Matrix elements for Ĵ2 and Ĵz operators over explicitly correlated Cartesian Gaussian functions

General formalism for evaluation of multiparticle integrals involving J?² and J?_z operators over explicitly correlated Cartesian Gaussian functions is presented. The integrals are expressed in terms of the general overlap integrals. An explicitly correlated Cartesian Gaussian function is a product of spherical orbital Gaussian functions, powers of the Cartesian coordinates of the particle, and exponential Gaussian factors, which depend on interparticular distances. This development is relevant to both adiabatic and nonadiabatic calculations of energy and properties of multiparticle systems. © 1995 John Wiley & Sons, Inc.     

Gaussian matrix elements of the operator X: Reduction to confluent hypergeometric functions

A new method has been proposed for calculating the Gaussian matrix elements of the interaction operator represented by an arbitrary degree of interelectron distance X. The method is based on the expansion of two-electron integrals as the sum of one-electron integrals which in turn admit compact operator representation in terms of confluent hypergeometric functions. The generating differential operator has been shown to be related to the modified Hermitian polynomials. The standard structure of the special functions encountered in this approach is useful in studying the analytical behavior of the integrals and makes it possible to obtain for these integrals recurrence relations, direct algebraic expressions in the forms of finite sum of confluent hypergeometric functions, integral representations, and asymptotic properties. Unlike the usual methods based on integral transformation of the interaction operator, the proposed approach has a wider field of application, and in addition, leads to compact and convenient analytical expressions. The idea of using differential properties of integrals to simplify the integrand structure gives the proposed approach a certain resemblance to that suggested by Boys but not developed in detail in his pioneer work.     

General quantum mechanical operators. An open-ended approach for one-electron integrals with Gaussian bases

A universal computational approach for evaluating integrals over gaussian basis functions for general operators of the form is presented. The implementation is open-ended with respect to the types of basis functions (s, p, d, f, g, h…) and with respect to the integers that specify the operator. These one-electron integrals comprise operators associated with electrical and magnetic properties of molecules and include those needed to find multipole polarizabilities, multipole susceptibilities, chemical shifts, and so on. The scheme also generates the usual kinetic, nuclear attraction, and overlap operators.     

The Hylleraas-CI method in molecular calculations: Two-electron integrals

In the Hylleraas-CI method, first proposed by Sims and Hagstrom, correlation factors of the type r are included into the configurations of a CI expansion. The computation of the matrix elements requires the evaluation of different two-, three-, and four-electron integrals. In this article we present formulas for the two-electron integrals over Cartesian Gaussian functions, the most used basis functions in molecular calculations. Most of the integrals have been calculated analytically in closed form (some of them in terms of the incomplete Gamma function), but in one case a numerical integration is required, although the interval for the integration is finite and the integrand well-behaved. We have also reported on partial and preliminary computations for the H₂ molecule using our four-center general formulas; a basis set of s- and p-type functions yielded at R = 1.4001 Å an energy of - 1.174380 a.u. to be compared with Kolos and Wolniewicz value of - 1.174475.     

Modified Gaussian functions and their use in quantum chemistry. I. Integrals

A modified Gaussian function g(u, v, w, a, R ) = const s(a, R ) is considered where l = u + v + w, s (a, R ) is a 1s-type Gaussian function centered at R , a is the coefficient in the exponent of the 1 s Gaussian function and X, Y, Z are components of R . General formulae are derived for overlap integrals, kinetic energy integrals, nuclear attraction integrals, and electron repulsion integrals, valid for any l. The formulae are much simpler than those derived by Huzinaga for Cartesian Gaussian functions.     

The Diversity of Difluoroacetylene Coordination Modes Obtained by Coupling Fluorocarbyne Ligands on Binuclear Manganese Carbonyl Sites

One Mn or two? The fluorocarbyne manganese carbonyl complexes [Mn(CF)(CO)_n] (n=3, 4) and [Mn₂(CF)₂(CO)_n] (n=4–7; see picture) have been investigated by density functional theory. In mononuclear complexes the CF ligand behaves very much like the NO ligand in terms of π‐acceptor strength. In binuclear complexes the two CF ligands couple in many of the low‐energy structures to form a bridging C₂F₂ ligand derived from difluoroacetylene.

