Diatomic Model Research Articles

Plasmon versus shake-up satellites in XPS spectroscopy of adsorbed species

Shake-up satellites found in photoemission spectroscopy from core levels of adsorbed atoms are considered using a simple diatomic molecule model. Time scales and the competition between surface plamons and shake-up for the satellite strength are considered.

A method for merging the results of separate least-squares fits and testing for systematic errors

A method is presented for merging the results of separate least-squares fits to obtain the most precise, single values for each of the molecular constants of a spectroscopic system. The output molecular constants and accompanying variance-covariance matrices from each of the separate fits are taken together as input to a correlated least-squares fit, which then yields the desired merged single values. The method is applied to a 1Σ- 1Σ diatomic model band system using well-defined synthetic data as an example. Other currently used, multistate methods of obtaining molecular constants for a spectroscopic system are special cases of the method described here. For the nonlinear least-squares fits required for multiplet states, the merge method is particularly advantageous. The properties of the merged values of the molecular constants and their standard errors are examined with regard to forming confidence limits for these estimates and it is shown that, in most spectroscopic applications, the confidence limits have their usual meaning. Furthermore, it is similarly shown that the confidence limits for the variance of the merged fit provide a simple and convenient test for the presence of relative systematic errors between the separate least-squares results being merged. Although the method is illustrated here by merged separate band-by-band results, it also can be applied to any type of spectroscopic results.

A new criterion for the mixed crystal behaviour in the diatomic linear chain model

Lucovsky, Brodsky, Burstein (LBB) have studied the behaviour of mixed crystals by setting up a criterion for the existence of local mode frequencies in real crystals starting from a diatomic linear chain model. This, while successfully predicting the one and two mode behaviour for some systems fails to predict the mixed mode behaviour. We propose a similar criterion for the existence of gap modes, by demanding that the gap mode predicted by the diatomic linear chain model should lie within the gap of the real three dimensional solid for its existence. It is shown that the gap modes for various systems calculated using this criterion are in reasonable agreement with the experimental values. The infrared behaviour of mixed crystals has to be determined by examining the existence of local as well as gap modes for the two end members of the system. This generalized new criterion successfully predicts the mixed mode behaviour of III–V mixed crystals besides predicting the one and two mode behaviour, observed in infrared absorption of mixed alkali halioes and III-V compounds.

Stark spectroscopy with the CO 2 laser: The ν3 band of CD 3F

The ν 3 band of CD 3F has been studied using coincidences with the 10.4 μm band of the CO 2 laser. The majority of the 150 resonances measured have been studied using the Lamb dip technique. These resonances have been analyzed, together with recent microwave results, to give the following vibration-rotation parameters and dipole moments in the ν 3 state. 12CD 3F ν 0 992.29882 (19) cm −1 B 20395.77 6 ± 0.08 MHz A-A 0 −35.73 ± 0.61 MHz D J 33.8 ± 0.5 kHz D JK 203.9 ± 7 kHz μ ν3 1.8964 ± 0.0015 D μ ν3 − μ 0 0.02617 ± 0.0005 D The change in dipole moment on excitation of ν 3 is much smaller than in 12CH 3F and 13CH 3F and cannot be explained by the diatomic model used previously. The model has been extended to allow for the change in ∂μ ∂q 3 caused by the increased mixing of ν 2 and ν 3 in the deuteride, and shows that this is the primary cause of the small dipole moment change.

Synthesis and structure of ammines of beryllium and magnesium borohydrides

1. It was shown by the method of IR spectroscopy and x-ray powder patterns that magnesium borohydride hexaammine and beryllium borohydride tetraammine have a “salt-like” structure, and their crystal lattices are constructed from the ions [Mg(NH3)6]2+, [Be(NH3)4]2+, and BH 4 − . 2. The parameters of the rhombic cells of [Mg(NH3)6](BH4)2 and [Be(NH3)4](BH4)2 were determined. 3. In the approximation of a “diatomic model” the valence force constants of the Be-N and Mg-N bonds were calculated, and the interatomic distances Be-N and Mg-N in the cations [Be(NH3)4]2+ and [Mg(NH3)6]2+ were determined.

