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Volume 39, Issue 6, 15 September 1963

Numerical Solutions of the Convolution‐Hypernetted Chain Integral Equation for the Pair Correlation Function of a Fluid. I. The Lennard‐Jones (12, 6) Potential
View Description Hide DescriptionThe integral equation of the convolution‐hypernetted chain approximation is solved numerically at five temperatures for the Lennard‐Jones (12, 6) potential. Due to an inconsistency in the approximation there are two equations of state associated with it. These are compared with each other, with other theories, and with the experimental argon equation of state. At high temperatures, the two equations of state agree both with experiment and with more a prioritheory at low densities and bracket them at higher densities, their mean giving a good representation of both. The integral equation is found to be singular at certain temperature—density points, their locus forming a dome shaped curve in the T—ρ plane. These points are shown to correspond to the limits of metastability in the van der Waals gas, the highest temperature point being the critical point. The nonsingular points below the critical point correspond either to a homogeneous gas (low densities) or liquid (high densities) phase. The former are in good agreement with the three‐term virial series, as expected, and the latter are a considerable improvement over it. The equation of state is a considerable improvement over that obtained from the Born—Green equation by Kirkwood, Lewinson, and Alder, the latter predicting negative pressures over most of the liquid range while these results show only positive pressures. The comparisons with experiment are obscured by the inadequacy of the (12, 6) potential function. The use of temperature ``compensated'' potential parameters (as determined by the second virial coefficient) indicates that the use of a more reasonable potential function in the theory will give a relatively good representation of the equation of state of a fluid over a wide range of temperatures and densities.

Numerical Solutions of the Convolution‐Hypernetted Chain Integral Equation for the Pair Correlation Function of a Fluid. II. The Hard‐Sphere Potential
View Description Hide DescriptionThe nonlinear integral equation of the hypernetted chain (HNC) approximation is solved for a fluid of hard spheres. An inconsistency introduced by the approximation results in a disagreement between two forms of the equation of state. The resulting pair of equations of state are compared with each other, with those obtained for the Born—Green and (approximate) Kirkwood equations by Kirkwood, Maun, and Alder, and with a reference isotherm made up of a combination of the five‐term virial series at low and intermediate densities and the gas‐like isotherm of molecular dynamics at high densities. The HNC equations of state are found to be a considerable improvement over those of the Born—Green and Kirkwood theories. At intermediate and high densities, the two HNC isotherms bracket the reference isotherm, their mean being a very good approximation to it. The HNC equation exhibits a tendency toward a gas—solid phase transition at extremely high densities. A transition is not actually predicted, however.

An Extended Hückel Theory. I. Hydrocarbons
View Description Hide DescriptionThe Hückel theory, with an extended basis set consisting of 2s and 2pcarbon and 1s hydrogen orbitals, with inclusion of overlap and all interactions, yields a good qualitative solution of most hydrocarbon conformational problems. Calculations have been performed within the same parametrization for nearly all simple saturated and unsaturated compounds,testing a variety of geometries for each. Barriers to internal rotation, ring conformations, and geometrical isomerism are among the topics treated. Consistent σ and π charge distributions and overlap populations are obtained for aromatics and their relative roles discussed. For alkanes and alkenes charge distributions are also presented. Failures include overemphasis on steric factors, which leads to some incorrect isomerization energies; also the failure to predict strain energies. It is stressed that the geometry of a molecule appears to be its most predictable quality.

Cross Sections of Ion—Permanent‐Dipole Reactions by Mass Spectrometry
View Description Hide DescriptionProton transfer is the most probable type of reaction observed in a number of ion—molecule interactions involving permanent dipoles. Integrated cross sections, which are unusually large, are described in terms of an ion—dipole pair oriented near the position of minimum energy at low relative velocity. As the relative velocity is increased, this alignment becomes less likely. The energy‐dependent cross section is eclipsed at high relative velocity by the ``hard'' cross section which does not vary with energy. Proton affinities are estimated for several alkyl cyanides. Some of the reported reactions appear to involve excited states of the primary ions.

Do the ``Ligand Field'' Parameters in Lanthanides Represent Weak Covalent Bonding?
View Description Hide DescriptionInstead of explaining the seven different f‐orbital energies or five different d‐orbital energies by parameters A_{n} ^{m} 〈r^{n} 〉 of the electrostaticligand fieldmodel, we propose to classify the energy levels according to the actual one‐electron energies and to interpret these quantities by the weak effects of σ antibonding on the partly filled shell. Calculations of the relative angular dependence Ξ^{2} of such effects are made in a simple model and compared with experimental data for nine‐ and eight‐coordinated lanthanide compounds. The agreement is judged to be much more satisfactory than when the electrostaticmodel is applied, and the number of freely chosen parameters is much smaller.

