Strong correlations via constrained-pairing mean-field theory

Møller-Plesset Adiabatic Connection at Large Coupling Strengths for Open-Shell Systems

We study the adiabatic connection that has as weak-coupling expansion the Møller-Plesset perturbation series, generalizing to the open-shell case previous closed-shell results for the large-coupling limit. We first focus on the hydrogen atom with fractional spins, providing results along the adiabatic connection from small to large coupling strengths. We reveal an intriguing phase diagram and an equation for the large-coupling leading order that has closed-form solutions for specific choices of its relevant quantum numbers. We then show that the hydrogen atom results provide variational estimates for the large-coupling leading terms for the general many-electron open-shell case in terms of functionals of the Hartree-Fock α-spin and β-spin densities.

Particle-Breaking Unrestricted Hartree-Fock Theory for Open Molecular Systems

We recently introduced the particle-breaking restricted Hartree-Fock (PBRHF) model, a mean-field approach to address the fractional charging of molecules when they interact with an electronic environment. In this paper, we present an extension of the model referred to as particle-breaking unrestricted Hartree-Fock (PBUHF). The unrestricted formulation contains odd-electron states necessary for a realistic description of fractional charging. Within the PBUHF parametrization, we use two-body operators as they yield convenient operator transformations. However, two-body operators can change only the particle number by two. Therefore, we include noninteracting zero-energy bath orbitals to generate a linear combination of even and odd electron states. Depending on whether the occupied or virtual orbitals of a molecule interact with the environment, the average number of electrons is either decreased or increased. Without interaction, PBUHF reduces to the unrestricted Hartree-Fock wave function.

Measuring Electron Correlation: The Impact of Symmetry and Orbital Transformations

In this perspective, the various measures of electron correlation used in wave function theory, density functional theory and quantum information theory are briefly reviewed. We then focus on a more traditional metric based on dominant weights in the full configuration solution and discuss its behavior with respect to the choice of the N-electron and the one-electron basis. The impact of symmetry is discussed, and we emphasize that the distinction among determinants, configuration state functions and configurations as reference functions is useful because the latter incorporate spin-coupling into the reference and should thus reduce the complexity of the wave function expansion. The corresponding notions of single determinant, single spin-coupling and single configuration wave functions are discussed and the effect of orbital rotations on the multireference character is reviewed by analyzing a simple model system. In molecular systems, the extent of correlation effects should be limited by finite system size and in most cases the appropriate choices of one-electron and N-electron bases should be able to incorporate these into a low-complexity reference function, often a single configurational one.

Particle-Breaking Hartree-Fock Theory for Open Molecular Systems

In this work we present the particle-breaking Hartree-Fock (PBHF) model which is a mean-field approach to open molecular systems. The interaction of a system with the environment is parametrized through a particle-breaking term in the molecular Hamiltonian. The PBHF wave function is constructed through an exponential unitary transformation of a Slater determinant with a given number of electrons. We consider only the closed-shell formalism. The parametrization results in a linear combination of Slater determinants with different numbers of electrons, i.e., the PBHF wave function is not an eigenfunction of the number operator. As a result, the density matrix may have fractional occupations which are, because of the unitary parametrization, always between 0.0 and 2.0. The occupations are optimized simultaneously with the orbitals, using the trust-region optimization procedure. In the limit of a particle-conserving Hamiltonian, the PBHF optimization will converge to a standard Hartree-Fock wave function. We show that the average number of electrons may be decreased or increased depending on whether the particle-breaking term affects occupied or virtual orbitals.

Correlation-driven phenomena in periodic molecular systems from variational two-electron reduced density matrix theory

Correlation-driven phenomena in molecular periodic systems are challenging to predict computationally not only because such systems are periodically infinite but also because they are typically strongly correlated. Here, we generalize the variational two-electron reduced density matrix (2-RDM) theory to compute the energies and properties of strongly correlated periodic systems. The 2-RDM of the unit cell is directly computed subject to necessary N-representability conditions such that the unit-cell 2-RDM represents at least one N-electron density matrix. Two canonical but non-trivial systems, periodic metallic hydrogen chains and periodic acenes, are treated to demonstrate the methodology. We show that while single-reference correlation theories do not capture the strong (static) correlation effects in either of these molecular systems, the periodic variational 2-RDM theory predicts the Mott metal-to-insulator transition in the hydrogen chains and the length-dependent polyradical formation in acenes. For both hydrogen chains and acenes, the periodic calculations are compared with previous non-periodic calculations with the results showing a significant change in energies and increase in the electron correlation from the periodic boundary conditions. The 2-RDM theory, which allows for much larger active spaces than are traditionally possible, is applicable to studying correlation-driven phenomena in general periodic molecular solids and materials.

