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The rotational excitation rates of NCCN species are studied for its collision with hydrogen (H2) in temperatures ranging from 1 to 100 K. Such collisions can occur in the interstellar medium with H2 in either para (p-) or ortho (o-) state, of which the p-H2 state can be approximated via its collision with He (using a scaling factor) or with a reduced rigid rotor-H2 surface (by averaging over various orientations of H2). In the current work, a four-dimensional (4D) ab initio potential energy surface (PES) is considered to study the collision dynamics of H2 in both p- and o-states and the results are compared with previous approximations. The 4D surface is constructed using the explicitly correlated coupled-cluster method CCSD(T)-F12b with the augmented triple zeta basis AVTZ and then fitted into an artificial neural networks (NN) model to augment the surface and account for missing data points. The radial coefficients are obtained from this NN fitted 4D PES via a least square fit over two spherical harmonics functions. The cross sections (σ) are computed using the close-coupling (CC) method (until 230 cm−1) for both p- and o-H2 collisions, and the rates are obtained by Boltzmann distribution over the translational energy of H2 until 100 K. The o-H2 rates are found to be higher by 25%–30% and 10%–20% compared to the p-H2 rates for Δj = 2 and higher order transitions, respectively. The coupled-state/centrifugal sudden approximated rates are also computed and found to have deviations as large as 40% when compared to CC rates, thus making quantitative descriptions unreliable. [ABSTRACT FROM AUTHOR]

Spherical harmonics provide a smooth, orthogonal, and symmetry-adapted basis to expand functions on a sphere, and they are used routinely in physical and theoretical chemistry as well as in different fields of science and technology, from geology and atmospheric sciences to signal processing and computer graphics. More recently, they have become a key component of rotationally equivariant models in geometric machine learning, including applications to atomic-scale modeling of molecules and materials. We present an elegant and efficient algorithm for the evaluation of the real-valued spherical harmonics. Our construction features many of the desirable properties of existing schemes and allows us to compute Cartesian derivatives in a numerically stable and computationally efficient manner. To facilitate usage, we implement this algorithm in sphericart, a fast C++ library that also provides C bindings, a Python API, and a PyTorch implementation that includes a GPU kernel. [ABSTRACT FROM AUTHOR]

Machine learning frameworks based on correlations of interatomic positions begin with a discretized description of the density of other atoms in the neighborhood of each atom in the system. Symmetry considerations support the use of spherical harmonics to expand the angular dependence of this density, but there is, as of yet, no clear rationale to choose one radial basis over another. Here, we investigate the basis that results from the solution of the Laplacian eigenvalue problem within a sphere around the atom of interest. We show that this generates a basis of controllable smoothness within the sphere (in the same sense as plane waves provide a basis with controllable smoothness for a problem with periodic boundaries) and that a tensor product of Laplacian eigenstates also provides a smooth basis for expanding any higher-order correlation of the atomic density within the appropriate hypersphere. We consider several unsupervised metrics of the quality of a basis for a given dataset and show that the Laplacian eigenstate basis has a performance that is much better than some widely used basis sets and competitive with data-driven bases that numerically optimize each metric. Finally, we investigate the role of the basis in building models of the potential energy. In these tests, we find that a combination of the Laplacian eigenstate basis and target-oriented heuristics leads to equal or improved regression performance when compared to both heuristic and data-driven bases in the literature. We conclude that the smoothness of the basis functions is a key aspect of successful atomic density representations. [ABSTRACT FROM AUTHOR]

Three- and four-center Coulomb integrals in the solid spherical harmonic Gaussian basis are solved by expansion in terms of two-center integrals. The two-electron Gaussian product rule, coupled with the addition theorem for solid spherical harmonics, reduces four-center Coulomb integrals into a linear combination of two-center Coulomb integrals and one-center overlap integrals. With this approach, three- and four-center Coulomb integrals can be reduced to the same form of two-center integrals. Resulting two-center Coulomb integrals can be further simplified into a simpler form, which can be related to the Boys function. Multi-center Coulomb integrals are solved hierarchically: simple two-center Coulomb integrals are used for calculation of more complicated two-center Coulomb integrals, which are used in the calculation of multicenter integrals. [ABSTRACT FROM AUTHOR]

