Second-order Reduced Density Matrix Research Articles

Entanglement entropy of the maximum geminal of the BCS ground state

From the viewpoint of the Bose–Einstein condensation (BEC) of the fermion system, the maximum geminal of the second-order reduced density matrix of the superconducting state exactly corresponds to the Cooper pair. In this paper the entanglement entropy (EE) for the maximum geminal of the BCS ground state is evaluated. The EE behaves logarithmically with respect to the number of the maximum geminal. Furthermore, the disappearance point of superconductivity is defined on the basis of the fermion BEC. In the superconducting ground state, almost all electrons in the energy width of the gap parameter near the Fermi level are condensed as a maximum geminal. They suddenly change to normal electrons with a finite gap of the EE at the disappearance point like a first-order phase transition.

Excited States by Coupling Piris Natural Orbital Functionals with the Extended Random-Phase Approximation.

In this work, we explore the use of Piris natural orbital functionals (PNOFs) to calculate excited-state energies by coupling their reconstructed second-order reduced density matrix with the extended random-phase approximation (ERPA). We have named the general method PNOF-ERPA, and specific approaches are referred to as PNOF-ERPA0, PNOF-ERPA1, and PNOF-ERPA2, according to the way the excitation operator is built. The implementation has been tested in the first excited states of H2, HeH+, LiH, Li2, and N2 showing good results compared to the configuration interaction (CI) method. As expected, an increase in accuracy is observed on going from ERPA0 to ERPA1 and ERPA2. We also studied the effect of electron correlation included by PNOF5, PNOF7, and the recently proposed global NOF (GNOF) on the predicted excited states. PNOF5 appears to be good and may even provide better results in very small systems, but including more electron correlation becomes important as the system size increases, where GNOF achieves better results. Overall, the extension of PNOF to excited states has been successful, making it a promising method for further applications.

Cluster decomposition principle and two-electron wave function of the Cooper pair in the BCS superconducting state

We present the explicit forms of the maximum eigenvalue and the corresponding eigenfunction for the second-order reduced density matrix (RDM2) of the BCS superconducting state (SS). Using these quantities, we deal with two topics in the present paper. As the first topic, it is shown that the cluster decomposition principle holds in the BCS-SS. This proof gives a theoretical foundation that the abnormal density can be chosen as the order parameter of the SS. As the second topic, it is shown that such an eigenfunction is spin singlet and spatially extends isotopically, and further that the mean distance of two electrons which consists of the above eigenfunction is in a good agreement with Pippard’s coherence length. This means that maximum geminal of the RDM2 of the BCS-SS can be regarded as the Cooper pair itself which are condensed to the same energy level in a number of

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.

Rigorous Lower Bounds for the Ground State Energy of Molecules by Employing Necessary N-Representability Conditions.

Electronic structure calculations, in particular the computation of the ground state energy, lead to challenging problems in optimization. These problems are of enormous importance in quantum chemistry for calculations of properties of solids and molecules. Minimization methods for computing the ground state energy can be developed by employing a variational approach, where the second-order reduced density matrix defines the variable. This concept leads to large-scale semidefinite programming problems that provide a lower bound for the ground state energy. Upper bounds of the ground state energy can be calculated for example with the Hartree-Fock method or numerically more exact for a given basis set by full CI. However, Nakata et al. ( J. Chem. Phys.200111482828292) observed that due to numerical errors the semidefinite solver produced erroneous results with a lower bound significantly larger than the full CI energy. For the LiH, CH-, NH-, OH, OH-, and HF molecules violations within one mhartree were observed. We applied the software VSDP which takes all numerical errors due to floating-point arithmetic operations into consideration. For two test libraries VSDP provides tight rigorous error bounds lower than full CI energies reported with an accuracy of 0.1 to 0.01 mhartree. Only little computation work must be spent in order to compute close rigorous error bounds for the ground state energy.

NBO 7.0: New vistas in localized and delocalized chemical bonding theory.

We briefly outline some leading features of the newest version, NBO 7.0, of the natural bond orbital (NBO) wavefunction analysis program. Major extensions include: (1) a new NPEPA module implementing Karafiloglou's "polyelectron population analysis" in the NBO framework; (2) new RDM2 program infrastructure for describing electron correlation effects based on full evaluation of the second-order reduced density matrix; (3) improved convex-solver implementation of natural resonance theory (NRT), allowing a greatly expanded range of applications and associated "resonance NBO" (RNBO) visualization of chemical reactivity; (4) a variety of other improvements in well-established NBO algorithms. We also provide brief introduction to the new NBOPro@Jmol utility program, a plugin to the Jmol chemical structure viewer that serves as a convenient tool to provide on-demand NBO descriptors or orbital visualizations for a broad variety of chemical inquiries in research or classroom applications. © 2019 Wiley Periodicals, Inc.

