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We employ an extended corresponding states theory for the description of liquid phase molar densities, ρ, and molar density isotope effects (IE's), and vapor pressures and vapor pressure IE's. In extended corresponding states, the conditions for liquid–vapor coexistence are given in terms of the critical properties of the fluid plus an additional parameter (e.g. the Pitzer acentric factor). Corresponding states theory is normally presented in its classical version, but thermodynamic IE's are quantum effects. We have chosen to introduce the quantization required to rationalize vapor–liquid equilibrium (VLE) isotope effects semi-empirically via the IE's on critical temperature, Δ T C = T ′ C − T C , critical pressure, Δ P C = P ′ C − P C , and critical density, Δ ρ C = ρ ′ C − ρ C . The primes refer to the lighter isotopomer. We limit attention to cubic or “almost cubic” equations of state (EOS), and point out useful correlations between critical temperature IE's and vapor pressure IE's in the near-critical region. When combined with EOS, such correlations allow the estimation of the other critical property IE's, and thence estimation of molar density IE's over a broad orthobaric liquidus range ( 0.5 T ′ R = T / T ′ C 1 ) . Using a new modification of the Van der Waals EOS we find that liquid molar density IE's correlate quite well with the critical property isotope effects alone, while rationalization of vapor pressure IE's requires the addition of an isotope effect on the acentric factor.

The self diffusion coefficient of methane in silicalite-1 is influenced by the flexibility of the lattice unlike the self diffusion coefficient of methane in the cation-free zeolite of type A. In the present paper, besides the influence of lattice vibrations on this process, the influence of internal vibrations of the methane molecule and the applicability of several spherical models of this molecule are examined. The method of moments [Chem. Phys. Lett. 198 (1992) 283] is generalized to anisotropic diffusion.

Experimental data on vapor phase second virial coefficient isotope effects (VCIEs) are reviewed and then interpreted using the general theory of isotope effects. Useful correlations are developed between −Δ(ℬ − bo)/(ℬ − bo) = (−VCIE) and [ln(fc/fg)]*, where [ln(fc/fg)]* is the reference condensed phase reduced isotopic partition function ratio, and ℬ is the second virial coefficient, bo = 2πσ3/3, σ is the Lennard-Jones size parameter, and Δ denotes an isotopic difference, light-heavy. [ln(fc/fg)]* can be straightforwardly obtained from measurements of vapor pressure isotope effects for TR = T/TCRITICAL < 0.7. We show (−VCIE) = ln(fp/fg2) where ln(fp/fg2) is the reduced isotopic partition function ratio associated with the equilibrium between isolated gas-phase monomer species and interacting pairs. At temperatures well removed from crossovers in ln(fp/fg2) or [ln(fc/fg)]*, ln(fp/fg2) = (0.4 ± 0.2) [ln(fc/fg)]*.

The guest–host and the guest–guest energy coupling in a zeolite are examined for both the vibrating and the rigid lattice. Although the effect of the vibrating lattice acting as a thermalizing heat bath is stronger than the mutual thermalization of diffusing guest molecules, as can be seen, for example, from the decay of the kinetic energy auto correlation function this mutual thermalization also leads to remarkable relaxation effects. The influence of lattice vibrations on the self-diffusion of the guest molecules is investigated. Even in the studied case of small zeolite cavities (cation-free LTA zeolite) with windows of the size of a guest molecule (methane), the influence of the lattice vibrations on the self-diffusion coefficient is negligible. Variations of interaction parameters and temperatures as well as of the elastic force constants in the lattice lead to the conclusion that the diffusion coefficient is quite stable to the influence of lattice vibrations.

The influence of lattice vibrations on the diffusion of methane in a cation-free zeolite of structure Type LTA is examined. It is shown that contrary to earlier published results the self-diffusion coefficients obtained with flexible and with rigid lattices are practically the same. This finding is true over a wide range of temperatures and for different interaction parameters. The reason why earlier papers did not state this independence of D on the lattice vibrations is explained.