The use of a non-spherical reference potential in statistical mechanical perturbation theory

A statistical mechanical perturbation theory for the pair correlation function and thermodynamic properties of molecular fluids is presented in which the reference potential function is non-spherical. With this choice the short-range molecular repulsive forces can be properly taken into consideration and attractive forces, such as those resulting from electric moments, treated as the perturbation. Calculations are presented for the first-order perturbation term to the Helmholtz free energy due to quadrupolar forces in models of liquid nitrogen and chlorine, and due to dipolar forces in liquid hydrogen chloride. For these calculations the rigid diatomic model and its modification appropriate to heteronuclear molecules were used for the reference potentials. It is found that the lowest-order perturbation terms here are proportional to the second power of the dipole or quadrupole moments, and not the fourth power as had been found previously using a spherical reference potential function. This second-order d...

Construction of linearized kinetic models for gaseous mixtures and molecular gases

A simple method of construction of linearized kinetic models is proposed which is based upon the equivalence of moments of the Nth-order modeled collision operator and the moments of the full collision operator calculated with the Nth-order approximation to the distribution function. This method is used to construct a third-order model of the linearized cross-collision operator for a general multicomponent monatomic gaseous mixture, not restricted to Maxwell molecules. In addition, this method is used to extend the Hanson-Morse diatomic gas model of the Wang Chang-Uhlenbeck collision operator to a polyatomic gas. Also presented is a new method of formally extending the Gross-Jackson modeling procedure to multicomponent monatomic mixtures. It is shown that the method of constructing mixture models based on the equivalence of moments is identical to this new procedure. In addition, the Gross-Jackson modeling procedure recently developed for mixtures by Boley and Yip is analyzed. It is shown that, under a certain weak restriction, their method is equivalent to the simpler Gross-Jackson procedure — and leads to the same expression for the Nth-order model, except for the manner of specification of the constant in the diagonal approximation.

Infrared surface modes in small NiO particles

Small particle specimens of NiO (powder and arc-produced smoke) were studied using infrared spectroscopy. Size distribution and particle configuration were established using an electron microscope. The importance of surface modes (polaritons) as well as their dependence on particle shape was established in the interpretation of the spectra. Chain formation in the smoke suggested cylinder calculations which showed the presence of both bulk and surface modes, in agreement with measurements. In both samples particle shape effects broaden the spectral features in the vicinity of the surface mode. Because of large errors in calculating surface modes from optical constants derived from a diatomic model, it was also shown that accurately measured optical constants are required in the study of the surface mode problem.

T4Dependence of the Nuclear-Quadrupole Resonance Frequencies in Solids

Nuclear-quadrupole-resonance frequencies are shown to have a ${T}^{4}$ dependence which becomes dominant at sufficiently low temperatures; the mechanism is discussed using a diatomic linear-chain model. The uniform-translational character of the acoustic phonons is shown to produce a ${T}^{4}$-dependent Doppler shift which is too small to account for the experimental results. The nonuniform $q$-dependent character of the acoustic branch is shown to induce modulations of the librational and vibrational coordinates at the low frequencies of the acoustic phonons. A Debye term is added to the familiar Bayer expression; both terms are shown to be produced by the modulation of the same coordinates, except that they occur at different frequencies.

Interactions between Oppositely Charged Nonspherical Ions. I. Potential Energy Function

In this paper we apply the theory of Steele to the problem of determining the pairwise interactions between oppositely charged nonspherical ions imbedded in a vacuum. The diatomic model of Sweet and Steele was extended by superimposing at either end of each linear, symmetric rod a Coulombic center characterized by a r−1 distance dependence. With respect to the consequent charge—charge interactions, a value of the Onsager coupling parameter was chosen which allowed a more detailed study of the relative importance of Lennard-Jones versus Coulombic forces; in effect, the ions were assumed to be only partially charged. Coefficients in the expansion of the potential energy function were calculated numerically, and changes in the profiles of the resulting curves [Ul,l′,m(r) as a function of interionic distance] were noted for different values of R/σ (where R is the length of the rod and σ is the usual Lennard-Jones range parameter). The trends observed were compared with those found by Sweet and Steele in their study of the diatomic model.