Nonadiabatic Effects in the High‐Energy Scattering of Normal Helium Atoms
View Description Hide DescriptionThere is a large discrepancy (∼9 eV) between the calculated adiabatic electronic energy of the ground state (X ^{1}Σ_{ g } ^{+}) of the system He_{2} and the effective scattering potential deduced from experiments with high‐energy (2 keV) atomic beams, at small interatomic distances (R=0.53 Å). It is shown that at least a significant part (25%) of this discrepancy arises from nonadiabatic effects of high relative angular velocity of the atoms in the collisions typical of the experiment. A perturbation‐theory treatment is included to give an estimate of the over‐all magnitude of the effect; but a variational calculation is employed with the nonadiabatic Hamiltonian to produce the minimum estimates of nonadiabatic energy shifts to which we refer above. The theory correctly predicts the observed approximate agreement between adiabatic energies and the experimental potentials at larger distances (R=1.06 Å). A method of extending the variational calculation to include continuum contributions is described; it employs a pseudo‐Hamiltonian to simplify the calculation. Arguments connected with this formulation are advanced to suggest that when continuum contributions are taken into account it may be possible to account for all or most of the observed discrepancy.

Observation of Fluorescence in Ketone Solutions of Europium and Terbium Salts
View Description Hide DescriptionRelatively long‐lived fluorescence has been observed in simple ketone solutions of europium and terbium salts at room temperature without any deliberate formation of chelates or other complexes. Emission and absorption spectra of the solutions are presented, as well as the effects of dilution and water content on their fluorescent output. The temperature behavior of the fluorescent output and lifetime is also shown.

Effects of Isotopic Labeling and Molecular Structure on Probability of Ionization by Electron Impact
View Description Hide DescriptionIsotope effects have been reported on probability of ionization of molecules by photons, by excited rare‐gas atoms, and by beta rays. Such effects furnish evidence that ionization occurs largely as a secondary process in competition with nonionizing processes following primary excitation to an energy level above the ionization potential.Measurements of total ion current produced in a mass‐spectrometer source show similar effects for the case of ionization by 70‐V electrons. Experimental support has been obtained also for the corollary that probability of ionization depends upon molecular structure and is not simply, as has been postulated, a constitutive property of the atoms.

Pressure‐Induced Shifts of Molecular Lines in Emission and in Absorption
View Description Hide DescriptionPressure‐induced shifts of lines in the 2–0 band of HCl have been measured both in emission and in absorption. Within the accuracy of the measurements, no consistent difference between absorption and emission was found. The result casts doubt on the validity of some static theories of line shape in their present form.

Magnetic Characteristics of Tb–Y and Ho–Y Solid Solutions
View Description Hide DescriptionMagnetization—temperature data are presented for temperatures ranging from 4.2° to 300°K for Tb–Y and Ho–Y solid solutions in which the Y content varies from 0 to 90 at. %. Incorporation of Y into the Tb lattice suppresses the tendencies for both ferromagnetic and antiferromagnetic ordering. The former in particular is affected so that alloys containing 70% Tb or less no longer exhibit ferromagnetism in the absence of an applied field. The Néel point (T_{N} ) is lowered when Ho is diluted with Y but the temperature of its lower magnetic transition point (sometime called its Curie point) is essentially unchanged. is observed to vary linearly with composition in both the Tb–Y and Ho–Y systems. The paramagnetic behavior of both systems is in general accordance with that expected on the basis of the simple molecular field picture. The magnetization—temperature behavior of Tb_{70}Y_{30} is anomalous, resembling that of Dy_{90}Y_{10} and Dy_{95}Y_{5}. Under certain circumstances these alloys appear to be magnetically two phase.

Analysis of the Electronic Spectra of Neodymium Ethylsulfate
View Description Hide DescriptionThe polarized absorption spectra of single crystals of neodymium ethylsulfate [Nd(C_{2}H_{5}SO_{4})_{3}·9H_{2}O] have been studied between 11 000 cm^{—1} and 28 500 cm^{—1} at 77° and at 4.2°K. From the 77°K polarized absorptionspectrum of crystals of different thicknesses, it is possible to identify transitions from all the Stark components of the ground level ^{4} I _{9/2} to many of the Stark components of excited SLJ levels of the 4f ^{3} configuration. The electronic transitions particularly of the ^{4} I _{9/2} ground level are reliably singled out from among vibronic transitions. The experimental assignment based on selection rules and the number of transitions observed is further verified by a comparison of the experimental levels with the ``free‐ion'' spectrum predicted on the basis of the parameters F _{2}=331.33 cm^{—1}, F _{4}=47.956 cm^{—1}, F _{6}=5.313 cm^{—1}, and ζ=880.11 cm^{—1}.
A first‐order crystalline‐field splitting calculation based on the parameters B _{2} ^{0}=58.4 cm^{—1}, B _{4} ^{0}=—68.2 cm^{—1}, B _{6} ^{0}=—42.7 cm^{—1}, and B _{6} ^{6}=595 cm^{—1} gives a predicted splitting to within a mean deviation of 4 cm^{—1}.

