Hartree-Fock approximation trial wavefunctions

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. 

Undoubtedly, the methods most widely used to solve the Schrodinger equation are those based on the approach originally proposed by Hartree [1] and Fock [2]. Hartree-Fock (HF) theory is the simplest of the ab initio or "first principles" quantum chemical theories, which are obtained directly from the Schrodinger equation without incorporating any empirical considerations. In the HF approximation, the n-electron wavefunction is built from a set of n independent one-electron spin orbitals which contain both spatial and spin components. The HF trial wavefunction is taken as a single Slater determinant of spin orbitals. 

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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