Coupled multiconfiguration Hartree Fock

Notice that in this MCSCF result, the multiplier of a is equal to the expectation value of the perturbation operator H,. We have thus obtained an analytical expression from which to determine the desired first- and second- order response properties. This analytical approach for determining the second-order properties is referred to as the coupled multiconfiguration Hartree-Fock (CMCHF) approach (Dalgaard and Jorgensen, 1978). 

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. 

However, until today no systematic comparison of methods based on MpUer-Plesset perturbation (MP) and Coupled Cluster theory, the SOPPA or multiconfigurational linear response theory has been presented. The present study is a first attempt to remedy this situation. Calculations of the rotational g factor of HF, H2O, NH3 and CH4 were carried out at the level of Hartree-Fock (SCF) and multiconfigurational Hartree-Fock (MCSCF) linear response theory, the SOPPA and SOPPA(CCSD) [40], MpUer-Plesset perturbation theory to second (MP2), third (MP3) and fourth order without the triples contributions (MP4SDQ) and finally coupled cluster singles and doubles theory. The same basis sets and geometries were employed in all calculations for a given molecule. The results obtained with the different methods are therefore for the first time direct comparable and consistent conclusions about the performance of the different methods can be made. 

Table II shows the QMC calculated energies of the iron atom [57] compared with LDA [94] and coupled cluster [89] calculations. These were systematic calculations of the 3d atom with Ne-core scalar relativistic pseudopotentials derived within the multiconfiguration Hartree-Fock,...

The essentials of the SSEA for numerically compufed energy-normalized N-elecfron wavefunctions were published in 1994 by Mercouris ef al. [54], The firsf application was not only to the multiphoton ionization of H (whose specfrum is known exactly analytically), as a test case, but also to the multiphoton detachment of the four-electron Li negative ion, with two free channels, Li ls 2s S and ls 2p P°. Li (or Be) is the first system of fhe Periodic Table for which the proper description of the zero-order electronic structure requires a multiconfigurational Hartree-Fock (MCHF) description. In the context of the review of the SSEA, we also discuss briefly the formulation of the problem in terms of the full atom-EMF interaction,Vext(f), which is computationally convenient as well as necessary for certain problems involving, say, off-resonance coupling of Rydberg states, for which use of just the electric dipole term is inadequate [55-57]. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDHF) is the Hartree-Fock approximation for the time-dependent Schrodinger equation. CPHF stands for coupled perturbed Hartree-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. 

DFT is the modern alternative to the wave-function based ab initio methods and allows to obtain accurate results at low computational cost, that also helps to understand the chemical origin of the effect. DFT, like Hartree-Fock (HF) methods, exploit molecular symmetry which is crucial in the case of computational studies of the JT effect. It also includes correlation effects into the Hamiltonian via the exchange-correlation functional. HF and many-body perturbation methods are found to perform poorly in the analysis of JT systems for obvious reasons, at contrast to the methods based on DFT, or multiconfigurational SCF and coupled cluster based methods [73]. The later are very accurate but have some drawbacks, mainly the very high computational cost that limits the applications to the smaller systems only. Another drawback is the choice of the active space which involves arbitrariness. 

Analytical second derivatives for closed-shell (or unrestricted Hartree-Fock (UHF)) SCF wavefunctions are used routinely now. The extension to the MCSCF case is relatively new, however. In contrast to the first derivatives, the coupled perturbed SCF equations have to be solved in order to calculate the second and third energy derivatives. The closed-shell case is relatively straightforward, and will be discussed. The multiconfigurational formalism is... 

In this substection we will shortly discuss the computational methods used for calculation of the spin-spin coupling constants. Two main approaches available are ab initio theory from Hartree-Fock (or self-consistent field SCF) technique to its correlated extensions, and density function theory (DFT), where the electron density, instead of the wave function, is the fundamental quantity. The discussion here is limited to the methods actually used for calculation of the intermolecular spin-spin coupling constants, i. e. multiconfigurational self consistent field (MCSCF) theory, coupled cluster (CC) theory and density functional theory (DFT). For example, the second order polarization propagator method (SOPPA) approach is not... 

Thus, as with SCF wavefunctions, the first derivative of an MC-SCF energy does not need the derivative of the wavefunction. However, to proceed to second derivatives, it is necessary to solve the MC-CHF equations (multiconfiguration coupled Hartree-Fock). Unlike SCF, however, not only the orbital changes are required but also the changes in the configuration weights. Thus the set of equations has the structure... 

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