Monte Carlo study of electron correlation functions for small molecules

A study is made of electron-electron correlation functions for use in trial wave functions for small molecules. New forms are proposed that have only a few variational parameters, and these parameters have physical meanings that are easily discerned. Total energies for H₂, LiH and Li₂ computed using these correlation functions are presented, and comparison is made with previous forms, including the Jastrow-Pade form often used in Monte Carlo studies. We further treat the possibility that correlation depends not only on the separation of a pair of electrons but also on the location of the electron pair relative to the nuclei — indicative of a density-dependent or many body correlation effect. Our results indicate that such a many-body correlation effect is weakly present.This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098     

Extremal electron pairs

Extremal pair functions for an n-electron wave function of a closed-shell state are defined as linear combinations of spin-orbital-product pair functions that make some functionals (e.g., r²₁₂ or r⁻¹₁₂) extremal. They are related to the natural spin geminals in the uncorrelated limit and are useful both for an analysis of wave functions in view of an understanding of the chemical bond and for the treatment of electron correlation. Numerical examples are shown and discussed for He₂ as well as the 10-electron systems Ne, HF, H₂O, NH₃, and CH₄. © 1996 John Wiley & Sons, Inc.     

Calculation of transition density matrices by nonunitary orbital transformations

An efficient scheme for calculating one- and two-electron transition density matrices for two wave functions is described. The method applies to CAS (complete active space) wave functions and certain multireference CI expansions. The orbital sets of the two wave functions are not assumed to be equal. They are transformed to a biorthonormal basis, and the corresponding transformation of the CI coefficients is carried out directly, using the one-electron coupling coefficients.     

Calculation of molecular quadrupole moments and a demonstration of the importance of overlap densities in the theory of polyatomic molecules

A computational method for calculating quadrupole moments from molecular wave functions in a Slater orbital basis set is described. Using both IEHT and CNDO wave functions quadrupole moments for a series of polyatomic molecules are calculated. They are compared with experimental results and the IEHT wave functions are found to give agreement with experiment while CNDO wave functions do not. The importance of bicentric densities (overlap densities) in the calculation of multipole moments is shown. This is followed by a discussion of the usefulness of these wave functions for a quantitative characterization of the electronic structure of large molecules.     

On the SCF calculation of excited states: Singlet states in the two-electron problem

The problem of determining SCF wave functions for excited electronic states is examined for singlet states of two-electron systems using a Lowdin natural orbital transformation of the full CI wave function. This analysis facilitates the comparison of various SCF methods with one another. The distribution of the full CI states among the natural orbital MCSCF states is obtained for the S states of helium using a modest Gaussian basis set. For SCF methods that are not equivalent to the full CI wave functions, it is shown that the Hartree-Fock plus all single excitation wave functions are equivalent to that of Hartree-Fock plus one single excitation. It is further shown that these wave functions are equivalent to the perfect pair or TCSCF wave functions in which the CI expansion coefficients are restricted to have opposite signs. The case of the natural orbital MCSCF wave function for two orbitals is examined in greater detail. It is shown that the first excited state must always be found on the lower natural orbital MCSCF CI root, thus precluding the use of the Hylleras-Undeim-MacDonald (HUM) theorem in locating this state. It is finally demonstrated that the solution obtained by applying the HUM theorem (minimizing the upper MCSCF CI root with respect to orbital mixing parameters) is an artifact of the MCSCF method and does not correspond to any of the full CI states.     

Electronic correlation studies. III. Self-correlated field method. Application to 2S ground state and 2P excited state of three-electron atomic systems

The self-correlated field method is based on the insertion in the group product wave function of pair functions built upon a set of correlated “local” functions and of “nonlocal” functions. This work is an application to three-electron systems. The effects of the outer electron on the inner pair are studied. The total electronic energy and some intermediary results such as pair energies, Coulomb and exchange “correlated” integrals, are given. The results are always better than those given by conventional SCF computations and reach the same level of accuracy as those given by more laborious methods used in correlation studies.     

