The evaluation of non-local pseudopotentials within the QMC solid codes is dealt with in detail in chapter . Optimising wavefunctions with respect to these non-local pseudopotentials is discussed in chapter . In chapters and , DMC calculations involving a non-local pseudopotential to describe silicon are reported. The use of such non-local pseudopotentials within DMC is a problem that is yet to be completely resolved. Here we discuss the approximations used later in the thesis to try and deal with this problem.
Consider Eq.(), in which a pseudopotential is used to describe the valence electrons in a system;
We can divide the full valence Hamiltonian, , into a local and non-local part
where includes the kinetic energy, the local part of the pseudopotential and the Coulomb interaction, and W includes the non-local part of the pseudopotential. The diffusion-drift equation for f, Eq.(), can be split into two terms representing the local and non-local parts of the Hamiltonian
The first three terms on the left hand side of Eq.() can be interpreted as a local diffusion, drifting and branching process. However, the fourth term represents the operation of the non-local pseudopotential on the unknown wavefunction, , producing non-local branching. In the DMC calculations described in chapters and , we have used the ``locality approximation'' which was introduced by Christiansen[38, 39, 40] and recently applied by Mitas[41, 42] to the problem of the unknown wavefunction in Eq.(). In this approximation, the non-local pseudopotential acts not on the unknown wavefunction, , but on the guiding wavefunction, ,
One is then free to impose the fixed-node approximation in the same way as before. In fact, when using the above local model potential, it is important that the fixed node approximation is applied as this steers the random walk away from the nodes of , where there will generally be divergences in the local model potential. Without the fixed node approximation, these divergences would cause large fluctuations in the population of diffusing particles.
It should be noted that within the ``locality approximation'', it is no longer generally true that the DMC estimate of the energy is an upper bound to the true groundstate energy. It must always be less than the VMC energy and it has been shown,  that the DMC energy converges quadratically to the exact groundstate energy as the guiding function approaches the true groundstate wavefunction.