Application of Maclaurin Series in Relating Interatomic Potential Functions: A Review

This paper provides a short review on the application of Maclaurin series in relating potential functions within the same category of interatomic interaction. The potential functions covered are those commonly adopted in computational chemistry softwares. While various mathematical approaches have been employed in generating relationships amongst potential functions, the use of Maclaurin series has been prevalent recently due to the increasing application of polynomial series-type potential functions. In the case of covalent bond-stretching, the Maclaurin series for the exponential function is used to transform the Morse potential into the polynomial series potential, and vice versa. For covalent bond-bending, Maclaurin series for the sine and cosine functions were employed to extract polynomial angle series potential from the Fourier series and harmonic cosine potential functions, and vice versa. Finally, both the exponential and the 1/(1 – x) expressions in Maclaurin series were used in obtaining the exact relationship for the repulsive terms between two potential 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.     

Infinite square well with a sinusoidal bottom: a candidate for the Klauder phenomenon?

In contrast to a recent observation, we notice that a particle in a box with any cosine bottom does not show up the Klauder phenomenon when the perturbation is gradually reduced to zero. Both perturbative and variational approaches have been pursued. The case of a harmonic oscillator perturbed by a similar potential is additionally studied. No peculiarity is observed anywhere in this case too. Possible reasons behind the phenomenon are sought to rationalize our findings.     

The effect of dipole-dipole interaction on zero-THz frequency polarisation

The effect of intramolecular dipole-dipole interaction on polarisation at frequencies from 0 – 10¹² Hz in dipolar media is investigated using a numerical solution of the Sturm-Liouville eigenvalue problem to which the governing Smoluchowski equation reduces. The results of Budó for the long-wavelength limit are extended to all frequencies up to the far infrared, where the predominant inertial effects are treated using a preliminary first-order correction. The numerical method of Pryce may be used to investigate much more realistic interaction potentials than the simple harmonic and cosine form considered by Budó. These simple forms are, however, sufficient to indicate that the integrated absorption intensity per molecule in the far infrared is very sensitive to dipole-dipole interaction and relative hydrodynamic drag on both interacting groups. Applications of the theory to the disordered solid and nematic phases are considered.     

How Accurate Are Approximate Methods for Evaluating Partition Functions for Hindered Internal Rotations?

The accuracy of several low-cost methods (harmonic oscillator approximation, CT-Comega, SR-TDPPI-HS, and TDPPI-HS) for calculating one-dimensional hindered rotor (1D-HR) partition functions is assessed for a test set of 644 rotations in 104 organic molecules, using full torsional eigenvalue summation (TES) as a benchmark. For methods requiring full rotational potentials, the effect of the resolution at which the rotational potential was calculated was also assessed. Although lower-cost methods such as Pitzer's Tables are appropriate when potentials can be adequately described by simple cosine curves, these were found to show large errors (as much as 3 orders of magnitude) for non-cosine curve potentials. In those cases, it is found that the TDPPI-HS method in conjunction with a potential compiled at a resolution of 60 degrees offers the best compromise between accuracy and computational expense. It can reproduce the benchmark values of the partition functions for an individual mode to within a factor of 2; its average error is just of a factor of 1.08. The corresponding error in the overall internal rotational partition functions of the molecules studied is less than a factor of 4 in all cases. Excellent cost-effective performance is also offered by CT-Comega, which requires only the geometries, energies, and frequencies of the distinguishable minima in the potential. With this method the geometric mean error in individual partition functions is 1.14, the maximum error is a modest 2.98 and the resulting error in the total 1D-HR partition function of a molecule is less than a factor of 5 in all cases.     

The hyperbolic chemical bond: Fourier analysis of ground and first excited state potential energy curves of HX (X = H-Ne)

RHF/aug-cc-pVnZ, UHF/aug-cc-pVnZ, and QCISD/aug-cc-pVnZ, n = 2-5, potential energy curves of H2 X (1) summation g (+) are analyzed by Fourier transform methods after transformation to a new coordinate system via an inverse hyperbolic cosine coordinate mapping. The Fourier frequency domain spectra are interpreted in terms of underlying mathematical behavior giving rise to distinctive features. There is a clear difference between the underlying mathematical nature of the potential energy curves calculated at the HF and full-CI levels. The method is particularly suited to the analysis of potential energy curves obtained at the highest levels of theory because the Fourier spectra are observed to be of a compact nature, with the envelope of the Fourier frequency coefficients decaying in magnitude in an exponential manner. The finite number of Fourier coefficients required to describe the CI curves allows for an optimum sampling strategy to be developed, corresponding to that required for exponential and geometric convergence. The underlying random numerical noise due to the finite convergence criterion is also a clearly identifiable feature in the Fourier spectrum. The methodology is applied to the analysis of MRCI potential energy curves for the ground and first excited states of HX (X = H-Ne). All potential energy curves exhibit structure in the Fourier spectrum consistent with the existence of resonances. The compact nature of the Fourier spectra following the inverse hyperbolic cosine coordinate mapping is highly suggestive that there is some advantage in viewing the chemical bond as having an underlying hyperbolic nature.     

