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This paper reports a theory of inner sphere electron transfer reactions for which, in addition to the electron transfer, there is bond formation and/or destruction. We assume that steps in the reaction sequence which involve bond synthesis take place in intermediate activated complex states. The problem of inner sphere electron transfer is large and complicated. The theory described applies to one aspect of the problem, namely, the dynamics of simultaneous electron transfer and bond break in the transition state complex. However, we assume the intermediate complex exists in steady state equilibrium with the reactants. The theory developed in this paper reveals features important in the unimolecular decay of the transition state. In particular, the importance of the Franck-Condon restrictions on bond forming and breaking steps is demonstrated.

In previous papers of this series (Breyer, Bauer, and Hacobian 1954 ; Breyer and Bauer 1956) it was shown that the A.C. polarographic behaviour of organic com- pounds can be explained by postulating adsorption of the substances at the electrode. Empirical and theoretical expressions for the dependence of the alternating current on the bulk concentration of depolarizer were qualitatively correlated. In the present paper, the theoretical treatment is developed further, permitting the evaluation of the adsorption coefficients of the oxidized and reduced forms at the electrode surface.

Sampling techniques provide a practical means of obtaining cross-correlation functions. In this paper, the correlation function is described by sums of the form Z = \begin{equation*}Z = \Sigma^{N}_{j=1}X_{j}Y_{j}\end{equation*}. A general expression is derived for the probability density function of the random variable Z under the condition that Xj and Yj are stationary, jointly Gaussian random processes with nonzero means and unit variances.

The paper contains an investigation of convergence and asymptotic behaviour of the generating functionals on mass shell in the Quantum Field Theory of one real, scalar, selfinteracting field. Also the relations between the Hilbert space fo entire functionals and the various spaces of exponentially bounded entire functionals are treated in some detail. There result various estimates for the functional series and double series representing vectors, operators and their functional derivatives.

In a previous paper, the validity of gradient corrections to the Dirac-Slater exchange energy was discussed within the density functional theory, for a local potential. This local potential, by definition, generated the exact charge density in the (infinite) system considered. The present work considers the full Hartree-Fock theory within the same density functional framework. In particular, using a perturbation expansion of the Dirac density matrix, we obtain the exchange energy as an expansion in the displaced charge Δ H . F . ( r ) = ρ H . F . ( r ) − ρ 0 , , p0 being the average electron density and p H.F. (r) the Hartree-Fock spatially dependent density. In contrast to the local potential theory, a basic integral equation must be solved to express the above relation in explicit form. No exact solution of this equation exists to date, but using the work of Kleinman and Langreth on the vertex function we are able to find an explicit approximation for the exchange energy to second-order in Δ H.F. The difficulties encountered in defining gradient series in a local potential theory are not removed in the full Hartree-Fock theory. However, it is possible to sum sub-series of gradient terms to infinite order to give a useful generalization of Dirac-Slater theory. Although it is known in metals that correlation effects have a major influence on the Hartree-Fock results, the results obtained here may well be useful in atomic and molecular problems with a sufficiently large number of electrons, as well as in non-metallic crystals. The influence of electronic screening in the case of metals is briefly referred to.

A non-empirical valence-electron method suitable for molecular orbital calculations on large molecules containing heavy atoms is described. The method uses a minimum STO basis for the valence orbitals and accounts for core-valence interactions by means of a simple pseudopotential which affords corrections to the diagonal F matrix elements. All integrals involving one and two centres are included, and these are evaluated theoretically. The method produces electronic energies and electron density properties which are comparable with those of corresponding all-electron ab-initio calculations. In this paper two versions of the method are described, the basic non-parametric method and the parametric method which involves a simple modification for the two-centre integrals involving atoms of different size or electronegativity. Results of calculations on first-row diatomic molecules are described. It is hoped that the method will be suitable for calculations on larger molecules and molecules containing heavier atoms.

