First-order Density Matrix Research Articles

Preliminary Assessment of the First-Order Density Matrix in Quantum Monte Carlo from Density Matrix Theory.

The trial wave function commonly used in the quantum Monte Carlo method consists of the product of up-spin and down-spin Slater determinants, allowing accurate calculations of multielectronic properties, although it is not antisymmetric under the exchange of electrons with opposite spins. An alternative description that overcomes these limitations using the Nth-order density matrix was already presented. This study introduces two new strategies based on the Dirac-Fock density matrix for QMC that still fully preserve antisymmetry and electron indistinguishability. Simulations are performed for the ground and excited states of He, Li, and Be showing that the present formulation and the conventional separation of spins are appropriate for a correct description of these systems, except for singlet excited states of the He and Be atoms, and that a part of the antisymmetry (antiparallel spins) can be neglected.

A fast approximate extension of the interacting quantum atoms energy decomposition to excited states.

An approach is developed for the fast calculation of the interacting quantum atoms energy decomposition (IQA) from the information contained in the first order reduced density matrix only. The proposed methodology utilizes an approximate exchange-correlation density from Density Matrix Functional Theory without the need to evaluate the correlation-exchange contribution directly. Instead, weight factors are estimated to decompose the exact Vxc into atomic and pairwise contributions. In this way, the sum of the IQA contributions recovers the energy obtained from the electronic structure calculation. This method can, hence, be applied to obtain atomic contributions in excited states on the same footing as in their ground states using any method that delivers the reduced first-order density matrix. In this way, one can locate chromophores from first principles quantum chemical calculations. Test calculations on the ground and excited states of a set of small molecules indicate that the scaled atomic contributions reproduce vertical electronic transition energies calculated exactly. This approach may be useful to extend the applicability of the IQA approach in the study of large photochemical systems especially when the calculations of the second order reduced density matrices is prohibitive or not possible.

Are Multicentre Bond Indices and Related Quantities Reliable Predictors of Excited-State Aromaticity?

Systematic scrutiny is carried out of the ability of multicentre bond indices and the NOEL-based similarity index dAB to serve as excited-state aromaticity criteria. These indices were calculated using state-optimized complete active-space self-consistent field wavefunctions for several low-lying singlet and triplet states of the paradigmatic molecules of benzene and square cyclobutadiene and the inorganic ring S2N2. The comparison of the excited-state indices with aromaticity trends for individual excited states suggested by the values of magnetic aromaticity criteria show that whereas the indices work well for aromaticity reversals between the ground singlet and first triplet electronic states, addressed by Baird’s rule, there are no straightforward parallels between the two sets of data for singlet excited states. The problems experienced while applying multicentre bond indices and dAB to singlet excited states are explained by the loss of the information inherently present in wavefunctions and/or pair densities when calculating the first-order density matrix.

A perturbation-based super-CI approach for the orbital optimization of a CASSCF wave function.

A perturbation theory-based algorithm for the iterative orbital update in complete active space self-consistent-field (CASSCF) calculations is presented. Following Angeli et al. (J. Chem. Phys. 2002, 117, 10525), the first-order contribution of singly excited configurations to the CASSCF wave function is evaluated using the Dyall Hamiltonian for the determination of a zeroth-order Hamiltonian. These authors employ an iterative diagonalization of the first-order density matrix including the first-order correction arising from single excitations, whereas the present approach uses the single-excitation amplitudes directly for the construction of the exponential of an anti-Hermitian matrix resulting in a unitary matrix which can be used for the orbital update. At convergence, the single-excitation amplitudes vanish as a consequence of the generalized Brillouin's theorem. It is shown that this approach in combination with direct inversion of the iterative subspace (DIIS) leads to very rapid convergence of the CASSCF iteration procedure. © 2019 Wiley Periodicals, Inc.

A Quasi-Atomic Analysis of Three-Center Two-Electron Zr-H-Si Interactions.

A comprehensive analysis of the bonding structure of the disilyl zirconocene amide cation {Cp2Zr[N(SiHMe2)2]}+ is conducted by application of an intrinsic orbital localization method that yields quasi-atomic orbitals (QUAOs). An emphasis is placed on describing a previously characterized three-center two-electron interaction between zirconium, hydrogen, and silicon that presents structural and spectroscopic features similar to that of agostic bonds. Expressions of the first-order density matrix in terms of the QUAOs yields bond orders (BOs), kinetic bond orders (KBOs), and the extent of transfer of charge that are useful to determine the electronic nature of the Zr-H-Si bond. The interactions between the QUAOs demonstrate the importance of vicinal interactions in the stabilization of the molecule. In addition, the evolution of the QUAOs during reactions with Lewis bases reveals the role of the Zr-H-Si interaction in facilitating the reaction.

