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.     

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     

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.     

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.     

Transient electric birefringence of colloidal particles immersed in shear flow. Part II. The initial response under the action of a rectangular electric pulse and the behavior at a low alternating electric field

The time-dependent rotational diffusion equation for rigid macromolecules in solution has been approximately solved for two cases in order to extend the electric birefringence technique to streaming-electric birefringence. One is for the initial period through the application of a rectangular electric pulse to the solution immersed in a low shear flow. The purpose of this is expansion of the distribution function into a function series made by the product of the powers of reduced time (= Thetat) and hydrodynamic field alpha (= G Theta , G: velocity gradient, Theta: rotary diffusion constant) and a surface harmonic P(i)(j)cos jphi. The solution for the build-up process at arbitrary electric field strength is found, but is limited to low hydrodynamic fields. The other is for the response when an alternating electric field is applied to the solution in a shear flow. Here, instead of reduced time, the maximum electric field E(0) is chosen as a parameter for the expansion. The expressions for the intensity of the transmitted light through crossed Nicols are derived in two optical systems where the polarizer is set at an angle of 45 degrees and 0 degrees to the direction of the electric field. The results in the former case show that we can determine four parameters, the ratio of velocity gradient to rotary diffusion constant, the axial ratio of a particle, the anisotropy of electric polarizability, and the optical anisotropy factor, from four values observed in two optical systems, namely, two light intensities before applying an electric field and two initial slopes of the build-up after applying an electric field. On the other hand, when a low alternating electric field with extremely high frequency is applied, the build-up of the light intensity in the former case is the same as that of electric birefringence for pure induced dipole orientation. The build-up for the latter optical system is the same as the expression for pure induced dipole orientation of Eq. (38) shown in a previous work.     

Ab initio study of electron transport in 4-(3-nitro-4-tetrafluorophenylthiolate-ethynyl, phenylethynyl) benzenethiolate

The electron transport of the 4-(3-nitro-4-tetrafluorophenylthiolate-ethynyl, phenylethynyl) benzenethiolate (S-FNPPB-o) molecule assembled in two Au (111) electrodes, was studied using two approaches: in the first approximate approach an electric field was applied to the pure molecule attached to two thiolate ends fixed, and in the second approach we used the nonequilibrium Green′s function formalism (NEGF) coupled to DFT to calculate the I-V curve and the voltage dependence of the transmission function in the extended system, molecule plus electrodes. By applying an electric field to the pure molecule plus thiolate ends fixed, and visualizing the changes in the spatial distribution of the frontier molecular orbitals, we can expect based on the continuity of the conduction pathway in electron transport, that if electron transport occurs through the frontier orbitals, only the LUMO orbital would create an open channel for electron transport due to its delocalized nature and large orbital density at the thiolate groups. The NEGF calculations indicate that at applied voltages lower than ±0.8 V, the current is related to transmission values through the tails of the broad LUMO orbital, and since this orbital is the one closer to the Fermi energy, and we observed very low current values in this region, higher current values at positive bias than at negative bias. As the voltage exceeds ±0.8 V the current increases from the contribution of more states from the broadened part of the transmission function from the LUMO orbital, and when the voltage approaches ±2 V, the LUMO + 1 orbital enters into the bias window and the current increases again.     

M?ssbauer spectroscopic study of compound ES of cytochrome c peroxidase.

M?ssbauer spectra of Compound ES of cytochrome c peroxidase have been observed over a range of temperature and applied magnetic field. These have been interpreted in terms of a crystal field model of the iron site in which the iron is assumed to be in the Fe(IV) state with unpaired spin S = 1. Detailed least-squares fitting of the spectra fitting of the spectra indicates that both the electric field gradient choice of a single parameter, the axial crystal field, the magnetic properties are well reproduced. The model also provides the observed positive sign for the electric field gradient interaction, but overestimates its magnitude. This apparent discrepnancy may be caused by the presence of significant electronic charge in filled bonding orbitals, a feature which is in keeping with expected covalent charge compensation of the extreme oxidation state. There is no evidence in the M?ssbauer spectra of interaction between the iron and the ESR-visible free radical. This suggests they are well separated.     

