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.     

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.     

Comment on “numerical treatment of two-center overlap integrals”

The subject paper by H. Safouhi (J Mol Model 12:213–220, 2006) presents a scheme based on a nonlinear convergence acceleration transformation for the numerical evaluation of two-center overlap integrals of Slater-type orbitals. In this comment we argue that there is no reason to adopt such an approach, as well-known methods for these integrals are substantially superior both in speed and in accuracy.     

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.     

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.     

Addition theorems for Slater-type orbitals and their application to multicenter multielectron integrals of central and noncentral interaction potentials

By the use of complete orthonormal sets of psi(alpha)-ETOs (alpha=1, 0, m1, m2,...) introduced by the author, new addition theorems are derived for STOs and arbitrary central and noncentral interaction potentials (CIPs and NCIPs). The expansion coefficients in these addition theorems are expressed through the Gaunt and Gegenbauer coefficients. Using the addition theorems obtained for STOs and potentials, general formulae in terms of three-center overlap integrals are established for the multicenter t-electron integrals of CIPs and NCIPs that arise in the solution of the N-electron atomic and molecular problem (2hthN) when a Hylleraas approximation in Hartree-Fock-Roothaan theory is employed. With the help of expansion formulae for translation of STOs, the three-center overlap integrals are expressed through the two-center overlap integrals. The formulae obtained are valid for arbitrary quantum numbers, screening constants and location of orbitals.     

Synthesis of new 2-phosphono-alpha-D-glycoside derivatives by stereoselective oxa-Michael addition to a D-galacto derived enone

The synthesis of new 2-phosphono-alpha-D-glycoside derivatives by stereoselective oxa-Michael addition to an enone derived from D-galactal and containing a phosphonate group is described. Retro-Michael reactions were prevented by tandem acetylation to trap the unstable enolic intermediates. The stereochemistry of the addition products was established by NOESY experiments and explained with molecular mechanics (MM) and density functional theory (DFT) calculations.     

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.     

Applications of natural orbitals for chemical valence in a description of bonding in conjugated molecules

Natural orbitals for chemical valence (NOCV) were used to describe bonding in conjugated pi-electron molecules. The 'single' C-C bond in trans-1,3-butadiene, 1,3-butadiene-1,1,4,4-tetra-carboxilic acid, 1,3,5,7-octatetraene, and 11-cis-retinal was characterized. In the NOCV framework, the formation of the sigma-bond appears as the sum of two complementary charge transfer processes from each vinyl fragment to the bond region, and partially to the other fragment. The formation of the pi-component of the bond is described by two pairs of NOCV representing the transfer of charge density from the neighboring 'double' C-C bonds. The NOCV eigenvalues and the related fragment-fragment bond multiplicities were used as quantitative measures of the sigma- and pi- contributions. The sigma-component of the 'single' C-C bonds appears to be practically constant in the systems analyzed, whereas the pi-contributions increase from butadiene (ca. 7.5%) to retinal (ca. 14%).     

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.     

Limit theorems for the population size of a birth and death process allowing catastrophes

The linear birth and death process with catastrophes is formulated as a right continuous random walk on the non-negative integers which evolves in continuous time with an instantaneous jump rate proportional to the current value of the process. It is shown that distributions of the population size can be represented in terms of those of a certain Markov branching process. The ergodic theory of Markov branching process transition probabilities is then used to develop a fairly complete understanding of the behaviour of the population size of the birth-death-catastrophe process.Research done while on leave at Colorado State University from the University of Western Australia and partially supported by N.S.F. grant DMS-8501763     

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) .     

QSAR of the testosterone binding globulin affinity by means of correlation weighting of local invariants of the graph of atomic orbitals

Numerical values of the testosterone binding globulin affinity have been modeled as a mathematical function of molecular structure in two versions of molecular structure elucidation: first, by hydrogen-filled molecular graphs (HFG); second, by the so-called graphs of atomic orbitals (GAO). Increased orders of Morgan extended connectivity in the HFG and GAO have been examined as local invariants. Using optimisation of the correlation weights of the above-mentioned invariants, quantitative structure–activity relationships (QSAR) have been obtained. Best statistical characteristics of these QSARs are derived in the case of the Morgan extended connectivity of first order in the GAO. They are as follows: n = 11, r² = 0.6540, s = 0.824, F = 17 (training set); n = 9, r² = 0.8791, s = 0.388, F = 51 (test set).     

