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Volume 48, Issue 8, 15 April 1968

Electron Spin Resonance Studies of Free Radicals in γ‐Irradiated Amino Acid Crystals: α‐Amino Isobutyric Acid and l‐Lysine·HCl·2H_{2}O
View Description Hide DescriptionSingle crystals of α‐amino isobutyric acid and l‐lysine monohydrochloride dihydrate were irradiated at 77°K and electron spin resonance spectra were observed in the temperature range between 77° and 298°K. It was confirmed that the stable radicals, (CH_{3})_{2}ĊCOOH and H_{2}N(CH_{2})_{3}CH_{2}ĊHCOOH, formed from these two amino acids, respectively, are not direct products of irradiation but are formed thermally by an inter‐ (or intra‐) molecular reaction involving intermediate radicals in the crystals. The intermediate radicals formed at 77°K are similar in these materials and are believed to be involved in the mechanism of amino‐group abstraction. The radical H_{2}N(CH_{2})_{3}CH_{2}ĊHCOOH in llysine·HCl·2H_{2}O exhibits different ESR spectra at different temperatures corresponding to different conformations in the crystal lattice.

Exact Calculation of the Partition Function for a Model of Two‐Dimensional Polymer Crystallization by Chain Folding
View Description Hide DescriptionA model of two‐dimensional polymercrystallization by chain folding is treated by equilibrium statistical mechanics. The Laplace transform of the partition function with respect to the length of the polymer chain is obtained in exact analytical form. The model leads to thermodynamically well‐defined chain‐folded crystals. Under certain circumstances in the limit of an infinitely long polymer chain, the model shows a second‐order phase transition from an “extended‐chain” crystal to the chain‐folded crystal.

Diffusion and Heterogeneous Reaction. IX. Theoretical Analysis of Nonsteady‐State Kinetics
View Description Hide DescriptionThe theoretical analysis of a transient method is presented for the quantitative study of heterogeneous reaction rates and gaseous, multicomponent diffusion coefficients. From observations of the rate of decay of reactant after its source of supply has been shut off, the kinetics of chemical reaction on the catalyticsurface may be obtained. This experimental approach is particularly applicable to the study of the diffusional transport of highly reactive species such as free radicals and atoms.

Vibrational Relaxation in the Tetrachlorides of C, Si, Ge, Sn, and Ti
View Description Hide DescriptionResults of attenuation measurements in CCl_{4}, SiCl_{4}, GeCl_{4}, SnCl_{4}, and TiCl_{4} indicate that thermal relaxation is responsible for the large attenuations observed in these substances. Comparison of relaxation times in the vapor and liquid phases suggests that the collision efficiencies are considerably smaller in the liquids and decrease through the series in the order of increasing force constants. This result is in general agreement with the suggestion by Madigosky and Litovitz that attractive forces do not enhance the collision efficiency in the liquid owing to close molecular packing. However, the theoretical reduction in collision efficiency suggested by these investigators is markedly different from that observed for SnCl_{4}.

Frequency Spectrum of Hydrogenous Molecular Solids by Inelastic Neutron Scattering. Hexagonal H_{2}O Ice
View Description Hide DescriptionA simple method of relating the scatteredneutronenergy distribution to the molecular frequency spectrum in hydrogenous solids is given. The method is approximate because of the neglect of translation‐rotation couplings, but should be of practical interest for systems where translational and librational frequencies do not appreciably overlap. The approach is applied to cold‐neutron‐scattering data from hexagonal ice at 150°K. The resulting thermodynamic frequency distribution is used to compute moments, specific heat, and root‐mean‐square amplitude of vibrations. Comparison is made with optical, thermodynamic, x‐ray, and neutron measurements.

Electron Spin Resonance Studies of Carbon‐13 Splittings in a Series of Phenoxy Radicals
View Description Hide DescriptionWe have observed carbon‐13 splittings in the electron spin resonance spectra of a series of phenoxy radicals. Through use of the Karplus–Fraenkel equation we were able to calculate π‐electron spin densities in the phenolic aromatic ring. Values for and were determined for the carbon–oxygen bond.

Magnetic Structure of TbAg_{2}
View Description Hide DescriptionNeutron‐diffraction measurements have revealed that TbAg_{2} having the tetragonal CaC_{2}‐type structure becomes antiferromagnetic below the Néel temperature of 35°K. The ordered magnetic structure consists of the ferromagnetic sheets which are perpendicular to the axis (face‐centered description), and the moment directions of the adjacent ferromagnetic sheets are opposite to one another. All moments are aligned in the direction of the axis, and the saturation moment per Tb is 8.95 ± 0.05 Bohr magnetons which is essentially equal to the ordered moment of the free Tb^{3 +} ion. This ordered structure is identical to the commensurable magnetic structure of TbAu_{2} at temperatures below 42.5°K, but the incommensurable transverse‐wavelike spin alignment of TbAu_{2} found in the 42.5°–55°K range was not detectable in TbAg_{2}. Also, no detectable moment is observed for Ag in TbAg_{2}.