     

Ladder operators for the Kratzer oscillator and the Morse potential

Taking a close look at the Infeld–Hull ladder operators for the Kratzer oscillator system, V(x) = [x² + β(β ? 1)x^?2]/2, we deduce and explicitly construct energy‐raising and ‐lowering operators for the generalized Morse potential system V(z) = (Ae^?4αz ? Be^?2αz)/2, through a canonical transformation that exists between the two systems. For the Morse potential system, we obtain a system of raising and lowering operators P_±(n) (n = 0, 1, 2, 3, … , n_max) with the specific property that P_±(n)Φ_n = c_±(n)Φ_n±1, where Φ_n denotes the nth energy eigenfunction. While P_?(0) annihilates the ground‐state Φ₀, the operator P₊(n_max), instead of annihilating the highest bound‐state Φ, actually knocks it out of the L₂ space spanned by the discrete bound states and becomes inadmissible. Yet, raising and lowering operators _± with proper end‐of‐spectrum behavior (i.e., _?|0〉 = 0 and ₊|n_max〉 = 0) can be constructed in a straightforward way in the energy representation. We show that the operators ₊, _?, and ₀ (where ₀ ≡ (1/2)[ ₊, _?]) form a su(2) algebra only if we restrict them to the (N ? 1)‐dimensional subspace spanned by the lowest (N ? 1) basis vectors, but not in the full (N + 1)‐dimensional space spanned by the discrete bound states [N ≡ n_max ≡ integral part of (1/2)(B/(2α ) ? 1)]. Realization of this su(2) algebra in the position representation (when restricted to the (N ? 1)‐dimensional subspace) is also given. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Symmetry-adapted integrals over many-electron basis functions and operators

We consider integrals over symmetry-adapted basis functions that involve the coordinates of more than one electron. We focus on basis functions that can be written as products of one-electron functions and (say) a two-electron function. We show first that the two-electron parts of the basis functions can be absorbed into the operator, resulting in an integral over only one-electron basis functions, but a more complicated many-electron operator. We then prove a general formula for expressing such integrals in terms of symmetry-distinct integrals only. Received: 16 June 2000 / Accepted: 10 July 2000 / Published online: 19 January 2001     

Molecular correlation integrals. Two center one electron integrals containing a function of interparticle distance in one center expansion approximation

Two-center one-electron integrals needed in certain molecular correlated wave function calculations, using one-center expansion approximation, have been studied. The form of the basic correlated function used in this study is The parent integral is expressed in terms of an angular integral, and an auxiliary radial integral depending upon the variables r₁, r₂, and r₁₂. Several analytical formulas, and a recursive formula are derived for the auxiliary integral, and other related integrals. All these formulas are given in computationally useful forms. Logical flow charts and FORTRAN programs were constructed for computing the basic integrals discussed in the paper. Numerical values of some integrals, thus obtained, are tabulated for comparisons.     

Fourier transform of spherical Laguerre Gaussian functions and its application in molecular integrals

The Fourier transform of the spherical Laguerre Gaussian‐type function (LGTF), L ${}_{\text{n}}^{\text{l}\text{+½}}$ (αr²)r^lY_lm( r̂ )e, was derived. Applying the Fourier transform convolution theorem, the basic two‐center integrals over the general two‐electron irregular solid harmonic operator, Y_LM( r̂ ₁₂)/r (which becomes Coulomb repulsion, spin–other‐orbit interaction or spin–spin interaction when L=0, 1, or 2, respectively) as well as the overlap were evaluated analytically. These basic integral results generate the two‐electron integrals of the Coulomb type, hybrid type, and exchange type as well as that of three‐ and four‐center. The formulas obtained, which are general for electronic wave functions of unrestricted quantum numbers n, l, and m, are expressed explicitly in terms of nuclear spherical LGTFs of internuclear geometrical variables. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 265–273, 1999     

Recurrence relations for the evaluation of electron repulsion integrals over spherical Gaussian functions

Recurrence relations are derived for the evaluation of two-electron repulsion integrals (ERIs) over Hermite and spherical Gaussian functions. Through such relations, a generic ERI or ERI derivative may be reduced to “basic” integrals, i.e., true and auxiliary integrals involving only zero angular momentum functions. Extensive use is made of differential operators, in particular, of the spherical tensor gradient ??(?). Spherical Gaussians, being nonseparable in the x, y, and z coordinates, were not included in previous formulations. The advantages of using spherical Gaussians instead of Cartesian or Hermite Gaussians are briefly discussed. © 1993 John Wiley & Sons, Inc.     

