Ian Mills

Anharmonic force constant calculations

The relationship of the anharmonic force constants in curvilinear internal coordinates to the observed vibration-rotation spectrum of a molecule is reviewed. A simplified method of setting up the required non-linear coordinate transformations is described: this makes use of an L tensor, which is a straightforward generalization of the L matrix used in the customary description of harmonic force constant calculations. General formulae for the L tensor elements, in terms of the familiar L matrix elements, are presented. The use of non-linear symmetry coordinates and redundancies are described. Sample calculations on the water and ammonia molecules are reported.

Quantities, units, and symbols in physical chemistry

Historical introduction Physical quantities and units Tables of physical quantities Definitions and symbols for units Recommended mathematical symbols Fundamental physical constants Conversion of units References Greek alphabet Index of symbols Pressure conversion factors Energy conversion factors.

Force Constants and Dipole‐Moment Derivatives of Molecules from Perturbed Hartree–Fock Calculations. I

General expressions for the force constants and dipole‐moment derivatives of molecules are derived, and the problems arising in their practical application are reviewed. Great emphasis is placed on the use of the Hartree–Fock function as an approximate wavefunction, and a number of its properties are discussed and re‐emphasised. The main content of this paper is the development of a perturbed Hartree–Fock theory that makes possible the direct calculation of force constants and dipole‐moment derivatives from SCF–MO wavefunctions. Essentially the theory yields ∂φi / ∂RJα, the derivative of an MO with respect to a nuclear coordinate.

Coriolis interactions about X-Y axes in symmetric tops

Selection rules and matrix elements are derived for Coriolis interactions between vibrational levels due to rotation about (x, y) axes in symmetric top molecules. The theory is developed in detail for the case of interaction between an A1 and an E species vibrational level in a C3v molecule; perturbations to both the positions and the intensities of the rovibration transitions in the spectrum are considered. A computer program has been written which calculates exactly the perturbed spectrum of two interacting rovibration bands according to this model, the results being presented directly by a graph plotter connected to the computer. This has been used to interpret perturbations observed in two pairs of interacting fundamentals in the spectrum of CH3F (ν2 - ν5 and ν3 - ν6) and one pair in CD3Cl (ν2 - ν5). The resulting analysis of the observed spectrum leads to new values for some vibration-rotation interaction constants and also leads to a unique determination of the sign relationship between the dipole moment derivatives in each pair of interacting normal vibrations. These sign relations are summarized in Figs. 8, 12, and 15.

On the relationship of normal modes to local modes in molecular vibrations

A simple model for the effective vibrational hamiltonian of the XH stretching vibrations in H2O, NH3 and CH4 is considered, based on a morse potential function for the bond stretches plus potential and kinetic energy coupling between pairs of bond oscillators. It is shown that this model can be set up as a matrix in local mode basis functions, or as a matrix in normal mode basis functions, leading to identical results. The energy levels obtained exhibit normal mode patterns at low vibrational excitation, and local mode patterns at high excitation. When the hamiltonian is set up in the normal mode basis it is shown that Darling-Dennison resonances must be included, and simple relations are found to exist between the xrs , gtt , and Krrss anharmonic constants (where the Darling-Dennison coefficients are denoted K) due to their contributions from morse anharmonicity in the bond stretches. The importance of the Darling-Dennison resonances is stressed. The relationship of the two alternative representations of...

Rotational Analysis of the 2600 angstrom Absorption System of Benzene

The 2600A absorption system of benzene has been examined with very high resolving power, and the rotational fine structure partially analysed by comparison with computed contours. Vibronic bands involving degenerate e2g vibrations have contours characteristic of the vibrational angular momentum, and Coriolis coefficients £ have been determined for the lowest e2g vibrations in ground and B2u excited states. Band contours thus provide an additional criterion for checking vibrational assignments. In particular, values of £ determine the separation between band maximum and origin which can thus be calculated, and the vibrational analysis consequently refined in certainty and precision. Improved values of several fundamentals have been obtained, some new vibrational assignments made, and some previous ones rejected. Some important anharmonic constants have also been obtained. Vibronic relative intensities are briefly discussed. The rotational and vibrational evidence together make it certain that the equilibrium configurations of the carbon skeleton in benzene are exactly planar and hexagonal in both the ground and excited states, point group Z)6A. The rotational constants then give an estimate of the increase in G—C distance on excitation of + 0*038 A, in excellent agreement with estimates from other sources. The electronic origin of the system is revised: T00 = 38086*1 cm

