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The adiabatic correction to the Born‐Oppenheimer approximation has been evaluated as a function of internuclear separation for the diatomic molecules in the isotopic exchange reaction HX+HD=DX+H2, where X=D, Li, or Cl. The sum of the isotope‐dependent adiabatic correction and the isotope‐independent Born‐Oppenheimer electronic energy is the effective potential energy for nuclear motion if nonadiabatic effects are neglected. The resulting shifts, from the Born‐Oppenheimer values, of the equilibrium internuclear separations and of other vibrational constants are calculated. It is found that these shifts are so small that they have negligible effect on calculated values of the isotopic exchange equilibrium constants. The only important contribution of the adiabatic correction to the equilibrium constants arises from the fact that the ΔE of the isotopic exchange reactions between translationless, rotationless, vibrationless reactants and corresponding products no longer equals zero.

A simple formula is derived for the nuclear mass dependence of the adiabatic correction to the Born–Oppenheimer approximation for molecules with zero total angular momentum in nondegenerate electronic states. The importance of this result to the calculation of isotopic exchange equilibrium constants is pointed out.

Corrections to the Born‐Oppenheimer electronic energy are calculated for several first and second row diatomic molecules. These corrections are evaluated in the adiabatic approximation at the experimental internuclear distance and thereby constitute the leading terms in the non‐Born‐Oppenheimer molecular energy. The corrections to the electronic energy are functions of the nuclear masses and are found to produce significant electronic isotope effects on the equilibrium constants of certain exchange reactions. In particular, at 300 °K the electronic isotope effect on the equilibrium constant of the reaction HX + HD = DX + H2 where X is Li, B, N, or F is found to range from 1.029 for Li to 1.101 for B. By partitioning the correction to the Born‐Oppenheimer electronic energy into a sum of atomic and overlap terms, the electronic isotope effect is shown to be dependent upon certain characteristics of the molecules, the single most important of which is the X atom electronic angular momentum.

The vibrational Hamiltonian of a molecule in the harmonic approximation, H = (1/2) J (gijpipj + fijqiqj), has been divided into a diagonal part (terms with i=j) and an off‐diagonal part (i≠j), which is regarded as the perturbation. The vibrational partition function of the molecule is then calculated by Schwinger perturbation theory as the partition function of the unperturbed problem, corresponding to a collection of oscillators with frequencies 2πνi′ = (fiigii)1/2, plus perturbation correction terms which are calculated to second order. With the usual assumptions of isotope effect calculations that the molecular translations and rotations are classical and separable from the vibrations, the perturbation formulation of the vibrational partition function is easily transformed into a perturbation theory formulation of (reduced) isotopic partition function ratios. If, for example, the molecular potential function is expressed in terms of the displacements of bond stretches and bond angle bends from their respective equilibrium values, the unperturbed partition function ratio corresponds to the isotope effect expected for noninteracting bond‐stretch and bond‐angle‐bend oscillators. Detailed comparison is made for a number of molecular systems of perturbation theory calculations of partition functions and isotopic partition function ratios with exact calculations carried out by actually obtaining the normal mode vibrational frequencies of the vibrational Hamiltonian. Good agreement is found. The utility of the perturbation theory formulation resides in the fact that it permits one to look at isotope effects in a very simple manner; some demonstrations are given.

Ideal gas phase isotopic partition function ratios involving both deuterium and tritium substitution for hydrogen and also heavy atom substitution in H2, H2O, H2S, H2Se, and NH3 have been calculated at a number of temperatures. The results have been least squares fitted to a five term series in powers of (300/T). Various correction factors to the harmonic approximation were considered in the calculation of the partition function ratios. It is demonstrated that the tabulated ratios can be used to calculate equilibrium constants for isotopic exchange reactions.

We have made use of a semiempirical treatment to calculate the energies of the molecular orbitals for the ground state and the first few excited states of permanganate, chromate, and perchlorate ions. The calculation of the excitation energies is in agreement with the qualitative features of the observed spectra, i.e., absorption in the far ultraviolet for ClO4−, two strong maxima in the visible or near ultraviolet for MnO4− and CrO4= with the chromate spectrum displaced toward higher energies. An approximate calculation of the relative f‐values for the first transitions in CrO4= and MnO4− is also in agreement with experiment.The data on the absorption spectra of permanganate ion in different crystalline fields is interpreted in terms of the symmetries of the excited states predicted by our calculations.

The anharmonicity correction to the zero‐point energy change in the triatomic dihydride isotopic self‐exchange equilibria, H2X+D2X=2HDX, is discussed. It is shown that the inclusion of the usually ignored term G0 in the expression for the anharmonic correction to the zero‐point energy (G0 + 14Σxij) causes the anharmonicity correction to the zero‐point energy change to be quite small for the cases considered (X is O, S, Se), but when this term is ignored the correction is found to be nonnegligible. Equilibrium constants are calculated, and it is found that the agreement between theory and experiment is considerably improved by including the G0 factor.

