Volume 38, Issue 6, 15 March 1963

Magnetic Susceptibility of Diatomic Molecules
View Description Hide DescriptionA variation—perturbation method is presented for the calculation of the magnetic susceptibilitytensor χ from a single‐determinant ground‐state wavefunction for a ^{1}Σ molecule. The diamagnetic contribution χ^{ d } is obtained directly by first‐order perturbation theory and the paramagnetic contribution χ^{ p } is determined by extremizing a functional corresponding to the second‐order energy in an external magnetic field. By an appropriate choice of perturbation function and zero‐order Hamiltonian, the functional is expressed in a relatively simple form requiring only one‐electron matrix elements. The importance of employing Hartree—Fock functions as the zero‐order solution is stressed and an analysis given of the order of the errors in the variation—perturbation treatment. Explicit comparisons of the present method with simplified forms of the variational procedure demonstrate that the neglect of electron exchange or use of an ``average'' perturbation function can introduce significant errors in the results.
Calculations are made of the magnetic susceptibilitytensor for a series of diatomic molecues. Ransil's minimal basis set functions serve as unperturbed wavefunctions and a four‐term polynomial is used to represent orbital perturbations in the second‐order treatment. The values obtained for χ^{ d }, χ^{ p }, and χ of H_{2}, Li_{2}, N_{2}, F_{2}, LiH, HF, LiF, and CO are presented. Comparisons with the available magnetic data for these molecules demonstrates that the variation—perturbation method is a useful tool for ab initiosusceptibility calculations. For the more difficult paramagnetic term χ^{ p }, criteria for the convergence of the second‐order energy are discussed and applied to the molecules considered. The magnetic anisotropy (Δχ=χ⊥—χ∥) is evaluated theoretically for hydrogen and found to be in reasonable agreement with experiment. The inaccuracies arising in Δ_{χ} values for more complex species (through differencing in a purely theoretical treatment) are overcome by the inclusion of rotational magnetic moment data to obtain relatively reliable results. To examine the effect of the vibrational state on the susceptibility of a molecule, an exploratory calculation is made for LiH, which demonstrates that small sut significant deviations from the equilibrium‐distance value are to be expected. Finally, some indication is given of the additional experimental and theoretical work that is required in this field.

Molecular Orbital Theory of Diamagnetic Polarization. III. Anisotropic Properties of the Carbonate and Nitrate Ions
View Description Hide DescriptionMany conjugated molecules have anisotropicdiamagnetic susceptibilities (with the axis of high diamagnetism perpendicular to the molecular plane) which cannot be explained in terms of interatomic ring currents as for aromatic hydrocarbons. The carbonate and nitrate ions are well‐known examples. In this paper, the approximate molecular orbital theory developed in Part I is applied to these systems and predicts considerable anisotropy of the correct sign for both, mainly on the central atom. A corresponding calculation of the C^{13}chemical shift of CO_{3} ^{2—} provides an interpretation of the NMRanisotropy observed by Lauterbur [Phys. Rev. Letters 1, 343 (1958)].

Screening Parameters for Two‐, Three‐, and Four‐Electron Systems
View Description Hide DescriptionA method described earlier for calculating screening parameters in two‐electron atoms has been modified and extended to three‐ and four‐electron systems. The angular and distance parameters describing the interelectron potential are fixed by taking them at their most probable values as determined by a distribution function of the form exp (—V/V _{0}), where V _{0} is the value of the interelectron potential V for these most probable values. This leads to four simultaneous algebraic equations in four unknowns. The accuracy of this simple, empirical treatment is on the order of a few hundredths of an atomic unit.

Relaxation Effects in the Transport Properties of a Gas of Rough Spheres
View Description Hide DescriptionThe rough‐sphere model is investigated in some detail from the point of view of the formal kinetic theory of polyatomic molecules developed by Wang Chang and Uhlenbeck, and by Taxman. The purpose is to clarify the sources of some discrepancies between the known results for the transport properties of a rough‐sphere gas and the results recently obtained by Mason and Monchick in an approximate treatment of the formal kinetic theory, in which the corrections for inelastic collisions are given in terms of relaxation times. It is found that the deviations of the transport coefficients of rough spheres from those of smooth spheres can be understood in first approximation as the result of two effects: an enhancement of the backward and sideward scattering of rough spheres over that for smooth spheres, and an apparent resonant exchange of internal energy when two rough spheres collide. Since these effects are, for the most part, peculiar to rough spheres, it is concluded that the deviations found between the rough‐sphere model and the approximate theory are not to be expected for real molecules.

