Direct solution of H 2 + Schrödinger equation using the hyperspherical coordinate

Summary By introducing a Gaussian factor to describe the fact that the nuclei in H ₂ ⁺ vibrate around a fixed point, we have modified the method of hyperspherical harmonics recently proposed by us. The modified method has been applied to solve the three-body Schrödinger equation for H ₂ ⁺ directly without recourse to the Born-Oppenheimer approximation and the calculations yield well-converged ground-state energies. These are the first-reported results obtained for H ₂ ⁺ by the method of hyperspherical harmonics. With 25 hyperspherical harmonics and 40 generalized-Laguerre functions, we obtain a ground-state energy of –0.5945 au, which is close to the exact value of –0.5971 au. A detailed presentation of the method of modified hyperspherical harmonics is presented.     

Use of computer simulation to determine the triplet distribution function in dense fluids

In this paper a new method is described for determining the triplet distribution function g₃ from computer simulations of dense fluids. The method involves expressing g₃ in terms of hyperspherical coordinates (?,θ,φ) and then expanding g₃ in spherical harmonics of the angular variables θ and φ. We first test the convergence of the spherical harmonic expansion by applying the method to the superposition approximation for g₃. We then show how the expansion coefficients may be obtained from an analysis of computer simulation data. Lastly, we report new results for g₃ obtained by applying the method to data from a molecular dynamics simultion of the Lennard-Jones fluid.     

Expansion of potential 1/r12 of c‐spherical harmonics

The expansion coefficient C^D_|L| of Coulomb potential 1/r₁₂ of atomic system in hyper‐spherical harmonics is derived and the explicit expression is given.     

Maxwell–Cartesian spherical harmonics in multipole potentials and atomic orbitals

 The nature of the Maxwell–Cartesian spherical harmonics S ⁽ⁿ⁾ _K and their relation to tesseral harmonics Y _nm is examined with the help of “tricorn arrays” that display the components of a totally symmetric Cartesian tensor of any rank in a systematic way. The arrays show the symmetries of the Maxwell–Cartesian harmonic tensors with respect to permutation of axes, the traceless properties of the tensors, the linearly independent subsets, the nonorthogonal subsets, and the subsets whose linear combinations produce the tesseral harmonics. The two families of harmonics are related by their connection with the gradients of 1/r, and explicit formulas for the transformation coefficients are derived. The rotational transformation of S ⁽ⁿ⁾ _K functions is described by a relatively simple Cartesian tensor method. The utility of the Maxwell–Cartesian harmonics in the theory of multipole potentials, where these functions originated in the work of Maxwell, is illustrated with some newer applications which employ a detracer exchange theorem and make use of the partial linear independence of the functions. The properties of atomic orbitals whose angular part is described by Maxwell–Cartesian harmonics are explored, including their angular momenta, adherence to an Uns?ld-type spherical symmetry relation, and potential for eliminating an angular momentum “contamination” problem in Cartesian Gaussian basis sets. Received: 9 July 2001 / Accepted: 7 September 2001 / Published online: 19 December 2001     

Equivalent spherical harmonics

It is known that the 2l+1 spherical harmonics Y_lm can be transformed into cyclically equivalent orbitals, of which only a few examples have so far been given explicitly. In this article the totality of such cyclic sets is derived. It is demonstrated that other kinds of equivalent spherical harmonics do not exist. Finally a set of five equivalent d orbitals related to icosahedral symmetry is introduced.     

Theoretical studies of structural and electronic properties of AlnAsn clusters

We present theoretical results of size dependent structural, electronic, and optical properties of ligand‐free stoichiometric Al_nAs_n clusters of zinc‐blende modification. The investigation is done using a simplified parametrized linear combination of atomic orbital–density functional theory‐local density approximation–tight‐binding (LCAO–DFT–LDA–TB) method and consider clusters with n up to around 100. Initial structures have assumed as spherical parts of infinite zinc‐blende structure and then allowed to relax to the closest local‐energy‐minimum structure. We analyze the radial distributions of atoms, Mulliken populations, electronic energy levels (in particular, HOMO and LUMO), bandgap, and stability as a function of size and composition. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

A generalized formula for the energies of alternant molecular orbitals. II. Heteronuclear molecules