Absorption of twisted light by a mesoscopic atomic target

The excitation of a hydrogen-atom target by a twisted Bessel light beam is investigated. The atoms are assumed to have a Gaussian spatial distribution in the target. Theoretical analysis is performed within a nonrelativistic framework using a first-order perturbation approach and density matrix formalism. By using this theory, we derive the expressions for excitation cross sections and for alignment parameters of the excited atomic state. In particular, we make calculations for the transition caused by the interaction of Bessel beams with the atomic target. For this transition, we analyze the population of magnetic sublevels for the excited state and study how it is affected by the projection of the total angular momentum of incident light. The calculations indicate that the projection of the total angular momentum of the incident Bessel beam affects the alignment of atoms for sufficiently small targets with size less than 200 nm. This can be observed experimentally by measuring the linear polarization of the subsequent fluorescent light.

Oxidation States from Wave Function Analysis.

We introduce a simple and general scheme to derive from wavefuntion analysis the most appropriate atomic/fragment electron configurations in a molecular system, from which oxidation states can be inferred. The method can be applied for any level of theory for which the first-order density matrix is available, and unlike others, it is not restricted to transition metal complexes. The method relies on the so-called spin-resolved effective atomic orbitals which for the present purpose is extended here to deal with molecular fragments/ligands. We describe in detail the most important points of the new scheme, in particular the hierarchical fragment approach devised for practical applications. A number of transition metal complexes with different formal oxidation states and spin states and a set of organic and inorganic compounds are provided as illustrative examples of the new scheme. Challenging systems such as transition state structures are also tackled on equal footing.

Bond metallicity measures

It was recently proposed that the metallicity of chemical bond can be determined by the electron density divided by its Laplacian, evaluated at the bond critical point, ρ(rbcp)/∇2ρ(rbcp). We define a new, dimensionless, form of this bond metallicity measure. We explain why these bond metallicity measures work by showing that the bond metallicity is inversely related to “nearsightedness” of the first-order density matrix. These bond metallicity measures are also related to various measures of electron localization functions and atomic shell structure that have been proposed in the literature. The local electronic temperature provides a common basis for understanding the links between most of the electron localization measures that appear in the literature. High local temperature is associated with nearsightedness of the first-order density matrix, high electron localization, and small values of our bond metallicity.

Bond fukui indices: Comparison of frozen molecular orbital and finite differences through mulliken populations

Bond Fukui functions and matrices are introduced for ab initio levels of theory using a Mulliken atoms in molecules model. It is shown how these indices may be obtained from first-order density matrix derivatives without need for going to second-order density matrices as in a previous work. The importance of taking into account the nonorthogonality of the basis in ab initio calculations is shown, contrasting the present results with previous work based on Hückel theory. It is shown how the extension of Fukui functions to Fukui matrices allows getting more insight into the nature of bond Fukui functions. All presently introduced indices respect the necessary normalization conditions and include the classical single atom condensed Fukui functions.

A proposed family of variationally correlated first-order density matrices for spin-polarized three-electron model atoms

In early work of March and Young (Phil Mag 4:384, 1959), it was pointed out for spin-free fermions that a first-order density matrix (1DM) for $$N-1$$ particles could be constructed from a 2DM ( $$\Gamma $$ ) for $$N$$ fermions divided by the diagonal of the 1DM, the density $$n(\mathbf{r}_1)$$ , as $$2\Gamma (\mathbf{r}_1,\mathbf{r}^{\prime }_2;\mathbf{r}_1,\mathbf{r}_2)/n(\mathbf{r}_1)$$ for any arbitrary fixed $$\mathbf{r}_1$$ . Here, we thereby set up a family of variationally valid 1DMS constructed via the above proposal, from an exact 2DM we have recently obtained for four electrons in a quintet state without confining potential, but with pairwise interparticle interactions which are harmonic. As an indication of the utility of this proposal, we apply it first to the two-electron (but spin-compensated) Moshinsky atom, for which the exact 1DM can be calculated. Then the 1DM is found for spin-polarized three-electron model atoms. The equation of motion of this correlated 1DM is exhibited and discussed, together with the correlated kinetic energy density, which is shown explicitly to be determined by the electron density.