Multiconfiguration Pair-Density Functional Theory

Kohn-Sham density functional theory with the available exchange-correlation functionals is less accurate for strongly correlated systems, which require a multiconfigurational description as a zero-order function, than for weakly correlated systems, and available functionals of the spin densities do not accurately predict energies for many strongly correlated systems when one uses multiconfigurational wave functions with spin symmetry. Furthermore, adding a correlation functional to a multiconfigurational reference energy can lead to double counting of electron correlation. Multiconfiguration pair-density functional theory (MC-PDFT) overcomes both obstacles, the second by calculating the quantum mechanical part of the electronic energy entirely by a functional, and the first by using a functional of the total density and the on-top pair density rather than the spin densities. This allows one to calculate the energy of strongly correlated systems efficiently with a pair-density functional and a suitable multiconfigurational reference function. This article reviews MC-PDFT and related background information.

An efficient method for strongly correlated electrons in one dimension

The one-particle reduced density matrix functional theory in its natural orbital functional (NOF) version is used to study strongly correlated electrons. We show the ability of the Piris NOF 7 (PNOF7) to describe non-dynamic correlation effects in one-dimensional (1D) systems. An extensive study of 1D systems that includes Hydrogen (H) chains and the 1D Hubbard model with periodic boundary conditions is provided. Different filling situations and large sizes with up to 122 electrons are considered. Compared to quasi-exact results, PNOF7 is accurate in different correlation regimes for the 1D Hubbard model even away from the half-filling, and maintains its accuracy when the system size increases. The symmetric and asymmetric dissociations of the linear H chain composed of 50 atoms are described to remark the importance of long-range interactions in presence of strong correlation effects. Our results compare remarkably well with those obtained at the density-matrix renormalization group level of theory.

Combining Wave Function Methods with Density Functional Theory for Excited States

We review state-of-the-art electronic structure methods based both on wave function theory (WFT) and density functional theory (DFT). Strengths and limitations of both the wave function and density functional based approaches are discussed, and modern attempts to combine these two methods are presented. The challenges in modeling excited-state chemistry using both single-reference and multireference methods are described. Topics covered include background, combining density functional theory with single-configuration wave function theory, generalized Kohn-Sham (KS) theory, global hybrids, range-separated hybrids, local hybrids, using KS orbitals in many-body theory (including calculations of the self-energy and the GW approximation), Bethe-Salpeter equation, algorithms to accelerate GW calculations, combining DFT with multiconfigurational WFT, orbital-dependent correlation functionals based on multiconfigurational WFT, building multiconfigurational wave functions from KS configurations, adding correlation functionals to multiconfiguration self-consistent-field (MCSCF) energies, combining DFT with configuration-interaction singles by means of time-dependent DFT, using range separation to combine DFT with MCSCF, embedding multiconfigurational WFT in DFT, and multiconfiguration pair-density functional theory.

Iterative stochastic subspace self-consistent field method

We propose a new self-consistent field (SCF) algorithm based on an iterative, partially stochastic "Divide & Conquer"-type approach. This new SCF algorithm is a simple variant of the usual SCF procedure and can be easily implemented in parallel. A detailed description of the algorithm is reported. We illustrate this new method on one-dimensional hydrogen chains and three-dimensional hydrogen clusters. Graphical Abstract Stochastic partition of the molecular orbitals.

Projector Quantum Monte Carlo Method for Nonlinear Wave Functions

We reformulate the projected imaginary-time evolution of the full configuration interaction quantum Monte Carlo method in terms of a Lagrangian minimization. This naturally leads to the admission of polynomial complex wave function parametrizations, circumventing the exponential scaling of the approach. While previously these functions have traditionally inhabited the domain of variational Monte Carlo approaches, we consider recent developments for the identification of deep-learning neural networks to optimize this Lagrangian, which can be written as a modification of the propagator for the wave function dynamics. We demonstrate this approach with a form of tensor network state, and use it to find solutions to the strongly correlated Hubbard model, as well as its application to a fully periodic ab initio graphene sheet. The number of variables which can be simultaneously optimized greatly exceeds alternative formulations of variational Monte Carlo methods, allowing for systematic improvability of the wave function flexibility towards exactness for a number of different forms, while blurring the line between traditional variational and projector quantum Monte Carlo approaches.

A Deterministic Projector Configuration Interaction Approach for the Ground State of Quantum Many-Body Systems

In this work we propose a novel approach to solve the Schrödinger equation which combines projection onto the ground state with a path-filtering truncation scheme. The resulting projector configuration interaction (PCI) approach realizes a deterministic version of the full configuration interaction quantum Monte Carlo (FCIQMC) method [Booth, G. H.; Thom, A. J. W.; Alavi, A. J. Chem. Phys. 2009, 131, 054106]. To improve upon the linearized imaginary-time propagator, we develop an optimal projector scheme based on an exponential Chebyshev expansion in the limit of an infinite imaginary time step. After writing the exact projector as a path integral in determinant space, we introduce a path filtering procedure that truncates the size of the determinantal basis and approximates the Hamiltonian. The path filtering procedure is controlled by one real threshold that determines the accuracy of the PCI energy and is not biased toward any determinant. Therefore, the PCI approach can equally well describe static and dynamic electron correlation effects. This point is illustrated in benchmark computations on N2 at both equilibrium and stretched geometries. In both cases, the PCI achieves chemical accuracy with wave functions that contain less than 0.5% determinants of full CI space. We also report computations on the ground state of C2 with up to quaduple-ζ basis sets and wave functions as large as 200 million determinants, which allow a direct comparison of the PCI, FCIQMC, and density matrix renormalization group (DMRG) methods. The size of the PCI wave function grows modestly with the number of unoccupied orbitals, and its accuracy may be tuned to match that of FCIQMC and DMRG.

