Polynomial Forms of Typical Interatomic Potential Functions

The use of polynomial functionals for describing two-body interactions in computational chemistry softwares has been surveyed and found to be prevalent. In this paper, Binomial and Maclaurin series expansions are used for expressing typical interatomic potential functions – such as Lennard-Jones, Morse, Rydberg and Buckingham potential – in a generic polynomial function, with the coefficients presented in a tabular format. Theoretical plots of these potential functions and their corresponding polynomial forms show increasing correlation with the order of polynomial, thereby validating the obtained polynomial’s coefficients. Conversely, a polynomial functional obtained by curve-fitting of experimental data can be converted into Morse, Rydberg and Buckingham potentials by using the generated table.     

Mathematical Connections Between Bond-Stretching Potential Functions

Mathematical connections are useful in enabling a set of parametric data from a chemical bond-stretching potential function to be applied in a computational chemistry software that adopts a different potential function. This paper establishes connections between four potential energy functions in stretching and compression of covalent bonds. The potential functions that are mathematically connected are: (i) harmonic potential, (ii) polynomial series potential, (iii) Morse potential, and (iv) Murrell–Mottram potential. Two methods are employed in obtaining the relationships between the four potential functions. The expansion approach enables the relationships to be made at large bond-stretching, whilst the differential approach allows for the connections to be made only at infinitesimal bond-stretching. For verification, parametric data from the Murrell–Mottram potential is converted to parametric data of the harmonic, polynomial series and Morse potentials. For comparison, the bond-stretching energies for these functions are plotted. Discrepancy between the Morse and the Murrell–Mottram potentials at large bond-stretching is discussed in terms of the assumed infinitesimal deformation.     

Expansion formulae for two-center charge densities of integer and noninteger n generalized exponential type orbitals with hyperbolic cosine and their use in evaluation of multicenter multielectron integrals

The expansion formulae are derived for the two-center charge densities of integer and noninteger n generalized exponential type orbitals with hyperbolic cosine (GETOHC) in terms of corresponding charge densities of generalized exponential type orbitals (GETO) presented in our previous paper. The general formulae obtained for the GETOHC charge densities are utilized for the evaluation of multicenter multielectron integrals appearing in the Hartree–Fock–Roothaan (HFR) and explicitly correlated theories when the GETOHC are employed as basis functions.     

Mathematical Relationships Between Bond-Bending Force Fields

Molecular mechanics softwares adopt various set of force field functions. In some cases, reliable data from one set of force field parameters cannot be used in a software that adopts another set of force field. Using mathematical approach, exact relationships between parameters from three bond-bending force fields, namely the (i) harmonic cosine angle, (ii) polynomial series, and (iii) Fourier series, are herein developed. Parameters from these three potential functions are further related to the approximate form, the harmonic angle function, which is valid for small change in chemical bond angle.     

Split series potential energy function

A split series potential energy function is developed herein such that it can be primarily reduced into generalized forms of Lennard-Jones, Morse, Buckingham, Linnett, and anything in between them by altering the type-parameters. Unlike the previous approach whereby the type-parameters act as weighting parameter between the inverse power and the exponential decay terms, the current approach attaches the type-parameters as indices to the denominator of every summation term. A change of the index shifts the series summation to mimic a curve that is representative of an inverse power, an exponential index, or a weighted proportion between the two. Comparison with the Lennard-Jones (12–6) and Exponential-6 (both ξ = 13.772 and ξ = 12) in the case of uncharged non-bonded interactions by using type-parameters (p, q) = (0, 0) and (p, q) = (1, 0) respectively in the model gives very good correlation. Comparison with the Morse potential in the case of bonded interaction by using (p, q) = (1, 1) gives excellent agreement. All results show that the use of only five expansion terms is sufficient for providing good agreement with the classical potentials. Better fit to experimental data and ab initio results are expected with the incorporation of coefficients in every term of the series expansion. The proposed model is useful for modeling the transition state from solid to fluid and vice versa.     