Order indices of density matrices for finite systems

The definition of order indices for density matrices is extended to finite systems. This makes it possible to characterize the level of ordering in such finite systems as macromolecules, nanoclusters, quantum dots, or trapped atoms. The general theory is exemplified by explicit calculations of the order index for the first-order density matrix of bosonic atoms confined in a finite box at zero temperature.

Nuclear motion effects on the density matrix of crystals: An ab initio Monte Carlo harmonic approach

In the frame of the Born-Oppenheimer approximation, nuclear motions in crystals can be simulated rather accurately using a harmonic model. In turn, the electronic first-order density matrix (DM) can be expressed as the statistically weighted average over all its determinations each resulting from an instantaneous nuclear configuration. This model has been implemented in a computational scheme which adopts an ab initio one-electron (Hartree-Fock or Kohn-Sham) Hamiltonian in the CRYSTAL program. After selecting a supercell of reasonable size and solving the corresponding vibrational problem in the harmonic approximation, a Metropolis algorithm is adopted for generating a sample of nuclear configurations which reflects their probability distribution at a given temperature. For each configuration in the sample the "instantaneous" DM is calculated, and its contribution to the observables of interest is extracted. Translational and point symmetry of the crystal as reflected in its average DM are fully exploited. The influence of zero-point and thermal motion of nuclei on such important first-order observables as x-ray structure factors and Compton profiles can thus be estimated.

Wigner distribution function for a harmonically trapped gas of ideal fermions and bosons at arbitrary temperature and dimensionality

We present a new, simple closed form expression for the exact Wigner function of an ideal gas of harmonically trapped fermions or bosons at arbitrary temperature and dimensionality. One of the primary advantages of our expression is that it provides a clean separation of the spatial and momentum variables, which can significantly simplify analytical calculations involving the Wigner function. Our analysis is based on the explicit Fourier transform of the exact first-order density matrix, which affords an exceedingly simple derivation of the Wigner function. We demonstrate the utility of our approach by presenting several analytical results which may be trivially derived from the Wigner function, but otherwise require more involved calculations. We also examine the temperature and dimensionality dependence of the Fermi Wigner function in detail, thereby pointing out some subtle issues regarding the applicability of the Thomas–Fermi approximation to the Wigner function in the limit of a ‘macroscopic’ number of particles, and finite-temperatures.

Relation between the Slater–Kohn–Sham orbitals generated by the one-body potential V(r) of DFT and the Löwdin natural orbitals for inhomogeneous electron liquids

The Löwdin natural orbitals (NO) are defined as those orbitals which bring the first-order density matrix of a correlated electron assembly into diagonal form. Another one-particle density matrix with the same diagonal elements is the single-particle idempotent Dirac density matrix generated by the one-body potential of density functional theory. Here, we compare the off diagonal form of γ expanded in terms of the Slater–Kohn–Sham (SKS) orbitals generated by V(r) with the NO expansion of Löwdin for general inhomogeneous electron liquids. In particular, the equation of motion of the correlated γ is corrected from that of γ s , both now containing the one-body potential V(r). To illustrate the theory presented here, we first construct an approximate, albeit accurate, correlated 1DM for the ground state of the He atom and display connections between the resulting NOs and the SKS orbitals. The second example we discuss, but now quite briefly, is that of the inhomogeneous electron liquid in crystalline Si, where the NO expansion is available from the literature.

Equation of motion of the correlated first-order density matrix for the ground-state of the Hookean atom with two electrons

The ground-state wave function Ψ for a given force constant k = 1/4 a.u. of the two-electron Hookean atom is known in exact analytical form. Here the corresponding first-order density matrix γ(r, r′) is studied, particular attention being focussed on its equation of motion. The exact form which results from the known Ψ is displayed, and given a physical interpretation. Harmonic confined model two-electron atoms with arbitrary interaction u(r 12) are also briefly referred to in the present context.

Differential equation for the Dirac single-particle first-order density matrix in terms of the ground-state electron density

The March-Suhai (MS) partial differential equation for the Dirac density matrix {gamma}{sub s}(r,r'), proved for one- and two-level occupancies, involves both the ground-state density n(r), with its low-order derivatives, and the positive definite kinetic energy density t{sub s}(r). Here, we examine the relation between the equation of motion for {gamma}{sub s}(r,r'), with input now being the one-body potential of density-functional theory, and the MS equation. The important link is the differential virial theorem, which can be used to remove t{sub s}(r) from the MS differential equation. For multiple occupancy, the Pauli potential enters in an important manner. In one dimension, however, the appearance of the Pauli potential can be avoided, obtaining a necessary condition for {gamma}{sub s}(x,x{sup '}) to satisfy for arbitrary level occupancy, in the form of a MS-type differential equation.