This paper proposes the first linear scaling implementation for the domain decomposition approach of the polarizable continuum model (ddPCM) for the computation of the solvation energy and forces. The ddPCM-equation consists of a (non-local) integral equation on the van der Waals or solvent accessible surface of the solute's cavity resulting in a dense solution matrix, and, in turn, one matrix–vector multiplication has a quadratic arithmetic complexity with respect to the number of atoms of the solute molecule. The use of spherical harmonics as basis functions makes it natural to employ the fast multipole method (FMM) in order to provide an asymptotically linear scaling method. In this paper, we employ the FMM in a non-uniform manner with a clusterization based on a recursive inertial bisection. We present some numerical tests illustrating the accuracy and scaling of our implementation. [ABSTRACT FROM AUTHOR]

Periodic nanoparticle arrays have attracted considerable interest recently since the lattice effect can lead to spectrally narrow resonances and tune the resonance position in a broad range. Multipole decomposition is widely used to analyze the role of the multipoles in the resonance excitations, radiation, and scattering of electromagnetic waves. However, previous studies have not addressed the validity and accuracy of the multipole decomposition around the lattice resonance. The applicability of the exact multipole decomposition based on spherical harmonics expansion has not been demonstrated around the lattice resonance with the strong multipole coupling. This work studies the two-dimensional periodic arrays of both plasmonic and dielectric nanospheres and compares the multipole decomposition results with the analytic ones around their lattice resonances. We study both the effective polarizabilities of multipoles and the scattering spectra of the structures. The analytical results are calculated from the coupled dipole–quadrupole model. This study demonstrates that the exact multipole decomposition agrees well with the numerical simulation around lattice resonances. Only a small number of multipoles are required to represent the results accurately. [ABSTRACT FROM AUTHOR]

We propose an optimization method for the inverse structural design of self-assembly of anisotropic patchy particles. The anisotropic interaction can be expressed by the spherical harmonics of the surface pattern on a patchy particle, and thus, arbitrary symmetries of the patch can be treated. The pairwise interaction potential includes several to-be-optimized parameters, which are the coefficients of each term in the spherical harmonics. We use the optimization method based on the relative entropy approach and generate structures by Brownian dynamics simulations. Our method successfully estimates the parameters in the potential for the target structures, such as square lattice, kagome lattice, and dodecagonal quasicrystal. [ABSTRACT FROM AUTHOR]

A representation of the three-dimensional potential energy surface (PES) of the H2O–H van der Waals dimer is presented. The H2O molecule is treated as a rigid body held at its experimentally determined equilibrium geometry, with the OH bond length set to 1.809 650 34 a0 and the HOH bond angle set to 1.824 044 93 radians. Ab initio calculations are carried out at the coupled-cluster single, double, and perturbative triple level, with scalar relativistic effects included using the second-order Douglas–Kroll–Hess approximation. The ab initio calculations employ the aug-cc-pVnZ-DK series of basis sets (n = D, T, Q), which are recontracted versions of the aug-cc-pVnZ basis sets that are appropriate for relativistic calculations. The counterpoise method is used to reduce the basis set superposition error; in addition, results obtained using the aug-cc-pVTZ-DK and aug-cc-pVQZ-DK basis sets were extrapolated to the complete basis set (CBS) limit. The PES is based on calculations carried out at 1054 symmetry-unique H2O–H geometries for which the distance R between the H-atom and the H2O center of mass ranges from R = 2.5–9.0 Å. The reproduction of the PES along the orientational degrees of freedom was performed using Lebedev quadrature and an expansion in spherical harmonics. The mean absolute error of the reproduced PES is <0.02 cm−1 for R ≥ 3.0 Å and <0.21 cm−1 for R between 2.5 and 3.0 Å. The global minimum for the CBS PES is a coplanar H2O–H geometry, with R = 3.41 Å, in which the angle formed between the H2O C2 symmetry axis and the H-atom is 122.25°; the CBS binding energy for this geometry is 61.297 cm−1. In addition, by utilizing the symmetry of the H2O molecule, the spherical harmonic expansion was simplified with no loss in accuracy and a speedup of ∼1.8 was achieved. The reproduced PES can be used in future molecular dynamics simulations. [ABSTRACT FROM AUTHOR]

