Two-body Density Research Articles

The [formula omitted] asymmetric nuclear matter two-body density distributions versus those of [formula omitted

The theoretical computations of the electron–nucleus scattering can be improved, by employing the asymmetric nuclear matter (ASM) two-body density distributions (TBDD). But, due to the sophistications of the calculations, the TBDD with arbitrary isospin asymmetry have not yet been computed in the Fermi Hypernetted Chain (FHNC) or the Monte Carlo (MC) approaches. So, in the present work, we intend to find the ASMTBDD, in the states with isospinT, spin S and spin projection Sz, in the Lowest Order Constrained Variational (LOCV) method. It is demonstrated that, at small relative distances, independent of the proton to neutron ratio β, the state-dependent TBDD have a universal shape. Expectedly, it is observed that, at low (high) β values, the nucleons prefer to make a pair in the T=1(0) states. In addition, the strength of the tensor-dependent correlations is investigated, using the ratio of the TBDD in the TSSz=010 state with θ=π∕2 and that of θ=0. The mentioned ratios peak at r∼0.9fm, considering different β values. It is hoped that, the present results could help a better reproduction of the experimental data of the electron–nucleus scattering.

Structural Origin of Shear Viscosity of Liquid Water.

The relation between the microscopic structure and shear viscosity of liquid water was analyzed by calculating the cross-correlation between the shear stress and the two-body density using the molecular dynamics simulation. The slow viscoelastic relaxation that dominates the steady-state shear viscosity was ascribed to the destruction of the hydrogen-bonding network structure along the compression axis of the shear distortion, which resembles the structural change under isotropic hydrostatic compression. It means that the shear viscosity of liquid water reflects the anisotropic destruction-formation dynamics of the hydrogen-bonding network.

Exact mean-energy expansion of Ginibre's gas for coupling constants Γ=2×(oddinteger).

Using the approach of a Vandermonde determinant to the power Γ=Q^{2}/k_{B}T expansion on monomial functions, a way to find the excess energy U_{exc} of the two-dimensional one-component plasma (2DOCP) on hard and soft disks (or a Dyson gas) for odd values of Γ/2 is provided. At Γ=2, the present study not only corroborates the result for the particle-particle energy contribution of the Dyson gas found by Shakirov [Shakirov, Phys. Lett. A 375, 984 (2011)10.1016/j.physleta.2011.01.004] by using an alternative approach, but also provides the exact N-finite expansion of the excess energy of the 2DOCP on the hard disk. The excess energy is fitted to the ansatz of the form U_{exc}=K^{1}N+K^{2}sqrt[N]+K^{3}+K^{4}/N+O(1/N^{2}) to study the finite-size correction, with K^{i} coefficients and N the number of particles. In particular, the bulk term of the excess energy is in agreement with the well known result of Jancovici for the hard disk in the thermodynamic limit [Jancovici, Phys. Rev. Lett. 46, 386 (1981)10.1103/PhysRevLett.46.386]. Finally, an expression is found for the pair correlation function which still keeps a link with the random matrix theory via the kernel in the Ginibre ensemble [Ginibre, J. Math. Phys. 6, 440 (1965)10.1063/1.1704292] for odd values of Γ/2. A comparison between the analytical two-body density function and histograms obtained with Monte Carlo simulations for small systems and Γ=2,6,10,... shows that the approach described in this paper may be used to study analytically the crossover behavior from systems in the fluid phase to small crystals.

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.

Semiclassics in a system without classical limit: The few-body spectrum of two interacting bosons in one dimension.

We present a semiclassical study of the spectrum of a few-body system consisting of two short-range interacting bosonic particles in one dimension, a particular case of a general class of integrable many-body systems where the energy spectrum is given by the solution of algebraic transcendental equations. By an exact mapping between δ-potentials and boundary conditions on the few-body wave functions, we are able to extend previous semiclassical results for single-particle systems with mixed boundary conditions to the two-body problem. The semiclassical approach allows us to derive explicit analytical results for the smooth part of the two-body density of states that are in excellent agreement with numerical calculations. It further enables us to include the effect of bound states in the attractive case. Remarkably, for the particular case of two particles in one dimension, the discrete energy levels obtained through a requantization condition of the smooth density of states are essentially in perfect agreement with the exact ones.