Partitioning of the molecular density matrix over atoms and bonds

A double-index atomic partitioning of the molecular first-order density matrix is proposed. Contributions diagonal in the atomic indices correspond to atomic density matrices, whereas off-diagonal contributions carry information about the bonds. The resulting matrices have good localization properties, in contrast to single-index atomic partitioning schemes of the molecular density matrix. It is shown that the electron density assigned to individual atoms, when derived from the density matrix partitioning, can be made consistent with well-known partitions of the electron density over atom in the molecule basins, either with sharp or with fuzzy boundaries. The method is applied to a test set of about 50 molecules, representative for various types of chemical binding. A close correlation is observed between the trace of the bond matrices and the shared electron density index.

Additive and non-additive information channels in orbital communication theory of the chemical bond

The molecular communication system in the resolution of the basis functions χ = {χ X} contributed by the molecule constituent atoms {X}, the key concept of the Orbital Communication Theory (OCT) of the chemical bond, is introduced and its information-theoretic (IT) bond descriptors are summarized. The additive and non-additive components of these molecular information channels are identified. The former involve only the internal (one-center) communications {X → X} between the basis functions χ X of each bonded atom X, determined by the associated (diagonal) block P(χ X|χ X) of the molecular conditional probabilities, which are responsible for the intra-atom promotion to its effective valence state. The latter accordingly involve only the external (two-center) communications between the contributed AO of each pair of bonded atoms, {Y → X and X → Y}, generated by the off-diagonal blocks of conditional probabilities P(χ X|χ Y) and P(χ Y|χ X),X ≠ Y, respectively, which are responsible for the inter-atomic bonding effects in the molecule. Both these probability scatterings ultimately determine the resultant multiplicities of the system chemical bonds. The input-ensemble average value of the channel conditional-entropy, which measures its communication “noise” due to electron delocalization via all chemical bonds, measures the IT-covalency in the molecule, while the complementary descriptor of the ensemble average value of the network mutual-information (information-capacity) reflects the electron localization effects and measures the system IT-ionic component. The illustrative example of the localized chemical bond originating from the interaction between two atomic orbitals is reexamined in some detail and the bond-weighted ensemble approach to chemical interactions in diatomic molecular fragments is discussed within the standard Restricted Hartree-Fock theory. In diatomic systems such treatment exactly reproduces the familiar bond index of Wiberg and provides its resolution into the complementary IT-covalent and IT-ionic components. The operator formulation of the probability-scattering phenomena in molecules is given and the probability-amplitude channel defined by the first-order density matrix is introduced. The AIM internal and external eigenvalue problems of this Charge-and-Bond-Order matrix are introduced and a similar approach to probability propagation matrices/operators is suggested.

Density-matrix theory for the ground state of spin-compensated harmonically confined two-electron model atoms with general interparticle repulsion

For model two-electron atoms with harmonic confinement, the correlated first-order density matrix can be expressed in terms of the relative motion wave function ${\ensuremath{\Psi}}_{R}(r)$. Here we demonstrate that the probability density $P(r)$ associated with this wave function is directly related to the x-ray scattering factor $f(G)$. This latter quantity, in turn, is determined by the ground-state electron density $n(r)$. The Euler-Lagrange equation of the resulting density-matrix theory is thereby shown to take the form of a third-order integro-differential equation for $n(r)$ in which the probability density $P(r)={\ensuremath{\Psi}}_{R}^{2}(r)$ also appears. For two specific choices of the interaction between the two fermions under consideration, the above integro-differential equation derived here is shown to lead back to known linear homogeneous differential equations for the electron density. Finally, it is emphasized that specific equations summarized here will apply directly to theoretical study of the nonrelativistic ground-state electron density $n(r,Z)$ in the He-like ions with atomic number $Z$.

Correlated and idempotent Dirac first-order density matrices with identical diagonal Fermion density: a route to extract a one-body potential energy in TDDFT

After a brief introduction to the use of the idempotent Dirac first-order density matrix (DM), its time-dependent generalization is considered. Special attention is focused on the equation of motion for the time-dependent DM, which is characterized by the one-body potential V(r, t) of time-dependent density functional theory. It is then shown how the force –∇V(r, t) can be extracted explicitly from this equation of motion. Following a linear-response treatment in which a weak potential V(r, t) is switched on to an initially uniform electron gas, the non-linear example of the two-electron spin-compensated Moshinsky atom is a further focal point. We demonstrate explicitly how the correlated DM for this model can be constructed from the idempotent Dirac DM, in this time-dependent example.