Lorentz Center - Electronic Structure beyond Density Functional Theory from 12 Jul 2004 through 16 Jul 2004
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### Electronic Structure beyond Density Functional Theoryfrom 12 Jul 2004 through 16 Jul 2004

The interplay between correlation and disorder in Ga1-xMnxAs

The reduced density matrix method for electronic structure calculations, and a new approach for fitting a potential energy surface

Bastiaan J. Braams

Department of Mathematics and Computer Science

Emory University, Atlanta, GA

The presentation will combine the two subjects indicated in the title. The first topic is a renewal of the use of the second order reduced density matrix (2-RDM) for electronic structure calculations. Previous explorations, going back to the early 1970's and culminating in recent work of M. Nakata et al [1], imposed the well-known P, Q, and G conditions of Coleman and Garrod and Percus. These conditions provide an accuracy that compares favourably with Hartree-Fock, but that is not impressive by the standards of other conventional ab initio methods. In our recent paper [2] we impose additional conditions towards N-representability. These conditions (which we call T1 and T2) are of semidefinite form just as are the P, Q, and G conditions, and they still involve only the 1-RDM and 2-RDM; there is no need for an approximate reconstruction of higher-order RDM's in our approach. We study a variety of molecules, extending the list of Nakata et al., and obtain in all cases an accuracy that is better than that of CISD or CCSD(T) on the same model space using full CI as the benchmark. I will review the reduced density matrix method and the T1 and T2 conditions, and discuss the results, including some more recent work.

The second topic is the construction of global fits for the potential energy surface (PES) of several molecular systems, among them CH5+, H3O2-, H4O2, H5O2+, and C3H3O [3,4]. Part of the novelty of our approach, and key to the success, is to use a functional form that is invariant under the complete symmetry group of permutations of like nuclei. This is technically quite difficult if one goes beyond the first few orders in an expansion, and we rely on the mathematical theory of invariants of finite groups and on a computational algebra system to help generate the codes. The accuracy of these global fits is superb. For example, for H5O2+ all the way up to dissociation into H2O and H3O+ (at energy around 120000/cm) we have an rms error in the fit less than 40/cm. The fitted potential is evaluated on a millisecond timescale, so we can do many long MD or QMC calculations at essentially ab initio accuracy without anywhere near the cost that is normally associated with ab initio MD, or even with a Car-Parrinello treatment. As shown by our treatment of C3H3O (including six different break-up and reaction channels) the work is immediately relevant to the evaluation of ab initio cross-sections for reactions in combustion. I will present the mathematics behind these fitting functions and discuss prospects for further applications.

[1] M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa, "Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm", J. Chem. Phys. 114 (2001) 8282-8292.

[2] Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton and J. K. Percus, "The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions", J. Chem. Phys. 120 (2004) 2095-2104.

[3] A. B. McCoy, B. J. Braams, A. Brown, X. C. Huang, Z. Jin, and J. M. Bowman, "Ab initio diffusion Monte Carlo calculations of the quantum behavior of CH5+ in full dimensionality", J. Phys. Chem., in press.

[4] X. Huang, B. J. Braams, S. Carter, and J. M. Bowman, "Quantum Calculations of Vibrational Energies of H3O2- on an ab Initio Potential", J. Amer. Chem. Soc., in press.

The interplay between correlation and disorder in Ga1-xMnxAs

L. Chioncel, A.I. Lichtenstein and M.I. Katsnelson

The behavior of correlated electrons in disordered systems is investigated using a combination between the coherent potential approximation (CPA) and the dynamical mean field theory (DMFT). We carried out finite temperature model calculations for doped magnetic semiconductor Ga1-xMnxAs, in the framework of the developed CPA+DMFT approach. The magnetic transition temperature T-c as a function of carrier density and Mn concentration is discussed.

# 'Self-Interaction Correction in Multiple Scattering Theory'

Martin Lueders

We propose a simplified version of self-interaction corrected local spin-density (SIC-LSD) approximation, based on multiple scattering theory, which implements self-interaction correction locally, within the KKR method. The multiple scattering aspect of this new SIC-LSD method allows for a straightforward generalization to the coherent potential approximation (CPA). This facilitates applications of the SIC to alloys and pseudoalloys which could describe disordered local moment systems, as well as intermediate valences. This new method is a first step towards a dynamical SIC, a method which will allow for dynamical valence and spin fluctuations. As a demonstration of the method, we study the well-known $\alpha$-$\gamma$ phase transition in Ce, where we also explain how SIC operates in terms of multiple scattering theory.

