Restricted Hartree-Fock theory operator

Restricted Hartree-Fock theory (RHF), 23, 234-236 energy, 35 operator, 35 Retinal imine, 270, 272 Rhodopsin, 270, 272... 

For most applications, the external potential operator v reduces to a local potential v(r), and the Hohenberg-Kohn theory is valid for the ground state. The present derivation follows exactly the logic of standard Hartree-Fock theory. It is not restricted to ground states and remains valid for fractional occupation numbers. 

Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

One of the strengths of CC theory is its ability to handle orbitals that are not variationally optimum for the problem of interest. In Cl methods, if one uses a set of orbitals that is not optimum, one usually introduces significant contributions from other references, i.e., making the problem a multireference problem. In CC theory this is not the case, as the exponential wave operator effectively rotates the orbitals, and the results from a nonoptimum set of orbitals are usually very good. This feature has several practical implications. For several years, a procedure termed quasi-restricted Hartree-Fock... 

While the expansion of hpq follows readily from the development for the Dirac operator above, the electron-electron interaction integrals must be considered separately. We also want to develop a Kramers-restricted Dirac-Hartree-Fock (KR-DHF) theory, but first we develop expressions for the general, Kramers-unrestricted case. In the developments below we follow the practice of giving only the basis function index in the integrals. 

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