Correlated Density Matrix Research Articles

Spectroscopic accuracy directly from quantum chemistry: Application to ground and excited states of beryllium dimer

We combine explicit correlation via the canonical transcorrelation approach with the density matrix renormalization group and initiator full configuration interaction quantum Monte Carlo methods to compute a near-exact beryllium dimer curve, without the use of composite methods. In particular, our direct density matrix renormalization group calculations produce a well-depth of D(e) = 931.2 cm(-1) which agrees very well with recent experimentally derived estimates D(e) = 929.7±2 cm(-1) [J. M. Merritt, V. E. Bondybey, and M. C. Heaven, Science 324, 1548 (2009)] and D(e) = 934.6 cm(-1) [K. Patkowski, V. Špirko, and K. Szalewicz, Science 326, 1382 (2009)], as well the best composite theoretical estimates, D(e) = 938±15 cm(-1) [K. Patkowski, R. Podeszwa, and K. Szalewicz, J. Phys. Chem. A 111, 12822 (2007)] and D(e) = 935.1±10 cm(-1) [J. Koput, Phys. Chem. Chem. Phys. 13, 20311 (2011)]. Our results suggest possible inaccuracies in the functional form of the potential used at shorter bond lengths to fit the experimental data [J. M. Merritt, V. E. Bondybey, and M. C. Heaven, Science 324, 1548 (2009)]. With the density matrix renormalization group we also compute near-exact vertical excitation energies at the equilibrium geometry. These provide non-trivial benchmarks for quantum chemical methods for excited states, and illustrate the surprisingly large error that remains for 1 ¹Σ(g)⁻ state with approximate multi-reference configuration interaction and equation-of-motion coupled cluster methods. Overall, we demonstrate that explicitly correlated density matrix renormalization group and initiator full configuration interaction quantum Monte Carlo methods allow us to fully converge to the basis set and correlation limit of the non-relativistic Schrödinger equation in small molecules.

MEMORIAL TRIBUTE TO MANFRED L. RISTIG (1935–2011)

Manfred Ristig was a leading contributor to the advancement of microscopic theory of strongly interacting quantum many-body systems for over four decades. This retrospective on his life and scientific career pays tribute to his pivotal role in the development of correlated wavefunction approaches to quantitative ab initio description of quantum fluids, nuclear systems and condensed matter more generally. Highlights include his contributions to the formulation of Fermi hypernetted chain theory and correlated density matrix theory. Special attention is given to Ristig's seminal work of recent years, which has yielded rich insights into the interplay of exchange effects (arising from quantum statistics) and the strong interactions between constituent bosons or fermions.

NATURAL ORBITALS REPRESENTATION AND FERMI SEA DEPLETION IN FINITE NUCLEI AND NUCLEAR MATTER

The natural orbitals (NOs) and natural occupation numbers (NON) of various N = Z, sp and sd shell nuclei are calculated by applying a correlated one-body density matrix (OBDM). The correlated density matrix has been evaluated by considering central correlations of Jastrow type and an approximation named factor cluster expansion. The correlation effects on NOs, NON and the Fermi sea depletion (FSD) are discussed and analyzed. In addition, an approximate expression for the correlated OBDM of the nuclear matter has been used for the evaluation of the relative momentum distribution and FSD. We found that the value of FSD is higher in closed shell nuclei compared to open shell ones and it is lower compared to the case of nuclear matter. This statement could be confirmed by relevant experimental studies.

MP2 energy and density for large molecular systems with internal error control using the Divide-Expand-Consolidate scheme

Divide-Expand-Consolidate (DEC) is a local correlation method where the inherent locality of the electron correlation problem is used to express the correlated calculation on a large molecular system in terms of small independent fragment calculations employing small subsets of local HF orbitals. A crucial feature of the DEC scheme is that the sizes of the local orbital spaces are determined in a black box manner during the calculation. In this way it is ensured that the correlation energy has been determined to a predefined precision compared to a conventional calculation. In the present work we apply the DEC scheme to calculate the correlation energy as well as the electron density matrix for the insulin molecule using second order Møller-Plesset (MP2) theory. This is the first DEC calculation on a molecular system which is too large to be treated using a conventional MP2 implementation. The fragmentation errors for the insulin DEC calculation are carefully analyzed using internal consistency checks. It is demonstrated that size-intensive properties are determined to the same precision for small and large molecules. For example, the percentage of correlation energy recovered and the error per electron in the correlated density matrix depend only on the predefined precision and not on the molecular size.

