W. Poelmans

Maximum probability domains for Hubbard models

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
A1

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

Reproducibility in density functional theory calculations of solids

K. Lejaeghere, G. Bihlmayer, T. Björkman, P. Blaha, S. Blügel, V. Blum, D. Caliste, I.E. Castelli, S.J. Clark, A. Dal Corso, S. de Gironcoli, T. Deutsch, J.K. Dewhurst, I. Di Marco, C. Draxl, M. Dułak, O. Eriksson, J.A. Flores-Livas, K.F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Grånäs, E.K.U. Gross, A. Gulans, F. Gygi, D.R. Hamann, P.J. Hasnip, N.A.W. Holzwarth, D. Iușan, D.B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. Küçükbenli, Y.O. Kvashnin, I.L.M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordström, T. Ozaki, L. Paulatto, C.J. Pickard, W. Poelmans, M.I.J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunström, A. Tkatchenko, M. Torrent, D. Vanderbilt, M.J. van Setten, V. Van Speybroeck, J.M. Wills, J.R. Yates, G.-X. Zhang, S. Cottenier
Science
351 (6280), 1415-aad3000-7
2016
A1

Abstract 

The widespread popularity of density functional theory has given rise to an extensive range of dedicated codes for predicting molecular and crystalline properties. However, each code implements the formalism in a different way, raising questions about the reproducibility of such predictions. We report the results of a community-wide effort that compared 15 solid-state codes, using 40 different potentials or basis set types, to assess the quality of the Perdew-Burke-Ernzerhof equations of state for 71 elemental crystals. We conclude that predictions from recent codes and pseudopotentials agree very well, with pairwise differences that are comparable to those between different high-precision experiments. Older methods, however, have less precise agreement. Our benchmark provides a framework for users and developers to document the precision of new applications and methodological improvements.

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

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
A1

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

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

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
A1

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.

CheMPS2: Improved DMRG-SCF routine and correlation functions

S. Wouters, W. Poelmans, S. De Baerdemacker, P.W. Ayers, D. Van Neck
Computer Physics Communications
191, 235-237
2015
A1

Abstract 

CheMPS2, our spin-adapted implementation of the density matrix renormalization group (DMRG) for ab initio quantum chemistry (Wouters et al., 2014), has several new features. A speed-up of the augmented Hessian Newton–Raphson DMRG self-consistent field (DMRG-SCF) routine is achieved with the direct inversion of the iterative subspace (DIIS). For extended molecules, the active space orbitals can be localized by maximizing the Edmiston–Ruedenberg cost function. These localized orbitals can be ordered according to the topology of the molecule by approximately minimizing the bandwidth of the exchange matrix with the Fiedler vector. The electronic structure can be analyzed by means of the two-orbital mutual information, spin, spin-flip, density, and singlet diradical correlation functions.

CheMPS2: a free open-source spin-adapted implementation of the density matrix renormalization group for ab initio quantum chemistry

S. Wouters, W. Poelmans, P.W. Ayers, D. Van Neck
Computer Physics Communications
185 (6), 1501-1514
2014
A1

Abstract 

The density matrix renormalization group (DMRG) has become an indispensable numerical tool to find exact eigenstates of finite-size quantum systems with strong correlation. In the fields of condensed matter, nuclear structure and molecular electronic structure, it has significantly extended the system sizes that can be handled compared to full configuration interaction, without losing numerical accuracy. For quantum chemistry (QC), the most efficient implementations of DMRG require the incorporation of particle number, spin and point group symmetries in the underlying matrix product state (MPS) ansatz, as well as the use of so-called complementary operators. The symmetries introduce a sparse block structure in the MPS ansatz and in the intermediary contracted tensors. If a symmetry is non-abelian, the Wigner-Eckart theorem allows to factorize a tensor into a Clebsch-Gordan coefficient and a reduced tensor. In addition, the fermion signs have to be carefully tracked. Because of these challenges, implementing DMRG efficiently for QC is not straightforward. Efficient and freely available implementations are therefore highly desired. In this work we present CheMPS2, our free open-source spin-adapted implementation of DMRG for ab initio QC. Around CheMPS2, we have implemented the augmented Hessian Newton-Raphson complete active space self-consistent field method, with exact Hessian. The bond dissociation curves of the 12 lowest states of the carbon dimer were obtained at the DMRG(28 orbitals, 12 electrons, DSU(2)=2500)/cc-pVDZ level of theory. The contribution of 1s core correlation to the X1Σ+g bond dissociation curve of the carbon dimer was estimated by comparing energies at the DMRG(36o, 12e, DSU(2)=2500)/cc-pCVDZ and DMRG-SCF(34o, 8e, DSU(2)=2500)/cc-pCVDZ levels of theory.