Using molecular models with the hydrogen bond as the only contributing interaction, the results from existing theories for the excess energies of mixtures with an associating component are compared with each other and with the experimental results for alcohol + alkane systems. After excluding the Redlich-Kister theory as inappropriate, these comparisons are made very effectively using plots of mixture property against composition parameter as then, for each of the Kretschmer-Wiebe or Barker models, one curve gives to a good approximation the mixing property for all systems for any values of the hydrogen bond free energy and enthalpy parameters. The results from the Kretschmer-Wiebe model are essentially the same as those from one of the two Barker models considered. ��� For a given temperature and alkane solvent the experimental excess free energies can also be approximately summarized by one parent mixture property curve. As shown previously this is also true for the excess enthalpies so that comparison of theory and experiment is made very easy by this approach as the results for individual systems do not have to be considered. The theoretical results are in qualitative agreement with the experimental results but give poor values for the excess energies. As deduced from excess enthalpies the extent of hydrogen bonding as a function of alcohol concentration differs from that deduced by i.r. measurements, as discussed fully in the following paper, but the theoretical results cannot be made to agree with either of these results. Reported results for more complex models also appear to reflect the quantitative failings of the simple models.

This paper reviews some of the methods by which electron correlation can be included in calculations on large conjugated molecules. The methods discussed are applied to benzene and the linear polyenes.

The theory of self-nucleation in a supersaturated vapour has been developed with considerable success by Becker and D�ring (1935), but little attention has hitherto been given to condensation at a solid surface. A theory is given for the nucleation of liquid condensates on a plane surface and at lines and corners formed by the intersection of plane surfaces. It is suggested that the line and corner sites are representative of typical features of real crystalline surfaces. The equilibrium properties and free energies of formation of embryos at these sites, which are required for the nucleation theory, have been given in the previous paper. The theory shows that the supersaturation for which nucleation proceeds at a just observable rate on a plane surface increases rapidly as the contact angle increases from zero. Unless an edge free energy is introduced, no supersaturation is necessary for nucleation at a line or corner of intersection of plane surfaces when the contact angle 1s less than a threshold value which depends on the angles between the planes. As the contact angle Increases above the threshold value the supersaturation required for observable condensation increases considerably. The effect of contamination of the substrate surfaces considered and the suggestions are made that, ~f a small amount of soluble contamination is present it is likely to become localized at surface steps and corners, sensitizing them as nucleation sites, and that microscopic observation of incipient condensation on solid surfaces may provide a tool for the investigation of surface structure.

The absorption band centred at c. 77 cm-1 in gaseous mixtures of CF3I and N(CH3)3 previously reported and attributed to the N-I stretching mode of the complex CF3I-N(CH3)3 has been carefully re-examined. This band is of interest as an example of a low frequency ?dissociative type? vibrational mode of a weak molecular complex. The band is asymmetric and apparently structureless with a half intensity width at room temperature of 28-30 cm-1. The width of the band may be accounted for as arising from transitions vi + vi+1 where vi is the vibrational quantum number of the N-I stretching mode with vi up to c. 10 making appreciable contribution to the intensity on the low wave-number side. Centrifugal distortion in the complex is considered. Centrifugal stretching and consequent weakening of the bond may shift the band envelope 2-3 cm-1 to lower wave numbers. Assessment of these and other factors affecting the band shape suggest that the fundamental frequency is probably c. 90 cm-1. The band shape of the vibrational mode of the complex at c. 272 cm-1 is briefly discussed. Many of the considerations presented in this paper should apply to vibration-rotation band shapes in other weak molecular complexes. Some general consequences of anharmonicity for the interpretation of the spectra of weak molecular complexes are discussed.

