Variational optimization of second order density matrices for electronic structure calculation

The exponential growth of the dimension of the exact wavefunction with the size of a chemical system makes it impossible to compute chemical properties of large chemical systems exactly. A myriad of ab initio methods that use simpler mathematical objects to describe the system has thrived on this realization, among which the variational second order density matrix method. The aim of my thesis has been to evaluate the use of this method for chemistry and to identify the major theoretical and computational challenges that need to be overcome to make it successful for chemical applications. The major theoretical challenges originate from the need for the second order density matrix to be N-representable: it must be derivable from an ensemble of N-electron states. Our calculations have pointed out major drawbacks of commonly used necessary N-representability conditions, such as incorrect dissociation into fractionally charged products and size-inconsistency. We have derived subspace energy constraints that fix these problems, albeit in an ad-hoc manner. Additionally, we have found that standard constraints on spin properties cause serious problems, such as false multiplet splitting and size-inconsistency. The subspace constraints relieve these problems as well, though only in the dissociation limit. The major computational challenges originate from the method’s formulation as a vast semidefinite optimization problem. We have implemented and compared several algorithms that exploit the specific structure of the problem. Even so, their slow speed remains prohibitive. Both the second order methods and the zeroth order boundary point method that we tried performed quite similar, which suggests that the underlying problem responsible for their slow convergence, ill-conditioning due to the singularity of the optimal matrix, manifests itself in all these algorithms even though it is most explicit in the barrier method. Significant progress in these theoretical and computational aspects is needed to make the variational second order density matrix method competitive to comparable wavefunction based methods

Exchange-Correlation Energy from Pairing Matrix Fluctuation and the Particle-Particle Random Phase Approximation

We formulate an adiabatic connection for the exchange-correlation energy in terms of pairing matrix fluctuation. This connection opens new channels for density functional approximations based on pairing interactions. Even the simplest approximation to the pairing matrix fluctuation, the particle-particle Random Phase Approximation (pp-RPA), has some highly desirable properties. It has no delocalization error with a nearly linear energy behavior for systems with fractional charges, describes van der Waals interactions similarly and thermodynamic properties significantly better than particle-hole RPA, and eliminates static correlation error for single-bond systems. Most significantly, the pp-RPA is the first known functional that has an explicit and closed-form dependence on the occupied and unoccupied orbitals and captures the energy derivative discontinuity in strongly correlated systems. These findings illlustrate the potential of including pairing interactions within a density functional framework

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

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.Comment: 28 pages, 10 figure

Variational density matrix optimization using semidefinite programming

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.Comment: corrected typos, added do

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 (T1, T2) 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 BeB+ is studied. The extension to polyatomic molecules, where more subsystem choices can be identified, is also discussed.Comment: published version;added reference

Many-electron expansion: A density functional hierarchy for strongly correlated systems

Density functional theory (DFT) is the de facto method for the electronic structure of weakly correlated systems. But for strongly correlated materials, common density functional approximations break down. Here, we derive a many-electron expansion (MEE) in DFT that accounts for successive one-, two-, three-, ... particle interactions within the system. To compute the correction terms, the density is first decomposed into a sum of localized, nodeless one-electron densities (ρ_{i}). These one-electron densities are used to construct relevant two- (ρ_{i}+ρ_{j}), three- (ρ_{i}+ρ_{j}+ρ_{k}), ... electron densities. Numerically exact results for these few-particle densities can then be used to correct an approximate density functional via any of several many-body expansions. We show that the resulting hierarchy gives accurate results for several important model systems: the Hubbard and Peierls-Hubbard models in 1D and the pure Hubbard model in 2D. We further show that the method is numerically convergent for strongly correlated systems: applying successively higher order corrections leads to systematic improvement of the results. MEE thus provides a hierarchy of density functional approximations that applies to both weakly and strongly correlated systems.National Science Foundation (U.S.) (NSF (CHE-1464804))David & Lucile Packard Foundation (grant

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

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.Comment: 19 pages, 7 figures, 3 tables v2: corrected typo in Eq. (52