Addition theorems for Slater-type orbitals in momentum space and their application to three-center overlap integrals

Using addition theorems for complete orthonormal sets of exponential type orbitals in the momentum representation introduced by the author, the addition theorems are established for Slater type orbitals in momentum space. With the help of these addition theorems, the general series expansion formulae in terms of the product of two-center overlap integrals are established for the three-center overlap integrals that arise in the solution of atomic and molecular problems occurring when explicitly correlated methods are employed. The formulae obtained for addition theorems and three-center overlap integrals are valid for arbitrary location and parameters of orbitals.     

Calculation of molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals and basic one-center Coulomb integrals

The recurrence relations are established for the basic one-center Coulomb integrals over Slater-type orbitals (STOs). These formulae and the recurrence relations for basic overlap integrals are utilized for the calculation of multicenter electron-repulsion integrals. The calculations of multicenter electron-repulsion integrals are performed by the use of translation formulae for STOs obtained from the Lambda and Coulomb Sturmian exponential-type functions (ETFs). It is shown that these integrals show a faster convergence rate in the case of Coulomb Sturmian ETFs. The accuracy of the results is quite high for the quantum numbers of STOs and for the arbitrary values of internuclear distances and screening constants of atomic orbitals.     

Addition and expansion theorems for complete orthonormal sets of exponential-type orbitals in coordinate and momentum representations

Analytical properties of new complete orthonormal sets of Psi(alpha) exponential-type orbitals (Psi(alpha)-ETOs where alpha=1, 0, -1, -2, em leader ), introduced by the author as finite linear combinations of Slater-type orbitals (STOs), are studied. Addition and expansion theorems for Psi(alpha)-ETOs are obtained in both coordinate and momentum representations. Using expressions of Psi(alpha)-ETOs in terms of STOs, the new methods are suggested to calculate multicenter multielectron integrals over STOs.     

Use of Ψ<Superscript>α</Superscript>-ETOs in the unified treatment of electronic attraction,electric field and electric field gradient multicenter integrals of screened Coulomb potentials over Slater orbitals

In this study, using complete orthonormal sets of -ETOs (where =1, 0, –1, –2, ...)introduced by the author, a large number of series expansion formulae for the multicenter electronic attraction (EA), electric field (EF) and electric field gradient (EFG) integrals of the Yukawa-like screened Coulomb potentials (SCPs) is presented through the new central and noncentral potentials and the overlap integrals with the same screening constants. The final results obtained are valid for arbitrary locations of STOs and their parameters.An erratum to this article can be found at     

Computation of multicenter overlap integrals with Slater-type orbitals using Ψ<Superscript>α</Superscript>-ETOs

Multicenter overlap integrals appearing in the evaluation of multicenter–multielectron integrals of central and noncentral interaction potentials are calculated using complete orthonormal sets of -ETOs (=1, 0, –1, –2, ...). The final results are expressed in terms of two-center overlap integrals between STOs. The convergence of the series is tested by calculating concrete cases for arbitrary quantum numbers, screening constants and location of STOs.     

One-range addition theorems for derivatives of Slater-type orbitals

Using addition theorems for STOs introduced by the author with the help of complete orthonormal sets of -ETOs (Guseinov II (2003) J Mol Model 9:190–194), where =1, 0, –1, –2, ..., a large number of one-range addition theorems for first and second derivatives of STOs are established. These addition theorems are especially useful for computation of multicenter-multielectron integrals over STOs that arise in the Hartree–Fock–Roothaan approximation and also in the Hylleraas function method, which play a significant role for the study of electronic structure and electron–nuclei interaction properties of atoms, molecules, and solids. The relationships obtained are valid for arbitrary quantum numbers, screening constants and location of STOs.     

Evaluation of overlap integrals with integer and noninteger n Slater-type orbitals using auxiliary functions

The series expansion formulae are derived for the overlap integrals with arbitrary integer n and noninteger n* Slater-type orbitals (ISTOs and NISTOs) in terms of a product of well-known auxiliary functions A(sigma) and B (k). The series becomes an ordinary closed expression when both principal quantum numbers n* and n'* of orbitals are integer n*= n and n'*= n'. These formulae are especially useful for the calculation of overlap integrals for large quantum numbers. Accuracy of the results is satisfactory for values of integer and noninteger quantum numbers up to n= n'=60, n*= n'*<33 and for arbitrary values of screening constants of orbitals and internuclear distances.     

