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The Many-Body Schrödinger Equation

 

The aim of an ab initio method is to find the solution to the many-body Schrödinger equation for the system being studied. The first simplification that we may make en route to this goal is the Born Oppenheimer approximation [6], whereby the electronic and nuclear degrees of freedom are separated. The justification for this is that the electrons are much less massive than the nuclei but experience similar forces and therefore the electrons will respond almost instantaneously to the movement of the nuclei. Thus, the energy for a given nuclear configuration will be that of the ground state of the electrons in that configuration. The equation we must solve is therefore,

equation153

where tex2html_wrap_inline2437 is the many body wavefunction for the N electronic eigenstates, an anti-symmetric function of the electronic coordinates tex2html_wrap_inline2441 , and E is the total energy. The Hamiltonian tex2html_wrap_inline2445 is given by

equation157

where tex2html_wrap_inline2447 is the external potential imposed by the nuclear configuration tex2html_wrap_inline2449 and tex2html_wrap_inline2451 is the electron - electron interaction given by the Hartree term tex2html_wrap_inline2453 .

In principle, this equation may be solved to arbitrary accuracy by representing tex2html_wrap_inline2437 as a direct product wavefunction and diagonalising the Hamiltonian. However, the cost of this calculation scales exponentially with the number of electrons in the system and is intractable for all but the smallest of systems. Other approaches are available for the solution of this equation by minimisation of

equation174

The variational principle states that the ground state is that given by minimisation of E over all possible tex2html_wrap_inline2457 . Quantum Monte Carlo techniques may be used to evaluate tex2html_wrap_inline2459 [7] for a given wavefunction and to perform the minimisation over possible wavefunction configurations [8, 9, 10]. However, the cost of such calculations is still prohibitively high for complex systems such as those studied in this thesis. Clearly, another approach is needed to this problem.


next up previous contents
Next: Density Functional Theory Up: The Total Energy Pseudopotential Previous: Introduction

Matthew Segall
Wed Sep 24 12:24:18 BST 1997