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.     

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.     

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.     

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 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.     

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     

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.     

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.     

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) .     

Evaluation of two-center Coulomb and hybrid integrals over complete orthonormal sets of $$ {\Psi^{\rm{\alpha }}} - {\hbox{ETO}} $$ using auxiliary functions

Journal of Molecular Modeling - By the use of ellipsoidal coordinates, the two-center Coulomb and hybrid integrals over complete orthonormal sets of $$ {\Psi^{\rm{\alpha }}} - {\hbox{exponential}}...     

Analytical expressions on harmonic oscillators for variational path integrals

In this study, we derive analytical expressions for one-dimensional harmonic oscillators for variational path integrals (VPIs). A Gaussian-type trial wavefunction is adopted. Total and potential energies of the system are analytically expressed both for the continuous time and an approximate discretised VPIs. Obtained expressions are numerically verified using molecular dynamics calculations. Convergence properties regarding the projection time and the Trotter number are discussed.     

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.     

Effects of embryotrophic factors on the embryogenesis and organogenesis of mouse embryos in vitro

In order to study embryogenesis and organogenesis in vitro, two cell mouse embryos were cultured with alpha-MEM supplemented 10% FCS and embryotrophic factors (ETFs). The ETFs were separated from the conditioned medium of a SKG-II-SF cell line derived from a human uterine cervical epidermoid carcinoma. IL-1 beta, IL-6, IL-8, EGF, GH, PDGF-AB, basic FGF, VEGF were also detected in the conditioned media of this cell line. Using ETFs and a 10% FCS supplemented culture medium, 23% of the mouse two cell stage embryos developed to the bilaminar disc stage, 13% to the trilaminar germ disc stage, 9% were observed with blood islets in the yolk sac, and the heart beat was noted in 7% (28 embryos) of the embryos. Furthermore, primordial organs, such as the liver, heart, kidney, notochord, retina-like structure, etc. were observed. Usually, structures associated with the primordial streak stage (bilaminar germ disc embryo) developed in vitro using ETFs from two cell stage embryos. These closely resemble structures found at the same stage in the normal embryo in vivo. After the primordial streak stage, the cultured embryos showed no resemblance to the same stage in normal embryos. None of the external appearances of these embryos appeared normal. On the other hand, trilaminar disc stage embryos never developed when using only a 10% FCS supplemented culture system.     

Organogenesis of heart-vascular system derived from mouse 2 cell stage embryos and from early embryonic stem cells in vitro

Regenerative medical treatment with embryonic stem cells (an ES cell) is a goal for organ transplantation. Structures that are tubular in nature (i.e. blood capillaries) were induced from early embryonic stem (EES) cells in vitro using embryotrophic factor (ETFs). In addition, cardiac muscle cells could be identified as well. However, differentiation of EES cells into a complete cardiovascular system was difficult because 3 germ layer primordial organs are directed embryologically in various ways and it is not possible to guide only cardiovascular organs. Thus, we introduced ETFs after the formation of an embryoid body and were successful in cloning cell clusters that beat, thus deriving only cardiovascular organs. The application of this to the treatment of various cardiovascular diseases is promising.     

The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells

Medial entorhinal cortex layer II stellate cells display subthreshold oscillations (STOs). We study a single compartment biophysical model of such cells which qualitatively reproduces these STOs. We argue that in the subthreshold interval (STI) the seven-dimensional model can be reduced to a three-dimensional system of equations with well differentiated times scales. Using dynamical systems arguments we provide a mechanism for generations of STOs. This mechanism is based on the “canard structure,” in which relevant trajectories stay close to repelling manifolds for a significant interval of time. We also show that the transition from subthreshold oscillatory activity to spiking (“canard explosion”) is controlled in the STI by the same structure. A similar mechanism is invoked to explain why noise increases the robustness of the STO regime. Taking advantage of the reduction of the dimensionality of the full stellate cell system, we propose a nonlinear artificially spiking (NAS) model in which the STI reduced system is supplemented with a threshold for spiking and a reset voltage. We show that the synchronization properties in networks made up of the NAS cells are similar to those of networks using the full stellate cell models. In memory of Angel A. Alonso     

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.     