Fluidity and Prefreezing Cluster Formation in Liquid Metals. Aluminum
View Description Hide DescriptionThe viscosity‐density relation, (η/ρ)^{⅓}=a+b/T due to Mitra is applicable to normal liquids over wide ranges of temperature, and obtainable accuracy is comparable to that of Andrade's well‐known equation connecting the quantities. Liquid metals, however, show anomalous viscosity on approaching the freezing point. Deviation from Andrade's equation in such liquids has been explained by McLaughlin and Ubbelohde as due to the formation of clusters on the approach of the freezing point. It is shown here that a similar analysis is possible starting from Mitra's relation as well. Deviations from Mitra's relation and the prefreezing phenomenon in the viscosity of molten aluminum are examined in terms of the McLaughlin—Ubbelohde theory. Starting from the Mitra relation, cluster parameters for aluminum are evaluated for the first time.

Statistical Computation of Configuration and Free Volume of a Polymer Molecule with Solvent Interaction
View Description Hide DescriptionA lattice model of a polymer molecule with excluded nearest‐neighbor contacts is of particular interest since it can serve as a model for polymer systems in which the interactions between the polymer and its environment are incorporated. These interactions result in the partial overlapping of polymer segments, and consequently the volume fraction of detached polymer segments exceeds the volume fraction of connected polymer segments. It is found that the contraction in volume per segment is independent of the number of polymer segments n. The mean‐square end‐to‐end distance is found to be approximately proportional to n ^{γ} for sufficiently large n, where γ is the mean‐square polymer index, in agreement with Wall et al. The value of γ is shown to be considerably lower than the previous estimates of this parameter for a three‐dimensional lattice with excluded nearest‐neighbor contacts. The decrease in the value of γ from previous estimates is due to the cloverlike structures which result from the addition of a single step to a permissible configuration.

Influence of Vibration—Rotation Interaction on the True Vibrational Wavefunctions
View Description Hide DescriptionA method is suggested for obtaining the true vibrational wavefunctions corrected for vibration—rotation interaction by the application of perturbation theory. For this purpose the harmonic oscillator wave‐functions have been considered to be perturbed by the potentialwhere U(r) is the true potential, V _{0}(r) is the harmonic oscillator potential and [ℏ ^{2} J (J+1) /2μr ^{2}], the vibration—rotation interaction term.
Expressions are given for the integralswhich occur in the transition‐moment matrix elements for electronic bands. Similar integrals occurring in the dipole‐moment matrix elements for vibration—rotation bands, can also be evaluated by using these wavefunctions.

Collision Lifetimes and the Thermodynamics of Real Gases
View Description Hide DescriptionThe perfect gas law p/kT=n=Σ_{ ini } (where n_{i} is in molecules per cm^{3}) is inadequate for describing real gases because of the interactions during collisions. By a simple intuitive argument, these interactions can be taken into account exactly if you know the collision lifetimes. The product of collision rate and lifetime gives the concentration of transient collision complexes, which must be considered in the perfect gas law along with the stable species. As a result, the complete virial expansion is obtained, in both quantal and classical mechanics. The argument leads further to a new form for the partition function which includes the continuum as well as bound states. From this all the thermodynamic functions can be obtained.

Triplet Spin Exchange in Some Ion Radical Salts
View Description Hide DescriptionStudies of the effects of spin exchange on the electron paramagnetic resonance spectra of the ion radical salts (φ_{3}PCH_{3})^{+}(TCNQ)_{2} ^{‐}, (φ_{3}AsCH_{3})^{+}(TCNQ)_{2} ^{‐}, (Cs^{+})_{2}(TCNQ)_{3} ^{=} and (morpholinium)^{+}(TCNQ)^{‐} have been carried out. Exchange parameters have been determined from all three limiting approximations relating to line width and line separation of the exchange‐modified Bloch equations. While in general the exchange frequency ν can be expressed as ν=ν_{0} exp (— ΔE/kT), different values of ν_{0} and ΔE for a given material are obtained depending upon the limiting approximation employed. This ``anomalous'' exchange‐narrowing behavior is discussed in terms of the current model for these triplet‐state systems with particular emphasis upon the possible complicating features presented by the presumed existence of triplet excitons.