Calculation of multicenter electric field gradient integrals over Slater-type orbitals using unsymmetrical one-range addition theorems

The electric field induced within a molecule by its electrons determines a whole series of important physical properties of the molecule. In particular, the values of the gradient of this field at the nuclei determine the interaction of their quadrupole moments with the electrons. Using unsymmetrical one-range addition theorems introduced by one of the authors, the sets of series expansion relations for multicenter electric field gradient integrals over Slater-type orbitals in terms of multicenter charge density expansion coefficients and two-center basic integrals are presented. The convergence of the series is tested by calculating concrete cases for different values of quantum numbers, parameters and locations of orbitals.     

New complete orthonormal sets of hyperspherical harmonics and their one-range addition and expansion theorems

In this study, the complete orthonormal sets of - momentum space orbitals (where α=1,0,−1, −2,...) obtained from the -ETO in coordinate representation (I.I. Guseinov, J. Mol. Model., 9 (2003) 135) are reduced to the complete orthonormal sets of hyperspherical harmonics (HSH) by means of a Fock transformation of the radial momentum to an angular variable. It is shown that the group of transformations is the four-dimensional rotation group O(4) and, therefore, the HSH presented in this work are the complete orthonormal sets of functions. For these functions, the one-range addition and expansion theorems are obtained. The formulae for HSH and their addition and expansion theorems derived in this work can be used to evaluate the multicenter integrals that arise when exponential-type basis functions are used in atomic and molecular calculations.     

Summation theorems for flux and concentration control coefficients of dynamic systems

Metabolic control analysis (MCA) was developed to quantify how system variables are affected by parameter variations in a system. In addition, MCA can express the global properties of a system in terms of the individual catalytic steps, using connectivity and summation theorems to link the control coefficients to the elasticity coefficients. MCA was originally developed for steady-state analysis and not all summation theorems have been derived for dynamic systems. A method to determine time-dependent flux and concentration control coefficients for dynamic systems by expressing the time domain as a function of percentage progression through any arbitrary fixed interval of time is reported. Time-dependent flux and concentration control coefficients of dynamic systems, provided that they are evaluated in this novel way, obey the same summation theorems as steady-state flux and concentration control coefficients, respectively.     

Analytical evaluation of Coulomb potential generated by multielectron molecule at arbitrary positions in space using one-range addition theorems of Slater type orbitals

By the use of unsymmetrical one-range addition theorems for Slater type orbitals (STO) and Coulomb potential introduced by the author, the analytical formulae in terms of two- and three-center nuclear attraction integrals, and linear combination coefficients of molecular orbitals are derived for the potential produced by the charges of molecule. These formulae can be useful for the study of interaction between atomic-molecular systems containing any number of closed and open shells when the STO are used in the combined Hartree-Fock-Roothaan (HFR) theory suggested by the author. It should be noted that the symmetry of the potential obtained is the same as the symmetry of the molecule. As an example of application, the calculations have been performed for the potential produced by the ground state of BH ₃ molecule( ( 1a₁ )²( 2a₁ )²( 1e_x )²( 1e_y )²,¹A₁ ) \left( {{{\left( {1{a_1}} \right)}^2}{{\left( {2{a_1}} \right)}^2}{{\left( {1{e_x}} \right)}^2}{{\left( {1{e_y}} \right)}^2},{}^1{A_1}} \right) .     