Using the method of alternant molecular orbitals (AMO ), it is shown that the energies of AMOS (E_kσ) for an arbitrary heteronuclear alternant system, having a singlet ground state, are connected with the energies of MOS (e_{k(k )}) obtained by means of the conventional Hartree–Fock (HF ) method (SCF -LCAO -MO -PPP ) via the formula: In the general case, the determination of the correlation corrections δ_i,kσ is connected with the solving of a complicated system of integral equations, which is considerably simplified if the Hubbard approximation is accepted for the electron interaction. The energy spectrum of a chain with two atoms in the elementary cell (AB)_n is considered as an example. It is shown that if nontrivial solutions exist (δ_i,kσ ≠ 0), the correlation correction for AMOS of different spin are different (δ_i,kσ ≠ δ_i,kβ), from which it follows, that the width of the energy gap ΔE_∞ for AMOS with different spin is different: ΔE_∞,α ≠ ΔE_∞,β.     

Advancing the computational methodology of rigid rod and semiflexible polymer systems: A new solution to the wormlike chain model with rod‐coil copolymer calculations

A wormlike chain model for rod type blocks in a rod‐coil diblock copolymer is implemented in the self‐consistent field theory (SCFT) formalism. A pseudo‐spectral method is used to solve for the single‐chain partition function of this copolymer system. Orientation degrees of freedom are discretized using Lebedev sphere rules such that orientation integrations are carried out through a Lebedev quadrature, an approach not used previously in tandem with the pseudo‐spectral method. Phase behavior in the rigid‐rod limit as a function of rod segment volume fraction, Flory–Huggins interaction parameter χ , degree of polymerization N , and rod contour length ratio β are examined in detail in one and two dimensions. Examples extending to three dimensions are included. Semiflexible behavior via the rod bending rigidity κ is explored. An approximation is used for rigid‐rods that do not need spherical harmonics leading to increased speed in finding equilibrium morphologies. The results show that standing vertical structures may be more easily produced with rigid‐rod blocks compared to coil‐coil lamellae, an important feature in nanolithographic applications. Suggestions are made for using the model in future molecular orientation studies where the model can be used with inverse search methods to measure the values of the model parameters for the real systems. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 29–39     

Transferable integrals in a deformation density approach to crystal calculations. I. Crystal harmonics and their properties

The term “crystal harmonic” is introduced to denote a symmetrized plane wave in the special case where the wave vector is a reciprocal lattice vector. Crystal harmonics, thus defined, have the translational symmetry of the lattice, and they also have the transformation properties of the irreducible representations of the crystal's point group. An expansion is derived expressing crystal harmonics in terms of spherical Bessel functions and in terms of the functions ??_??,ξ (eigenfunctions of L² which are also basis functions for IRS of the crystal's point group). A sum rule for the functions ??_??,ξ is derived. Methods are given for expanding periodic functions of special symmetry in terms of crystal harmonics. Methods are also presented for calculating matrix elements of the potential in a crystal using crystal harmonics as a basis and for transforming to a STO basis. It is shown that the invariant component of the product of two crystal harmonics can be expressed as a sum of a few invariant crystal harmonics, and expressions for the coefficients in the sum are derived. Orthogonality with respect to summation over networks of points and normalization are also discussed. The properties mentioned above are illustrated in detail in the case of cubic crystals with point group O_h.     

The ORAC Assay: Mathematical Analysis of the Rate Equations and Some Practical Considerations

ORAC (oxygen radical absorbance capacity), a method widely used for measuring the total antioxidant capacity of biological samples, can also be used for the determination of the relative reactivity of an antioxidant compound (XH) by examining the dependence of the rate of consumption of the probe (PH) on the concentration of XH; initial conditions are chosen in such a way that the rate of consumption of the starting reactants may be assumed to follow a drastically simplified kinetic scheme, and the steady‐state approximation for the concentration of the azo compound peroxyl (ROO^•) radical is invoked to simplify the analysis. Here we first attempted to find an analytical solution to the coupled first‐order ordinary differential equations (ODEs) of the minimal ORAC kinetic system, applying Lie symmetry group theory without any precondition. However, the Lie symmetry transformations applied to the Chini equation, which appeared after mathematical transformations, showed that the form of the coefficients of the Chini equation precluded the analytical solution of the minimal ORAC kinetic system through symmetry reduction. Consequently, an approximate analytical solution was sought, valid for the case when the bimolecular rate constant of XH with ROO^• (i.e., k_x ) was much larger than that of PH with ROO^• (i.e., k_p ). Using numerical solutions of the original set of ODEs of the ORAC kinetic system, the quality of the approximate solution was inspected under conditions that correspond to those employed in several ORAC methods together with a low initial concentration of the azo compound radical initiator. The simulations allowed us to conclude that the approximate analytical solution of the ODEs of the minimal ORAC kinetic system was not entirely devoid of academic interest, but its applicability was restricted to conditions where both k_x ≫ k_p and the initial concentration of XH was higher than that of PH.     

