# Showing papers in "Molecular Physics in 1970"

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TL;DR: In this paper, a direct difference method for the computation of molecular interactions has been based on a bivariational transcorrelated treatment, together with special methods for the balancing of other errors.

Abstract: A new direct difference method for the computation of molecular interactions has been based on a bivariational transcorrelated treatment, together with special methods for the balancing of other errors. It appears that these new features can give a strong reduction in the error of the interaction energy, and they seem to be particularly suitable for computations in the important region near the minimum energy. It has been generally accepted that this problem is dominated by unresolved difficulties and the relation of the new methods to these apparent difficulties is analysed here.

19,483 citations

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TL;DR: In this article, a unified treatment of non-radiative decay processes in large molecules which involve either electronic relaxation between two electronic states or unimolecular rearrangeme(s) is presented.

Abstract: In this paper we present a unified treatment of non-radiative decay processes in large molecules which involve either electronic relaxation between two electronic states or unimolecular rearrangeme...

2,060 citations

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TL;DR: In this article, the authors derived the correct quantum-mechanical form for the vibration-rotation bamiltonian of a linear molecule by an extension of the method of Wilson and Howard.

Abstract: The correct quantum-mechanical form for the vibration-rotation bamiltonian of a linear molecule is derived by an extension of the method of Wilson and Howard. The absence of the third rotational degree of freedom complicates the relations obeyed by the angular momentum components in this hamiltonian. Calculations of energy levels can be simplified by transforming to the ‘isomorphic’ hamiltonian introduced by Hougen. The application of this isomorphic transformation to other operators is illustrated with the dipole moment operator.

328 citations

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TL;DR: In this article, the exchange of protons between sites with different chemical shift is studied by means of N.M. relaxation time measurements (T 1ρ ) in the presence of an r.f. magnetic field.

Abstract: The exchange of protons between sites with different chemical shift is studied by means of N.M.R. relaxation time measurements (T 1ρ ) in the presence of an r.f. magnetic field. The system studied is chair-to-chair isomerization in cyclohexane. The exchange rate and chemical shift are measured in the temperature range 215°k to 250°k. The method is compared with other N.M.R. techniques for studying chemical exchange and is found to have advantages particularly at high exchange rates.

248 citations

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TL;DR: In this paper, the Percus-Yevick hard-sphere radial distribution function, g(R), for R ˇ-5σ is analyzed in terms of complex variables.

Abstract: Explicit analytical expressions, written in terms of complex variables and suitable for rapid computer evaluation, are presented for the Percus-Yevick hard-sphere radial distribution function, g(R), for R ˇ- 5σ. Some effects of truncating g(R) to unity past R = 5 σ are discussed.

213 citations

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191 citations

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TL;DR: In this article, the vibrational components of the NH3 bands were calculated from vibrational functions appropriate to the best one-dimensional potential curves for the inversion mode, and the experimental data were fitted by an expression: the Franck-Condon approximation in which the Q 2 matrix element is neglected gives a poor fit to the data.

Abstract: The intensities of the vibrational components of the A ← X and B ← X band systems of NH3 have been calculated from vibrational functions appropriate to the best one-dimensional potential curves for the inversion mode. For the A ← X system the experimental data is fitted by an expression: The Franck-Condon approximation in which the Q 2 matrix element is neglected gives a poor fit to the data. We deduce that the electronic transition moment increases as the planar molecule is bent. For the B ← X system, which is a forbidden electronic transition, the experimental data are fitted well by the matrix element 2 which arises from the leading term in the Taylor expansion of the electronic transition moment. Using the same potential functions for ND3, and a reduced mass based on a pure bending mode, gives good agreement for the intensities of the corresponding ND3 bands. New experimental values for the extinction coefficients of the A ← X system have been obtained.

160 citations

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TL;DR: In this article, the E.S.R. spectra of substituted ferrocenes have been measured at 20° and 77°k, respectively, and it has been shown that Fe(C5H5)2 + has the orbitally degenerate 2 E 2g [(e 2g )3(a 1g 2 ] ground state configuration.