Modified Hodgkin-Huxley gating kinetics of sodium activation in giant axons of squid (Doryteuthis bleekeri)

The probabilities m of the sodium activation gate being open are shown to fit experimentally-determined running integrals Qg of recordings of the colchicine-sensitive fraction of the asymmetry current, within the Hodgkin-Huxley framework that the gate could have only two conformations, open and closed. Using the Hodgkin-Huxley framework, we are obliged to assume that the transition velocities, alpha m and beta m, between the open and closed gates depend not only on membrane potentials V but also on the time after a potential step was externally applied. We introduce the following functions of alpha m and beta m. (sequence in text) where VH, td and tau p stand for holding potential, constant delay time of 10 microseconds, and transit time of the transition velocity of alpha m (or beta m) from its initial value alpha om (or beta om) to its final steady value alpha infinity m (or beta infinity m), respectively. The transit time tau p was found to be potential-dependent; typically it was 30 microseconds at -20 mV, and 100 microseconds at 20-40 mV. The values of alpha infinity m, alpha om, beta infinity m and beta om were found to be in reasonable agreement with those obtained by others, under the Hodgkin-Huxley assumption that the gate followed first-order kinetics. The requirement of new parameters, tau p and td, in the transition velocities was discussed in a relation to a membrane model where a voltage-receptor and a sodium channel macromolecule are spatially separated but functionally connected through underlying cytoskeletons (Matsumoto, 1984).     

Techniques for obtaining analytical solutions to the multicylinder somatic shunt cable model for passive neurones.

The somatic shunt cable model for neurones is extended to the case in which several equivalent cylinders, not necessarily of the same electrotonic length, emanate from the cell soma. The cable equation is assumed to hold in each cylinder and is solved with sealed end conditions and a lumped soma boundary condition at a common origin. A Green's function (G) is defined, corresponding to the voltage response to an instantaneous current pulse at an arbitrary point along one of the cylinders. An eigenfunction expansion for G is obtained where the coefficients are determined using the calculus of residues and compared with an alternative method of derivation using a modified orthogonality condition. This expansion converges quickly for large time, but, for small time, a more convenient alternative expansion is obtained by Laplace transforms. The voltage response to arbitrary currents injected at arbitrary sites in the dendritic tree (including the soma) may then be expressed as a convolution integral involving G. Illustrative examples are presented for a point charge input.     

Numerical treatment of two-center overlap integrals

Among the two-center integrals occurring in the molecular context, the two-center overlap integrals are numerous and difficult to evaluate to a level of high accuracy. The analytical and numerical difficulties arise mainly from the presence of the spherical Bessel integrals in the analytic expressions of these molecular integrals. Different approaches have been used to develop efficient algorithms for the numerical evaluation of the molecular integrals under consideration. These approaches are based on quadrature rules, Levin’s u transform, or the epsilon-algorithm of Wynn. In the present work, we use the nonlinear transformation of Sidi. This transformation is shown to be highly efficient in improving the convergence of highly oscillatory integrals, and it has been applied to molecular multicenter integrals, namely three-center attraction, hybrid, two-, three-, and four-center two-electron Coulomb and exchange integrals over B functions and over Slater-type functions. It is also been shown that when evaluating these molecular multicenter integrals the transformation is more efficient compared with the methods cited above. It is now proven that the integrand occurring in the analytic expression of the two-center overlap integrals satisfies all the conditions required to apply the transformation. A highly accurate algorithm based on this transformation is now developed. Special cases are presented and discussed for a better optimization of the algorithm. The numerical results section illustrates clearly the high efficiency of our algorithm.     

Concentration distribution and spatial moments of moving macromolecules undergoing isomerization

Expressions for zeroth, first, and second spatial moments are obtained for diffusing macromolecules A and B that move due to an external field and undergo reversible isomerization, switching back and forth according to first-order kinetics. In addition, expressions for third and fourth moments are derived for the special case of equal diffusion coefficients, equal rate constants, and equal but opposite velocities. The initial conditions are arbitrary amounts of A or B concentrated in an infinitesimally narrow region. The moments are computed from derivatives of the Fourier-transformed concentration profiles of A and B. The moments are used in an expansion in term of Hermite polynomials, the Gram-Charlier expansion, to construct the concentration profiles of A or B or A and B together. The examples presented show that a few terms of the expansion, for which explicit expressions are given, can give tolerable accuracy if the velocity is not too large and the rate constants and diffusion coefficient are not too small. The expansion can be used to determine when the profiles are unimodal.     

