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.     

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.     

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.     

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.     

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) .     

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.     

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.     

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.     

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.     

Occurrence of bifurcated three-center hydrogen bonds in proteins

Analysis of 13 high-resolution protein X-ray crystal structures shows that 1204 (24%) of all the 4974 hydrogen bonds are of the bifurcated three-center type with the donor X-H opposing two acceptors A1, A2. They occur systematically in alpha-helices where 90% of the hydrogen bonds are of this type; the major component is (n + 4)N-H ... O = C(n) as expected for a 3.6(13) alpha-helix, and the minor component is (n + 4)N-H ... O = C(n + 1), as observed in 3(10) helices; distortions at the C-termini of alpha-helices are stabilized by three-center bonds. In beta-sheets 40% of the hydrogen bonds are three-centered. The frequent occurrence of three-center hydrogen bonds suggests that they should not be neglected in protein structural studies.     

生态位有关术语的定义及计算公式评述

生态位(niche)理论在种间关系、群落结构、种的多样性及种群进化的研究中已被广泛应用。但对生态位及有关术语诸如生态位宽度、生态位重叠、生态位大小的定义至今还比较混乱。对于它们的计测,虽已提出了许多公式,但对其在生态学上的合理性仍有争议,本文试就这方面的问题作一评述。     

Computer graphics presentations and analysis of hydrogen bonds from molecular dynamics simulation

To obtain better insights into the dynamic nature of hydrogen bonding, computer graphics representations were introduced as an aid for the analysis of molecular dynamics trajectories. A schematic representation of hydrogen bonding patterns is generated to reflect the frequency and the type of hydrogen bonding occurring during the simulation period. Various trajectory plots for monitoring geometrical parameters and for analyzing three-center hydrogen bonding were also generated. The methods proposed are applicable to a variety of biopolymers. In this study, hydrogen bonding in the d(G) · d(C)₆ system was examined. For the nucleic acid fragments examined, three-center hydrogen bonds can be classified as in-plane and major or minor groove types. The in-plane three-center hydrogen bond represents a stable state in which both bonds simultaneously satisfy the relaxed hydrogen bonding criteria for a measurable period. On the other hand, groove three-center hydrogen bonds behave as a transient intermediate state in a flip-flop hydrogen bonding system.     

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     

Erratum and answer to the comment by Harris on: Numerical treatment of two-center overlap integrals (J Mol Mod, 12,213–220, 2006)

In the present contribution, we present two numerical tables for overlap integrals over Slater type functions and over B functions using the method present in the paper (J Mol Model, 12:213–220, 2006) where there were typos in the data that led Harris to conclude that our method is flawed and not useful for serious work. The typos are corrected and the numerical tables are listed as well as values from the literature and values obtained using ACJU code in order to perform comparisons with regards to accuracy.     

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.     

Valence orbital response to conformers of n-butane

Individual outer valence orbital responses to rotations of the C–C central bond of butane (C₄H₁₀) are explored on the torsional potential energy surface. Orbital ionization energies, topologies and momentum distributions for the four most significant butane conformation are presented, as snapshots of the conformational variations. The analysis is based on quantum mechanically generated information from coordinate space and momentum space, a technique called dual space analysis (DSA). By comparison with experimental measurements of photo-electron spectra (PES) for energies and of electron momentum spectra (EMS) for energies and Dyson orbitals, we demonstrate that the individual outer valence orbitals of these conformers response differently to the rotations of the central C–C bond of n-butane. Orbital signatures of other higher energy conformations, such as orbitals la ₂ and 5a ₁ of conformation D (C_2v ), are identified. This finding indicates a co-existence of butane conformations, although the global minimum structure of anti-butane, A (C_2h ), is dominant. Orbital topology and electron charges redistribution during the transformation provide useful information on the chemical bonding and related chemical reactions.     

On the Occurrence of Three-Center Hydrogen Bonds in Cyclodextrins in Crystalline Form and in Aqueous Solution: Comparison of Neutron Diffraction and Molecular Dynamics Results

Abstract

Three-center (bifurcated) hydrogen bonds may play a role by serving as an intermediate state between different dynamically changing hydrogen bonding patterns. Hydrogen bonding configurations can be studied experimentally by neutron diffraction and theoretically by computer simulation techniques. Here, both methods are used to analyse the occurrence of three-center hydrogen bonds in crystals of cyclodextrins.

Almost all experimentally observed three-center hydrogen bonds in the crystal are reproduced in the molecular dynamics (MD) simulations, even as far as the detailed asymmetric geometry is concerned. On the basis of this result a MD simulation of cyclodextrin in aqueous solution is searched for the occurrence of three-center hydrogen bonds. Significant differences are found. In solution more different three-center hydrogen bonds per α-cyclodextrin molecule are observed than in the crystal but the population (existence as percent of the simulation period) of each three-center hydrogen bond is lower in solution than in crystal. These may indeed serve as intermediate states in the process of changing one hydrogen bonding pattern into another.     

Band Shape Analysis of Vibronic Spectra of Flexible Molecules in Solution

Abstract

The absorption and emission spectra of biphenyl in cyclohexane are measured. Calculated Franck-Condon overlap integrals are used to simulate measured absorption and emission spectra of biphenyl in cyclohexane. The vibronic theory with the Born-Oppenheimer and Condon approximation is used.     

Mechanical performance of rat, mouse and mole spring traps, and possible implications for welfare performance

Lethal spring traps are widely used for killing small mammals in the UK. Many require government approval, based primarily on humaneness. However, mole traps and break-back traps for rats and mice are exempt; those available vary widely in price and apparent quality. The EU is considering implementing a Trapping Directive that would alter UK legislation, and a recent report advised the EU that trapping legislation should cover all trapped species and encourage improvement of traps. Mechanical trap performance is often used as an indicator of welfare impact. We examined the mechanical evidence for scope to improve the welfare standards of rat, mouse and mole spring traps. We measured mechanical performance among a range of rat, mouse and mole traps. Impact momentum values varied 6-8 fold, and clamping force values 4-5.5 fold, among traps for killing each species. There was considerable overlap in the performance of rat and mouse traps. Trap-opening angle and spring type were related to impact momentum and clamping force in traps for both species. There was no relationship between price and mechanical performance in traps for any species, except talpa mole traps. We are unable to judge the direct welfare impact of the traps tested, but rather the potential welfare threat associated with their exemption from approval. The wide variation in mechanical performance in traps for each species, overlap in performance between rat and mouse traps and increasing availability of weaker plastic rodent traps indicate considerable scope for improving the humaneness of spring traps for rats, mice and moles. We conclude that all such traps should be subject to the UK approval process. New welfare categories might improve trap standards further. Our results could also help improve rodent trap design and assist consumers in selecting more powerful traps. Many thousands of rats, mice and moles might benefit.