Brecht Verstichel

Subsystem constraints in variational second order density matrix optimization: curing the dissociative behavior.

A previous study of diatomic molecules revealed that variational second-order density matrix theory has serious problems in the dissociation limit when the N-representability is imposed at the level of the usual two-index (P,Q,G) or even three-index (T(1),T(2)) conditions [H. Van Aggelen et al., Phys. Chem. Chem. Phys. 11, 5558 (2009)]. Heteronuclear molecules tend to dissociate into fractionally charged atoms. In this paper we introduce a general class of N-representability conditions, called subsystem constraints, and show that they cure the dissociation problem at little additional computational cost. As a numerical example the singlet potential energy surface of Be B(+) is studied. The extension to polyatomic molecules, where more subsystem choices can be identified, is also discussed.

Chemical verification of variational second-order density matrix based potential energy surfaces for the N2 isoelectronic series

A variational optimization of the second-order density matrix under the P-, Q-, and G-conditions was carried out for a set of diatomic 14-electron molecules, including N(2), O(2) (2+), NO(+), CO, and CN(-). The dissociation of these molecules is studied by analyzing several chemical properties (dipole moments, population analysis, and bond indices) up to the dissociation limit (10 and 20 A). Serious chemical flaws are observed for the heteronuclear diatomics in the dissociation limit. A careful examination of the chemical properties reveals that the origin of the dissociation problem lies in the flawed description of fractionally occupied species under the P-, Q-, and G-conditions. A novel constraint is introduced that imposes the correct dissociation and enforces size consistency. The effect of this constraint is illustrated with calculations on NO(+), CO, CN(-), N(2), and O(2)(2+).

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

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.

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

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.

Variational Optimization of the Second-Order Density Matrix Corresponding to a Seniority-Zero Configuration Interaction Wave Function.

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 P-, Q-, and 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, N2, and 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.

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

The variational determination of the two-particle density matrix is an interesting, but not yet fully explored technique that allows us 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.

Projector quantum Monte Carlo with matrix product states

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 wave functions 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 density-matrix renormalization group (DMRG) ground-state energy. For the J_1 − J_2 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 1 for the walkers. The computational cost of MPS-AFQMC is then quadratic in the bond dimension of the trial wave function, 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 wave function. We show that for the J_1 − J_2 model on two-dimensional square lattices, a linear extrapolation of the MPS-AFQMC energy with the discarded weight from the DMRG calculation allows us to remove the constrained-path bias. Extensions to other tensor networks are briefly discussed.

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

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.

Variational density matrix optimization using semidefinite programming

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.

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

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.