Time-dependent Hartree-Fock approximation

Lagrangian as a functional -(v,v). Note however that unlike functionals used in the Time-dependent Hartree Fock approximation (14), this Lagrangian is not complex analytic in the variables (v,v) separately. 

To properly describe electronic rearrangement and its dependence on both nuclear positions and velocities, it is necessary to develop a time-dependent theory of the electronic dynamics in molecular systems. A very useful approximation in this regard is the time-dependent Hartree-Fock approximation (34). Its combination with the eikonal treatment has been called the Eik/TDHF approximation, and has been implemented for ion-atom collisions.(21, 35-37) Approximations can be systematically developed from time-dependent variational principles.(38-41) These can be stated for wavefunctions and lead to differential equations for time-dependent parameters present in trial wavefunctions. 

The eikonal/time-dependent Hartree Fock approximation and extensions. 

P. Albertsen, P. Jorgensen, and D. L. Yeager, Mol. Phys, 41, 409 (1980). Frequency Dependent Polarizabilities in a Multiconfigurational Time Dependent Hartree-Fock Approximation. 

This approximation is better known as the time-dependent Hartree—Fock approximation (TDHF) (McLachlan and Ball, 1964) (see Section 11.1) or random phase approximation (RPA) (Rowe, 1968) and can also be derived as the linear response of an SCF wavefunction, as described in Section 11.2. Furthermore, the structure of the equations is the same as in time-dependent density functional theory (TD-DFT), although they differ in the expressions for the elements of the Hessian matrix E22. The polarization propagator in the RPA is then given as... 

At the SCF level all methods lead to the same expressions for the response functions as obtained in the random phase approximation, in Section 10.3, with the time-dependent Hartree-Fock approximation, in Chapter 11.1, or with SCF linear response theory. The QED and time-averaged QED method for an MCSCF energy was also shown to yield the same expressions as obtained from propagator or response theory in Sections 10.4 and 11.2. 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

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. 

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. 

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. 

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. 

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]... 

Approximate solutions of the time-dependent Schrodinger equation can be obtained by using Frenkel variational principle within the PCM theoretical framework [17]. The restriction to a one-determinant wavefunction with orbital expansion over a finite atomic basis set leads to the following time-dependent Hartree-Fock or Kohn-Sham equation ... 

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... 

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. 

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. 

It has recently been shown [ 12] that time-dependent or linear-response theory based on local exchange and correlation potentials is inconsistent in the pure exchange limit with the time-dependent Hartree-Fock theory (TDHF) of Dirac [13] and with the random-phase approximation (RPA) [14] including exchange. The DFT-based exchange-response kernel [15] is inconsistent with the structure of the second-quantized Hamiltonian. 

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. 

Before we do so it is worth-while to establish some conventions and terminology in this area. The obvious name for a model of electronic structure which has a time-dependent Hamiltonian and consists of a single determinant of orbit s and remains a single determinant at all times is the Time-Dependent Hartree-Fock (TDHF) model, and this is the terminology which will be used here. However, there is, particularly in the theoretical physics literature, another related usage. Because the use of perturbation theory is so much their stock-in-trade, many theoretical physicists use the term time-dependent Hartree-Fock to mean the first-order (in the sense of perturbation theory) approximation to what we will call the time-dependent Hartree-Fock model. 

Much more enigmatically, the first-order time-dependent self-consistent field approximation is also widely called the Random Phase Approximation (RPA). This terminology is entrenched and so, although the name time-dependent self-consistent perturbation theory is more descriptive and preferable to both time-dependent Hartree-Fock and RPA all are used more or less interchangeably. The evolution of the concept and phrase random phase approximation is sketched in Appendix 27.A to this chapter. 

The problem of a perturbation approximation to the solutions of the time-dependent Hartree-Fock equations presuppose a knowledge of what those equations are. We do not, as yet, know these equations but we can quickly derive... 

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