Volume 50, Issue 9, 01 May 1969

ESR Spectrum of the 9‐Molybdomanganate(IV) Ion in Dilute Single Crystal
View Description Hide DescriptionThe ESR spectrum of a single crystal of (NH_{4})_{6}[NI(IV)O_{6}Mo_{9}O_{26}]·8H_{2}O containing 1% (NH_{4})_{6}‐[Mn(IV)O_{6}Mo_{9}O_{26}]·8H_{2}O has been investigated. This is the first ESR report on quadrivalent manganese in a chemically well‐characterized environment. The site symmetry is , and anisotropic values, hyperfine splittings, and zero‐field splitting values were obtained by fitting the spectrum with an axial spin Hamiltonian.

Perturbed Hartree–Fock Theory. I. Diagrammatic Double‐Perturbation Analysis
View Description Hide DescriptionThe coupled Hartree–Fock scheme, for calculating atomic and molecular properties which are second order in some perturbing field, is studied with the help of a double‐perturbation expansion and diagrammatic techniques. The double‐perturbation expansion, with the external field as one perturbation and the difference between the true electron‐repulsion potential and the Hartree–Fock potential as the other , is compared with an iterative solution of the set of coupled Hartree–Fock equations for the first‐order perturbed orbitals. The term in the double‐perturbation expansion that is second order in and zeroth order in is shown to be the Dalgarno “uncoupled” second‐order energy. The coupled Hartree–Fock second‐order (in ) energy expression includes all terms first order in and some of the terms to all orders in , up to infinite order. It is shown how to separate the diagrams contributing to the CPHF energy from those that do not; in particular, the CPHF equations are composed of the so‐called “bubble diagrams” and their corresponding exchange diagrams. It is shown that each particle–hole pair, or “bubble,” is independent of all the other particle–hole pairs, in the sense that the total energy contribution from the diagram can be factored into contributions from the individual particle–hole pairs. It is shown that the first‐order perturbed wave‐function obtained from the coupled Hartree–Fock formalism includes effects due to all singly‐excited determinants and a certain class of linear combinations of doubly‐excited determinants, namely those singlet spin eigenfunctions which correspond to two electrons being excited without spin flips from Hartree–Fock unexcited orbitals to virtual orbitals; there exists a second, linearly independent, doubly excited singlet function which is not included in the CPHF wavefunction and does not contribute to the perturbation energy. The contribution to the second‐order energy from the doubly excited determinants amounts to about 10% of the total CPHF second‐order energy in the case of some electric and magnetic properties of H_{2} and 25% of the total second‐order energy corresponding to the electric dipole polarizability of the Be atom. The larger contribution of the terms involving doubly excited determinants for Be has its origin in the fact that a double excitation from the to the orbital gives a large fraction of the correlation energy between the electrons and that the single excitation from the to the orbital is the largest single contribution to the polarizability. An alternate iterative scheme for the solution of the coupled Hartree–Fock equations is derived. This alternate iterative scheme, which includes certain terms summed to infinite order, is expected to be a considerable improvement over the simple scheme which starts with the “uncoupled” approximation as the zeroth iteration. Comparisons of the different schemes for the properties considered in this paper are made to confirm this conclusion. The CPHF energy (second order in ) is partitioned approximately into a part due to correcting the unperturbed Hartree–Fock wavefunction for correlation and a part due to correcting the uncoupled Hartree–Fock perturbed wavefunction for correlation; the contribution from the former was shown to be about 10%–20% of that from the latter.

Decay of Correlations. II. Relaxation of Momentum Correlations and Momentum Distributions in Harmonic Oscillator Chains
View Description Hide DescriptionWe have studied the relaxation of the n‐particle momentum distribution function and the corresponding n‐particle Ursell function for both an isolated infinite chain of coupled harmonic oscillators and an infinite chain of coupled harmonic oscillators in contact with a heat bath. The time development of the Ursell function gives us the desired information on the decay of initial momentum correlations. For both of these systems we have found the asymptotic relaxation laws and as , where the superscript zero denotes the equilibrium distribution. The momentum correlation function thus goes to zero (no correlations) much more rapidly than the n‐particle distribution function approaches its equilibrium value. These results are identical with those obtained by us previously for initially correlated, noninteracting many‐particle systems in contact with a heat bath and by Glauber for the relaxation of spins in an Ising lattice. It is interesting to speculate on the reason for this identity of results in such apparently different relaxation models, but no answer can be provided as yet.

