Hartree-Fock reference/equations

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) ... 

Multireference There is no division into occupied and virtual orbitals, all orbitals appear on an equal footing in the ansatz (Equation 8). In particular, the Hartree-Fock reference has no special significance here. For this reason, we expect (and observe) the ansatz to be very well balanced for describing nondynamic correlation in multireference problems (see e.g., refs. 10-12). Conversely, the ansatz is inefficient for describing dynamic correlation, since to treat dynamic correlation one would benefit from the knowledge of which orbitals are in the occupied and virtual spaces. 

The main difference and the potential of this approach lies in the detail that Vxc(r) includes not only the exchange in the Hartree-Fock (HF) equations, but also the correlation (referred to all that is missed by the Hartree-Fock approach) components. In addition, the difference between the exact kinetic energy of the system and the one calculated from the KS orbitals are included. This method states that Vxc(r) is the best way to describe the fact that every electron aims to maximize the attraction from the nuclei and to minimize the repulsion from the rest of the electrons along its constant movement within an entity (atom or molecule). Vxc(r) describes the exchange correlation... 

For the numerical implementation of these formulae one can either use a perturbation expansion for the functions coupled Hartree-Fock (CHF) equations for each order.124 This procedure seems to be more advantageous for numerical calculations, but the resulting expressions are quite complicated. For this reason we do not reproduce them here, but refer to the original paper.124 It seems to be an acceptable compromise to use the second order CHF equations for the dynamic as and (Js, but use simple perturbation theory for the dynamic ys using as unperturbed wave functions the results of the solutions of the first and second order CHF equations. One should point out, however, that in our calculations124 the second numerical derivatives... 

Though used in some semiempirical applications by Paldus and Cizek [11] and one ab initio study [12] (see later), the CCD equations were not implemented into general purpose programs until 1978 by me and Purvis [5] and Pople et al. [13]. This general implementation included allowing for the open-shell case subject to an unrestricted Hartree-Fock reference function. 

Approximate many-electron wave functions are then constructed from the Hartree-Fock reference and the excited-state configurations via some sort of expansion (e.g., a linear expansion in Cl theory, an exponential expansion in CC theory, or a perturbative power series expansion in MBPT). When all possible excitations have been incorporated (S, D, T,. .., for an -electron system), one obtains the exact solution to the nonrelativistic electronic Schrodinger equation for a given AO basis set. This -particle limit is typically referred to as the full Cl (FCI) limit (which is equivalent to the full CC limit). As Figure 5 illustrates, several WFT methods can, at least in principle, converge to the FCI limit by systematically increasing the excitation level (or perturbation order) included in the expansion technique. 

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 discussed above, it is impossible to solve equation (1-13) by searching through all acceptable N-electron wave functions. We need to define a suitable subset, which offers a physically reasonable approximation to the exact wave function without being unmanageable in practice. In the Hartree-Fock scheme the simplest, yet physically sound approximation to the complicated many-electron wave function is utilized. It consists of approximating the N-electron wave function by an antisymmetrized product4 of N one-electron wave functions (x ). This product is usually referred to as a Slater determinant, OSD ... 

What does this mean We have replaced the non-local and therefore fairly complicated exchange term of Hartree-Fock theory as given in equation (3-3) by a simple approximate expression which depends only on the local values of the electron density. Thus, this expression represents a density functional for the exchange energy. As noted above, this formula was originally explicitly derived as an approximation to the HF scheme, without any reference to density functional theory. To improve the quality of this approximation an adjustable, semiempirical parameter a was introduced into the pre-factor Cx which leads to the Xa or Hartree-Fock-Slater (HFS) method which enjoyed a significant amount of popularity among physicists, but never had much impact in chemistry,... 

In the literature we may find the procedure for creating localized Hartree-Fock orbitals via an energy minimization based on a Cl procedure employing monoexcitations (see for instance Reference [24]). The scheme starts from a set of given (guess) orbitals and solves iteratively the Hartree-Fock equations via the steps ... 

In most work reported so far, the solute is treated by the Hartree-Fock method (i.e., Ho is expressed as a Fock operator), in which each electron moves in the self-consistent field (SCF) of the others. The term SCRF, which should refer to the treatment of the reaction field, is used by some workers to refer to a combination of the SCRF nonlinear Schrodinger equation (34) and SCF method to solve it, but in the future, as correlated treatments of the solute becomes more common, it will be necessary to more clearly distinguish the SCRF and SCF approximations. The SCRF method, with or without the additional SCF approximation, was first proposed by Rinaldi and Rivail [87, 88], Yomosa [89, 90], and Tapia and Goscinski [91], A highly recommended review of the foundations of the field was given by Tapia [71],... 

The density functional theory (DFT) [32] represents the major alternative to methods based on the Hartree-Fock formalism. In DFT, the focus is not in the wavefunction, but in the electron density. The total energy of an n-electron system can in all generality be expressed as a summation of four terms (equation 4). The first three terms, making reference to the noninteracting kinetic energy, the electron-nucleus Coulomb attraction and the electron-electron Coulomb repulsion, can be computed in a straightforward way. The practical problem of this method is the calculation of the fourth term Exc, the exchange-correlation term, for which the exact expression is not known. 

Recall that the core-valence separation in molecules is described in real space [83], as any atom-by atom or bond-by-bond partitioning of a molecule is inherently a real-space problem. Equation (10.6) does indeed refer to a partitioning in real space (as opposed to the usual Hartree-Fock orbital space), both for ground-state isolated atoms or ions and for atoms embedded in a molecule, with N = 2c for first-row elements. 

The set of molecular orbitals leading to the lowest energy are obtained by a process referred to as a self-consistent-field or SCF procedure. The archetypal SCF procedure is the Hartree-Fock procedure, but SCF methods also include density functional procedures. All SCF procedures lead to equations of the form. 

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. 

Presented in this chapter is a verbal and pictorial description of Hartree-Fock MO theory. No equations will be given but reference will be made to appropriate parts of Appendix A where more details may be found. 

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