Canonical pair condensation in a flat-band BCS superconductor

The standard approach of the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity is to introduce a self-consistent mean-field approximation, and a variational ansatz for the many-body ground state. The resulting mean-field Hamiltonian no longer commutes with the total number operator, and the variational search takes place in Fock space rather than in a Hilbert space of states with fixed number of particles. This is a disadvantage when studying small systems where the canonical ensemble predictions differ from the corresponding grand-canonical results. To remedy this, alternative approaches such as Richardson’s method have been put forward. Here, we derive the exact many-body ground state of a model Hamiltonian corresponding to the deep-BCS or flat-band regime, without having to resort to Richardson’s set of coupled nonlinear equations. This allows to write the exact many-body ground state in a way that makes the difference with the BCS variational wave function particularly clear. We show that the exact wave function consists of a superposition of many-pair states in such a way that the mean-field averaging corresponds to a summation over these many-pair states. This explains why many expectation values calculated with the BCS variational wave function give the same result as when calculated with the exact wave function, even though these wave functions are different. In the canonical (fixed-number) approach, pairing is investigated using the second-order reduced density matrix and calculating its largest eigenvalue. When interpreted as the order parameter of the superconducting state, this can be compared directly to the behavior of the mean-field gap. Finally, we show that a clear difference between the canonical approach and the BCS grand canonical estimates appears when evaluating pair condensate fluctuations as well as the pair entanglement entropy.

Symmetry dependence and universality of practical algebraic functionals in density-matrix-functional theory

When dealing with a fully symmetrical ground state, the symmetry dependence of the universal Hohenberg-Kohn energy functional $F[\ensuremath{\gamma}]$ of the first-order reduced density matrix (RDM) $\ensuremath{\gamma}$ can be conveniently neglected. The situation changes drastically in the case of the dissociation of a symmetrical molecule with the state crossing, in the course of which the potential energy curve of the initial non-fully symmetrical ground state is eventually crossed with that of the fully symmetrical state. In this case, as is demonstrated in the present paper, the second-order RDM ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ in the representation of the natural orbitals (NOs) is symmetry dependent. Since ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ is the goal in the design of ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$ as a functional of NO occupations ${n}$, which is part of a practical density matrix functional $F[\ensuremath{\gamma}],\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$ must also depend on the symmetry, especially the irreducible representation of the symmetry group. The result has immediate implications for study of structural (or phase) transitions based on a single symmetry-independent functional. The demonstration is given in the minimal-base model of the dissociation of the prototype ${\mathrm{H}}_{4}$ molecule in the rhombic structure.

A non-JKL density matrix functional for intergeminal correlation between closed-shell geminals from analysis of natural orbital configuration interaction expansions

Almost all functionals that are currently used in density matrix functional theory have been created by some a priori ansatz that generates approximations to the second-order reduced density matrix (2RDM). In this paper, a more consistent approach is used: we analyze the 2RDMs (in the natural orbital basis) of rather accurate multi-reference configuration interaction expansions for several small molecules (CH4, NH3, H2O, FH, and N2) and use the knowledge gained to generate new functionals. The analysis shows that a geminal-like structure is present in the 2RDMs, even though no geminal theory has been applied from the onset. It is also shown that the leading non-geminal dynamical correlation contributions are generated by a specific set of double excitations. The corresponding determinants give rise to non-JKL (non Coulomb/Exchange like) multipole-multipole dispersive attractive terms between geminals. Due to the proximity of the geminals, these dispersion terms are large and cannot be omitted, proving pure JKL functionals to be essentially deficient. A second correction emerges from the observation that the "normal" geminal-like exchange between geminals breaks down when one breaks multiple bonds. This problem can be fixed by doubling the exchange between bond broken geminals, effectively restoring the often physically correct high-spin configurations on the bond broken fragments. Both of these corrections have been added to the commonly used antisymmetrized product of strongly orthogonal geminals functional. The resulting non-JKL functional Extended Löwdin-Shull Dynamical-Multibond is capable of reproducing complete active space self-consistent field curves, in which one active orbital is used for each valence electron.

