Volume 46, Issue 10, 15 May 1967
Index of content:

Analysis of Elastic‐Scattering Data in the Semiclassical Approximation
View Description Hide DescriptionThe usefulness and accuracy of several approximations in the semiclassical derivation of the differential elastic scattering cross section are evaluated for the purpose of defining the best way of analyzing experimental data to obtain potential parameters.

On the Separation of the Broadening‐Relaxation Time and Molecular Concentration from Pure‐Rotational Spectroscopic Intensity Data
View Description Hide DescriptionThe broadening‐relaxation time τ and the molecular concentration N of the absorbing species become separable at the transition frequency if a new intensity coefficient Γ is substituted for the conventional Beer's Law intensity coefficient γ. This investigation reveals that Γ generally can be factored into two measurable terms, η and φ. η is shown to be linear with molecular concentration and independent of relaxation time and φ is a dimensionless function of the broadening‐relaxation time. The essence of this separation is that it is possible to fix to a constant value the number of radiation excited transitions each molecule will undergo during the molecular rotational relaxation time. When the number of transitions per molecule is the same for all relaxation times, the amount of radiation absorbed during the relaxation time is proportional to the molecular concentration of the absorbing species. The theory has been subjected to experimental examination. Several of the results are presented and discussed. This intensity coefficient gives rise to a new line shape, which is compared with the conventional line shape.

Functional Integral Representation of Nonequilibrium Statistical Mechanics
View Description Hide DescriptionA functional integral representation of the Green's function of the Liouville equation is presented, and it is shown how the weak‐coupling expansion and the binary‐collision expansion are generated from the functional integral. Furthermore, when the interaction between particles is represented as a sum of short‐ and long‐ranged components, it is possible to generate a new ``mixed'' expansion. It is shown how this mixed expansion may be used to derive a kinetic equation descriptive of the liquid phase. Under the conditions that successive correlated binary encounters may be neglected and that the effects of the weak interaction on the dynamics of the quasibinary encounter are neglected, the Rice—Allnatt equation is recovered. A discussion of the implications of the derivation is given along with a brief description of those diagrams of the Prigogine theory which are summed in the Rice—Allnatt approximation.

Molecular Structure of PClF_{4}: Infrared Spectrum, Low‐Temperature Raman Spectrum, and Gas‐Phase Dipole Moment: Pentacoordinated Molecules. IX
View Description Hide DescriptionThe vapor‐state infrared spectrum (2000–250 cm^{−1}) and liquid‐state Raman displacements (Δv=50–1200 cm^{−1}) of PClF_{4} are reported. The vapor‐state electric dipole moment of PClF_{4} was obtained from a study of the temperature variation of the dielectric constant over the range −68° to −2°. The resulting value is 0.78±0.01 D. The induced polarization is 13.7±0.4 cc. Interpretation of the vibrational data as well as the dielectric data and ^{35}Cl pure quadrupole resonance frequency support the conclusion that the structure of PClF_{4} is a trigonal bipyramid (C _{2v } point group) with the chlorine atom located at an equatorial site.

Potential Field and Molecular Vibrations of the Trigonal Bipyramidal Model AX_{3}YZ: Pentacoordinated Molecules. X
View Description Hide DescriptionThe secular equations with angular dependence are derived in terms of the FG matrix system representing the normal vibrations associated with the trigonal bipyramidal model of C _{3v } symmetry. A normal‐coordinate analysis of PCl_{4}F and CF_{3}PCl_{4} both of C _{3v } symmetry is given based on recent infrared and Raman data. The symmetry coordinates adequately represent the normal modes except for ν_{4}, ν_{6}, and ν_{7} where appreciable mixing is indicated.