Spin- and isospin-dependent momentum distributions in Fermi liquids at nonzero temperatures

We explore the structure of momentum distributions of Fermi liquids such as completely polarized 3He, unpolarized liquid 3He, and nuclear matter at nonzero temperatures. The study employs correlated density matrix theory and adapts the algorithm to deal with spin- and isospin-dependent correlations. The analysis is based on the factor decomposition of the one-body reduced density matrix. The decomposition permits to distinguish between statistical correlations and dynamic (direct) correlations. Together with the concept of renormalized fermions the formal results open the pathway to investigate the thermal boundaries of normal Fermi phases within the correlated density matrix theory. We also discuss possible transitions from normal phases to anomalous fermion phases triggered by statistical correlations or by periodic phase–phase structures.

RENORMALIZED BOSONS AND FERMIONS

Correlated Density Matrix (CDM) theory permits formal analyses of microscopic properties of strongly correlated quantum fluids and liquids at nonzero temperatures. Equilibrium properties, thermodynamic potentials, correlation and structure functions can be studied formally as well as numerically within the CDM algorithm. Here we provide the essential building blocks for studying the radial distribution function and the single-particle momentum distribution of the ingredients of the quantum systems. We focus on the statistical properties of correlated fluids and introduce the concept of renormalized bosons and fermions. These entities carry the main statistical features of the correlated systems such as liquid 4 He through their specific dependence on temperature, particle number density, and wavenumber encapsulated in their effective masses. The formalism is developed for systems of bosons and of fermions. Numerical calculations for fluid 4 He in the normal phase demonstrate the power of the renormalization concept. The formalism is further extended to analyze the Bose-Einstein condensed phases and gives a microscopic understanding of Tisza's two-fluid model for the normal and superfluid density components.

Quantum entanglement and teleportation using statistical correlations

A study of quantum teleportation using two and three-particle correlated density matrix is presented. A criterion based on standard quantum statistical correlations employed in the many-body virial expansion is used to determine the extent of entanglement for a 2N-particle system. A relation between the probability and statistical parameters is established using the correlated density matrices for the particles.

FLUID HELIUM-4 IN THERMAL EQUILIBRIUM

The microscopic quantum structure of fluid 4 He may be clearly revealed by a proper decomposition of its spatial correlations into quantum statistical components and direct quantum correlations. This decomposition permits to elucidate the competition between the short-ranged statistical (or particle exchange) correlations and the quantum correlations brought about by the existing strong interparticle repulsion at short relative particle-particle distances. The appropriate method of choice is provided by correlated density-matrix (CDM) theory. It does not only permit a detailed formal analysis of this competition but also allows for a quantitative numerical computation of correlation functions, structure functions, and momentum and energy distributions. The theoretical CDM results for 4 He are, so far as possible, compared with results from path-integral Monte Carlo (PIMC) calculations and with available experimental results. Reported are CDM results on relevant structure functions, correlation functions in coordinate space, kinetic energy distributions, and gross quantitities such as the exchange energy for fluid 4 He . The calculations are performed for normal helium at various temperatures in the range T BE = 2.17 K ≤ T < 14 K .

STRONGLY CORRELATED BOSONS AT NONZERO TEMPERATURES

We investigate the structure of strongly correlated normal Bose liquids and fluids by employing correlated density-matrix (CDM) theory, a generalization of correlated basis functions (CBF) theory to nonzero temperatures. The formalism is applied in a study of structural properties of various correlation functions that characterize the one-body and two-body reduced density matrix elements of a boson system and the associated Fourier transforms such as the static structure function, the exchange structure function, the momentum distribution, and related thermodynamic quantities. We perform a numerical analysis for liquid para-hydrogen, close to the triple point, supercritical 4 He gas at temperatures T ≥ 12 K , and fluid and liquid 4 He in the normal phase below T = 12 K . The results reveal that H 2 and 4 He , at T > 12 K , are typical quantum Boltz-mann systems following classical statistics. In contrast, liquid 4 He exhibits non-classical effects of particle exchange correlations, when the temperature is lowered toward the Bose-Einstein transition regime.

Comment on “Momentum density and spatial form of correlated density matrix in model two-electron atoms with harmonic confinement”

In a recent paper A. Akbari, N. H. March, and A. Rubio [Phys. Rev. A 76, 032510 (2007)] investigated the one-electron reduced density matrix and the momentum density of several two-electron model atoms, including Hooke's atom. The method used by the authors for deriving an integral form of the momentum density is in fact the most suited for deriving a closed-form expression of the exact reciprocal form factor, in a special case. This function is important inasmuch as it reflects the off-diagonal side of the exact reduced density matrix.