Comment on “Nonuniqueness of algebraic first-order density-matrix functionals”

Wang and Knowles (WK) [Phys. Rev. A 92, 012520 (2015)] have given a counterexample to the conventional in reduced density-matrix functional theory representation of the second-order reduced density matrix (2RDM) ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ in the basis of the natural orbitals as a function ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$ of the orbital occupation numbers (ONs) ${n}_{i}$. The observed nonuniqueness of ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ for prototype systems of different symmetry has been interpreted as the inherent inability of ON functions to reproduce the 2RDM, due to the insufficient information contained in the 1RDM spectrum. In this Comment, it is argued that, rather than totally invalidating ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$, the WK example exposes its symmetry dependence which, as well as the previously established analogous dependence in density functional theory, is demonstrated with a general formulation based on the Levy constrained search.

Analytic second-order energy derivatives in natural orbital functional theory

The analytic energy gradients in the atomic orbital representation have recently been published (J. Chem. Phys. 146, 014102, 2017) within the framework of the natural orbital functional theory (NOFT). We provide here an alternative expression for them in terms of natural orbitals, and use it to derive the analytic second-order energy derivatives with respect to nuclear displacements in the NOFT. The computational burden is shifted to the calculation of perturbed natural orbitals and occupancies, since a set of linear coupled-perturbed equations obtained from the variational Euler equations must be solved to attain the analytic Hessian at the perturbed geometry. The linear response of both natural orbitals and occupation numbers to nuclear geometry displacements need only specify the reconstruction of the second-order reduced density matrix in terms of occupation numbers.

Time-dependent geminal method applied to laser-driven beryllium

We introduce the time-dependent geminal method, in which the total wave function is written as an antisymmetrized product of time-dependent geminals. A geminal is a two-electron orbital depending on the coordinates of two electrons, and each geminal is expanded as a sum of products of time-dependent one-electron orbitals. The equation of motion for the geminal coefficients similar to the time-dependent Hartree-Fock equation is derived. The evaluation of the largest eigenvalues of the second-order reduced density matrix is proposed as a way to measure the extent of the intergeminal correlation in a time-dependent wave function. Using the time-dependent geminal method, we simulate the evolution of the time-dependent wave function of a beryllium atom exposed to an intense laser pulse at two different wavelengths, 400 and 10 nm. The results are compared to those obtained by the time-dependent Hartree-Fock method and by the multiconfiguration time-dependent Hartree-Fock method.

Basic Variables to Be Reproduced in the First-Principles Theory for Superconductors: Fluctuation of the Particle Number

We show that the diagonal elements of the second-order reduced density matrix (RDM2) can be chosen as basic variables for describing the superconducting state, instead of the off-diagonal elements of the RDM2 that are usually adopted as basic variables in the density functional scheme. The diagonal elements of the RDM2 are called pair-density (PD), which is explicitly related to the fluctuation of the particle number of the system. In this paper, we argue that the fluctuation of the particle number can become an indication of the superconducting state, and that the density functional scheme in which the PD is chosen as a basic variable would be a promising first-principles theory for superconductors.

Separation of dynamic and nondynamic correlation

The account of electron correlation and its efficient separation into dynamic and nondynamic parts plays a key role in the development of computational methods. In this paper we suggest a physically-sound matrix formulation to split electron correlation into dynamic and nondynamic parts using the two-particle cumulant matrix and a measure of the deviation from idempotency of the first-order density matrix. These matrices are applied to a two-electron model, giving rise to a simplified electron correlation index that (i) depends only on natural orbitals and their occupancies, (ii) can be straightforwardly decomposed into orbital contributions and (iii) splits into dynamic and nondynamic correlation parts that (iv) admit a local version. These expressions are shown to account for dynamic and nondynamic correlation in a variety of systems containing different electron correlation regimes, thus providing the first separation of dynamic and nondynamic correlation using solely natural orbital occupancies.

Kohn-Sham exchange-correlation potentials from second-order reduced density matrices.

We describe a practical algorithm for constructing the Kohn-Sham exchange-correlation potential corresponding to a given second-order reduced density matrix. Unlike conventional Kohn-Sham inversion methods in which such potentials are extracted from ground-state electron densities, the proposed technique delivers unambiguous results in finite basis sets. The approach can also be used to separate approximately the exchange and correlation potentials for a many-electron system for which the reduced density matrix is known. The algorithm is implemented for configuration-interaction wave functions and its performance is illustrated with numerical examples.