Interatomic exchange and σ-bonding of outer 3d and 4s-orbitals

A valence bond treatment is applied to a diatomic model involving a σ-bond between an outer sulphur orbital ϕ = 3 d, 4 s and a fluorine sp-hybrid. Interatomic exchange terms are included and the results discussed from the point of view of excited orbitals size and of hybridization in fluorine. In the case of 4 s orbitals it is concluded that the electrostatic approach is fully adequate to discuss orbital size and energy. When states involving d c-orbitals are concerned interatomic exchange terms with fluorine 2 s orbital are very important, while those with 1 s orbital are not of great relevance. For a reasonable amount of sp hybridization in fluorine the size of the d-function is strongly reduced from the free atom value.

Intermolecular Potential and Lattice Dynamics of the CO2 Crystal

The lattice dynamics of solid CO2 has been treated throughout the Brillouin zone. The molecular librational degrees of freedom have been treated by the use of Eulerian displacement coordinates. The potential model assumes the molecule represented by two centers of interaction (diatomic model) and each molecular pair interaction term is written as a sum of 6–12 terms between interaction centers on different molecules. The three parameters of the potential are obtained by fitting to the five optic modes, from the crystal energy, and by invoking the condition of zero uniform stress. It is found that the interaction between translational and librational degrees of freedom is large inside the Brillouin zone. The low temperature specific heat was calculated and found to agree well with experimental data. In an appendix, the applicability of the classical harmonic oscillator treatment to small molecular solids is discussed, and it is concluded that this approach is justified for solid CO2.

Saxon-hutner theorem in the coherent potential approximation

We examine the application of the coherent potential approximation to the one-dimensional disordered diatomic Kronig-Penney model, for which the Saxon-Hutner theorem is known to be exactly true. It seems remarkable that this theorem also holds for the coherent potential approximation.

Statistical Mechanics of Linear Molecules. VI. Solutions of the Percus–Yevick Integral Equation for a Hard-Core Model

The Ornstein–Zernike integral equation relating direct and indirect correlation functions has been generalized to linear molecules and solved numerically for a hard-core diatomic model with the aid of the Percus–Yevick approximation. The generalized PY equation was decomposed into a set of coupled integral equations by expanding the correlation functions in series of spherical harmonics. Iterative solutions of the Fourier-transformed form of these expressions were obtained after truncating the set. Correlation functions were calculated in this way as a function of density for two values of the molecular length and then used to evaluate compressibility factors from both the pressure and compressibility equations. It is found that the pair correlations at moderate densities obtained from the PY equation do not differ greatly from those calculated previously from density series. In addition, the various equations of state for this and similar hard-core nonspherical models do not differ greatly from one another or from the hard-sphere results, when the data are plotted as a function of reduced density using a reducing factor proportional to the molecular volume. A brief discussion is given of the limitations imposed upon these results by the mathematical approximations introduced.

Semiclassical Theory for Molecular Auto-Ionization

Diatomic molecular autoionization model, calculating limit of high vibrational and electronic principal quantum numbers

Harmoniques des vibrations de valence CH dans les alcanes et les halogénures de t‐butyle

AbstractThe first and second harmonic CH and CD valence vibrations of alkanes and t‐butyl halides have been measured. A model is proposed for the methyl and methylene groups, which allows the calculation of the anharmonicity constants of every vibrational mode and of the couplings between the modes. The already known large values (−100 to −150 cm−1) for the anharmonicity constants that couple the symmetrical with the asymmetrical modes are confirmed. As compared with lower harmonics combination bands become predominant for the higher ones; they allow the application of a diatomic model to estimate higher harmonics and the energy of dissociation.