Momentum density and spatial form of correlated density matrix in model two-electron atoms with harmonic confinement

This work has being funded by the EC Network of Excellence NANOQUANTA Grant No. NMP4-CT-2004-500198 , the Spanish Ministry of Education, the SANES Grant No. NMP4-CT-2006- 017310 , DNA-NANODEVICES Grant No. IST-2006- 029192 , and NANO-ERA Chemistry projects, the University of the Basque Country EHU/UPV SGIker Arina , and the Basque Country Government.

THE PHYSICS OF LIQUID PARA-HYDROGEN

Macroscopic systems of hydrogen molecules exhibit a rich thermodynamic phase behavior. Due to the simplicity of the molecular constituents a detailed exploration of the thermal properties of these boson systems at low temperatures is of fundamental interest. Here, we report theoretical and experimental results on various spatial correlation functions and corresponding distributions in momentum space of liquid para-hydrogen close to the triple point. They characterize the structure of the correlated liquid and provide information on quantum effects present in this Bose fluid. Numerical calculations employ Correlated Density-Matrix (CDM) theory and Path-Integral Monte-Carlo(PIMC)simulations. A comparison of these theoretical results demonstrates the accuracy of CDM theory. This algorithm therefore permits a fast and efficient quantitative analysis of the normal phase of liquid para-hydrogen. We compare and discuss the theoretical results with available experimental data.

RADIAL DISTRIBUTION AND LIQUID STRUCTURE FUNCTION FOR LIQUID PARA-HYDROGEN AT LOW TEMPERATURES

We present theoretical results for the radial distribution function g(r) and the static liquid structure function S(k) of liquid para-hydrogen at low temperatures. The results have been obtained via quantum Monte Carlo Path Integral simulations, classical Monte Carlo calculations, and correlated density matrix theory.

Density dependence of the momentum distributions in liquidpara-hydrogen

Momentum distributions of liquid para-hydrogen were determined by means of inelastic neutron scattering under applied pressures ranging from 1 to 80 bars, at T=16.5 K. The data processing procedure involves the parametrization of the dynamic structure factor and yields a set of momentum distributions as functions of the density. The results depict significant pressure dependences for all single-particle quantities such as the momentum distributions and average kinetic energies as well as for the final-state effects. The obtained results enable us to quantify the departure of the momentum distributions from classical Maxwell-Gauss shape. Such observations are then rationalized with the help of calculations that were carried out in terms of the correlated density matrix formalism.

Bose–Einstein Condensation and the λ Transition in Liquid Helium

Integrating seminal ideas of London, Feynman, Uhlenbeck, Bloch, Bardeen, and other illustrious antecessors, this paper continues the development of an ab initio theory of the λ transition in liquid 4He. The theory is based upon variational determination of a correlated density matrix suitable for description of both normal and superfluid phases, within an approach that extends to finite temperatures the very successful correlated wave-functions theory of the ground state and elementary excitations at zero temperature. We present the results of a full optimization of a correlated trial form for the density matrix that includes the effects both of temperature-dependent dynamical correlations and of statistical correlations corresponding to thermal phonon/roton and quasiparticle/hole excitations—all at the level of two-point descriptors. The optimization process involves constrained functional minimization of the associated free energy through solution of a set of Euler–Lagrange equations, consisting of a generalized paired-phonon equation for the structure function, an analogous equation for the Fourier transform of the statistical exchange function, and a Feynman equation for the dispersion law of the collective excitations. Violation of particle-hole exchange symmetry emerges as an important aspect of the transition, along with broken gauge symmetry. In conjunction with a semi-phenomenological study in which renormalized masses are introduced for quasiparticle/hole and collective excitations, the results suggest that a quantitative description of the λ transition and associated thermodynamic quantities can be achieved once the trial density matrix is modified—notably through the addition of three-point descriptors—to include backflow effects and allow for ab initio treatment of important variations in effective masses.

Explorations of the Microstructure of Crystalline Quantum Many-Body Systems

We undertake a formal analysis of crystallographic point and space group symmetries in the one-body density and in the two-body density of strongly correlated many-body systems. The mathematical tools presented in this article are put to a numerical test in an exact Fourier path integral Monte Carlo simulation of solid natural argon at the triple point temperature in the hexagonal close-packed structure. We point out the experimental significance of this system in particular and, more generally, briefly describe how quantities computed within the approach taken in the present work may be directly related to experimentally measurable quantities. We also address the potential relevance of the techniques presented and tested in this paper to the liquid-to-solid phase transition. The group-theoretical formalism developed in this work is of general validity and might be a valuable means also for other condensed matter theories, e.g., for applications of correlated basis functions and correlated density matrix theories of the ground state and of the collective excitations of quantum many-body systems to quantum crystals.

