Two-electron Reduced Density Matrices Research Articles

On the notion of strong correlation in electronic structure theory.

Strong correlation has been said to have many faces, and appears to have many synonyms of questionable suitability. In this work we aim not to define the term once and for all, but to highlight one possibility that is both rigorously defined and physically transparent, and remains so in reference to molecules and quantum lattice models. We survey both molecular examples - hydrogen systems (Hn, n = 2, 4, 6), Be2, H-He-H, and benzene - and the half-filled Hubbard model over a range of correlation regimes. Various quantities are examined including the extent of spin symmetry breaking in correlated single-reference wave functions, energetic ratios inspired by the Hubbard model and the Virial theorem, and metrics derived from the one- and two-electron reduced density matrices (RDMs). The trace and the square norm of the cumulant of the two-electron reduced density matrix capture what may well be defined as strong correlation. Accordingly, strong correlation is understood as a statistical dependence between two electrons, and is distinct from the concepts of "correlation energy" and more general than entanglement quantities that require a partitioning of a quantum system into distinguishable subspaces. This work enables us to build a bridge between a rigorous and quantifiable regime of strong electron correlation and more familiar chemical concepts such as anti-aromaticity in the context of Baird's rule.

Toward more accurate adiabatic connection approach for multireference wavefunctions.

A multiconfigurational adiabatic connection (AC) formalism is an attractive approach to compute the dynamic correlation within the complete active space self-consistent field and density matrix renormalization group (DMRG) models. Practical realizations of AC have been based on two approximations: (i) fixing one- and two-electron reduced density matrices (1- and 2-RDMs) at the zero-coupling constant limit and (ii) extended random phase approximation (ERPA). This work investigates the effect of removing the "fixed-RDM" approximation in AC. The analysis is carried out for two electronic Hamiltonian partitionings: the group product function- and the Dyall Hamiltonians. Exact reference AC integrands are generated from the DMRG full configuration interaction solver. Two AC models are investigated, employing either exact 1- and 2-RDMs or their second-order expansions in the coupling constant in the ERPA equations. Calculations for model molecules indicate that lifting the fixed-RDM approximation is a viable way toward improving the accuracy of existing AC approximations.

Influence of the N-representability conditions on the variational determination of two-electron reduced density matrices for ground and excited N-electron states in the doubly occupied configuration interaction space.

In this work, we analyze the effectiveness of different sets of well-known necessary N-representability conditions that are imposed on the variational determination of two-electron reduced density matrices corresponding to the ground and excited states of N-electron Hamiltonians within doubly occupied configuration interaction subspaces. The two-electron reduced density matrix elements are determined so that they are suitably related to those of higher-order reduced density matrices and other associated matrices, up to a given order, constituting positive semidefinite matrices. These conditions up to two-, three-, and four-order have been sequentially imposed on the variational calculation of the expectation values of the dispersion operator corresponding to several Hamiltonian models, obtaining the electronic spectra. These results have been contrasted with those arising from the full configuration interaction method, illuminating the sufficiency/insufficiency of the variational N-representability conditions applied.

Efficient Computation of Two-Electron Reduced Density Matrices via Selected Configuration Interaction.

We create an approach to efficiently calculate two-electron reduced density matrices (2-RDMs) using selected configuration interaction wavefunctions. This is demonstrated using the specific example of Monte Carlo configuration interaction (MCCI). The computation of the 2-RDMs is accelerated by using ideas from fast implementations of full configuration interaction (FCI) and recent advances in implementing the Slater-Condon rules using hardware bitwise operations. This method enables a comparison of MCCI and truncated CI 2-RDMs with FCI values for a range of molecules, which includes stretched bonds and excited states. The accuracy in energies, wavefunctions, and 2-RDMs is seen to exhibit a similar behavior. We find that MCCI can reach sufficient accuracy of the 2-RDM using significantly fewer configurations than truncated CI, particularly for systems with strong multireference character.

