Approximate Density Matrix Research Articles

Spectral and optical properties of a [formula omitted] molecular complex confined in InAs/GaAs Quantum Camels: A spatial finite element method treatment

The evolution of eigenstates and related optical absorption (sum of linear and nonlinear contributions) of a singly ionized double donor (D2+) confined in InAs/GaAs (quantum-well-dot nanostructure) quantum camels were theoretically investigated. Within the effective mass and the compact density matrix approximations, the energy levels and optical absorption were characterized as a function of a hydrostatic pressure field applied, the sample temperature, and the wetting layer thickness via a spatial finite element method analysis. Results show that an increase of hydrostatic pressure (sample temperature) between 0 to 30 kbar (4 to 300 K) leads to a stabilization (destabilization) of the energy levels and a decrease (increase) of the absorption peak value with a red-shift (blue-shift). On the other hand, an increase in the wetting layer thickness leads to a reduction in the energy levels, and a monotonous evolution from quantum-dot states to wetting-layer states, especially in the excited ones, is seen. In addition, an increase in the absorption peak value is reported, which can be intended as an electron mobility enhancement. The donor-electron interactions reduce the wavefunction’s spatially-extended feature, enhancing the absorption peak in the analyzed |1〉→|6〉 transition. In the |1〉→|2〉 optical transition, the resonant peak can be tuned within the band ≈25–48 meV, slightly beyond the Terahertz frequency range.

Unified construction of relativistic Hamiltonians.

It is shown that the four-component (4C), quasi-four-component (Q4C), and exact two-component (X2C) relativistic Hartree-Fock equations can be implemented in a unified manner by making use of the atomic nature of the small components of molecular 4-spinors. A model density matrix approximation can first be invoked for the small-component charge/current density functions, which gives rise to a static, pre-molecular mean field to be combined with the one-electron term. As a result, only the nonrelativistic-like two-electron term of the 4C/Q4C/X2C Fock matrix needs to be updated during the iterations. A "one-center small-component" approximation can then be invoked in the evaluation of relativistic integrals, that is, all atom-centered small-component basis functions are regarded as extremely localized near the position of the atom to which they belong such that they have vanishing overlaps with all small- or large-component functions centered at other nuclei. Under these approximations, the 4C, Q4C, and X2C mean-field and many-electron Hamiltonians share precisely the same structure and accuracy. Beyond these is the effective quantum electrodynamics Hamiltonian that can be constructed in the same way. Such approximations lead to errors that are orders of magnitude smaller than other sources of errors (e.g., truncation errors in the one- and many-particle bases as well as uncertainties of experimental measurements) and are, hence, safe to use for whatever purposes. The quaternion forms of the 4C, Q4C, and X2C equations are also presented in the most general way, based on which the corresponding Kramers-restricted open-shell variants are formulated for "high-spin" open-shell systems.

Intensity-dependent nonlinear optical properties in an asymmetric Gaussian potential quantum well-modulated by external fields

In this paper, the effects of external electric, magnetic and non-resonant intense laser fields on the nonlinear optical rectification (NOR), second-harmonic (SH), and third harmonic (TH) generation in a GaAs quantum well with asymmetrical Gaussian potential are theoretically investigated. Firstly, the energy eigenvalues and eigenfunctions of a single electron confined in the structure are obtained by using the diagonalization method within the framework of the effective-mass and parabolic band approaches. Then, using these energy eigenvalues and eigenfunctions, expressions derived within the compact density matrix approximation have been employed to calculate the coefficients of the nonlinear optical response in the structure. The obtained simulation results show that the influence of the external fields leads to significant changes in the coefficients of nonlinear optical rectification, second and third harmonic generation in the system. As a result, it has been seen that the amplitude and position of the peaks of nonlinear optical rectification, second and third harmonic coefficients can be controlled by changing the applied external fields.

