Bipolar Expansion Research Articles

Extension of the PM3 Method on s,p,d Basis. Test Calculations on Organochromium Compounds

A model for estimation of the two-electron interaction in molecules is proposed. It is based on the bipolar expansion of the Ohno potential which is considered as a universal effective potential of the two-electron interaction. It allows to estimate efficiently the two-electron fockian terms taking into implicit account the dynamic electron correlation. The formulas obtained do not depend explicitly on the orbital quantum number of the basis AO's, and this approach may be used to extend the semiempirical NDDO-type methods to the s,p,d basis. In this work, this approach was applied to the semiempirical PM3 model, and the test calculations were performed on organochromium compounds, taking into explicit account the chromium d orbitals. The calculated thermodynamic, structural, and electron properties of above 30 organochromium compounds of different classes (sandwich complexes, carbonyls, nytrosyls, and mixed derivatives) stay in good agreement with the experimental data.

Photofragment μ-v-j correlation measured by 1+n′ resonance-enhanced multiphoton ionization: Selective probing of bipolar moments and detection of chiral dynamics

A practical procedure is described to measure photofragment μ-v-j correlations using polarized 1+n′ resonance-enhanced multiphoton ionization with a time-of-flight mass spectrometer detector. Following the theory of Dixon [R. N. Dixon, J. Chem. Phys. 85, 1866 (1986)], the correlations are expressed as the moments of a bipolar harmonic expansion of the correlated angular distribution of photofragment velocity and angular momentum (v and j) about the parent molecule transition dipole, μ. At a fixed detection geometry and on a single rotational transition, polarization control of the dissociating or probing light permits selective determination of targeted moments of the bipolar harmonic expansion. The velocity-dependent spherical tensor moments of the angular momentum distribution depend upon these bipolar moments and are given for a general experimental geometry and for general elliptical polarization of the probing light. Several practical experimental geometries are described that isolate and measure targeted bipolar moments. The bipolar moments that can be measured using elliptical probe polarizations are described. A method for analyzing the symmetries of dissociation dynamics is proposed and used to identify the moments unique to the dynamics of chiral systems.

Semiempirical calculation of two-electron integrals using bipolar expansion of the Ohno potential

Bipolar expansion of the Ohno potential as a method of calculating two-center Coulomb integrals that appear in the NDDO approximation is generalized to one-center two-electron integrals. A unified semiempirical scheme is suggested for estimating two-electron interactions in molecules. This scheme can be readily extended to arbitrary Slater basis sets (including the s,p,d-orbitals) and involves no a priori data on the valent states of atoms. In this work, the scheme is employed to extend the semiempirical PM3 method to the s,p,d-basis set. The efficiency of the method is proven by test calculations of 24 chromium compounds (π-complexes, carbonyls, isocyanides, etc.).

The resistivity and mobility functions for a model system of two equal-sized proteins in a lipid bilayer

We have calculated the mobility and resistivity functions for two equal-sized cylinders moving in a thin viscous sheet surrounded by a lower-viscosity fluid. These functions describe the hydrodynamic interactions between proteins embedded in a lipid bilayer surrounded by an aqueous solution. For a protein of radius a embedded in a biological membrane of thickness h and viscosity μ, a key parameter λ = μh/aμ′, which characterizes the viscosity ratio between the bilayer and surounding solution, is O(100). The method of solution for the hydrodynamic interactions differs depending on the separation distance, r, between the cylinders. When r = O(a), the particles are in the near-field regime, and the solution of the Stokes equations is divided between an inner and outer domain based on the asymptotically large value of λ. The inner solution neglects the flow in the lower-viscosity fluid and is solved numerically using a bipolar expansion. The outer solution is based on the fluid flows in all phases, but the cylinders are approximated by net forces. When r [Gt ] O(a), the particles are in the far-field regime, and we use the method of reflections to solve for the hydrodynamic interactions. The uniformly valid approximations, constructed from a combination of the near-field and far-field solutions, agree with the analytic solutions obtained within the lubrication and far-field regimes. Our results show that the hydrodynamic interactions between the cylinders are long range, perarting on a lengthscale of O(λ). The range of the hydrodynamic interactions is much longer than that of non-hydrodynamic interparticle forces, suggesting that hydrodynamic interactions will be significant determinants of the structure and dynamics of biological membranes.

Angular correlations between recoil velocity and angular momentum vectors in molecular photodissociation

A technique has been developed for determining the angular correlation between a photofragment’s angular momentum vector J, its recoil velocity vector v, and the transition dipole moment of the parent molecule μp . Doppler profile spectroscopy used in conjunction with laser-induced fluorescence probing by polarized light can be used to determine the correlations. The pairwise correlations between these vectors as well as their triple correlation are discussed for limiting cases using a classical approach as well as for the general case using a quantum approach based on density matrices. The current formulations differ in two ways from the recent approach of Dixon, who used a bipolar expansion of the correlated velocity and angular momentum distributions. The physical basis for the influence of the vector correlations on the Doppler profile is somewhat more transparent in the current formulations, and the direct connection between the measured correlations and the t-matrix elements occurring in the theory of Balint-Kurti and Shapiro for the photodissociation of a triatomic molecule is also demonstrated.