Abstract: E.S.R. spectra of cations of substituted ferrocenes have been measured at 20° and 77°k. The g values are highly anisotropic, with g ¶ = 4·35 and g⊥=1·26 for Fe(C5H5)2 +. Owing to a fast spin-lattice relaxation the E.S.R. signals are much broader and less intense at 77° than at 20°k. A theoretical analysis shows that Fe(C5H5)2 + has the orbitally degenerate 2 E 2g [(e 2g )3(a 1g 2 ] ground-state configuration. The combined action of the spin-orbit interaction H so and a low-symmetry perturbation H 1 splits the fourfold degenerate level into two Kramers doublets. Since H so and H 1 are of the same order of magnitude, the orbital momentum contribution to the g values is only partly quenched and the g values deviate strongly from the free-electron value. The predicted splitting of the Kramers doublets is small and this explains the fast relaxation. The e 2g molecular orbitals are essentially localized on the iron atom. Substitution influences primarily the symmetry of the complex; it has no great influence on...

137 citations

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TL;DR: In this article, the authors derived analytical predissociation linewidths from the scattering solutions to the two-state curve crossing problem, and showed that linwidths in each case take the form Γ = Γo cos2 η, where Γ 0 is related to the Landau-Zener transition probability and η is a phase integral between one of the classical turning points of the known bound state curve, and the turning point of the continuum potential.

Abstract: Analytical predissociation linewidths are derived from the scattering solutions to the two-state curve crossing problem. Four cases are considered: ‘inner’ and ‘outer’ crossings in the ‘weak’ or ‘strong’ coupling limits. It is shown that linewidths in each case take the form Γ = Γo cos2 η, where Γ0 is related to the Landau-Zener transition probability and η is a phase integral between one of the classical turning points of the known bound state curve, and the turning point of the continuum potential. Γ0 is a decreasing function of energy in the weak interaction limit, whereas for strong interaction, it is an increasing function of energy. The outer crossing, strong coupling case is shown to provide an interpretation of the characteristic appearance of the visible spectra of ICl and IBr.

115 citations

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TL;DR: In this paper, general harmonic and hybrid orbital force fields have been calculated for methyl chloride, bromide and iodide using the best available frequency, Coriolis coefficient, and centrifugal distortion data, including the recent 13C frequencies for the chloride and ionide.

Abstract: General harmonic and hybrid orbital force fields have been calculated for methyl chloride, bromide and iodide using the best available frequency, Coriolis coefficient, and centrifugal distortion data, including the recent 13C frequencies for the chloride and iodide. Anharmonicity corrections were made to the fundamental frequencies using Dennison's rule and x values of 0·04 for CH stretching, 0·02 for CH bending, and 0·01 for CX stretching modes. These were the most reasonable values which did not seriously over-compensate for the product rule deficiencies in the 12C and 13C frequencies. The 13C and CD3X frequency information was input in the form of frequency shifts from the 12CH3X frequencies. All 12 parameters in the most general harmonic force fields are determined with significance for the first time, although the previously uncertain parameters F12 and F13 are only determined with good precision when the 13C frequency data are utilized in addition to the other information. The signs of F12, F45 and ...

110 citations

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TL;DR: The theory of rotational-pucker-vibrational transitions in the vibrational spectrum of cyclobutane is reviewed in this article, in terms of two slightly different puckering potential functions for the ground and the excited vibrational states.

Abstract: The theory of rotational-pucker-vibrational transitions in the vibrational spectrum of cyclobutane is reviewed. Puckering sideband structure on the 1453 cm-1 v 14 infra-red fundamental of C4H8 has been observed and analysed, in terms of two slightly different puckering potential functions for the ground and the excited vibrational states. The results have been fitted to quartic-quadratic potential functions in the puckering coordinate, with a barrier to inversion of 503 cm-1 (1·44 kcal mole-1 = 6·02 kJ mole-1) in the ground state and 491 cm-1 in the excited state ν 14 = 1. For reasonable assumptions about the reduced mass, the equilibrium dihedral angle of the C4 ring is determined to be about 35°, in agreement with previous estimates. Ueda and Shimanouchi's observations on the 2878 cm-1 C4H8 band have been re-analysed, and puckering sidebands have also been observed and analysed for the 1083 cm-1 v 14 infra-red fundamental of C4D8. Pure puckering transitions have been observed in the Raman spectrum of C4...