The electronic correlation problem and loge localization

The model of complete loge localization is employed here to develop a practical method for handling correlation effects in atomic and molecular many electron systems. Intraloge correlation is dealt with by an independent variational treatment of pair functions which are continuous and vanish outside a given loge. It is shown that in the context of the model it is possible to compute pair correlation energies for localized single and double bonds in molecules by evaluating only modified atomic integrals. We bypass in this manner the evaluation of multicenter integrals necessary in other formalisms. In addition, the corrections to the model are discussed and in particular it is shown that part of the interloge correlation effects are already described by the loge localized wave function.     

Computational study of the structures of gaseous helium-3 at low temperature

A computational study of gaseous helium-3 at T=5.23 K, for number densities rho N<0.0021 A(-3), analyzing the different pair and triplet structures in both the r and the k spaces, is presented. Structures in r space (i.e., instantaneous, total continuous linear response, and centroids) are determined via path-integral Monte Carlo simulations in the canonical ensemble by utilizing the Aziz-Slaman and the SAPT2 interatomic potentials. Additional results obtained with the application of two closures for triplets in r space, the Kirkwood superposition approximation and the Jackson-Feenberg convolution, are also reported. Besides, an analysis of the nonsuitability of quantum hard spheres for describing this system is included. The pair structures in k space are fixed via Ornstein-Zernike schemes appropriate for dealing with quantum diffraction effects in fluids. The effect on the quality of the computed isothermal compressibilities brought about by increasing the sample size in the simulations and by the subsequent application of a grand ensemble correction to the asymptotic behavior of the canonical pair radial correlation functions is also investigated. Furthermore, it is demonstrated analytically that the methods of classical statistical mechanics for dealing with the higher-order direct correlation functions remain fully valid for studying the higher-order correlations of path-integral centroids. By taking advantage of this result, the triplet structure factors for the centroid (also for the instantaneous) correlations are computed by following a number of distinct closures and methods that involve triplet direct correlation functions. The latter computations are intended to explore an alternative scheme to the very expensive fixing of triplet structure factors through direct path integral simulations, an alternative which is expected to yield the main features of these triplet quantities for this gas. Comparison with experiment is made wherever possible, and the results presented allow one to explain the substantial structural features existing in gaseous helium-3.     

A formal mathematical legalization of the “exponential” debye—hückel theory

By linearization of the Poisson—Boltzmann equation in a different way from that usually followed, the negativity of the pair correlation functions of like-charged ions can be avoided. The expression thus obtained for the pair correlation functions yields values very close to the exponential Debye—Hückel (DHX) approximation. Since these new pair correlation functions leave the DH charge distribution analytically intact, their use does not violate the electroneutrality condition which the DHX does.     

The loge theory as a starting point for variational calculations. I. General formalism

It is shown that any expectation value of any observable associated with a molecule is the sum of loge contributions and of loge pair contributions. This result provides a rigorous theoretical basis for the study of additive properties of molecules. It is demonstrated that molecular wave functions (exact or approximate) can be expressed as a sum of functions corresponding to the various electronic events. Furthermore any of these event functions can be expressed in terms of correlated loge functions. This expression suggests many kinds of variational procedures of calculating wave functions (known methods and new ones). The case in which noncorrelated completely localized loge functions are used is discussed. If continuous functions are used the variational equation reduces to a sum of independent variational equations, each one corresponding to a particular electronic event. This is not so when discontinuous functions are used or when a delocalized function is added to replace the correlation interloge function. The noncorrelated completely localized loge model is analyzed in more detail. It is seen that local spin operators can be introduced and that each event density operator is the product of the loge density operators. Therefore that model is an independent loge model. The corresponding generalized self-consistent field equations are derived. This treatment helps us to understand how a localized state of a molecule can produce an ion containing a delocalized region, a phenomenon which is sometimes at the origin of some misunderstanding in photoelectron spectroscopy. Finally it is seen how virtual loge functions can be introduced to describe excited states.     