Molecular Vibrations of the Trigonal Bipyramidal Model MXY_{2}Z_{2}: Pentacoordinated Molecules. XI
View Description Hide DescriptionA vibrational analysis of the trigonal bipyramidal model of C _{2v } symmetry neglecting anharmonicity is given in terms of the FG matrix system. The angular dependence of the secular equations is derived in expanded form. A normal‐coordinate analysis of CH_{3}PF_{4}, PClF_{4}, and PCl_{2}F_{3} based on recent vibrational data provides a partial description of the potential function for this symmetry. The resulting force constants show that equatorial P–F bonds are considerably ``stronger'' than axial P–F bonds in these molecules.

Test of Laplace Transform Technique for Energy‐Level Density Calculations
View Description Hide DescriptionThe series obtained by Haarhoff and Thiele for the density of vibrational states by the Laplace transform expansion technique has been generalized to obtain the density of vibration—rotation states. The resulting series expression is compared at several energies with the exact count and with a compact formula due to Haarhoff. Two cases are considered: a ``large'' molecule with many vibrations and rotations and a ``small'' molecule with few vibrations and rotations. The series expression gives a much better approximation than Haarhoff's formula for the ``large'' molecule, but there is little to choose between the two in the case of the ``small'' molecule.

Velocity Dependence of the Collisional Excitation Transfer Processes of a Resonant Type
View Description Hide DescriptionThe velocity dependence or the temperature dependence of the excitation transfer processes were discussed within the limit of the straightline trajectory and the impactparameter method. It has been shown that the velocity dependence corresponds directly with the radial part of the interaction in some cases. In the discrete—discrete resonant case, the cross section σ is proportional to v ^{−2/(n−1)} and in the discrete—continuum case it is proportional to v ^{−2/(2n−1)}, where v is the relative incident velocity of the atom and n is the inverse exponent of the radial part of the interaction.

Radio‐Frequency Spectra of KF and KCl by the Molecular‐Beam Electric‐Resonance Method
View Description Hide DescriptionThe molecular‐beam electric‐resonance method has been used to obtain quadrupole‐coupling constants eq(v)Q, spin—rotation constants c, and tensorial and scalar spin—spin interaction constants d_{T} and d_{S} for ^{39}K^{19}F, ^{39}K^{35}Cl, and ^{39}K^{37}Cl in several vibrational states. For ^{39}K^{35}Cl and ^{39}K^{37}Cl also the electric dipole moments have been determined. For the first five measured vibrational levels of ^{39}K^{19}F and ^{39}K^{35}Cl, the values of the quadrupole coupling constants and dipole moments can be written as a polynomial of second degree in (v+½).

Physical Properties of Aromatic Hydrocarbons. II. Solidification Behavior of 1,3,5‐Tri‐α‐Naphthylbenzene
View Description Hide DescriptionThe kinetics of solidification of 1,3,5‐tri‐α‐naphthylbenzene have been studied from 25° above the glass temperature (69°C) almost to the crystal thermodynamic melting point (199°C). The crystal growth rate has been analyzed using current theories of crystallization and a mechanism for crystal growth has been proposed. It has been demonstrated that mass transport in crystal growth and viscous flow do not have the same temperature dependence. The morphology of the solid phase is discussed. Pertinent parameters pertaining to the glassy state of this material are also reported.

EPR Spectrum of Cr^{3+}: K_{2}NaCo(CN)_{6}
View Description Hide DescriptionThe EPRspectrum of Cr^{3+} in K_{2}NaCo (CN)_{6} has been examined at room temperature and K‐band microwave frequency. The spin Hamiltonian is assumed to have the form . A general analysis is given for fitting this spin Hamiltonian when the orientation of the crystal‐field axes is unknown. The application of this analysis to the experimental data yields g=1.992±0.001, D=0.022±0.001 cm^{−1}, E=0.003±0.001 cm^{−1}.

