Variational determination of the two-particle density matrix : the case of doubly-occupied space

The world at the level of the atom is described by the branch of science called quantum mechanics. The crown jewel of quantum mechanics is given by the Schrödinger equation which describes a system of indistinguishable particles, that interact with each other. However, an equation alone is not enough: the solution is what interests us. Unfortunately, the exponential scaling of the Hilbert space makes it unfeasible to calculate the exact wave function. This dissertation concerns itself with one of the many ab initio methods that were developed to solve this problem: the variational determination of the second-order density matrix. This method already has a long history. It is not considered to be on par with best ab initio methods. This work tries an alternative approach. We assume that the wave function has a Slater determinant expansion where all orbitals are doubly occupied or empty. This assumption drastically reduces the scaling of the N-representability conditions. The downside is that the energy explicitly depends on the used orbitals and thus an orbital optimizer is needed. The hope is that by using this approximation, we can capture the lion's share of the static correlation and that any missing dynamic correlation can be added through perturbation theory. We developed an algorithm based on Jacobi rotations. The scaling is much more favorable compared to the general case. The method is then tested on a array of benchmark systems

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

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, D$_{\mathsf{SU(2)}}$=2500)/cc-pVDZ level of theory. The contribution of $1s$ core correlation to the $X^1\Sigma_g^+$ bond dissociation curve of the carbon dimer was estimated by comparing energies at the DMRG(36o, 12e, D$_{\mathsf{SU(2)}}$=2500)/cc-pCVDZ and DMRG-SCF(34o, 8e, D$_{\mathsf{SU(2)}}$=2500)/cc-pCVDZ levels of theory.Comment: 16 pages, 13 figure

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 $6\times6$ 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 $2\times2$ and $3\times3$ 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 $6\times6$ 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.Comment: 26 pages, 10 figures, published versio

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

Direct variational determination of the two-electron reduced density matrix for doubly occupied-configuration-interaction wave functions: The influence of three-index N -representability conditions

This work proposes the variational determination of two-electron reduced density matrices corresponding to the ground state of N-electron systems within the doubly occupied-configuration-interaction methodology. The P, Q, and G two-index N-representability conditions have been extended to the T1 and T2 (T2′) three-index ones and the resulting optimization problem has been addressed using a standard semidefinite program. We report results obtained from the doubly occupied-configuration-interaction method, from the two-index constraint variational procedure and from the two- and three-index constraint variational treatment. The discussion of these results along with a study of the computational cost demanded shows the usefulness of our proposal.Fil: Alcoba, Diego Ricardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Torre, Alicia. Universidad del País Vasco; EspañaFil: Lain, Luis. Universidad del País Vasco; EspañaFil: Massaccesi, Gustavo Ernesto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires; ArgentinaFil: Oña, Ofelia Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas; ArgentinaFil: Honore, Eduardo Misael. Universidad de Buenos Aires; ArgentinaFil: Poelmans, Ward. University of Ghent; BélgicaFil: Van Neck, Dimitri. University of Ghent; BélgicaFil: Bultinck, Patrick. University of Ghent; BélgicaFil: De Baerdemacker, Stijn. University of Ghent; Bélgic

Polynomial scaling approximations and dynamic correlation corrections to doubly occupied configuration interaction wave functions

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.Fil: Van Raemdonck, Mario. Ghent University; BélgicaFil: Alcoba, Diego Ricardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Poelmans, Ward. Ghent University; BélgicaFil: De Baerdemacker, Stijn. Ghent University; BélgicaFil: Torre, Alicia. Universidad del País Vasco; EspañaFil: Lain, Luis. Universidad del País Vasco; EspañaFil: Massaccesi, Gustavo Ernesto. Universidad de Barcelona. Facultad de Física. Departamento de Física Fomental; EspañaFil: Van Neck, D.. Ghent University; BélgicaFil: Bultinck, P.. Ghent University; Bélgic

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 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 {\it et al.}, Phys. Rev. A 73, 062505 (2006)], while a more recent study explored so-called subsystem constraints [N. Shenvi {\it 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 {\it 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.Comment: 10 pages, 2 figure

Social and non-social autism symptom and trait domains are genetically dissociable

The core diagnostic criteria for autism comprise two symptom domains – social and communication difficulties, and unusually repetitive and restricted behaviour, interests and activities. There is some evidence to suggest that these two domains are dissociable, yet, this hypothesis has not been tested using molecular genetics. We test this using a GWAS of a non-social autistic trait, systemizing (N = 51,564), defined as the drive to analyse and build systems. We demonstrate that systemizing is heritable and genetically correlated with autism. In contrast, we do not identify significant genetic correlations between social autistic traits and systemizing. Supporting this, polygenic scores for systemizing are significantly positively associated with restricted and repetitive behaviour but not with social difficulties in autistic individuals. These findings strongly suggest that the two core domains of autism are genetically dissociable, and point at how to fractionate the genetics of autism

Social and non-social autism symptoms and trait domains are genetically dissociable

Abstract: The core diagnostic criteria for autism comprise two symptom domains – social and communication difficulties, and unusually repetitive and restricted behaviour, interests and activities. There is some evidence to suggest that these two domains are dissociable, though this hypothesis has not yet been tested using molecular genetics. We test this using a genome-wide association study (N = 51,564) of a non-social trait related to autism, systemising, defined as the drive to analyse and build systems. We demonstrate that systemising is heritable and genetically correlated with autism. In contrast, we do not identify significant genetic correlations between social autistic traits and systemising. Supporting this, polygenic scores for systemising are significantly and positively associated with restricted and repetitive behaviour but not with social difficulties in autistic individuals. These findings strongly suggest that the two core domains of autism are genetically dissociable, and point at how to fractionate the genetics of autism