Hartree-Fock approximation trial wave function

The trial wave function in the Hartree-Fock approximation takes the form of a single Slater determinant ... 

Now we are ready to start the derivation of the intermediate scheme bridging quantum and classical descriptions of molecular PES. The basic idea underlying the whole derivation is that the experimental fact that the numerous MM models of molecular PES and the VSEPR model of stereochemistry are that successful, as reported in the literature, must have a theoretical explanation [21], The only way to obtain such an explanation is to perform a derivation departing from a certain form of the trial wave function of electrons in a molecule. QM methods employing the trial wave function of the self consistent field (or equivalently Hartree-Fock-Roothaan) approximation can hardly be used to base such a derivation upon, as these methods result in an inherently delocalized and therefore nontransferable description of the molecular electronic structure in terms of canonical MOs. Subsequent a posteriori localization... 

To solve the corresponding Eq. (6.8), we have at our disposal the variational and the perturbation methods. The latter should have a reasonable starting point (i.e., an unperturbed system). This is not the case in the problem that we want to consider at the moment. Thus, only the variational method remains. If so, a class of the trial functions should be proposed. In this chapter, the trial wave function will have a very specific form, bearing significant importance for the theory. We mean here the so-called Slater determinant, which is composed of molecular orbitals. At a certain level of approximation, each molecular orbital is a parking place for two electrons. We wUl now learn on how to get the optimum molecular orbitals (using the Hartree-Fock method). Despite some quite complex formulas, which will appear below, the main idea behind them is extremely simple. It can be expressed in the following way. 

In the Hartree approximation, discussed above, the trial wave function did not satisfy the antisymmetry, imposed by the Pauli exclusion principle. Therefore, it seems more appropriate to consider a trial wave function which is a Slater determinant of single particle wave functions. This approximation is known as the Hartree-Fock approximation. The resulting Hartree-... 

Because of the Pauli principle antisymmetry requirement, the ground-state wave function has nodal surfaces in 3n-dimensional space, and to ensure that the walkers converge to the ground-state wave function, one must know the locations of these nodes and must eliminate any walker that crosses a nodal surface in the simulation. In the fixed-node (FN) DQMC method, the nodes are fixed at the locations of the nodes in a known approximate wave function for the system, such as found firom a large basis-set Hartree-Fock calculation. This approximation introduces some error, but FN-DQMC calculations are variational. (In practice, the accuracy of FN-DQMC calculations is improved by a procedure called importance sampling. Here, instead of simulating the evolution of with t, one simulates the evolution off, where / = where is a known accurate trial variation function for the ground state.)... 

We will begin this chapter by constructing determinantal trial functions from the Hartree-Fock molecular orbitals, obtained by solving Roothaan s equations. It will prove convenient to describe the possible N-electron functions by specifying how they differ from the Hartree-Fock wave function Fo Wave functions that differ from Tq by w spin orbitals are called n-tuply excited determinants. We then consider the structure of the full Cl matrix, which is simply the Hamiltonian matrix in the basis of all possible N-electron functions formed by replacing none, one, two,... all the way up to N spin orbitals in Section 4.2 we consider various approximations to the full Cl matrix obtained by truncating the many-electron trial function at some excitation level. In particular, we discuss, in some detail, a form of truncated Cl in which the trial function contains determinants which differ from To by at most two spin orbitals. Such a calculation is referred to as singly and doubly excited Cl (SDCI). 

The failure to properly reproduce the particle-particle coalescence asymptotics bears upon the rates of convergence of the computed energies and other observables to their complete-basis-set (CBS) limits [1]. Whereas in practice this convergence is sufficiently rapid for the solutions of the Hartree-Fock equations [2, 3], obtaining accurate approximations to correlated electronic wave-functions is much more difficult [4, 5]. In order to alleviate this problem, two distinct strategies have been developed, namely inclusion of a correlation factor in the trial function [6-8] and extrapolation to the CBS limit [9, 10]. Successful implementations of the latter approach hinge upon understanding how the approximate wavefunction approaches its exact counterpart as the size of the basis set increases. 

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