Magnetic Structure of DyC_{2}
View Description Hide DescriptionThe compound, DyC_{2}, having the tetragonal body‐centered CaC_{2}‐type structure has been shown by neutron diffraction to exhibit the magnetic spin alignment of a linear, transverse wave mode below the Néel temperature of 59°K. This static moment wave is propagating along the axis and is polarized in the c‐axis direction. The root‐mean‐square and maximum saturation moments per Dy are 8.37 and 11.8 Bohr magnetons, respectively, the latter being considerably larger than the ordered moment of the free Dy^{3 +} ion, 10.0 . The wavelength of the moment wave is about 1.3 times the spacing and is practically temperature independent. An additional, coexisting spin alignment with a very small moment appears to take place below 31°K. The crystallographic parameters in the range 300°–5°K are also presented. The thermal‐neutron coherent scattering amplitude of Dy is established as (1.70 ± 0.01) × 10^{− 12} cm.

Zeeman Effect in the (0, 0) Band of OD
View Description Hide DescriptionThe rotational structure of the (0, 0) band of the system of OD has been photographed with and without magnetic field. The Zeeman splittings are clearly resolved in some of the lines whereas in the other cases only a shift in the intensity maxima is observed. These splittings and shifts have been measured and correlated with the predictions of Hill and Van Vleck. The polarization characteristics have also been studied.

ESR Studies of Radiation‐Induced Niobium Centers in Nb_{2}O_{5}–Na_{2}O–SiO_{2} Glasses
View Description Hide DescriptionElectron spin resonance(ESR) studies have been made of gamma‐irradiated Nb_{2}O_{5}–Na_{2}O–SiO_{2}glasses. The samples received a gamma‐irradiation dose of 2 × 10^{7} R at liquid‐nitrogen temperature, and were investigated at that temperature. A broad spectrum which is characterized by an axially symmetric g tensor and hyperfine interaction with the nucleus, is identified as due to an Nb^{4 +} center formed in a niobium–oxygen unit. A separate, narrow spectrum located in approximately the center of the broad spectrum can be separated into three distinct spectra: a sharp central spectrum, an asymmetric spectrum on the high‐field side, a 10‐line spectrum. The sharp central and asymmetric spectra are identified as due to two different centers formed in silicon–oxygen units. A hole in an NbO_{6} unit is proposed as a model for the 10‐line spectrum.

NMR Study of Ferroelectric LiNbO_{3} and LiTaO_{3}. II
View Description Hide DescriptionA study of the ^{7}Li nuclear magnetic resonance in a LiTaO_{3}single crystal is made and the electric field gradient at the lithium site is determined. Electric‐field‐gradient calculations indicate that the net charge on the tantalum is +1.21 while that on the oxygen is −0.74. This shows that the tantalate ion is predominantly convalently bonded. The ^{93}Nb resonance in a LiNbO_{3}single crystal is also studied and the quadrupole coupling constant is found to be 22.02 Mc/sec.

Evidence for Structural Transformation in Liquid Octyl Alcohols from PVT Studies
View Description Hide DescriptionThe densities of 2‐ and 3‐octanol and 2‐ and 5‐methyl‐3‐heptanol have been measured over a wide range of temperature and pressure. The isochoric loci of 2‐ and 3‐octanol show definite inflections; those of the latter two compounds are linear in the range of measurement. The data are discussed in terms of a structural transformation involving H‐bonded linear chains, dimer rings, and monomer and it is proposed that the inflections characterize the transition from chains to rings and monomer. The proposition is supported by reference to dielectric data and the difference between the two sets of compounds is discussed.

Raman Spectrum of Crystalline Sodium Nitrate
View Description Hide DescriptionRaman scattering from crystalline NaNO_{3} has been observed in all polarization orientations. Except for small intensity contributions in some orientations which are probably due to depolarization effects in the birefringent crystal, the polarizability activities of both internal and external modes follow the expected symmetry selection rules. From relative intensity considerations, the bands at 185 and 98 cm^{− 1} have been designated as the Raman‐active librational and translational lattice modes, respectively. This assignment has been confirmed by the comparison of the lattice spectra of Na^{14}NO_{3} and Na^{15}NO_{3} at 35°K, where an isotope shift is observed only on the lower frequency band. No evidence of mixing between these lattice modes has been observed.