Evaluation of a single-monochromator diode array spectroradiometer for sunbed UV-radiation measurements

The suitability of a new technology single-monochromator diode array spectroradiometer for UV-radiation safety measurements, in particular for sunbed measurements, was evaluated. The linearity, cosine response, temperature response, wavelength scale, stray-light and slit function of the spectroradiometer were determined and their effects on the measurement accuracy evaluated. The main error sources were stray-light and nonideal cosine response, for which correction methods are presented. Without correction, the stray-light may reduce the accuracy of the measurement excessively, particularly in the UV-B range. The expanded uncertainty of the corrected UV measurements is estimated to be 14%, which is confirmed with the comparative measurements carried out with a well-characterized double-monochromator spectroradiometer. The measurement accuracy is sufficient for sunbed measurements, provided that all corrections described above have been done and the user of the instrument has a good understanding of the instrument's operating principles and potential error sources. If these requirements are met, the tested spectroradiometer improves and facilitates market surveillance field measurements of sunbeds.     

Non-perturbative theory of harmonic generation under a high-intensity laser field

We briefly review the theory of harmonic generation. The harmonic distributions with and without the incoherent part are discussed. The time-dependent Schrödinger equation (TDSE) for He with a model potential is solved on a numerical grid. The effects of intermediate-state resonance and of the ionization of He upon the harmonic generation are investigated. The photoelectron spectrum under resonant conditions is obtained in order to verify our arguments of resonant harmonic generation. We also evaluate the contribution of He⁺ when the laser intensity is sufficiently high for most of the atoms to be ionized during the laser pulse. We find that the harmonics with order higher than 13 are due to the ion when the photon energy is 5 eV, while for photon energy 2 eV the atom produces up to about the 49th harmonic.     

Overtone intensities in resonance Raman scattering

A model calculation of resonance Raman scattering tensors has been carried out for a diatomic molecule with a harmonic potential for the ground state and a linear repulsive potential for the excited state. Expressions for scattering tensors have been obtained by using a series of recurrence formulas induced from the Green function of the nuclear hamiltonian of the repulsive potential of the excited state and nuclear wavefunctions for the harmonic potential. The relative scattering intensities of overtones depend on the gradient of the repulsive potential curve and are interpreted in terms of the overlap integrals between nuclear wavefunctions of upper and lower states.     

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.     

On Feasibility of “Spectroscopic” Calculations of XH Bond Dissociation Energies for Polyatomic Molecules from Fundamental Vibration Frequencies Using the Anharmonic Molecular Model

This paper discusses the applicability of the variational technique using a minimal Morse-harmonic basis set to calculations of the fundamental spectrum and the potential function parameters for polyatomic molecules. The potential function is assumed to be the sum of the Morse function for XH bonds and the harmonic function for the skeletal and deformation vibrations. The initial approximation for the potential function is found by ab initio calculations and refined by solving the inverse mechanical problem (selecting the scaling indices). The thus selected harmonic part of the potential function gives equally good agreement between the experimental and calculated transition frequencies in both harmonic and anharmonic approximations. The anharmonic (Morse) term of the potential function (bond dissociation energy) is selected by solving the inverse mechanical problem until the best agreement between the experimental and calculated CH bond stretching frequencies has been achieved. Problem solving ends with the construction of a transmission curve in the IR spectrum. Variations of the dipole moment of the molecule induced by vibrations are found by ab initio calculations.     

Algebraic approximation to the Franck–Condon factors for the Morse oscillator

An algebraic approach is proposed to calculate the Franck–Condon factors for the Morse potential of diatomic molecules. The Morse oscillator is approximated by means of a fourth-order anharmonic oscillator. In the second-quantized formalism, this anharmonic Hamiltonian is diagonalized by way of the Bogoliubov–Tyablikov transformation. The Franck–Condon factors are estimated using the harmonic frequency equivalent and the recurrence relations for the Franck–Condon factors of the harmonic oscillator. Overlap integrals are shown for three band systems and compared with values calculated with an RKR potential. Excellent agreement is achieved.     