Statistical Mechanics of Linear Molecules. V. Pressure Virial Coefficients for the Hard-Core Model

The formalism necessary to evaluate the pressure equation of state for nonspherical hard-core molecules is developed. By making use of previous results, exact second and third pressure virial coefficients are calculated for a diatomic hard-core model. Also, the fourth pressure virial is calculated in the Percus–Yevick approximation. These results are compared with the corresponding compressibility virials and with virial coefficients calculated from the scaled particle theory as applied to spherocylindrical hard-core molecules.

Viscous shock layer at the stagnation point with nonequilibrium air chemistry

A finite-difference method and a nonlinear overrelaxation method are investigated for solving the viscous shock layer at the stagnation point of a blunt body. An air gas model is employed with finite reaction rates and accurate thermodynamic and transport properties. For a body with a 1-in. nose radius and at a velocity of 20 kfps, the present results at 100, 150, 200, and 250 kft show that boundary-layer theory with the inviscid edge flow in chemical equilibrium is appropriate for some altitude below 150 kft. When the altitude is 250 kft, the effects of shock slip must be included in the viscous shock-layer solution. For this case, the air is only slightly dissociated and ionized. The present results, with a seven-species air model, are in general agreement with the diatomic air model results of Cheng and Chung.

A generalized diatomic Kronig-Penney model

Kerner showed that for a lattice of A-atoms with B-atoms, periodically interspread, the Saxon-Hutner theorem does not hold. Here the equations for determining one- electron energy levels in a generalized diatomic lattice of rectangular potential barriers is given, which has been obtained by the method of the transfer matrix. Numerical work makes clear that though the polarity criterion restores the Saxon-Hutner theorem for Kerner's model, it would not do so for more general lattices. Explicit examples are given.

Quantization of the Kramers and Pauli model

The years ago Kramers and pauli quantized the diatomic molecular model which bears their names, obtaining the formula which has placed so large a part in hand spectroscopy. It is sometimes pointed out in modern text-books that these results, founded on the old mechanics, do not offer a very far-reaching analogy to those which the wave mechanics give for the real diatomic molecule. I think the reason will be seen on closely examining the classical calculation, either in the original papers, or in Born's "Atom-mechanik," Without adequate cause, all types of motion save one were rejected in the quantization; all that remained was the simple type of motion which, apart from the gyroscope's axial spin consists merely of a steady precession of the body as a whole about the invariable axis. It will here be shown that in reality we can have Kramers and Pauli models performing a very different type of motion, while completely satisfying the old-quantum equations (which were determined in the original papers). In these motions the gyroscope presesses not only about the invariable axis but also about the nuclear axis. Kemble proposed that in a modified model, having an elastically mounted gyroscope, notions of this more complex type should be possible as quantized States, though his actual treatment is confined to motions of the simpler type. As stated above, we shall see that the original model can have the more complex type of motion, in strictly quantized states. Kemble also proposed an energy formula for the corresponding motions of the real molecule; it is with this and with the well-established molecular formulæ of the wave mechanics that results obtained from the model are to be compared. For specifying orders of magnitude it is convenient to think what happens when the mass of the nuclei is made very great, their angular momentum retaining the same order of magnitude as that of the gyroscope. Then the existence of quantized motions other than those prescribed by Kramers and Pauli is not surprising; for by analogy with the vectorial conception of the real molecular (described, notably, by Hund and Weizel), we should expext, in these circumstances, to find models in whose quantized states the nuclei, if of great mass, will rotate about the invariable axis very s;lowly in comparison with the gyroscope's precession about the nuclear axis. For the real molecule, wave mechanics equations such as Kronig's (22) will hold with an accuracy increasing with the nuclear mass, in the sense that though the main rotational terms will became smaller, the neglected perturbations (if so they may be called) with diminish far more rapidly.