Hamiltonian matrix, electron correlation

The principal semi-empirical schemes usually involve one of two approaches. The first uses an effective one-electron Hamiltonian, where the Hamiltonian matrix elements are given empirical or semi-empirical values to try to correlate the results of calculations with experiment, but no specified and clear mathematical derivation of the explicit form of this one-electron Hamiltonian is available beyond that given above. The extended Hiickel calculations are of this type. 

The local ground-state correlation potential is defined in RDFT as the functional derivative of Eq.(7) with respect to p. When infinitesimal variation of occupation numbers is allowed, a more practical definition follows from the fact that the unsymmetrical energy formuala used to construct Eq.(7) is itself a Landau functional of the occupation numbers [19]. Correlation energies of Landau quasiparticles, expressed as diagonal elements of a one-electron Hamiltonian matrix, are defined by differentiating with respect to occupation numbers to give... 

Most of the variational treatments of spin-orbit interaction utilize one-component MOs as the one-particle basis. The SOC is then introduced at the Cl level. A so-called SOCI can be realized either as a one- or two-step procedure. Evidently, one-step methods determine spin-orbit coupling and electron correlation simultaneously. In two-step procedures, typically different matrix representations of the electrostatic and magnetic Hamiltonians are chosen. 

Another promising development is the semi-empirical adjustment of Hamiltonian matrix elements to account for the effects of correlation of the electron core and for basis set limitations within this core " As currently... 

To apply these equations, we need the wavefuncdons m> in order to get the dipole moment transition elements and the frequencies spectral series, where only the ground state need be near-exact. This is done by diagonalizing the Hamiltonian matrix formed from a large number of basis functions (which implicitly include the interelectronic coordinate and thus electron correlation). We do this for each symmetry state that is involved. All the ensuing eigenvalues and eigenvectors are then used in the sum-over-states expressions. For helium we require S, P, and D states and for H2 (or D2) E, II, and A states. 

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