 

Given the equality i h i = (i h i), these integrals have been written here as ha- The Coulombic and exchange integrals expressed in spinorbitals are denoted Jij and Kij and expressed in orbitals as Jij and K-ij. 

Both formulae (8.35) and (8.36) may give different results, because in the first, no double occupancy is assumed (we will discuss this further on p. 372). 

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We note that in the Hartree-Fock method exchange is treated exactly but correlations are neglected completely. The Hartree-Fock total energy is an upper bound to the true ground-state total energy, while the LSD and GGA energies are not. 

And we would alreatty expect full additivity, since the electrons in the Hartree-Fock method are treated as independent. Yet independent does not mean noninteracting . The reason for the non-additivity is that for each electron we need to calculate its effective interaction with all the electrons, hence we would get too much repulsion. Of course, the total energy, and not the sum of the orbital energies, is the most valuable. Yet in many quantum chemical problems we interpret orbital energy lowering as energetically profitable. And it turns out that such an interpretation has an approximate justification. Works by Fraga, Politzer and Rue-... 

In the Hartree-Fock method (closed shell, U = sum of the orbital energies for the doubly occupied orbitals), the total electronic energy of a system is ... 

Y -Hab(Hbb -E)- Hbd Haa -Hab(Hbb -E)"Hh If the Hamiltonian is a one-electron Hamiltonian, for example the Fock operator, the partitioning is done by basis functions, since the latter are usually centered on the atomic nuclei, which belong to donor (d), bridge (b) or acceptor (a). In the Hartree-Fock case, the total wave function is a Slater determinant. There may be problems with symmetry breaking in the symmetric case. Cl that includes the two localized solutions can solve this problem [29-31]. The problem is that the Hartree-Fock method gives energy advantage to a localized state, which holds true also in the unsymmetric case. 

Tab. 12.1. Total interaction energies Ei of the multiply-hydrogen-bonded complex in Fig. 12.2 in kJ mol. BHLYP is a hybrid density functional which features 50% admixture of exact Hartree-Fock-type exchange, AMI and PM3 are semi-empirical models, which are an efficient approximation to the Hartree-Fock method, and CCSD is a coupled-cluster model.

In the quantitative development of the structure in the self-consistent field approximation (S.C.F.) using the Hartree-Fock method the energy Ei is made up of three terms, one for the mean kinetic energy of the electron in ipi, one for its mean potential energy in the field of the nuclei, and a correction term for the effect of all the other electrons. The total energy... 

SIC-OEP-KLI approach with the Hartree-Fock method, we found the SIC-OEP-KLI method as inadequately describing the unpaired pair of electrons, since this method overestimates the total energy of the system for large confinement radii, while it underestimates the one-electron and two-electron contributions to the energy. For small confinement radii, the SIC-OEP-KLI approach underestimates the total energy in particular for very small confinement radii the two-electron contributions show large discrepancies from those results obtained with the Hartree-Fock method. 

In Sections 3.1.5 and 3.1.6 we discuss two important alternative methods to intraorbit optimization. Both are based on the use of local-scaling transformations in order to produce sets of transformed orbitals which are then directly employed in the calculation of the total energy. In the non-variational case, we deal with arbitrary orbitals which are locally-scaled in order to yield the Hartree-Fock one-particle density, which we assume to be known beforehand. In the second method, the final density is optimized by energy minimization. But as in the previous case, locally-scaled transformed orbitals are used in the energy calculation. 

Hybrid Methods. - During the last few years, so-called hybrid methods which combine the density-functional and the Hartree-Fock methods have become increasingly popular, first of all for calculating total energies and their changes due to structural changes or chemical reactions. In this subsection we shall briefly review their foundations. 

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