Decay Rate of Correlated Real-Space Delocalization Measures: Insights into Chemical Bonding and Mott Transitions from Hydrogen Chains

We study in this contribution the spatial decay rate of real-space localization and delocalization indices in correlated systems. To that end, we examine Hubbard and quantum chemical models of simple cyclic hydrogen chains, showing that all descriptors of delocalization converge quickly toward the infinite chain limits. It is then shown that the localization index may be understood as a generalization of the standard order parameter in Mott insulator transitions and that the origin of the enigmatic sigmoidal profile of delocalization indices in chemical bond-breaking processes lies in the nonlinear mapping between intersite distances and correlation parameters. Although the long-range asymptotic decay of delocalization indices is exponential, we show that as the correlation parameter decreases quantum mechanical interference sets in and a switch to an oscillating pattern, related to core chemical concepts such as resonance or mesomerism, appears.

Theory and practice of uncommon molecular electronic configurations

The electronic configuration of the molecule is the foundation of its structure and reactivity. The spin state is one of the key characteristics arising from the ordering of electrons within the molecule's set of orbitals. Organic molecules that have open-shell ground states and interesting physicochemical properties, particularly those influencing their spin alignment, are of immense interest within the up-and-coming field of molecular electronics. In this advanced review, we scrutinize various qualitative rules of orbital occupation and spin alignment, viz., the aufbau principle, Hund's multiplicity rule, and dynamic spin polarization concept, through the prism of quantum mechanics. While such rules hold in selected simple cases, in general the spin state of a system depends on a combination of electronic factors that include Coulomb and Pauli repulsion, nuclear attraction, kinetic energy, orbital relaxation, and static correlation. A number of fascinating chemical systems with spin states that fluctuate between triplet and open-shell singlet, and are responsive to irradiation, pH, and other external stimuli, are highlighted. In addition, we outline a range of organic molecules with intriguing non-aufbau orbital configurations. In such quasi-closed-shell systems, the singly occupied molecular orbital (SOMO) is energetically lower than one or more doubly occupied orbitals. As a result, the SOMO is not affected by electron attachment to or removal from the molecule, and the products of such redox processes are polyradicals. These peculiar species possess attractive conductive and magnetic properties, and a number of them that have already been developed into molecular electronics applications are highlighted in this review. WIREs Comput Mol Sci 2015, 5:440-459. doi: 10.1002/wcms.1233 For further resources related to this article, please visit the WIREs website.

Chemistry in one dimension

Orbital basis functions in a one-dimensional triatomic molecule ABC.

Optimal composition of atomic orbital basis sets for recovering static correlation energies

Static correlation energies (Estat) are calculated in a range of basis sets for a chemically diverse collection of atoms and molecules. The reliability of a basis set in capturing Estat is assessed according to the following: mean and maximum absolute deviations from near-exact Estat estimates, monotonic convergence to the complete basis set limit, and ability to capture Estat accurately independent of changes in geometry, molecular size, and electronic configuration. Within the polarization and correlation-consistent basis set series, triple-ζ basis sets are the smallest that can reliably capture Estat. The cc-pVTZ basis set performs particularly well, recovering Estat to chemical accuracy for all atoms and molecules in our data set. A series of customized basis sets are constructed by stripping polarization functions from, and swapping polarization functions among, existing basis sets. Basis sets without polarization functions are incapable of accurately recovering Estat. Basis sets with a near-complete set of s, p, and d functions can approach chemical accuracy in maximum absolute error. However, this may be achieved at lower computational cost by using a well balanced triple-ζ basis set including f functions, along with a smaller number of s, p, and d functions. Recommended basis sets for calculating Estat with increasing accuracy at increasing computational cost are 6-311G(2d,2p), cc-pVTZ, and cc-pVQZ stripped of g functions.

Emergence of excited-state plasmon modes in linear hydrogen chains from time-dependent quantum mechanical methods

Explicitly time-dependent configuration-interaction theory is used to predict a new type of plasmonic behavior in linear hydrogen chains. After an intense ultrashort laser pulse brings the system into a broad superposition of excited states, the electronic dipole of the entire chain oscillates coherently, and the system is predicted to emit radiation at energies significantly lower than the first absorption band. A simple classical model accurately predicts the energy of this plasmon resonance for different hydrogen chain lengths and electron densities, demonstrating that collective, free-electron-like behavior can arise in chains of as few as 20 hydrogen atoms. The excitation mechanism for this plasmonic resonance is a highly nonlinear, multiphoton process, different from the linear excitation of ordinary surface plasmons.