Fock operator diagonal matrix elements

Practical calculations require approximations in the self-energy operator. Perturbative improvements to Hartree-Fock, canonical orbital energies can be generated efficiently by neglecting off-diagonal matrix elements of the selfenergy operator in this basis. Such diagonal, or quasiparticle, approximations simplify the Dyson equation to the form... 

In practice, approximations in the self-energy operator are needed. Efficient, perturbative improvements to Koopmans results may be produced by the neglect of off-diagonal matrix elements of the self-energy operator in the canonical, Hartree-Fock basis. Such diagonal approximations in the selfenergy, which are also known as quasiparticle methods, yield an especially simple form of the Dyson equation. 

Diagonal matrix elements of the P3 self-energy approximation may be expressed in terms of canonical Hartree-Fock orbital energies and electron repulsion integrals in this basis. For ionization energies, the bottleneck arithmetic operation has a scaling factor of O V, where O is the number of occupied orbitals and V is the number of virtual orbitals. Electron repulsion integrals in the Hartree-Fock... 

Here F(N) is simply the N-electron ground state Hartree-Fock operator. Similarly one may obtain the off diagonal matrix elements between two states N-1, iA> and n-1, jB>. These are ... 

This is an occupied-virtual off-diagonal element of the Fock matrix in the MO basis, and is identical to the gradient of the energy with respect to an occupied-virtual mixing parameter (except for a factor of 4), see eq. (3.67). If the determinants are constructed from optimized canonical HF MOs, the gradient is zero, and the matrix element is zero. This may also be realized by noting that the MOs are eigenfunctions of the Fock operator, eq. (3.41). 

NBO analysis can be used to quantify this phenomenon. Since tire NBOs do not diagonalize the Fock operator (or tire Kohn-Sham operator, if the analysis is carried out for DFT instead of HF), when the Fock matrix is formed in the NBO basis, off-diagonal elements will in general be non-zero. Second-order perturbation tlieory indicates that these off-diagonal elements between filled and empty NBOs can be interpreted as the stabilization energies... 

The evaluation of the matrix elements of the Hartree-Fock operator is usually carried out with a number of approximations. The diagonal, one-electron terms,... 

It appears, therefore, that it is possible to obtain accurate expectation values of the spin-orbit operators for diatomic molecules. Matcha et a/.112-115 have provided general expressions for the integrals involved and from their work Hall, Walker, and Richards116 derived the diagonal one-centre matrix elements of the spin-other-orbit operator for linear molecules. Provided good Hartree-Fock wavefunctions are available, these should be sufficient for most calculations involving diatomic molecules. 

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