Variational determination of the two-electron reduced density matrix within the doubly occupied configuration interaction framework: Treatments of triplet N-electron systems.

In this work, we perform variational calculations of two-electron reduced density matrices corresponding to open-shell N-electron systems within the framework of the doubly occupied configuration interaction treatment, traditionally limited to studies of closed-shell systems. This has allowed us to provide a satisfactory description of molecular systems in triplet states following two methods. One of them adds hydrogen atoms at an infinite distance of the triplet system studied, constituting a singlet supersystem. The energies and reduced density matrices of the triplet system are obtained by removing the contributions of the added atoms from the singlet supersystem results. The second procedure involves variational determination of the two-electron reduced density matrices corresponding to the triplet systems by means of adequate couplings of basis-set functions. Both models have been studied by imposing N-representability conditions on the reduced density matrix calculations. Results obtained from these methods for molecular systems in triplet ground states are reported and compared with those provided by benchmark methods.

Efficient Adiabatic Connection Approach for Strongly Correlated Systems: Application to Singlet-Triplet Gaps of Biradicals.

Strong electron correlation can be captured with multireference wave function methods, but an accurate description of the electronic structure requires accounting for the dynamic correlation, which they miss. In this work, a new approach for the correlation energy based on the adiabatic connection (AC) is proposed. The ACn method accounts for terms up to order n in the coupling constant, and it is size-consistent and free from instabilities. It employs the multireference random phase approximation and the Cholesky decomposition technique, leading to a computational cost growing with the fifth power of the system size. Because of the dependence on only one- and two-electron reduced density matrices, ACn is more efficient than existing ab initio multireference dynamic correlation methods. ACn affords excellent results for singlet–triplet gaps of challenging organic biradicals. The development presented in this work opens new perspectives for accurate calculations of systems with dozens of strongly correlated electrons.

Variational determination of ground and excited-state two-electron reduced density matrices in the doubly occupied configuration space: A dispersion operator approach.

This work implements a variational determination of the elements of two-electron reduced density matrices corresponding to the ground and excited states of N-electron interacting systems based on the dispersion operator technique. The procedure extends the previously reported proposal [Nakata et al., J. Chem. Phys. 125, 244109 (2006)] to two-particle interaction Hamiltonians and N-representability conditions for the two-, three-, and four-particle reduced density matrices in the doubly occupied configuration interaction space. The treatment has been applied to describe electronic spectra using two benchmark exactly solvable pairing models: reduced Bardeen-Cooper-Schrieffer and Richardson-Gaudin-Kitaev Hamiltonians. The dispersion operator combined with N-representability conditions up to the four-particle reduced density matrices provides excellent results.

Angular-Momentum Extrapolations to the Complete Basis Set Limit: Why and When They Work.

The leading L-3 dependence of the errors in the energies computed with nuclei-centered basis sets comprising functions with angular momenta not exceeding L is rigorously proven for the 1Σ states of linear molecules and molecular ions with arbitrary even numbers of electrons. This major expansion of the domain of applicability over that offered by the routinely cited Hill asymptotic expression, which is valid only for the helium isoelectronic series, is accomplished with a formalism in which the off-diagonal cusp conditions for the one- and two-electron reduced density matrices play the central role. Despite being provided by these results with theoretical foundations more solid than ever before, the angular-momentum extrapolations to the complete basis set limit appear to work more by happenstance than mathematical rigor due to the poorly predictable variability in the prefactor multiplying the L-3 term and the far from negligible contributions from the terms involving higher powers of L-1.