Absorption and Rotation of Light by Helical Polymers: The Effect of Chain Length
View Description Hide DescriptionThe absorption of light and the rotation of polarized light near an absorption band is discussed for finite polymers. Only zero‐order wavefunctions and first‐order energies are considered for the polymer, which is treated as a rigid array of identical groups. The length dependence of the optical properties is calculated. The resulting equations are applied to the polypeptide α helix and 3_{10} helix, polyadenylic acid, and dye complexes with these molecules.

Triplet—Triplet Annihilation and Delayed Fluorescence in Molecular Aggregates
View Description Hide DescriptionThis paper deals with triplet—triplet annihilation in pure and mixed organic crystals. In crystals containing a small concentration of impurity traps, triplet excitation migration may proceed from trap to trap on a time scale which is short compared with the long triplet state lifetime but which is long compared with the normal fluorescence lifetime. Nearest‐neighbor and long‐range mutual annihilation of two triplets may then take place giving rise to delayed fluorescence. The rates of long‐range triplet excitation migration and annihilation show a concentration dependence, a temperature dependence, and a solvent dependence. Providing the triplet—triplet annihilation rate is not too fast, the intensity of the delayed fluorescence can be shown to depend upon the square of the intensity of the exciting light. This expectation is borne out by experiments, briefly reported here, on delayed fluorescence in dilute isotopic mixed crystals. In crystals containing high concentrations of such impurity traps, or in pure crystals, the annihilation rate becomes extremely rapid and this mechanism effectively quenches phosphorescence in many, but not all, classes of pure organic crystals. The kinetics of the over‐all process are discussed in both the limits of fast and slow annihilation rates. A theoretical investigation of the origin of the annihilation matrix element is carried out, and it is shown that exchange interactions play the largest role in determining annihilation rates. Past work [H. Sponer, Y. Kanda, and L. A. Blackwell, J. Chem. Phys. 29, 721 (1958); N. W. Blake and D. S. McClure, ibid., p. 722] on delayed fluorescence of assumed pure naphthalene crystals containing very small amounts of β‐methyl naphthalene (traps) can be understood within the framework of this paper. The need in organic crystals for the major delayed fluorescence mechanism to be based upon ionization and electron trapping seems now to be considerably lessened.

Interactions in Polar Media. I. Interparticle Interaction Energy
View Description Hide DescriptionUsing permanent plus induced charge distributions on particles a Hartree method is used to compute the polar contribution to the interparticle interaction enerby. Each particle may be a single molecule or a coll?ction of molecules, a whole electrode or an entire medium for example. The total interparticle interaction energy is taken to be the sum of this polar term and of an interparticle electron correlation term. The second of these includes effects due to interparticle London dispersion and exchange repulsion forces. A symmetry property associated with a quantum mechanical operator characterizing the induced charge distribution is obtained in a compact way using a projection operator. The polar term is also calculated from classical electrostatics and shown to be the same as the one above. The present work is a generalization of earlier treatments in which the particles were single molecules interacting via permanent and induced dipoles.
An application of this work is the more general formulation of theories associated with polar interactions in condensed phases, such as the theory of electron transfer rates in solution and at electrodes, and the theory of electronic spectral shifts and band broadening of polar solutes in polar media.

Ring Currents in Pentacyclic Hydrocarbons
View Description Hide DescriptionMcWeeny's modification of London's molecular orbital theory of π‐electron ring currents has been used to calculate the individual currents in the 15 pentacyclic hydrocarbons. The results, useful in themselves for the calculation of NMRchemical shifts, are also significant in the light of what they shed on the question of the range of validity of the assumption that the current in each ring is approximately equal to the benzene ring current.

Crystallization Wave Propagation in Supercooled Phosphorus
View Description Hide DescriptionThe propagation, through long tubes, of crystallization waves in supercooled phosphorus is experimentally investigated. The propagating temperature fields associated with such waves are measured and the existence of steady‐state fields demonstrated. The crystallization wave propagates as a cloud of crystallites which individually grow and agglomerate during a slushlike interval to finally form the polycrystallinesolid.