# Is all-electron QMC for heavy atoms feasible?

A. Ma, M.D. Towler, N. Drummond and R. Needs

We investigate whether QMC methods can be reasonably applied to calculate the atomic energies of noble gas atoms He, Ne, Ar, Kr and Xe. In previous works, QMC methods have been applied to study atoms up to atomic number Z = 10 [1, 2], but very few calculations have been reported for heavier atoms. Variational Monte Carlo (VMC) and di®usion Monte Carlo (DMC) calculations are performed by using numerical orbitals calculated on a radial grid and orbitals expanded in a Gaussian basis set. We also study the e®ect of the cusp radius on the calculated energy and variance, as well as the relativistic corrections to the atomic energies using perturbation theory. The study is carried out using CASINO [3].

References

[1] C.E. Campbell and E. Krotschecke and T. Pang, Phys. Rep., 223, 1 (1992).

[2] C.J. Huang and C.J. Umrigar and M.P. Nightingale, J. Chem. Phys., 107, 3007 (1997)

[3] R.J. Needs and M.D. Towler and N.D. Drummond and P.R.C. Kent, CASINO version 1.7 User Manual, University of Cambridge (2004)

# "First-Principles Phase Diagram for Ce-Th System"

Alex Landa and Per Soderlind (1), Andrei Ruban and Levente Vitos (2),

Leonid Pourovskii (3)

{\it Ab initio} total energy calculations based on the exact muffin-tin orbitals (EMTO) theory are used to determine the high pressure and low temperature phase diagram of Ce and Th metals as well as the Ce$_{43}$Th$_{57}$ disordered alloy. The compositional disorder for the alloy is treated in the framework of the coherent potential approximation (CPA). Equation of state for Ce, Th and Ce$_{43}$Th$_{57}$ has been calculated up to 1 Mbar in good comparison with experimental data: upon compression the Ce-Th system undergoes crystallographic phase transformation from an fcc to a bct structure and the transition pressure increases with Th content in the alloy.

(1) Physics and Advances Technologies, Lawrence Livermore National Laboratory, University of California, P.O. Box 808, Livermore, CA 94550

(2) Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, SE-10044, Stockholm, Sweden

(3) Electronic Structure of Materials, Department of Theoretical Physics, University of Nijmegen, 6525ED, Nijmegen, The Netherlands

Using Zero Temperature Quantum Monte Carlo for Impurity Problems Applied to Dynamical Mean Field Theory

This work was supported by the Emmy Noether program of the DFG.

M. Feldbacher, K. Held and F.F. Assaad

Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart Universität Würzburg, Institut fïr Theoretische Physik I, Am Hubland, 97074 Würzburg

In recent years there has been a revival of interest in Kondo-like physics, in particular in quantum dot systems and in connection with the dynamical mean field theory (DMFT). The numerical solution of the underlying Anderson impurity models is, however, limited: In the Numerical Renormalization Group treatment the effort grows exponentially with the number of orbitals, allowing not more than two interacting orbitals; the Hirsch-Fye Quantum Monte Carlo (QMC) algorithm on the other hand scales like $T^{-3}$ ($T$: temperature) and quickly becomes too expensive in CPU time. This limitation is especially severe when DMFT is used to model materials with strong electron correlations where, in order to observe the physics of interest, low temperatures need to be achieved.

We propose a projective QMC algorithm for the Anderson impurity model which converges rapidly to the ground state. With this new impurity solver we study the Mott-Hubbard metal-insulator transition in the Hubbard model, demonstrating that it gives reliable $T=0$'' DMFT results.

## Finite-temperature effects on correlation of electrons in quantum dots

Tapio Rantala

The properties of few-electron quantum dots, e.g. at heterojunction interfaces, are important to understand for the development of novel semiconductor technology. The size and shape of the confining potential, and the effective mass of charge carriers can be adjusted. Thus, the quantum dots are convenient for device design and fascinating for theoretical studies.

We apply path-integral Monte Carlo simulation method to investigate the properties of a two electron quantum dot. We evaluate the one-electron distributions and two-electron correlation functions, and temperature effects on both. Furthermore, we resolve the finite-temperature mixed states to contributing pure states, and by that, we are able to consider the transition energies, and thus, the optical responce of charge carriers. Also, the correlation effect on transitions is discussed.

Increasing temperature broadens the one-electron distributions, as expected, the effect being smaller for two correlated electrons than for a single one.  The simulated one and two electron distributions are also compared to those from experiments and other theoretical (0 K) methods.

GW self-consistent calculations

V.A. Popa, G. Brocks, P.J. Kelly

The GW approximation has been very successful in describing single particle excitations in inorganic semiconductors and insulators and correcting the notorious underestimation by LDA of fundamental band gaps. LDA wave functions and eigenvalues are used as starting point for GW calculations. When the system under consideration is wrongly predicted by LDA to be metallic, the standard perturbative approach of GW is no longer effective, because of the spuriously large screening. We propose as a solution to this problem to open a gap in the LDA energy spectrum and iterate the GW procedure on eigenvalues to self-consistency. In addition, the influence of semicore polarization and the effect of iterating the quasiparticle wave functions are investigated. The different schemes are compared by analyzing the results obtained for conventional semiconductors whose fundamental band gaps are well known from experiment.