Analysis of shear viscosity and viscoelastic relaxation of liquid methanol based on molecular dynamics simulation and mode-coupling theory.

The role of the prepeak structure of liquid methanol in determining its shear viscosity was studied by means of molecular dynamics (MD) simulation and mode-coupling theory (MCT). The autocorrelation function of the shear stress and the intermediate scattering functions at both the prepeak and the main peak were calculated from the MD trajectories. Their comparison based on MCT suggests that the viscoelastic relaxation in the ps regime is affected by the slow structural dynamics at the prepeak. On the other hand, the MCT for molecular liquids based on the interaction-site model (site-site MCT) fails to describe the coupling between the prepeak dynamics and shear stress. The direct evaluation of the coupling between the two-body density and the shear stress reveals that the viscoelastic relaxation is actually affected by the prepeak dynamics, although the coupling is not captured by the site-site MCT. The site-site MCT works well for a model methanol without partial charges, suggesting that the failure of the site-site MCT originates from the existence of a hydrogen-bonding network structure.

Interaction cross sections and matter radii of oxygen isotopes using the Glauber model

Using the Coulomb modified correlation expansion for the Glauber model $S$ matrix, we calculate the interaction cross sections of oxygen isotopes ($^{16--26}\mathrm{O}$) on $^{12}\mathrm{C}$ at 1.0 GeV/nucleon. The densities of $^{16--26}\mathrm{O}$ are obtained using (i) the Slater determinants consisting of the harmonic oscillator single-particle wave functions (SDHO) and (ii) the relativistic mean-field approach (RMF). Retaining up to the two-body density term in the correlation expansion, the calculations are performed employing the free as well as the in-medium nucleon-nucleon ($NN$) scattering amplitude. The in-medium $NN$ amplitude considers the effects arising due to phase variation, higher momentum transfer components, and Pauli blocking. Our main focus in this work is to reveal how could one make the best use of SDHO densities with reference to the RMF one. The results demonstrate that the SDHO densities, along with the in-medium $NN$ amplitude, are able to provide satisfactory explanation of the experimental data. It is found that, except for $^{23,24}\mathrm{O}$, the predicted SDHO matter rms radii of oxygen isotopes closely agree with those obtained using the RMF densities. However, for $^{23,24}\mathrm{O}$, our results require reasonably larger SDHO matter rms radii than the RMF values, thereby predicting thicker neutron skins in $^{23}\mathrm{O}$ and $^{24}\mathrm{O}$ as compared to RMF ones. In conclusion, the results of the present analysis establish the utility of SDHO densities in predicting fairly reliable estimates of the matter rms radii of neutron-rich nuclei.

Inelastic Longitudinal C2 Form Factors with Core-Polarization Effects for 10B Nucleus

An effective two-body density operator for point nucleon system folded with the full two-body correlations (which include the tensor correlations and short-range correlations), which take account of the effect of the strong tensor force and the strong short-range repulsion in the nucleon–nucleon forces, is produced and used to derive an explicit form for ground-state two-body charge density distributions. The inelastic longitudinal form factors C2 are calculated using this transition charge density with excitation of the levels in 10B (J π , T):(1+, 0) at E x = 0.718 MeV, (1+, 0) at E x = 2.154 MeV, (2+, 0) at E x = 3.587 MeV, (3+, 0) at E x = 4.774 MeV, (2+, 0) at E x = 5.920 MeV, and (4+, 0) at E x = 6.025 MeV. In this work, the core-polarization transition density is evaluated by adopting the shape of Tassie model together with the derived form of the ground-state two-body charge density distributions with the effect of higher occupation probabilities η. It is noticed that the core-polarization effects which represent the collective modes are essential in obtaining a remarkable agreement between the calculated inelastic longitudinal F(q)s and those of the experimental data.