We present numerical results for the dipole induced by interactions between a hydrogen molecule and a hydrogen atom, obtained from finite-field calculations in an aug-cc-pV5Z basis at the unrestricted coupled-cluster level including all single and double excitations in the exponential operator applied to a restricted Hartree–Fock reference state, with the triple excitations treated perturbatively, i.e., UCCSD(T) level. The Cartesian components of the dipole have been computed for nine different bond lengths r of H2 ranging from 0.942 a.u. to 2.801 a.u., for 16 different separations R between the centers of mass of H2 and H between 3.0 a.u. and 10.0 a.u., and for 19 angles θ between the H2 bond vector r and the vector R from the H2 center of mass to the nucleus of the H atom, ranging from 0° to 90° in intervals of 5°. We have expanded the interaction-induced dipole as a series in the spherical harmonics of the orientation angles of the H2 bond axis and of the intermolecular vector, with coefficients DλL(r, R). For the geometrical configurations that we have studied in this work, the most important coefficients DλL(r, R) in the series expansion are D01(r, R), D21(r, R), D23(r, R), D43(r, R), and D45(r, R). We show that the ab initio results for D23(r, R) and D45(r, R) converge to the classical induction forms at large R. The convergence of D45(r, R) to the hexadecapolar induction form is demonstrated for the first time. Close agreement between the long-range ab initio values of D01(r0 = 1.449 a.u., R) and the known analytical values due to van der Waals dispersion and back induction is also demonstrated for the first time. At shorter range, D01(r, R) characterizes isotropic overlap and exchange effects, as well as dispersion. The coefficients D21(r, R) and D43(r, R) represent anisotropic overlap effects. Our results for the DλL(r, R) coefficients are useful for calculations of the line shapes for collision-induced absorption and collision-induced emission in the infrared and far-infrared by gas mixtures containing both H2 molecules and H atoms. [ABSTRACT FROM AUTHOR]

We show that generalized spherical harmonics are well suited for representing the space and orientation molecular density in the resolution of the molecular density functional theory. We consider the common system made of a rigid solute of arbitrary complexity immersed in a molecular solvent, both represented by molecules with interacting atomic sites and classical force fields. The molecular solvent density ρ(r, Ω) around the solute is a function of the position r ≡ (x, y, z) and of the three Euler angles Ω (ϑ, ∅, ψ) describing the solvent orientation. The standard density functional, equivalent to the hypernetted-chain closure for the solute-solvent correlations in the liquid theory, is minimized with respect to ρ(r, Ω). The up-to-now very expensive angular convolution products are advantageously replaced by simple products between projections onto generalized spherical harmonics. The dramatic gain in speed of resolution enables to explore in a systematic way molecular solutes of up to nanometric sizes in arbitrary solvents and to calculate their solvation free energy and associated microscopic solvent structure in at most a few minutes.We finally illustrate the formalism by tackling the solvation of molecules of various complexities in water. [ABSTRACT FROM AUTHOR]

An extended E ⊗ e Jahn-Teller Hamiltonian is presented for the case where the (slow) nuclear motion extends far from the symmetry point and may be described approximately as motion on a sphere. Rather than the traditional power series expansion in the displacement from theC3v symmetry point, an expansion in the spherical harmonics is employed. Application is made to the vibrational Jahn-Teller effect in CH3XH, with X = S, O, where the equilibrium CXH angles are 83° and 72°, respectively. In addition to the symmetry-required conical intersection (CI) at the C3v symmetry point, ab initio calculations reveal sets of six symmetry-allowed vibrational CIs in each molecule. The CIs for each molecule are arranged differently in the large-amplitude space, and that difference is reflected in the infrared spectra. The CIs inCH3SHare found in both eclipsed and staggered geometries, whereas those for CH3OH are found only in the eclipsed geometry near the torsional saddle point. This difference between the two molecules is reflected in the respective high-resolution spectra in the CH stretch fundamental region. [ABSTRACT FROM AUTHOR]

The surface structure and its fluctuation of spherical micelles were investigated using a series of density correlation functions newly defined by spherical harmonics and Legendre polynomials based on the molecular dynamics calculations. To investigate the influence of head-group charges on the micelle surface structure, ionic sodium dodecyl sulfate and nonionic octaethyleneglycol monododecylether (C12E8) micelles were investigated as model systems. Large-scale density fluctuations were observed for both micelles in the calculated surface static structure factor. The area compressibility of the micelle surface evaluated by the surface static structure factor was tens-of-times larger than a typical value of a lipid membrane surface. The structural relaxation time, which was evaluated from the surface intermediate scattering function, indicates that the relaxation mechanism of the long-range surface structure can be well described by the hydrostatic approximation. The density fluctuation on the two-dimensional micelle surface has similar characteristics to that of three-dimensional fluids near the critical point. [ABSTRACT FROM AUTHOR]