A nonrandom method for approximating multidimensional integrations is compared to traditional ``random'' Monte Carlo techniques for determining reaction cross sections from quasiclassical trajectories. Reaction cross sections are calculated for the H + H2 collision system to determine the relative rates of convergence for each method as a function of the number of trajectories employed in the calculation. Estimates of the maximum probable error for the nonrandom integrations of a given number of trajectories compare favorably with the errors expected for Monte Carlo integrations and seem to imply that nonrandom integrations are more accurate than Monte Carlo calculations. However, the error of a Monte Carlo integration is easily estimated while that of a nonrandom integration is not. Thus the nonrandom method's advantage in accuracy may be overcome by other characteristics of the Monte Carlo method. For classical trajectory calculations, the nonrandom method should be considered as an alternative, but not...

Recent calculations of the binding energy of the nitrogen molecule carried out by Kopineck are examined by means of molecular orbital calculations. The Hellmann‐Kopineck procedure of omitting certain permutations in a valence bond procedure is investigated. It is concluded that the Kopineck calculation should not be used as an argument favoring the 9.8 ev value for the binding energy (D00) of the nitrogen molecule.

Interaction of the hindered translations and rotations with the internal vibrations in the condensed phase leads to isotope‐dependent shifts of the internal frequencies. This isotope dependence is a necessary consequence of the fact that the coordinates representing the translations and rotations of the molecule as a whole are, in general, isotope dependent. This external—internal interaction is investigated and the XH2 system is employed as a simple example to demonstrate the nature of the interaction. It is shown that the experimentally determined vapor pressures of the isotopic ethylenes may be rationalized when the external—internal interaction is taken into account. A force field based on a simple cell model for liquid ethylene is obtained which yields good agreement with the experimentally determined vapor‐pressure isotope effects. The force field, while not unique, fits (and is in fact partially based on) other available data on liquid ethylene.

Two methods are proposed for calculating zero-point energies of molecules. The first makes use of the fact that one can easily write down the quantum mechanical HAMILTONian for a vibrating system. The zero-point energy can then be obtained by a perturbation scheme without solving the secular equation. The second method requires a knowledge of the normal modes and frequencies of a reference molecule, but then enables one to calculate isotope effects by a perturbation scheme. The methods are applied to some examples and the convergence of the perturbation series is investigated. The approximate validity of the law of the mean for the isotope effect on zero-point energies is explored within the framework of the methods.

A semiempirical treatment, which gives satisfactory agreement between calculated and observed spectra for chromate and permanganate ions, has been employed to calculate the excited state energies and transition moments for the related molecules of the chemical formulas CrO3X— and CrO2X2. The results show agreement with experimental data which unfortunately are not extensive for these molecules. The calculations indicate an absorption for SO2Cl2 in the farther ultraviolet in agreement with experiment. The qualitative significance of the results, so far as the relative importance of σ and π bonding, is discussed.

The theoretical formulation of the unimolecular decomposition of excited (active) molecules developed by Marcus is applied to the numerical calculation of rate constants for such processes in the threshold region. The Marcus formulation for calculating numbers of vibrational energy states and densities of such states is briefly investigated. Threshold‐region rate constants are investigated as a function of threshold energy and of molecular size. Decompositions of excited molecule ions in the ion source of a mass spectrometer are discussed in terms of these rate constants. It is found that this more exact theory of unimolecular dissociations removes some of the difficulties found in an earlier approach to the theory of mass spectra. Some interesting predictions of the present formulation are pointed out.

Exact classical calculations are carried out on energy transfer from the translational degree of freedom to the vibrational and rotational degrees of freedom in nonreactive atom–diatomic‐molecule collisions. The diatomic molecule is either a harmonic or a Morse oscillator and the interaction potential is exponentially repulsive in the interatomic distances. The exact calculations are used to test the validity of a number of approximations which have been applied in theoretical treatments of this type of system. It is shown that the dynamic interaction between molecular rotational and vibrational motion during collision can introduce important differences between noncolinear and colllinear collisions; as a result, great care must be exercised in drawing conclusions from calculations on collinear systems.