# Diffusion Monte Carlo Study of Hydrogen Bonded Model Systems

Martin Fuchs, Alexander Badinski, Claudia Filippi [*], Joel Ireta, and Matthias Scheffler

Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany

[*] Instituut Lorentz for Theoretical Physics, Univ. Leiden, Leiden, The Netherlands

Hydrogen bonds play a key role in the structure and functionality of biomolecules. Density-functional theory (DFT) can be a useful tool for describing such systems. Still, the accuracy of DFT is limited due to its in practice approximate account of exchange-correlation. Quantum-Monte-Carlo (QMC) calculations may help to identify (and correct) such shortcomings, even in larger systems where traditional quantum chemical methods are impractical. Applying the Diffusion Monte Carlo (DMC) method we calculate model systems with medium-weak H-bonds: (i) Di-ammonia and formamide-water, where we examine equilibrium states and proton transfer; (ii) formamide chains, where we address the H-bonds' cooperativity dependent on the chain length; (iii) N-dimethylformamide, where the H-bonds need not be aligned and e.g. the Perdew-Burke-Ernzerhof gradient functional gives an unusually large underestimate of the H-bond strength. For these systems, our QMC results agree well with available post Hartree-Fock quantum chemical data. Furthermore we (iv) evaluate the H-bonding and stacking interactions in the adenine-thymine nucleic acid-base pair, where for stacked conformations DFT fails qualitatively yet also Moeller-Plesset (MP2) data differ from those obtained in a Coupled Cluster approach. In this case our QMC results conform with those from Coupled Cluster, suggesting that a reliable description of non-covalent interactions requires a fully correlated approach. The present study shows that QMC can be a productive tool for benchmarking standard ab initio total energy approaches also for non-covalently bonded systems.

# Electron localization in QMC

Anthony Scemama, Patrick Chaquin, Michel Caffarel, Andreas Savin,

Laboratoire de Chimie Théorique, UMR 7616 du CNRS,

Université Pierre et Marie Curie Paris VI,

Case 137, 4, place Jussieu 75252 PARIS Cedex 05, France}

We introduce two new methods to localize electrons within the QMC framework. For a N electron system, the first method consists in finding the volume which maximizes a given probability P(m) of finding m electrons inside the volume and N-m outside it. The second method is based upon a local function, the Electron Pair Localization Function, depending on the electron-electron distances and their spins.

The two major advantages of these methods are

i.) the simplicity and generality of their definition;

ii.) the possibility of calculating them with quantum Monte Carlo at various levels of accuracy (Hartree-Fock, MCSCF, VB, DFT, VMC, DMC...).

Some applications of these localization functions to simple molecular systems are presented. Results illustrate that these functions are simple and practical tools for visualizing electronic localization in molecules.

# Diffusion Monte-Carlo for high-pressure silicon phases

Richard G. Hennig [a], Cyrus J. Umrigar [b], and John W. Wilkins [a]

[a] Department of Physics, Ohio State University

[b] Cornell Theory Center, Cornell University

Silicon displays at least 11 crystallographic phases at pressures up to 250 GPa. Under pressure the coordination number increases from four (diamond) to twelve (hcp and fcc). While density functional calculations (DFT) correctly predict the structural sequence, they underestimate defect formation energies by more than 1 eV compared to diffusion Monte-Carlo calculations (DMC) [1]. To understand these discrepancies we perform DMC calculations for high-pressure silicon phases. Our DMC reproduces the experimental cohesive energy for the diamond structure if zero-point motion is taken into account. The DMC examines the accuracy of DFT for the silicon phases: diamond, beta-tin, BC8, and simple hexagonal. LDA predicts energies 0.6 eV/atom too low. Relative to the energy of the diamond ground state, the other phases are systematically too low in both LDA and GGA. The discrepancy increases for higher energy structures. The largest error is observed for the beta-Sn structure; LDA overestimates the stability by 50% and GGA by 30% relative to DMC. This is consistent with the finding the Needs group [1]: current approximations of DFT underestimate the formation energy of excited configurations relative to DMC. The LDA behaviour is because LDA tends to favor uniform densities, and the high pressure phases have higher coordination and therefore more uniform densities than diamond.

[1] W.-K. Leung, R. J. Needs, G. Rajagopal, S. Itoh, and S. Ihara. Phys. Rev. Lett. 83, 2351 (1999).

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