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      

Computation of electron repulsion integrals using the rys quadrature method

Following an earlier proposal to evaluate electron repulsion integrals over Gaussian basis functions by a numerical quadrature based on a set of orthogonal polynomials (Rys polynomials), a computational procedure is outlined for efficient evaluation of the two-dimensional integrals I_x, I_y, and I_z. Compact recurrence formulas for the integrals make the method particularly fitted to handle high-angular-momentum basis functions. The technique has been implemented in the HONDO molecular orbital program.     

Multicenter integrals over polarization potential operators

Analytic expressions for multicenter integrals over the general one‐particle operator x^n′y^l′z^m′| r |^−k(1−exp(−αr²))ⁿ(n′, m′, l′, n≥0, k>2, α>0), employing Cartesian Gaussians, are presented. While until now only P. Schwerdtfeger and H. Silberbach (Phys Rev A 1988, 37, 2834) have succeeded in finding such expressions, using a Laplace transform, we shall show that one can also get them according to the method of L. E. McMurchie and E. R. Davidson (J Comp Phys 1978, 26, 218; J Comp Phys 1981, 44, 289). ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 403–416, 1999     

Products of class operators of the symmetric group

A combinatorial derivation of the product of the class of three cycles, [(1)^N?3(3)]_N with an arbitrary class operator of the symmetric group S_N is presented. The form of this result suggests a conjecture concerning the expression of the general class operator product in terms of a relatively small number of reduced class coefficients. The conjecture is applied to the determination of the products of [(1)^N?4(4)]_N, [(1)^N?4(2)²]_N, and [(1)^N?5(5)]_N with arbitrary class operators. General expressions for the reduced class coefficients of the simplest type are obtained.     

Oscillator and hydrogenic matrix elements by operator algebra

Occupation number representation of the two-dimensional harmonic oscillator and some operator formulae are used in a simple algebraic derivation of complicated integrals. The calculation of full oscillator- and radial integrals of r?^w and exp (\documentclass{article}\pagestyle{empty}\begin{document}$ (iw\hat{\varphi})$\end{document}), where w is an arbitrary positive or negative integer, are performed by an integral transform, leading to a generalized Gauss matrix element. Thus it is possible, because of the back transformation, to derive from one generalized Gauss matrix element all matrix elements which are permitted by the selection rules. Some integrals of r?^w and exp (\documentclass{article}\pagestyle{empty}\begin{document}$ (iw\hat{\varphi})$\end{document}), Laguerre polynomials, and Bessel functions are completely new. For the already known integrals, the mathematical labour is considerably reduced. The relation between the two-dimensional oscillator and the hydrogen atom and their angular momentum properties are discussed. A survey on the various methods applied to the oscillator problem, from complex integration to noncompact Lie groups, and a comprehensive bibliography on this important spectroscopic field are given.     

Analytic evaluation of multicenter integrals for gaussian-type orbitals

This work contains the evaluation of multicenter integrals with Cartesian Gaussian functions occurring in ||||² These integrals have to be used if it is necessary to calculate the lower bounds for eigenvalues with the method of the minimization of the variance [1], Considering the varianceF() = ||H||² - (H!| )², the integrals from (HY, Y) are well known in contrast to those for ||H ||².     

Comparative study of unconventional 1s basis functions for the 1Σ ground state of H2 and He

We investigated various nonstandard 1s basis functions (generalized Slater-Gaussian, ellipsoidal Gaussian, floating spherical and ellipsoidal Gaussian, rational function, Hulthén approximation, two-Slater-type orbital, generalized Guillemin–Zener function, and various noninteger-n elliptical orbitals) for approximating the ¹Σ ground state of H₂ and He₂⁺⁺. A CI trial wave-function including Σ_g-type MO's is adopted and molecular integrals are evaluated numerically. The energy improvement on the 1s STO is small except for noninteger-n orbitals which closely approach the “SCF limit”.     

Glass Transition of Low‐Dimensional Polystyrene

Summary: A unified model is developed for the finite size‐effect on the glass‐transition temperature of polymers, T_g(D), where D denotes the diameter of particles or thickness of films. In terms of this model, T_g depends on both the size and interface conditions. The predicted results are consistent with the experimental evidence for polystyrene (PS) particles and films with different interface situations.

T_g(D) function of free‐standing PS films.