Electron Correlation and Separated‐Pair Approximation. An Application to Berylliumlike Atomic Systems
View Description Hide DescriptionA method is developed for determining, with the help of the variation principle, the best antisymmetrized product of separated geminals for a polyatomic system. The strongly orthogonal geminals are determined directly in natural form and the natural orbitals are expressed in terms of Slater‐type atomic orbitals. An application to the berylliumlike atomic systems uniformly recovers 90% of the correlation energy. Correction for defects in the shell of the calculated wavefunctions gives an estimate of 91% for the correlation energy recovered by the best possible separated‐pair approximation, corresponding to an absolute error of about 5 kcal in beryllium. The correlation energy recovered is analyzed by partitioning the energy expression into intra‐ and intergeminal contributions, and both are further resolved into natural orbital contributions. The analysis makes it possible to assess the roles played by the various natural orbitals in lowering the energy and shows certain types of intergeminal as well as intrageminal interactions to be negligible. It also permits the comparison with calculations by other methods. The introduction of additional terms containing interelectronic distances is found to be less efficient than the further addition of natural orbitals.

Electron Correlation and Augmented Separated‐Pair Expansion
View Description Hide DescriptionAs a refinement over the separated‐pair approximation, the augmented separated‐pair expansion is developed. The wavefunction is expanded in a series of augmented separated‐pair configurations. The leading term is the separated‐pair approximation. Each of the higher terms is an antisymmetrized product of separated space geminals and spin harmonics. The spin harmonics are eigenfunctions of and and, moreover, span irreducible one‐dimensional representations of the geminal subgroups. The separated space geminals are constructed from the natural orbitals of the separated‐pair approximation. It is expected that only relatively few augmented separated‐pair configurations contribute significantly and that they can be readily identified. The additional correlation energy recovered by the augmented separated‐pair expansion beyond the separated‐pair approximation can be classified as correlation between different strongly orthogonal geminals. It can be analyzed by a partitioning of the energy in terms of contributions from the various separated‐pair configurations.

Electron Correlation and Augmented Separated‐Pair Expansion in Berylliumlike Atomic Systems
View Description Hide DescriptionThe method of the augmented separated‐pair expansion, developed in the preceding paper, is applied to the berylliumlike atomic systems from Li^{−} to Ne^{6 +}. The variational calculation recovers 94% of the total correlationenergy, corresponding to an absolute error of about 2.5 kcal in beryllium. The wavefunctions are given explicitly and contain 28 separated‐pair configurations, the only ones out of several hundred investigated which contribute more than 0.01 kcal. The intergeminal correlationenergy is analyzed in terms of contributions from various separated‐pair configurations and the latter are ordered according to their importance. The variation of the intergeminal correlation with nuclear change is discussed and the relation to second‐order perturbation theory is investigated.

Configuration‐Interaction Study of the H_{3} ^{–} System. I. 1 Orbitals
View Description Hide DescriptionA configuration‐interaction (CI) calculation on the ground state of H_{3} ^{–} has been carried out using a minimal basis set made up of a 1 Slater‐type orbital (STO) centered on each nucleus. The most stable nuclear geometry of H_{3} ^{–} is a linear symmetric arrangement of the nuclei with a separation of 2.2098 a.u. and the corresponding value of the energy is −1.581076 a.u. The system is unstable relative to H_{2} + H^{–}. The energy of the system when the nuclei occupy the corners of an equilateral triangle, and the reaction H_{2} + H^{–} have also been studied. No experimental data are available for H_{3} ^{–} but the equation calculated for the interaction of H_{2} with H^{–} as a function of distance agrees fairly well with equations derived from scattering experiments.

Lowest Ionization Potentials of Some Nitrogen Heterocyclics
View Description Hide DescriptionThe photionization potentials of the highest occupied orbital of pyrazine (9.29 eV), pyrimidine (9.35 eV), pryidazine (8.71 eV), naphthalene (8.15 eV), quinoline (8.62 eV), isoquinoline (8.55 eV), quinoxaline (9.02 eV), cinnoline (8.95 eV), and phthalazine (9.22 eV) are determined. In addition, estimates of the second ionization potentials of cinnoline (∼9.1 eV) and phthalazine (∼9.3 eV) are obtained from the analyses of the spectral characteristics of the photoionization yield curves. A discussion of the observed and estimated values of the ionization potentials in terms of the type of the ionized electron ( vs nonbonding) is given. It seems that for all the molecules studied, except pyridazine and phthalazine, the electron has a lower ionization potential than the nonbonding electron. The interchange of the type of the ionized electron in pyridazine and phthalazine is caused mainly by the strong repulsive interaction between the two lone pairs of electrons on the adjacent nitrogen atoms in these molecules as well as by the stabilization of the electrons in these systems.

