Intracule functional models. V. Recurrence relations for two-electron integrals in position and momentum space

The approach used by Ahlrichs [Phys. Chem. Chem. Phys., 2006, 8, 3072] to derive the Obara-Saika recurrence relation (RR) for two-electron integrals over Gaussian basis functions, is used to derive an 18-term RR for six-dimensional integrals in phase space and 8-term RRs for three-dimensional integrals in position or momentum space. The 18-term RR reduces to a 5-term RR in the special cases of Dot and Posmom intracule integrals in Fourier space. We use these RRs to show explicitly how to construct Position, Momentum, Omega, Dot and Posmom intracule integrals recursively.     

A family of intracules, a conjecture and the electron correlation problem

We present a radical approach to the calculation of electron correlation energies. Unlike conventional methods based on Hartree-Fock or density functional theory, it is based on the two-electron phase-space information in the Omega intracule, a three-dimensional function derived from the Wigner distribution. Our formula for the correlation energy is isomorphic to the Hartree-Fock energy expression but requires a new type of four-index integral. Preliminary results, obtained using a model that is based on the known correlation energies of small atoms, are encouraging.     

Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density

We construct improved quantum Monte Carlo estimators for the spherically and system-averaged electron pair density (i.e., the probability density of finding two electrons separated by a relative distance u), also known as the spherically averaged electron position intracule density I(u), using the general zero-variance zero-bias principle for observables, introduced by Assaraf and Caffarel. The calculation of I(u) is made vastly more efficient by replacing the average of the local delta-function operator by the average of a smooth nonlocal operator that has several orders of magnitude smaller variance. These new estimators also reduce the systematic error (or bias) of the intracule density due to the approximate trial wave function. Used in combination with the optimization of an increasing number of parameters in trial Jastrow-Slater wave functions, they allow one to obtain well converged correlated intracule densities for atoms and molecules. These ideas can be applied to calculating any pair-correlation function in classical or quantum Monte Carlo calculations.     

Intracule densities in the strong-interaction limit of density functional theory

The correlation energy in density functional theory can be expressed exactly in terms of the change in the probability of finding two electrons at a given distance r(12) (intracule density) when the electron-electron interaction is multiplied by a real parameter lambda varying between 0 (Kohn-Sham system) and 1 (physical system). In this process, usually called adiabatic connection, the one-electron density is (ideally) kept fixed by a suitable local one-body potential. While an accurate intracule density of the physical system can only be obtained from expensive wavefunction-based calculations, being able to construct good models starting from Kohn-Sham ingredients would highly improve the accuracy of density functional calculations. To this purpose, we investigate the intracule density in the lambda --> infinity limit of the adiabatic connection. This strong-interaction limit of density functional theory turns out to be, like the opposite non-interacting Kohn-Sham limit, mathematically simple and can be entirely constructed from the knowledge of the one-electron density. We develop here the theoretical framework and, using accurate correlated one-electron densities, we calculate the intracule densities in the strong interaction limit for few atoms. Comparison of our results with the corresponding Kohn-Sham and physical quantities provides useful hints for building approximate intracule densities along the adiabatic connection of density functional theory.     

Visualizing molecular wavefunctions using Monte Carlo methods

Using explicitly correlated wavefunctions and variational Monte Carlo we calculate the electron density, the electron density difference, the intracule density, the extracule density, two forms of the kinetic energy density, the Laplacian of the electron density, the Laplacian of the intracule density, and the Laplacian of the extracule density on a dense grid of points for the ground state of the hydrogen molecule at three internuclear distances (0.6, 1.4, 8.0). With these values we construct a contour plot of each function and describe how it can be used to visualize the distribution of electrons in this molecule. We also examine the influence of electron correlation on each expectation value by calculating each function with a Hartree–Fock wavefunction and then comparing these values with our explicitly correlated values. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Anisotropic electronic intracule densities for diatomics

The electronic intracule density, a three-dimensional contraction of the spinless electron pair density, is the probability density function for an interelectronic vector. A computationally efficient algorithm for the evaluation of the basic two-electron intracule integral for GTO s is presented. In order to provide an initial understanding of the topography of intracule distributions, anisotropic intracule densities for the X¹Σ ground states of the H₂ and N₂ molecules are reported and analyzed.     

Intracule functional models. Part III. The dot intracule and its Fourier transform

The dot intracule D(x) of a system gives the Wigner quasi-probability of finding two of its electrons with u.v = x, where u and v are the interelectronic distance vectors in position and momentum space, respectively. In this paper, we discuss D(x) and show that its Fourier transform d(k) can be obtained in closed form for any system whose wavefunction is expanded in a Gaussian basis set. We then invoke Parseval's theorem to transform our intracule-based correlation energy method into a d(k)-based model that requires, at most, a one-dimensional quadrature.     

