Hartree-Fock equation, time-dependent

The time-dependent Hartree-Fock equation is expressed formally by... 

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... 

A nice development of the theory is given in a book by McWeeny, which shows that the assumption of a single determinant form for leads to the time-dependent Hartree-Fock equations... 

Using the ground-state density matrix as an input, the CEO procedure - computes vertical transition energies and the relevant transition density matrices (denoted electronic normal modes ( v)mn = g c j Cn v ), which connect the optical response with the underlying electronic motions. Each electronic transition between the ground state and an electronically excited state v) is described by a mode which is represented hy K x K matrix. These modes are computed directly as eigenmodes of the linearized time-dependent Hartree—Fock equations of motion for the density matrix (eq A4) of the molecule driven by the optical field. 

Additionally, the Fock operator which is used in the following Schrodinger eqn (35) contains the terms that account for the Coulomb and exchange interactions. It is defined similar to eqn (33), but now these two operators are time-dependent as well. As in static Hartree-Fock theory it is assumed, that we can write the wavefunction as a single Slater determinant like in eqn (31), and therefore the time-dependent Hartree-Fock equation is... 

In analogy to the derivation of the coupled Hartree-Fock equations one can then derive the time-dependent Hartree-Fock equations for the first-order expansion coefficients... 

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. 

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. 

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. 

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 ... 

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. 

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 equation (17) 5 is defined as the overlap of two electronic determinantal wave functions S = z,R, P z,R,P ) and the energy is E = Pl[/2Mi + (z,7 ,Pl/7eiecl, ./ ,E)/(z,7 ,PIz,7 ,PX This level of theory can be characterized as fully non-linear time-dependent Hartree-Fock for quantum electrons and classical nuclei. It has been applied to a great variety of problems involving ion-atom [12,14,15,23-25], and ion-molecule reactive collisions... 

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. 

The basic response equation of TDDFT has the same form as that of time-dependent Hartree-Fock theory, or of the Random Phase Approximation, i.e.,... 

In this section we will introduce some wavefunction-based methods to calculate photoabsorption spectra. The Hartree-Fock method itself is a wavefunction-based approach to solve the static Schrodinger equation. For excited states one has to account for time-dependent phenomena as in the density-based approaches. Therefore, we will start with a short review of time-dependent Hartree-Fock. Several more advanced methods are available as well, e.g. configuration interaction (Cl), multireference configuration interaction (MRCI), multireference Moller-Plesset (MRMP), or complete active space self-consistent field (CASSCF), to name only a few. Also flavours of the coupled-cluster approach (equations-of-motion CC and linear-response CQ are used to calculate excited states. However, all these methods are applicable only to fairly small molecules due to their high computational costs. These approaches are therefore discussed only in a more phenomenological way here, and many post-Hartree-Fock methods are explicitly not included. 

Consistent with time-independent Hartree-Fock theory the main approximation in time-dependent Hartree-Fock theory is, that the system is represented by a single Slater determinant, which now is composed of time-dependent single-particle wavefunctions. The time-dependent Schrodinger equation that has to be solved is given in eqn (1). The time-dependent Hamiltonian consists of a static Hamiltonian and an additional time-dependent operator describing the time-dependent perturbation, e.g. an electric field, which is a sum of time-dependent single-particle potentials ... 

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... 

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