Effects of Electrolytes on Manipulation of the Stationary Phase in Electrochemically Modulated Liquid Chromatography

Abstract

Electrolytes were found to have an important influence on the cell constant in electrochemically modulated liquid chromatography. The dependence of cell constant on electrolyte type did not reflect any relation with regard to electrolyte conductivity values. However, electrolytic species of larger ionic sizes result in significantly lower cell constants, and vice versa. It was also found that the cell constant is exponentially dependent on electrolyte concentration, with higher electrolyte concentrations resulting in decreased cell constants. When a potential ramp is applied to the working electrode, a steep change in the potential of the working electrode towards the final potential takes place directly after the application of the potential ramp. The change in the potential of the working electrode then follows an exponential decay isotherm, which depends on both electrolyte type and concentration.

     

Studies on some exponential‐screened coulomb potentials

The generalized pseudospectral method is used to study the bound‐state spectra of some of the exponentially screened Coulomb potentials, namely, the exponential cosine screened Coulomb (ECSC) and general exponential screened Coulomb (GESC) potential, with special emphasis on higher states and stronger interaction. Eigenvalues accurate up to 11 significant figures are obtained through a nonuniform optimal spatial discretization of the radial Schrödinger equation. All the 55 eigenstates of ECSC potential with n ≤ 10 and 36 eigenstates of GESC potential with n ≤ 8 are considered for arbitrary values of the screening parameter, covering a wide range of interaction. Excited states as high as up to n = 18 have been computed with high accuracy for the first time. Excellent agreement with the literature data has been observed in all cases. All the GESC eigenstates are calculated with much greater accuracy than the existing methods available in literature. Many l ≠ 0 states of this potential are reported here. In both cases, a detailed variation of energies with respect to the parameters in potential is monitored. © 2012 Wiley Periodicals, Inc.     

Critical parameters and spherical confinement of H atom in screened Coulomb potential

Critical parameters in three screened potentials, namely, Hulthén, Yukawa, and exponential cosine screened Coulomb potential are reported. Accurate estimates of these parameters are given for each of these potentials, for all states having . Comparison with literature results is made, wherever possible. Present values compare excellently with reference values; for higher n, ?, our results are slightly better. Some of these are presented for first time. Further, we investigate the spherical confinement of H atom embedded in a dense plasma modeled by an exponential cosine screened potential. Accurate energies along with their variation with respect to box size and screening parameter are calculated and compared with reference results in literature. Sample dipole polarizabilities are also provided in this case. The generalized pseudospectral method is used for accurate determination of eigenvalues and eigenfunctions for all calculations. © 2016 Wiley Periodicals, Inc.     

Coefficient interrelatedness among polynomial potential functions of diatomic molecules

It is shown that the diatomic potential energy functions of Dunham, SPF and Ogilvie can be easily converted from one another when their coefficients are related. Through Maclaurin expansion and comparison of terms, the coefficients can be related by using the Pascal Triangle. In this paper, the coefficients were related up to the tenth order of δr/R for HX (X = H, Ga, Cl, I). Comparison of all three potential energy curves shows very good agreement for r ≤ 1.5R, thereby verifying the formulated relations. Observation of the plotted potential energy curves for r > 1.5R shows that the difference of the three potential function definitions is not reflected as any consistent trend arising from the related potential functions.     

Mathematical Optimization of in vivo NMR Chemistry Through the Fast Padé Transform: Potential Relevance for Early Breast Cancer Detection by Magnetic Resonance Spectroscopy

Mathematical advances in signal processing through the fast Padé transform (FPT) can greatly improve the information extracted via in vivo nuclear magnetic resonance (NMR) chemistry. The FPT is a frequency-dependent, non-linear rational polynomial approximation of the exact Maclaurin series, which dramatically improves resolution and signal-to-noise ratio in a stable manner with robust error analysis and provides precise numerical data for all the peak parameters (position, height, width and phase) for every true resonance including those that are weak and/or overlapping. The concentrations of many of the chemical constituents of tissues can thereby be accurately determined. These advantages of the FPT are particularly germane for in vivo NMR detection and quantification of a number of molecular markers of breast cancer, such as phosphocholine, as well as lactate, which cannot be assessed using standard Fourier data analytical techniques applied to in vivo NMR in the clinical setting.     

