One-particle Density Research Articles

A well-scaling natural orbital theory

We introduce an energy functional for ground-state electronic structure calculations. Its variables are the natural spin-orbitals of singlet many-body wave functions and their joint occupation probabilities deriving from controlled approximations to the two-particle density matrix that yield algebraic scaling in general, and Hartree-Fock scaling in its seniority-zero version. Results from the latter version for small molecular systems are compared with those of highly accurate quantum-chemical computations. The energies lie above full configuration interaction calculations, close to doubly occupied configuration interaction calculations. Their accuracy is considerably greater than that obtained from current density-functional theory approximations and from current functionals of the one-particle density matrix.

Quantum Correlations of Ideal Bose and Fermi Gases in the Canonical Ensemble

We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with $N$ particles. The pair distribution function $g^{(2)}(r)$ of fermions clearly exhibits antibunching with $g^{(2)}(0)=0$ due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with $g^{(2)}(0)\approx 2$, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature $T_0$, an off-diagonal long-range order develops in the one-particle density matrix to reach $g^{(1)}(r)=1$ at $T=0$, and the pair correlation starts to decrease towards $g^{(2)}(r)\approx 1$ at $T=0$. The results for $N\rightarrow \infty$ are seen to converge to those of the grand canonical ensemble obtained by assuming the average $\langle\hat\psi({\bf r})\rangle$ of the field operator $\hat\psi({\bf r})$ below $T_0$. This fact justifies the introduction of the "anomalous" average $\langle\hat\psi({\bf r})\rangle\neq 0$ below $T_0$ in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.

Analytic energy derivatives for the calculation of the first-order molecular properties using the domain-based local pair-natural orbital coupled-cluster theory

The domain-based local pair-natural orbital coupled-cluster (DLPNO-CC) theory has recently emerged as an efficient and powerful quantum-chemical method for the calculation of energies of molecules comprised of several hundred atoms. It has been demonstrated that the DLPNO-CC approach attains the accuracy of a standard canonical coupled-cluster calculation to about 99.9% of the basis set correlation energy while realizing linear scaling of the computational cost with respect to system size. This is achieved by combining (a) localized occupied orbitals, (b) large virtual orbital correlation domains spanned by the projected atomic orbitals (PAOs), and (c) compaction of the virtual space through a truncated pair natural orbital (PNO) basis. In this paper, we report on the implementation of an analytic scheme for the calculation of the first derivatives of the DLPNO-CC energy for basis set independent perturbations within the singles and doubles approximation (DLPNO-CCSD) for closed-shell molecules. Perturbation-independent one-particle density matrices have been implemented in order to account for the response of the CC wave function to the external perturbation. Orbital-relaxation effects due to external perturbation are not taken into account in the current implementation. We investigate in detail the dependence of the computed first-order electrical properties (e.g., dipole moment) on the three major truncation parameters used in a DLPNO-CC calculation, namely, the natural orbital occupation number cutoff used for the construction of the PNOs, the weak electron-pair cutoff, and the domain size cutoff. No additional truncation parameter has been introduced for property calculation. We present benchmark calculations on dipole moments for a set of 10 molecules consisting of 20-40 atoms. We demonstrate that 98%-99% accuracy relative to the canonical CCSD results can be consistently achieved in these calculations. However, this comes with the price of tightening the threshold for the natural orbital occupation number cutoff by an order of magnitude compared to the DLPNO-CCSD energy calculations.

Manifestly Hermitian semiclassical expansion for the one-particle density matrix of a two-dimensional Fermi gas

The semiclassical $\hbar$-expansion of the one-particle density matrix for a two-dimensional Fermi gas is calculated within the Wigner transform method of Grammaticos and Voros, originally developed in the context of nuclear physics. The method of Grammaticos and Voros has the virture of preserving both the Hermiticity and idempotency of the density matrix to all orders in the $\hbar$-expansion. As a topical application, we use our semiclassical expansion to go beyond the local-density approximation for the construction of the total dipole-dipole interaction energy functional of a two-dimensional, spin-polarized dipolar Fermi gas. We find a {\em finite}, second-order gradient correction to the Hartree-Fock energy, which takes the form $\varepsilon (\nabla \rho)^2/\sqrt{\rho}$, with $\varepsilon$ being small ($|\varepsilon| \ll1$) and negative. We test the quality of the corrected energy by comparing it with the exact results available for harmonic confinement. Even for small particle numbers, the gradient correction to the dipole-dipole energy provides a significant improvement over the local-density approximation.

