CECAM/Psi-k Workshop

Diffusion Monte Carlo

CECAM, Lyon, 19-21 September 2002

Abstracts for Friday, 20 September 2002

A method for performing QMC calculations where the computational time required to evaluate the local energy of a configuration of electron coordinates scales linearly with the number of electrons will be presented. Truncated, maximally localized Wannier functions are chosen to represent the single particle orbitals in the Slater determinant part of the many-body wavefunction. This choice or orbitals yields increasingly sparse Slater determinants as the system size is increased. As the evaluation of the Slater determinant and its derivatives is typically the most time intensive part of a QMC calculation, the ability to use sparse operations yields a near linear scaling of the computational time required to evaluate the total energy of a single configuration of electron coordinates. Recent advances to this approach using non-orthogonal Wannier functions will be presented.

The application of these QMC techniques to the evaluation of a variety of optical properties of silicon quantum dots will be presented. The benchmark level of accuracy that can be achieved within the fixed node, diffusion quantum Monte Carlo approach enables QMC calculations to be used as a tool for evaluating the accuracy of more conventional electronic structure techniques. We present a comparison of the size dependence of optical gaps predicted by QMC, density functional and semi-empirical techniques. We also present results of a study of the effects of structural relaxations of the excited state geometry of silicon quantum dots, the Stokes shift, to predict the energy difference between optical absorption and emission in quantum dots.

We use a number of electronic structure methods, including DFT, post-SCF, and QMC to study a range of cluster systems (ionic hydrogen clusters, silicon clusters), surfaces in cluster model (Si(001)), and biological molecules (porphyrines). It is shown that the correlation energy is responsible for ordering of the different potential energy minima in these systems and that the energy differences in ordering may be surprisingly large. The importance of a proper treatment of electronic correlation will be demonstrated on systems where mean-field treatment leads to a total failure of the system description. In the opposite limit, we will discuss on the example of the Si(001) surface whether a correlated method can discern unequivocally an energy difference between two structures (buckled or symmetric dimers on the Si(001) surface) separated in energy by approximately 0.1 eV. These systems will serve to highlight the strengths and limitation.

Jellium systems provide a suitable playground to check the predictions of many-body theories by QMC computations. I present Fixed-Node Diffusion MC results for homogeneous and inhomogeneous systems.

In the case of homogeneous systems, the focus is on the paramagnetic to ferromagnetic transition and on the Wigner crystallization. I present new results, and I discuss the comparison with previous computations.

In the case of inhomogeneous systems, I present results for isolated and for interacting jellium spheres. In the case of isolated spheres, total energy, electron and spin density have bee computed for a wide range of sizes and background densities. The QMC results are compared to the predictions of recent approximations for the exchange and correlation energy proposed by J.P. Perdew and co-workers. Interacting jellium spheres provide benchmark results for non-local correlations in valence electron systems.

Recently, Yan et al. [1] cast doubt on the accuracy of all existing Quantum Monte Carlo studies of jellium surfaces. One possible source of inaccuracy in previous calculations may be an insufficiently detailed treatment of the finite-size errors related to the Coulomb interaction. We have been studying jellium slabs using a different implementation of this interaction, and present the preliminary results of our investigations here.

1. Z. Yan, J. P. Perdew, S. Kurth, C. Fiolhais, and L. Almeida, Phys. Rev. B 61, 2595 (2000).

The most important sources of errors in diffusion Monte Carlo (other than the obvious statistical error) are probably:

1. Time-step error and fixed-node error See my talk.

2. Population control error. It is well known that it is inversely proportional to the size of the target population. In practice it is frequently, but not always, negligible compared to other errors. The error can be estimated using the method of Nightingale and Blote and of Hetherington (also described in my 1993 paper with Nightingale and Runge) even when it is much smaller than the statistical error. This is useful when one is forced to use a small target population.

3. Pseudopotential error. What is the justification for using DFT or HF pseudopotentials in QMC? Claudia Filippi, and, Richard Needs and coworkers have independent evidence that HF pseudopotentials work better in QMC than LDA ones. The Cambridge group is building a library of HF pseudopotentials for use in QMC. How convincing is the evidence? Is there any advantage to using OPM/OEP pseudopotentials instead of HF? Could one add some approximate correlation to either HF or OEP to further improve? What are the prospects for more fundamentally correct approaches, such as that in the Acioli-Ceperley paper, or the semiempirical construction of Lee-Kent-Towler-Needs-Rajagopal? Acioli-Ceperley did not do anything beyond Li. LKTNR had good results for Si at least. Does it help to use core-polarization potentials?

4. Pseudopotential locality error in DMC. Most people just seem to assume that this is small, but what is the evidence for that? I have tested the locality error for the Si₂ dimer by using the same single determinant in conjunction with a sequence of seven progressively better Jastrow functions. The difference in the energy obtained from the 0^th-order Jastrow (no Jastrow) and the 6^th-order Jastrow is 14 mH, which is certainly not negligible. Claudia had, prior to me, observed a 13 mH difference between the energies obtained with an e-e Jastrow and an e-e-n Jastrow, using a different pseudopotential. What is the experience of other researchers?

5. Finite size errors. This error can be considerable for periodic systems. The CASINO code developed by the Needs group implements the Model Periodic Coulomb (MPC) interaction (in addition to the standard Ewald interaction) which reduces the Coulomb component of this error. Ceperley's group has developed the twist-averaged boundary condition method. How well do these work?

Jeff Grossman in a very recent JCP has performed a systematic study of the accuracy of DMC. He calculated the atomization energies of 55 molecules. He finds that the mean absolute deviation from experiment is 2.9 kcal/mol and the largest deviation is 13 kcal/mol and discusses the likely causes of these errors. What would it take to do much better than this?

People who would like to contribute to the discussion should come prepared, if possible, with a couple of viewgraphs. It would he helpful, but not necessary, for me to know before hand who would like to contribute.