Geometric constraints on two-electron reduced density matrices

For many-electron systems, the second-order reduced density matrix (2-RDM) provides sufficient information for characterizing their properties of interests in physics and chemistry, ranging from total energy, magnetism, quantum correlation and entanglement to long-range orders. Theoretical prediction of the structural properties of 2-RDM is an essential endeavor in quantum chemistry, condensed matter physics and, more recently, in quantum computation. Since 1960s, enormous progresses have been made in developing RDM-based electronic structure theories and their large-scale computational applications in predicting molecular structure and mechanical, electrical and optical properties of various materials. However, for strongly correlated systems, such as high-temperature superconductors, transition-metal-based biological catalysts and complex chemical bonds near dissociation limit, accurate approximation is still out of reach by currently most sophisticated approaches. This limitation highlights the elusive structural feature of 2-RDM that determines quantum correlation in many-electron system. Here, we present a set of constraints on 2-RDM based on the basic geometric property of Hilbert space and the commutation relations of operators. Numerical examples are provided to demonstrate the pronounced violation of these constraints by the variational 2-RDMs. It is shown that, for a strongly correlated model system, the constraint violation may be responsible for a considerable portion of the variational error in ground state energy. Our findings provide new insights into the structural subtlety of many-electron 2-RDMs.

Long-range-corrected multiconfiguration density functional with the on-top pair density.

We propose a multiconfiguration density functional combining a short-range density functional approximation with a novel long-range correction for dynamic correlation effects. The correction is derived from the adiabatic connection formalism so that the resulting functional requires access only to one- and two-electron reduced density matrices of the system. In practice, the functional is formulated for wavefunctions of the complete active space (CAS) type and the short-range density functional part is made dependent on the on-top pair density via auxiliary spin densities. The latter allows for reducing the self-interaction and the static correlation errors without breaking the spin symmetry. We study the properties and the performance of the non-self-consistent variant of the method, termed lrAC0-postCAS. Numerical demonstration on a set of dissociation energy curves and excitation energies shows that lrAC0-postCAS provides accuracy comparable with more computationally expensive ab initio rivals.

Sparse non-orthogonal wave function expansions from the extension of the generalized Pauli constraints to the two-electron reduced density matrix.

Generalized Pauli constraints (GPCs) impose constraints in the form of inequalities on the natural orbital occupation numbers of the one electron reduced density matrix (1-RDM), defining the set of pure N-representable 1-RDMs, or 1-RDMs that can be derived from an N-electron wave function. Saturation of these constraints is termed "pinning" and implies a significant simplification of the N-electron wave function as the number of Slater determinants required to fully describe the system is reduced. Recent research has shown pinning to occur for the ground states of atoms and molecules with N = 3 and r = 6, where N is the number of electrons and r is the number of spin orbitals. For N = 4 and r = 8, however, pinning occurs not to the GPCs but rather to inequalities defining the pure N-representable two-electron reduced density matrices (2-RDMs). Using these more general inequalities, we derive a wave function ansatz for a system with four electrons in eight spin orbitals. We apply the ansatz to the isoelectronic series of the carbon atom and the dissociation of linear H4 where the correlation energies are recovered to fractions of a kcal/mol. These results provide a foundation for further developments in wave function and RDM theories based on "pinned" solutions, and elucidate a fundamental physical basis for the emergence of non-orthogonal bases in electronic systems of N ≥ 4.

Modeling core-level excitations with variationally optimized reduced-density matrices and the extended random phase approximation.

The information contained within ground-state one- and two-electron reduced-density matrices (RDMs) can be used to compute wave functions and energies for electronically excited states through the extended random phase approximation (ERPA). The ERPA is an appealing framework for describing excitations out of states obtained via the variational optimization of the two-electron RDM (2-RDM), as the variational 2-RDM (v2RDM) approach itself can only be used to describe the lowest-energy state of a given spin symmetry. The utility of the ERPA for predicting near-edge features relevant to x-ray absorption spectroscopy is assessed for the case that the 2-RDM is obtained from a ground-state v2RDM-driven complete active space self-consistent field (CASSCF) computation. A class of killer conditions for the CASSCF-specific ERPA excitation operator is derived, and it is demonstrated that a reliable description of core-level excitations requires an excitation operator that fulfills these conditions; the core-valence separation (CVS) scheme yields such an operator. Absolute excitation energies evaluated within the CASSCF/CVS-ERPA framework are slightly more accurate than those obtained from the usual random phase approximation (RPA), but the CVS-ERPA is not more accurate than RPA for predicting the relative positions of near-edge features. Nonetheless, CVS-ERPA is established as a reasonable starting point for the treatment of core-level excitations using variationally optimized 2-RDMs.