Optical properties of spherical quantum dot in the presence of donor impurity under the magnetic field

In this study, the binding energies of an on-center and off-center hydrogenic donor impurity in a spherical quantum dot with finite confinement potential under an applied external magnetic field are investigated in detail by using the matrix diagonalization method in the effective mass approximation. Depending on the calculated energies and wave functions, the linear, third-order nonlinear and total optical absorption coefficients between the ground and several excited states are examined by the iterative method in the compact density matrix approximation for the two-level system. The combined effect of donor position, magnetic field and dot size on binding energies and optical absorption coefficients is observed. The results show that these effects cause significant changes on donor binding energy and optical absorption coefficients.

Time dependent reduced density matrix functional theory at strong correlation: insights from a two-site Anderson impurity model.

The one-body density matrix has recently attracted considerable attention as a promising key quantity for the description of systems out of equilibrium. Its time evolution is given in terms of the two-body density matrix, and thus the central challenge is to find approximations to the latter. An extra layer of difficulty is added when dealing with strong electron correlations. In this work, we explore precisely this regime by looking at the two-site Anderson impurity model as a case study. To address the system's dynamics, we use an adiabatic approximation based on the exact ground-state two-body density matrix. We find that this adiabatic extension does not reproduce the exact results even for a slow switch-on of the external perturbation, and we trace back this behavior to the lack of an accurate imaginary part of the adiabatic approximation to the two-body density matrix. The attempt to restore an approximate imaginary part through a Hilbert transform of the real part works well only for very short times, but quickly deteriorates for longer times, with the one-body density matrix being pushed out of its N-representability domain. Our results thus pose an important constraint on practical prescriptions to perform the time evolution of the one-body density matrix.

Superposition of waves or densities: Which is the nature of chemical resonance?

Resonance is a fundamental and widely used concept in chemistry, but there exist two distinct theories of chemical resonance, based on quite different and incompatible premises: the wave-function-based resonance theory (WFRT), assuming the superposition of wave functions, versus the density-matrix-based resonance theory (DMRT), which interprets the resonance phenomenon as the superposition of density matrices. The latter theory, best known to the chemistry community as the natural resonance theory (NRT), has received much more popularity than the WFRT. In this contribution, the DMRT is shown to be inherently inadequate: (i) the exact density matrix expansion is mathematically impossible unless unphysical negative weights are introduced; (ii) any approximate density matrix representing the resonance hybrid lacks the idempotent property. Therefore, the validity of the NRT ansatz should be seriously questioned. The WFRT seems the only reasonable explanation of resonance so far, and has been shown to provide valuable insights into diverse chemical problems.

Approximate density matrix functionals applied to hetero-atomic bond dissociation

A two-orbital two-electron diatomic model resembling LiH is used to investigate the differences between the exact L\"owdin-Shull and approximate Hartree-Fock-Bogoliubov and Baerends-Buijse density matrix functionals in the medium- to long-distance dissociation region. In case of homolytic dissociation (one electron on each atom), the approximate functionals fail to generate the correct energy due to a compromise between the Hartree-Fock component (which favors partial charge transfer) and the strong correlation component (which hampers charge transfer). The exact functional is able to generate the physically correct answer by enforcing the equi-charge distribution of the bonding and antibonding orbitals. Besides, the approximate functionals also have issues in correctly describing heterolytic dissociation (two electrons on one atom) due to the strong correlation component hampering charge transfer. In this work, we propose a new scheme in which the homolytic dissociation problem for approximate functionals is avoided by adding a Lagrange multiplier that enforces equi-charge distribution of the bonding and antibonding orbitals. The symmety based nature of the findings implies that they are most likely transferable to other cases in which one uses an approximate one-particle method in conjunction with a symmetrical particle-hole correction factor.

