One-particle Density Matrix Research Articles

Nonspherical model density matrices for Rung 3.5 density functionals

"Rung 3.5" exchange-correlation functionals for Kohn-Sham density functional theory depend linearly on the nonlocal one-particle density matrix of the noninteracting Kohn-Sham reference system. Rung 3.5 functionals also require a semilocal model for the one-particle density matrix. This work presents new model density matrices for Rung 3.5 functionals. The resulting functionals give reasonable predictions for total energies, molecular thermochemistry and kinetics, odd-electron bonds, and conjugated polymer bandgaps. Global-hybrid-like combinations of semilocal and Rung 3.5 exchange, and empirical density matrix models, also show promise.

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.

Quantum continuum mechanics in a strong magnetic field

We extend a recent formulation of quantum continuum mechanics [J. Tao et. al, Phys. Rev. Lett. {\bf 103}, 086401 (2009)] to many-body systems subjected to a magnetic field. To accomplish this, we propose a modified Lagrangian approach, in which motion of infinitesimal volume elements of the system is referred to the "quantum convective motion" that the magnetic field produces already in the ground-state of the system. In the linear approximation, this approach results in a redefinition of the elastic displacement field $\uv$, such that the particle current $\jv$ contains both an electric displacement and a magnetization contribution: $\jv=\jv_0+n_0\partial_t \uv+\nabla \times (\jv_0\times \uv)$, where $n_0$ and $\jv_0$ are the particle density and the current density of the ground-state and $\partial_t$ is the partial derivative with respect to time. In terms of this displacement, we formulate an "elastic approximation" analogous to the one proposed in the absence of magnetic field. The resulting equation of motion for $\uv$ is expressed in terms of ground-state properties -- the one-particle density matrix and the two-particle pair correlation function -- and in this form it neatly generalizes the equation obtained for vanishing magnetic field.

Finite-temperature properties of one-dimensional hard-core bosons in a quasiperiodic optical lattice

We investigate the properties of impenetrable bosons confined in a one-dimensional lattice at finite temperature in the presence of an additional incommensurate periodic potential. Relying on the exact Fermi-Bose mapping, we study the effects of temperature on the one-particle density matrix and related quantities such as the momentum distribution function and the natural orbitals. We found evidence of a finite-temperature crossover related to the zero-temperature superfluid-to-Bose-glass transition that induces a delocalization of the lowest natural orbitals.

An efficient linear scaling procedure for constructing localized orbitals of large molecules based on the one-particle density matrix

We have developed a linear-scaling algorithm for obtaining the Boys localized molecular orbitals from the one-particle density matrix. The algorithm is made up of two steps: the Cholesky decomposition of the density matrix to obtain Cholesky molecular orbitals and the subsequent Boys localization process. Linear-scaling algorithms have been proposed to achieve linear-scaling calculations of these two steps, based on the sparse matrix technique and the locality of the Cholesky molecular orbitals. The present algorithm has been applied to compute the Boys localized orbitals in a number of systems including α-helix peptides, water clusters, and protein molecules. Illustrative calculations demonstrate that the computational time of obtaining Boys localized orbitals with the present algorithm is asymptotically linear with increasing the system size.

Isotropic contact forces in arbitrary representation: Heterogeneous few-body problems and low dimensions

The Bethe-Peierls asymptotic approach which models pairwise short-range forces by contact conditions is introduced in arbitrary representation for spatial dimensions less than or equal to 3. The formalism is applied in various situations and emphasis is put on the momentum representation. In the presence of a transverse harmonic confinement, dimensional reduction toward two-dimensional (2D) or one-dimensional (1D) physics is derived within this formalism. The energy theorem relating the mean energy of an interacting system to the asymptotic behavior of the one-particle density matrix illustrates the method in its second quantized form. Integral equations that encapsulate the Bethe-Peierls contact condition for few-body systems are derived. In three dimensions, for three-body systems supporting Efimov states, a nodal condition is introduced in order to obtain universal results from the Skorniakov Ter-Martirosian equation and the Thomas collapse is avoided. Four-body bound state eigenequations are derived and the 2D '3+1' bosonic ground state is computed as a function of the mass ratio.

On the optimal symmetric purification scheme of the one-particle density matrix

Density matrix purification is an efficient way of avoiding the expensive cubic scaling diagonalization in self-consistent field calculations. Although there are a number of different algorithms suggested to reduce the number of matrix multiplications for purification, there is no rigorous mathematical proof which scheme is optimal. In this Letter, we show analytically that the repeated application of the fifth-order Holas polynomial throughout all iterations is an optimal scheme that reduces the error symmetrically for both occupied and virtual occupations, and either the use of lower/higher-order polynomials throughout or mixed use of polynomials of different degree at different iteration results in higher cost.