Second virial coefficient of gaseous HF and rotational inelastic scattering for HF-HF collisions

A recently determined intermolecular potential for HF has been used to calculate the second virial coefficient of the gas in the temperature range 275-9000 K and the rotational inelastic cross sections for HF-HF collisions at a collision energy of 300 cm-1. The virial coefficients calculated in the temperature range 292·5-329 K are in better agreement with an empirical estimate made by Redington than values calculated from other potentials. We have shown that at least seven terms in the bipolar expansion of the potential are necessary to obtain converged scattering cross sections. However, computational limitations have prevented us from concluding that even the seven term expansion has fully converged. We predict large cross sections for transitions which are dipolar coupled (for example (0, 0) →(1, 1)) and some that are not (for example (0, 1) →(1, 1)). The transition (0, 0) →(0, 2) is predicted to have a very small cross section.

The intermolecular potential of HF

The functional form of separate contributions to the intermolecular potential between small dipolar molecules has been analysed using the HF intermolecular potential as an example. The SCF energy was partitioned into first order electrostatic and exchange terms and higher order terms, called the polarization energy, which can be obtained by iterating the wavefunction to self consistency. A comparison was made for each term between a bipolar expansion about a single centre in each molecule and spherically symmetric site-site models. The overlap dependent part of the electrostatic energy could be combined with the exchange energy to give a single term which we call the repulsion energy and this was most accurately represented by a six term bipolar expansion. The polarization energy has a long range (R -6) component and an overlap dependent component and both could be separately represented by a two-term bipolar expansion. A semiempirical estimate was made of the dispersion energy and the total potential predicts a geometry of the HF dimer in good agreement with experiment. The predicted binding energy of the dimer is 17 kJ mol-1 which is slightly smaller than a previous empirical estimate (23 kJ mol-1).

Bipolar expansion of tensor fields: Recurrence relations and analytic construction of form factors

Bipolar expansions of tensor under transformations of arbitrary translation and scaling of coordinates are obtained. The judicious choice of a coupled representation renders the expression compact and transparent. The radical coefficient functions (form factors) in this expansion can be given as integral transforms. They are shown to satisfy four‐term recurrence relations, which together with an appropriate exploitation of the underlying symmetry result in a significant simplification of the original problem. A computational scheme is also worked out with which the radial form factors can be constructed analytically in terms of Gauss hypergeometric functions.

Convergence of the completely separated bipolar expansion of r

AbstractA proof of the convergence of the double infinite series form of the radial factor in the bipolar expansion of r is given. It consists of demonstrating the validity of the required term‐by‐term integrations.

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

AbstractA 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 qp = {np, lp, mp}. 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 σnlm(A, B) depend on the charge distribution (χAχB); they have the character of overlap integrals and are of the form

Bipolar Expansion for r12nYlm(θ12, φ12)

Explicit formulas for the radial functions Vl1l2l3l(n)(r1, r2, R) in the bipolar expansion for r12nYlm(θ12, φ12), r12nYlm(θ12, φ12) = Σ(2λ + 1)1/2(2l3 + 1)1/2cλ( l1m1)c3l(λ, m − m1; l2m2) × Yl1m1(θ1, φ1)Yl2m2(θ2,φ2) Yl3m−m1−m2(θR, φR)Vl1l2l3l(n)(r1, r2, R), where r12 = r1 − r2 − R, are derived with the use of the theory of generalized functions and Fourier transforms. When n ≤ − 4 and n − l is odd, there are delta-function terms. In this approach the delta-function terms and the four-region form of the expansion are obtained from a single, unified formula valid in all regions. Recurrence formulas for the Vl1l2l3l(n) are given.

Evaluation in closed form of integrals of products or spherical bessel functions related to the bipolar expansion

The bipolar expansion obtained by Ruedenberg by a Fourier transform method leads to radial functions which are integrals of a product of three spherical Bessel functions. It is shown that these integrals can be evaluated in closed form by means of the theory of generalized functions.

Convergence Properties of Certain Formulas for Multicenter Electron Repulsion Integrals Obtained from the Bipolar Expansion of r12−1

Convergence Properties of Certain Formulas for Multicenter Electron Repulsion Integrals Obtained from the Bipolar Expansion of r12−1

New Aspects of the Bipolar Expansion and Molecular Multicenter Integrals

Standard bipolar expansions contain radial functions Jl1l2 L(r1, r2, R) which depend on three variables, r1, r2, R. Using representations of Bessel functions as series of Laguerre polynomials, the J's are expanded in terms of functions of the individual variables. As a result, the inverse distance between two points is brought into the form r12−1 = ∑ nlllml ∑ n2l2m2 Fn1l1m1, n2l2m2(R, θ, Ψ) Yl1m1(θ1, φ1)fn1l1(r1)Yl2m2(θ2, θ2)fn2l2(r2). This modified bipolar expansion is used to derive expressions for multicenter electron repulsion and nuclear attraction integrals. The method is particularly suitable for Gaussian orbitals expressed with spherical harmonics and yields compact expressions directly. For Slater-type orbitals, the multicenter energy integrals appear as series involving only integrals of the two-center overlap type. The one-center and multipole limits of the bipolar expansion are examined.

Bipolar Expansion of Screened Coulomb Potentials, Helmholtz' Solid Harmonics, and their Addition Theorems

Helmholtz' solid harmonics (spherical Bessel functions × spherical surface harmonics) are generated from fundamental spherical waves through a ladder procedure using raising and lowering operators. The addition theorem for them and the bipolar expansion formula of a screened Coulomb potential are derived. A method for evaluating two-center integrals with a general potential is given in the last section. This is new and useful in practical calculations. A key function Jlnm(x)=12 ∫ −11 dμ eiμxPlm(μ)Pnm(μ),which appears in all the essential results, is studied in detail.

Bipolar Expansion of Coulombic Potentials. Addenda

Bipolar Expansion of Coulombic Potentials. Addenda