Two-body Density Matrices Research Articles

Applications of Time-Dependent Density-Matrix Approach

The equations of motion for reduced density matrices form a coupled chain known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. To close the coupled chain at the two-body level, approximations for a three-body density matrix with one-body and two-body density matrices are needed. The time-dependent density-matrix theory (TDDM) assumes that the three-body density matrix is given by the antisymmetrized products of the one-body and two-body density matrices. In this review the truncation schemes of the BBGKY hierarchy beyond TDDM are discussed and a formulation for the study of excited states which is derived from the time-dependent approach is explained. The truncation schemes and the formulation for excited states are applied to the Lipkin model and the Hubbard model to corroborate their validity. Two realistic applications of the TDDM approaches are also presented. One is the dipole and quadrupole excitations of $^{40}$Ca and $^{48}$Ca and the other the fusion reactions of $^{16}$O + $^{16}$O.

Truncation scheme of time-dependent density-matrix approach III

The time-dependent density-matrix theory (TDDM) where the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for reduced density matrices is truncated by approximating a three-body density matrix with one-body and two-body density matrices is applied to the Lipkin model. It is shown that in the large $N$ limit the ground state in TDDM approaches the exact solution. Various truncation schemes for the three-body density matrix are also tested for an extended three-level Lipkin model.

Analytical energy gradients for explicitly correlated wave functions. I. Explicitly correlated second-order Møller-Plesset perturbation theory.

We present the theory and algorithms for computing analytical energy gradients for explicitly correlated second-order Møller-Plesset perturbation theory (MP2-F12). The main difficulty in F12 gradient theory arises from the large number of two-electron integrals for which effective two-body density matrices and integral derivatives need to be calculated. For efficiency, the density fitting approximation is used for evaluating all two-electron integrals and their derivatives. The accuracies of various previously proposed MP2-F12 approximations [3C, 3C(HY1), 3*C(HY1), and 3*A] are demonstrated by computing equilibrium geometries for a set of molecules containing first- and second-row elements, using double-ζ to quintuple-ζ basis sets. Generally, the convergence of the bond lengths and angles with respect to the basis set size is strongly improved by the F12 treatment, and augmented triple-ζ basis sets are sufficient to closely approach the basis set limit. The results obtained with the different approximations differ only very slightly. This paper is the first step towards analytical gradients for coupled-cluster singles and doubles with perturbative treatment of triple excitations, which will be presented in the second part of this series.

Evolution from few- to many-body physics in one-dimensional Fermi systems: One- and two-body density matrices and particle-partition entanglement

We study the evolution from few- to many-body physics of fermionic systems in one spatial dimension with attractive pairwise interactions. We determine the detailed form of the momentum distribution, the structure of the one-body density matrix, and the pairing properties encoded in the two-body density matrix. From the low- and high-momentum scaling behavior of the single-particle momentum distribution we estimate the speed of sound and Tan's contact, respectively. Both quantities are found to be in agreement with previous calculations. Based on our calculations of the one-body density matrices, we also present results for the particle-partition entanglement entropy, for which we find a logarithmic dependence on the total particle number.

Truncation scheme of time-dependent density-matrix approach II

A truncation scheme of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for reduced density matrices, where a three-body density matrix is approximated by two-body density matrices, is improved to take into account a normalization effect. The truncation scheme is tested for the Lipkin model. It is shown that the obtained results are in good agreement with the exact solutions.

Evolution from few- to many-body physics in one-dimensional Fermi systems: One- and two-body density matrices and particle-partition entanglement

Evolution from few- to many-body physics in one-dimensional Fermi systems: One- and two-body density matrices and particle-partition entanglement

Some universal properties of density matrices for finite nuclei (bound systems)