Thermalization after photoexcitation from the perspective of optical spectroscopy

We analyze the thermalization of a photoexcited charge carrier coupled to a single branch of quantum phonons within the Holstein model. To this end, we calculate the far-from-equilibrium time evolution of a pure many-body state and compare it with predictions of the thermal Gibbs ensemble. We show that at strong enough carrier excitation, the nonequilibrium system evolves towards a thermal steady state. Our analysis is based on two classes of observables. First, the occupations of fermionic momenta, which are the eigenvalues of the one-particle density matrix, match in the steady state the values in the corresponding Gibbs ensemble. This indicates thermalization of static fermionic correlations on the entire lattice. Second, the dynamic current-current correlations, including the time-resolved optical conductivity, also take the form of their thermal counterparts. Remarkably, both static and dynamic fermionic correlations thermalize with identical temperatures. Our results suggest that the subsequent relaxation processes, observed in time-resolved ultrafast spectroscopy, may be efficiently described by applying quasithermal approaches, e.g., multi-temperature models.

Natural ionization orbitals for interpreting electron detachment processes.

A compact orbital representation of ionization processes is described utilizing the difference of calculated one-particle density matrices. Natural orbital analysis involving this difference density matrix simplifies interpretation of electronic detachment processes and allows differentiation between one-electron transitions and shake-up/shake-off transitions, in which one-electron processes are accompanied by excitation of a second electron into the virtual orbital space.

A Theoretical Study of the Atomic Properties for Subshells of N + and O +2 Using Hartree-Fock Approximation

In this research, we calculated the atomic properties of systems have been studied (N+ and O+2 ) for intra-shells (1s, 2s and 2p) using Hartree-Fock wave function. These properties included, one-particle radial density function, one-particle and inter-particle expectation values, inter-particle density function and expectation values of energies. All these atomic properties increase with atomic number, have highest values in 1s shell and lowest values in 2p shell. All results are obtained numerically by using the computer program (MathCad 14) because it able to calculation and plot functions. All atomic properties are calculated in atomic units.

Comparison of the accuracy of approximate methods TrESP and TrCAMM for evaluation of pigment coupling in light-harvesting complexes

Due to the large size of light-harvesting complexes, approximate methods are commonly used for calculating the coupling energies between their constituents. Two approximate methods are studied in this work: TrESP and the recently suggested TrCAMM method, based on the expansion of the one-particle density matrix. The quality of approximation of the electrostatic potential in the framework of these two approaches has been compared for two biological pigments as an example—chlorophyll a and bacteriochlorophyll a. It has been shown that both approaches provide high accuracy of approximation of the matrix elements of the electrostatic potential operator. In addition, it has been demonstrated that symmetrization of the one-particle density matrix in the framework of the TrCAMM method significantly improves the calculation accuracy and makes the computational procedure unambiguous.

Efficient Temperature-Dependent Green's Function Methods for Realistic Systems: Using Cubic Spline Interpolation to Approximate Matsubara Green's Functions.

The popular, stable, robust, and computationally inexpensive cubic spline interpolation algorithm is adopted and used for finite temperature Green's function calculations of realistic systems. We demonstrate that with appropriate modifications the temperature dependence can be preserved while the Green's function grid size can be reduced by about 2 orders of magnitude by replacing the standard Matsubara frequency grid with a sparser grid and a set of interpolation coefficients. We benchmarked the accuracy of our algorithm as a function of a single parameter sensitive to the shape of the Green's function. Through numerous examples, we confirmed that our algorithm can be utilized in a systematically improvable, controlled, and black-box manner and highly accurate one- and two-body energies and one-particle density matrices can be obtained using only around 5% of the original grid points. Additionally, we established that to improve accuracy by an order of magnitude, the number of grid points needs to be doubled, whereas for the Matsubara frequency grid, an order of magnitude more grid points must be used. This suggests that realistic calculations with large basis sets that were previously out of reach because they required enormous grid sizes may now become feasible.

Communication: Generalized canonical purification for density matrix minimization.

