First-order Reduced Density Matrix Research Articles

Extending Conceptual Density Functional Theory toward First-Order Reduced Density Matrices: An Open Subsystems Viewpoint on the Fukui Matrix.

As a matrix extension of the Fukui function, a reactivity descriptor grounded within Conceptual Density Functional Theory, the Fukui matrix extends Frontier Molecular Orbital Theory to correlated regimes with its eigendecomposition in Fukui occupations and Fukui naturals. Despite successful applications, the questions remain as to whether replacing a quantity derived from a purely density-based framework by its matrix extension is theoretically well-founded and what chemical information is contained in the corresponding eigendecomposition. In this study, we show that the matrix extension of the Fukui function is only well-defined if one also generalizes the external potential to become nonlocal, leading to the introduction of Conceptual First-Order Reduced Density Matrix Functional Theory. By interpreting the Anderson impurity model from an interacting open subsystem perspective, we show how Fukui occupations and Fukui naturals reflect the influence of an increasing (static) correlation and which characteristic patterns we should expect within a molecular context. This study represents a step in generalizing Conceptual Density Functional Theory beyond its density-based perspective.

Relativistic reduced density matrix functional theory.

As a new approach to efficiently describe correlation effects in the relativistic quantum world we propose to consider reduced density matrix functional theory, where the key quantity is the first-order reduced density matrix (1-RDM). In this work, we first introduce the theoretical foundations to extend the applicability of this theory to the relativistic domain. Then, using the so-called no-pair (np) approximation, we arrive at an approximate treatment of the relativistic effects by focusing on electronic wavefunctions and neglecting explicit contributions from positrons. Within the np approximation the theory becomes similar to the nonrelativistic case, with as unknown only the functional that describes the electron-electron interactions in terms of the 1-RDM. This requires the construction of functional approximations, and we therefore also present the relativistic versions of some common RDMFT approximations that are used in the nonrelativistic context and discuss their properties.

Natural range separation of the Coulomb hole

A natural range separation of the Coulomb hole into two components, one of them being predominant at long interelectronic separations (hcI ) and the other at short distances (hcII ), is exhaustively analyzed throughout various examples that put forward the most relevant features of this approach and how they can be used to develop efficient ways to capture electron correlation. We show that hcI , which only depends on the first-order reduced density matrix, can be used to identify molecules with a predominant nondynamic correlation regime and differentiate between two types of nondynamic correlation, types A and B. Through the asymptotic properties of the hole components, we explain how hcI can retrieve the long-range part of electron correlation. We perform an exhaustive analysis of the hydrogen molecule in a minimal basis set, dissecting the hole contributions into spin components. We also analyze the simplest molecule presenting a dispersion interaction and how hcII helps identify it. The study of several atoms in different spin states reveals that the Coulomb hole components distinguish correlation regimes that are not apparent from the entire hole. The results of this work hold out the promise to aid in developing new electronic structure methods that efficiently capture electron correlation.

Multiconfigurational Effects on the Density Coherence.

This paper is a study of density coherences in multiconfiguration self-consistent field theory and Kohn-Sham (KS) density functional theory. We visualize and compare the nondiagonal elements of the first-order reduced density matrix in the electronic coordinate representation. This is the electronic density coherence, and it also appears in Hartree-Fock (HF) theory in the integrand of the exchange integral. The density coherence is calculated as a function of the internuclear distance for three diatomic molecules (H2, F2, and HF) using both restricted and unrestricted KS and HF theory, as well as the complete active space self-consistent field method. We identify a group of closely associated peaks corresponding to the coherence of electrons on opposite sides of the center of the molecule, and we call this the exchange massif. We find that Slater-determinant methods with a higher percentage of HF exchange tend to underestimate the density coherence at the exchange massif. We explain the trends in terms of a multireference diagnostic and the number of unpaired electrons.