Orientational potentials extracted from protein structures improve native fold recognition

We develop coarse-grained, distance- and orientation-dependent statistical potentials from the growing protein structural databases. For protein structural classes (alpha, beta, and alpha/beta), a substantial number of backbone-backbone and backbone-side-chain contacts stabilize the native folds. By taking into account the importance of backbone interactions with a virtual backbone interaction center as the 21st anisotropic site, we construct a 21 x 21 interaction scheme. The new potentials are studied using spherical harmonics analysis (SHA) and a smooth, continuous version is constructed using spherical harmonic synthesis (SHS). Our approach has the following advantages: (1) The smooth, continuous form of the resulting potentials is more realistic and presents significant advantages for computational simulations, and (2) with SHS, the potential values can be computed efficiently for arbitrary coordinates, requiring only the knowledge of a few spherical harmonic coefficients. The performance of the new orientation-dependent potentials was tested using a standard database of decoy structures. The results show that the ability of the new orientation-dependent potentials to recognize native protein folds from a set of decoy structures is strongly enhanced by the inclusion of anisotropic backbone interaction centers. The anisotropic potentials can be used to develop realistic coarse-grained simulations of proteins, with direct applications to protein design, folding, and aggregation.     

Donnan membrane equilibrium,sedimentation equilibrium,and coil expansion of DNA in salt solutions

This is a review of applications of the McMillan-Mayer-Hill virial theory and the ionic double-layer theory to dilute colloidal solutions, in particular, solutions of DNA. Interactions of highly charged colloidal rods are developed in terms of the second virial coefficients between two rods, and between one rod and one small co-ion. The relevant cluster integrals are evaluated with interaction potentials based on the Poisson-Boltzmann equation. The treatment is extended to the intrachain repulsion responsible for the statistical swelling of coiled DNA (excluded volume effect). The theory is compared with three sets of experimental data: The salt distribution in Donnan membrane equilibria of DNA-salt solutions, sedimentation equilibria of short DNA fragments at different ionic strengths, and the intrinsic viscosity of T7 DNA in NaCl solutions. In all cases the theory agrees well with the experiments. The agreement is not convincing for the sedimentation equilibrium at low ionic strength, because here the experimental DNA concentration is too high for the truncated dilute solution expansion of the DNA-salt repulsion.     

Design of an optimal Chebyshev-expanded discrimination function for globular proteins

We describe the construction of a scoring function designed to model the free energy of protein folding. An optimization technique is used to determine the best functional forms of the hydrophobic, residue-residue and hydrogen-bonding components of the potential. The scoring function is expanded by use of Chebyshev polynomials, the coefficients of which are determined by minimizing the score, in units of standard deviation, of native structures in the ensembles of alternate decoy conformations. The derived effective potential is then tested on decoy sets used conventionally in such studies. Using our scoring function, we achieve a high level of discrimination between correct and incorrect folds. In addition, our method is able to represent functions of arbitrary shape with fewer parameters than the usual histogram potentials of similar resolution. Finally, our representation can be combined easily with many optimization methods, because the total energy is a linear function of the parameters. Our results show that the techniques of Z-score optimization and Chebyshev expansion work well.     

An activity coefficient model for proteins

Modeling of the properties of biochemical components is gaining increasing interest due to its potential for further application within the area of biochemical process development. Generally protein solution properties such as protein solubility are expressed through component activity coefficients which are studied here. The original UNIQUAC model is chosen for the representation of protein activity coefficients and, to the best of our knowledge, this is the first time it has been directly applied to protein solutions. Ten different protein-salt-water systems with four different proteins, serum albumin, alphacymotrypsin, beta-lactoglobulin and ovalbumin, are investigated. A root-mean-squared deviation of 0.54% is obtained for the model by comparing calculated protein activity coefficients and protein activity coefficients deduced from osmotic measurements through virial expansion. Model predictions are used to analyze the effect of salt concentrations, pH, salt types, and temperature on protein activity coefficients and also on protein solubility and demonstrate consistency with results from other references. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 65-71, 1997.