Hartree-Fock approach multiconfiguration

Hurley proposed a simple, sufficient condition for the applicability of the Hellmann-Feynman theorem. " If within a variational framework, the family of trial functions is invariant to changes in parameter a, then the Hellmann-Feynman theorem is satisfied by the optimum trial function. In variational approaches involving Lagrange multipliers, for example, in the Hartree-Fock and multiconfigurational self-consistent field methods, Hurley s condition is fulfilled. ... 

The one-particle basis functions x are often referred to as atomic orbitals (AOs). The MO coefficients C are obtained by solving an electronic structure problem simpler than that of Eq. (2), such as the independent particle (Hartree-Fock) approximation, or using a multiconfigurational Hartree-Fock approach. This has the advantage that these approximations generally... 

In principle, the correspondence between the two theories is not complete, because scattering theory is the more general formulation. For our purposes, however, the fact that the applications to atomic physics obtained by both methods are quite consistent with each other is an important and useful conclusion. The same result and connections have been obtained independently by Komninos and Nicolaides [378]. Both [373] and [378] noted that the derivation of MQDT from Wigner s scattering theory establishes its basic structure and theorems without special assumptions about the asymptotic forms of wavefunctions. The approach of Komninos and Nicolaides [378] is designed for applications involving Hartree-Fock and multiconfigurational Hartree-Fock bases. In the present exposition, we follow the approach and notation of Lane [379] and others [380, 381], who exploit the analytic K-matrix formalism and include photon widths explicitly when interferences occur. 

D. L. Yeager and P. Jorgensen, Chem. Phys, Lett., 6S, 77 (1979). A Multiconfigurational Time-Dependent Hartree-Fock Approach. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

Approximations have been reviewed in the case of short deBroglie wavelengths for the nuclei to derive coupled quantal-semiclassical computational procedures, by choosing different types of many-electron wavefunctions. Time-dependent Hartree-Fock and time-dependent multiconfiguration Hartree-Fock formulations are possible, and lead to the Eik/TDHF and Eik/TDMCHF approximations, respectively. More generally, these can be considered special cases of an Eik/TDDM approach, in terms of a general density matrix for many-electron systems. 

The Cl procedure just described uses a fixed set of orbitals in the functions An alternative approach is to vary the forms of the MOs in each determinantal function O, in (1.300), in addition to varying the coefficients c,. One uses an iterative process (which resembles the Hartree-Fock procedure) to find the optimum orbitals in the Cl determinants. This form of Cl is called the multiconfiguration SCF (MCSCF) method. Because the orbitals are optimized, the MCSCF method requires far fewer configurations than ordinary Cl to get an accurate wave function. A particular form of the MCSCF approach developed for calculations on diatomic molecules is the optimized valence configuration (OVC) method. 

C. Froese Fischer, Self-Consistent-Field (SCF) euid Multiconfiguration (MC) Hartree-Fock (HF) Methods in Atomic Calculations Numerical Integration Approaches, Comput. Phys. Rep. 3 (1986) 273-325. 

As fully numerical Multiconfiguration Dirac-Hartree-Fock programs (MCDHF) [83-85] became available a rigorous approach was undertaken to systematically improve orbital spaces and the correlation treatment in hyperfine structure calculations. 

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