Center for Molecular Modeling - P. Bultinck

Variational Hirshfeld Partitioning: General Framework and the Additive Variational Hirshfeld Partitioning Method

toon — Thu, 22 Jan 2026 11:24:40 +0000

F. Heidar-Zadeh, C. Castillo-Orellana, M. Van Zyl, L. Pujal, T. Verstraelen, P. Bultinck, E. Vohringer-Martinez, P. W. Ayers

Journal of Chemical Theory and Computation

20(22), pp.9939--9953

2024

DOI

https://doi.org/10.1021/acs.jctc.4c01077

Constrained iterative Hirshfeld charges: A variational approach

leen — Mon, 09 Jan 2023 09:55:34 +0000

L. Pujal, M. Van Zyl, E. Vohringer-Martinez, T. Verstraelen, P. Bultinck, P.W. Ayers, F. Heidar-Zadeh

Journal of Chemical Physics

Volume 156, Issue 19

2022

Abstract

We develop a variational procedure for the iterative Hirshfeld (HI) partitioning scheme. The main practical advantage of having a variational framework is that it provides a formal and straightforward approach for imposing constraints (e.g., fixed charges on certain atoms or molecular fragments) when computing HI atoms and their properties. Unlike many other variants of the Hirshfeld partitioning scheme, HI charges do not arise naturally from the information-theoretic framework, but only as a reverse-engineered construction of the objective function. However, the procedure we use is quite general and could be applied to other problems as well. We also prove that there is always at least one solution to the HI equations, but we could not prove that its self-consistent equations would always converge for any given initial pro-atom charges. Our numerical assessment of the constrained iterative Hirshfeld method shows that it satisfies many desirable traits of atoms in molecules and has the potential to surpass existing approaches for adding constraints when computing atomic properties.

Published under an exclusive license by AIP Publishing.

DOI

http://dx.doi.org/10.1063/5.0089466

Information-Theoretic Approaches to Atoms-in-Molecules: Hirshfeld Family of Partitioning Schemes

toon — Mon, 26 Nov 2018 15:34:51 +0000

F. Heidar-Zadeh, P.W. Ayers, T. Verstraelen, I. Vinogradov, E. Vohringer-Martinez, P. Bultinck

Journal of Physical Chemistry A

112 (17) 4219-4245

2018

Abstract

Many population analysis methods are based on the precept that molecules should be built from fragments (typically atoms) that maximally resemble the isolated fragment. The resulting molecular building blocks are intuitive (because they maximally resemble well-understood systems) and transferable (because if two molecular fragments both resemble an isolated fragment, they necessarily resemble each other). Information theory is one way to measure the deviation between molecular fragments and their isolated counterparts, and it is a way that lends itself to interpretation. For example, one can analyze the relative importance of electron transfer and polarization of the fragments. We present key features, advantages, and disadvantages of the information-theoretic approach. We also codify existing information-theoretic partitioning methods in a way, that clarifies the enormous freedom one has within the information-theoretic ansatz.

DOI

http://dx.doi.org/10.1021/acs.jpca.7b08966

Method for making 2-electron response reduced density matrices approximately N-representable

caitlin — Thu, 01 Mar 2018 08:49:44 +0000

C. Lanssens, Paul W. Ayers, D. Van Neck, S. De Baerdemacker, K. Gunst, P. Bultinck

Journal of Chemical Physics

148, 8, 084104

2018

Abstract

In methods like geminal-based approaches or coupled cluster that are solved using the projected Schrödinger equation, direct computation of the 2-electron reduced density matrix (2-RDM) is impractical and one falls back to a 2-RDM based on response theory. However, the 2-RDMs from response theory are not $N$-representable. That is, the response 2-RDM does not correspond to an actual physical $N$-electron wave function. We present a new algorithm for making these non-$N$-representable 2-RDMs approximately $N$-representable, i.e., it has the right symmetry and normalization and it fulfills the $P$-, $Q$-, and $G$-conditions. Next to an algorithm which can be applied to any 2-RDM, we have also developed a 2-RDM optimization procedure specifically for seniority-zero 2-RDMs. We aim to find the 2-RDM with the right properties which is the closest (in the sense of the Frobenius norm) to the non-$N$-representable 2-RDM by minimizing the square norm of the difference between this initial response 2-RDM and the targeted 2-RDM under the constraint that the trace is normalized and the 2-RDM, $Q$-matrix, and $G$-matrix are positive semidefinite, i.e., their eigenvalues are non-negative. Our method is suitable for fixing non-$N$-representable 2-RDMs which are close to being $N$-representable. Through the $N$-representability optimization algorithm we add a small correction to the initial 2-RDM such that it fulfills the most important $N$-representability conditions.