It is possible to associate with every atom or molecule in a liquid its own atomic stress tensor. These atomic stress tensors can be used to describe liquids' structures and to investigate the connection between structural and dynamic properties. In particular, atomic stresses allow to address atomic scale correlations relevant to the Green-Kubo expression for viscosity. Previously correlations between the atomic stresses of different atoms were studied using the Cartesian representation of the stress tensors or the representation based on spherical harmonics. In this paper we address structural correlations in a 3D model binary liquid using the eigenvalues and eigenvectors of the atomic stress tensors. This approach allows to interpret correlations relevant to the Green-Kubo expression for viscosity in a simple geometric way. On decrease of temperature the changes in the relevant stress correlation function between different atoms are significantly more pronounced than the changes in the pair density function. We demonstrate that this behaviour originates from the orientational correlations between the eigenvectors of the atomic stress tensors.We also found correlations between the eigenvalues of the same atomic stress tensor. For the studied system, with purely repulsive interactions between the particles, the eigenvalues of every atomic stress tensor are positive and they can be ordered: λ1 ≥ λ2 ≥ λ3 ≥ 0. We found that, for the particles of a given type, the probability distributions of the ratios (λ2/λ1) and (λ3/λ2) are essentially identical to each other in the liquids state. We also found that λ2 tends to be equal to the geometric average of λ1 and λ3. In our view, correlations between the eigenvalues may represent "the Poisson ratio effect" at the atomic scale. [ABSTRACT FROM AUTHOR]

We introduce a new basis function (the spherical Gaussian) for electronic structure calculations on spheres of any dimension D. We find general expressions for the one- and two-electron integrals and propose an efficient computational algorithm incorporating the Cauchy-Schwarz bound. Using numerical calculations for the D = 2 case, we show that spherical Gaussians are more efficient than spherical harmonics when the electrons are strongly localized. [ABSTRACT FROM AUTHOR]

Next-generation molecular force fields deliver accurate descriptions of non-covalent interactions by employing more elaborate functional forms than their predecessors. Much work has been dedicated to improving the description of the electrostatic potential (ESP) generated by these force fields. A common approach to improving the ESP is by augmenting the point charges on each center with higher-order multipole moments. The resulting anisotropy greatly improves the directionality of the non-covalent bonding, with a concomitant increase in computational cost. In this work, we develop an efficient strategy for enumerating multipole interactions, by casting an efficient spherical harmonic based approach within a particle mesh Ewald (PME) framework. Although the derivation involves lengthy algebra, the final expressions are relatively compact, yielding an approach that can efficiently handle both finite and periodic systems without imposing any approximations beyond PME. Forces and torques are readily obtained, making our method well suited to modern molecular dynamics simulations. [ABSTRACT FROM AUTHOR]

Hyper-Rayleigh scattering (HRS) experiments and quantum chemical calculations are combined to investigate the second-order nonlinear optical responses of a series of reference molecules, namely, carbon tetrachloride, chloroform, trichloroacetonitrile, acetonitrile, and dichloromethane. The multipolar decomposition of the first hyperpolarizability tensor through the use of the spherical harmonics formalism is employed to highlight the impact of the symmetry of the molecular scatterers on their nonlinear optical responses. It is demonstrated that HRS is a technique of choice to probe the molecular symmetry of the compounds. Coupled-cluster calculations performed at the coupled-cluster level with singles, doubles, and perturbative triples in combination with highly extended basis sets and including environment effects by using the polarizable continuum model qualitatively reproduce the molecular first hyperpolarizabilities and depolarization ratios of the molecular scatterers. [ABSTRACT FROM AUTHOR]

We introduce a novel interpolation strategy, based on nonequispaced fast transforms involving spherical harmonics or Wigner functions, for efficient calculations of powder spectra in (nuclear) magnetic resonance spectroscopy. The fast Wigner transform (FWT) interpolation operates by minimizing the time-consuming calculation stages, by sampling over a small number of Gaussian spherical quadrature (GSQ) orientations that are exploited to determine the spectral frequencies and amplitudes from a 10-70 times larger GSQ set. This results in almost the same orientational averaging accuracy as if the expanded grid was utilized explicitly in an order of magnitude slower computation. FWT interpolation is applicable to spectral simulations involving any time-independent or time-dependent and noncommuting spin Hamiltonian. We further show that the merging of FWT interpolation with the well-established ASG procedure of Alderman, Solum and Grant [J. Chem. Phys. 134, 3717 (1986)] speeds up simulations by 2-7 times relative to using ASG alone (besides greatly extending its scope of application), and between 1-2 orders of magnitude compared to direct orientational averaging in the absence of interpolation. Demonstrations of efficient spectral simulations are given for several magic-angle spinning scenarios in NMR, encompassing half-integer quadrupolar spins and homonuclear dipolar-coupled 13C systems. [ABSTRACT FROM AUTHOR]