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TL;DR: In this paper, the rotational constants of the first singlet excited state of five monosubstituted and five p-disubstitized benzenes were compared.

Abstract: The results of a comparison of the rotational constants of the first singlet excited state of five monosubstituted and five p-disubstituted benzenes show that (a) In the monosubstituted benzenes, it is reasonable to assume a common model in which, on electronic excitation, all the bonds in the ring expand as in the 1 B 2u state of benzene but that the carbon-substitutent bond contracts, the contraction being in the order N > O ≈ Br ≈ Cl > F and of approximate magnitudes 0·08, 0·04, 0·04, 0·04, 0·02 A respectively. There is also a tendency for the ring angle adjacent to the substituent to open by a few degrees. (b) In the p-disubstituted benzenes there is a slight shortening of the molecules along the long axis compared to the geometry of an excited state in which the two ends of the molecule and the C2C3 bond behave as in the corresponding monosubstituted benzenes. This effect is almost independent of the substituents, except in the case of -NH2 where it is largest, and probably results from the ring diff...

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TL;DR: The Gibbs-Bogoliubov inequality is used to develop a first-order perturbation theory that provides an upper bound on the free energy.

Abstract: The Gibbs-Bogoliubov inequality is used to develop a first-order perturbation theory that provides an upper bound on the free energy. Charged systems as well as a system of Lennard-Jones particles are discussed, and detailed numerical estimates of the bounds are presented.

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TL;DR: The determination of expressions for derivatives of the SCF energy with regard to both linear and non-linear parameters in the LCAO formulation enables powerful minimization methods to be used to derive optimum values of these parameters as mentioned in this paper.

Abstract: The determination of expressions for derivatives of the SCF energy with regard to both linear and non-linear parameters in the LCAO formulation enables powerful minimization methods to be used to derive optimum values of these parameters

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TL;DR: In this article, the paramagnetic term of the nuclear magnetic shielding constant, σ p, was calculated from the chemical shift of nuclear resonance, which provides a general method for estimating spin-rotation interaction constants.

Abstract: Spin-rotation interaction constants, c, are calculated from the paramagnetic term of the nuclear magnetic shielding constant, σ p, using a relationship originally due to Ramsey. Calculated values show excellent agreement with experimental determinations from molecular beam measurements. Since σ p can be easily estimated from the chemical shift of the nuclear resonance this provides a general method for estimating spin-rotation interaction constants. Chemical shift anisotropies allow the components c ⊥ and c ‖ of the spin-rotation interaction tensor to be determined. Generally both components have the same sign and are of similar magnitude. The r.m.s. value of the spin-rotation interaction constant, required to calculate nuclear spin-lattice relaxation times, is not expected to differ appreciably from average values obtained from molecular beam measurements or magnetic shielding. Values for the molecular angular velocity correlation time, τ sr, calculated directly from nuclear spin-lattice relaxation times...

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TL;DR: In this article, the symmetry groups of non-rigid molecules are formulated as products of subgroups of invariant torsional subgroups and cyclic groups, and a correlation table is drawn up linking the representations of the representations with those of the molecular symmetry group.

Abstract: The possibility of formulating the symmetry groups of non-rigid molecules (as defined by Longuet-Higgins) as products of subgroups is investigated. For the most general case of non-rigidity, no such formulation is possible. When the non-rigidity results from internal rotation about one or more axes, the molecular symmetry group may be written as a semi-direct product of an invariant torsional subgroup, which is a direct product of cyclic groups, and a ‘frame subgroup’, which is isomorphous with the point group of the ‘frame’ formed by detaching all the nuclei undergoing internal rotation. A correlation table may be drawn up linking the representations of the torsional and frame subgroups with those of the molecular symmetry group. This table may be used to classify the various factors of the molecular wave-function, without the need to perform the tedious construction of the molecular symmetry group. The method is extended to molecules containing rotating groups which in turn contain more rotating groups ...

