One-electron Density Matrix Research Articles

New tools for the systematic analysis and visualization of electronic excitations. II. Applications.

The excited states of a diverse set of molecules are examined using a collection of newly implemented analysis methods. These examples expose the particular power of three of these tools: (i) natural difference orbitals (the eigenvectors of the difference density matrix) for the description of orbital relaxation effects, (ii) analysis of the one-electron transition density matrix in terms of an electron-hole picture to identify charge resonance and excitonic correlation effects, and (iii) state-averaged natural transition orbitals for a compact simultaneous representation of several states. Furthermore, the utility of a wide array of additional analysis methods is highlighted. Five molecules with diverse excited state characteristics are chosen for these tasks: pyridine as a prototypical small heteroaromatic molecule, a model system of six neon atoms to study charge resonance effects, hexatriene in its neutral and radical cation forms to exemplify the cases of double excitations and spin-polarization, respectively, and a model iridium complex as a representative metal organic compound. Using these examples a number of phenomena, which are at first sight unexpected, are highlighted and their physical significance is discussed. Moreover, the generality of the conclusions of this paper is verified by a comparison of single- and multireference ab initio methods.

Extension of many-body theory and approximate density functionals to fractional charges and fractional spins

The exact conditions for density functionals and density matrix functionals in terms of fractional charges and fractional spins are known, and their violation in commonly used functionals has been shown to be the root of many major failures in practical applications. However, approximate functionals are designed for physical systems with integer charges and spins, not in terms of the fractional variables. Here we develop a general framework for extending approximate density functionals and many-electron theory to fractional-charge and fractional-spin systems. Our development allows for the fractional extension of any approximate theory that is a functional of G(0), the one-electron Green's function of the non-interacting reference system. The extension to fractional charge and fractional spin systems is based on the ensemble average of the basic variable, G(0). We demonstrate the fractional extension for the following theories: (1) any explicit functional of the one-electron density, such as the local density approximation and generalized gradient approximations; (2) any explicit functional of the one-electron density matrix of the non-interacting reference system, such as the exact exchange functional (or Hartree-Fock theory) and hybrid functionals; (3) many-body perturbation theory; and (4) random-phase approximations. A general rule for such an extension has also been derived through scaling the orbitals and should be useful for functionals where the link to the Green's function is not obvious. The development thus enables the examination of approximate theories against known exact conditions on the fractional variables and the analysis of their failures in chemical and physical applications in terms of violations of exact conditions of the energy functionals. The present work should facilitate the calculation of chemical potentials and fundamental bandgaps with approximate functionals and many-electron theories through the energy derivatives with respect to the fractional charge. It should play an important role in developing accurate approximate density functionals and many-body theory.

Nonstationarity and related measures for time-dependent hartree-fock and multiconfigurational models

Based on an earlier article (Eberly and Singh, Phys. Rev. D 1973, 7, 359) and related works on short-time evolution, this article proposes a many-electron formulation for the nonstationarity degree which can be assigned to quantum system at each time point. The key measure introduced, , is a nonstationarity index that can be thought of as an inverse nominal lifetime at each instance of time. The index is directly computed from the time derivative of one-electron density matrix and is a size-consistent quantity. In this article, the approach is developed for the time-dependent Hartree–Fock (TDHF), single-excitation (TDCIS), and time-dependent full configuration interaction (TDFCI) models. As a rule, nonstationarity effects are more pronounced in correlated electron systems, and a joint analysis of and the multiconfigurational character of wave functions apparently provide a deeper insight into dynamical molecular processes. The performed calculations on small molecules in laser fields show a preference for the TDCIS model when comparing TDCIS and TDHF with the “exact” TDFCI model. © 2013 Wiley Periodicals, Inc.

Analysis of Excitonic and Charge Transfer Interactions from Quantum Chemical Calculations.

A procedure for a detailed analysis of excited states in systems of interacting chromophores is proposed. By considering the one-electron transition density matrix, a wealth of information is recovered that may be missed by manually analyzing the wave function. Not only are the position and spatial extent given, but insight into the intrinsic structure of the exciton is readily obtained as well. For example, the method can differentiate between excitonic and charge resonance interactions even in completely symmetric systems. Four examples are considered to highlight the utility of the approach: interactions between the nπ* states in a formaldehyde dimer, excimer formation in the naphthalene dimer, stacking interaction in an adenine dimer, and the excitonic band structure in a conjugated phenylenevinylene oligomer.