An expansion for four-center integrals over Slater-type orbitals

A new expression is given for the electron repulsion integral over Slater-type orbitals on four different centers. It is based on the asymptotic expansion derived from the bipolar expansion of a previous paper. The expression has the form where q_p = {n_p, l_p, m_p}. Both F and σ are closed expressions. The quantity F is a combination of incomplete gamma functions, Laguerre polynomials and spherical harmonics. It depends upon the relative coordinates of a point P on the AB axis and a point Q on the CD axis. The functions σ_nlm(A, B) depend on the charge distribution (χ_Aχ_B); they have the character of overlap integrals and are of the form      

Structure and associated properties of concentrated SiO2 sols in dilute salt solutions

Using the Verwey-Overbeek potential (VO) function the various liquid-state properties of SiO₂ sols in dilute salt solutions have been evaluated under the mean spherical model approximation (MSMA). The structure factors of these SiO₂ sols predicted by this model are compared with results obtained from small-angle neutron scattering experiments by Ramsay et al. Fourier transformation of these structure factors have been performed to obtain the radial distribution functions (RDF), and from these RDF's we computed coordination numbers of the sol particles. The interparticle distanced _c of sol particles has been obtained from the peak position in structure factorS(k) by using the Bragg's equation. The surface potential _s of the oxide sols has been determined from the amplitude (A) of the VO potential. The present calculations clearly indicate some sort of ordering in the sols system. It is gratifying to note that the present theoretical calculations could reproduce the available observed results very satisfactorily with respect to structure factor and other data.     

On a family of differential approximations of the radiative transfer equation

The radiative transfer equation (RTE) arises in a variety of applications and is challenging to solve numerically due to its integro-differential form and high dimension. For highly forward-peaked media, it is even more difficult to solve RTE since accurate numerical solutions require a high resolution of the direction variable. For this reason, various approximations of RTE have been proposed in the literature. In this paper, we study a family of differential approximations of the RTE in three spatial variables. We explain the idea of constructing the differential approximations, and comment on the usefulness of the approximations.     

Shell Structures in Molecular Orbital Energy Diagrams for “Small” Fullerene Cages: Free-Electron Versus Generator Orbital Models

The nearly spherical nature of small fullerene cages C _N (N<70) suggests that their molecular orbital (MO) energy diagrams should show a shell structure. Although group theoretical analysis of the Hückel-type energies for icosahedral and other highly symmetric cages confirms this assumption, this has not been established for fullerene cages in general. This work presents a simple computational algorithm based upon the canonical orthogonalization of generator orbitals (GOs) to analyze the -MO energy diagrams for any fullerene cage C _N , and demonstrates the validity of a shell structure in these diagrams. Results are compared to simple central force (spherical) models for the calculations of -MO energies in fullerene cages. The GO approach provides a ready assignment of the -MOs to individual spherical harmonics and allows valuable interpretations of various physical phenomena.     

Equation of state based on the weeks-chandler-andersen perturbation theory

An equation of state based on the Weeks-Chandler-Andersen (WCA) perturbation theory utilizing the simplified random-phase approximation (RPA) term is p of state is applied successfully to calculate the P-V-T relationship for a spherical molecule, argon.For nonspherical molecules, the effects of anisotropic interactions are treated empirically. The calculated P-V-T relationships and saturated properties for nonspherical and nonpolar molecules agree well with experimental data. The potential parameters for nonpolar substances are well correlated with the acentric factors.     