DOI

https://doi.org/10.1063/1.4994618

Atom and Bond Fukui Functions and Matrices: A Hirshfeld-I Atoms-in-Molecule Approach

wim — Thu, 12 Jan 2017 09:31:48 +0000

O.B. Ona, O. De Clercq, D.R. Alcoba, A. Torre, L. Lain, D. Van Neck, P. Bultinck

ChemPhysChem

17 (18), 2881–2889

2016

Abstract

The Fukui function is often used in its atom-condensed form by isolating it from the molecular Fukui function using a chosen weight function for the atom in the molecule. Recently, Fukui functions and matrices for both atoms and bonds separately were introduced for semiempirical and ab initio levels of theory using Hückel and Mulliken atoms-in-molecule models. In this work, a double partitioning method of the Fukui matrix is proposed within the Hirshfeld-I atoms-in-molecule framework. Diagonalizing the resulting atomic and bond matrices gives eigenvalues and eigenvectors (Fukui orbitals) describing the reactivity of atoms and bonds. The Fukui function is the diagonal element of the Fukui matrix and may be resolved in atom and bond contributions. The extra information contained in the atom and bond resolution of the Fukui matrices and functions is highlighted. The effect of the choice of weight function arising from the Hirshfeld-I approach to obtain atom- and bond-condensed Fukui functions is studied. A comparison of the results with those generated by using the Mulliken atoms-in-molecule approach shows low correlation between the two partitioning schemes.

DOI

http://dx.doi.org/10.1002/cphc.201600433

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O-a_et_al-2016-ChemPhysChem.pdf

Performance of Shannon-entropy compacted N-electron wave functions for configuration interaction methods

wim — Mon, 01 Aug 2016 10:10:59 +0000

D.R. Alcoba, A. Torre, L. Lain, G. Massaccesi, O.B. Ona, P.W. Ayers, M. Van Raemdonck, P. Bultinck, D. Van Neck

Theoretical Chemistry Accounts

135 (6), 153

2016

Abstract

The coefficients of full configuration interaction wave functions (FCI) for N-electron systems expanded in N-electron Slater determinants depend on the orthonormal one-particle basis chosen although the total energy remains invariant. Some bases result in more compact wave functions, i.e. result in fewer determinants with significant expansion coefficients. In this work, the Shannon entropy, as a measure of information content, is evaluated for such wave functions to examine whether there is a relationship between the FCI Shannon entropy of a given basis and the performance of that basis in truncated CI approaches. The results obtained for a set of randomly picked bases are compared to those obtained using the traditional canonical molecular orbitals, natural orbitals, seniority minimising orbitals and a basis that derives from direct minimisation of the Shannon entropy. FCI calculations for selected atomic and molecular systems clearly reflect the influence of the chosen basis. However, it is found that there is no direct relationship between the entropy computed for each basis and truncated CI energies.

DOI

http://dx.doi.org/10.1007/s00214-016-1905-x

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16-TheorChemAcc-135-153-Alcoba.pdf

When is the Fukui Function Not Normalized? The Danger of Inconsistent Energy Interpolation Models in Density Functional Theory

toon — Thu, 30 Jun 2016 08:08:45 +0000

F. Heidar-Zadeh, R.A. Miranda-Quintana, T. Verstraelen, P. Bultinck, P.W. Ayers, A. Buekenhoudt

Journal of Chemical Theory and Computation (JCTC)

12 (12), 5777–5787

2016

Abstract

When one defines the energy of a molecule with a noninteger number of electrons by interpolation of the energy values for integer-charged states, the interpolated electron density, Fukui function, and higher-order derivatives of the density are generally not normalized correctly. The necessary and sufficient condition for consistent energy interpolation models is that the corresponding interpolated electron density is correctly normalized to the number of electrons. A necessary, but not sufficient, condition for correct normalization is that the energy interpolant be a linear function of the reference energies. Consistent with this general rule, polynomial interpolation models and, in particular, the quadratic E vs N model popularized by Parr and Pearson, do give normalized densities and density derivatives. Interestingly, an interpolation model based on the square root of the electron number also satisfies the normalization constraints. We also derive consistent least-norm interpolation models. In contrast to these models, the popular rational and exponential forms for E vs N do not give normalized electron densities and density derivatives.

