(4.2) 
(4.3) 
The use of separate determinants for differing spins results in a trial function that is not antisymmetric with respect to interchange of opposite spin electrons, but gives the correct expectation values for spinindependent operators. [23] The Jastrow factor, , in the most general case, is a function of all electron and atom positions. Jastrow functions are discussed in the next section.
The determinantal part of the wavefunction is usually obtained from a selfconsistent method such as LDADFT or HF. [37,38] For most of the calculations in this thesis a single product of up and downspin determinants of orbitals, , is used to form the wavefunctions. In principle, configurationinteraction or multiconfiguration calculations can be used to provide multiple determinants, including the coefficients ,^{4.3} but many applications of QMC have demonstrated that a high accuracy may usually be obtained using only a single determinant.
The orbitals, , are expanded in a computationally convenient basisset. In principle the orbitals could be expanded in a basis that is optimal for QMC. The key criterion would be the speed of evaluation of the value and first and second derivatives of the orbitals at arbitrary points in space. However, to be obtainable from within a selfconsistent scheme the basis must be convenient for solving the singleparticle equations. Consequently, radial numerical grids, Gaussian functions [39] and planewaves [40] are typically used for atomic, molecular and continuum systems respectively.^{4.4}
A further consideration is that the orbitals in the Slater determinants must be purely real functions. In supercell (periodic boundary condition) calculations, this is achieved by choosing sets of bloch orbitals such that simple linear combinations of the orbitals yield real functions.
The number of forms of Jastrow factor that have enjoyed repeated use in electronic structure applications is limited. In this section, three forms are presented, two of which are used in subsequent calculations. A thorough study of the efficiency of different forms would be difficult to produce due to the computational cost involved. Alexander and Coldwell [41] reviewed 118 trial wavefunction forms for the He, Li and Be atoms, but did not consider other species, or molecular and continuum systems.
Although the quality of a Jastrow function can be judged by the VMC energies obtained for candidate systems, compromises must inevitably be made. Benchmark accuracy Jastrow factors for molecular and continuum systems typically consist of very large numbers of terms and parameters. The increase in accuracy per additional term (of a given kind) inevitably reduces. Consequently, a wavefunction obtaining of the correlation energy and 30 parameters may be preferable to one obtaining of the correlation energy with 200 parameters. For use as a guiding DMC wavefunction, the simpler wavefunction would be preferable due to the lower computational cost. For VMC calculations, the preference would depend on the specific application and desired final accuracy.
(4.4) 
(4.5) 
(4.6)  
(4.7) 
This functional form has yielded wavefunctions of sufficient accuracy to perform DMC calculations in several systems, but has not been systematically tested for accuracy and convergence of VMC energies.
Although successfully used in several calculations, the functional form has two undesirable features. The correlation function, , is long ranged implying that the sum over electrons and ions is not readily cutoff and will increase in cost with system size. For certain values of , the long range of the exponential term requires the use of Ewald summation (section 4.6.1) in supercell calculations. As this sum involves all pairs of electrons, this operation would become costly.
Additionally, the function is not linear in the parameters, which is desirable to avoid full recalculation of the function as the parameters are changed during optimisation. This increases the computational cost of the optimisation (see section 4.7.1).
(4.8) 
The electronelectron dependent part of the Jastrow function consists
of cuspsatisfying part and a separate power expansion, both of which
are constrained to be short ranged. For the cuspsatisfying part, the
following form was chosen:
The parameters , are different for spin parallel and antiparallel electron pairs, to model the additional Pauli repulsion experienced by parallel spin electrons. These functions satisfy the electronelectron cusp condition and are continuous in value and first derivative. The expansion range, , is chosen to be less than or equal to the WignerSeitz radius in supercell calculations avoiding the need to perform an Ewald sum. A further advantage of the Chebyshev expansion is that unlike the terms in a simple power expansion, the terms carry equal weight over their range and consequently exhibit greater stability during optimisation.
In applications to bulk carbon and silicon, wavefunctions obtaining of the fixednode DMC correlation energy were obtained. The high accuracy of these wavefunctions indicates that the limited range (form) of the correlation functions does not severely limit the accuracies obtained.
A general power expansion, in and , about an atom over a
range must satisfy similar conditions to Williamson's function,
above: the function must go to zero at and be continuous in value
and first derivative. The atom centred power expansion in electronion
distance, and electronelectron distance contains 5
independent series: [48,49]
(4.13)  
(4.14) 
The overall Jastrow factor consists of sets of the above functions supplemented by the electronelectron cuspsatisfying term, equation 4.10, and the homogeneous dependent term, equation 4.11. This is shown schematically in figure 4.1.

For the main applications in this thesis, only the terms were used as sufficient accuracy was obtained and the full series had yet to be implemented. In tests on the ground state of an allelectron oxygen atom, using exact HF orbitals, of the experimental correlation energy was obtained using all the terms, approximately more than a wavefunction using only the terms. This accuracy is similar to that obtained in pseudopotential calculations of oxygen atoms, demonstrating the importance of the additional correlation terms when treating core electrons.