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 single-electron picture based on the multiconfiguration time-dependent Hartree–Fock method: application to the anisotropic ionization and subsequent high-harmonic generation of the CO molecule

The mechanisms of anisotropic near-IR tunnel ionization and high-order harmonic generation (HHG) in a CO molecule are theoretically investigated by using the multiconfiguration time-dependent Hartree–Fock (MCTDHF) method developed for the simulation of multielectron dynamics of molecules. The multielectron dynamics obtained by numerically solving the equations of motion (EOMs) in the MCTDHF method is converted to a single orbital picture in the natural orbital representation where the first-order reduced density matrix is diagonalized. The ionization through each natural orbital is examined and the process of HHG is classified into different optical paths designated by a combinations of initial, intermediate and final natural orbitals. The EOMs for natural spin-orbitals are also derived within the framework of the MCTDHF, which maintains the first-order reduced density matrix to be a diagonal one throughout the time propagation of a many-electron wave function. The orbital dependent, time-dependent effective potentials that govern the dynamics of respective time-dependent natural orbitals are deduced from the derived EOMs, of which the temporal variation can be used to interpret the motion of the electron density associated with each natural spin-orbital. The roles of the orbital shape, multiorbital ionization, linear Stark effect and multielectron interaction in the ionization and HHG of a CO molecule are revealed by the effective potentials obtained. When the laser electric field points to the nucleus O from C, tunnel ionization from the C atom side is enhanced; a hump structure originating from multielectron interaction is then formed on the top of the field-induced distorted barrier of the HOMO effective potential. This hump formation, responsible for the directional anisotropy of tunnel ionization, restrains the influence of the linear Stark effect on the energy shifts of bound states.

Performance of the Density Matrix Functional Theory in the Quantum Theory of Atoms in Molecules

The generalization to arbitrary molecular geometries of the energetic partitioning provided by the atomic virial theorem of the quantum theory of atoms in molecules (QTAIM) leads to an exact and chemically intuitive energy partitioning scheme, the interacting quantum atoms (IQA) approach, that depends on the availability of second-order reduced density matrices (2-RDMs). This work explores the performance of this approach in particular and of the QTAIM in general with approximate 2-RDMs obtained from the density matrix functional theory (DMFT), which rests on the natural expansion (natural orbitals and their corresponding occupation numbers) of the first-order reduced density matrix (1-RDM). A number of these functionals have been implemented in the promolden code and used to perform QTAIM and IQA analyses on several representative molecules and model chemical reactions. Total energies, covalent intra- and interbasin exchange-correlation interactions, as well as localization and delocalization indices have been determined with these functionals from 1-RDMs obtained at different levels of theory. Results are compared to the values computed from the exact 2-RDMs, whenever possible.

Time-dependent density functional theory based Ehrenfest dynamics

Time-dependent density functional theory based Ehrenfest dynamics with atom-centered basis functions is developed in present work. The equation of motion for electrons is formulated in terms of first-order reduced density matrix and an additional term arises due to the time-dependence of basis functions through their dependence on nuclear coordinates. This time-dependence of basis functions together with the imaginary part of density matrix leads to an additional term for nuclear force. The effects of the two additional terms are examined by studying the dynamics of H(2) and C(2)H(4), and it is concluded that the inclusion of these two terms is essential for correct electronic and nuclear dynamics.

Determination of Energies and Electronic Densities of Functional Groups According to Partitionings in the Physical Space

This work formulates our previously reported partitionings of the first-order reduced density matrix and the molecular electronic energy using for both quantities an identical mathematical framework. The procedure provides a consistent and rigorous scheme for extending our algorithms to unions of atomic domains, in order to describe molecular fragments which can be identified as functional groups. Numerical determinations, performed in several series of organic compounds and clusters, support the reliability of our methodology to describe properties of atomic groups.

Description of functional groups by means of domain-restricted reduced density matrices

This report constitutes an application of our previous theoretical works on partitionings of the first-order reduced density matrix according to the atomic domains defined in the theory of atoms in molecules. The numerical determinations obtained reveal that the domain-restricted reduced density matrices, which are the tools resulting from the former treatments, are suitable devices to describe chemical features of molecular fragments. We have focused attention on a study of functional groups in several series of organic compounds confirming the usefulness of these tools.

Decomposition of the First-Order Reduced Density Matrix: An Isopycnic Localization Treatment

In this work, we propose a partitioning of the first-order reduced density matrix corresponding to an N-electron system into first-order reduced density matrices associated with regions defined in the real space (regional matrices). The treatment is based on an isopycnic orbital localization transformation that provides regional matrices that are diagonalized by identical localized orbitals, having many attributes associated with chemical concepts (appropriate localization in space, high transferability, etc.). Although the obtained numerical values are similar to those arising from previous studies, their interpretation is more rigorous and the computational cost is much lower.