Relativistic Effect and Nonconservative Spin–Orbital Forces in the Radiative Transition of Molecules
View Description Hide DescriptionA treatment is given of a one‐electron orbital model for a many‐electron molecule, in which each electron is allowed to interact with the over‐all orbital and spin magnetic fields, as well as the (Coulomb) electric field of the rest of the electrons and nuclei. It is shown that when redundancy is properly taken care of, by introducing a factor of for the mutual‐magnetic vector potential between the electrons, the subsequent reduction of Dirac's equation reproduces all of Darwin's orbit–orbit, spin–own‐orbit, spin–other‐orbit, and spin–spin interactions, etc., given by the Breit–Pauli approximation. The above treatment is extended to a system of Dirac electrons interacting with the (time‐dependent) electromagnetic field of radiation, in which field–field interaction in the form of (radiation), is also included. After integration over the photon space, effective transition operators for the large‐component spinors are obtained. When the non‐Hermitian part of the “Dirac Hamiltonian” for the large‐component spinors is not neglected as was customarily done, it is shown that the effective transition operators are already complete, in the sense that they already contain the relativistic‐correction‐of‐mass terms, nonconservative force terms, and corrections due to operation on small‐component spinors. This is shown from the Hermiticity of the operators as well as from explicit derivations. In explicit derivations, the origin of each of the effective transition operators is traced and tabulated. It is shown that for a spinless electron, or a Dirac electron described by an equation linear in time, and by a four‐component spinor, the use of the transition dipole‐length operator is justified. However, the straightforward use of the transition dipole‐length operator on the large‐component spinors, approximated by Breit–Pauli eigenstates, would neglect not only corrections due to small‐component spinors but also the “ordinary” spin radiation term due to the direct spin interaction with the magnetic field of radiation. This interaction is shown to come from a nonconservative force acting on the electron, which is not derivable from a vector potential. All other externalelectromagnetic forces derivable from a vector potential are shown to contribute to radiation in a similar manner, and such contributions are obtainable by analogy to the classical Lagrangian–Hamiltonian formalism for conservative forces. The internal orbital and spin magnetic forces, such as spin–own‐orbit interaction, are shown to contribute to radiation through the non‐Hermitian part of the “Dirac Hamiltonian.” Although these forces can be represented by “effective” vector potentials, it is emphasized that the latter are not usable as (the external) vector potentials in the relativistic Lagrangian, and in the Dirac equation. The magnitudes of the effective transition operators obtained in our treatment are estimated (Table I, see also Table II) for a constant vector potential of radiation over the molecule, and for a (linearly) position‐dependent vector potential. For application to single–triplet transitions, the relative importance of the spin‐dependent transition operators, coming from nonconservative forces, non‐Hermitian terms, and from corrections for small‐component spinors is examined. Explicit expressions of these operators for different propagation and polarization directions are also given (Table III). For example, spin–other‐orbit interaction is shown to contribute to radiative transitions between states of the same parity. The two‐electron radiative transition operators, due to the inclusion of the mutual electromagnetic interaction of the electrons, may be of interest in cooperative optical phenomena, double excitations, and in the radiative contribution to van der Waals forces. The inclusion of the radiative transition operators from the non‐Hermitian term also offers several alternative two‐photon transition possibilities.

Theory of Spin–Lattice Relaxation in Classical Liquids
View Description Hide DescriptionA set of coupled differential equations is obtained which represents an exact solution for the high‐temperature spin autocorrelation function for spins in a liquid whose motion is governed by a classical isotropic rotational diffusionequation with a single rotational diffusion constant, . If the diffusion is rapid, i.e., if is large compared to the spin–lattice interaction, , then these equations can be solved by means of a perturbation expansion in . In this case, the dominant terms correspond to those in the well‐known Redfield theory; in the absence of spin degeneracy the spectrum consists of Lorentzian lines whose widths are of the order of where , and whose frequencies are shifted by an amount of the order of from the Zeeman frequency, where is a characteristic spectral frequency difference. The present theory introduces a number of corrections: The linewidth should be corrected by terms of the order of and ; the frequency shift should be corrected by terms of the order of . Furthermore, a number of weak auxiliary Lorentzian lines at frequencies of the order of from the Zeeman frequencies must be included; these lines have intensities which are of the order of below that of the principal “Redfield lines” and their widths are . The superposition of these auxiliary lines on the Redfield lines gives rise to unsymmetrical, non‐Lorentzian lines, but in the region , where this perturbation expansion is valid, the auxiliary lines contribute little to the central part of the composite lines, but they play a significant role in the wings. The coupled differential equations have been reformulated in order to treat the problem of slow diffusion,. In this case the spin Hamiltonian is diagonalized at each molecular orientation and the diffusion jumps between orientations are treated as a perturbation.