Stockholder projector analysis: A Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A previously introduced partitioning of the molecular one-electron density matrix over atoms and bonds [D. Vanfleteren et al., J. Chem. Phys. 133, 231103 (2010)] is investigated in detail. Orthogonal projection operators are used to define atomic subspaces, as in Natural Population Analysis. The orthogonal projection operators are constructed with a recursive scheme. These operators are chemically relevant and obey a stockholder principle, familiar from the Hirshfeld-I partitioning of the electron density. The stockholder principle is extended to density matrices, where the orthogonal projectors are considered to be atomic fractions of the summed contributions. All calculations are performed as matrix manipulations in one-electron Hilbert space. Mathematical proofs and numerical evidence concerning this recursive scheme are provided in the present paper. The advantages associated with the use of these stockholder projection operators are examined with respect to covalent bond orders, bond polarization, and transferability.

Fast orbital localization scheme in molecular fragments resolution

The method for localizing orbitals on a set of predefined molecular fragments is introduced. Regional localized molecular orbitals (RLMO) are obtained through block diagonalization of the one-electron density matrix and further refinement of the resulting eigenvectors. The algorithm is fast and reliable, as is illustrated by a few examples. Potential applications range from conceptual insight into a chemical bonding to reduced scaling computational techniques. RLMOs are particularly well suited for fragmentation computational methods and for exploiting the locality of electronic correlation in post-HF methods.

A new parametrizable model of molecular electronic structure

A new electronic structure model is developed in which the ground state energy of a molecular system is given by a Hartree-Fock-like expression with parametrized one- and two-electron integrals over an extended (minimal + polarization) set of orthogonalized atom-centered basis functions, the variational equations being solved formally within the minimal basis but the effect of polarization functions being included in the spirit of second-order perturbation theory. It is designed to yield good dipole polarizabilities and improved intermolecular potentials with dispersion terms. The molecular integrals include up to three-center one-electron and two-center two-electron terms, all in simple analytical forms. A method to extract the effective one-electron Hamiltonian of nonlocal-exchange Kohn-Sham theory from the coupled-cluster one-electron density matrix is designed and used to get its matrix representation in a molecule-intrinsic minimal basis as an input to the parametrization procedure--making a direct link to the correlated wavefunction theory. The model has been trained for 15 elements (H, Li-F, Na-Cl, 720 parameters) on a set of 5581 molecules (including ions, transition states, and weakly bound complexes) whose first- and second-order properties were computed by the coupled-cluster theory as a reference, and a good agreement is seen. The model looks promising for the study of large molecular systems, it is believed to be an important step forward from the traditional semiempirical models towards higher accuracy at nearly as low a computational cost.

Fast density matrix-based partitioning of the energy over the atoms in a molecule consistent with the hirshfeld-I partitioning of the electron density

For the Hirshfeld-I atom in the molecule (AIM) model, associated single-atom energies and interaction energies at the Hartree-Fock level are efficiently determined in one-electron Hilbert space. In contrast to most other approaches, the energy terms are fully consistent with the partitioning of the underlying one-electron density matrix (1DM). Starting from the Hirshfeld-I AIM model for the electron density, the molecular 1DM is partitioned with a previously introduced double-atom scheme (Vanfleteren et al., J Chem Phys 2010, 132, 164111). Single-atom density matrices are constructed from the atomic and bond contributions of the double-atom scheme. As the Hartree-Fock energy can be expressed solely in terms of the 1DM, the partitioning of the latter over the AIM naturally leads to a corresponding partitioning of the Hartree-Fock energy. When the size of the molecule or the molecular basis set does not grow too large, the method shows considerable computational advantages compared with other approaches that require cumbersome numerical integration of the molecular energy integrals weighted by atomic weight functions.

Topological Analysis of the Molecular Charge Density and Impact Sensitivy Models of Energetic Molecules