Open Access version available at UGent repository

Critical analysis of the accuracy of models predicting or extracting liquid structure information

M. Van Houteghem, A. Ghysels, T. Verstraelen, W. Poelmans, M. Waroquier, V. Van Speybroeck
Journal of Physical Chemistry B
118 (9), 2451–2470
2014
A1

Abstract 

This work aims at a critical assessment of properties predicting or extracting information on the density and structure of liquids. State-of-the-art NVT and NpT molecular dynamics (MD) simulations have been performed on five liquids: methanol, chloroform, acetonitrile, tetrahydrofuran and ethanol. These simulations allow the computation of properties based on first principles, including the equilibrium density and radial distribution functions (RDFs), characterizing the liquid structure. Refinements have been incorporated in the MD simulations by taking into account Basis Set Superposition Errors (BSSE). An extended BSSE model for an instantaneous evaluation of the BSSE corrections has been proposed, and their impact on the liquid properties has been assessed. If available, the theoretical RDFs have been compared with the experimentally derived RDFs. For some liquids significant discrepancies have been observed and a profound but critical investigation is presented to unravel the origin of these deficiencies. This discussion is focused on tetrahydrofuran where the experiment reveals some prominent peaks completely missing in any MD simulation. Experiments providing information on liquid structure consist mainly of neutron diffraction measurements offering total structure factors as the primary observables. The splitting of these factors in reciprocal space into intra- and intermolecular contributions is extensively discussed, together with their sensitivity in reproducing correct RDFs in coordinate space.

Variational optimization of the 2DM: approaching three-index accuracy using extended cluster constraints

B. Verstichel, W. Poelmans, S. De Baerdemacker, S. Wouters, D. Van Neck
European Physical Journal B
87(3), 59
2014
A1

Abstract 

The reduced density matrix is variationally optimized for the two-dimensional Hubbard model. Exploiting all symmetries present in the system, we have been able to study 6 × 6 lattices at various fillings and different values for the on-site repulsion, using the highly accurate but computationally expensive three-index conditions. To reduce the computational cost we study the performance of imposing the three-index constraints on local clusters of 2 × 2 and 3 × 3 sites. We subsequently derive new constraints which extend these cluster constraints to incorporate the open-system nature of a cluster on a larger lattice. The feasibility of implementing these new constraints is demonstrated by performing a proof-of-principle calculation on the 6 × 6 lattice. It is shown that a large portion of the three-index result can be recovered using these extended cluster constraints, at a fraction of the computational cost.

Extensive v2DM study of the one-dimensional Hubbard model for large lattice sizes: Exploiting translational invariance and parity

B. Verstichel, H. van Aggelen, W. Poelmans, S. Wouters, D. Van Neck
Computational and Theoretical Chemistry
1003 (2013), 12-21
2013
A1

Abstract 

Using variational density matrix optimization with two- and three-index conditions we study the one-dimensional Hubbard model with periodic boundary conditions at various filling factors. Special attention is directed to the full exploitation of the available symmetries, more specifically the combination of translational invariance and space-inversion parity, which allows for the study of large lattice sizes. We compare the computational scaling of three different semidefinite programming algorithms with increasing lattice size, and find the boundary point method to be the most suited for this type of problem. Several physical properties, such as the two-particle correlation functions, are extracted to check the physical content of the variationally determined density matrix. It is found that the three-index conditions are needed to correctly describe the full phase diagram of the Hubbard model. We also show that even in the case of half filling, where the ground-state energy is close to the exact value, other properties such as the spin-correlation function can be flawed.

Open Access version available at UGent repository

Variational two-particle density matrix calculation for the Hubbard model below half filling using spin-adapted lifting conditions

B. Verstichel, H. van Aggelen, W. Poelmans, D. Van Neck
Physical Review Letters
108 (21), 213001
2012
A1

Abstract 

The variational determination of the two-particle density matrix is an interesting, but not yet fully explored technique that allows to obtain ground-state properties of a quantum many-body system without reference to an N-particle wave function. The one-dimensional fermionic Hubbard model has been studied before with this method, using standard two- and three-index conditions on the density matrix [J. R. Hammond et al., Phys. Rev. A 73, 062505 (2006)], while a more recent study explored so-called subsystem constraints [N. Shenvi et al., Phys. Rev. Lett. 105, 213003 (2010)]. These studies reported good results even with only standard two-index conditions, but have always been limited to the half-filled lattice. In this Letter we establish the fact that the two-index approach fails for other fillings. In this case, a subset of three-index conditions is absolutely needed to describe the correct physics in the strong-repulsion limit. We show that applying lifting conditions [J.R. Hammond et al., Phys. Rev. A 71, 062503 (2005)] is the most economical way to achieve this, while still avoiding the computationally much heavier three-index conditions. A further extension to spin-adapted lifting conditions leads to increased accuracy in the intermediate repulsion regime. At the same time we establish the feasibility of such studies to the more complicated phase diagram in two-dimensional Hubbard models.

Open Access version available at UGent repository

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