Crystal Structure of Paramagnetic DyMn_{2}O_{5} at 298°K
View Description Hide DescriptionParamagnetic DyMn_{2}O_{5}crystallizes in the orthorhombic system with lattice constantsa=7.2940±0.0008, b=8.5551±0.0008, and c=5.6875±0.0008 Å, in the space group Pbam(D _{2h } ^{9}). There are four DyMn_{2}O_{5} per unit cell. The crystal structure was solved by a combination of three‐dimensional Patterson and Fourier series and refined by the method of least squares. The integrated intensities of 5094 reflections were measured by PEXRAD, with 976 symmetry‐independent structure factors significantly above background. The final agreement factor between measured and calculated structure factors is 0.0401. There are two independent Mn atoms. One is pentacoordinated, with a slightly distorted tetragonal pyramidal configuration: two basal Mn–O distances are 1.926±0.006 Å, the other two are 1.911±0.005 Å, and the apical Mn–O bond, nearly normal to the basal plane, is 2.020±0.005 Å. The other Mn occupies a slightly compressed octahedron, with four planar Mn–O bonds of 1.940±0.004 Å and two, normal to this plane, of 1.873±0.004 Å. The octahedral Mn environment closely resembles that in β‐MnO_{2}. The Dy is surrounded by a somewhat distorted square‐antiprismatic polyhedron of oxygen atoms, the Dy–O distance ranging from 2.341±0.005 to 2.469±0.005 Å, the average being 2.390 Å. The thermal vibrations are significantly anisotropic. The room‐temperature paramagnetism could mask an antiferromagnetic ordering of the octahedral Mn atoms.

Radial Distributions of Ions for a Primitive Model of an Electrolyte Solution
View Description Hide DescriptionRadial distribution functions are computed for the primitive model of an electrolytic solution using the Percus—Yevick and convolution‐hypernetted‐chain integral equations. The results are compared with the Debye—Hückel theory. The Debye—Hückel approximation appears to be particularly bad when the density is great enough that the hard core plays a significant role in determining the radial distribution functions.

Analytical Treatment of Excluded Volume. I. Dimensions of Real Polymer Chains
View Description Hide DescriptionExcluded‐volume interactions in long chains are treated in terms of Gaussian probabilities for intersegmental contacts. Thus for a chain of n segments and end‐to‐end distance h the energy factor,is evaluated by linearly expanding and averaging—sequentially for l=1, 2, ···, n—the interactions of segments l with those following after, k. Accordingly, for each l exp[—βΣ_{ k }δ(h_{lk} )] into exp[—βΣ_{ k } f ^{ n—l }(0_{ lk }  h)], where f ^{ n—l }(0_{ lk }  h) is the probability, conditional to h, for a contact among segments l and k. The probability refers to a fictitious chain n in which only the segments from l+1 to n are still interacting (— the ``real section''). Thereafter it is assumed that the contact probabilities can be represented by the Gaussian expressions obtaining for random‐flight chains, with the links representing the real section taken as expanded as the most probable end‐to‐end distance of a corresponding real chain—viz., a chain consisting of r=n—l−1 interacting segments. On this basis an integral equation is derived for the expansion coefficient α_{ n }=h ^{*}/h ^{*0}, h ^{*}, and h ^{*0} being respectively the most probable value of the end‐to‐end distance of chain n, in the presence and absence of interactions. (The integral form of the equation results from the number of interacting segments in the fictitious chains varying from r=n to r=0.) The line obtained for α_{ n } ^{2} vs n initially rises like Fixman's but, beyond α_{ n } ^{2}≈4, diverges from it, the asymptotic limit being α_{ n } ^{2}∝n ^{0.236} (vs Fixman's ∝n ^{0.33} and Flory's ∝n ^{0.2}).
The results also show that the excluded‐volume effect is essentially contributed by contacts among segment pairs separated by a vanishingly small fraction of the real chain section, (k—l) ^{*}«r ^{*}. It seems therefore that the distribution of the internal distances h_{lk} can be treated in Gaussian terms, in spite of the distribution of the end‐to‐end distance in a real chain being non‐Gaussian. On the same grounds it is concluded that a similar treatment of the 2‐dimensional chain, leading to α_{ n } ^{2}∝n ^{0.5}, should fail, for in this case (k—l) ^{*}≈r ^{*}.