Experimental design for estimating integrals by numerical quadrature, with applications to pharmacokinetic studies

Many experimental situations, including bioavailability studies, require the estimation of integrals by numerical quadrature applied to dependent variable observations with measurement error. A strategy is described for selecting values for the independent variable (e.g. time). The strategy minimizes the expectation of the square of the difference between the exact integral and the quadrature approximation. This approach was applied to simulated pharmacokinetic problems, including the estimation of bioavailability. Results indicate that the procedure is potentially useful in reducing the variance of resulting estimates and that it appears to be robust with respect to prior assumptions about model parameter values.     

Structural and redox relationships between Paracoccus denitrificans, porcine and human electron-transferring flavoproteins.

Electron-transferring flavoprotein (ETF) was purified from the bacterium Paracoccus denitrificans and the structural and redox relationships to the porcine and human ETFs were investigated. The three proteins have essentially identical subunit masses and the alpha-helix content of the bacterial and porcine ETFs are very similar, indicating global structural similarity. An anti-(porcine ETF) polyclonal antibody that crossreacts with the human large and small subunits also crossreacts strongly with the large subunit of Paracoccus ETF. However, crossreactivity with the small subunit is very weak. Nonetheless, an amino-terminal peptide and four internal peptides of the small bacterial subunit show extensive sequence identity with the human small subunit. Local similarities in environment are also indicated by the intrinsic tryptophan fluorescence emission spectra of porcine and Paracoccus ETFs. Although the visible spectra of porcine and Paracoccus ETFs are virtually identical, flavin fluorescence in the bacterial protein is only 15% that of the mammalian protein. Further, the circular dichroic spectrum of the flavin in the bacterial protein is significantly more intense, suggesting that the microenvironment of the isoalloxazine ring is different in the two proteins. Enzymatic or photochemical reduction of Paracoccus ETF rapidly yields an anionic semiquinone; formation of the fully reduced flavin in the bacterial ETF is very slow. The spacing of the oxidation-reduction potentials of the flavin couples in the bacterial ETF is essentially identical to that in procine ETF as judged from the disproportionation equilibrium of the bacterial ETF flavin semiquinone. Together, the enzymatic reduction and disproportionation equilibria suggest that the flavin potentials of the two ETFs must be very close. The data indicate that the structural properties of the bacterial and mammalian proteins and the thermodynamic properties of the flavin prosthetic group of the proteins are very similar.     

New approach for the establishment of mouse early embryonic stem cells and induction of their differentiation

Eleven early embryonic stem (EES) cell lines were established using a new novel method. Two cell stage embryos from the ddY mouse strain were cultured in alpha-MEM supplemented with 10% fetal calf serum (FCS) and embryotrophic factors (ETFs) and allowed to develop to the trilaminal germ disc embryonic stage. Only small round cells (EES cells) were isolated by the colony isolating technique and subsequently cultured in the same medium containing the ETFs and leukemia inhibitory factors (LIF-10 ng/ml). The newly established embryonic stem (ES) cells isolated from inner cell mass of blastocysts differentiated from two cell stage embryo in culture. The EES and ES cell lines were maintained in an undifferentiated state using Ham's F12 medium supplemented with 10% FCS and 1 ng/ml of LIF. The EES cells maintained their normal genetic and morphological features as well as their potential to differentiate into a broad spectrum of cell types as well as their ability to contribute to all cell lineages in chimeric mice. Moreover, these cell lines changed and differentiated into various kinds of cells by removing LIF and by the addition of ETFs to the vitro culture system. All 11 EES cell lines and 3 ES cell lines formed embryoid bodies; however, cell line EES-4 formed tube-like structures which extended, anastomosed with each other, and finally formed networks when the LIF were absent. Primitive germ organ-like structures composed of 3 germ layers were recognized in the cultures following the administration of ETFs. In conclusion, the new method devised by us is a novel, easy and reliable technique for establishing EES cell lines.