The dielectric properties of three primary and three secondary long-chain aliphatic alcohols have been investigated within the temperature range of -20 to 70 �C. The experimental results are discussed in relation to the theory of Sack on dielectric absorption in linear polar chains and to the conclusions reached in earlier papers on the dielectric properties of these compounds. The apparent activation energies in the primary alcohols are approximately three times those in the secondary alcohols. This indicates either different mechanisms of absorption in the two types of alcohols or, if the mechanisms are the same, significant differences in the energy barriers involved.

In the tunnel model, a fluid is regarded as being composed of lines of molecules which move almost one-dimensionally in lines or tunnels, the walls of which are formed by neighbouring lines of molecules. This model has been used, with good success, to describe the properties of classical hard spheres and classical simple liquids. For quantum fluids the energy levels of a one-dimensional line of molecules must be calculated. This has been done only for a system of hard spheres. For more complicated systems the energy levels must be approximated. In this paper the tunnel model is applied to liquid 4He at absolute zero and to liquid H2. The lowest energy level has been approximated and the hard-sphere energy levels are used to estimate the higher energy levels. The agreement with experiment is generally good except in the critical region.

A previous publication described some electrical properties of crystalline n-primary alcohols of 16 to 26 carbon atoms. The present paper deals with an extension of the work to the lower homologues down to n-octyl alcohol, and includes the results of measurements on both the liquid and crystalline materials. To enable their more extensive investigation, the frequency range of measurement has been extended down to 10-4 c/s. by the use of D.C. methods.The alcohols of the present series resemble the higher homologues in giving two regions of dielectric absorption which overlap to an increasing extent with decreasing molecular weight. The absorption region at lower frequencies merges with the loss due to D.C. conductivity, the latter being much larger than for other long-chain compounds so far investigated.A feature of the results is the variation in the electrical properties of different samples of the same alcohol. This may be due to variations in the relative proportions of α- and β- crystalline modifications present in different samples.The results suggest that the absorption and the D.C. conductivity both depend on the presence in the crystals of hydrogen-bond chains. The high frequency absorption appears to be related to that in secondary alcohols and is probably associated with reversal of the hydrogen-bond chains by cooperative rotation of the hydroxyl group about the C-O bonds. The latter mechanism, together with proton transition, could lead to migration of the protons, and this may account for the D.C. conductivity. The absorption at low frequencies is attributed to a Maxwell-Wagner type of mechanism resulting from the situation of the conducting hydrogen-bond chains in a non-conducting matrix.

This paper presents a theoretical analysis of the Joule-Thomson cooling of normal hydrogen and deuterium from an initial temperature of 64 �K. Below 70 atm the cooling is greater for H2 than for D2, but at higher pressures it is less. It is found that the difference at low pressures is due to the lower heat capacity of H2, associated with the more quantized rotation of its molecules. The difference at high pressures is due to the less negative enthalpy of imperfection of H2, and this in turn can be attributed to the higher zero-point energy for translational motion in the lighter isotope. Both differences can be accounted for quantitatively.

In this paper we give the results of computing the third virial coefficient and the cohesive energy of the crystal for argon taking into account the higher-order multipole terms in the long-range three- body interaction as recently calculated by Bell. The Barker-Pompe potential has been used as the two-body potential function. We find that the third virial coefficient values for argon computed with this more complete non-additive energy function agree very much better with the experimental values than when only the triple-dipole term is used. This is particularly true at lower temperatures. The results also show that better agreement would be obtained if some form of repulsive non- addivity were included in the computation. For the cohesive energy of the crystal we find that the dipole-dipole-quadrupole energy is one-third as large as the triple-dipole energy and so cannot be neglected in these lattice computations. Furthermore, we find that these higher- order three-body forces do not stabilize the face-centred-cubic lattice for argon, the hexagonal-close-packed lattice having a slightly lower energy.