Density functional, density matrix functional, and the virial theorem

On the basis of the virial theorem for finite motion of interacting electrons in a static external field, it is shown that ground-state energy is determined evidently by the one-particle density matrix. The exact expression for ground-state energy is found and can be applied to various problems, in particular, in studying the electronic structure of atoms, molecules, and other bound complexes.

Virial theorem, one-particle density matrix, and equilibrium condition in an external field

Based on the analysis of the virial theorem and stress tensor, it is shown that the average energy of the Coulomb system under a static external field in a given volume is completely defined by a one-particle density matrix. In this case, a one-particle density matrix satisfies a closed equation expressing the equilibrium condition in a given static external field. An expression for the pressure of a heterogeneous system of interacting particles is found.

Analytic energy gradient in combined time-dependent density functional theory and polarizable force field calculation

Formulas for evaluating analytic energy gradient are derived for combined time-dependent density functional theory (TDDFT) and polarizable force field methods that incorporate dipole polarizability tensors and linearly induced point dipoles. The Z-vector method for determining relaxed one-particle difference density matrix in regular TDDFT methods is extended to include induced dipoles. The analytic gradient of the mutual polarization energy of the force field and the TDDFT excited state can be formulated by using the TDDFT difference density-induced dipoles and the transition state density-induced dipoles. All the forces and torques involving induced dipoles can be efficiently evaluated using standard electrostatic formulas as if the induced dipoles were permanent dipoles. The formulas are given in the most general form and are applicable to various flavors of polarizable force fields. Implementation and tests with a polarizable five-point water model show that the formulas are rigorous. The carbonyl vibration modes and infrared spectrum intensities of a cluster formed by acetone and two water molecules are studied.

Combining the advantages of semi-direct schemes and linear-scaling self-consistent field methods

A new stored-integral self-consistent field (SCF) scheme is presented for both Hartree–Fock (HF) and density-functional theory (DFT), which combines the advantages originally attributed to the direct and also linear-scaling SCF methods with those of traditional indirect or semi-direct approaches. By storing the density-screened two-electron integrals in the first time-consuming SCF iteration, recalculation of integrals can be almost avoided in subsequent iterations as the one-particle density matrix is shown to be quite representative for the one in later iterations. For the dominant exchange-type contractions, speed-ups of typically 1.5–3.7 are observed.

Rung 3.5 density functionals

This work proposes a new term on the "Jacob's ladder" of approximate exchange-correlation functionals in Kohn-Sham density functional theory. In these Rung 3.5 functionals, the exchange-correlation energy density at a point depends linearly (rather than quadratically) on the nonlocal Kohn-Sham one-particle density matrix in a finite neighborhood around the point. These functionals are intermediate in complexity between the semilocal approximations of Rungs 1-3, and fully nonlocal Rung 4 approximations such as global hybrids. Rung 3.5 functionals built on the model for exchange in [B. G. Janesko, J. Chem. Phys. 131, 234111 (2009)] predict molecular thermochemistry and kinetics with accuracy intermediate between their "parent" semilocal functional and the corresponding global hybrid. The best Rung 3.5 functional presented here gives mean absolute errors of 5.7 kcal/mol for G3/99 thermochemistry, 5.2 kcal/mol for HTBH38/04 hydrogen-transfer reaction barriers, and 5.7 kcal/mol for NHTBH38/04 nonhydrogen-transfer reaction barriers, while incorporating only two empirical parameters.

Response Calculations with an Independent Particle System with an Exact One-Particle Density Matrix

We use the natural orbitals to define an independent particle system, from which the exact one-particle density matrix can be obtained with an ensemble of degenerate determinantal ground states. Also defining explicit phases for the orbitals, and admitting functionals that are dependent on those phases, time-dependent equations for the orbitals and occupation numbers are obtained from an action principle. The wrong polarizability and lack of double excitations of straightforward adiabatic time-dependent density matrix functional theory are then corrected, and the important symmetry χ(ω)=χ{*}(-ω), lost in previous ad hoc improvements, is restored. The extension of the response calculations beyond the occupied-virtual pairs, which are the only ones admitted in time-dependent density functional theory, leads to greatly improved response properties.

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.