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

The “Fermi hole” and the correlation introduced by the symmetrization or the anti-symmetrization of the wave function

The impact of the antisymmetrization is often addressed as a local property of the many-electron wave function, namely that the wave function should vanish when two electrons with parallel spins are in the same position in space. In this paper, we emphasize that this presentation is unduly restrictive: we illustrate the strong non-local character of the antisymmetrization principle, together with the fact that it is a matter of spin symmetry rather than spin parallelism. To this aim, we focus our attention on the simplest representation of various states of two-electron systems, both in atomic (helium atom) and molecular (H2 and the π system of the ethylene molecule) cases. We discuss the non-local property of the nodal structure of some two-electron wave functions, both using analytical derivations and graphical representations of cuttings of the nodal hypersurfaces. The attention is then focussed on the impact of the antisymmetrization on the maxima of the two-body density, and we show that it introduces strong correlation effects (radial and/or angular) with a non-local character. These correlation effects are analyzed in terms of inflation and depletion zones, which are easily identifiable, thanks to the nodes of the orbitals composing the wave function. Also, we show that the correlation effects induced by the antisymmetrization occur also for anti-parallel spins since all Ms components of a given spin state have the same N-body densities. Finally, we illustrate that these correlation effects occur also for the singlet states, but they have strictly opposite impacts: the inflation zones in the triplet become depletion zones in the singlet and vice versa.

Dynamic pair correlations and superadiabatic forces in a dense Brownian liquid

We study dynamic two-body correlation functions, i.e. the two-body density, the current-density correlator or van Hove current, and the current-current correlator in Brownian dynamics computer simulations of a dense Lennard-Jones bulk liquid. The dynamic decay of the correlation shells of the two-body density is examined in detail. Inner correlation shells decay faster than outer correlation shells, whereas outer correlation shells remain stable for increasing times. Within a dynamic test particle picture the mechanism is assumed to be triggered by the dislocation of the self particle, which releases the confinement of the surrounding correlation shells. We present a division of the van Hove current into an adiabatic and a superadiabatic contribution. The magnitude of the adiabatic van Hove current is found to exceed that of the total van Hove current, which is consistent with dynamic density functional theory overestimating the speed of the dynamics. The direction of the superadiabatic van Hove current opposes that of the total van Hove current. The current-current correlator reveals detailed insight in the collisions of the particles. We find a large static nearest-neighbor peak, which results from colliding particles and different dynamic peaks, that are attributed to consecutive collisions.

Study of the neon interaction cross section using the Glauber model

Working within the framework of the Coulomb-modified correlation expansion for the Glauber model S-matrix, we calculate the interaction cross section ( $\sigma_{I}$ ) of neon isotopes, 17-32Ne, on 12C at 240 MeV/nucleon. The calculations involve i) up to the two-body density term in the correlation expansion, and ii) the single Gaussian approximation for the nucleon-nucleon amplitude. The colliding nuclei are described with Slater determinants consisting of the harmonic oscillator single-particle wave functions. The sole input of the density of each colliding nucleus, the oscillator constant, is fixed from the respective root-mean-square (rms) radius calculated using the relativistic mean-field approach. It is found that the calculated results for $\sigma_{I}$ generally provide fairly good agreement with the experimental data except for 31Ne, for which the required rms neutron radius comes closer to the one obtained earlier using the extended (halo-like) neutron density distribution. This finding is also supported by our predicted differential cross section of 31Ne on 12C at 240 MeV/nucleon. However, as expected, the results of the present analysis are unable to discriminate between the halo and non-halo structure of 31Ne. In conclusion, our results suggest that the present calculations can be considered as a good starting point to predict the rms matter radii of exotic neutron-rich nuclei.

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.

Uncertainty product of an out-of-equilibrium many-particle system

In the present work we show, analytically and numerically, that the variance of many-particle operators and their uncertainty product for an out-of-equilibrium Bose-Einstein condensate (BEC) can deviate from the outcome of the time-dependent Gross-Pitaevskii dynamics, even in the limit of infinite number of particles and at constant interaction parameter when the system becomes 100% condensed. We demonstrate our finding on the dynamics of the center-of-mass position--momentum uncertainty product of a freely expanding as well as of a trapped BEC. This time-dependent many-body phenomenon is explained by the existence of time-dependent correlations which manifest themselves in the system's reduced two-body density matrix used to evaluate the uncertainty product. Our work demonstrates that one has to use a many-body propagation theory to describe an out-of-equilibrium BEC, even in the infinite particle limit.

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.