A study of the partitioning of the first-order reduced density matrix according to the theory of atoms in molecules

This work describes a simple spatial decomposition of the first-order reduced density matrix corresponding to an N-electron system into first-order density matrices, each of them associated to an atomic domain defined in the theory of atoms in molecules. A study of the representability of the density matrices arisen from this decomposition is reported and analyzed. An appropriate treatment of the eigenvectors of the matrices defined over atomic domains or over unions of these domains allows one to describe satisfactorily molecular properties and chemical bondings within a determined molecule and among its fragments. Numerical determinations, performed in selected molecules, confirm the reliability of our proposal.

Improving the orbital-free density functional theory description of covalent materials

The essential challenge in orbital-free density functional theory (OF-DFT) is to construct accurate kinetic energy density functionals (KEDFs) with general applicability (i.e., transferability). During the last decade, several linear-response (LR)-based KEDFs have been proposed. Among them, the Wang-Govind-Carter (WGC) KEDF, containing a density-dependent response kernel, is one of the most accurate that still affords a linear scaling algorithm. For nearly-free-electron-like metals such as Al and its alloys, OF-DFT employing the WGC KEDF produces bulk properties in good agreement with orbital-based Kohn-Sham (KS) DFT predictions. However, when OF-DFT, using the WGC KEDF combined with a recently proposed bulk-derived local pseudopotential (BLPS), was applied to semiconducting and metallic phases of Si, problems arose with convergence of the self-consistent density and energy, leading to poor results. Here we provide evidence that the convergence problem is very likely caused by the use of a truncated Taylor series expansion of the WGC response kernel. Moreover, we show that a defect in the ansatz for the first-order reduced density matrix underlying the LR KEDFs limits the accuracy of these KEDFs. By optimizing the two free parameters involved in the WGC KEDF, the two-body Fermi wave vector mixing parameter gamma and the reference density rho* used in the Taylor expansion, OF-DFT calculations with the BLPS can achieve semiquantitative results for nine phases of bulk silicon. These new parameters are recommended whenever the WGC KEDF is used to study nonmetallic systems.

The structure of the second-order reduced density matrix in density matrix functional theory and its construction from formal criteria.

Some formal requirements for the second-order reduced density matrix are discussed in the context of density matrix functional theory. They serve as a basis for the ad hoc construction of the second-order reduced density matrix in terms of the first-order reduced density matrix and lead to implicit functionals where the occupation numbers of the natural orbitals are obtained as diagonal elements of an idempotent matrix the elements of which represent the variational parameters to be optimized. The numerical results obtained from a first realization of such an implicit density matrix functional give excellent agreement with the results of full configuration interaction calculations for four-electron systems like LiH and Be. Results for H2O taken as an example for a somewhat larger molecule are numerically less satisfactory but still give reasonable occupation numbers of the natural orbitals and indicate the capability of density matrix functional theory to cope with static electron correlation.

Coupled cluster approach to electron densities

The electron charge density of closed-shell systems is calculated within the framework of the orthogonally spin-adapted coupled cluster (CC) approach with singles and doubles (CCSD). To avoid a nonterminating series in cluster amplitudes when relying on the direct calculation of expectation values, we employ the linear response equations which yield expressions with finitely many terms. For the first-order properties, we derive an alternative form of these equations, which is amenable to the large-scale computation of properties. In particular, the first-order reduced density matrix corresponding to the CC wavefunction is generated as the expectation value of the orbital replacement operators. To illustrate the effects of correlation we generate the difference density maps between the CC and restricted Hartree–Fock (RHF) wavefunctions for the P4 model of the (H 2) 2 system. These results are compared with the corresponding full configuration interaction (FCI) densities, representing the exact solution for the ab initio model employed, as well as with the CI truncated at the doubly excited level (CISD). It is shown that the CCSD provides by far the superior results.

Local approximation of the correlation energy functional in the density matrix functional theory.

