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Trial Wavefunctions


To perform a VMC calculation using the algorithm outlined in figure gif, one has to chose the form of the trial wavefunction, tex2html_wrap_inline6049 . This trial wavefunction should contain as much knowledge of the physics of the system being studied as possible. The choice of tex2html_wrap_inline6049 will completely determine the values of all the observables, such as the energy, obtained from the calculation.

For a bosonic system the many-body wavefunction is a symmetric function of the coordinates of the particles. McMillan, in his study of the ground state of liquid tex2html_wrap_inline6217 He by the VMC method [20], used a many-body wavefunction given by a Jastrow function [21],


where u(r) is a two-body function chosen to minimise the energy of the state. The function, u(r), was chosen so as to enhance the probability of pairs of tex2html_wrap_inline6217 He atoms being separated by a distance which minimises their interaction energy. The price to be paid for this is that the kinetic energy is increased due to the confinement, but the total energy is still reduced.

For fermionic systems the many-body wavefunction is antisymmetric under particle exchange. The simplest antisymmetric function one can choose is the Slater determinant, often referred to as the Hartree-Fock approximation.




For solids the single particle orbitals, tex2html_wrap_inline6225 are normally taken from either density-functional-theory, local-density-approximation calculations (DFT-LDA) or Hartree-Fock (HF) calculations. For all-electron atomic calculations the orbitals used are generally those obtained from some minimisation scheme [22]. The use of a separate determinant for up- and down-spin electrons means that the wavefunction is not antisymmetric on exchange of opposite spin electrons, however, this form gives the same expectation value as long as the operator is spin-independent[23]. The advantage of using two smaller determinants rather than one larger one is that it is computationally more efficient.

In this thesis, we adopt the definition of electron correlation as any further electron-electron interaction beyond that described by the exchange interaction in Hartree-Fock theory. According to this definition, the above form of fermionic wavefunction, Eq.(gif), contains no correlation. In order to introduce correlation we multiply by a Jastrow factor which is symmetric under the exchange of particles, giving a wavefunction of the form


Two forms of the Jastrow factor are commonly used:


for solids, and


for atoms. The ratio of the two parameters (A/F) and the value of a are chosen such that the electron-electron ``cusp'' conditions [24] are obeyed, that is


The value of b can be chosen variationally. For solids the standard choice for fixing the remaining degree of freedom in the u function is made by considering the long-range behaviour of u [25, 26]. More optimised choices for this degree of freedom are discussed in chapter gif. For atoms this extra degree of freedom is used to either minimise the energy or the variance of the energy.

Recently more sophisticated Jastrow factors have been used. For atoms [22] this has been done by making the Jastrow factor a function of the electron-nucleus distance as well as the inter-electron distance. Similar schemes have also been implemented for solids [27, 1].

For solids it is found to be beneficial to introduce a one-body, tex2html_wrap_inline6237 function which attempts to reverse the effect of the Jastrow factor on the charge density. As the Jastrow function introduces an extra repulsion between electrons, this has the effect of smearing out the charge density. i.e. in regions where the original charge density is high, the effect of the Jastrow function is to reduce it and vice-versa. Methods for constructing the tex2html_wrap_inline6237 function and more optimised forms of Jastrow function are discussed in chapter gif.

Thus the final form of the fermionic wavefunction is


This is referred to as the Hartree-Fock-Jastrow-Chi trial wavefunction.

next up previous contents
Next: Evaluating the Local Energy Up: Variational Quantum Monte Carlo Previous: Variational Quantum Monte Carlo

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996