The diffusion coefficient for a cation-free zeolite of type LTA is examined employing a widely used set of interaction parameters. A comparison with earlier results that have been obtained with an alternative parameter set enables conclusions concerning the parameter dependence of the observed physical properties. The diffusion behaviour of the guest molecules is illustrated by representations of concentration profiles and the propagator.

Diffusion of methane in LTA type zeolites has been calculated from molecular dynamics simulations. Taking into account the polarization interaction of the exchangeable cations in the zeolite NaCaA with the methane molecules, the self-diffusion coefficients in NaCaA decrease by one order of magnitude in comparison with its cation-free analogue ZK4. The order of magnitude of the calculated values is in agreement with NMR data.

Literature data on H/D isotope effects on polarizability are reviewed. An argument based on perturbation theory correlates the isotopic differences with electronic transition moments via vibrational averaging. The contribution of the second moment isotope effect is especially important. The approach is first illustrated with a calculation on H2/D2 and then employed to correlate differential refractive index data on about twenty other H/D isotopic pairs. A table of bond polarizability isotope effects is presented which permits the estimation of polarizability isotope effects for various kinds of H/D substituted compounds. Coupling these results with known isotope effects on vibrational amplitude permits calculation of vapor phase virial coefficient isotope effects, while combina­tion with refractive index data on liquids yields reasonable values for the sum of the isotope effects on molar volume and electronic second moment. However the quantitative correlation of the polarizability isotope effect with isotope effects on vapor pressure or thermodynamic activity is less direct.

The multiple signal classification method (MUSIC) for spectral estimation is applied to data generated for three cases of a simple model system, based on five harmonic frequencies. The results are compared to the exact spectrum as well as to the results obtained with the Fourier transform method. The effect of the parameters required to be set in the application of the MUSIC methods as well as the effect of signal-to-noise ratio on the spectral estimate and its error are studied. Spectral resolution is discussed. It is found here that MUSIC tends to yield very good estimates of the frequencies but that relative amplitudes of lines and line shapes are not generally estimated.

Histories of a molecular dynamics simulation of a chain of fifty anharmonically coupled atoms (taken as a prototype for a solid or a surface) are examined to obtain power spectra corresponding to a given k value. Two methods for obtaining the power spectra from the atom velocities are studied: (1) the method which involves computing the appropriate velocity-velocity autocorrelation function (ACF) and then taking its Fourier transform and (2) the direct method which involves finding the absolute value squared of the Fourier transform of the appropriate velocity. The long (in time) simulations necessary to obtain reasonably reproducible power spectrum parameters are rationalized in terms of a previous theoretical study of autocorrelation functions by Zwanzig and Ailawadi. The method of choice for obtaining power spectra is the use of long trajectories coupled with truncation of the ACF prior to its Fourier transform in such a way that the tail of the ACT is disregarded.

This chapter introduces the basic principles used in applying isotope effects to studies of the kinetics and mechanisms of enzyme catalyzed reactions. Following the introduction of algebraic equations typically used for kinetic analysis of enzyme reactions and a brief discussion of aqueous solvent isotope effects (because enzyme reactions universally occur in aqueous solutions), practical examples illustrating methods and techniques for studying enzyme isotope effects are presented. Finally, computer modeling of enzyme catalysis is briefly discussed.

The application of isotope effect studies to problems in geochemistry and geology, meteorology, environmental studies, archeology and paleobotany, food authentication, medical diagnostics and various other practical areas is described using numerous examples.

Isotope effects on the PVT properties of non-ideal gases and isotope effects on condensed phase physical properties such as vapor pressure, molar volume, heats of vaporization or solution, solubility, etc., are treated in some thermodynamic detail. Both pure component and mixture properties are considered. Numerous examples of condensed phase isotope effects are employed to illustrate theoretical and practical points of interest.

In this chapter we discuss practical techniques and instrumentation used in experimental measurements of kinetic and equilibrium isotope effects. After describing methods to determine IE’s on rate constants, brief treatments of mass spectrometry and isotope ratio mass spectrometry, NMR measurements of isotope effects, the use of radio-isotopes, techniques to determine vapor pressure and other equilibrium IE’s, and IE’s in small angle neutron scattering are presented.

Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion.

The theoretical framework needed for interpretation of kinetic isotope effects on unimolecular reactions is reviewed. Application to the satisfactory rationalization of the theoretically puzzling ”mass independent isotope effect” observed for oxygen isotope fractionation in extraterrestrial samples is described.

The Born–Oppenheimer approximation is introduced and discussed. This approximation, which states the potential energy surface on which the molecule vibrates/rotates is independent of isotopic substitution, is of central importance in the theory of isotope effects.

In corresponding states (CS) theory the PVT properties of fluids are expressed in terms of the critical constants and one or more additional parameters. In this chapter the use of CS theory to correlate isotope effects on the physical properties of fluids is explored.

A brief introduction deals with the time period from Dalton to the discovery of isotopes by Soddy and Fajans in the early twentieth century which was soon followed by the invention of the mass spectrograph (1922). The next section covers the period from 1922 to the discovery of deuterium by Urey and his colleagues. It includes a discussion of isotope effects in spectroscopy, particularly band spectra of diatomic molecules, and also discusses the discovery of the important stable isotopes in the second row of the periodic table. It ends with the discovery of deuterium, probably the most “popular” isotope for isotope effect studies. The chapter ends with a short description of the “apparatus” of theory and experimentation available for isotope effect work at the time of the discovery of deuterium.

Some of the successes and several of the inadequacies of transition state theory (TST) as applied to kinetic isotope effects are briefly discussed. Corrections for quantum mechanical tunneling are introduced. The bulk of the chapter, however, deals with the more sophisticated approach known as variational transition state theory (VTST).

This chapter describes a number of examples of kinetic isotope effects on chemical reactions of different types (simple gas phase reactions, SN2 and E reactions in solution and in the gas phase, α and β secondary isotope effects, etc.). These examples are used to illustrate many aspects of the measurement, interpretation, and theoretical calculation of KIE’s. The chapter concludes with an example of an harmonic semiclassical calculation of a kinetic isotope effect.

We have performed classical Molecular Dynamics studies of laser induced heating and desorption of adsorbed water molecules on a model metal surface. The study focuses on the excitation of water modes. The laser field is simulated as an oscillatory time-dependent external electrical field that interacts with the molecular dipoles. Unrealistically high field strengths of the order of 1010 V/m were used in order to observed significant energy absorption and also molecular desorption on the time scale from 10 to 20 ps. Frequencies of 400, 1700, and 3750 cm−1 were chosen to excite the molecule in the librational, bending, and stretching modes, respectively. The rates for desorption of isolated molecules are substantially lower for excitation in the bending and stretching modes than for excitation in the librational modes. At higher adsorbate coverages, intermolecular energy transfer plays a major role in the behavior of molecules in the adsorbate layer.

Variational calculations have been carried out on the ammonia molecule using two-dimensional potential energy functions. The two dimensions used are the inversion and symmetrical stretching coordinates. If the potential function includes a term to describe the interaction between these two coordinates, the results are as good as those obtained in other studies using a six-dimensional potential function. The barrier height for the new potential function is 1857.5 cm−1.

Molecular dynamics (MD) is applied to simulate methane self-diffusion in a model zeolite ZK4. It is found that the mutual interaction of the adsorbed molecules is sufficient to guarantee their thermalization, so that in first-order considerations, the energy exchange with the zeolite framework may be neglected.

The disappearance of the rotational rainbow in the scattering of unexcited diatomic molecules from a smooth rigid surface due to initial vibrational and rotational excitation and surface corrugation has been studied by classical trajectory calculations. The ratio of vibrational to rotational excitation in initially unexcited molecules strongly increases with increasing impact energy and increasing surface hardness.

The photoredox reaction transients of ferrioxalate in water have been studied by means of time-resolved EXAFS and ultrafast optical transient spectroscopy. The transient spectra and kinetics have been measured from the femtosecond to millisecond range, and the Fe−O bond lengths of the ferrioxalate redox reaction transients have been determined with 2 ps time resolution and 0.04 A accuracy. These data in conjunction with quantum-chemistry DFT and UHF calculations were used to formulate a mechanism for the Fe(III) to Fe(II) redox reaction where dissociation precedes electron transfer. In addition, radical scavenging experiments support the mechanism proposed.