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TL;DR: In this article, the McWeeny ring current theory is tested against a set of consistent, accurate experimental chemical shifts for 85 protons in 16 planar unsubstituted condensed benzenoid hydrocarbons, and a regression equation is derived relating quantities calculated from the theory to experimental τ values for the nonhindered protons.

Abstract: The McWeeny ‘ring current’ theory is tested critically against a set of consistent, accurate experimental chemical shifts for 85 protons in 16 planar unsubstituted condensed benzenoid hydrocarbons, and a regression equation is derived relating quantities calculated from the theory to experimental τ values for the non-hindered protons. Systematic deviations of different types of protons from this line are discussed, as also are sigma-bond anisotropy effects and discrepancies observed for overcrowded protons. Some of the assumptions about the geometry of planar benzenoid hydrocarbons normally made in these calculations are relaxed. The McWeeny theory gives a good account of the chemical shifts of non-hindered protons in planar molecules, and the predictions of the theory are not significantly altered if the calculations are based on experimental x-ray geometry. Sigma-bond anisotropy effects are unimportant for non-hindered protons, and, at most, make only a partial contribution to the down-field shifts of h...

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TL;DR: In this article, it was shown that scaled particle theory is not thermodynamically consistent for mixtures of different shapes and sizes of hard spheres, cubes, tetrahedra, ellipsoids and cylinders.

Abstract: The excess volume and Gibbs free energy predicted by the scaled particle theory have been calculated for five binary systems of hard spheres, cubes, tetrahedra, ellipsoids and cylinders where the particles of the mixtures have the same shape but different sizes. At a fixed pressure, the excess volume was dependent on the shape of the particles involved, the temperature, pressure and size ratio of the particle. There appeared to be no equivalent sphere which could describe the equation of state and the mixing properties of the systems of non-spherical particles. The difficulties of extending the scaled particle theory to mixtures of different shaped particles are discussed and it is shown that the scaled particle theory is not thermodynamically consistent for these systems.

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TL;DR: In this paper, the angular distribution and intensity of photoelectrons ejected from diatomic molecules are considered and the potential information contained in rotational fine structure in molecular photoelectron spectroscopy is investigated.

Abstract: A previously developed theory of the angular distribution and intensity of photoelectrons ejected from diatomic molecules is used to consider the potential information contained in rotational fine structure in molecular photoelectron spectroscopy. It is shown that resolution of such structure would permit the evaluation of contributions corresponding to the terms in a one-centre expansion of the molecular orbital from which the photoelectron is ejected. The case of H2 is analysed in detail, and the analysis compared with that of Tully, Berry and Dalton.

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TL;DR: In this article, a complete hamiltonian for a translating, rotating, vibrating molecule in the presence of a constant external electromagnetic field is derived starting with the Breit equation reduced to a non-relativistic form, correct to order c -2.

Abstract: A complete hamiltonian for a translating, rotating, vibrating molecule in the presence of a constant external electromagnetic field is derived starting with the Breit equation reduced to a non-relativistic form, correct to order c -2. A number of new terms appear, which have not been obtained in less precise derivations. These include mass polarization corrections to the orbital Zeeman, spin-orbit and orbit-orbit interactions together with spin-vibration, orbit-vibration and orbit-rotation interactions. In addition there is a vibrational Zeeman interaction which in certain circumstances may be of the same order of magnitude as rotational Zeeman terms. The final hamiltonian provides a starting point for future investigations of the possible effects of these new interactions on the microwave and radiofrequency spectra of both open and closed-shell molecules.

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TL;DR: The physical properties of a fluid in which there is only a two-body potential, uαβ, can be expressed in terms of the total correlation function, h 12, which is a sum of all connected graphs with root-points on molecules 1 and 2, whose links are f bonds, where fαβ = exp (-uαβ/κT) -1.