A simplified data treatment for dead-end and high conversion polymerization kinetics

A simplified approximation method for the treatment of dead-end and high conversion polymerization kinetics is presented. The method is based on the treatment of dead-end polymerization first described by Tobolsky. In appropriate circumstances, by contrast with Tobolsky's method, this method provides measurements of k_d and k_p/k_t^1/2 without recourse to the measurement of the monomer conversion at infinite time. Kinetic studies of free radical polymerizations are normally confined to measurements of initial rates. At low conversions the predictions of the general mechanism for chain-growth polymerization involving initiation, propagation, and termination steps are generally obeyed. Thus the polymerization rate should be first order in the vinyl monomer and half-order in the initiator concentrations. At high conversions, however, large deviations which can be ascribed to various effects can occur; for example, (1) the effect of the increasing viscosity of the polymerization medium on the termination rate constant k_t, and possibly also on the propagation rate constant k_p, which have been considered by North¹ and Cardenas and O'Driscoll,² or (2) depletion of the initiator as the polymerization progresses. This depletion will occur in all polymerizations but its significance will depend on the magnitude of the rate constant for initiator decomposition (k_d) and the period of polymerization. Appropriate conditions will lead to limiting monomer conversion even after infinite polymerization time; this phenomenon has been called dead-end polymerization by Tobolsky.³ Free radical polymerizations to high conversion are particularly important in the industrial context when initial kinetics are obviously inadequate. Suitable treatment of the conversion/time relationship is highly desirable. Senogles and Woolf⁴ have examined the polymerization of n-lauryl methacrylate at 60°C with 2-azobisisobutyronitrile as initiator under dead-end conditions. Here we propose a modification of Tobolsky's treatment of such polymerizations by using an approximation for the exponential decay in the initiator concentration. This method permits easy manipulation of the experimental data and the estimation of values for the kinetic parameters in favorable circumstances without recourse to the measurement of the conversion at infinite time or the evaluation of complicated functions of the monomer conversion. The method thus allows the duration of the laboratory experimentation to be significantly shortened and the complexity of the subsequent data analysis to be considerably reduced.     

Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. I. Three-electron integrals

A method to evaluate the nonrelativistic electron-repulsion, nuclear attraction and kinetic energy three-electron integrals over Slater orbitals appearing in Hylleraas-CI (Hy-CI) electron structure calculations on atoms is shown. It consists on the direct integration over the interelectronic coordinate r _ij and the sucessive integration over the coordinates of one of the electrons. All the integrals are expressed as linear combinations of basic two-electron integrals. These last are solved in terms of auxiliary two-electron integrals which are easy to compute and have high accuracy. The use of auxiliary three-electron ones is avoided, with great saving of storage memory. Therefore this method can be used for Hy-CI calculations on atoms with number of electrons N ≥ 5. It has been possible to calculate the kinetic energy also in terms of basic two-electron integrals by using the Hamiltonian in Hylleraas coordinates, for this purpose some mathematical aspects like derivatives of the spherical harmonics with respect to the polar angles and recursion relations are treated and some new relations are given.     

Flow-alignment and rheology of polymer melts: Computation of the single-link orientational distribution function

The single-link orientational distribution function and the space-averaged stresses in the fluid are computed for the case of steady shear flow of polymer melts. The computation is achieved with Galerkin's method with spherical harmonics and Euler polynomials as trial functions. The stress components become power functions of shear rate when the latter is large. The single-link orientational distribution function f solves the Fokker-Planck equation subject to a boundary condition for f at the chain ends. A solution is obtained for every shear rate and ratio of the orientational and one-dimensional diffusion coefficient. It is demonstrated that the Fokker-Planck equation with appropriate boundary condition is useful in order to predict the flow-alignment and stresses in good agreement with experimental data as well as with recent results of a nonequilibrium molecular dynamics computer simulation on polymer melts.     

Steps toward a high precision evaluation of the hyperfine interactions in neutral muonic helium

The solution of the Schrödinger equation of the neutral muonic helium is sketched by an eigenfunction expansion method: the eigenfunctions of the two Coulombic centres problem (of chargesZ ₁=2,Z ₂=–1) are used to expand the wave function. Our Born expansion method is a generalization of the Born-Oppenheimer approximation to a system in which the two centres (He, ^–) do not have a stable equilibrium distance. The adiabatic approximation is solved, upper-lower bounds on the eigenvalue are given for a number of states. The hyperfine energy corrections are calculated in general terms and are given numerically for the ground state and for the first muonically and electronically excited states in the frames of the adiabatic approximation. Our best value fails to give the observed hyperfine splitting of the ground state by some 5 × 10^–4 (2 MHz).     

The surface charge density/surface potential relationship and the Poisson-Boltzmann equation

A procedure is described which provides an approximate analytic expression for the relationship between the surface charge density and the surface potential of a spherical or cylindrical colloidal particle in a general type of electrolyte. The first approximation is found and the approximation error is given. Also derived are the fourth approximation for a symmetric z-z type electrolyte and the second approximation for a symmetric z-z, 2z-2z type electrolyte.