On the effectiveness of exponential type orbitals with hyperbolic cosine functions in atomic calculations

Unconventional basis functions, constructed from exponential type orbitals (ETOs) with hyperbolic cosine functions, are applied to Roothaan-Hartree-Fock calculations of atoms within the minimal basis sets framework. The most popular ETOs, Slater type orbitals, B functions and \(\psi ^{(\alpha ^*)}\) functions with \(\alpha ^*=2\), and two types of hyperbolic cosine functions, \(\cosh (\beta r)\) and \(\cosh (\beta r+\gamma )\), are used in this work. The performance of the present basis functions is investigated and compared to the conventional double-zeta Slater-type basis set and numerical Hartree-Fock results. The improvement in the atomic energies clearly demonstrates how the accuracy increases when we move from ETO to ETO with hyperbolic cosine basis functions. The resulting improved minimal basis sets can also be useful in molecular calculations.     

Use of noninteger n‐generalized exponential type orbitals with hyperbolic cosine in atomic calculations

The efficiency of noninteger n‐generalized exponential type orbitals (NGETO) r^n*?1 e with hyperbolic cosine (HC) cosh (βr^μ) as radial basis functions in atomic ground state total energy calculations is studied. By the use of these functions, the combined Hartree‐Fock‐Roothaan calculations have been performed for some closed and open shell neutral atoms and their anions and cations with Z ≤ 21. The performance of new basis functions within the minimal basis framework has been compared with numerical Hartree‐Fock (NHF) results. Our total energy values are significantly close to NHF results. The presented minimal basis total energies obtained from the noninteger NGETO with HC are notably better than minimal basis functions total energies previously reported in the literature. It is found that the accuracy of new noninteger NGETO with HC almost correspond to the accuracy of the conventional double‐zeta functions. All the nonlinear parameters are fully optimized. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Calculations of D‐wave bound states and resonance states of the screened helium atom using correlated exponential wave functions

We have investigated the effects of screened Coulomb (Yukawa) potentials on the bound ^1,3D states and the doubly excited ^1,3 D^e resonance states of helium atom using highly correlated exponential basis functions. The Density of resonance states are calculated using stabilization method. Highly correlated exponential basis functions are used to consider the correlation effect between the charged particles. A total of 18 resonances (nine each for ¹ D^e and ³ D^e states) below the n = 2 He ⁺ threshold has been calculated. For each spin states, this includes four members in the 2pnp series, three members in the 2snd series, and two members in 2pnf series. The resonance energies and widths for various screening parameters ranging from infinity to a small value for these ^1,3 D^e resonance states are reported along with the bound‐excited 1s3d ^1,3 D state energies. Overall behavior of the spectral profile of 1s3d ¹D state of helium atom due to electron‐electron and electron‐nucleus screening are also presented. Accurate resonance energies and widths are also reported for He in vacuum. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Simultaneous quantification of promazine hydrochloride and its sulfoxide in pharmaceutical preparations.

The use of derivative UV-spectrophotometry is proposed for the simultaneous quantification of promazine hydrochloride in the presence of sulfoxide, and vice versa. For this purpose, mathematical parameters were established for generating derivative spectra of analytes. The determination of promazine was made using the first-order derivative (deltalambda = 10 nm, second polynomial degree) at 268 nm. The quantification of sulfoxide was achieved by applying third-derivative spectra (deltalambda = 14 nm, sixth polynomial degree) based on measurements of the amplitude at 342 - 344 nm. An elaborated method was successfully used to determine analytes in commercial promazine pharmaceuticals. The obtained results agreed well with those obtained by the HPLC method.     

Properties of hydrogen molecular ion with static screened coulomb and exponential cosine screened coulomb potentials

We have investigated the properties of hydrogen molecular ion (H) interacting with screened Coulomb potentials. Two types of potentials have been considered, namely, static screened Coulomb potential and exponential cosine‐screened Coulomb potential. Simulating the localized motion of the nuclei by using high powers of the internuclear coordinate in the generalized Hylleraas‐type wave function we have been able to obtain reasonably accurate binding energies for 1¹S(J = 0, ν = 0) and 2¹S(J = 0, ν = 1) states for various values of the screening parameter within the frame work of Ritz variational method. Borromean bindings are found to exist in the ion for a wide range of the screening parameter. Furthermore, expectation values for various operators and cusp condition have been calculated. Results for the unscreened case are in nice agreement with some of the accurate results available in the literature. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Investigation of electronic transport properties of lanthanides in liquid phase

