Quantum Monte Carlo (MgO)

Quantum Monte Carlo (QMC) calculations are a promising alternative to quantum chemistry methods to achieve high accuracy in first principles quantum mechanics calculations. One of the attractive features of QMC is its benign scaling with respect to the number of atoms in the system, with the cost of evaluating the total energy of a system containing N atoms being proportional to N³. In fact, it is often useful to refer to the cost of performing one 'QMC step', which involves the evaluation of a trial wavefunction. The trial wavefunction is usually constructed as the product of a Slater determinant of single particles orbitals times a Jastrow factor. Most of the time in the evaluation of the trial wavefunction goes into the evaluation of the single particle orbitals, proportional to N in number, for each electron in the system, also proportional to N, so that the cost of one evaluation is proportional to N² (in fact, the cost raises to N³ if plane-waves are used to describe the single particle orbitals). However, if the single particles orbitals can be effectively localised, then only a finite number of them will have to be computed, and the cost of evaluating the trial wavefunction is reduced to be only proportional to N.
We have recently developed a technique which allows an efficient localisation of the single particle orbitals, and therefore introduce 'linear scaling' in QMC calculations. The algorithm is based on the idea that any non-singular linear combination of single particle orbitals only changes the value of the Slater determinant by a constant, which is irrelevant for QMC calculations. The linear combinations do not need to preserve the orthogonality of the orbitals, which allows very effective localisations.

The figure shows a comparison of the 'exact' QMC energy of a 64 atoms MgO crystal, evaluated with extended orbitals, and that obtained from non-orthogonal localised orbitals, as a function of the radius where the orbitals have been truncated to zero. We see that the exact energy is already achieved with the modest cutoff of 6 Bohr's. The calculations were performed using Materials Chemistry consortium time.

The paper ( D. Alfe` and M. J. Gillan, "Linear-scaling Quantum Monte Carlo with non-orthogonal localized orbitals", Journal of Physics: Condensed Matter, 16, L305-L311 (2004) ) was listed among the 24 top papers (among more than 1600) in 2004. The work was also highligthed by a report in CSAR Focus: ( D. Alfe` and M. J. Gillan, N. D. Drummond, M. D. Towler and R. J. Needs, "New avenues for quantum Monte Carlo Techniques", CSAR Focus, 12, 7-9 (2004) )

Dario Alfe` and Mike Gillan (UCL)