Excited state bond orders calculated in the previous paper (McCoy and Ross 1962) are used in a necessarily approximate attempt to predict the locations of intersections between potential energy surfaces for all the lower excited states of benzene, naphthalene, azulene, and anthracene, and thus to investigate the mechanism of degradation of electronic excitation energy. It is concluded that excited states commonly, but not invariably, intersect in such a way that there is no barrier to passage from the zero-point level of one state to some state below it. Quite often, however, tunnelling, as suggested by Robinson (1961), is necessary to effect the passage. Diagrams are presented showing the preferred routes for descent through the singlet and triplet states. The radiative properties of the compounds considered are then successfully correlated with the distances through which tunnelling needs to take place, and an approximate empirical relationship emerges in which the tunnelling rate (multiplied by 106 if there is a spin change) decreases exponentially with barrier width. The mechanism of tunnelling is briefly discussed.

The construction and testing is described of photometric apparatus for the comparison of electric birefringences caused in fluid dielectrics by sinusoidally varying applied voltages. The sensitivity displayed is of the order required for measurement of the small incremental Kerr constants shown by very dilute solutions. Acetoxime, cyclohexanone oxime, and the α- and β-isomers of benzaldoxime have therefore been examined as solutes in benzene at concentrations low enough for monomeric forms to predominate in the dissolved species. No marked differences between the α- and β-benzaldoximes have emerged. Certain conformational indications relevant to the C=NOH group are noted at the end of the paper.

Reliable measurements of the small incremental Cotton-Mouton constants encountered at low solute concentrations with solutions in which the magnetic birefringences of solute and solvent do not differ widely cannot be made with equipment at present available. Five such systems are examined in this paper. It is shown that the larger incrementa easily observable among stronger solutions can be fitted to empirical equations containing ascending powers of the concentrations and hence estimates of ΔC/w2 as w2 approaches zero become accessible. Values of ∞(mC2) obtained by this device agree reasonably with others directly drawn from much weaker solutions of the same solutes in carbon tetrachloride or cyclohexane, i.e, in media for which C1 = 0.

Results of a VESCF treatment of pyrazole, its anion, and cation are reported. A comparison is made of the chemical properties of pyrazole and those predicted from the calculated π-electron densities. An ambiguity in the comparison owing to the rapid tautomerization of pyrazole is emphasized. The calculated π-electron distribution in the anion supports the suggestion made in previous papers that the relative electronegativities of carbon and tertiary nitrogen reverse when their π-electron densities exceed 1.2. The dipole moment is predicted to be about 2.5 D for pyrazole, in agreement with observations in solution. Theoretical values of ionization potentials are also presented. An analysis is made of VESCF matrix elements and some empirical guides as to suitable values of coulomb and resonance parameters for five-membered ring nitrogen heterocycles are suggested.

Some earlier theories in which attempts have been made to allow for the influence of molecular shape on the static dielectric constant of a liquid are shown to be inaccurate. A general equation is derived for the dielectric constant of a liquid, and this is then applied to a molecular model consisting, in the first instance, of an ellipsoid uniformly polarized in a principal direction, and secondly, to an optically anisotropic ellipsoid ; in both cases the surroundings are assumed to form a continuum. The results of this more general approach applied to several substances are more satisfactory than those obtained by the original Onsager theory. The theory is also applied to mixtures, and in the special case of a dilute solution in a non-polar solvent, the equation of Ross and Sack is obtained when the ellipsoid is isotropic. A correlation has been noted in the discrepancies found when the dipole moment of a substance is calculated by means of observations on the pure liquid and on dilute solutions in a non-polar solvent, using the equations derived in the present paper.

This paper presents the experimental results on the velocity of sound, densities, and heat capacities of eight organic liquids at 25�, 35�, and 45�C. Using Eyring's equation, the free volumes have been calculated from the sound velocity data. For pure liquids, a quantity Cv* = (Cv)L- (Cv)g- Cstr called the residual heat capacity is found to be linearly dependent on free volume. Analysis of the data for 34 liquids shows that a plot of residual heat capacity against the free volume gives a series of straight lines differing in slopes for different groups of liquids such as hydrocarbons, halogen-substituted hydrocarbons, alcohols, etc. This behaviour is ascribed as being due to different degrees of rotational freedom of molecules in these liquids.