Molecular dynamics modelling of an electrical-driven linear nanopump

The dynamics of an electrical-driven linear nanopump, consisting of a carbon nanotube, a C60⁺ molecule and a graphene sheet, has been simulated via the application of the molecular dynamics method. In this nanopump, the nanotube and the graphene sheet are used as the sleeve of the pump and the boundary between the two sides of the nanopump, respectively. By exposing the nanopump to an external alternative electric field, the C60⁺ molecule will be oscillating linearly in the nanotube. We found that the linear oscillating motion of the C60⁺ molecule causes the gas atoms to flow through the nanotube, and a density gradient is generated between the two sides of the nanopump. Also, it was observed that the frequency of the external alternative electric field affected the pump performance in the generation of the density gradient amount. The maximum performance occurred at a specific frequency of the electric field. This specific frequency can be computed by an analytical formula for given materials and temperatures. Moreover, the results indicate that the length of the nanotube can affect the gas pumping.     

Simultaneous analysis of the cardiac electric and magnetic fields using the scalar multipole expansion

A theoretical analysis of electric and magnetic fields of the heart, based solely upon the scalar multipole expansion, is carried out in order to gain an insight into the interrelation of the data contained in electro- and magnetocardiological measurements. The usual multipole expansion is applied for the electric field, however corresp[nding equivalent multipoles are formulated as idealized generators, having not only flow sources, but also vortex sources of the field. Furthermore, the magnetic field in a homogeneous infinite volume conductor is expressed as a sum of two series, the first being the usual multipole expansion of the nonvortex component of the magnetic field, and the second being a sequence of magnetic fields set up by the aforementioned electric multipole generators reduced to axial form. The former term is uniquely defined by the electric multipole components, but the latter reflects properties of the cardiogenerator that can be revealed only by means of magnetic measurements. Features of the electric and magnetic multipole components as integral characteristics of the cardiogenerator are discussed and concepts of the magnetic centre and magnetic axis of the cardiogenerator are proposed. The analysis is illustrated by examples of simple generator configurations.     

Dissipation of Energy of an AC Electric Field in a Semi-Infinite Electron Plasma with Mirror Boundary Conditions

Absorption of the electromagnetic energy in a semi-infinite electron plasma is calculated for an arbitrary degree of the electron gas degeneracy. Absorption is determined by solving the boundary-value problem on the oscillations of electron plasma in a half-space with mirror boundary conditions for electrons. The Vlasov?Boltzmann kinetic equation with the Bhatnagar–Gross–Krook collision integral for the electron distribution function and Maxwell’s equation for the electric field are employed. The electron distribution function and the electric field inside plasma are searched for in the form of expansions in the eigenfunctions of the initial set of equations. The expansion coefficients are found for the case of mirror boundary conditions. The contribution of the plasma surface to absorption is analyzed. Cases with different degrees of electron gas degeneracy are considered. It is shown that absorption of the electromagnetic energy near the surface depends substantially on the ratio between the electric field frequency and the volumetric electron collision frequency.     

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.     

Concerning the width of spark channels with different polarities in submicrosecond sliding discharges in noble gases

Previously, the parameters of submicrosecond (with a duration of <200 ns) multichannel high-current discharges sliding along a ceramic surface in Ne, Ar, and Xe were studied only for the negative polarity of the applied voltage. The experimental data indicate that the channels expand in the transverse direction mainly due to electron drift from the channel surface layer under the action of the electric field perpendicular to the channel axis and subsequent gas ionization by these electrons. To investigate mechanisms for the channel development in a sliding discharge—in particular, to determine the contribution of electron drift—it is necessary to carry out experiments similar to those performed earlier for the opposite polarity of the applied voltage. Here, the results of measurements of the widths of the spark channels of negativeand positive-polarity sliding discharges excited in Ne, Ar, and Xe at pressures of 30 and 100 kPa are presented and discussed. It is shown that, depending on the pressure and sort of gas, the averaged optical width of positive-polarity channels is smaller by a factor of 1.27–1.60 than that of negative-polarity channels. The experimental data are analyzed using the theory of propagation of ionization waves with different polarities in gases. Analysis has shown that electron diffusion contributes insignificantly to channel expansion and that, for both polarities, the channel expansion rate exceeds the electron drift velocity in the transverse electric field near the channel. In the framework of the so-called approximation of nonlocalized initial conditions, the measured ratio between of the widths of negativeand positive-polarity channels and their relation to the electron mobility are explained by the channel expansion governed by both electron drift and primary free electrons produced by a short-term source in a narrow region ahead of the front of the expansion wave. Numerical simulations show that the width of this region is comparable with that of the wave front and is more than one order of magnitude smaller than the observed channel radius. Gas photoionization by the channel radiation can serve as a source of primary electrons.     