Functional <i>N</i>-Representability in 2-Matrix, 1-Matrix, and Density Functional Theories

The N-representability conditions on the reduced second-order reduced density matrix (2-RDM), impose restrictions not only in the context of reduced density matrix theory (RDMT), but also on functionals advanced in one-matrix theory such as natural orbital functional theory (NOFT), and on functionals depending on the one-electron density such as those of density functional theory (DFT). We review some aspects of the applications of these N-representability conditions in these theories and present some conclusions.

The second-order reduced density matrix method and the two-dimensional Hubbard model

The second-order reduced density matrix method (the RDM method) has performed well in determining energies and properties of atomic and molecular systems, achieving coupled-cluster singles and doubles with perturbative triples (CCSD (T)) accuracy without using the wave-function. One question that arises is how well does the RDM method perform with the same conditions that result in CCSD (T) accuracy in the strong correlation limit. The simplest and a theoretically important model for strongly correlated electronic systems is the Hubbard model. In this paper, we establish the utility of the RDM method when employing the P,Q,G,T1 and T2′ conditions in the two-dimensional Hubbard model case and we conduct a thorough study applying the 4×4 Hubbard model employing a coefficients. Within the Hubbard Hamiltonian we found that even in the intermediate setting, where U/t is between 4 and 10, the P, Q, G, T1 and T2′ conditions reproduced good ground state energies.

On the size-consistency of the reduced-density-matrix method and the unitary invariant diagonal N-representability conditions

A promising variational approach for determining the ground state energy and its properties is by using the second-order reduced density matrix (2-RDM). However, the leading obstacle with this approach is the N-representability problem. By employing a subset of conditions (typically the P, Q, G, T1 and T2′ conditions) results comparable to those of CCSD(T) can be achieved. However, these conditions do not guarantee size-consistency. In this work, we show that size-consistency can be satisfied if the 2-RDM satisfies the following conditions: (i) the 2-RDM is unitary invariant diagonal N-representable; (ii) the 2-RDM corresponding to each (unspecified) subsystem is the eigenstate of the number of corresponding electrons; and (iii) the 2-RDM satisfies at least one of the P, Q, G, T1 and T2′ conditions. This is the first time that a computationally feasible (though demanding) sufficient condition for the RDM method that guarantees size-consistency in all chemical systems has been published in the literature.

Density Functional Approach for Calculating the Diagonal Elements of the Second-Order Reduced Density Matrix: Restrictive Conditions on the Kinetic Energy Functional

In order to develop the density functional approach for calculating the diagonal elements of the second-order reduced density matrix, we present the restrictive conditions on the kinetic energy functional. These conditions are useful for devising the approximate form of the kinetic energy functional, which is a key issue for this approach.

Information carriers and (reading them through) information theory in quantum chemistry

This Perspective discusses the reduction of the electronic wave function via the second-order reduced density matrix to the electron density ρ(r), which is the key ingredient in density functional theory (DFT) as a basic carrier of information. Simplifying further, the 1-normalized density function turns out to contain essentially the same information as ρ(r) and is even of preferred use as an information carrier when discussing the periodic properties along Mendeleev's table where essentially the valence electrons are at stake. The Kullback-Leibler information deficiency turns out to be the most interesting choice to obtain information on the differences in ρ(r) or σ(r) between two systems. To put it otherwise: when looking for the construction of a functional F(AB) = F[ζ(A)(r),ζ(B)(r)] for extracting differences in information from an information carrier ζ(r) (i.e. ρ(r), σ(r)) for two systems A and B the Kullback-Leibler information measure ΔS is a particularly adequate choice. Examples are given, varying from atoms, to molecules and molecular interactions. Quantum similarity of atoms indicates that the shape function based KL information deficiency is the most appropriate tool to retrieve periodicity in the Periodic Table. The dissimilarity of enantiomers for which different information measures are presented at global and local (i.e. molecular and atomic) level leads to an extension of Mezey's holographic density theorem and shows numerical evidence that in a chiral molecule the whole molecule is pervaded by chirality. Finally Kullback-Leibler information profiles are discussed for intra- and intermolecular proton transfer reactions and a simple S(N)2 reaction indicating that the theoretical information profile can be used as a companion to the energy based Hammond postulate to discuss the early or late transition state character of a reaction. All in all this Perspective's answer is positive to the question of whether an even simpler carrier of information than the electron density function ρ(r) can be envisaged: the shape function, integrating to 1 by construction fulfils this role. On the other hand obtaining the information (or information difference) contained in one (or two) systems from ρ(r) or σ(r) can be most efficiently done by using information theory, the Kulback-Leibler information deficiency being at the moment (one of) the most advisable functionals.