Investigation of linear optical absorption coefficients in core-shell quantum dot (QD) luminescent solar concentrators (LSCs)

The interlevel absorption coefficient of CdSe/ZnS and ZnS/CdSe core-shell Quantum Dot (QD) in luminescent solar concentrators (LSCs) is reported. By considering the quantum confinement effects, the wave functions and eigenenergies of electrons in the nonperturebative system consists of a core-shell QD have been numerically calculated under the frame work of effective-mass approximation by solving a three-dimensional Schrӧdinger equation. And then the absorption coefficient is obtained under density matrix approximation considering in the polymer sheets of the concentrator including the core-shell QDs. The effect of the hetero-structure geometry upon the energy spectrum and absorption coefficient associated to interlevel transitions was also considered. The results show that the core-shell QDs can absorb the photons with higher energy in solar spectrum as compared to the inverted core-shell. And with a small shell layer diameter, the core-shell QDs produce larger linear absorption coefficients and consequently higher efficiency values, however it is inversed for inverted core-shell QDs. The work described here gives a detailed insight into the promise of QD-based LSCs and the optoelectronic devices applications.

Multiconfigurational Second-Order Perturbation Theory with Frozen Natural Orbitals Extended to the Treatment of Photochemical Problems.

A new flavor of the frozen natural orbital complete active space second-order perturbation theory method (FNO-CASPT2, Aquilante et al., J. Chem. Phys. 131, 034113) is proposed herein. In this new implementation, the virtual space in Cholesky decomposition-based CASPT2 computations (CD-CASPT2) is truncated by excluding those orbitals that contribute the least toward preserving a predefined value of the trace of an approximate density matrix, as that represents a measure of the amount of dynamic correlation retained in the model. In this way, the amount of correlation included is practically constant at all nuclear arrangements, thus allowing for the computation of smooth electronic states surfaces and energy gradients-essential requirements for theoretical studies in photochemistry. The method has been benchmarked for a series of relevant biochromophores for which large speed-ups have been recorded while retaining the accuracy achieved in the corresponding CD-CASPT2 calculations. Both vertical excitation energies and gradient calculations have been carried out to establish general guidelines as to how much correlation needs to be retained in the calculation for the results to be consistent with the CD-CASPT2 findings. Our results feature errors within a tenth of an eV for the most difficult cases and have been validated to be used for gradient computations where an up to 3-fold speed-up is observed depending on the size of the system and the basis set employed.

The Coulomb, exchange, and correlation components of the electron-electron repulsion in harmonium atoms.

Highly accurate Coulomb, exchange, and correlation components of the electron-electron repulsion energies of the three-electron harmonium atoms in the (2)P- and (4)P+ states are obtained for 19 values of the confinement strength ω ranging from 10(-3) to 10(3). The computed data are consistent with their ω → 0 and ω → ∞ asymptotics that are given by closed-form algebraic expressions. Robust approximants that accurately reproduce the actual values of the energy components while strictly conforming to these limits are constructed, opening an avenue to stringent tests capable of predicting the performance of electronic structure methods for systems with varying extents of the dynamical and nondynamical electron correlation. The values of the correlation components, paired with the computed 1-matrices are expected to be particularly useful in the context of benchmarking of approximate density matrix functionals.

Multipolar Ewald methods, 1: theory, accuracy, and performance.

The Ewald, Particle Mesh Ewald (PME), and Fast Fourier–Poisson (FFP) methods are developed for systems composed of spherical multipole moment expansions. A unified set of equations is derived that takes advantage of a spherical tensor gradient operator formalism in both real space and reciprocal space to allow extension to arbitrary multipole order. The implementation of these methods into a novel linear-scaling modified “divide-and-conquer” (mDC) quantum mechanical force field is discussed. The evaluation times and relative force errors are compared between the three methods, as a function of multipole expansion order. Timings and errors are also compared within the context of the quantum mechanical force field, which encounters primary errors related to the quality of reproducing electrostatic forces for a given density matrix and secondary errors resulting from the propagation of the approximate electrostatics into the self-consistent field procedure, which yields a converged, variational, but nonetheless approximate density matrix. Condensed-phase simulations of an mDC water model are performed with the multipolar PME method and compared to an electrostatic cutoff method, which is shown to artificially increase the density of water and heat of vaporization relative to full electrostatic treatment.

Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time-Dependent RDMFT (TD-RDMFT).

Recent advances in reduced density matrix functional theory (RDMFT) and linear response time-dependent reduced density matrix functional theory (TD-RDMFT) are reviewed. In particular, we present various approaches to develop approximate density matrix functionals which have been employed in RDMFT. We discuss the properties and performance of most available density matrix functionals. Progress in the development of functionals has been paralleled by formulation of novel RDMFT-based methods for predicting properties of molecular systems and solids. We give an overview of these methods. The time-dependent extension, TD-RDMFT, is a relatively new theory still awaiting practical and generally useful functionals which would work within the adiabatic approximation. In this chapter we concentrate on the formulation of TD-RDMFT response equations and various adiabatic approximations. None of the adiabatic approximations is fully satisfactory, so we also discuss a phase-dependent extension to TD-RDMFT employing the concept of phase-including-natural-spinorbitals (PINOs). We focus on applications of the linear response formulations to two-electron systems, for which the (almost) exact functional is known.

On the inherited “purity” of certain extrapolated density matrices

The method of Löwdin transform – inverse Löwdin transform is extended to approximate density matrix extrapolation in cases where the reference density matrix in not “pure” that is, if it fails the test of idempotency. It is shown that a matrix expression provides a scalar measure of the overall level of impurity and for large, sparse matrices of approximate macromolecular fragment-based quantum chemistry methods the one-step Diophantine density matrix purification method is recommended if a charge-preserving improvement of the extrapolated density matrix is needed.

Excitation energies from range-separated time-dependent density and density matrix functional theory

Time-dependent density functional theory (TD-DFT) in the adiabatic formulation exhibits known failures when applied to predicting excitation energies. One of them is the lack of the doubly excited configurations. On the other hand, the time-dependent theory based on a one-electron reduced density matrix functional (time-dependent density matrix functional theory, TD-DMFT) has proven accurate in determining single and double excitations of H(2) molecule if the exact functional is employed in the adiabatic approximation. We propose a new approach for computing excited state energies that relies on functionals of electron density and one-electron reduced density matrix, where the latter is applied in the long-range region of electron-electron interactions. A similar approach has been recently successfully employed in predicting ground state potential energy curves of diatomic molecules even in the dissociation limit, where static correlation effects are dominating. In the paper, a time-dependent functional theory based on the range-separation of electronic interaction operator is rigorously formulated. To turn the approach into a practical scheme the adiabatic approximation is proposed for the short- and long-range components of the coupling matrix present in the linear response equations. In the end, the problem of finding excitation energies is turned into an eigenproblem for a symmetric matrix. Assignment of obtained excitations is discussed and it is shown how to identify double excitations from the analysis of approximate transition density matrix elements. The proposed method used with the short-range local density approximation (srLDA) and the long-range Buijse-Baerends density matrix functional (lrBB) is applied to H(2) molecule (at equilibrium geometry and in the dissociation limit) and to Be atom. The method accounts for double excitations in the investigated systems but, unfortunately, the accuracy of some of them is poor. The quality of the other excitations is in general much better than that offered by TD-DFT-LDA or TD-DMFT-BB approximations if the range-separation parameter is properly chosen. The latter remains an open problem.

Open-shell reduced density matrix functional theory

Open-shell reduced density matrix functional theory is established by investigating the domain of the exact functional. For spin states that are the ground state, a particularly simple set is found to be the domain. It cannot be generalized to other spin states. A number of conditions satisfied by the exact density matrix functional is formulated and tested for approximate functionals. The exact functional does not suffer from fractional spin error, which is the source of the static correlation error in dissociated molecules. We prove that a simple approximation (called the Buijse-Baerends functional, Müller or square root functional) has a non-positive fractional spin error. In the case of the H atom the error is zero. Numerical results for a few atoms are given for approximate density and density matrix functionals as well as a recently developed range-separated combination of both.