A local approximation formula of the correlation energy functional E(c) in terms of the first-order reduced density matrix (1-RDM) is presented. With the contracted Schrödinger equation the principal dependence of E(c) on the natural occupation numbers n(i) is identified. Using the effective mass theory, E(c) is expressed as a functional of the local density and the local variable, J = SUM (i)[square root of (n(i)(1-n(i))] /phi(i)/(2), where phi(i) are the natural spin orbitals. This local approximation satisfies the homogeneous coordinate scaling relation, gives the exact result for a one-electron system, and is almost free from the exchange energy error. It reproduced about 90% of the correlation energies of atoms and molecules.

Correlation energy functional in the density-matrix functional theory

A new systematic method is presented to construct a correlation energy functional in the density-matrix functional theory. A new exact relation, particle-hole symmetry, is presented, which states that the exact functional gives the same correlation energy for the first-order reduced density matrix (1-RDM) and the corresponding hole 1-RDM. Nakatsuji's density equations of the first and the second orders together with the decoupling approximation of the 3- and 4-RDMs are solved numerically to examine the properties of the correlation energy functional. This functional, defined as the solution of the set of equations, satisfies Levy's homogeneous coordinate scaling and the particle-hole symmetry and reproduces about 95% of the correlation energies of several molecules. By expanding these equations in perturbation series the leading term of the correlation energy functional is identified. Numerical analysis shows that this simplified functional, which contains one fitting parameter, reproduces the energies of several molecules accurately. The direction of further study is discussed.

A strictly variational procedure for cluster embedding based on the extended subspace approach

Even if an isolated defect results only in a local perturbation of the electron density, the wave function and the first-order reduced density matrix may still exhibit a long-range response to the defect. We present an axiomatic approach to the construction of a general-purpose embedding scheme which is able to cope with this problem. We start from a list of requirements, which we consider pertinent to an accurate embedding technique, and we proceed to demonstrate that the extended subspace approach recently proposed by Head and Silva [J. Chem. Phys. 104, 3244 (1996)] is the minimal realization of such an embedding scheme. The variational principle, strict fulfillment of the Pauli exclusion principle, a finite dimensional parameter space, and the possibility to perform the minimization by a standard SCF (self-consistent field) procedure are the key requirements which lead to a constrained SCF procedure. Self-embedding consistency and local completeness of the Hilbert space can then be realized by a mathematically very simple construction principle for the active subspace which can be formulated independent of any basis set. We analyze the spatial structure of the resulting minimal orbital space by means of tight-binding model Hamiltonians. For metal systems, we find active and frozen constrained SCF spaces to necessarily interlock in a strong and complicated fashion.

Natural resonance theory: I. General formalism

We present a new quantum-mechanical resonance theory based on the first-order reduced density matrix and its representation in terms of natural bond orbitals (NBOs). This “natural” resonance theory (NRT) departs in important respects from the classical Pauling-Wheland formulation, yet it leads to quantitative resonance weights that are in qualitative accord with conventional resonance theory and chemical intuition. The NRT variational functional leads to an optimal resonance-weighted approximation to the full density matrix, combining the “single reference” limit of weak delocalization (incorporating diagonal population changes only) with the full “multireference” limit of strong delocalization (incorporating off-diagonal couplings between resonance structures. The NRT variational functional yields an error measure that serves as an intrinsic criterion of accuracy of the resonance-theoretic description. The NRT program structure, algorithms, and numerical characteristics are described in supplementary material, and detailed chemical applications are presented in two companion papers. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 593–609, 1998

The AGP model for fermion systems

A mathematically rigorous treatment of the antisymmetrized geminal power (AGP) model is given. The model is advocated as an extremely flexible tool for studying a wide variety of fermion systems, characterized by a single Slater determinant, on the one hand, or Yang's most highly correlated wave function, on the other. When the first-order reduced density matrix of the actual system has eigenvalues which are evenly degenerate, the model can treat one-particle properties exactly. A formula is obtained for the precise energy of an AGP state in the Born-Oppenheimer approximation. An estimate is given of the error which occurs when the second-order density matrix of an AGP state is approximated by the widely used expression involving the so-called anomalous Green's function. © 1997 John Wiley & Sons, Inc.