On the Evaluation of Two-Center Overlap Integrals over Integer and Noninteger n-Slater-Type Orbitals

In this study, two-center overlap integrals over Slater-type orbitals (STOs) with integer and noninteger principal quantum numbers in unaligned coordinate systems have been calculated using formulas for overlap integrals in aligned coordinate systems obtained by the author's previous work (T. Özdogan and M. Orbay, Int. J. Quant. Chem. 87 (2002) 15). The obtained results for integer case have been found to be in excellent agreement with the prior literature. On the other hand, to the best of authors knowledge, because of the scarcity of the literatures on the use of noninteger n-STOs in unaligned coordinate systems, the results for noninteger case have been tested with the limit of integer case, and good agreement has been obtained too.     

On Evaluation of Overlap Integrals with Noninteger Principal Quantum Numbers

By use of complete orthonormal sets of ψ^α exponential-type orbitals (ψ^α-ETOs,α=1,0,-1,-2,...) the series expansion formulas for the noninteger n Slater-type orbitals (NISTOs) in terms of integer n Slater-type orbitals (ISTOs) are derived. These formulas enable us to express the overlap integrals with NISTOs through the overlap integrals over ISTOs with the same and different screening constants. By calculating concrete cases the convergence of the series for arbitrary values of noninteger principal quantum numbers and screening constants of NISTOs and internuclear distances is tested. The accuracy of the results is quite high for quantum numbers, screening constants and location of STOs.     

Calculation of Arbitrary Overlap Integrals over Slater Type Orbitals Using Basic Overlap Integrals

The recurrence relations are presented for the calculation of basic overlap integrals, by making use of which other overlap integrals are calculated analytically. These recurrence relations are especially useful for the calculation of any overlap integral for large quantum numbers. For the arbitrary values of screening constants of atomic orbitals and internuclear distances an accuracy of the computer results is satisfactory for the values of principal quantum numbers of Slater functions up to 50.     

On the Calculation of Two-Center Overlap Integrals with STO's Using Zeta-Function Method and Multipole Moments

One electron two-center overlap integrals are re-derived using Barnett's -function method and Guseinov electric-multipole-moments translation technique. To evaluate these integrals a computer program has been constructed. The obtained results are in good agreement with literature.     

Integrals in the theory of four-particle atomic-molecular systems

Explicit expressions for four-particle Coulomb interaction integrals and overlap integrals for exponential basis functions dependent on five interparticle distances have been obtained. These formulas significantly simplify calculations in comparison with the general algorithm and are free from uncertainties arising in its application. The values of the integrals under consideration are calculated for a number of points forbidden in the general algorithm. The results obtained are applicable to analysis of four-particle integrals of the general form and calculation of four-particle atomic-molecular systems with partial allowance for the correlation of particle motion.     

On the Evaluation Overlap Integrals with the Same and Different Screening Parameters Over Slater Type Orbitals via the Fourier-Transform Method

In this paper, derivation of analytical expressions for overlap integrals with the same and different screening parameters of Slater type orbitals (STOs) via the Fourier-transform method is presented. Consequently, it is relatively easy to express the Fourier integral representations of the overlap integrals with same and different screening parameters mentioned as finite sums of Gegenbauer, Gaunt, binomial coefficients, and STOs.     

Analytical evaluation of cylindrical exponential integrals arising in radiative transfer

New analytical results are presented performing to cylindrical exponential integral (CEI) functions for integer and noninteger values of parameter n. These integrals are often employed of two-dimensional radiative transfer in an absorbing-emitting medium and determination of the radiative flux in cylindrical media. The simple and efficient algorithm for the calculation of these functions is developed. The series expansion relations established in this work are accurate enough in the whole range of parameters.     

Fractional Fokker-Planck equation for fractal media

We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski equation). In this paper fractional Fokker-Planck equation for fractal media is derived from the fractional Chapman-Kolmogorov equation. Using the Fourier transform, we get the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives. The Fokker-Planck equation for the fractal media is an equation with fractional derivatives in the dual space.     

Time-Dependent First Integrals,Nonlinear Dynamical Systems,and Numerical Integration

Many nonlinear dynamical systems expressed as autonomous systems of first-order ordinary differential equations admit first integrals and explicitly time-dependent first integrals. Under numerical integration these first integrals should be preserved. We discuss this case for explicitly time-dependent first integrals.     

Calculation of Electric Mult ipole Moment Integrals Using Translation Formulas for Slater-Type Orbitals

Using translation formulas for Slater-type orbitals the infinite series through the overlap integrals are derived for the electric multipole moment integrals. By the use of the derived expressions the electric multipole moment integrals, and therefore, the electric properties of molecules can be evaluated most efficiently and accurately. The convergence of the series is tested by calculating concrete cases. An accuracy of for 10^-5 the computer results is obtained for 1 ≤ v ≤ 5, and for the arbitrary values of internuclear distances and screening constants of atomic orbitals.     

Evaluation of one-centre electron interaction integrals over slater type atomic orbitals

Simplified expressions for one-centre electron interaction integrals in general and Slater-Condon parameters as well as repulsion integrals of the type (aa|aa) in particular have been obtained over Slater-type atomic orbitals. Results calculated using these expressions are in agreement with those due to other authors.     

Determination of Domain of Attraction by Short-Time Overlap Dynamics in Neural Networks

Since the overlap dynamics of spin-glass-like neural network is solely governed by such two parameters, the symmetry of the net's connections and the stabilities of the embedded patterns, we propose that the domain of attraction of a network could be determined by a short-time (at least the first two-step) dynamics, in which both parameters begin to be visible.The first step overlap dynamics Q calculated by probability theory and the second step overlap dynamics for pseudo-inverse-model is obtained by extended Krauth's theory. The calculated domain of attraction for this model is in good agreement with the results obtained by computer simulation.     