Bridging continuum and statistical thermodynamics via equations of state and the density of states

The connection between molecular force fields and equations of state (EoS) is typically established at the level of predicted quantities, e.g., by comparing simulation data and EoS data. In this paper we show how an EoS can be used to extract the density of states (Omega) of a system thus establishing a deeper connection between EoSs and statistical thermodynamics. We also show how such an EoS Omega can be used to aid molecular simulation methods designed to map out Omega (like the multicanonical approach). Central to the implementation of these ideas is the fact that the configurational Omega is related to thermodynamic properties accessible by an EoS via Boltzmann's equation. Sample calculations are presented for the Omega relevant to isothermal-isobaric and grand canonical ensemble simulations using the hard-sphere system and the Lennard-Jones system as model fluids, and the Carnahan-Starling EoS and a cubic EoS, respectively, as thermodynamic models.     

The evaluation of electronic extracule and intracule densities and related probability functions in terms of Gaussian basis functions

The evaluation of the basic two-electron integrals involved in the calculation of extracule and intracule densities is described. Expressions are given for the evaluation of the related spherically averaged, longitudinal, and transverse probability functions from wave functions constructed from Gaussian basis sets. All results are expressed in closed analytical forms which are suited to efficient coding. Given that certain pair densities can be related to experimental scattering cross sections, the formulae reported herein will facilitate further comparison between experiment and theory and lead to a more comprehensive understanding of the electronic structures of molecules.     

On the extremum property of electron density functions in position and momentum spaces

For statistically uncorrelated wave functions, the electron density ρ(r) is known to be an extremum function of the generalized electron-pair density g(q;a,b), which smoothly connects ρ(r), the electron-pair intracule (relative motion) density h(u), and the electron-pair extracule (center-of-mass motion) density d(R). The present systematic examination of the numerical Hartree–Fock results of moments rⁿ, uⁿ, Rⁿ and density values at the origin ρ(0), h(0), d(0) for 102 neutral atoms and 96 singly charged ions suggests that ρ(r) is local maximum for a small q and is local minimum for a large q of the function g(q;a,b). Analogous results are obtained for the momentum-space counterparts Π(p) and .     

Application of Wigner and Husimi intracule based electron correlation models to excited states

A new approach to the electron correlation problem based on phase space intracules derived from the Wigner distribution is applied to excited states. The computed electron correlation energy reduces the mean absolute error in the prediction of the excitation energies of 55 atomic excited states from 0.65 eV for unrestricted Hartree-Fock to 0.32 eV. This compares favorably to a mean absolute deviation of 0.52 eV for second order Moller-Plesset perturbation theory and 0.35 eV for the Lee-Yang-Parr functional. An analogous correlation model based on the Husimi distribution is developed. Predicted correlation energies and excitation energies from this model are significantly worse than for the Wigner intracule based model. Alternative correlation kernels may be more suitable for the Husimi intracule based approach.     

Scattering of fast electrons and X-rays from molecules: CH4 and C2H2

Self-consistent-field (SCF ) wave functions are used to calculate cross sections for the elastic and inelastic scattering of fast electrons and x-rays from CH₄ and C₂H₂ molecules. The effects of basis set choice and free rotation on these cross sections are investigated. The utility of an approximate scheme to correct SCF inelastic cross sections for the effects of electron correlation is examined. The probability density for the interelectronic distance, or radial intracule density, is obtained and discussed.     

Non-Born-Oppenheimer treatment of the H2 Hookean molecule

We show that the exact non-Born-Oppenheimer Schrodinger equation for the Hookean diatomic molecule H2 (a two-proton, two-electron system where the electron-proton interaction is harmonic while the proton-proton and electron-electron interactions are Coulombic) can be decoupled into equations describing the relative motion of the electrons, the relative motion of nuclei, the motion of a collective mode representing a three-dimensional harmonic oscillator, and the motion of a free particle expressed as a linear combination of the individual center-of-mass coordinates of the nuclei and electrons. Analytic solutions to the relative motion of electrons can be readily obtained for the given values of the harmonic coupling constant. However, exact analytic solutions to the equation for the relative motion of the nuclei cannot be obtained simultaneously due to the fact that the harmonic constants in these two equations are coupled. For this reason, we present for the relative nuclear motion approximate analytic wave functions, one of them obtained variationally and the other by a series solution where the coefficients are determined recursively. We also explore a variational solution to the Taylor-series expansion of the nuclear interaction potential. Properties of the electronic and nuclear intracule densities are examined at different values of the coupling constant. An interesting result of the present non-Born-Oppenheimer treatment of this harmonic model is the fact that the relative nuclear motion occurs in a highly correlated regime. This leads in a natural way to a spatial localization of the nuclei akin to Wigner electronic crystallization.     