Universal Parameter for the Extrapolation of Force Constants, Mean‐Square Quantities, and Bond Energies of Halides
View Description Hide DescriptionA general formula of the type , which is simultaneously applicable to force constants, the reciprocals of the mean‐square amplitude quantities , bonddissociation as well as thermochemicalbondenergies of halides, has been presented. , moreover, is given by the expression where and are the Pauling electronegativities of the atoms forming the M–X bond, and where is a constant for each of the aformentioned molecular parameters and may have a nonzero value only in the fluorides. The values for the three parameters studied were found to be 0, − 11, and − 15 or − 30, respectively. The choice of the latter depends on whether , or , except for the NaX series for which .

Nitrogen Quadrupole Coupling Constants in Ethyl Cyanide; Coupling Constant of the Cyano Group
View Description Hide DescriptionAn investigation of the hyperfine structure of the microwave transitions of ethyl cyanide has led to the following values of the nitrogen quadrupole coupling constants: , , and . These results indicate the existence of cylindrical symmetry about the CN bond with a quadrupole coupling constant along the bond axis of . The coupling constants in ethyl cyanide and several other cyano compounds have been described by an extended Hückel scheme.

Diffuse Interface in a Critical Fluid Mixture
View Description Hide DescriptionThe interface between the two fluids in equilibrium at temperature below the critical (consolute) temperature of a binary mixture in a gravitational field has been observed to expand into a transition layer that thickens, while the coexisting phases become more similar and the interfacial energy vanishes as . The effective thickness of the transition layer has been deduced from reflectivity measurements as a function of light wavelength and temperature. The shape of the interface concentration distribution function has been found from the dependence of the reflectivity to be close to an error functon and to a function recently proposed by Fisk and Widom and distinguishable from the classical hyperbolic tangent distribution. In the range where and the appropriately defined thickness is comparable with the correlation length of the bulk fluids for corresponding . The critical index , together with the critical index for the coexistence curve, and for the surface tension, can be combined to test various theoretical relations amongst the critical exponents and theories of the critical interface.

Interfacial Tension of Near‐Critical Cyclohexane–Methanol Mixtures
View Description Hide DescriptionWe have measured the capillary length for the interface between the coexisting phases in the vicinity of the critical consolute temperature of mixtures of cyclohexane and methanol. The data on the capillary length were combined with values of the difference in density between the coexisting phases calculated from the coexisting compositions to obtain the interfacial tension . Our measurements of for this system combine with the only existing measurements of the effective thickness of a critical interface in a new test of theories of critical interfaces. Data include .

Electron Spin Resonance of a Sigma‐Electron Radical in X‐Ray‐Irradiated Malonamide
View Description Hide DescriptionTwo radical species were formed when malonamide crystals were irradiated with x ray. One was a π‐electron radical of common occurrence, NH_{2}COĊHCONH_{2}, and the other was interpreted as a σ‐electron radical of the formula NH_{2}COĊH_{2}COṄH. The ESR spectra of the latter were studied in detail and the assignments were substantiated by ESR spectra of deuterated malonamide and ^{15}N‐malonamide.

Cluster Distributions on a One‐Dimensional Lattice
View Description Hide DescriptionAn exact and comprehensive calculation of monomer distributions on a one‐dimensional Ising lattice is presented using the matrix method. The monomer sequence probabilities are expressed in terms of three parameters: the first nearest‐neighbor interaction energy between bound monomers, the lattice length, and the intrinsic binding constant of a monomer to a lattice site. Graphical illustrations are given as fractional quantities which can be directly converted to concentrations for experimental systems. An expression of the average length of monomer sequences and their standard deviation is presented.