The orientational correlation functions measured in the time-resolved second-harmonic generation (TRSHG) and time-resolved sum-frequency generation (TRSFG) experiments are derived. In the laboratory coordinate system, the (l=1,3 and m=0,2) correlation functions, where the Ylm are spherical harmonics, describe the orientational relaxation observables of molecules at interfaces. A wobbling-in-a-cone model is used to evaluate the correlation functions. The theory demonstrates that the orientational relaxation diffusion constant is not directly obtained from an experimental decay time in contrast to the situation for a bulk liquid. Model calculations of the correlation functions are presented to demonstrate how the diffusion constant and cone half-angle affect the time-dependence of the signals in TRSHG and TRSFG experiments. Calculations for the TRSHG experiments on Coumarin C314 molecules at air-water and air-water-surfactant interfaces are presented and used to examine the implications of published experimental results for these systems. [ABSTRACT FROM AUTHOR]

We present a method for describing and comparing global electrostatic properties of biomolecules based on the spherical harmonic decomposition of electrostatic potential data. Unlike other approaches our method does not require any prior three dimensional structural alignment. The electrostatic potential, given as a volumetric data set from a numerical solution of the Poisson or Poisson–Boltzmann equation, is represented with descriptors that are rotation invariant. The method can be applied to large and structurally diverse sets of biomolecules enabling to cluster them according to their electrostatic features. [ABSTRACT FROM AUTHOR]

The vibrational energy levels of the H3O2- anion have been calculated using a rigorous quantum dynamics method based on an accurate ab initio potential energy surface. The eigenvalue problem is solved using the two-layer Lanczos iterative diagonalization algorithm in a mixed grid/nondirect product basis set, where the system Hamiltonian is expressed in a set of orthogonal polyspherical coordinates. The lowest 312 vibrational energy levels in each inversion symmetry, together with a comparison of fundamental frequencies with previous quantum dynamics calculations, are reported. Finally, a statistical analysis of nearest level spacing distribution is carried out, revealing a strongly chaotic nature. [ABSTRACT FROM AUTHOR]

The vibrational energy levels of the H3O2- anion have been calculated using a rigorous quantum dynamics method based on an accurate ab initio potential energy surface. The eigenvalue problem is solved using the two-layer Lanczos iterative diagonalization algorithm in a mixed grid/nondirect product basis set, where the system Hamiltonian is expressed in a set of orthogonal polyspherical coordinates. The lowest 312 vibrational energy levels in each inversion symmetry, together with a comparison of fundamental frequencies with previous quantum dynamics calculations, are reported. Finally, a statistical analysis of nearest level spacing distribution is carried out, revealing a strongly chaotic nature. [ABSTRACT FROM AUTHOR]