Construction of Fermi Potentials from Electronic Wave Functions.

The Fermi potential, vF(r), is the nonclassical part of the multiplicative effective potential appearing in the one-particle Schrödinger-type equation for the square root of the electron density. The usual way of constructing vF(r) by inverting that equation produces unsatisfactory results when applied to electron densities expanded in Gaussian basis sets. We suggest a different method that is based on an explicit formula for vF(r) in terms of the interacting one- and two-electron reduced density matrices of the system. This method is exact in the basis-set limit and yields accurate approximations to the basis-set-limit vF(r) when applied to reduced density matrices represented in terms of finite basis sets. Illustrative applications involve atomic and molecular wave functions generated at various levels of ab initio theory. It is also shown how to construct the Pauli and exchange-correlation potentials of any system starting with only vF(r).

Correlation Energy from the Adiabatic Connection Formalism for Complete Active Space Wave Functions.

Recently, the adiabatic connection (AC) formula for the electron correlation energy has been proposed for a broad class of multireference wave functions (Pernal, K. Electron Correlation from the Adiabatic Connection for Multireference Wave Functions. Phys. Rev. Lett. 2018, 120, 013001). We show that the AC formula used together with the extended random phase approximation (ERPA) can be efficiently applied to complete active space (CAS) wave functions to recover the remaining electron correlation. Unlike most of the perturbation theory approaches, the proposed AC-CAS method does not require construction of higher than two-electron reduced density matrices, which offers an immediate computational saving. In addition, we show that typically the AC-CAS systematically reduces the errors of both the absolute value of energy and of the energy differences (energy barrier) upon enlarging active spaces for electrons and orbitals. AC-CAS consistently gains in accuracy from including more active electrons. We also propose and study that the performance of the AC0 approach resulting from the first-order expansion of the AC integrand at the coupling constant equal to 0. AC0 avoids solving the full ERPA eigenequation, replacing it with small-dimension eigenproblems, while retaining good accuracy of the AC-CAS method. Low computational cost, compared to AC-CAS or perturbational approaches, makes AC0 the most efficient ab initio approach to capturing electron correlation for the CAS wave functions.

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.

Electron Correlation from the Adiabatic Connection for Multireference Wave Functions

An adiabatic connection (AC) formula for the electron correlation energy is derived for a broad class of multireference wave functions. The AC expression recovers dynamic correlation energy and assures a balanced treatment of the correlation energy. Coupling the AC formalism with the extended random phase approximation allows one to find the correlation energy only from reference one- and two-electron reduced density matrices. If the generalized valence bond perfect pairing model is employed a simple closed-form expression for the approximate AC formula is obtained. This results in the overall M^{5} scaling of the computation cost making the method one of the most efficient multireference approaches accounting for dynamic electron correlation also for the strongly correlated systems.

Reduction of Electronic Wave Functions to Kohn-Sham Effective Potentials.

A method for calculating the Kohn-Sham exchange-correlation potential v(XC)(r) from a given electronic wave function is devised and implemented. It requires on input one- and two-electron reduced density matrices and involves construction of the generalized Fock matrix. The method is free from numerical limitations and basis-set artifacts of conventional schemes for constructing v(XC)(r) in which the potential is recovered from a given electron density, and is simpler than various many-body techniques. The chief significance of this development is that it allows one to directly probe the functional derivative of the true exchange-correlation energy functional and to rigorously test and improve various density-functional approximations.

