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Electronic Structure beyond Density Functional Theory |
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. http://pubs3.acs.org/acs/journals/doilookup?in_doi=10.1021/jp0487096 [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. http://pubs3.acs.org/acs/journals/doilookup?in_doi=10.1021/ja049801i 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). [Back] |