Benchmarking the variational cluster approach by means of the one-dimensional Bose-Hubbard model

Convergence properties of the variational cluster approach with respect to the variational parameter space, cluster size, and boundary conditions of the reference system are investigated and discussed for bosonic many-body systems. Specifically, the variational cluster approach is applied to the one-dimensional Bose-Hubbard model, which exhibits a quantum phase transition from Mott to superfluid phase. In order to benchmark the variational cluster approach, results for the phase boundary delimiting the first Mott lobe are compared with essentially exact density matrix renormalization group data. Furthermore, static quantities, such as the ground state energy and the one-particle density matrix are compared with high-order strong coupling perturbation theory results. For reference systems with open boundary conditions the variational parameter space is extended by an additional variational parameter which allows for a more uniform particle density on the reference system and thus drastically improves the results. It turns out that the variational cluster approach yields accurate results with relatively low-computational effort for both the phase boundary as well as the static properties of the one-dimensional Bose-Hubbard model, even at the tip of the first Mott lobe where correlation effects are most pronounced.

Continuum mechanics for quantum many-body systems: Linear response regime

We derive a closed equation of motion for the current density of an inhomogeneous quantum many-body system under the assumption that the time-dependent wave function can be described as a geometric deformation of the ground-state wave function. By describing the many-body system in terms of a single collective field we provide an alternative to traditional approaches, which emphasize one-particle orbitals. We refer to our approach as continuum mechanics for quantum many-body systems. In the linear response regime, the equation of motion for the displacement field becomes a linear fourth-order integro-differential equation, whose only inputs are the one-particle density matrix and the pair correlation function of the ground-state. The complexity of this equation remains essentially unchanged as the number of particles increases. We show that our equation of motion is a hermitian eigenvalue problem, which admits a complete set of orthonormal eigenfunctions under a scalar product that involves the ground-state density. Further, we show that the excitation energies derived from this approach satisfy a sum rule which guarantees the exactness of the integrated spectral strength. Our formulation becomes exact for systems consisting of a single particle, and for any many-body system in the high-frequency limit. The theory is illustrated by explicit calculations for simple one- and two-particle systems.

Electronic excitation energies in solution at equation of motion CCSD level within a state specific polarizable continuum model approach

We present a study of excitation energies in solution at the equation of motion coupled cluster singles and doubles (EOM-CCSD) level of theory. The solvent effect is introduced with a state specific polarizable continuum model (PCM), where the solute-solvent interaction is specific for the state of interest. Three definitions of the excited state one-particle density matrix (1PDM) are tested in order to gain information for the development of an integrated EOM-CCSD/PCM method. The calculations show the accuracy of this approach for the computation of such property in solution. Solvent shifts between nonpolar and polar solvents are in good agreement with experiment for the test cases. The completely unrelaxed 1PDM is shown to be a balanced choice between computational effort and accuracy for vertical excitation energies, whereas the response of the ground state CCSD amplitudes and of the molecular orbitals is important for other properties, as for instance the dipole moment.

Application of Large-Scale First Principles Quantum Mechanical Calculations With the ONETEP Program To Biophysical Problems

Properties and processes at the molecular level are determined by the interactions between electrons and atomic nuclei whose motions are governed by the theory of quantum mechanics. Therefore, an unbiased and highly accurate approach for molecular simulations can be provided by quantum mechanical calculations from “first principles”, which do not rely on empirical parameters. Density Functional Theory (DFT) is such an approach and is widely used because of its computational efficiency but the computational effort of conventional DFT increases with the third power of the number of atoms. As a result it is practically not feasible even on supercomputers to perform DFT calculations with more than a few hundred atoms. Novel reformulations of DFT based on the one-particle density matrix can lead to computational effort which increases linearly with the number of atoms and hence overcome this length-scale problem. I will briefly describe how this is achieved in our ONETEP linear-scaling DFT program which is designed to achieve the same high level of accuracy as conventional cubic-scaling approaches. Then I will give an overview of current applications we are performing with ONETEP, including quantum mechanical simulations of entire proteins with thousands of atoms participating in protein-protein and protein-ligand complexes.[1] C.-K. Skylaris, P. D. Haynes, A. A. Mostofi and M. C. Payne. J. Chem. Phys. 122 (2005) 084119.[2] L. Heady, M. Fernandez-Serra, R. L. Mancera, S. Joyce, A. R. Venkitaraman, E. Artacho, C.-K. Skylaris, L. Colombi Ciacchi and M. C. Payne. J. Med. Chem. 49 (2006) 5141.[3] C.-K. Skylaris, P. D. Haynes, A. A. Mostofi and M. C. Payne. J. Phys. Condens. Matter 20 (2008) 064209.

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.

Bogolubov–Hartree–Fock mean field theory for neutron stars and other systems with attractive interactions

A simplification of the Bogolubov–Hartree–Fock theory, which is a natural generalization of the traditional Hartree–Fock theory, is derived. This simplification allows to express the pairing interaction in terms of the one-particle density matrix for systems interacting by attractive pair potentials, such as the Newtonian gravitational potential.