The properties of one-body and two-body density matrices of a nucleus as a nonrelativistic system are studied. Unlike the usual procedure applied in the theory of infinite systems, for a nucleus of A nucleons these quantities are determined as the expectation values of A-body multiplicative operators that depend on the relative coordinates and momenta (Jacobi variables) and affect the intrinsic wave functions. The translational invariance of these operators is thus ensured. An algebraic technique based on the Cartesian representation is developed for handling such operators. Within this technique, the coordinate and momentum operators are linear combinations of production and annihilation operators \(\hat \vec a^ + \) and \(\hat \vec a\) with commutation relations for bosons (oscillators). Each of the multiplicative operators reduces to the normally ordered product of two exponents, one of which depends on set \(\left\{ {\hat \vec a^ + } \right\}\) only and is located to the left of the second, which depends on \(\left\{ {\hat \vec a} \right\}\) only. This procedure offers a new view of the origin of the so-called Tassie–Barker factors and other model-independent results. In addition, the proposed method yields a fair description of the available experimental data and can easily be extended to investigate the role of short-range nucleon-nucleon correlations within these translationally invariant calculations of nucleon density and momentum distributions in light nuclei.

Two-body dissipation effects on the synthesis of superheavy elements

To investigate the two-body dissipation effects on the synthesis of superheavy elements, we calculate low-energy collisions of the $N=50$ isotones ($^{82}$Ge, $^{84}$Se, $^{86}$Kr and $^{88}$Sr) on $^{208}$Pb using the time-dependent density-matrix theory (TDDM). TDDM is an extension of the time-dependent Hartree-Fock (TDHF) theory and can determine the time evolution of one-body and two-body density matrices. Thus TDDM describes both one-body and two-body dissipation of collective energies. It is shown that the two-body dissipation may increase fusion cross sections and enhance the synthesis of superheavy elements.

Combining the Complete Active Space Self-Consistent Field Method and the Full Configuration Interaction Quantum Monte Carlo within a Super-CI Framework, with Application to Challenging Metal-Porphyrins.

A novel stochastic Complete Active Space Self-Consistent Field (CASSCF) method has been developed and implemented in the Molcas software package. A two-step procedure is used, in which the CAS configuration interaction secular equations are solved stochastically with the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) approach, while orbital rotations are performed using an approximated form of the Super-CI method. This new method does not suffer from the strong combinatorial limitations of standard MCSCF implementations using direct schemes and can handle active spaces well in excess of those accessible to traditional CASSCF approaches. The density matrix formulation of the Super-CI method makes this step independent of the size of the CI expansion, depending exclusively on one- and two-body density matrices with indices restricted to the relatively small number of active orbitals. No sigma vectors need to be stored in memory for the FCIQMC eigensolver--a substantial gain in comparison to implementations using the Davidson method, which require three or more vectors of the size of the CI expansion. Further, no orbital Hessian is computed, circumventing limitations on basis set expansions. Like the parent FCIQMC method, the present technique is scalable on massively parallel architectures. We present in this report the method and its application to the free-base porphyrin, Mg(II) porphyrin, and Fe(II) porphyrin. In the present study, active spaces up to 32 electrons and 29 orbitals in orbital expansions containing up to 916 contracted functions are treated with modest computational resources. Results are quite promising even without accounting for the correlation outside the active space. The systems here presented clearly demonstrate that large CASSCF calculations are possible via FCIQMC-CASSCF without limitations on basis set size.

Novel methods for configuration interaction and orbital optimization for wave functions containing non-orthogonal orbitals with applications to the chromium dimer and trimer.

A novel algorithm for performing configuration interaction (CI) calculations using non-orthogonal orbitals is introduced. In the new algorithm, the explicit calculation of the Hamiltonian matrix is replaced by the direct evaluation of the Hamiltonian matrix times a vector, which allows expressing the CI-vector in a bi-orthonormal basis, thereby drastically reducing the computational complexity. A new non-orthogonal orbital optimization method that employs exponential mappings is also described. To allow non-orthogonal transformations of the orbitals, the standard exponential mapping using anti-symmetric operators is supplemented with an exponential mapping based on a symmetric operator in the active orbital space. Expressions are obtained for the orbital gradient and Hessian, which involve the calculation of at most two-body density matrices, thereby avoiding the time-consuming calculation of the three- and four-body density matrices of the previous approaches. An approach that completely avoids the calculation of any four-body terms with limited degradation of convergence is also devised. The novel methods for non-orthogonal configuration interaction and orbital optimization are applied to the chromium dimer and trimer. For internuclear distances that are typical for chromium clusters, it is shown that a reference configuration consisting of optimized singly occupied active orbitals is sufficient to give a potential curve that is in qualitative agreement with complete active space self-consistent field (CASSCF) calculations containing more than 500 × 10(6) determinants. To obtain a potential curve that deviates from the CASSCF curve by less than 1 mHartree, it is sufficient to add single and double excitations out from the reference configuration.