THE TRANSVERSE ISING MODEL BY CBF

The method of correlated basis functions (CBF) is applied at the variational level to the transverse Ising model in two and three spatial dimensions (D=2,3). The model consists of Pauli spins arranged on a simple square or cubic lattice, experiencing nearest-neighbor interactions through their x components and subject to a transverse field in the z direction of strength λ. Working at zero temperature, full optimization of a Hartree-Jastrow trial wave function is performed by solving two Euler-Lagrange equations, namely a renormalized Hartree equation for the order parameter characterizing the ferromagnetic phase and a paired-magnon equation for the optimal two-spin spatial distribution function. The optimized trial wave function yields a second-order transition with a numerically determined critical coupling of λc=3.14(D=2) or λc= 5.10(D=3). Numerical results have been obtained for the magnetization order parameter, the energy per spin and its potential component, the static structure function at zero wave number, and the magnon energy gap corresponding to a Feynman description of the elementary excitations. Correlated density matrix theory provides for a natural extension of this approach to finite temperature.

Toward a microscopic theory of the λ transition in liquid 4He

A microscopic many-body analysis of Bose–Einstein condensation in a strongly interacting system of identical bosons is presented in the framework of correlated density matrix theory. With the aid of hypernetted-chain techniques and a replica ansatz for the entropy, the free energy is constructed for a trial density matrix incorporating temperature-dependent two-point dynamical and statistical correlations. The free energy decomposes naturally into contributions from phonon excitations and from two types of quasiparticle excitations (identified as “holes” and “particles”), in addition to a component that becomes the ground-state energy at zero temperature. The subsequent analysis is conducted in terms of two order parameters: A condensation strength Bccand an exchange strength M. The former measures the breaking of gauge symmetry associated with the development of off-diagonal long-range order that signals Bose condensation; the latter characterizes the violation of particle-hole exchange symmetry. A description of exchange-symmetry breaking is formulated in terms of an analogy with the behavior of a diamagnetic material in a magnetic field, and a physically plausible model for the coupling of the order parameters Bccand M is proposed. The “particle” and “hole” excitation branches coincide in the normal phase, but follow different dispersion relations in the condensed phase, where exchange symmetry is broken. In a first application of the theory to the λ transition in liquid4He, phonon effects (dominant at very low temperatures) are neglected, and simple parametrized forms are assumed for the dynamical correlations and for the hole spectrum, which determines the remaining statistical correlations. Numerical results are reported for the two order parameters and for the quasihole and quasiparticle energies, as functions of temperature through the condensation point. The calculated specific heat shows the characteristic λ shape. Exchange symmetry breaking reduces the Bose–Einstein temperature from that of the ideal Bose gas to a predicted value near 2.2 K.

CCSD(T) expectation value calculations of first-order properties

An expectation value approach to calculations of first-order properties using the non-iterative, triple-excitation amplitudes in the coupled cluster wave function is exploited. Three methods are suggested and analysed using the many body perturbation theory (MBPT) expansion arguments. The first method, in which non-iterative triple-excitation amplitudes are used in the expression for the expectation values, makes the wave function accurate through the second order of MBPT. In the second method, which is an extension of the first, effects of triple-excitation amplitudes are coupled with single- and double-excitation amplitudes. The correlated density matrix equivalent through the fourth order to that obtained when CCSDT-la amplitudes are used is employed in the third method. The suggested methods are tested on dipole moment and polarizability calculations for several diatomic closed-shell molecules and are compared to other related approaches.

Correlated one-body density matrix of boson superfluids

The correlated density matrix theory is employed and further developed to analyze the one-body density matrix ρ1(|r1-r2|) of the normal and superfluid phases of a strongly interacting Bose system at non-zero temperature. The approach continues the formal development described in an earlier article and is based on a suitable trial ansatz for the many-body density matrixW(R, R′)∼Φ(R) Q(R, R′) Φ(R′) with the wave function Φ and incoherence factorQ incorporating the essential statistical and dynamical correlations. Special attention is given to the appearance of off-diagonal long-range order in function ρ1(|r1-r2|) and its relation to the condensation strength Bcc characterizing the degree of coherence in the superfluid phase. We derive a number of structural relations that have counterparts in known results on ρ1 in the Jastrow variational theory of the Bose ground state. We discuss Bose-Einstein condensation and make contact to Landau's phenomenological theory of continuous phase transitions. Numerical estimates are presented on the condensation strength and the condensate fraction of liquid4He as functions of the temperature.