Molecular Orbital Studies of Excited States of HeH
View Description Hide DescriptionThe total energy of HeH was computed using a method of molecular orbital calculation in which the total wavefunction is taken as a linear combination of configurations, each a properly antisymmetrized product of flexible one‐electron functions in an elliptic coordinate system. We have examined the lowest ^{2}Π state of this system and both of the ^{2}Σ excited states which arise from the interaction of ground‐state He and a 2p or 2s H atom. The lowest ^{2}Π state of HeH and the first excited ^{2}Σ state are found to be bonding with calculated dissociation energies of 1.935 and 2.336 eV, respectively. A comparison of these results with those previously reported for the ground state of HeH and the molecular ion HeH^{+} indicates that the charge distribution about the helium atom is similar in all of these systems and it is predicted that, with the single exception of the ground state, all of the doublet states of HeH up to the ionization limit of the H atom are bonding and can be represented by very similar potential energy curves.

Radiation Decomposition of Aqueous Nitrite Solution Containing Mercuric Salts
View Description Hide DescriptionAqueous sodium nitrite solutions containing mercuric salts give mercurous salts when irradiated with Co^{60} γ rays. The G value of mercurous salt attains to such a high value over a wide range of solute concentration that the accepted radical yields cannot account for.
In order to settle the apparent contradiction it is suggested that in a spur several excited water molecules may be produced and that they can react with nitrite ions to form excited complexes which can reduce mercuric into mercurous salt.

Photolysis of Aqueous Nitrite Solution Containing Mercuric Salts
View Description Hide DescriptionPhotolysis of aqueous sodium nitrite plus mercuric salt solutions has been studied with a high‐pressure mercury lamp. The irradiation of mercuric salt solution does not give any noticeable amount of mercurous salt whereas the admixture of sodium nitrite and mercuric salt does.
Two possible explanations are suggested for the reduction of mercuric salt; one is that the absorption of light quantum by nitrite ions induces electron transfer from the ions to water molecules in the hydration shell and that the expanded electron of the excited ions reduce mercuric to mercurous. An alternative is that nitrite ions are excited to their low‐lying electronic states in which they can reduce mercuric salts directly.

Photolysis of Aqueous Azide Solution Containing Mercuric Salts
View Description Hide DescriptionThe previous study of photolysis of sodium azide plus mercuric chloride solution has been extended to the solutions of sodium azide plus several mercuric salts. The yield of photolysis is given as a function of solute concentrations. From the measurement of light intensity it is found that the photolysis is not a chain reaction as stated in the previous paper.

Raman Spectral Studies of Aqueous Solutions of Selenic Acid
View Description Hide DescriptionRelative integrated Raman intensities of aqueous solutions of selenic acid have provided concentrations of the species H2SeO4, HSeO4 ^{—}, and SeO4 ^{2—}. When water is added to supercooled selenic acid at 25°C, the molecular H2SeO4 concentration, [H2SeO4], which is approximately 18 mole liter^{—1} in the pure acid, decreases rapidly and approaches zero, as the stoichiometric concentration of selenic acid, C H2SeO4 , approaches 11–12M. Simultaneously, the HSeO4 ^{—} concentration, [HSeO4 ^{—}], increases, but it falls below the stoichiometric water molarity C H2O as dilution progresses. When C H2SeO4 = C H2O = 14.5M, [HSeO4 ^{—}]<C H2O, indicating that the reaction H2SeO4+H2O = HSeO4 ^{—}+OH3 ^{+} is incomplete. With further dilution, [HSeO4 ^{—}] attains a maximum value of about 12.9 mole liter^{—1}, at C H2SeO4 = 13.5M, and near C H2SeO4 = 11–12M, where [H2SeO4]≈0, the absence of intense characteristic Raman lines of SeO4 ^{2—} indicates that [H2SeO4 ^{—}]≈[OH3 ^{+}]. Below C H2SeO4 = 11–12M, Raman lines of SeO4 ^{2—} appear, as [HSeO4 ^{—}] continues to decrease. When C H2SeO4 has decreased to approximately 4.2M, [SeO4 ^{2—}] attains a maximum value of about 0.91 mole liter^{—1}, and the concentration at which [SeO4 ^{2—}] is maximal, compares favorably with the concentration corresponding to a maximum in the electrical conductivity, C H2SeO4 = 3.7M.
Fundamental vibrational frequencies of H2SeO4, HSeO4 ^{—}, and SeO4 ^{2—} have been assigned according to structures of C 2v , C s , and Td symmetry, respectively, and those assignments are compared with vibrational assignments of the closely related species H2SO4, HSO4 ^{—}, and SO4 ^{2—}. Other comparisons between Raman spectra of nearly pure selenic and sulfuric acids, however, reveal additional frequencies at 932 and ∼1200 cm^{—1}, respectively, produced by intermolecular vibrations. Such additional vibrations suggest the existence of extended, strongly hydrogen‐bonded networks, in which symmetric vibrations of the SeO2 or SO2 groups of individual H2SeO4 or H2SO4 molecules are out‐of‐phase in neighboring molecules. In addition, another frequency of aqueous solutions of selenic acid at ∼800 cm^{—1}, obtained from careful analysis of band contours, presumably is also produced by intermolecular vibrations between OH3 ^{+} and HSeO4 ^{—}, in accordance with the low electrical conductivity and the high viscosity of solutions in which those ions are predominant.

