Quantum Monte Carlo

Introduction

Most of the methods currently used to study the electronic properties of real solids are based on density-functional theory. Although such methods are exact in principle, they rely in practice on approximations to the unknown exchange-correlation functional, which accounts for the complicated correlated motion of the many interacting electrons. The standard local-density and generalised-gradient approximations are very successful on the whole, but fail in a number of important cases. Examples include the overestimation of the cohesive energies of simple solids, the underestimation of s to d transfer energies in transition metal atoms, and difficulties in describing van der Waals bonding. In strongly correlated systems, in particular, it seems unlikely that simple approximate exchange-correlation energy functionals will ever provide a satisfactory quantitative description of the physics.

The quantum Monte Carlo (QMC) methods we use tackle the full many-electron Schrodinger equation for a real solid directly and do not rely on approximate exchange-correlation energy functionals. QMC methods have already been shown capable of calculating cohesive energies, s to d transfer energies, and van der Waals bonding energies. Being non-perturbative and based on full many-electron wave functions, they should also prove useful in studies of strongly correlated systems such as transition metal oxides and high-temperature superconductors.

Scientific Highlights

The CCP9 QMC group has been running for almost 10 years and is among the world leaders in the field. Highlights of our work include the first QMC studies of correlation energy densities in solids [1, 2, 3], the first QMC calculations of the exchange-correlation energy of the relativistic three-dimensional electron gas [4], the first thorough investigations of the accuracy of QMC methods for calculating excitation energies of real solids [5, 6], and the first QMC study of a defect in a real solid [7]. This latter project, which was done in collaboration with researchers from Hitachi Ltd., exposed serious flaws in the density-functional results for the technologically important silicon self-interstitial defect, emphasising the value of QMC when accurate results are required. Our leading position in the field of QMC simulations for solids was recently confirmed by the editors of Reviews of Modern Physics and Advances in Chemical Physics, who commissioned us to write major reviews of the subject, the first of which has now been published [8].

Plans for Future Work

During the next few years we believe that QMC methods will begin to enter the mainstream of electronic structure theory; they will become viewed as a standard part of the electronic structure tool box and will be used by a much wider range of groups interested in a much wider range of problems. For this to happen, the process of doing QMC simulations needs to be made less painful and less time consuming. We aim to lead this broadening process by developing our QMC methods and programs to the stage where any competent electronic structure theorist can use QMC more or less as a black box. We have already made a good deal of progress on this front and have recently made our main QMC code, named CASINO, freely available for academic use (contact Richard Needs for details). The coordinating role of CCP9 has been very important to this project.

In addition, we wish to carry on widening the range of existing applications of QMC. The three projects described below are good examples of the sorts of applications we would like to tackle.

Spin-states of Cytochrome P450: Members of the cytochrome P450 superfamily of enzymes are responsible for the metabolism of a large number of compounds including drugs. The catalytic reaction occurs at a single haem group. Experimentally, it is found that in ligandfree P450 (the resting state) the Fe³⁺ ion of the haem moeity adopts a low-spin state, while on binding of a substrate molecule it adopts a high-spin state. This change in spin character promotes the transfer of an electron in the system. This view is widely accepted, but if we are to use first-principles computational techniques to study such reactions then we have to be sure that our chemical models are accurate. Even the most advanced first-principles work involves two major simplifications: (1) studying only a relatively small part of the system around the haem group, and (2) using DFT or Hartree-Fock methods. Recently, there has been a vigorous debate about the effects of these simplifications [11]. The relative energies of the different spin states vary by several eV depending on whether DFT or Hartree-Fock methods are used, indicating that the treatment of electron correlation is very important. The role of QMC calculations is primarily to establish the correct energies of the different spin states for a realistic model of the haem moeity. We hope to be able to deduce which of the more approximate theories is most suitable for this problem, and to infer whether the model of the haem moeity is adequate or whether the effects of the wider environment must be considered more carefully.

Conductivities from QMC calculations: The effects of correlation on electron transport are among the most studied but least understood phenomena in electronic structure and quantum many-body theory. One of the main theoretical difficulties sounds almost trivial: how can you tell if a material is an insulator or a metal? An experimentalist might answer the question by observing the behaviour of the conductivity as the temperature is reduced. An alternative approach is to study the behaviour of the conductivity as the frequency is reduced at very low temperature; in an ideal metal (in the absence of impurity scattering), the imaginary part of the zero-temperature optical conductivity σ(ω) diverges as ω → 0, while in an insulator it does not. Kohn [9] showed that the strength of this divergence - the quantity that characterises the metallic or insulating behaviour - is related to the change in the total energy of a finite periodic system as the boundary conditions are "twisted". This insight has been used to calculate conductivies for various model systems [10], but has never before been applied to a realistic solid. We believe, however, that it can be applied within QMC simulations, and propose to carry out the first QMC calculations of the conductivities of several simple solids, including aluminium and carbon.

Surface Chemistry: Our recent work on self-interstitial defects in silicon [7] showed that conventional methods such as Kohn-Sham DFT within the local density and generalised gradient approximations are often surprisingly inaccurate when comparing very different types of chemical bonding. This observation has serious implications for much theoretical work on surfaces. We are currently using QMC to calculate the surface energies of jellium (about which there is a current controversy [13]) and Al(111). This will allow us to make an initial estimate of the accuracy of conventional techniques used to study surfaces, but the more ambitious project envisaged here will tell us much more. It is now well established experimentally [14] that at reasonably high coverages Na adsorbs substitutionally on both the Al(001) and Al(111) surfaces. In other words, the Na atoms move into the top layer of the surface, taking the place of Al atoms. Since the energetics that drive this process are subtle, involving the comparison of Na atoms and Al atoms bonded in very different ways, substitutional absorption is a prime candidate for study with QMC methods.

References

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List of Publications, 1997-2001