Abstract: The physical properties of a fluid in which there is only a two-body potential, uαβ , can be expressed in terms of the total correlation function, h 12, which is a sum of all connected graphs with root-points on molecules 1 and 2, whose links are f bonds, where fαβ = exp (-uαβ/κT) - 1. It is shown that the total correlation function in the presence of a weak three-body potential, uαβγ , is h 12*, where h 12* is the sum of all two-body connected graphs in which each f bond is replaced in turn by an f* bond, where and where ✶ is a sub-set of the elementary graphs each of which contains one fαβγ link. We call this sub-set the line-irreducible graphs, and its leading term is a graph discussed by Rushbrooke and Silbert. The three-body potential is set equal to the dipole-dipole-dipole potential of Axilrod and Teller, and the analytic properties and numerical values of the first term ✶1 examined in detail. Other effective potentials have been defined and the relations between them are elucidated. In particular ...

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TL;DR: In this paper, the radiative lifetimes of the ππ′ phosphorescent state of quinoxaline and a number of aromatic hydrocarbons were calculated using an atoms in molecules approach.

Abstract: A calculation is presented of the radiative lifetimes of the ππ′ phosphorescent state of quinoxaline and a number of aromatic hydrocarbons. For the matrix elements of the spin-orbit hamiltonian an ‘atoms in molecules’ approach is used and only the one-electron one-centre terms in the expansion of these matrix elements in integrals over AO's are retained. This approach is similar to that used by Goodman and Krishna for the nπ′ states of some aza-aromatics. For quinoxaline a lifetime τ=0·33 s is obtained. For aromatic hydrocarbons the phosphorescence is attributed to ‘intensity stealing’ from π→σ′ and σ→π′ transitions in which an electron is transferred between the σ and π system, in agreement with the out-of-plane polarization of phosphorescence observed experimentally. It is found that interactions involving the sigma orbitals of the C-C bonds are an order of magnitude more effective than those involving C-H orbitals. The calculated radiative lifetimes are: benzene 23·2 s (77 s), naphthalene 3·3 s (11 s),...

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TL;DR: In this article, self-consistent field molecular orbitals are used to describe two interacting conjugated molecules with overlapping p orbitals, and the total interaction energy is obtained as an expansion in powers of overlap Srr between interacting atomic orbitals and of the net electronic charge Δ qr on the atoms.

Abstract: Self-consistent field molecular orbitals are used to describe two interacting conjugated molecules with overlapping p orbitals. Perturbation theory is used to determine intermolecular orbitals which could serve as starting point for a self-consistent field calculation for the composite system. The total interaction energy is obtained as an expansion in powers of the overlap Srr between interacting atomic orbitals and of the net electronic charge Δ qr on the atoms. The energy expression contains an important term for net charge interactions. Expressions are also obtained when one of the interacting molecules is excited. Configuration interaction is examined in this case and the behaviour of singlets and triplets compared.

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TL;DR: In this article, the authors used the penetration part of the two-electron coulomb integral to estimate the exchange energy of two hydrogen atoms in 2s, 2p or hybrid valence states.

Abstract: Several methods of estimating exchange energies have been tested for the interaction of two hydrogen atoms in 2s, 2p or hybrid valence states. The simplest relationship previously used, X = KS 2/R, does not give an accurate picture of the dependence on internuclear distance. A two-term expression X = S 2(AR -1 + BR -2) is considerably better but fails for the case of exchange between orthogonal orbitals (S = 0). An alternative expression in which C 0 - C is the penetration part of the two-electron coulomb integral, is just as accurate, contains no more constants and holds equally well for orthogonal and non-orthogonal orbitals. C 0 - C is a better measure of the net overlap of two orbitals than is the overlap integral S, because the latter is zero for orbitals which are orthogonal but nevertheless overlap. We have found no method of deducing the constants K, A or B other than by calculating the exchange energy exactly at one or two values of R.

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TL;DR: In this paper, a quantum theory for the differential and total elastic and reactive scattering cross sections for an electronically adiabatic bimolecular exchange reaction, with the restriction that the three atoms are constrained to move on a straight line, but with the whole system free to rotate in three dimensions.