The relative rates of reaction of H2 and HD with chlorine atoms have been measured over the temperature range of 243–350°K. In this temperature interval, the ratio of the second‐order rate constants, R=kH2/ΣkHD, is equal to (1.24∓0.03) exp (490±6/RT). A search was made for HD after an unequilibrated mixture of H2 and D2 was half converted to hydrogen chloride by photochemical reaction with Cl2. From the failure to detect 0.02% HD in the unreacted hydrogen, a lower limit is set for the ratio of the rate constants of the reactions H+Cl2→ lim k16HCl+Cl and H+HCl→ lim k17H2+Cl.It is shown that the pre‐exponential factors in R for the HD experiments and in the analogous experiments on HT are in quantitative agreement with theoretical calculations for either linear or triangular transition states, subject to the sole restriction that k(HD+Cl→HCl+D) is approximately equal to k(HD+Cl→DCl+H). An intercomparison is made between the experimental difference in activation energies between H2, HD, HT, and D2 for reacti...

It is shown that secondary kinetic isotope effects, for all isotopes with the possible exception of hydrogen isotopes, are expected to be small unless the force constants involving the isotopically substituted positions are considerably different in the transition state from those in the reactant. Cases involving secondary hydrogen isotope effects are investigated in detail for certain model reactions. It is concluded that hydrogen isotope effects do not constitute an exception to the above rule.

The anharmonicity correction to the zero‐point energy of isotopic ammonias is theoretically evaluated. The G0 term, which is usually ignored, is found to make a significant contribution to this correction. The comparison of the corrections leads to the conclusion that the anharmonicity corrections to the isotopic self‐exchange equilibria among the ammonias are negligible. Other correction factors to the self‐exchange equilibrium constants calculated in the usual harmonic approximation are also considered and found to be quite negligible. The calculated self‐exchange equilibrium constants are compared with experiment and good agreement is found. Agreement is also found between theory and experiment for the isotopic exchange equilibrium constant between ammonia and hydrogen.

Model calculations, based on the statistical-mechanical formulation of isotope effects, have been used to predict how analyses of experimentally measured isotope effects may be used to gain information concerning the differences between the reactants and the transition state in a rate process or between the reactants and the products in an equilibrium process. It is shown that while isotope effects cannot be used generally to obtain reliable information about geometry differences, they can be used to determine or set limits on force constant differences and possibly to gain insight into the nature of reaction coordinates. Information of this type can be useful in the elucidation of reaction mechanisms. A method by which isotope effects for large molecular systems may be studied theoretically by working with smaller model molecules is presented.

The theory of electronic, vibrational, and rotational excitation of a molecule, specifically a diatomic molecule, due to β decay of a substituent atom is presented. The nature of the vibrational and rotational excitation for certain specific cases is mathematically investigated. The problem of dissociation arising from the excitation of rotations is discussed. It is shown that it is very difficult to obtain exact figures on the probability of dissociation due to the excitation of vibrations and rotations. Molecular orbital wave functions are employed to gain some insight into the electronic excitation of the daughter molecule. A very rough calculation is made of the probability of nondissociation in the decay of C14 labeled ethane.

Calculations have been carried out to demonstrate that there is usually very little quantum‐mechanical contribution to an isotope effect on a rate (equilibrium) in the absence of force constant changes, between reactant and transition state (product), at the position(s) of isotopic substitution. The usual upper limits to the deviations of such secondary isotope effects [(k1/k2)(ν1L‡/ν2L‡)−1 or K1/K2] from unity, at room temperature, have been estimated to be 1.5% per α deuterium, 0.2% per β deuterium, and even less for heavy‐atom isotopes or for deuteriums further removed from the reaction site. Some exceptional circumstances under which there may be large ``no‐force‐constant‐change'' isotope effects have been found. However, usually it is quite proper to interpret the existence of sizable quantum‐mechanical isotope effects as indicative of force constant changes at the isotopic position(s).

The relative rates of the reactions H+HCl→ lim k3H2+Cl and H+Cl2→ lim k6HCl+Cl, were determined in the temperature range of 0° to 62°C and found to be given by k3/k6=(0.143±0.033) exp−(1540±130/RT). The temperature‐independent factor in the above expression is interpreted in terms of the structure and the vibrational frequencies of the HCl2 transition state. The doubly degenerate bending frequency of this transition state is found to have a value of about 105 cm−1.

A numerical integration technique based upon the use of nonrandom number sequences is examined with test integrations of a simple, analytical function. A comparison of the nonrandom technique with the familiar Monte Carlo method shows that the error of the new method decreases faster as more points are used in the calculation. Moreover, the new method needs fewer points to calculate an integral to an accuracy of 10% than does the Monte Carlo method.