A Lagrangian formulation for the constrained search for the N-representable one-particle density matrix based on the McWeeny idempotency error minimization is proposed, which converges systematically to the ground state. A closed form of the canonical purification is derived for which no a posteriori adjustment on the trace of the density matrix is needed. The relationship with comparable methods is discussed, showing their possible generalization through the hole-particle duality. The appealing simplicity of this self-consistent recursion relation along with its low computational complexity could prove useful as an alternative to diagonalization in solving dense and sparse matrix eigenvalue problems.

Time-dependent Landauer—Büttiker formalism for superconducting junctions at arbitrary temperatures

We discuss an extension of our earlier work on the time-dependent Landauer- Buttiker formalism for noninteracting electronic transport. The formalism can without complication be extended to superconducting central regions since the Green's functions in the Nambu representation satisfy the same equations of motion which, in turn, leads to the same closed expression for the equal-time lesser Green's function, i.e., for the time-dependent reduced one-particle density matrix. We further write the finite-temperature frequency integrals in terms of known special functions thereby considerably speeding up the computation. Simulations in simple normal metal - superconductor - normal metal junctions are also presented.

Many-body ground state localization and coexistence of localized and extended states in an interacting quasiperiodic system

We study the localization problem of one-dimensional interacting spinless fermions in an incommensurate optical lattice, which changes from an extended phase to a nonergoic many-body localized phase by increasing the strength of the incommensurate potential. We identify that there exists an intermediate regime before the system enters the many-body localized phase, in which both the localized and extended many-body states coexist, thus the system is divided into three different phases, which can be characterized by normalized participation ratios of the many-body eigenstates and distributions of natural orbitals of the corresponding one-particle density matrix. This is very different from its noninterating limit, in which all eigenstaes undergo a delocaliztion-localization transtion when the strength of the incommensurate potential exceeds a critical value.

A Variational Framework for Spectral Approximations of Kohn–Sham Density Functional Theory

We reformulate the Kohn–Sham density functional theory (KSDFT) as a nested variational problem in the one-particle density operator, the electrostatic potential and a field dual to the electron density. The corresponding functional is linear in the density operator and thus amenable to spectral representation. Based on this reformulation, we introduce a new approximation scheme, termed spectral binning, which does not require smoothing of the occupancy function and thus applies at arbitrarily low temperatures. We prove convergence of the approximate solutions with respect to spectral binning and with respect to an additional spatial discretization of the domain.

Quantum Brownian motion induced by thermal noise in the presence of disorder

The motion of a quantum particle hopping on a simple cubic lattice under the influence of thermal noise and of a static random potential is expected to be diffusive, i.e., the particle is expected to exhibit “quantum Brownian motion,” no matter how weak the thermal noise is. This is shown to be true in a model where the dynamics of the particle is governed by a Lindblad equation for a one-particle density matrix. The generator appearing in this equation is the sum of two terms: a Liouvillian corresponding to a random Schrödinger operator and a Lindbladian describing the effect of thermal noise in the kinetic limit. Under suitable but rather general assumptions on the Lindbladian, the diffusion constant characterizing the asymptotics of the motion of the particle is proven to be strictly positive and finite. If the disorder in the random potential is so large that transport is completely suppressed in the limit where the thermal noise is turned off, then the diffusion constant tends to zero proportional to the coupling of the particle to the heat bath.

Mean-field evolution of fermionic systems

We study the dynamics of interacting fermionic systems, in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on the inital data and on the many-body interaction, the quantum evolution of the system is approximated by a time-dependent quasi-free state. In particular we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent Hartree-Fock equation. Our theorems allow to describe the dynamics of both pure states (zero temperature states) and mixed states (positive temperature states). Our results hold for all times, and give effective estimates on the rate of convergence towards the Hartree-Fock evolution. The results on pure states are based on joint works with N. Benedikter and B. Schlein, [5, 6]; while those on mixed states are based on a joint work with N. Benedikter, V. Jaksic, C. Saffirio and B. Schlein, [7].

Hermitian one-particle density matrix through a semiclassical gradient expansion

We carry out the semiclassical expansion of the one-particle density matrix up to the second order in We use the method of Grammaticos and Voros based on the Wigner transform of operators. We show that the resulting density matrix is Hermitian and idempotent in contrast with the well-known result of the semiclassical Kirzhnits expansion. Our density matrix leads to the same particle density and kinetic energy density as in the literature, and it satisfies the consistency criterion of the Euler equation. The derived Hermitian density matrix clarifies the ambiguity in the usefulness of gradient expansion approximations and might reignite the development of density functionals with semiclassical methods.

