Center for Molecular Modeling - B. Verstichel

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

Projector quantum Monte Carlo with matrix product states

seba — Tue, 24 Jun 2014 09:01:00 +0000

S. Wouters, B. Verstichel, D. Van Neck, G. K.-L. Chan

Physical Review B

90, 045104

2014

Abstract

We marry tensor network states (TNS) and projector quantum Monte Carlo (PMC) to overcome the high computational scaling of TNS and the sign problem of PMC. Using TNS as trial wavefunctions provides a route to systematically improve the sign structure and to eliminate the bias in fixed-node and constrained-path PMC. As a specific example, we describe phaseless auxiliary-field quantum Monte Carlo with matrix product states (MPS-AFQMC). MPS-AFQMC improves significantly on the DMRG ground-state energy. For the J1-J2 model on two-dimensional square lattices, we observe with MPS-AFQMC an order of magnitude reduction in the error for all couplings, compared to DMRG. The improvement is independent of walker bond dimension, and we therefore use bond dimension one for the walkers. The computational cost of MPS-AFQMC is then quadratic in the bond dimension of the trial wavefunction, which is lower than the cubic scaling of DMRG. The error due to the constrained-path bias is proportional to the variational error of the trial wavefunction. We show that for the J1-J2 model on two-dimensional square lattices, a linear extrapolation of the MPS-AFQMC energy with the discarded weight from the DMRG calculation allows to remove the constrained-path bias. Extensions to other tensor networks are briefly discussed.

Open Access version available at UGent repository

DOI

http://dx.doi.org/10.1103/PhysRevB.90.045104

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paper8.pdf

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

seba — Fri, 08 Nov 2013 07:54:37 +0000

B. Verstichel, W. Poelmans, S. De Baerdemacker, S. Wouters, D. Van Neck

European Physical Journal B

87(3), 59

2014

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.

DOI

http://dx.doi.org/10.1140/epjb/e2014-40788-x

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paper5.pdf

Extended random phase approximation method for atomic excitation energies from correlated and variationally optimized second-order density matrices

wim — Wed, 13 Feb 2013 08:33:06 +0000

H. van Aggelen, B. Verstichel, G. Acke, M. Degroote, P. Bultinck, P.W. Ayers, D. Van Neck

Computational and Theoretical Chemistry

1003 (2013), 50-54

2013

DOI

http://dx.doi.org/10.1016/j.comptc.2012.09.036

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13_comp_theoretical_chem_1003_50_Van Aggelen.pdf

The sharp-G N-representability condition

wim — Wed, 13 Feb 2013 08:30:45 +0000

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

Computational and Theoretical Chemistry

1003 (2013), 32-36

2013

Abstract

The G-condition for the N-representability of the two-electron reduced density matrix is tightened by replacing the semidefiniteness constraint with the true upper and lower bounds of the G-type Hamiltonian operator. The lower bound is not easily computed (in contrast to the sharp P- and Q-conditions), but maps onto a well-known integer programming problem. The sharp-G, sharp-P, and sharp-Q conditions are just three members of a much broader class of conditions based on exactly solvable model Hamiltonians.

DOI

http://dx.doi.org/10.1016/j.comptc.2012.09.029

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13_comp_theoretical_chem_1003_32_Johnson.pdf

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

wim — Wed, 13 Feb 2013 08:28:13 +0000

B. Verstichel, H. van Aggelen, W. Poelmans, S. Wouters, D. Van Neck

Computational and Theoretical Chemistry

1003 (2013), 12-21

2013

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

DOI

http://dx.doi.org/10.1016/j.comptc.2012.09.014

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13_comp_theoretical_chem_1003_12_Verstichel.pdf

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

michel — Tue, 17 Apr 2012 16:26:03 +0000

B. Verstichel, H. van Aggelen, W. Poelmans, D. Van Neck

Physical Review Letters

108 (21), 213001

2012

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

DOI

http://dx.doi.org/10.1103/PhysRevLett.108.213001

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12 phys. rev. lett. 108(21)213001 Verstichel.pdf

Considerations on describing non-singlet spin states in variational second order density matrix methods

wim — Wed, 18 Jan 2012 15:16:17 +0000

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

Journal of Chemical Physics

136, 014110

2012

Abstract

Despite the importance of non-singlet molecules in chemistry, most variational second order density matrix calculations have focused on singlet states. Ensuring that a second order density matrix is derivable from a proper N-electron spin state is a difficult problem because the second order density matrix only describes one- and two-particle interactions. In pursuit of a consistent description of spin in second order density matrix theory, we propose and evaluate two main approaches: we consider constraints derived from a pure spin state and from an ensemble of spin states. This paper makes a comparative assessment of the different approaches by applying them to potential energy surfaces for different spin states of the oxygen and carbon dimer. We observe two major shortcomings of the applied spin constraints: they are not size consistent and they do not reproduce the degeneracy of the different states in a spin multiplet. First of all, the spin constraints are less strong when applied to a dissociated molecule than when they are applied to the dissociation products separately. Although they impose correct spin expectation values on the dissociated molecule, the dissociation products do not have correct spin expectation values. Secondly, both under “pure spin state conditions” and under “ensemble spin state” conditions is the energy a convex function of the spin projection. Potential energy surfaces for different spin projections of the same spin state may give a completely different picture of the molecule's bonding. The maximal spin projection always gives the most strongly constrained energy, but is also significantly more expensive to compute than a spin-averaged ensemble. In the dissociation limit, both the problem of nondegeneracy of equivalent spin projections, size-inconsistency and unphysical dissociation can be corrected by means of subspace energy constraints.

DOI

http://dx.doi.org/10.1063/1.3672087

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12 J. Chem. Phys. 136 014110 VanAggelen.pdf

Variational determination of the second-order density matrix for the isoelectronic series of beryllium, neon, and silicon

wim — Mon, 03 Oct 2011 12:16:21 +0000

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

Physical Review A

80 (3), 032508

2009

Abstract

The isoelectronic series of Be, Ne and Si are investigated using a variational determination of the second-order density matrix. A semidefinite program was developed that exploits all rotational and spin symmetries in the atomic system. We find that the method is capable of describing the strong static electron correlations due to the incipient degeneracy in the hydrogenic spectrum for increasing central charge. Apart from the ground-state energy various other properties are extracted from the variationally determined second-order density matrix. The ionization energy is constructed using the extended Koopmans' theorem. The natural occupations are also studied, as well as the correlated Hartree-Fock-like single particle energies. The exploitation of symmetry allows to study the basis set dependence and results are presented for correlation-consistent polarized valence double, triple and quadruple zeta basis sets.

Open Access version available at UGent repository

DOI

http://dx.doi.org/10.1103/PhysRevA.80.032508

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09 phys. rev. A 80(3)032508 verstichel.pdf

Incorrect diatomic dissociation in variational reduced density matrix theory arises from the flawed description of fractionally charged atoms

wim — Mon, 03 Oct 2011 11:47:59 +0000

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

Physical Chemistry Chemical Physics (PCCP)

11 (27), 5558-5560

2009

Abstract

The behaviour of diatomic molecules is examined using the variational second-order density matrix method under the P, Q and G conditions. It is found that the method describes the dissociation limit incorrectly, with fractional charges on the well-separated atoms. This can be traced back to the behaviour of the energy versus the number of electrons for the isolated atoms. It is shown that the energies for fractional charges are much too low.

DOI

http://dx.doi.org/

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09 phys. chem. chem. phys 11(27)5558 van aggelen.pdf