Time-dependent Hartree-Fock approach

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 relativistic or non-relativistic random-phase approximation (RRPA or RPA)t is a generalized self-consistent field procedure which may be derived making the Dirac/Hartree-Fock equations time-dependent. Therefore, the approach is often called time-dependent Dirac/Hartree-Fock. The name random phase comes from the original application of this method to very large systems where it was argued that terms due to interactions between many alternative pairs of excited particles, so-called two-particle-two-hole interactions ((2p-2h) see below) tend to... 

On the other hand, the polarizability a and hyperpolarizabflity )3 related to the electro-optical effect of the SiC clusters are calculated using Hartree-Fock and time-dependent DFT approaches at A = 0.633 fim. The calculations are performed for an isolated cluster and then for the one embedded into polymer matrix. The effects of the surrounding media are taken into account via local field theory using the point-dipole approach. The obtained results are compared with the experimental data published recently (Boucle et al. 2006). The obtained polarizabilities a and hyperpolarizabilities )3(w 0,w) are siunmarized in O Table 18-3. Even with large differences observed on the local field calculations (see O Table 18-2) in the different... 

The simplest polarization propagator corresponds to choosing an HF reference and including only the h2 operator, known as the Random Phase Approximation (RPA). For the static case oj = 0) the resulting equations are identical to those obtained from a Time-Dependent Hartree-Fock (TDHF) analysis or Coupled Hartree-Fock approach, discussed in Section 10.5. 

The corrected Linear Response approach (cLR) consists in the use the TDDFT relaxed density and the corresponding apparent charges (7-38) into Eqs. (7-36) and (7-37) to obtain the first-order approximation to the state specific free energy of the excited state. The details of the implementation are described in Ref. [17], This corrected Linear Response computational scheme can be applied to the analogous of the Time Dependent Hartree-Fock approach either in the complete (Random Phase Approximation) or approximated (Tamm-Dancoff approximation or Cl singles, CIS) version. 

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. 

As a consequence, field methods, which consist of computing the energy or dipole moment of the system for external electric field of different amplitudes and then evaluating their first, second derivatives with respect to the field amplitude numerically, cannot be applied. Similarly, procedures such as the coupled-perturbed Hartree-Fock (CPHF) or time-dependent Hartree-Fock (TDHF) approaches which determine the first-order response of the density matrix with respect to the perturbation cannot be applied due to the breakdown of periodicity. 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

An approach briefly presented here is based on a combination of the eikonal (or short wavelength) approximation for nuclei, and time-dependent Hartree-Fock states for the many-electron system, in what we have called the Eikonal/ TDHF approach.[13] A similar description can be obtained with narrow wavepackets for the nuclear motions. Several other approaches have recently been proposed for doing first principles dynamics, a very active area of current research. [39, 11, 15]... 

TDDFRT presented in this section is also applied within the time-dependent hybrid approach. It parallels the corresponding approach in DFT and it combines TDDFRT with the time-dependent Hartree-Fock (TDHF) theory [10, 54]. Instead of a pure DFT xc potential vxca, the hybrid approach employs for the orbital (j)i(y in (7) an admixture of an approximate potential vxca with the exchange Hartree-Fock potential vxja for this orbital... 

UV spectra usually involve electronic state transitions, so that simple Hartree-Fock and DFT calculations often are not sufficient PCM has been recently extended also to multi-configurational (MC-SCF) calculations [113] and to time-dependent approaches, allowing for the description of excited states and then the prediction of the so-called solvatochromic effects on these spectra. Nuclear magnetic resonance (NMR) and electron spin resonance (EPR) spectra are even more influenced by solute-solvent interactions moreover, the interpretation of experimental data is often very difficult without the support of reliable ab initio calculation, especially for EPR which is usually applied to unstable radical species. 

For multi-electron systems, it is not feasible, except possibly in the case of helium, to solve the exact atom-laser problem in 3 -dimensional space, where n is the number of electrons. One might consider using time-dependent Hartree Fock (TDHF) or the time-dependent local density approximation to represent the state of the system. These approaches lead to at least njl coupled equations in 3-dimensional space which is much more attractive computationally. For example, in TDHF the wave function for a closed shell system can be approximated by a single Slater determinant of time dependent orbitals,... 

Nunes and Gonze [153] have recently extended DFPT to static responses of insulating ciystals for any order of perturbation theory by combining the variation perturbation approach with the modern theory of polarization [154]. There are evident similarities between this formalism and (a) the developments of Sipe and collaborators [117,121,123] within the independent particle approximation and (b) the recent work of Bishop, Gu and Kirtman [24, 155,156] at the time-dependent Hartree Fock level for one-dimensional periodic systems. 

As pointed out above, in the time-dependent case the correlation treatment cannot be based on time-dependent Hartree-Fock orbitals - at least not on the real frequency axis in the vicinity of poles of the response functions. Thus, the polarization of the wavefunction must be described through the variables of the correlation method, i.e. for the CC approach by means of the cluster amplitudes. This has important implications on the choice or suitability of correlation methods. As it is apparent from the sum-over-states expression for the -th response function [96]... 

Difficulties arise in the band structure treatment for quasilinear periodic chains because the scalar dipole interaction potential is neither periodic nor bounded. These difficulties are overcome in the approach presented in [115] by using the time-dependent vector potential, A, instead of the scalar potential. In that formulation the momentum operator p is replaced by tt =p + (e/c A while the corresponding quasi-momentum Ic becomes k = lc + (e/c)A. Then, a proper treatment of the time-dependence of k, leads to the time-dependent self-consistent field Hartree-Fock (TDHF) equation [115] ... 

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