Time-dependent Hartree-Fock methods

Schrddinger equation (280) represents a time-dependent scheme which is numerically much less involved than, e.g., the time-dependent Hartree-Fock method. Numerical results obtained with this scheme for atoms in strong laser pulses will be described in Sect. 8. 

Multiconfigurational time-dependent Hartree-Fock method Semiclassical method... 

M. Jaszunski, A. Rizzo, and D. L. Yeager,). Chem. Phys., 89, 3063 (1988). Application of the Multiconfiguration Time-Dependent Hartree-Fock Method to CH The Auxiliary Active Space. 

The effect of the solvent (acetone) on the first hyperpolarizability of p-nitroaniline has been investigated using supermolecule and semicontinuum approaches.Employing the time-dependent Hartree-Fock method, the projection of the first hyperpolarizability on the dipole moment increases by 18 and 37% when considering one or two acetone molecules forming H-bonds with the amino group, respectively while the use of a continuum model leads to an enhancement by more than a factor of 3. 

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. 

TDHF (time-dependent Hartree-Fock) ah initio method used for computing nonlinear optical properties... 

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. 

Time-dependent Hartree-Fock (TDHF) approximation. See also Multiconfiguration time-dependent Hartree (MCTDH) method... 

Currently the time dependent DFT methods are becoming popular among the workers in the area of molecular modelling of TMCs. A comprehensive review of this area is recently given by renown workers in this field [116]. From this review one can clearly see [117] that the equations used for the density evolution in time are formally equivalent to those known in the time dependent Hartree-Fock (TDHF) theory [118-120] or in its equivalent - the random phase approximation (RPA) both well known for more than three quarters of a century (more recent references can be found in [36,121,122]). This allows to use the analysis performed for one of these equivalent theories to understand the features of others. 

Molecular polarizabilities and hyperpolarizabilities are now routinely calculated in many computational packages and reported in publications that are not primarily concerned with these properties. Very often the calculated values are not likely to be of quantitative accuracy when compared with experimental data. One difficulty is that, except in the case of very small molecules, gas phase data is unobtainable and some allowance has to be made for the effect of the molecular environment in a condensed phase. Another is that the accurate determination of the nonlinear response functions requires that electron correlation should be treated accurately and this is not easy to achieve for the molecules that are of greatest interest. Very often the higher-level calculation is confined to zero frequency and the results scaled by using a less complete theory for the frequency dependence. Typically, ab initio studies use coupled-cluster methods for the static values scaled to frequencies where the effects are observable with time-dependent Hartree-Fock theory. Density functional methods require the introduction of specialized functions before they can cope with the hyperpolarizabilities and higher order magnetic effects. 

Norman and Jensen27 have implemented a method for obtaining second order response functions within the four component (relativistic) time-dependent Hartree-Fock scheme. Results are presented for the first order hyperpolarizabilities for second harmonic generation, />(—2o o),o ) for CsAg and CsAu. A comparison of the results with those of non-relativistic calculations implies that the nonrelativistic results are over-estimated by 18% and 66% respectively. In this method transitions that are weakly-allowed relativistically can lead to divergences in the frequency-dependent response, which would be removed if the finite lifetimes of the excited states could be taken into account. 

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]... 

In principle, the differentiation is either done numerically in the so-called finite-field methods, or in an analytical scheme, or a combination of both. Numerical finite-field calculations are limited to derivatives with respect to static fields. Since SFG is an optical process that involves dynamic oscillating fields, it becomes necessary to use an analytical approach, such as the time-dependent Hartree Fock (TDHF) method. 

From a very general point of view every ion-atom collision system has to be treated as a correlated many-body time-dependent quantum system. To solve this from an ab initio point of view is still impossible. So, one has to rely on various approximations. Nowadays the best method which can be applied to realistic collision systems (which we discuss here) is on the level of the non-selfconsistent time-dependent Hartree-Fock-Slater or, in the relativistic case, the Dirac-Fock-Slater method. Up-to-now no correlation beyond this approximation can be taken into account in the case of 3 or more electrons. (This is in accordance with the definition of correlation given by Lowdin [1] in 1956) In addition no QED contributions, i.e. no correction to the 1/r Coulomb interaction between the electrons, ever have been taken into account, although in very heavy collision systems this effect may become important. This will be discussed in section 5. A short survey of the theory used is followed by our results on impact parameter dependent electron transfer and excitation calculations of ion-atom and ion-solid collisions as well as first results of an ab initio calculation of MO X-rays in such complicated many particle scattering systems. 

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