Wireless excitation of quartz crystals immersed in an aqueous fluid

A novel toroidal coil geometry able to induce remote acoustic waves in quartz crystals has been evaluated for the development of (bio)sensors. Remote acoustic generation in air was obtained for two alternative toroidal coils, with corresponding electrical impedance changes of 40 Omega for a PDMS- and 140 Omega for a ferrite-supported toroid respectively. It was found that the range of remote acoustic generation relative to the spiral coil standard was much improved, increasing the axial separation of their resonant sensing element from 0.1 mm to 20 mm, thereby allowing electromagnetic wave penetration across glass walls and fluid media to be utilised. Consideration of the transduction mechanism, along with measured cyclic changes in acoustic signal as a function of rotation, indicated that the large PDMS toroidal coil produced an asymmetric electric field. It was shown for the first time that a quartz crystal blank fully immersed in an aqueous fluid could support chemically sensitive shear acoustic standing waves that were excited and detected remotely. A signal to noise ratio of 30 ratio 1 at 20.13 MHz was achieved by placing a ferrite supported toroidal coil on the lower side of a glass beaker containing a 12 x 0.25 mm AT crystal blank and 1 mL of water. This discovery allows wireless shear acoustic wave measurements to be performed with total separation between the electronic detection system and assays undertaken in fluidic systems.     

Intracule functional models. II. Analytically integrable kernels

We present, within the framework of intracule functional theory (IFT), a class of kernels whose correlation integrals can be found in closed form. This approach affords three major advantages over other kernels that we have considered previously; ease of implementation, computational efficiency, and numerical stability. We show that even the simplest member of the class yields reasonable estimates of the correlation energies of 18 atomic and 56 molecular systems and we conclude that this kernel class will prove useful in the development of future IFT models.     

Two-electron distribution functions and intracules

 Two-electron distribution functions and intracules are functions of electronic coordinates and occupy an important, and frequently overlooked, middle ground between the beguiling simplicity of electron densities and the bewildering complexity of wavefunctions. We survey the functions that have been considered by earlier workers and introduce two new ones, the Wigner intracule and the action intracule, that have not previously been discussed. To illustrate their usefulness, we consider the intracules of jellium, a few small atoms and the dissociating hydrogen molecule. Received: 26 July 2002 / Accepted: 20 October 2002 / Published online: 30 January 2003 Correspondence to: P.M.W. Gill e-mail: peter.gill@nott.ac.uk     

A simultaneous probability density for the intracule and extracule coordinates

We introduce the intex density X(R,u), which combines both the intracular and extracular coordinates to yield a simultaneous probability density for the position of the center-of-mass radius (R) and relative separation (u) of electron pairs. One of the principle applications of the intex density is to investigate the origin of the recently observed secondary Coulomb hole. The Hartree-Fock (HF) intex densities for the helium atom and heliumlike ions are symmetric functions that may be used to prove the isomorphism 2I(2R)=E(R), where I(u) is the intracule density and E(R) is the extracule density. This is not true of the densities that we have constructed from explicitly correlated wave functions. The difference between these asymmetric functions and their symmetric HF counterparts produces a topologically rich intex correlation hole. From the intex hole distributions (X(exact)(R,u)-X(HF)(R,u)), we conclude that the probability of observing an electron pair with a very large interelectronic separation increases with the inclusion of correlation only when their center-of-mass radius is close to half of their separation.     

Diagnosing broken ergodicity using an energy fluctuation metric

The Mountain and Thirumalai energy fluctuation metric, Omega(t), has been used to study the effective ergodicity of 60- and 256-atom binary Lennard-Jones mixtures in order to determine the reliability of the calculated diffusion constants at different energies. A plot of Omega(t) against 1time allows the identification of two distinct regimes: ergodic supercooled liquids, where Omega(t) approaches zero, and nonergodic glassy states, where Omega(t) asymptotically approaches a nonzero value on the molecular dynamics time scale. This approach seems to be more appropriate than attempting to define a threshold value for Omega(t)/Omega(0). The behavior of systems between these two limits, which are nonergodic on the time scale considered but may be approaching ergodicity, was examined for a range of simulation times. The calculated diffusion constants change as effective ergodicity is approached, moving closer to the Vogel-Tammann-Fulcher fit defined by higher-energy systems that are already considered to be effectively ergodic. Using the form of the decay of the metric as a measure of ergodicity, we have been able to reproduce the trend in fragility obtained by Sastry for a 256-atom system [Nature (London) 409, 164 (2001)], correcting some of our earlier results [J. Chem. Phys. 120, 8314 (2004)].