Abstract: A quantum mechanical theory is developed for the differential and total elastic and reactive scattering cross sections for an electronically adiabatic bimolecular exchange reaction, with the restriction that the three atoms are constrained to move on a straight line, but with the whole system free to rotate in three dimensions. The introduction of a set of natural collision coordinates, together with a vibrationally adiabatic approximation, is used to reduce the scattering problem to the solution of one-dimensional Schrodinger equations. Semi-classical expressions for the elastic and reactive phase shifts are derived. The partial wave summations that occur in the theory are evaluated by semi-classical techniques and elastic and reactive differential cross sections are calculated for three different kinds of potential surface. A feature of the calculations is the appearance of a new kind of rainbow, which is named a ‘cubic’ rainbow since it arises when the deflection function varies cubically with impact p...

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TL;DR: In this paper, the 0-0 band of a system of p-difluorobenzene has been rotationally analysed by the method of band contour analysis, and it has been shown to be a type B band requiring a 1 B 2u -1 Ag electronic assignment.

Abstract: The 0–0 band of the 2710 a system of p-difluorobenzene has been rotationally analysed by the method of band contour analysis. It has been shown to be a type B band requiring a 1 B 2u –1 Ag electronic assignment. The rotational origin is at 36837·9 ± 0·2 cm-1. Interpretation of the excited state rotational constants indicates a contraction of the C-F bonds compared with the ground state and a ring geometry which differs only slightly from that of the 1 B 2u state of benzene. There is no evidence for non-planarity of the excited state.

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TL;DR: The origin of the 2700 A system of chlorobenzene has been rotationally analyzed by the asymmetric rotor band contour method in this paper, and it is shown to be a type B band of a prolate asymmetric top which requires the electronic assignment of the system to be 1 B 2-1 A 1.

Abstract: The origin band of the 2700 A system of chlorobenzene has been rotationally analysed by the asymmetric rotor band contour method. It is shown to be a type B band of a prolate asymmetric top which requires the electronic assignment of the system to be 1 B 2-1 A 1. The excited state rotational constants are: A′=0·1797 ± 0·0001 cm-1, B′=0·05230 ± 0.00002 cm-1, C′=0.040512 ± 0.000007 cm-1. These can be interpreted in terms of a contraction of the C-Cl bond of about 0·04 A and an expansion of the benzene ring similar to that in the 1 B 2u excited state of benzene. The origin of the 2700 A band is shown to be at 37 048·2 ± 0·2 cm−1. Strong type A bands involving the b 2 vibration v 29 are also assigned from their rotational contours. The 291 level is shown to be in Fermi resonance with the 141201 level. Of the eight prominent sequence intervals associated with the origin band, five are interpreted with considerable certainty.

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TL;DR: In this paper, the authors considered the He-He, He-H(1s), H-H (1s) and H-1s)-H(2s interactions as model systems for studying how charge overlap effects in second-order dispersion energies vary as a function of the nature of the interacting species.

Abstract: The He-He, He-H(1s), H(1s)-H(1s) and H(1s)-H(2s) interactions are considered as model systems for studying how charge overlap effects in second-order dispersion energies vary as a function of the nature of the interacting species. The non-expanded second-order energy and the corresponding multipole R -1 expansions, through all powers of R -1, are obtained for each interaction using Unsold's average energy approximation. The results are used to discuss the limitations of the usefulness of the R -1 expansions.

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TL;DR: The gas-phase electron resonance spectrum of the CF radical in its X 2 π 3/2 electronic state is described in this paper, which gives values of the rotational constant B 0 and fine structure constant A which are in good agreement with those previously obtained from the ultra-violet emission spectrum.

Abstract: The gas-phase electron resonance spectrum of the CF radical in its X 2 π 3/2 electronic state is described. Analysis of the spectrum gives values of the rotational constant B 0 and fine structure constant A which are in good agreement with those previously obtained from the ultra-violet emission spectrum. The axial component of the 19F magnetic hyperfine interaction is derived, and significant contributions from the non-axial components are discussed. The electric dipole moment of CF is obtained from measurements of the first-order Stark effect.