Vibrational partition functions for H2O derived from perturbation-theory energy levels

Purely vibrational energy levels and partition functions are calculated using three different potential energy surfaces for the H₂O molecule. Results obtained with perturbation-theory, independent-normal-mode (INM), and harmonic approximations are compared with accurate values. For the cases considered here, the expected improvement that perturbation theory provides over the corresponding harmonic treatment is found to be substantial, while the INM approximation leads to results which are worse than the corresponding harmonic ones. In fact, we show that reliable partition functions for these potential surfaces can be obtained when resonance contributions are removed from the perturbation-theory treatment, and we propose a theoretical criterion for deciding when a particular interaction should be treated as resonant.     

Utilization of shape data in molecular mechanics using a potential based on spherical harmonic surfaces

This article introduces a novel potential function that allows the use of topographical information in molecular modeling. Quantitative shape data are provided by techniques such as electron-microscopy–based three-dimensional image reconstruction for large macromolecular assemblies. Such data can provide important constraints for molecular mechanics. We represent topographical data by spherical harmonic surfaces, first used by Max and Getzoff²¹ for displaying molecular surfaces. A simple harmonic potential is used to constrain atoms within these spherical harmonic surfaces. This potential was implemented in the yammp molecular mechanics package.²⁷ Implementation details and results of several test cases are discussed here. © 1994 by John Wiley & Sons, Inc.     

Nonreactive molecular dynamics force field for crystalline hexahydro-1,3,5-trinitro-1,3,5 triazine

An empirical nonreactive force field has been developed for molecular dynamics (MD)/Monte Carlo simulation of the formation, diffusion, and agglomeration of point defects in the crystal lattice of the alpha modification of hexahydro-1,3,5-trinitro-1,3,5 triazine (RDX) using flexible molecules. Bond stretching and angle bending are represented by Morse and harmonic functions, and torsion by a truncated cosine series. Nonbonded interactions, both inter- and intramolecular, are described by Buckingham potentials separately parametrized. Intermolecular electrostatic interactions are treated via a Coulomb term coupled with a smooth 15.0 A cutoff radius. Parameters were taken in part from earlier published works and were determined partly by fitting to known molecular and crystal properties of RDX. In MD simulations at constant pressure and temperature, the model was able to stabilize and maintain the correct crystal structure, symmetry, and molecular conformation of alpha-RDX. Vibrational frequencies, lattice binding energy and dimensions, coefficients of thermal expansion, and several unusually short intermolecular distances are all reproduced in satisfactory agreement with experimental data.     

Trapping of a particle in a short-range harmonic potential well

Eigenstates of a particle in a localized and unconfined harmonic potential well are investigated. Effects due to the variation of the potential parameters as well as certain results from asymptotic expansions are discussed.     

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.     

Theory of multidimensional wavepacket propagation

Ehrenfest's theorem implies that a gaussian remains a gaussian when propagated in a general (time-dependent and multidimensional) harmonic potential. We shall prove that the statement remains true with the replacement of a gaussian with a harmonic oscillator. For the one-dimensional case, this is implicit in the work of Meyer. Here, we prove it more generally for the multidimensional case. A complete, orthonormal, evolving basis of harmonic oscillator wavefunctions can then be constructed by using a local, time-dependent, harmonic approximation to the potential. An evolving wavepacket in the actual potential can be expanded in the basis set, and the coefficients of the expansion obey a set of coupled, linear, first-order differential equations. The theory has practical applications for processes such as Raman scattering, photodissociation, and other time-dependent processes that can benefit from multidimensional wavepacket propagation.     

New torsion potential expression for molecules without rotational symmetry

A new torsion potential function for bond rotations without rotational symmetry is proposed. This function is composed of a few Gaussian-type terms each corresponding to an eclipsed conformation of the 1,2 substituents of the C-C bonds. Different from the truncated Fourier series or the truncated cosine polynomial, it is easy to determine how many terms are needed to represent any type of torsion potential barrier at a glance using the Gaussian-type function. It could also intuitively deduce the physical meaning of the expansion parameters of the new torsion potential function, which corresponds to the barrier height, the dihedral defining the eclipsed conformations, and the size of the substituents, respectively. The new torsion potential function is also applied to the 1, 2-substituted haloethanes with satisfactory results, where three Gaussian-type terms corresponding to the fully eclipsed and the partially eclipsed conformations are needed.     

Franck–Condon factors for diatomic molecules with anharmonic corrections

Some of the band systems of several astrophysically important molecules are calculated and compared with the results obtained by calculations based on realistic Klein–Dunham and Rydberg–Klein–Rees potential functions. The Morse potential is approximated by means of a fourth-order anharmonic oscillator model. In the second-quantized formalism, the anharmonic Hamiltonian is diagonalized by using the Bogoliubov–Tyablikov transformation. The diagonalization process gives a shift in the frequency associated with each normal mode of harmonic vibration of the molecules presented here. The Franck–Condon factors are estimated using this new frequency within the framework of a harmonic oscillator.