Kinetic and dynamic probability-density-function descriptions of disperse turbulent two-phase flows.

This article analyzes the status of two classical one-particle probability density function (PDF) descriptions of the dynamics of discrete particles dispersed in turbulent flows. The first PDF formulation considers only the process made up by particle position and velocity Z(p)=(x(p),U(p)) and is represented by its PDF p(t; y(p),V(p)) which is the solution of a kinetic PDF equation obtained through a flux closure based on the Furutsu-Novikov theorem. The second PDF formulation includes fluid variables into the particle state vector, for example, the fluid velocity seen by particles Z(p)=(x(p),U(p),U(s)), and, consequently, handles an extended PDF p(t; y(p),V(p),V(s)) which is the solution of a dynamic PDF equation. For high-Reynolds-number fluid flows, a typical formulation of the latter category relies on a Langevin model for the trajectories of the fluid seen or, conversely, on a Fokker-Planck equation for the extended PDF. In the present work, a new derivation of the kinetic PDF equation is worked out and new physical expressions of the dispersion tensors entering the kinetic PDF equation are obtained by starting from the extended PDF and integrating over the fluid seen. This demonstrates that, under the same assumption of a Gaussian colored noise and irrespective of the specific stochastic model chosen for the fluid seen, the kinetic PDF description is the marginal of a dynamic PDF one. However, a detailed analysis reveals that kinetic PDF models of particle dynamics in turbulent flows described by statistical correlations constitute incomplete stand-alone PDF descriptions and, moreover, that present kinetic-PDF equations are mathematically ill posed. This is shown to be the consequence of the non-Markovian characteristic of the stochastic process retained to describe the system and the use of an external colored noise. Furthermore, developments bring out that well-posed PDF descriptions are essentially due to a proper choice of the variables selected to describe physical systems and guidelines are formulated to emphasize the key role played by the notion of slow and fast variables.

Communication: Exciton analysis in time-dependent density functional theory: How functionals shape excited-state characters.

Excited-state descriptors based on the one-particle transition density matrix referring to the exciton picture have been implemented for time-dependent density functional theory. State characters such as local, extended ππ(∗), Rydberg, or charge transfer can be intuitively classified by simple comparison of these descriptors. Strong effects of the choice of the exchange-correlation kernel on the physical nature of excited states can be found and decomposed in detail leading to a new perspective on functional performance and the design of new functionals.

Anisotropic pair correlations in binary and multicomponent hard-sphere mixtures in the vicinity of a hard wall: A combined density functional theory and simulation study

The fundamental measure approach to classical density functional theory has been shown to be a powerful tool to predict various thermodynamic properties of hard-sphere systems. We employ this approach to determine not only one-particle densities but also two-particle correlations in binary and six-component mixtures of hard spheres in the vicinity of a hard wall. The broken isotropy enables us to carefully test a large variety of theoretically predicted two-particle features by quantitatively comparing them to the results of Brownian dynamics simulations. Specifically, we determine and compare the one-particle density, the total correlation functions, their contact values, and the force distributions acting on a particle. For this purpose, we follow the compressibility route and theoretically calculate the direct correlation functions by taking functional derivatives. We usually observe an excellent agreement between theory and simulations, except for small deviations in cases where local crystal-like order sets in. Our results set the course for further investigations on the consistency of functionals as well as for structural analysis on, e.g., the primitive model. In addition, we demonstrate that due to the suppression of local crystallization, the predictions of six-component mixtures are better than those in bidisperse or monodisperse systems. Finally, we are confident that our results of the structural modulations induced by the wall lead to a deeper understanding of ordering in anisotropic systems in general, the onset of heterogeneous crystallization, caging effects, and glassy dynamics close to a wall, as well as structural properties in systems with confinement.

Band structure and density of states in FeAs-based superconductors

The generalized quantum Monte Carlo algorithm was used to obtain one-particle excitation spectrum and electron density of states for two-dimensional FeAs-clusters modeling iron-based superconductors within the limits of the full two-orbital model. The calculations were performed for clusters with sizes up to 10×10 FeAs-cells. The excitation spectra were reconstructed from Matsubara Green's function. The spectral density of states and the total density of states near the Fermi level were obtained. The data are in accordance with known experimental results. The influence of the cluster size, temperature, and the interaction strength on the density of states was analysed.