The density matrix renormalization group for ab initio quantum chemistry

During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. Its underlying wavefunction ansatz, the matrix product state (MPS), is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS, the rank of the decomposition, controls the size of the corner of the many-body Hilbert space that can be reached with the ansatz. This parameter can be systematically increased until numerical convergence is reached. The MPS ansatz naturally captures exponentially decaying correlation functions. Therefore DMRG works extremely well for noncritical one-dimensional systems. The active orbital spaces in quantum chemistry are however often far from one-dimensional, and relatively large virtual dimensions are required to use DMRG for ab initio quantum chemistry (QC-DMRG). The QC-DMRG algorithm, its computational cost, and its properties are discussed. Two important aspects to reduce the computational cost are given special attention: the orbital choice and ordering, and the exploitation of the symmetry group of the Hamiltonian. With these considerations, the QC-DMRG algorithm allows to find numerically exact solutions in active spaces of up to 40 electrons in 40 orbitals.Comment: 24 pages; 10 figures; based on arXiv:1405.1225; invited review for European Physical Journal

DMRG-CASPT2 study of the longitudinal static second hyperpolarizability of all-trans polyenes

We have implemented internally contracted complete active space second order perturbation theory (CASPT2) with the density matrix renormalization group (DMRG) as active space solver [Y. Kurashige and T. Yanai, J. Chem. Phys. 135, 094104 (2011)]. Internally contracted CASPT2 requires to contract the generalized Fock matrix with the 4-particle reduced density matrix (4-RDM) of the reference wavefunction. The required 4-RDM elements can be obtained from 3-particle reduced density matrices (3-RDM) of different wavefunctions, formed by symmetry-conserving single-particle excitations op top of the reference wavefunction. In our spin-adapted DMRG code chemps2 [https://github.com/sebwouters/chemps2], we decompose these excited wavefunctions as spin-adapted matrix product states, and calculate their 3-RDM in order to obtain the required contraction of the generalized Fock matrix with the 4-RDM of the reference wavefunction. In this work, we study the longitudinal static second hyperpolarizability of all-trans polyenes C$_{2n}$H$_{2n+2}$ [n = 4 - 12] in the cc-pVDZ basis set. DMRG-SCF and DMRG-CASPT2 yield substantially lower values and scaling with system size compared to RHF and MP2, respectively.Comment: 9 pages, 4 figure

Perturbations on the superconducting state of metallic nanoparticles: influence of geometry and impurities

The pair condensation energy of a finite-size superconducting particle is studied as a function of two control parameters. The first control parameter is the shape of the particle, and the second parameter is a position-dependent impurity introduced in the particle. Whereas the former parameter is known to induce strong fluctuations in the condensation energy, the latter control parameter is found to be a more gentle probe of the pairing correlations.Comment: Submitted for the special issue of the International Symposium on Small Particles and Inorganic Clusters XVI, at KULeuven (July 8-13, 2012). PACS: 74.78.Na, 74.20.F

Exact solution of the px+ipyp_x + ip_y pairing Hamiltonian by deforming the pairing algebra

The present paper makes a connection between collective bosonic states and the exact solutions of the $p_x + ip_y$ pairing Hamiltonian. This makes it possible to investigate the effects of the Pauli principle on the energy spectrum, by gradually reintroducing the Pauli principle. It also introduces an efficient and stable numerical method to probe all the eigenstates of this class of Hamiltonians.Comment: 18 pages, 10 figure

Faddeev Random Phase Approximation for Molecules

The Faddeev Random Phase Approximation is a Green's function technique that makes use of Faddeev-equations to couple the motion of a single electron to the two-particle--one-hole and two-hole--one-particle excitations. This method goes beyond the frequently used third-order Algebraic Diagrammatic Construction method: all diagrams involving the exchange of phonons in the particle-hole and particle-particle channel are retained, but the phonons are described at the level of the Random Phase Approximation. This paper presents the first results for diatomic molecules at equilibrium geometry. The behavior of the method in the dissociation limit is also investigated

The Dicke model as the contraction limit of a pseudo-deformed Richardson-Gaudin model

The Dicke model is derived in the contraction limit of a pseudo-deformation of the quasispin algebra in the su(2)-based Richardson-Gaudin models. Likewise, the integrability of the Dicke model is established by constructing the full set of conserved charges, the form of the Bethe Ansatz state, and the associated Richardson-Gaudin equations. Thanks to the formulation in terms of the pseudo-deformation, the connection from the su(2)-based Richardson-Gaudin model towards the Dicke model can be performed adiabatically

A new variational, information theory based atoms in molecules method

A new iterative Hirshfeld type AIM method, called Hirshfeld-I-Lambda, is presented. The weight function that defines the AIM is constructed by minimizing the information loss on formation of the molecule, with explicitly requiring that the promolecular densities integrate to the same number of electrons as the AIM densities constructed. The atoms defined by this AIM method are the ones that minimize the information lost upon formation of the molecule out of its isolated atoms. The resulting Hirshfeld-I-Lambda AIM scheme is examined and discussed

Inner products in integrable Richardson-Gaudin models

We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula. This framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.Comment: 21+16 pages, minor revisions compared to the previous versio

Read-Green resonances in a topological superconductor coupled to a bath

We study a topological superconductor capable of exchanging particles with an environment. This additional interaction breaks particle-number symmetry and can be modelled by means of an integrable Hamiltonian, building on the class of Richardson-Gaudin pairing models. The isolated system supports zero-energy modes at a topological phase transition, which disappear when allowing for particle exchange with an environment. However, it is shown from the exact solution that these still play an important role in system-environment particle exchange, which can be observed through resonances in low-energy and -momentum level occupations. These fluctuations signal topologically protected Read-Green points and cannot be observed within traditional mean-field theory.Comment: 7 pages, 4 figure