ESR Measurement of Quadrupole Coupling in N(SO_{3})_{2} ^{2−}
View Description Hide DescriptionThe so‐called second‐order transitions in ESR spectra, which involve a change in both electron and nuclear spin, are sensitive to electric quadrupole coupling. The quadrupole interaction for ^{14}N in N(SO_{3})_{2} ^{2−} was studied by observation of these transitions at 31.5 GHz in single crystals of γ‐irradiated potassium aminedisulfonate. The resulting spectra can be explained assuming the principal axis system of the field‐gradient tensor is the same as the hyperfine coupling tensor. The coupling constants then become and , where the and principal axes have direction cosines of (0.829, 0, 0.559) and (0, 1.0, 0) in an axis system.

Electronic De‐excitation of by Rare Gases—Vacuum‐Ultraviolet Flash Photolysis of O_{2}
View Description Hide DescriptionFar‐uv photolysis of O_{2} produces excited atoms whose relative quenching constants with argon, krypton, and neon are if is taken as unity. These values are in good agreement with Cvetanović's relative efficiencies [J. Chem. Phys. 45, 8, 2888 (1966) ].

Enthalpy of Sublimation of Zinc and Cadmium; Correlation of vs ; Comparison of Torsional and Knudsen Vapor Pressures
View Description Hide DescriptionThe vapor pressures of solid zinc and cadmium have been determined directly by measurements of the torsional recoil of a suspended effusion cell and indirectly by measurement of mass effusion . The results with in calories/mole·degree and and in atmospheres are as follows: For zinc (610°–690°K) the torsional recoil yields , at 645°K the mass effusion yields , at 645°K ; for cadmium (525°–590°K) the torsional recoil yields , at 555°K ; the mass effusion yields , at 555°K . In these equations the cited errors are standard deviations generated in least‐squares analyses. The measured pressures for zinc and cadmium are 1.075(± 0.01) and 1.063(± 0.01), respectively, times the equivalent mass effusion for molecular weight corresponding to monomer. This ratio appears to represent a demonstrable systematic difference between the two procedures. A brief survey of measurements for various material by others reveals that is generally greater than probably because the restituted momentum from the surroundings makes the former too high. A procedure based on is defined which recognizes systematic errors in lnp vs and which derives a value for free of the inconsistencies frequently introduced in averaged values of RT lnp (the so‐called third law procedure). For zero error in absolute entropies and for mass effusion, the results are for zinc and for cadmium.

Molecular‐Beam Measurements of the Emission Spectrum and Radiative Lifetime of N_{2} in the Metastable State
View Description Hide DescriptionElectron‐impact excitation of a molecular beam of nitrogen was used to produce the metastable state. This state, which lies 11.87 eV above the ground state, has recently been observed in a number of electron‐impact experiments. This paper reports the emission spectrum from the molecular beam, and identifies three forbidden electronic transitions, . Approximate absolute transition probabilities are determined. The radiative lifetime of the state is 270 ± 100 μsec, and the equilibrium internuclear distance is 1.16 ± 0.02 Å.

Alternant Molecular Orbitals without Spin Projections
View Description Hide DescriptionA new method for utilizing different orbitals for different spins is proposed which obviates the procedures usually employed to project states of desired multiplicities out of a single determinantal function. The method, which calculates an approximation to a spatial eigenfunction of the Hamiltonian rather than to the total wavefunction itself, serves to simplify calculations not only for alternant molecular orbitals, but also for wavefunctions of the valence‐bond and nonpaired spatial‐orbital type.

Hyperfine Structure of Sodium Iodide
View Description Hide DescriptionA molecular‐beam electric resonance spectrometer has been used to study radio‐frequency transitions between the hyperfine‐structure sublevels of the state of ^{23}Na^{127}I. Transitions for the first four vibrational states have been observed in very weak electric field and near zero magnetic field. The intramolecular interaction constants obtained for the ground vibrational state are: . The numbers in parentheses are the uncertainties in units of the last quoted digit. Constants for the excited vibrational states have been determined with less precision because fewer transitions were observed.