We investigated electrical transport properties of lanthanide series in liquid phase using the self-consistent approximation employed by Khajil and Tomak [Phys Stat. Sol. B, 134 (1), 321–324 (1986)]. We used our model potential by implemented ionic and atomic radius which is incorporated with the Charged Hard Sphere, One Component Plasma and General Mean Spherical Approximation reference systems to study the electronic transport properties like electrical resistivity (ρ), thermal conductivity (σ) and thermoelectric power (Q). The screening effect on aforesaid properties has been studied by using different screening functions. The correlations of our results and others data within addition experimental values are profoundly promising to the scientists working in this field.     

On the perturbation of the H-bonding interaction in ethylene glycol clusters upon hydration

Ab initio and density functional methods have been employed to study the structure, stability, and spectral properties of various ethylene glycol (EG(m)) and ethylene glycol-water (EG(m)W(n)) (m = 1-3, n = 1-4) clusters. The effective fragment potential (EFP) approach was used to explore various possible EG(m)W(n) clusters. Calculated interaction energies of EG(m)W(n) clusters confirm that the hydrogen-bonding interaction between EG molecules is perturbed by the presence of water molecules and vice versa. Further, energy decomposition analysis shows that both electrostatic and polarization interactions predominantly contribute to the stability of these clusters. It was found from the same analysis that ethylene glycol-water interaction is predominant over the ethylene glycol-ethylene glycol and water-water interactions. Overall, the results clearly illustrate that the presence of water disrupts the ethylene glycol-ethylene glycol hydrogen bonds.     

Stochastic searches for lactone and cycloalkene conformers

The stochastic search method coupled to MM3(92) has been used to locate as many conformers as possible for 6- to 11-membered ring lactones, trans-cycloalkenes, and cis-cycloalkenes. A comparison was carried out between the conformers of lactones and cycloalkenes of the same ring size for each force field. These comparisons were carried out by means of (1) conformational distances, defined as the rms deviation between the dihedral angles of the conformers being compared, and (2), substitution, in which a lactone was transformed into an olefin and vice versa, trying to keep as much as possible the initial geometry, followed by reoptimization. It is found that cycloalkenes and lactones share many common characteristics. The thermodynamic information provided by MM3 was used to study the dependence of conformer population upon (1) temperature and (2) total number of conformers. © 1993 John Wiley & Sons, Inc.     

The S-matrix version of the Hulthén-Kohn variational principle for quantum scattering: Finite element calculations for state-to-state reaction probabilities

Summary The finite element (FE) method can be applied to solve the reactive scattering problem of the A + BC AB + C type. We are using the S-matrix version of the Hulthén-Kohn variational principle for the two-dimensional collinear problem. The asymptotic wavefunction is described by analytical functions and the interaction part by FE functions.Two approaches are used and compared to each other: 1) FE functions are employed as the basis functions for the interior part of the potential; 2) the FE method is used in order to calculate eigenfunctions for the interior region of the potential and then these eigenfunctions are used as a new basis. In the second case two ways of choosing grid points are applied. First a grid is built that covers both reaction channels, secondly we considered the two reactions channels separately and two basis sets are used in the scattering calculations. This last version would be more helpful from the point of application to three-dimensional non-collinear problems. Results obtained by raising the number of elements, or by increasing the polynomial order, are compared. General formulas for the formfunctions in case of arbitrary order of the polynomials will be presented.Dedicated to the 60th birthday of Prof. Jan Linderberg (Aarhus)     

Accurate analytical expressions for partition functions for all values of temperature

We obtain accurate analytical expressions for partition functions valid for all values of temperature by the application of the Euler—Maclaurin summation formula to that part of the sum over the energies that is dominant at high values of T. Several useful properties of the Euler—Maclaurin summation formula are easily derived by means of an operator formalism. As an illustrative example we explicitly consider the rotational partition function for diatomic molecules and use the analytical expressions to introduce centrifugal distortion effects and to calculate the contribution of the rotational motion to the specific heat. The accuracy of the analytical expressions is tested by comparison with exact numerical calculations.