Density-matrix based determination of low-energy model Hamiltonians from ab initio wavefunctions.

We propose a way of obtaining effective low energy Hubbard-like model Hamiltonians from ab initio quantum Monte Carlo calculations for molecular and extended systems. The Hamiltonian parameters are fit to best match the ab initio two-body density matrices and energies of the ground and excited states, and thus we refer to the method as ab initio density matrix based downfolding. For benzene (a finite system), we find good agreement with experimentally available energy gaps without using any experimental inputs. For graphene, a two dimensional solid (extended system) with periodic boundary conditions, we find the effective on-site Hubbard U(∗)/t to be 1.3 ± 0.2, comparable to a recent estimate based on the constrained random phase approximation. For molecules, such parameterizations enable calculation of excited states that are usually not accessible within ground state approaches. For solids, the effective Hamiltonian enables large-scale calculations using techniques designed for lattice models.

New truncation scheme for a time-dependent density-matrix approach applied to the ground state ofO16

The ground state of $^{16}\mathrm{O}$ is calculated by using a time-dependent density-matrix approach derived from a new truncation scheme of the Bogoliubov--Born--Green--Kirkwood--Yvon hierarchy for reduced density matrices, where a three-body density matrix is approximated by an antisymmetrized product of two-body density matrices. The new scheme is compared with a simpler truncation scheme previously used for the calculation of the ground state of $^{16}\mathrm{O}$ where the three-body density matrix is neglected and only two-particle--two-hole elements of the two-body density matrix are considered. It is shown that the results obtained from the two truncation schemes agree well with the exact solution.

Truncation scheme of time-dependent density-matrix approach

A truncation scheme of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for reduced density matrices, where a three-body density matrix is approximated by the antisymmetrized products of two-body density matrices, is proposed. This truncation scheme is tested for three model Hamiltonians. It is shown that the obtained results are in good agreement with the exact solutions.

CHAOTIC PARAMETER λ IN HANBURY-BROWN–TWISS INTERFEROMETRY IN AN ANISOTROPIC BOSON GAS MODEL

Using one- and two-body density matrices, we calculate the spatial and momentum distributions, two-particle Hanbury-Brown–Twiss (HBT) correlation functions, and the chaotic parameter λ in HBT interferometry for the systems of boson gas within the harmonic oscillator potentials with anisotropic frequencies in transverse and longitudinal directions. The HBT chaotic parameter, which can be obtained by measuring the correlation functions at zero relative momentum of the particle pair, is related to the degree of Bose–Einstein condensation and thus the system environment. We investigate the effects of system temperature, particle number and the average momentum of the particle pair on the chaotic parameter. The value of λ decreases with the condensed fraction, f0. It is one for f0 = 0 and zero for f0 = 1. For a certain f0 between 0 and 1, we find that λ increases with the average momentum of the particle pair and decreases with the particle number of system. The results of λ are sensitive to the ratio, ν = ωz/ωρ, of the frequencies in longitudinal and transverse directions. They are smaller for larger ν when ωρ is fixed. In the heavy-ion collisions at the Large Hadron Collider (LHC) energy the large identical pion multiplicity may possibly lead to a considerable Bose–Einstein condensation. Its effect on the chaotic parameter in two-pion interferometry is worth considering in earnest.