Measurement of the Molecular Values in H_{2}O and D_{2}O and Hyperfine Structure in H_{2}O
View Description Hide DescriptionIndividual components of the Zeeman transitions in the rotational spectrum of H_{2}O and D_{2}O were resolved in fields of 25–30 kG. This allowed determination of the values with greater accuracy than previous measurements. The values obtained for H_{2}O were and . The values obtained for D_{2}O were . Analysis of the combined results indicates that the values for H_{2}O are positive and that the values are , and . Hyperfine structure on the 6_{16} − 5_{23} transition in H_{2}O was observed in a maser spectrometer, and the spin–rotation and spin–spin coupling constants were obtained. The measured value of the rotational transition frequency was 22 235 079.85 ± 0.05 kHz. The magnetic susceptibilityanisotropies and molecular quadrupole moments were determined in H_{2}O.

Decay of Correlations in Linear Systems
View Description Hide DescriptionThe conditions under which the decay of the pair correlation function at large distances is monotonic or oscillatory are investigated for one‐dimensional systems and discussed in detail for certain linear continuum and lattice models in which the molecules interact only with their nearest neighbors. In each case a locus is found in the pressure–temperature plane and in the density–temperature plane, such that in thermodynamic states of the one‐dimensional fluid that lie on one side of the locus, the decay of the correlation function is oscillatory, and in those that lie on the other side it is monotonic. At every temperature the decay is monotonic below a uniquely determined transition pressure or density. It is argued that such loci will likewise be found in real three‐dimensional systems, and that the critical point and a range of fluid states around the critical point, as well as states of the low‐pressure vapor, will lie in the region in which the correlation function at large distances is asymptotically positive and decays monotonically.

Absorption Spectrum of Np^{4+} in Single‐Crystal Thorium Oxide
View Description Hide DescriptionThe optical absorptionspectrum of Np^{4+} in single‐crystal ThO_{2} has been investigated at 4°K, 77°K, and room temperature between 0.3 and 2.5 μ. Infrared spectra were recorded between 2.5 and 300 μ. Intermediate‐coupling wavefunctions and eigenvalues for the ground‐state electronic configuration were generated by an IBM 360‐67 program using the combined Coulombic and spin–orbit Hamiltonian. The best over‐all agreement between calculated free‐ion levels and the observed spectrum of Np^{4+} are obtained with the Racah parameters, , , , and the spin–orbit parameter in cm^{−1}. A six‐parameter calculation that includes the effect of configuration interaction yields the parameters , , , , , and in cm^{−1}. Intermediate‐coupling wavefunctions were used to calculate operator equivalent factors. A first‐order crystal‐field analysis was carried out for the six lowest manifolds, where the separation of adjacent levels is large compared to the crystal‐field splitting of each level. For Np^{4+} in an eightfold cubic environment, the crystal‐field parameters and predict the positions of the crystal quantum states in good agreement with experiment.

Dipole Moments of the Lowest Singlet States in Phenol and Aniline by the Optical Stark Effect
View Description Hide DescriptionThe Stark effect in the rotational fine structure is used to measure the change in dipole moment on excitation in aniline and phenol. The spectra are in the 0–0 bands of the lowest‐lying singlet states. The results are for aniline and for phenol. Evidence is presented which indicates that the sign of is positive. If this is true, these results correlate well with the structural changes on excitation, namely, that the contraction along the in‐plane long axis is considerably greater in aniline than phenol. This correlation is strong indication of the extent of contribution of quinoidlike resonance structures to the electronic nature of the excited state.

Rotational Relaxation and Chemical Exchange
View Description Hide DescriptionThis article couples chemical exchange to rotational relaxation in liquids. We found that the relative rates of exchange and molecular rotation have a profound effect on relaxation behavior. Systems exchanging more slowly than the rate of molecular rotation can be distinguished from systems experiencing faster exchange. This leads to a novel class of experiments using molecular rotation as a frequency standard for the exchange process. Since rotational correlation times are typically 10^{−10}–10^{−12} sec. these experiments provide sensitivity on a time scale not easily achieved by other techniques. Analysis of these experiments is presented. Studies of NMR spin–lattice relaxation in iodine charge‐transfer complexes substantiate this analysis.