Important explosives of practical use are composed of nitroaromatic molecules. In this work, we optimized geometries and calculated the electron density of 17 nitroaromatic molecules using the Density Functional Theory (DFT) method. From the DFT one-electron density matrix, we computed the molecular charge densities, thus the electron densities, which were then decomposed into electric multipoles located at the atomic sites of the molecules using the distributed multipole analysis (DMA). The multipoles, which have a direct chemical interpretation, were then used to analyze in details the ground state charge structure of the molecules and to seek for correlations between charge properties and sensitivity of the corresponding energetic material. The DMA multipole moments do not present large variations when the size of the Gaussian basis set is changed; the largest variations occurred in the range 10-15% for the dipole and quadrupole moments of oxygen atoms. The charges on the carbon atoms of the aromatic ring of each molecule become more positive when the number of nitro groups increases and saturate when there are five and six nitro groups. The magnitude and the direction of the dipole moments of the carbon atoms, indicators of site polarization, also depend on the nature of adjacent groups, with the largest dipole value being for C-H bonds. The total magnitude of the quadrupole moment of the aromatic ring carbon atoms indicates a decrease in the delocalized electron density due to an electron-withdrawing effect. Three models for sensitivity of the materials based on the DMA multipoles were proposed. Explosives with large delocalized electron densities in the aromatic ring of the component molecule, expressed by large quadrupole values on the ring carbon atoms, correspond to more insensitive materials. Furthermore, the charges on the nitro groups also influence the impact sensitivity.

Variational method with scattering boundary conditions imposed by the Wigner distribution

A functional form for electrons scattering off localized potentials with boundary conditions imposed through the Wigner distribution function is formulated in a manner appropriate for the study of tunneling in nanoscale junctions. Variation of the functional with respect to the electronic density matrix leads to stationary or critical points respecting open-system boundary conditions. Examples are presented for single-electron and many- (noninteracting) electron scattering and, in both instances, the usual outcome of a one-electron density matrix comprised of scattering wave functions results. The examples highlight how the degrees of freedom associated with imposing scattering boundary conditions can be constrained and removed from a variational determination of the electronic wave functions on a scattering region. The form for the scattering functional is motivated in terms of the maximum entropy principle.

How to find an optimum cluster size through topological site properties: MoSxmodel clusters

Computational investigations in catalysis frequently use model clusters to represent realistically the catalyst and its reaction sites. Detailed knowledge of the molecular charge, thus electronic density, of a cluster would then allow physical and chemical insights of properties and can provide a procedure to establish their optimum size for catalyst studies. For this purpose, an approach is suggested to study model clusters based on the distributed multipole analysis (DMA) of molecular charge properties. After full density functional theory (DFT) geometry optimization of each cluster, DMA computed from the converged DFT one-electron density matrix allowed the partition of the corresponding cluster charge distribution into monopole, dipole, and quadrupole moments on the atomic sites. The procedure was applied to MoS2 model clusters Mo10S18, Mo12S26, Mo16S32, Mo23S48, and Mo27S54. This analysis provided detailed features of the charge distribution of each cluster, focused on the 1010 (Mo or metallic edge) and 1010 (sulfur edge) active planes. Properties of the Mo27S54 cluster, including the formation of HDS active surfaces, were extensively discussed. The effect of cluster size on the site charge distribution properties of both planes was evaluated. The results showed that the Mo16S32 cluster can adequately model both active planes of real size Mo27S54. These results can guide future computational studies of MoS2 catalytic processes. Furthermore, this approach is of general applicability.

A Density-Genetic Algorithm Method for Computing Electronic Structures of Doped and Undoped Polythiophene Oligomers: A Modified Su–Schrieffer–Heeger Hamiltonian-Based Study

We propose a genetic algorithm (GA)-driven density matrix method for calculating the equilibrium geometry and charge storage configurations of hole(bipolaron)-doped Polythiophene (PT) oligomers. A modified version of the Su–Schrieffer–Heeger Hamiltonian is used to describe the PT chain. A population of geometry strings are used to generate the corresponding PT-Hamiltonians which act as generators of corresponding unitary transformations which transform a single trial one-electron density matrix into a population of density matrices for different geometries. As the geometry strings evolve under the action of GA operators, the density matrices also evolve on the fitness landscape. Once the fitness reaches maximum, the optimum geometry string, the corresponding Hamiltonian, and the density matrix lead to equilibrium geometry, energy, charge distribution, band gap, Fermi energy, etc. The bipolaronic defect–induced conducting state is predicted to have a symmetric lattice-like admixture of “aromatic” and “quininoid” regions and is characterized by nonzero density of states at Fermi energy and low band gaps.

Communication: Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A double-atom partitioning of the molecular one-electron density matrix is used to describe atoms and bonds. All calculations are performed in Hilbert space. The concept of atomic weight functions (familiar from Hirshfeld analysis of the electron density) is extended to atomic weight matrices. These are constructed to be orthogonal projection operators on atomic subspaces, which has significant advantages in the interpretation of the bond contributions. In close analogy to the iterative Hirshfeld procedure, self-consistency is built in at the level of atomic charges and occupancies. The method is applied to a test set of about 67 molecules, representing various types of chemical binding. A close correlation is observed between the atomic charges and the Hirshfeld-I atomic charges.