This paper reports some measurements of the second virial coefficient B of cyclopropane in the temperature range 300 to 400°K . It also gives some values of B for cyclohexane and benzene, derived from critical analyses of the published vapour densities and P-V-T properties of these gases between 300 and 650°K . In each case the results have been fitted to the relation B = α + β/T + γ/T2 + δ/T3 The following conclusions can be drawn : (i) The values of B for benzene and cyclohexane are consistent with the effects of pressure on the enthalpies and heat capacities of the gases. (ii) cycloHexane and benzene show large and almost equal deviations from the law of corresponding states for monatomic gases, but cyclopropane shows a much smaller deviation. (iii) It is impossible to fit the second virial coefficients of benzene and cyclohexane to the theoretical curve for a Lennard-Jones (12,6) gas. But they can be fitted to the curve for a (28,7) gas, and the associated force constants are physically reasonable. It appears that the interaction potential for cyclopropane molecules is intermediate between the (12,6) and (28,7) potentials.

The magnetic susceptibilities of 2-hydroxynaphthaldehyde-(1)-[2-hydroxyanil] Cu(II), 5-nitrosalicylalanthranilic acid Cu(II), 5-bromosalicylalanthranilic acid Cu(II), 2-hydroxyformazylbenzene Cu(II), and 2-oarboxyformazylbenzene Cu(II), as well as their monopyridino-compounds, have been measured by the Gouy method at 25 �C. From the data of magnetic susceptibility, the effective magnetic moments were calculated per one copper atom in these chelates. Some of the pyridine-free complexes show magnetic moments that are smaller than the theoretical moment, 1.73 B.M., for one odd electron, while others have normal magnetic moments. Taking into account four other tri-coordinated copper chelates reported in a preceding paper, two alternative suggestions were made as follows. First, the subnormal moments resulting from the close distance of approach of two copper atoms in dimer molecules can be correlated with the planar configuration of chelate rings involving a copper atom. On the other hand, when a three-dimensional ring exists, the approach of two copper atoms to such a small distance as to cause an appreciable exchange effect is hindered by steric effects, resulting in the normal magnetic moment for this type of tri-coordinated complexes. Secondly, those pyridine-free tri-coordinated copper complexes showing normal moments have a carbonyl group, the oxygen atom of which may take part in the coordination about a copper atom.

The theory presented in an earlier paper is here developed as a method for the evaluation of' true dipole moments from data obtained from solutions in polar solvents. As illustrations, the new method is applied to values relating to solutions of several substances in chloroform, chlorobenzene, and nitrobenzene. The results, considering the nature of the problem, are satisfactory. The accuracy is greatest when the solute is highly polar and the dielectric constant of the solvent small. Aqueous solutions of pyridine and of two amino acids are also considered. In the latter cases especially, the predicted dipole moments agree most favourably with anticipated values obtained for the zwitterions by multiplying the electronic charge by the expected charge separation.

When nuclear emulsions are used in the investigation of the angular distributions of scattered particles those tracks which pass through the emulsion surface cannot be accepted for measurement and in consequence the observed angular distribution must be corrected to allow for this. In the present paper, formulae are developed for the case of particles recoiling from a mono-energetic beam of particles incident at any angle on the surface. The effect of small-angle scattering is neglected but on the basis of work done by Powell and Occhialini it is believed that the formulae are valid for recoil energies at least as great as 5 MeV. Curves are given from which the factor can be obtained in any given case.

The present paper describes some calculations of the speed of sound u in a theoretical Lennard-Jones and Devonshire liquid. It is found that u decreases as the temperature is raised, but increases if the liquid is compressed. When quantum effects are considered it is found that u decreases as the value of de Boer's quanta1 parameter Λ* is increased. All these effects have been observed in experiments on real liquids.