Theoretical investigation of electronic structure, electric field gradients, and photoemission of PuCoGa5and PuRhGa5superconductors

We report electronic structure calculations of PuCoGa${}_{5}$ and PuRhGa${}_{5}$ superconductors. The band structure, electric field gradients, and Fermi surface topologies were investigated using two methods of correlated band theory, namely the conventional static mean-field LDA$+$$U$ and the local density matrix approximation (LDMA). The latter is based on the Hubbard-I approximation and includes intra-atomic dynamical correlations in a self-consistent manner. Our results show that although LDA$+$$U$ and LDMA calculations do not significantly modify the Fermi surface topologies compared to LDA, they do lead to a substantial reduction of the $f$ character of the electronic states at the Fermi energy. The calculations indicate presence of a pseudogap in the electronic spectrum. We achieve a good agreement between calculated and experimental nuclear quadrupolar resonance frequencies and a fairly good agreement between calculated and experimental photoemission spectra. Our findings can be important for the theory of superconductivity in PuCoGa${}_{5}$ and related compounds.

Simulation of interaction of oriented J aggregates with resonance laser radiation

The interaction of laser radiation with single J aggregates of cyanine dyes is theoretically analyzed and numerically simulated. The quantum-mechanical calculations of the equilibrium geometry and the energies and intensities of the lowest singlet electronic transitions in pseudoisocyanine chloride and its linear (chain) oligomers are fulfilled. The data of these calculations can serve as parameters of the analyzed model of interaction of J aggregates with radiation in the one-particle density matrix approximation. This model takes into account relaxation processes, the annihilation of excitations at neighboring molecules, and inhomogeneous broadening. Assuming that the inhomogeneous broadening is absent, calculations demonstrate the existence of spatial bistability, molecular switching waves, and dissipative solitons. The effect of the inhomogeneous broadening and the radiation intensity on the effective coherence length in linear (chain) J aggregates is analyzed.

Double resonances upon three-photon ionization of the samarium atom

The manifestation of double resonances upon the three-photon ionization of the samarium atom has been studied in a frequency range of 17200–18400 cm−1. Thirty peaks caused by double resonances have been found in the dependence of the yield of Sm+ ions on the laser radiation frequency. The majority of these peaks greatly exceed in amplitude the peaks associated with ordinary two-photon resonances. The influence of double resonances on the three-photon ionization has been theoretically analyzed using the density matrix approximation. The calculations show that the probability of the three-photon resonance ionization can increase by more than two orders of magnitude due to the additional one-photon population of an intermediate level. The double resonance slows down the rate of increase in the probability of ionization with an increase in the laser radiation intensity compared to ordinary two-photon resonance.

A simple nonlocal model for exchange

This work presents a new nonlocal model for the exchange energy density. The model is obtained from the product of the Kohn-Sham one-particle density matrix used to construct exact [Hartree-Fock-like (HF)] exchange, and an approximate density matrix used to construct local spin-density approximation (LSDA) exchange. The proposed exchange energy density has useful formal properties, including correct spin and coordinate scaling and the correct uniform limit. It can readily be evaluated in finite basis sets, with a computational scaling intermediate between HF exchange and semilocal quantities such as the noninteracting kinetic energy density. Applications to representative systems indicate that its properties are typically intermediate between HF and LSDA exchange, and often similar to global hybrids of HF and LSDA exchange. The model is proposed as a novel "Rung 3.5" ingredient for constructing approximate exchange-correlation functionals.

An approximate density matrix energy functional

Several authors have shown that there exists an energy functional based on the one-matrix of an atomic or molecular ground state; this functional has been contrasted with energy functionals based on the two-matrix. Based on earlier work of Slater and Lowdin it is shown that the commutation rules satisfied by an approximate energy functional are just those satisfied by the exact functional. Thus there exists a known energy functional which is dependent on the one-matrix. For an idempotent density matrix this is the energy functional proposed originally by Slater.