Molecule-Optimized Basis Sets and Hamiltonians for Accelerated Electronic Structure Calculations of Atoms and Molecules

Molecule-optimized basis sets, based on approximate natural orbitals, are developed for accelerating the convergence of quantum calculations with strongly correlated (multireferenced) electrons. We use a low-cost approximate solution of the anti-Hermitian contracted Schrödinger equation (ACSE) for the one- and two-electron reduced density matrices (RDMs) to generate an approximate set of natural orbitals for strongly correlated quantum systems. The natural-orbital basis set is truncated to generate a molecule-optimized basis set whose rank matches that of a standard correlation-consistent basis set optimized for the atoms. We show that basis-set truncation by approximate natural orbitals can be viewed as a one-electron unitary transformation of the Hamiltonian operator and suggest an extension of approximate natural-orbital truncations through two-electron unitary transformations of the Hamiltonian operator, such as those employed in the solution of the ACSE. The molecule-optimized basis set from the ACSE improves the accuracy of the equivalent standard atom-optimized basis set at little additional computational cost. We illustrate the method with the potential energy curves of hydrogen fluoride and diatomic nitrogen. Relative to the hydrogen fluoride potential energy curve from the ACSE in a polarized triple-ζ basis set, the ACSE curve in a molecule-optimized basis set, equivalent in size to a polarized double-ζ basis, has a nonparallelity error of 0.0154 au, which is significantly better than the nonparallelity error of 0.0252 au from the polarized double-ζ basis set.

Excitation energies from extended random phase approximation employed with approximate one- and two-electron reduced density matrices

Starting from Rowe's equation of motion we derive extended random phase approximation (ERPA) equations for excitation energies. The ERPA matrix elements are expressed in terms of the correlated ground state one- and two-electron reduced density matrices, 1- and 2-RDM, respectively. Three ways of obtaining approximate 2-RDM are considered: linearization of the ERPA equations, obtaining 2-RDM from density matrix functionals, and employing 2-RDM corresponding to an antisymmetrized product of strongly orthogonal geminals (APSG) ansatz. Applying the ERPA equations with the exact 2-RDM to a hydrogen molecule reveals that the resulting (1)Σ(g)(+) excitation energies are not exact. A correction to the ERPA excitation operator involving some double excitations is proposed leading to the ERPA2 approach, which employs the APSG one- and two-electron reduced density matrices. For two-electron systems ERPA2 satisfies a consistency condition and yields exact singlet excitations. It is shown that 2-RDM corresponding to the APSG theory employed in the ERPA2 equations yields excellent singlet excitation energies for Be and LiH systems, and for the N(2) molecule the quality of the potential energy curves is at the coupled cluster singles and doubles level. ERPA2 nearly satisfies the consistency condition for small molecules that partially explains its good performance.

Photoexcited tautomerization of vinyl alcohol to acetylaldehydevia a conical intersection from contracted Schrödinger theory

The photoexcited tautomerization of vinyl alcohol to acetylaldehyde via a conical intersection is analyzed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions to the anti-Hermitian contracted Schrödinger equation (ACSE). The study utilizes the recent generalization of the ACSE method for the treatment of excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A, 2009, 80, 022507]. We computed absolute energies of the critical points as well as various intermediate points along the ground- and excited-state potential energy surface of vinyl alcohol and acetylaldehyde. The ACSE, MCSCF, second-order multireference many-body perturbation theory (MRMP2), and various coupled cluster methods all demonstrate the existence of a family of pathways from vinyl alcohol to acetylaldehyde via a conical intersection that are monotonically decreasing in energy. The conical intersection, proposed for the first time in this paper, is both structurally and energetically similar to the ground-state transition state. We observe a relationship between conical intersections and transition states both in this paper and in our previous work on bicyclobutane's ring conical intersection [J. W. Snyder, Jr. and D. A. Mazziotti, J. Chem. Phys., 2011, 135, 024107]. To treat multireference correlation, we seeded the ACSE with an initial 2-RDM guess from a multiconfiguration self-consistent field (MCSCF) calculation. The ACSE recovers more single-reference correlation energy than MRMP2 and more multireference correlation energy than comparable single-reference wave function methods. The 2-RDMs from the ACSE nearly satisfy all necessary N-representability conditions.