The Hamiltonian equations of motion for collinear‐colliding diatomic‐molecule—atom and diatomic‐molecule—diatomic‐molecule systems are numerically integrated to obtain vibrational energy‐transfer data. The diatomic molecules are taken to be either harmonic or Morse‐type oscillators, and the collision potential is either an exponentially repulsive potential or a Lennard‐Jones 6–12 potential.The results of the exact classical three‐body calculations are compared to results predicted by classical and semiclassical approximations. The ratios of approximate to exact energy‐transfer values are related to the value of a parameter which depends only upon the masses of the particles involved in the collision; this correlation is found to be relatively insensitive to the values of the collision energy and of the potential parameters over wide ranges.The exact classical energy‐transfer data for the four‐body systems are compared to the approximate and exact three‐body data. The validity of using three‐body approxima...

It is demonstrated that one can simplify statistical‐mechanical calculations of isotope effects on rates and on equilibria, without introducing significant error, by omitting large portions of the molecules involved. One can usually ``cut off'' parts of the molecules more than two bonds removed from the position(s) of isotopic substitution at which force‐constant changes occur between reactant and transition state in a rate process or between reactant and product in an equilibrium process. Sometimes more extreme cutoff is possible. The force‐constant and geometry changes at the isotopic position(s) must be the same in the cutoff and complete calculations. The success of the cutoff procedure focuses attention on the fact that experiments which determine isotope effects in the room‐temperature region and above can directly yield information regarding only changes occurring in the neighborhood of the isotopic position(s).

Linear and triangular structures for the H2Cl transition complex have been analyzed in terms of normal vibration theory and by an evaluation of the experimental data on the effect of hydrogen isotope substitution on the rate of reaction of hydrogen molecules and chlorine atoms. The frequencies of the bending vibrations are probably sufficiently small that their contribution to the relative rates can be expressed in terms of a small quantum correction of the order of (hc/kT)2(ωH2—ω2D,T/24). The near classical behavior of the bending frequencies for both linear and triangular structures of the H2Cl complex serves to reduce the ratio of the frequency factors, AH/AD,T, in the Arrhenius equation from that expected from structural considerations alone. Quantitative agreement is found between the calculated and experimentally determined frequency factor ratios for the triangular complex. The ``symmetrical'' stretching frequencies, for both linear and triangular complexes respectively, have been evaluated from th...

Equations relating dipole velocity and dipole length matrix elements have been tested for π electron systems treated by a molecular orbital method. Considerable improvement in the validity of the equations is noted when configuration interaction is included in the calculations.

The C13 kinetic isotope fractionation in the dehydration of formic acid by concentrated sulfuric acid was studied over the range --30.0 to 0.0 . The observed temperature dependence was examined by the gamma-barl' method of Bigeleisen and Wolfsberg (both with and without consideration of the effects of loss of bending frequencies) and by the semi-empirical" method of Yankwich, Weber, and Ikeda. Calculated values of the temperature-independent factor in the isotopic rate constant ratio lie within 1% of unity; the calculated sums of the differences upon activation of the diagonal cantesian force constants correspond especially well with the value expected for C- O bond rupture when bending frequency losses are taken into account, being near 7 md.A.-I.

Griffing1 computed the vertical excitation energy for the first electronic transition of benzene as a function of the carbon-carbon distance (rn) within the Sklar valence bond (SVB) and Goeppert-Mayer and Sklar molecular orbital (G1SMO) methods to evaluate the change on excitation of the carbon-carbon equilibrium internuclear separation (rn0) and of the corresponding stretching force constant, f. While ring expansion was correctly predicted, both methods appeared to predict that f increases on excitation, contrary to experiment. We shall demonstrate a) that Griffing's paradox may be resolved if terms higher than the second power in rn are included in the energy expressions for the states and b) that a Huckel calculation leads to results qualitatively similar to those obtained from the more elaborate calculations.

It is shown that the theoretical equation for the ratio of the rate constants of isotopic molecules is essentially independent of whether the crossing of the barrier is treated as a translation or a vibration. The temperature independent factor in this equation is evaluated for the case of substitution or displacement reactions. Calculations are presented for several cases involving the isotopes of carbon where there is simultaneous bond rupture and bond formation.

We present "exact" calculations, by the semiclassical method, of vibrational excitation of a harmonic diatomic molecule A-B, in its ground vibrational state, upon collinear collision with an atom C. Results are compared with those of first-order quantum mechanical time dependent perturbation methods and those of purely classical methods.

The high temperature limit of the reaction rate isotope effect is discussed. The Slater approach is compared in this respect to the transition state theory approach. There is no essential disagreement between the two approaches. The Slater assumption with respect to this limit for simple bond rupture represents a special case, however, and is not necessarily correct.

Theoretical considerations show that malonic acid −2C14 should decarboxylate precisely twice as fast as the isotopic isomer malonic acid −1C14 liberates C14O2. The calculations are supported by the experimental data of Ropp and Raaen. It is shown that these experiments are consistent with a C14 isotope effect twice that of the C13 one.