A Variation‐Perturbation Approach to the Interaction of Radiation with Atoms and Molecules
View Description Hide DescriptionA variation‐perturbation formulation for time‐dependent interactions is presented and applied to the interaction between atoms or molecules and a semiclassical radiation field. Details of the treatment through second order are given for the steady‐state problem and are compared with the standard perturbation‐theory results. It is demonstrated that the usual expression for the time‐dependent perturbationsolution, which consists of a sum over an infinite set of excited states (including any continuum), can be replaced by a small number of frequency‐dependent functions that satisfy certain time‐independent differential equations. Since an exact solution to these equations is very difficult for all but the simplest cases, an equivalent variational formulation of more general utility is introduced.
The induced time‐dependent electric and magnetic moments of atoms and molecules in an oscillating electromagnetic field are evaluated in terms of the frequency‐dependent functions. With the usual continuum medium assumptions, the macroscopic polarizations are determined from these moments, and simple formulas for the index of refraction and optical rotatory power are obtained. The method is illustrated by a simple application to the dynamic polarizability of the harmonic oscillator and by a more detailed examination of the dynamic polarizability and index of refraction of the nonrelativistic hydrogen atom. A comparison is made between an exact solution and a variational approximation for this problem. The excellent agreement between the two methods provides an indication of the power of the variational formulation. An indication of its utility for more general applications is given.

HFS Constants of OH Ground State
View Description Hide DescriptionFermi contact term at the proton in the OH ground state is pointed out to be negative either from a simple qualitative consideration or from an LCAO MO CI calculation. The calculated value agrees well with the experimental value revised recently by Radford. Other hfs constants, 〈1/r ^{3}〉_{H} and 〈2P _{2}(cos_{χ})/r ^{3}〉_{H}, are also calculated by a single configuration approximation with the use of an analytical Hartree—Fock orbital. The results agree quite well with the experiment, although the agreement is somewhat worse for 〈sin^{2} _{χ}/r ^{3}〉_{H}.

Destruction of Acetylene in Flames with Oxygen
View Description Hide DescriptionThe rate of consumption of acetylene in premixed flames is compared with the simultaneous rates of disappearance of oxygen, supposed to react by H+O_{2}⇄OH+O, and appearance of carbon dioxide, OH+CO→CO_{2}+H. The comparisons are consistent with a consumption of acetylene by O atoms but not by H, OH, or O_{2}. The destruction probably involves a direct attack of O on carbon in acetylene with a rate constant of 1 to 2×10^{13} cm^{3} mole^{—1} sec^{—1} and little temperature dependence.

Assignment of NMR Spectra with the Aid of Double‐Quantum Transitions
View Description Hide DescriptionThe analysis of complex nuclear magnetic resonance spectra usually requires the assignment of the observed transition to an energy‐level diagram. It is shown that double‐quantum transitions can be used to determine such an assignment. Small frequency shifts of the double‐quantum transitions have been observed and their dependence upon the strength of the applied radio‐frequency field measured and compared with theory. The double‐quantum method is compared to the double resonance method in making an energy‐level assignment using the 60‐Mc/sec spectrum of trivinylphosphine for the purposes of illustration. The analysis has been completed by an algebraic method and by iteration on a computer, and the results checked by recording the 100‐Mc/sec proton spectrum and the 19‐Mc/sec P^{31} spectrum. The three proton—proton couplings and the three P^{31}proton couplings all have the same sign.