Analytical Treatment of Excluded Volume. II. Quasi Random‐Flight Behavior
View Description Hide DescriptionA ``quasi'' random‐flight behavior of real chains of r segments is described by the following: The distribution of long statistical elements in typical chain configurations can be expressed by a product of functions, each describing an element independently of the others. This random property differs from true random flight in three aspects: (a) The elements' length depends on r, being as stretched as the most probable end‐to‐end distance of the chain. (b) The randomness holds only for elements constituting a large enough fraction of r. Its employment is restricted therefore to describing distances of correspondingly long enough sequences of j segments, or j/r> (j/r)_{min}. (c) The random behavior is characteristic of typical configurations, corresponding to a given mean state of the chain. It will lead therefore to a Gaussian distribution of distances h_{i} only if the specification of various h_{i} 's does not vary the shape, and the mean state, of the chain to any appreciable extent. This implies however that j is not too large, or j/r< (j/r)_{max}. This description, it is argued, appears to be self‐consistent. For: assuming that the mean density of contacts in the bounded range of (j/r) can be evaluated in terms of the random elements, computing on this basis (as fluctuations) the microscopic variation of the interaction energy, and, from it, the typical variation of the configurational probability, and judging from the results, it is concluded that the mean density of contacts can be indeed evaluated with random elements—as assumed to begin with. The contacts contributing to long‐range interactions in 3‐dimensional chains fall within the range bounded by (j/r)_{min} to (j/r)_{max}; their evaluation with Gaussian probabilities (Pt. I) will therefore be justified if the above description is correct. Examination of two‐dimensional chains shows that (j/r)_{max} is violated; a relatively simple modification of the treatment however suffices, leading to α_{ n } ^{2}∼n ^{0.434}. The behavior of four‐, and more‐, dimensional chains is also considered.

Inversion of Low‐Energy Experimental He–He^{+} Elastic Differential Scattering Sections to the ^{2}Σ_{ g } ^{+} and ^{2}Σ_{ u } ^{+} Potentials
View Description Hide DescriptionAn iterative technique coupled with the two‐state theory has been employed to obtain the He–He^{+}ground state and first excited state potentials from the 10‐ and 15‐eV experimental differential cross sections of Aberth and Lorents. This procedure is adaptable for the inversion of relative data. The resulting ungerade potential displays good agreement with the theoretical calculations of Reagen, Browne, and Matsen in the 0.8‐ to 2.5‐Å region, with the depth of the well found to be 2.34 eV from the 15‐eV data. The intermediate regions of both potentials show the dominant term of the power law to be approximately r^{−4}.

Electron Paramagnetic Resonance Studies of the Radiolysis of H_{2}O in the Solid State
View Description Hide DescriptionFree radicals formed by electron irradiation of H_{2}O at 77°K have been studied with the electron paramagnetic resonance(EPR) technique. In particular the spectrum of the OH radical is identified in irradiated CaSO_{4}·2H_{2}O and LiSO_{4}·H_{2}O and in irradiated hexagonal ice. The effective spin Hamiltonian is determined for OH in these crystals. For the ice case, the experimentally determined spectroscopic splitting tensor is shown to be in agreement with a simple theory based on crystal‐field theory. The experimentally determined hyperfine splitting of the OH radical is shown to be in agreement with that predicted by an LCAO MO calculation and verified by an EPR study of OH produced by microwave discharges in gases.
In the irradiated sulfates magnetic resonance signals from OH, H atoms, and some form of trapped electron were observed.
Radiation‐yield measurements for OH in ice at 77°K are given for the dose range 10^{4} to 10^{9} rad.