An effective energy gradient expression for divide-and-conquer second-order Møller–Plesset perturbation theory

We recently proposed a linear-scaling evaluation scheme for the second-order Mo̸ller-Plesset perturbation (MP2) energy based on the divide-and-conquer (DC) method [M. Kobayashi, Y. Imamura, and H. Nakai, J. Chem. Phys. 127, 074103 (2007)]. In this paper, we propose an approximate but effective expression for the first derivative of the DC-MP2 energy. The present scheme evaluates the one- and two-body density matrices, which appear in the MP2 gradient formula, in the DC manner; that is, the entire matrix is obtained as the sum of subsystem matrices masked by the partition matrix. Therefore, the method requires solving only the local Z-vector equations. Illustrative applications to three types of systems, peptides, Si surface model, and delocalized polyenes, reveal the effectiveness of the present method.

Time-Dependent Density-Matrix Approach to Collective Excitations of a Quantum Dot

We study collective excitations of a quantum dot consisting of two electrons using a time-dependent density-matrix approach. The advantages of the density-matrix approach are that one- and two-body observables are directly calculated using one- and two-body density matrices and that it has a clear relation to the time-dependent Hartree–Fock theory. Some low-lying spin modes associated with intrinsic transitions of a deformed configuration are predicted.

Formulation of functional theory for pairing with particle number restoration

The restoration of particle number within Energy Density Functional theory is analyzed. It is shown that the standard method based on configuration mixing leads to a functional of both the projected and non-projected densities. As an alternative that might be advantageous for mass models, nuclear dynamics and thermodynamics, we propose to formulate the functional in terms directly of the one-body and two-body density matrices of the state with good particle number. Our approach does not contain the pathologies recently observed when restoring the particle number in an Energy Density Functional framework based on transition density matrices and can eventually be applied with functionals having arbitrary density dependencies.

Trapped two-component Fermi gases with up to six particles: Energetics, structural properties, and molecular condensate fraction

We investigate small equal-mass two-component Fermi gases under external spherically symmetric confinement in which atoms with opposite spins interact through a short-range two-body model potential. We employ a non-perturbative microscopic framework, the stochastic variational approach, and determine the system properties as functions of the interspecies s-wave scattering length a, the orbital angular momentum L of the system, and the numbers N1 and N2 of spin-up and spin-down atoms (with N1-N2 =0 or 1 and N < 7, where N=N1+N2). At unitarity, we determine the energies of the five- and six-particle systems for various ranges r0 of the underlying two-body model potential and extrapolate to the zero-range limit. These energies serve as benchmark results that can be used to validate and assess other numerical approaches. We also present structural properties such as the pair distribution function and the radial density. Furthermore, we analyze the one-body and two-body density matrices. A measure for the molecular condensate fraction is proposed and applied. Our calculations show explicitly that the natural orbitals and the momentum distributions of atomic Fermi gases approach those characteristic for a molecular Bose gas if the s-wave scattering length a, a>0, is sufficiently small.

Ground state of a mixture of two bosonic Calogero-Sutherland gases with strong odd-wave interspecies attraction

A model of two Calogero-Sutherland Bose gases A and B with strong odd-wave AB attractions induced by a p-wave AB Feshbach resonance is studied. The ground state wave function is found analytically by a Bose-Bose duality mapping, which permits one to accurately determine static physical properties by a Monte Carlo method. The condensation of particles or particle pairs (molecules) is tested by analyzing the presence of the off-diagonal long-range order in one- or two-body density matrices. The p-wave symmetry of AB interaction makes possible quasi-condensation of type A particles at the Fermi momentum of the B component. The zero-temperature phase diagram is drawn in terms of densities and interaction strengths.

Ultracold atoms at unitarity within quantum Monte Carlo methods

Variational and diffusion quantum Monte Carlo (VMC and DMC) calculations of the properties of the zero-temperature fermionic gas at unitarity are reported. The ratio of the energy of the interacting to the non-interacting gas for a system of 128 particles is calculated to be 0.4517(3) in VMC and 0.4339(1) in the more accurate DMC method. The spherically-averaged pair-correlation functions, momentum densities, and one-body density matrices are very similar in VMC and DMC, but the two-body density matrices and condensate fractions show some differences. Our best estimate of the condensate fraction of 0.51 is a little smaller than values from other quantum Monte Carlo calculations.