Role of electron correlation in the quantum-mechanical calculations of the coulomb interaction energy in the DNA base pairs

The intermolecular electronic correlation contributions to the Coulomb component of the nucleic acid base interaction energy are estimated. The Coulomb energy is evaluated with the use of atomic monopoles, which are determined from the π-electronic densities calculated by the SCF method and by employing partially or completely optimized APSG wave functions. When the correlation is thus taken into account, a systematic decrease in atomic charges occurs; this effect is considerable only if an optimized orbital set is used. As a result, the Coulomb interaction energy due to the π-electronic atoms decreases from ?1.13 to ?0.85 kcal/mol for the AT pair and from ?7.15 to ?4.61 kcal/mol for the GC pair.     

Irreducible correlation functions of the S matrix in the coordinate representation: application in calculating Lorentzian half-widths and shifts

By introducing the coordinate representation, the derivation of the perturbation expansion of the Liouville S matrix is formulated in terms of classically behaved autocorrelation functions. Because these functions are characterized by a pair of irreducible tensors, their number is limited to a few. They represent how the overlaps of the potential components change with a time displacement, and under normal conditions, their magnitudes decrease by several orders of magnitude when the displacement reaches several picoseconds. The correlation functions contain all dynamical information of the collision processes necessary in calculating half-widths and shifts and can be easily derived with high accuracy. Their well-behaved profiles, especially the rapid decrease of the magnitude, enables one to transform easily the dynamical information contained in them from the time domain to the frequency domain. More specifically, because these correlation functions are well time limited, their continuous Fourier transforms should be band limited. Then, the latter can be accurately replaced by discrete Fourier transforms and calculated with a standard fast Fourier transform method. Besides, one can easily calculate their Cauchy principal integrations and derive all functions necessary in calculating half-widths and shifts. A great advantage resulting from introducing the coordinate representation and choosing the correlation functions as the starting point is that one is able to calculate the half-widths and shifts with high accuracy, no matter how complicated the potential models are and no matter what kind of trajectories are chosen. In any case, the convergence of the calculated results is always guaranteed. As a result, with this new method, one can remove some uncertainties incorporated in the current width and shift studies. As a test, we present calculated Raman Q linewidths for the N2-N2 pair based on several trajectories, including the more accurate "exact" ones. Finally, by using this new method as a benchmark, we have carried out convergence checks for calculated values based on usual methods and have found that some results in the literature are not converged.     

Méthode des Biorbitales: Application Préliminaire au Calcul de l'Energie de l'Etat Fondamental de l'Atome de Beryllium

The ground-state electronic energy of Be is calculated using the method of biorbitals (SCF –BI ). In this method the wave function is represented by an antisymmetrized product of identical pair functions. The basic set used to develop the biorbitals consists of the Watson s and p orbitals. The pair function is presumed to describe a singlet pair state. The energy associated with this function is minimized using a steepest descent procedure. A value of 0.0414 a.u. was found for the correlation energy, which is 44% of the total correlation energy. The SCF –BI method is compared with the CI method. The relationships are established between the expansion coefficients of both methods. The occupation numbers of orbitals are calculated.     

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.     

Low temperature second virial coefficients of H2-He obtained from quantum mechanical pair correlation functions

The previously tested ab initio potential of H₂-He has been used for calculating scattering wave functions, diagonal density matrix elements, pair correlation functions, and finally second virial coefficients, for temperatures up to 100 K. Qualitative agreement with experiment has been found at 90 K, but deviations increased with decreasing temperature up to more than a factor two at temperatures below 20 K. The most likely explanation of this strong effect is the neglect of the formation of dimer hydrogen at the very low temperatures. Some aspects concerning the reliability of the theoretical results will be discussed which are at least partly relevant for equivalent methods.     