The currently most efficient finite basis representation (FBR) method [Corey et al., in Numerical Grid Methods and Their Applications to Schrödinger Equation, NATO ASI Series C, edited by C. Cerjan (Kluwer Academic, New York, 1993), Vol. 412, p. 1; Bramley et al., J. Chem. Phys. 100, 6175 (1994)] designed specifically to deal with nondirect product bases of structures Φ n ¹ (s) f1 (u), Xm¹ (t) Φn¹(s)f1(u) etc., employs very special l-independent grids and results in a symmetric FBR. While highly efficient, this method is not general enough. For instance, it cannot deal with nondirect product bases of the above structure efficiently if the functions Φn¹(s) [and/or Xm¹(t)] are discrete variable representation (DVR) functions of the infinite type. The optimal-generalized FBR(DVR) method [V. Szalay, J. Chem. Phys. 105, 6940 (1996)] is designed to deal with general, i.e., direct and/or nondirect product, bases and grids. This robust method, however, is too general, and its direct application can result in inefficient computer codes [Czakó et al., J. Chem. Phys. 122, 024101 (2005)]. It is shown here how the optimal-generalized FBR method can be simplified in the case of nondirect product bases of structures Φn¹ (s) f1(u), Xm¹(t)Φn¹(s) f1 (u) etc. As a result the commonly used symmetric FBR is recovered and simplified nonsymmetric FBRs utilizing very special l-dependent grids are obtained. The nonsymmetric FBRs are more general than the symmetric FBR in that they can be employed efficiently even when the functions Φn¹ (s) [and/or Xm¹ (t) are DVR functions of the infinite type. Arithmetic operation counts and a simple numerical example presented show unambiguously that setting up the Hamiltonian matrix requires significantly less computer time when using one of the proposed nonsymmetric FBRs than that in the symmetric FBR. Therefore, application of this nonsymmetric FBR is more efficient than that of the symmetric FBR when one wants to diagonalize the Hamiltonian matrix either by a direct or via a basis-set contraction method. Enormous decrease of computer time can be achieved, with respect to a direct application of the optimal-generalized FBR, by employing one of the simplified nonsymmetric FBRs as is demonstrated in noniterative calculations of the low-lying vibrational energy levels of the H3+ molecular ion. The arithmetic operation counts of the Hamiltonian matrix vector products and the properties of a recently developed diagonalization method [Andreozzi et al., J. Phys. A Math. Gen. 35, L61 (2002)] suggest that the nonsymmetric FBR applied along with this particular diagonalization method is suitable to large scale iterative calculations. Whether or not the nonsymmetric FBR is competitive with the symmetric FBR in large-scale iterative calculations still has to be investigated numerically. [ABSTRACT FROM AUTHOR]

We present a method for computing bend energy levels of a five-atom molecule. We use polyspherical coordinates in terms of which both the kinetic energy operator and its associated volume element are simple. The basis functions we use are new. They are parity-adapted combinations of products of spherical harmonics and an associated Legendre function. We show that in the parity-adpated basis it is possible to evaluate matrix-vector products efficiently, despite the fact that the parity-adapted functions are not products of functions of a single variable. The method is applied to compute bend levels of methane. Within each parity block we use the symmetry-adapted Lanczos algorithm to compute levels labeled by irreps of a four-member group that contains the parity operator and the permutation of two of the four hydrogen atoms. In a future publication we shall use the bend wave functions as contracted basis functions to compute numerically exact vibrational levels of methane. [ABSTRACT FROM AUTHOR]

A partition of the molecular density for Slater basis sets (STO), which parallels one previously developed for Gaussian basis sets (GTO), is reported. The atomic fragments are expanded in spherical harmonics times radial factors. Each fragment contains all the one-center charge distributions centered in the atom plus the part of every two-center distribution assigned to the atom by the partition criterion. The performance of the procedure is analyzed, concluding that the analysis gives highly accurate representations of the molecular density at a very low cost. Moreover, the results of the analysis are illustrated with the study of the densities in CO and H[sub 2]O and the comparison of the atomic densities obtained from STO and GTO molecular calculations. [ABSTRACT FROM AUTHOR]

Studies the rapid and stable determination of rotation matrices between spherical harmonics by direct recursion. Definition of complex spherical harmonics; Recurrence relations for complex spherical harmonics; Rotation of complex spherical harmonics.

Studies the angular distributions of photoelectrons from fixed-in-space molecules of C[sub 3v] symmetry group for the particular case when the light beam in incoming parallel to the C[sub 3] rotational axis of the molecule. Basis of the consideration on the expansion of photoelectron wave functions in symmetry-adapted spherical harmonics.

Extends the minimal deformation criterion, previously proposed for the partition of the molecular density into atomic contributions. Expansion of the atomic contributions for any Gaussian basis set in a series of real spherical harmonics by radial factors; Determination of the spherical parts of the atomic clouds and the remaining ones.