Cholesky decomposition-based definition of atomic subsystems in electronic structure calculations

Decomposing the Hartree-Fock one-electron density matrix and a virtual pseudodensity matrix, we obtain an orthogonal set of normalized molecular orbitals with local character to be used in post-Hartree-Fock calculations. The applicability of the procedure is illustrated by calculating CCSD(T) energies and CCSD molecular properties in reduced active spaces.

High-performance ab initio density matrix renormalization group method: Applicability to large-scale multireference problems for metal compounds

This article presents an efficient and parallelized implementation of the density matrix renormalization group (DMRG) algorithm for quantum chemistry calculations. The DMRG method as a large-scale multireference electronic structure model is by nature particularly efficient for one-dimensionally correlated systems, while the present development is oriented toward applications for polynuclear transition metal compounds, in which the macroscopic one-dimensional structure of electron correlation is absent. A straightforward extension of the DMRG algorithm is proposed with further improvements and aggressive optimizations to allow its application with large multireference active space, which is often demanded for metal compound calculations. Special efficiency is achieved by making better use of sparsity and symmetry in the operator and wave function representations. By accomplishing computationally intensive DMRG calculations, the authors have found that a large number of renormalized basis states are required to represent high entanglement of the electron correlation for metal compound applications, and it is crucial to adopt auxiliary perturbative correction to the projected density matrix during the DMRG sweep optimization in order to attain proper convergence to the solution. Potential energy curve calculations for the Cr(2) molecule near the known equilibrium precisely predicted the full configuration interaction energies with a correlation space of 24 electrons in 30 orbitals [denoted by (24e,30o)]. The energies are demonstrated to be accurate to 0.6mE(h) (the error from the extrapolated best value) when as many as 10,000 renormalized basis states are employed for the left and right DMRG block representations. The relative energy curves for [Cu(2)O(2)](2+) along the isomerization coordinate were obtained from DMRG and other correlated calculations, for which a fairly large orbital space (32e,62o) is modeled as a full correlation space. The DMRG prediction nearly overlaps with the energy curve from the coupled cluster with singles, doubles, and perturbative triple [CCSD(T)] calculations, while the multireference complete active space self-consistent field (CASSCF) calculations with the small reference configuration (8e,8o) are found to overestimate the biradical character of the electronic state of [Cu(2)O(2)](2+) according to the one-electron density matrix analysis.

Real-Time Propagation of the Reduced One-Electron Density Matrix in Atom-Centered Orbitals: Application to Electron Injection Dynamics in Dye-Sensitized TiO2 Clusters

The ultrafast electron-transfer (ET) processes in three dye-sensitized TiO2 systems (pycooh−, catechol−, and alizarin−) are studied by using the real-time time-dependent density functional theory (RT-TDDFT). TiO2 cluster models are used to substitute TiO2 nanocrystals in order to check the quantum size effect on ET. The initial-state geometrical optimization for the individual constituents and coupled systems and the subsequent calculations for IR spectra and the density of states (DOS) are performed at the B3LYP/Lanl2dz theory level. The calculated IR spectra, the DOS, and the low-lying excited states reveal that the couplings between three dyes and TiO2 clusters are very strong so that an ultrafast electron injection from the excited dyes to TiO2 clusters is favored. By following the electronic motion of coupled systems after the photoexcitation of adsorbates in real time without allowing the nuclei to move, we predict an electronic injection time of a few femtoseconds for the present finite systems, which is slightly longer than the experimental measurements and other theoretical predications for the ET time on the same dye-sensitized bulk TiO2 systems due to the small clusters used in our simulation. We find that the ET time is appreciably dependent on the cluster size when the cluster is quite small. However, the size effects on ET time reduce dramatically as the cluster size reaches to a moderate middle size, for example, (TiO2)14. The electron−nuclear coupled movement does not play a significant role in the initial ET process in these three systems. The effects of different initial excited states on electronic dynamics are also discussed.

