P. Bultinck

Variational density matrix optimization using semidefinite programming

B. Verstichel, H. van Aggelen, D. Van Neck, P.W. Ayers, P. Bultinck

Computer Physics Communications

182 (9), 2025-2028

2011

A1

Abstract

We discuss how semidefinite programming can be used to determine the second-order density matrix directly through a variational optimization. We show how the problem of characterizing a physical or N-representable density matrix leads to matrix-positivity constraints on the density matrix. We then formulate this in a standard semidefinite programming form, after which two interior point methods are discussed to solve the SDP. As an example we show the results of an application of the method on the isoelectronic series of Beryllium.

Open Access version available at UGent repository

DOI

http://dx.doi.org/10.1016/j.cpc.2010.12.034

Stockholder Projector Analysis: a Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

D. Vanfleteren, D. Van Neck, P. Bultinck, P.W. Ayers, M. Waroquier

Journal of Chemical Physics

136, 014107

2012

A1

Abstract

A previously introduced partitioning of the molecular one-electron density matrix over atoms and bonds [D. Vanfleteren et al., J. Chem. Phys. 133, 231103 (2010)] is investigated in detail. Orthogonal projection operators are used to define atomic subspaces, as in Natural Population Analysis. The orthogonal projection operators are constructed with a recursive scheme. These operators are chemically relevant and obey a stockholder principle, familiar from the Hirshfeld-I partitioning of the electron density. The stockholder principle is extended to density matrices, where the orthogonal projectors are considered to be atomic fractions of the summed contributions. All calculations are performed as matrix manipulations in one-electron Hilbert space. Mathematical proofs and numerical evidence concerning this recursive scheme are provided in the present paper. The advantages associated with the use of these stockholder projection operators are examined with respect to covalent bond orders, bond polarization, and transferability.

DOI

http://dx.doi.org/10.1063/1.3673321

Variational second order density matrix study of F3−: Importance of subspace constraints for size-consistency

H. van Aggelen, B. Verstichel, P. Bultinck, D. Van Neck, P.W. Ayers, D.L. Cooper

Journal of Chemical Physics

134, 054115

2011

A1

Abstract

Variational second order density matrix theory under “two-positivity” constraints tends to dissociate molecules into unphysical fractionally charged products with too low energies. We aim to construct a qualitatively correct potential energy surface for F3− by applying subspace energy constraints on mono- and diatomic subspaces of the molecular basis space. Monoatomic subspace constraints do not guarantee correct dissociation: the constraints are thus geometry dependent. Furthermore, the number of subspace constraints needed for correct dissociation does not grow linearly with the number of atoms. The subspace constraints do impose correct chemical properties in the dissociation limit and size-consistency, but the structure of the resulting second order density matrix method does not exactly correspond to a system of noninteracting units. © 2011 American Institute of Physics

Open Access version available at UGent repository

DOI

http://dx.doi.org/10.1063/1.3532409

A primal-dual semidefinite programming algorithm tailored to the variational determination of the two-body density matrix

B. Verstichel, H. van Aggelen, D. Van Neck, P. Bultinck, S. De Baerdemacker

Computer Physics Communications

182 (6), 1235-1244

2011

A1

Abstract

The quantum many-body problem can be rephrased as a variational determination of the two-body reduced density matrix, subject to a set of N-representability constraints. The mathematical problem has the form of a semidefinite program. We adapt a standard primal–dual interior point algorithm in order to exploit the specific structure of the physical problem. In particular the matrix-vector product can be calculated very efficiently. We have applied the proposed algorithm to a pairing-type Hamiltonian and studied the computational aspects of the method. The standard N-representability conditions perform very well for this problem.
Keywords: Density matrix; Variational; Semidefinite programming

Open Access version available at UGent repository

DOI

http://dx.doi.org/10.1016/j.cpc.2011.02.005

A self-consistent Hirshfeld method for the atom in the molecule based on minimization of information loss

D. Ghillemijn, P. Bultinck, D. Van Neck, P.W. Ayers

Journal of Computational Chemistry

32, 1561-1567

2011

A1

Abstract

Based on the so-called Hirshfeld atom in the molecule scheme, a new AIM method is presented. The method is similar to the Hirshfeld-I scheme, with the AIM weight function being constructed by minimizing the information loss upon formation of the molecule, but now requiring explicitly that the promolecular densities integrate to the same number of electrons as the AIM densities. This new weight function leads to a new iterative AIM scheme, and the resulting operative scheme is examined and discussed. The final results indicate that the newly proposed method does not perform as well as the Hirshfeld-I method.

Open Access version available at UGent repository

DOI

http://dx.doi.org/10.1002/jcc.21734

The significance of parameters in charge equilibration models

T. Verstraelen, P. Bultinck, V. Van Speybroeck, P.W. Ayers, D. Van Neck, M. Waroquier

Journal of Chemical Theory and Computation (JCTC)

7 (6), 1750-1764

2011

A1

Abstract

Charge equilibration models such as the electronegativity equalization method (EEM) and the split charge equilibration (SQE) are extensively used in the literature for the efficient computation of accurate atomic charges in molecules. However, there is no consensus on a generic set of optimal parameters, even when one only considers parameters calibrated against atomic charges in organic molecules. In this work, the origin of the disagreement in the parameters is investigated by comparing and analyzing six sets of parameters based on two sets of molecules and three calibration procedures. The resulting statistical analysis clearly indicates that the conventional least-squares cost function based solely on atomic charges is in general ill-conditioned and not capable of fixing all parameters in a charge-equilibration model. Methodological guidelines are formulated to improve the stability of the parameters. Although in this case a simple interpretation of individual parameters is not possible, charge equilibration models remain of great practical use for the computation of atomic charges.

DOI

http://dx.doi.org/10.1021/ct200006e

Density functional theory study of La2Ce2O7: Disordered fluorite versus pyrochlore structure

D.E.P. Vanpoucke, P. Bultinck, S. Cottenier, V. Van Speybroeck, I. Van Driessche

Physical Review B

84, 054110

2011

A1

Abstract

The crystal structure of lanthanum cerium oxide (La2Ce2O7) is investigated using ab initio density functional theory calculations. The relative stability of fluorite- and pyrochlorelike structures is studied through comparison of their formation energies. These formation energies show the pyrochlore structure to be favored over the fluorite structure, apparently contradicting the conclusions based on experimental neutron and x-ray diffraction (XRD). By calculating and comparing XRD spectra for a set of differently ordered and random structures, we show that the pyrochlore structure is consistent with diffraction experiments. For these reasons, we suggest the pyrochlore structure as the ground-state crystal structure for La2Ce2O7. © 2011 American Physical Society