Adsorption of Barium on Tungsten: Measurements on Individual Crystal Planes
View Description Hide DescriptionThe adsorption of barium on the (110), (211), (100), (111), and (221) planes of a tungstenfield emitter was studied for both thermally equilibrated and unequilibrated adsorbate layers. At low coverage the equilibrated atom densities on the (110) and (221) planes are ∼100 times greater than on the (211) and (100) planes, while for equilibrated atom densities are essentially identical on all planes. Dipole moments are roughly the same on all planes, corresponding to q∼1.0 electron/atom. At high coverages (θ∼0.8) there is evidence for rearrangement on heating to produce pronounced work function minima on the close‐packed crystal planes.
These results are interpreted in terms of the previously suggested model of electropositive adsorption, with generally good agreement obtained. The influence of the small size of the substrate on the measurements is considered, and it is shown that these results should be identical to those on perfect macroscopic surfaces in most situations.

Study of the Electronic Structure of Molecules. I. Molecular Wavefunctions and Their Analysis
View Description Hide DescriptionIn this paper we have summarized the complete multiconfiguration self‐consistent‐field theory (CMC SCF LCAO MO or CMC for short) as given from the analysis of Veillard and the author (published elsewhere).
In the analysis of the CMC theory it is shown that the CMC technique allows determination of the pair and pair—pair correlation energy and retains the physical simplicity of the Hartree—Fock functions. The CMC theory differs from usual many‐body techniques in so far as it does not take the Hartree—Fock energy as zero‐order energy. It has been shown that the CMC wavefunction can be cast in the form of a new type of ``single determinant'' (the ``CMC determinant'') with elements which are different from the standard elements of the Slater determinant. The existence of the CMC determinant provides the formal justification for the use of a single determinant in semiempirical work aiming at exact prediction.
The analysis of the CMC wavefunctions is then continued in terms of a population analysis which to a large extent follows Mulliken's work. Then the energy expression of the SCF MO and CMC techniques are analyzed in terms of bond‐energy diagrams. These are special partitions of the total energy with immediate reference to structural chemistry. Finally, the bond‐energy diagrams are briefly discussed in their relation to bond‐energy transferability, vibrational analysis, and other chemical phenomena. This paper introduces both the notation and the terminology we shall use in describing all‐electron computations for the NH_{3}, HCl reactionsurfaces, and for pyrrole, pyridine, pyrazine, fluorobenzene, benzene, and a few other molecules presented in the following papers of this series.

Study of the Electronic Structure of Molecules. II. Wavefunctions for the NH_{3}+HCl→NH_{4}Cl Reaction
View Description Hide DescriptionAb initio computations are presented for the reaction NH_{3}+HCl→NH_{4}Cl. The two reactants are studied at a large number of positions and for each point an SCF LCAO MO wavefunction and the corresponding total energy are obtained. These results are then collected in an energy surface diagram. All the electrons of the system are considered and a contracted Gaussian basis set has been used in this work. This study considers the two reactants as one system and both charge transfer as well as hydrogen‐bonding characteristics are analyzed within the molecular‐orbitals framework instead of the valence‐bond approximation as customary in the past. The reaction has been analyzed for the case where the HCl approaches the NH_{3} molecule in a path normal to the plane of the three hydrogens of NH_{3}; the H atom of HCl is between the N atom and the Cl atom. The first stage of the reaction (NH_{3} and HCl at large distances) indicates mutual polarization of the two molecules. In this region the SCF LCAO MO approximation is rather poor since it can not take into consideration dispersive forces. With HCl closer to the ammonia molecule, there is first hydrogen bondingformation and partial charge transfer. When NH_{4}Cl is formed the molecule is stable with respect to the NH_{3} and HCl components by about 19 kcal/mole. No activation energy has been found in the reaction, contrary to previous assumptions. The computed geometry of the NH_{4}Cl molecule at the equilibrium configuration is discussed in this work.