Extended Basis‐Set LCSTO—MO—SCF Calculations on the Ground State of Carbon Dioxide
View Description Hide DescriptionExtended basis‐set SCF—STO—MO wavefunctions for the ground state of CO_{2} are reported. In particular, the effect of doubling the number of 2pσ basis functions and the effect of changing their orbital exponents has been investigated, together with the effect of addition of 3dσ_{C} basis functions. The effect of expanding the π basis set by addition of extra 2pπ_{C} basis functions and 3dπ_{O1, O2 } basis functions is also shown. A wave‐function using the best σ basis set together with the best π basis set used in these calculations would give a molecular energy of approximately 9 eV below that obtained from a `best atom' minimal basis‐set calculation. This is to be compared with the estimated Hartree—Fock molecular energy which is approximately 24 eV below the ``best atom'' minimal basis‐set value. This highlights the importance of the choice of basis functions and minimization of the total energy with respect to orbital exponents in the case that the number of basis functions out of which MO's can be constructed is limited.
The quadrupole moment of CO_{2} is computed from two of the wavefunctions reported.

Effect of Purification on the Semiconduction of Imidazole
View Description Hide DescriptionThe technique of measuring the temperature dependence of the electrical resistance of organic compounds directly in a zone‐refining tube has been explored with imidazole. The temperature dependence, determined as a function of the number of zone‐refining passes, became constant after 39 passes. The results were compared with those of imidazole purified by successive sublimations. The contribution of hydrogen bonding to the semiconduction of imidazole is discussed.

Modified Sum Rule for the Vibrational Frequencies of Symmetrical Isotopic Molecules
View Description Hide DescriptionA general method of applying the modified sum rule for the vibrational frequencies of symmetrical isotopic molecules has been derived. This rule states that the difference in the sum of the frequency parameters between two isotopic molecules A and A^{*} is the same for the appropriate sets I, II, ... of the symmetry species. The sets I, II, ... are selected in accordance with the interactions between the equivalent atoms involved in the isotopic change from A into A^{*}. Various sum‐rule relations are obtained for a given pair of isotopic molecules depending upon the interaction terms of the potential function. The proof of the rule is given by using the property of the transformation from the internal coordinates into the symmetry coordinates. The application of the rule is illustrated for ethylene, benzene, and their deuterated homologs.

Self‐Diffusion of Lead 210 in Single Crystals of Lead Sulfide as a Function of Stoichiometry and Doping Additions
View Description Hide DescriptionSelf‐diffusion of lead 210 in single crystals of lead sulfide has been studied in the temperature range of 500° to 800°C. Measurements were made on undoped crystals and on crystals doped with Bi_{2}S_{3} or Ag_{2}S under well‐defined sulfur pressures and under an inert gas. Results show that increasing the cation vacancy concentration by means of excess sulfur or by bismuth additions increases the diffusion coefficient, whereas decreasing the concentration of cation vacancies by means of excess lead or by silver additions also increases the diffusion coefficient over those of stoichiometric PbS. In all cases the electron concentration is accounted for according to:andassuming singly ionized interstitiallead atoms and singly ionized cation vacancies. The activation energies decrease in the order, stoichiometric PbS, lead excess PbS and sulfur excess PbS.
Results support a Frenkel disorder of the cation type.

Statistics of Random Placement, Subject to Restrictions, on a Linear Lattice
View Description Hide DescriptionBy iterative methods of solving difference equations (or by a Monte Carlo method), statistical parameters are calculated for a number of models concerned with linear lattices. These contain sites which may be in one or two states (empty or occupied), and the progressive occupation of the sites is arranged to occur at random, but subject to specified restrictions. Examples of applications are cited from the field of crystallization or chemical reactions of chain polymers, sorption in graphite, and the close packing of hard spheres in one dimension. Special interest attaches to limiting cases in which the lattice is made to go over into a continuous, infinite straight line. It is shown that when such a line is covered with straight‐line segments of constant length by placing these segments on the line at random (but avoiding overlap), the asymptotic packing density approached is approximately 0.748.

Radiative Lifetime of I_{2} Fluorescence,
View Description Hide DescriptionThe radiative lifetime of the fluorescence of molecular iodine vapor, , has been determined to be (7.2±1.0)×10^{—7} sec by using a phase fluorometer. Measurements on equilibrium vapor have been carried out over the pressure range 10^{—4} to 5 mm Hg. Results at the lowest pressures are considerably shorter than the true lifetime due to mixing of scattered, incident light with the fluorescence. This effect has been corrected for analytically. At higher pressures the measured lifetime is shortened by self‐quenching. A Stern—Volmer plot of the data yields a self‐quenching collision diameter for I_{2} of 11.3 Å. Measurements have also been carried out with a molecular beam of iodine vapor.