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Electronic Structure by Quantum Monte Carlo Methods

Lubos Mitas*

National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801

This is an abstract for a presentation given at the
Sixth Foresight Conference on Molecular Nanotechnology.
There will be a link from here to the full article when it is available on the web.


A prediction of electronic structure and related properties of molecular systems is one of the primary goals of chemical physics. Despite the decades of effort of several generations of scientists, efficient, accurate and robust theoretical methods remain on the forefront of current research. In many cases the progress is far from satisfactory and is often overshadowed by new experimental methods which provide higher accuracy or faster way to elucidate the electronic structure puzzles. With the onset of nanotechnology represented, e.g., by quantum dots or molecular carbon systems, the demands for high quality and predictive methods for determining important electronic structure parameters (such as binding and excitation energies, barriers, formation heats for molecular reactions, etc.) are ever increasing.

The traditional electronic structure methods can be divided roughly into two mainstream types of methodologies. The first one is the Density Functional Theory, based on a proof about the existence of a functional which can in principle provide exact ground state energies from one-electron density. In practice, the functional remains elusive and approximations, based on the theory of homogenous electron gas, are used instead. Although the method, being effectively a one-particle approach, is quite efficient, accuracy remains to be a problem. In many cases, errors of the order 10-30 % are common when compared with experiments and in some cases qualitatively incorrect results can be obtained.

The other main direction is based on expansions of wavefunctions in one-particle basis sets and is represented by Hartree-Fock and post-Hartree-Fock methods such as Configuration Interaction or Coupled Custer approaches. Although these approaches are formally exact, in practice, the compuatational demands prevent to achieve an accuracy of 1 kcal/mol for energy differences already for 10-20 valence electrons or so. Application of these methods to systems with periodic boundary conditions such as solids or surfaces is extremely demanding and significant progress is not in sight.

Over the last decade a new promising approach for the electronic structure calculations have emerged in the methodology known as quantum Monte Carlo (QMC). The QMC methods are based on using explictly correlated many-particle variational wavefunctions and stochastic techniques in solving the Schrödinger equation. In this contribution I will describe the basic steps and methods involved:

  1. construction of many-particle wavefunctions and their optimizations
  2. variational Monte Carlo
  3. and fixed-node diffusion Monte Carlo.

Applications will be demonstrated on calculations of several systems which will include molecular reactions, carbon and silicon clusters and insulating solids. The QMC determination of energy ordering for competing isomers of silicon and carbon clusters with up to 20 atoms represents the largest correlated wave function calculations of molecular systems to date which recovered more than 90 % of the valence correlation the energy. I will also show the first QMC calculations of the optical band gaps in insulating solids. The results suggest that overall QMC decreases errors of mean-field methods by a factor of 5 to 10 when compared with available experiments. On the other hand, it can be applied to 100-200 valence electrons enabling thus studies of systems which are out of reach of more traditional quantum chemistry approaches. Besides the high accuracy treatment of electron-electron correlation the QMC approach has several other attractive features:

  1. wide range of applicability
  2. favorable scaling in the number of valence electrons
  3. inherent scalability on parallel architectures.

These advantages together with the obtained results show that the QMC method is becoming a powerful new alternative for ab initio electronic structure calculations.

  • L. Mitas, Electronic structure by Quantum Monte Carlo: atoms, molecules and solids, Computer Physics Communications, 97, 107 (1996)
  • D.M. Ceperley and L. Mitas, Monte Carlo methods in quantum chemistry, Advances in Chemical Physics, Vol. XCIII, Ed. by I. Prigogine and S.A. Rice, Wiley, New York 1996, pp. 1-38.
  • J.C. Grossman and L. Mitas, High accuracy molecular heats of formation and reaction barriers: Essential role of electron correlation, Phys. Rev. Lett. 79, 4353 (1997)
  • J.C. Grossman, L. Mitas, Quantum Monte Carlo Determination of Electronic and Structural Properties of Si_n Clusters (n= < 20), Phys. Rev. Lett. 74, 1323 (1995)
  • J.C. Grossman, L. Mitas, and K. Raghavachari, Structure and Stability of Molecular Carbon: Importance of Electron Correlation, Phys. Rev. Lett. 75, 3870 (1995)
  • L. Mitas and R.M. Martin, Quantum Monte Carlo of nitrogen: atom, dimer, atomic and molecular solids, Phys. Rev. Lett. 72, 2438 (1994)

*Corresponding Address:
Lubos Mitas, National Center for Supercomputing Applications
University of Illinois at Urbana-Champaign
405 N. Mathews Ave., Urbana, IL 61801
Ph.: 217 244 1971, Fax: 217 244 2909


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