Skin Effects in a Dusty Plasma

A multifluid MHD model is applied to study the magnetic field dynamics in a dusty plasma. The motion of plasma electrons and ions is treated against the background of arbitrarily charged, immobile dust grains. When the dust density gradient is nonzero and when the inertia of the ions and electrons and the dissipation from their collisions with dust grains are neglected, we are dealing with a nonlinear convective penetration of the magnetic field into the plasma. When the dust density is uniform, the magnetic field dynamics is described by the nonlinear diffusion equations. The limiting cases of diffusion equations are analyzed for different parameter values of the problem (i.e., different rates of the collisions of ions and electrons with the dust grains and different ratios between the concentrations of the plasma components), and some of their solutions (including self-similar ones) are found. The results obtained can also be useful for research in solid-state physics, in which case the electrons and holes in a semiconductor may be analogues of plasma electrons and ions and the role of dust grains may be played by the crystal lattice and impurity atoms.     

Experimental study of electron vortex structures in an electrostatic plasma lens

Results are presented from experimental studies of electron vortex bunches in a cold ion-beam plasma consisting of strongly magnetized electrons and a beam of almost free positive ions. The existence of electron vortex bunches was detected from local minima of the electric potential on surfaces perpendicular to the magnetic field lines. It is found that the vortices have the form of magnetic-field-aligned filaments, in which electrons rotate with a velocity significantly exceeding both the velocity of the vortex as a whole and the electron velocity in the ambient plasma. It is shown that, in a sufficiently strong magnetic field, the accumulation of electrons in the vortices terminates when the condition for the longitudinal confinement of electrons by the electric field fails to hold.     

Energy of electrons emitted as a result of separation of surfaces

Experiments indicating acceleration of charged particles as a result of separation of solid surfaces are analyzed. As a possible mechanism of such acceleration, generation of surface charge on the separated surfaces of a cleaved ionic crystal is considered. The maximum electric field generated due to the charging of the separated surfaces and the energy of electrons accelerated in such a field are estimated. It is shown that, for the maximum attainable electric field, conditions are created for the generation of runaway electrons that, even at atmospheric pressure, electrons are accelerated to high energies, not experiencing collisions with gas particles.     

Note on the validity of constant field assumption in relation to the exact solution of Nernst--Planck equations

By means of the exact solution of the Nernst-Planck equations, it can be demonstrated that the constant electric field assumption is incompatible with the stationary flows of ions in the homogeneously charged membrane. The exact solution of the Nernst-Planck equation system is readily given by the power series expansion method. The constant field assumption is valid for membrane in which the fixed charge can move freely and neutralize the charge of permeating ions.The parabolical electric potential is shown to be incompatible with the homogeneously charged membranes including the neutral membranes.     

Geometric and electronic structures of new endohedral fullerenes: Eu@C72

The geometric and electronic structures of rare earth metallofullerenes Eu@C₇₂ were investigated using density functional theory (DFT) within a generalized gradient approximation (GGA). The geometric optimization revealed that the most favorable endohedral site for Eu is off-center along the C₂ axis on the σ_v plane pointing to the (5, 5) bond at the fusion of two pentagons. Calculations for electronic structures show that two 6s electrons in Eu transfer to the lowest-unoccupied-molecular orbitals of C₇₂ while 4f electrons remain in Eu.     

A theoretical interpretation of the variations of some physical parameters within the [2Fe-2S] ferredoxin group.

A model is proposed to explain the variation of some physical parameters within the reduced [2Fe-2S] ferredoxin group. According to this model, the main effects result from a variable mixing of some d orbitals of the Fe2+ ion owing to rhombic distortion of the active site having the same geometrical character, but different in intensity, for each protein. Some peculiar experimental results such as the axial electron paramagnetic resonance spectra of adrenal ferredoxin and Pseudomonas putida ferredoxin and the electric field gradient tensor of P. putida ferredoxin are explained without assuming properties drastically different from those of the other ferredoxins, as had been suggested in the literature.