Simulation of biological evolution and the NFL theorems

William Dembski (No free lunch: why specified complexity cannot be purchased without intelligence, 2002) claimed that the NFL theorems from optimization theory render darwinian biological evolution impossible. Häggström (Biology and Philosophy 22:217–230, 2007) argued that the NFL theorems are not relevant for biological evolution at all, since the assumptions of the NFL theorems are not met. Although I agree with Häggström (Biology and Philosophy 22:217–230, 2007), in this article I argue that the NFL theorems should be interpreted as dealing with an extreme case within a much broader context. This broader context is in fact relevant for scientific research of certain evolutionary processes; not in the sense that the theorems can be used to draw conclusions about any intelligent design inference, but in the sense that it helps us to interpret computer simulations of evolutionary processes. As a result of this discussion, I will argue that from simulations, we do not learn much about how complexity arises in the universe. This position is in contrast with certain claims in the literature that I will discuss.     

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.     

The supercritical birth,death and catastrophe process: Limit theorems on the set of non-extinction

The skip-free property of the super-critical linear birth and death process with a linear catastrophe component is exploited to obtain an almost sure convergence theorem for the population size without any extra moment assumptions. This completes some earlier work of the author. A proof is sketched of a central limit theorem for the logarithm of the population size. Identification of the form of the norming constants for this result requires estimates of the extinction probabilities as functions of the initial population size. This is related to moment conditions on the decrement distribution.Most of this research was carried out at Colorado State University with the partial support of N.S.F. grant DMS-8501763.     

Synthesis and characterization of polyacids from palm acid oil and sunflower oil via addition reaction

In this study aliphatic polyacids were synthesized using palm acid oil (PAO) and sunflower oil (SFO) via addition reaction technique. The synthesized materials were characterized using Fourier-transform infra-red (FTIR) spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-ToF-MS) and thermo-gravimetric analysis (TGA). Mixing formic acid and hydrogen peroxide with PAO or SFO at the ratio 3:10:1 produced the lowest iodine value of 10.57 and 9.24 respectively, indicating the increase in epoxidization of both oils. Adding adipic acid to the epoxidized oils at a ratio of 1:10 increases the acid values of SFO and PAO to 11.22 and 6.73 respectively. The existence of multi-acid groups present in synthesized polyacid was confirmed by MALD-ToF-MS. This feature indicates a possible value to the biomaterials development.     

Production and cytogenetics of Brassica campestris-alboglabra chromosome addition lines

 Four different Brassica campestris-alboglabra monosomic addition lines (AA+1 chromosome from C, 2n=21) were obtained after consecutive backcrosses between resynthesized B. napus (AACC, 2n=38) and the parental B. campestris (AA, 2n=20) accession. The alien chromosomes of B. alboglabra (CC, 2n=18) in the addition lines were distinguished by random amplified polymorphic DNA (RAPD) marker analysis and morphology of mitotic chromosomes. Four RAPD marker synteny groups were established, which represented the four different alien chromosomes of B. alboglabra in the four addition lines. Three of the four addition lines were identified to harbour chromosomes 4, 8 or 9 of B. alboglabra. Studies on meiotic pairing in the addition lines revealed intergenomic homoeology relationships among specific chromosome arms between the A- and C-genomes. The long arm of B. campestris chromosome 9 was homoeologous with the long arm of B. alboglabra chromosome 4, while its short arm with the short arms of B. alboglabra chromosomes 8 and 9. Such an intergenomic homoeology relationship supports the hypothesis that B. campestris and B. alboglabra share a common ancestor but that chromosomal rearrangements have occurred during the evolution of the two species. Intergenomic introgression was observed in the progenies of the addition lines. The introgression of an entire B. alboglabra marker synteny group into the B. campestris genome implied the possible occurrence of interspecific chromosomal substitution. Received: 30 May 1996 / Accepted: 18 October 1996     

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.