Evaluation of Multicenter Nuclear Attraction Integrals by the Use of Translation Formula for Slater Type Orbitals

By the use of translation formula for Slater type orbitals (STOs), three-center nuclear attraction integralsare represented in terms of two-center overlap and nuclear attraction integrals. The computing results for the formulapresented here has been tested under wide changes in molecular parameters and good convergence has been obtainedwith the prior literature.     

Integrals of the motion,green functions,and coherent states of dynamical systems

The connection between the integrals of the motion of a quantum system and its Green function is established. The Green function is shown to be the eigenfunction of the integrals of the motion which describe initial points of the system trajectory in the phase space of average coordinates and moments. The explicit expressions for the Green functions of theN-dimensional system with the Hamiltonian which is the most general quadratic form of coordinates and momenta with time-dependent coefficients is obtained in coordinate, momentum, and coherent states representations. The Green functions of the nonstationary singular oscillator and of the stationary Schrödinger equation are also obtained.     

Unified treatment of one-range addition theorems for integer and non-integer n-STO, -GTO and -generalized exponential type orbitals with hyperbolic cosine in position, momentum and four-dimensional spaces

Simpler formulas are derived for one-range addition theorems for the integer and noninteger n generalized exponential type orbitals, momentum space orbitals, and hyperspherical harmonics with hyperbolic cosine (GETO HC, GMSO HC, and GHSH HC) in position, momentum and four-dimensional spaces, respectively. The final results are expressed in terms of one-range addition theorems of complete orthonormal sets of ψα -exponential type orbitals, α - momentum space orbitals and z α -hyperspherical harmonics. We notice that the one-range addition theorems for integer and noninteger n-Slater type orbitals and Gaussian type orbitals in position, momentum and four dimensional spaces are special cases of GETO HC, GMSO HC, and GHSH HC. The theorems presented can be useful in the accurate study of the electronic structure of atomic and molecular systems.     

A New Expression for Multidimensional Franck-Condon Integrals

A new closed form expression for the Franck-Condon integrals for overlap between arbitrary multidimensional harmonic oscillators has been exactly derived by employing the generating functions method. The calculations lead to the deduction of some general rules whereby an arbitrary multidimensional Franck-Condon integral can be expressed as sums of products of the Hermite polynomials. Copyright 1999 Academic Press.     

Monte Carlo study of dipolar and quadrupolar hard prolate spherocylinders

Monte Carlo simulations of hard prolate spherocylinders (HPSs) with embedded dipole or quadrupole moment are reported for two elongations (L?=?0.5 and 1) and several values of the packing fraction. The MC values of the residual internal and Helmholtz energy and compressibility factor were determined. Our work represents the first simulation study of dipolar HPSs focused on the determination of the thermodynamic properties. In the case of quadrupolar HPSs, our results enlarge the range of state conditions for which the simulation data are available. The obtained MC data were used for a test of the perturbation theory of polar non-spherical molecule fluids. In order to evaluate the perturbation contributions containing the two-particle integrals, the values of the shape integrals (evaluated recently for dipolar and quadrupolar hard prolate spherocylinders) were employed and we were allowed to avoid the use of the similarity between Kihara and Gaussian overlap models. Fair agreement between the simulation data and the theoretical predictions was reached.     

Overlap distribution in random and designed heteropolymers

We study the overlap between low-energy states in lattice models of heteropolymers with contact interactions. The overlap distribution gives information on the degree of correlation in the energy landscape. Designed sequences have rather correlated energy landscapes, which favor fast folding kinetics. Chains with random interactions have much less correlated energy landscapes. It is indeed believed that the mean-field theory for this model coincides with the Random Energy Model, whose different low-energy states are completely unrelated. This picture has been supported by numerical studies of maximally compact configurations. Without applying this constraint, we find that the overlap distribution is indeed bimodal as expected, but it has a broad peak at large overlap, indicating a non-vanishing width for the valleys of low-energy states. This feature probably plays an important role in the kinetics of the model. It is not evident that the range of such correlations shrinks to zero for large systems. The range of the correlations seems to be influenced by the number of contacts per residue in the ground state: the smaller this quantity, the larger the correlations. Received 16 August 2000     

Fractional dynamics of systems with long-range space interaction and temporal memory

Field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a fractional generalization of the Ginzburg-Landau and nonlinear Schrödinger equations. As another example, dynamical equations for particle chains with power-law interaction and memory are considered in the continuous limit. The obtained fractional equations can be applied to complex media with/without random parameters or processes.     

Static and thermophysical properties of chalcogenide crystals with NaCl structure

In the present communication an analysis of interionic potentials in fourteen chalcogenide crystals has been performed. This interionic potential has been used to predict the values of cohesive energy, isothermal bulk modulus and the pressure derivatives of bulk modulus in the solids under study. The many body interaction (MBI) effects have been taken into account within the framework of Hafemeister Zarht potential. Instead of using BM potential the Hafemeister-Zarht (HZ) type short range overlap potential has been considered between nearest as well as between next nearest neighbour ions. The short-range interactions, effective up to second neighbours are treated by considering the hardness parameter as an ionic property. The hardness parameter ρ_ij is evaluated using the data on overlap integrals. The results achieved in the present study are generally in good agreement with available experimental data. Values of cohesive energy, bulk modulus and its pressure derivatives calculated by previous investigators have also been shown for the sake of comparison.