Tests of an approximate scaling principle for dynamics of classical fluids

We used molecular dynamics computer simulations to test an approximate scaling principle that conjectures that two equilibrium atomic liquids have very similar dynamical properties if they have the same density and similar static pair correlation functions when the length scales of the two liquids are adjusted appropriately, even if they have different interatomic potentials and different temperatures. The simulations were performed on two types of model atomic liquids at various temperatures at the same density. In the first type, the interatomic potential is the Lennard-Jones potential (LJ). In the second type, the interatomic potential is the repulsive part of the Lennard-Jones potential (RLJ). We identified pairs of systems that have very similar pair correlation functions despite the fact that they had different potentials. Each pair consisted of an LJ liquid at a specific temperature and a corresponding RLJ liquid at a lower temperature. We compared various time correlation functions and transport coefficients of the two systems in each pair. Many dynamical properties are very similar in each pair, in accordance with the approximate scaling principle, whereas others are significantly different. The results indicate that certain dynamical properties are very insensitive to large changes in the interatomic potential that leave the pair correlation function largely unchanged, whereas other dynamical properties are much more sensitive to such changes in the potential. The transport coefficients for diffusion and viscosity are among the dynamical properties that are insensitive to such changes in the potential, and this may be part of the reason transport properties of many fluids have been calculated or rationalized in terms of a simple hard sphere model of liquids.     

Dynamic Viscoelastic and Multiple Scattering Effects in Fiber Suspensions

This study considers the most fundamental problem of 2‐D acoustic scattering in fiber suspensions. It treats the interaction of a plane compressional sound wave with a cluster of two flexible fibers submerged in a boundless viscous fluid medium. The dynamic viscoelastic properties of the fibers and the viscosity of the surrounding fluid are rigorously taken into account in the solution of the problem. The translational addition theorem for cylindrical wave functions, the Havriliak‐Negami model for viscoelastic material behaviors and the appropriate wave field expansions and the pertinent boundary conditions are employed to develop a closed‐form solution in the form of an infinite series. The prime objectives are to investigate the influence of dynamic viscoelastic properties of fiber material as well as multiple scattering interaction effects on acoustic scattering and its associated quantities. The analytical results are illustrated with a numerical example in which two identical viscoelastic fibers are insonified by a plane sound wave at broadside/end‐on incidence. The backscattering form function amplitude and the spatial distribution of the total acoustic pressure are numerically evaluated and discussed for representative values of the parameters characterizing the system. The effects of incident wave frequency, angle of incidence, material properties, and proximity of the two fibers are examined. A limiting case involving a pair of rigid cylinders in an ideal fluid is considered, and fair agreement with a well‐known solution is established.     

A partially projected wave function for radicals

A partially projected wave function for odd electron systems with quantum number M=1/2, containing μu spin functions α and μ spin functions α, with fractional spin component αS_z=1/2 and 3/2 are derived from the totally projected wave function. To obtain these wave functions new symmetry relations between Sanibel coefficients for the odd electron case have been found, as well as the relations between primitive spin functions and their spin permutations. The wave function for the doublet state is shown not to contain contamination of the quadruplet state, and the wave function for the quadruplet does not have contamination of the duplet. Both wave functions exhibit equal forms except in the signs of their summation terms. The number of primitive spin functions depends on the number of electrons (n_s), it grows linearly as n_s=(N+3)/2. It can be considered as a generalization of the half projected Hartree–Fock wave function to the odd electron case. The HPHF wave function is defined for even electron systems and consists of only two Slater determinants, it has been shown to introduce some correlation effects and it has been successfully applied to calculate the low-lying excited states of molecules. Therefore, this investigation is the first step to propose a method to calculate the excited states of radicals when other methods are impracticable. This revised version was published online in July 2006 with corrections to the Cover Date.     

The structure of hyperspherical fluids in various dimensions

The structure of hard, hyperspherical fluids in dimension one, two, three, four, and five has been examined by calculating the pair correlation function using a Monte Carlo simulation. The pair correlation functions match known results in one, two, and three dimensions. The contact value of the pair correlation functions in all the different dimensions agrees well with the theory of Song, Mason, and Stratt [J. Phys. Chem. 93, 6916 (1989)]. The decrease in ordering as the dimension is increased is readily apparent in the structure of the pair correlation function.