Several fast algorithms for the approximation of particle–particle interactions by means of multipole expansions in spherical harmonics have appeared recently. In this letter we present a redefinition of the solid spherical harmonics that is real and gives simple expressions for the evaluation of the functions and their derivatives. Application to the recursive bisection method [J. M. Pérez-Jordá and W. Yang, Chem. Phys. Lett. 247, 484 (1995)] greatly improves its performance. © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In developing a pseudospectral transform between a nondirect product basis of spherical harmonics and a direct product grid, Corey and Lemoine [J. Chem. Phys. 97, 4115 (1992)] generalized the Fourier method of Kosloff and the discrete variable representation (DVR) of Light by introducing more grid points than spectral basis functions. Assuming that the potential energy matrix is diagonal on the grid destroys the variational principle in the Fourier and DVR methods. In the present article we (1) demonstrate that the extra grid points in the generalized discrete variable representation (GDVR) act as dealiasing functions that restore the variational principle and make a pseudospectral calculation equivalent to a purely spectral one, (2) describe the general principles for extending the GDVR to other nondirect product spectral bases of orthogonal polynomials, and (3) establish the relation between the GDVR and the least squares method exploited in the pseudospectral electronic structure and adiabatic pseudospectral bound state calculations of Friesner and collaborators. © 1995 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A comparison of Percus–Yevick–Pynn–Lado model theory and a density functional (DF) theory of nonuniform fluids of nonspherical particles is performed. The DF used is a new generalization of Tarazona’s theory. The conclusion is that DF theory provides a preferable route to describe the system under consideration. Its accuracy can be improved with better approximation for the direct correlation function (DCF) for bulk system. © 1995 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The fast multipole method (FMM) has become an important alternative to traditional methods such as the Ewald method for computing the long-range interactions necessary to simulate charged or dipolar systems. In this paper, we present an improvement of this method, which we shall call the very fast multipole method (VFMM). The VFMM is shown to be a factor of about 1.2 faster than the FMM for two-dimensional systems and a factor about 2–3 times faster for three-dimensional systems without losing any accuracy for the worst case error. [ABSTRACT FROM AUTHOR]

The Percus–Yevick (PY) equation for inhomogeneous hard linear molecules is presented and solved. Calculations of the density profile are shown. Despite earlier failures of the PY approximation to predict some kinds of angular properties in homogeneous systems, here we find predictions which are qualitatively correct and quantitatively acceptable. In particular, we find that the molecules near the wall lie parallel to the wall. [ABSTRACT FROM AUTHOR]

Reports on the derivation of the infinite hierarchy of differential-recurrence relations for ensemble averages of the spherical harmonics pertaining to the noninertial rotational Brownian motion of an ensemble of polar and anisotropically polarizable molecules in a strong external direct current electric field. Use of averaging of the underlying Langevin equation.

The stationary version of the linear Boltzmann equation (the Lorentz model) is studied in the half-space x>0, x=0 being a grey surface of reflection coefficient κ (0≤κ≤1). The spherical harmonics method and Marshak-type boundary conditions are used to find expressions for n(K)(x), the particle density in the so-called PK approximation (K=1,3,5). The results given by the P5 approximation accord very well with recent numerical computations [R. G. Cole, T. Keyes, and V. Protopopescu, J. Chem. Phys. 81, 2771 (1984)]. The desideratum in this context, an accurate expression for the particle density, is produced by grafting the desirable features of n(5)(x) on an almost exact expression derived, for the special case of a black surface (κ=0), by J. LeCaine [Phys. Rev. 72, 564 (1947)]. [ABSTRACT FROM AUTHOR]

The authors discuss the article "Effective field parameters in iron Mössbauer spectroscopy." Topics include the use of magnetically ordered powder sample Mössbauer spectrum to determine local hyperfine field direction in the coordinate system determined by electric field gradient (efg) tensor, the use of Mössbauer isotopes, and Hamiltonian elements of the excited and ground states. Also mentioned the Clebsch-Gordon coefficients which are the vector spherical harmonics.

A simple model for α- and β-rotational relaxation (detected through the behavior of the lth order spherical harmonics) in supercooled liquids predicts that the relaxation frequency of the latter is proportional to l(l+1) and that of the former (confirmed or at least suggested experimentally) is independent of l. This has been taken to mean that the β relaxation takes place by means of small angular jumps (diffusion) and the α relaxation by means of long angular jumps. However, because the amplitude of the α relaxation decreases rapidly with increasing l, despite the fact that the α-relaxation frequency is independent of l, the motion may be diffusional in accord with the observations of Spiess et al. This is discussed in terms of a simple model. [ABSTRACT FROM AUTHOR]

General formulas are derived for the two-center Coulomb and hybrid integrals for real and complex STOs with respect to the common molecular coordinate system. The angular and radial parts of these integrals are expressed through the spherical harmonics and the auxiliary functions ∼(Gq-NN’), respectively. [ABSTRACT FROM AUTHOR]