DOI

http://dx.doi.org/10.1021/acs.jctc.6b00494

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Maximum probability domains for Hubbard models

claeys — Tue, 01 Mar 2016 09:55:04 +0000

G. Acke, S. De Baerdemacker, P. Claeys, M. Van Raemdonck, W. Poelmans, D. Van Neck, P. Bultinck

Molecular Physics

114 (7-8), 1392-1405

2016

Abstract

The theory of maximum probability domains (MPDs) is formulated for the Hubbard model in terms of projection operators and generating functions for both exact eigenstates as well as Slater determinants. A fast MPD analysis procedure is proposed, which is subsequently used to analyse numerical results for the Hubbard model. It is shown that the essential physics behind the considered Hubbard models can be exposed using MPDs. Furthermore, the MPDs appear to be in line with what is expected from Valence Bond (VB) Theory-based knowledge.

Open Access version available at UGent repository

DOI

http://dx.doi.org/10.1080/00268976.2016.1153742

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MolPhys.00268976.2016.1153742.pdf

Polynomial scaling approximations and Dynamic Correlation Corrections to Doubly Occupied Configuration Interaction wave functions

Ward — Thu, 09 Jul 2015 15:26:35 +0000

M. Van Raemdonck, D. Alcoba, W. Poelmans, S. De Baerdemacker, A. Torre, L. Lain, G. Massaccesi, D. Van Neck, P. Bultinck

Journal of Chemical Physics

143 (10), 104106

2015

Abstract

A class of polynomial scaling methods that approximate Doubly Occupied Configuration Interaction (DOCI) wave functions and improve the description of dynamic correlation is introduced. The accuracy of the resulting wave functions is analysed by comparing energies and studying the overlap between the newly developed methods and full configuration interaction wave functions, showing that a low energy does not necessarily entail a good approximation of the exact wave function. Due to the dependence of DOCI wave functions on the single-particle basis chosen, several orbital optimisation algorithms are introduced. An energy-based algorithm using the simulated annealing method is used as a benchmark. As a computationally more affordable alternative, a seniority number minimising algorithm is developed and compared to the energy based one revealing that the seniority minimising orbital set performs well. Given a well-chosen orbital basis, it is shown that the newly developed DOCI based wave functions are especially suitable for the computationally efficient description of static correlation and to lesser extent dynamic correlation.

Open Access version available at UGent repository

DOI

http://dx.doi.org/10.1063/1.4930260

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Variational optimization of the second order density matrix corresponding to a seniority-zero configuration interaction wave function

Ward — Thu, 09 Jul 2015 15:20:53 +0000

W. Poelmans, M. Van Raemdonck, B. Verstichel, S. De Baerdemacker, A. Torre, L. Lain, G. Massaccesi, D. Alcoba, P. Bultinck, D. Van Neck

Journal of Chemical Theory and Computation (JCTC)

11 (9), 4064–4076

2015

Abstract

We perform a direct variational determination of the second-order (two-particle) density matrix corresponding to a many-electron system, under a restricted set of the two-index $N$-representability $\mathcal{P}$-, $\mathcal{Q}$-, and $\mathcal{G}$-conditions. In addition, we impose a set of necessary constraints that the two-particle density matrix must be derivable from a doubly-occupied many-electron wave function, i.e.\ a singlet wave function for which the Slater determinant decomposition only contains determinants in which spatial orbitals are doubly occupied. We rederive the two-index $N$-representability conditions first found by Weinhold and Wilson and apply them to various benchmark systems (linear hydrogen chains, He, $\text{N}_2$ and $\text{CN}^-$). This work is motivated by the fact that a doubly-occupied many-electron wave function captures in many cases the bulk of the static correlation. Compared to the general case, the structure of doubly-occupied two-particle density matrices causes the associate semidefinite program to have a very favorable scaling as $L^3$, where $L$ is the number of spatial orbitals. Since the doubly-occupied Hilbert space depends on the choice of the orbitals, variational calculation steps of the two-particle density matrix are interspersed with orbital-optimization steps (based on Jacobi rotations in the space of the spatial orbitals). We also point to the importance of symmetry breaking of the orbitals when performing calculations in a doubly-occupied framework.

DOI

http://dx.doi.org/10.1021/acs.jctc.5b00378

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15-JCTC-11(9)4064-Poelmans.pdf