Time-dependent Hartree-Fock TDHF

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. 

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. 

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 metric term Eq. (2.8) is important for all cases in which the manifold M has non-zero curvature and is thus nonlinear, e.g. in the cases of Time-Dependent Hartree-Fock (TDHF) and Time-Dependent Multi-Configurational Self-Consistent Field (TDMCSCF) c culations. In such situations the metric tensor varies from point to point and has a nontrivial effect on the time evolution. It plays the role of a time-dependent force (somewhat like the location-dependent gravitational force which arises in general relativity from the curvature of space-time). In the case of flat i.e. linear manifolds, as are found in Time-Dependent Configuration Interaction (TDCI) calculations, the metric is constant and does not have a significant effect on the dynamics. 

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. 

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... 

Time-dependent Hartree-Fock (TDHF) theory... 

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,... 

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. 

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. 

Integration of the system of equations (9) yields trajectories of classical nuclei dressed with END. This approach can be characterized as being direct, and non-adiabatic or as fully non-linear time-dependent Hartree-Fock (TDHF) theory of quantum electrons and classical nuclei. This simultaneous dynamics of electrons and nuclei driven by their mutual instantaneous forces requires a different approach to the choice of basis sets than that commonly encountered in electronic structure calculations with fixed nuclei. This aspect will be further discussed in connection with applications of END. 

In time-dependent Hartree-Fock (TDHF) and TDDFT calculations, the selfinteractions for unoccupied orbitals are automatically corrected. Here, in order to remove the SIEs of occupied orbitals, the following modified A matrix is adopted to estimate SIC core-excitation energies ... 

The time-dependent Hartree-Fock (TDHF) method has a lengthy history as a technique for calculating the electronic excitation energies and transition moments of molecular systems. There are two ways to formulate TDHF theory that look quite different but are in fact equivalent. The formulation used below... 

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