The calculation of force constants and normal coordinates—IV XH4 and XH3 molecules

Abstract Normal coordinate calculations of XH 4 and XH 3 molecules are reviewed and discussed. It is shown that for most of these molecules the true values of the force constants in the most General Harmonic Force Field can be uniquely determined only by making use of vibration-rotation interaction constants. It is emphasized that without these extra data the GFF is not determined. The results are compared with various model force fields for these molecules.

Redefinition of the kilogram, ampere, kelvin and mole: a proposed approach to implementing CIPM recommendation 1 (CI-2005)

The International System of Units (SI) is founded on seven base units, the metre, kilogram, second, ampere, kelvin, mole and candela corresponding to the seven base quantities of length, mass, time, electric current, thermodynamic temperature, amount of substance and luminous intensity. At its 94th meeting in October 2005, the International Committee for Weights and Measures (CIPM) adopted a recommendation on preparative steps towards redefining the kilogram, ampere, kelvin and mole so that these units are linked to exactly known values of fundamental constants. We propose here that these four base units should be given new definitions linking them to exactly defined values of the Planck constant h, elementary charge e, Boltzmann constant k and Avogadro constant NA, respectively. This would mean that six of the seven base units of the SI would be defined in terms of true invariants of nature. In addition, not only would these four fundamental constants have exactly defined values but also the uncertainties of many of the other fundamental constants of physics would be either eliminated or appreciably reduced. In this paper we present the background and discuss the merits of these proposed changes, and we also present possible wordings for the four new definitions. We also suggest a novel way to define the entire SI explicitly using such definitions without making any distinction between base units and derived units. We list a number of key points that should be addressed when the new definitions are adopted by the General Conference on Weights and Measures (CGPM), possibly by the 24th CGPM in 2011, and we discuss the implications of these changes for other aspects of metrology.

ℓ-Resonance perturbations in infrared perpendicular bands

Abstract The effects of l-type resonance on rovibrational bands in infrared spectra are reviewed. Observed spectra are compared with computer-simulated spectra obtained by solving the Hamiltonian matrix numerically and calculating the true (perturbed) wavenumber and intensity of each line in the band. The most obvious effects in the spectra are shown to result from intensity perturbations rather than line-shifts; in oblate symmetric tops the Q branch structure near the band center may show anomalies due to l-resonance even at quite low resolution and even when the accidental resonance is not very exact. Numerical values of l-doubling constants are obtained for several cyclopropane bands by comparing observed band contours at about 0.2-cm −1 resolution with computed contours. Although the constants are not determined with great precision, the sign of the l-doubling constants is determined unambiguously.

The calculation of force constants and normal co-ordinates—II: Methyl fluoride

Abstract The mathematical difficulties which can arise in the force constant refinement procedure for calculating force constants and normal co-ordinates are described and discussed. The method has been applied to the methyl fluoride molecule, using an electronic computer. The best values of the twelve force constants in the most general harmonic potential field were obtained to fit twenty-two independently observed experimental data, these being the six vibration frequencies, three Coriolis zeta constants and two centrifugal stretching constants D J and D JK , for both CH 3 F and CD 3 F. The calculations have been repeated both with and without anharmonicity corrections to the vibration frequencies. All the experimental data were weighted according to the reliability of the observations, and the corresponding standard errors and correlation coefficients of the force constants have been deduced. The final force constants are discussed briefly, and compared with previous treatments, particularly with a recent Urey-Bradley treatment for this molecule.