Fourier–Legendre expansion of the one-electron density matrix of ground-state two-electron atoms

The density matrix rho(r,r(')) of a spherically symmetric system can be expanded as a Fourier-Legendre series of Legendre polynomials P(l)(cos theta=rr(')rr(')). Application is here made to harmonically trapped electron pairs (i.e., Moshinsky's and Hooke's atoms), for which exact wavefunctions are known, and to the helium atom, using a near-exact wavefunction. In the present approach, generic closed form expressions are derived for the series coefficients of rho(r,r(')). The series expansions are shown to converge rapidly in each case, with respect to both the electron number and the kinetic energy. In practice, a two-term expansion accounts for most of the correlation effects, so that the correlated density matrices of the atoms at issue are essentially a linear functions of P(l)(cos theta)=cos theta. For example, in the case of Hooke's atom, a two-term expansion takes in 99.9% of the electrons and 99.6% of the kinetic energy. The correlated density matrices obtained are finally compared to their determinantal counterparts, using a simplified representation of the density matrix rho(r,r(')), suggested by the Legendre expansion. Interestingly, two-particle correlation is shown to impact the angular delocalization of each electron, in the one-particle space spanned by the r and r(') variables.

Real-Time Propagation of the Reduced One-Electron Density Matrix in Atom-Centered Orbitals: Application to Multielectron Dynamics of Carbon Clusters Cn in the Strong Laser Pulses

We present a time-dependent density functional theory (TDDFT) study on the electron dynamics of small carbon clusters C(n) (n = 9, 10) exposed to a linearly polarized (LP) or circularly polarized (CP) oscillating electric field of ultrafast laser with moderate laser intensity. The multielectron dynamics is described by propagating the reduced one-electron density matrix in real-time domain. The high harmonic generation (HHG) spectra of emission as well as the time evolution of atomic charges, dipole moments and dipole accelerations during harmonic generation are calculated. The microscopic structure-property correlation of carbon chains is characterized. It is found that the electron responses of C(n) to the laser field oscillation become nonadiabatic as the field intensity is larger than 1.4 x 10(13) W/cm(2). The nonadiabatic multielectron effect is displayed by an explicit fluctuation on the induced atomic charges and the instantaneous dipole acceleration and by observing the additional peaks other than those predicted from the spectral selection rule in HHG spectra of C(n) as well. The origin of these additional peaks is elucidated. The atomic charges of C(n) in LP and CP laser pulses experience different type of oscillations as expected. In the linear structure C9, the atomic charges at the two ends experience larger amplitude oscillations than those near the chain center whereas the induced charges on each atom of C10 experience the equal amplitude oscillations in the CP laser pulse.

THEORETICAL STUDIES ON THE LASER-DRIVEN MANY-ELECTRON DYNAMICS OF FINITE-SIZE BORON-NITRIDE NANOTUBES

We present a theoretical study on the electronic dynamics of finite-size Boron-Nitride Nanotubes (BNNTs) driven by the linearly polarized (LP) and circularly polarized (CP) intense laser fields. The many-electron dynamics is described by propagating the reduced one-electron density matrix in real-time domain. The high harmonic generation (HHG) spectra and the time evolution of charge density during harmonic generation are demonstrated. Comparison is made with Carbon Nanotubes (CNTs). The selection rules (SRs) for HHG of finite-size nanotubes are established. One of the important consequences of our results is that HHG of finite-size nanotubes are dependent not only on the molecular point-groups but also on the strength of inter chain interactions. The characteristic difference in nonlinear optical properties between finite and infinitely extended zigzag and armchair BNNTs is significant. Another important consequence is that the laser pulse can drive the electrons to move along or normal to tube axis and produce a charge current on the tube wall.

Real-time propagation of the reduced one-electron density matrix in atom-centered Gaussian orbitals: Application to absorption spectra of silicon clusters

We solve the time-dependent density functional theory equation by propagating the reduced one-electron density matrix in real-time domain. The efficiency of several standard solvers such as the short-iterative Krylov-subspace propagator, the low-order Magnus integration method with the matrix polynomial (MP) or Chebyshev matrix polynomial (CMP) expansion of the evolution operator, and Runge-Kutta algorithm are assessed. Fast methods for summing MP and CMP are implemented to speed the calculation of the matrix exponential. It is found that the exponential propagators can tolerate large time step size and retain the computational accuracy whereas the Krylov-subspace algorithm is a little inferior for a larger time step size compared with the second-order Magnus integration method with the MP/CMP expansion of the evolution operator in both weak and intense fields. As an application, we calculate the absorption spectra of hydrogen-passivated silicon nanoparticles Si(29)H(x). The popular hybrid and generalized gradient approximation exchange-correlation functionals are applied. We find that the experimental spectra can be reproduced by using B3LYP and that the silicon particles with sizes of 1 nm and the optical excitations at 3.7, 4.0, and 4.6 eV may consist of 29 Si atoms surrounded by 24 hydrogen atoms.