The equilibrium isotope effect (EIE) for the interconversion of the two chair isotopomers of 1-trideutero-1,3,3-trimethylcyclohexane was predicted using geometry and vibrational force constants derived from electronic structure theory at HF, B3LYP, and MP2 levels as input for the program THERMISTP. Agreement between theory and previously reported NMR results is very good (experimental K(eq) = 1.042 +/- 0.001 vs K(eq) = 1.0409 at MP2/6-311G* level, K(eq) = 1.0503 at HF/6-311G*, and K(eq) = 1.0417 at B3LYP/6-311G* level, all at 17 degrees C). In order to investigate the origin of this isotope effect, the calculated EIEs for the monodeuterated isotopomers were analyzed. It has been shown that the hydrogen atom on an axial methyl group which is unusually close to its counterpart on the other axial methyl is responsible for the large (steric) isotope effect in the compound studied.

Publisher Summary This chapter discusses the diffusion and relaxation processes of different nonspherical guest molecules in zeolites by using molecular dynamics (MD) simulations with rigid and flexible lattice and molecules. Interrelations among such processes are studied by different tools, such as correlation functions. The results show that there exist cases where a strong mutual influence can be found and others in which there is only small influence.

Using molecular dynamics computer simulations (MD) the dynamics of kinetic processes in zeolites is discussed on a molecular level. Small changes in lattice parameters can cause large changes in the diffusion coefficient. This has been examined in various simulations employing different interaction parameters. The presence of Na + , Ca ++ cations is shown to reduce the self-diffusivity dramatically. The influence of lattice vibrations on the selfdiffusion coefficient is found to depend strongly on the structure. Isotopic substitution of methane is examined. The propagator P(r → , t ) illustrates the diffusion behaviour of the guest molecules. The process of transport diffusion, has been studied by different kinds of nonequilibrium simulations.

We report computer simulations of the adsorption process of water on a platinum (100) surface. At very low impact energies the sticking coefficient approaches unity in agreement with experimental evidence. The sticking coefficient for the bare surface decreases strongly with increasing impact energy already in the energy range

The reaction path for cycloaddition of acrylonitrile to allene was computed using the AM1 Hamiltonian with limited configuration interaction (C.I. = 4). Cycloaddition is found to proceed stepwise via a biradical intermediate, in confirmation of the conclusion reached by Dolbier and Dai on the basis of the intramolecular and intermolecular H/D kinetic isotope effects on the reaction. A transition state theory calculation of the ratio of the two isotopomeric products obtained from cycloaddition of acrylonitrile with gem-dideuteroallene is in good agreement with the experimental observations. A parallel set of computations on the cycloaddition of fluoroethylene to allene yielded similar results.

One-electron properties calculated from SCF Xα SW/MT and near-HF functions for water are compared. A regional orbital analysis is used to assess the differences between the functions.

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.

Calculations are carried out of the rotational–vibrational energy levels of H2O and D2O for J≤10 by a variational method. The full Watson Hamiltonian is employed, with the potential function in valence displacement coordinates and with the integrations over normal coordinates carried out by Gauss–Hermite quadrature. The basis set consists of products of vibrational functions and symmetric top functions; the vibrational functions diagonalize the Hamiltonian for J=0. Comparison is made with experiment, and also results obtained with different force fields are compared. The mixing of different vibrational functions into a wave function for a given rotational–vibrational state is studied; mixing is found to be quite prevalent.

Observed fundamentals in infrared spectra of pure liquids are red-shifted by the so-called "dielectric effect". This dielectric shift is an excited state effect, and the observed fundamentals must be corrected before one deduces the isotope effects on the zero-point vibrational energy which are needed in the theoretical evaluation of vapor pressure isotope effects. A simple formula is applied to calculate the dielectric shift, which requires only the molar concentration and the integrated absorption coefficient for the fundamental.

Computational methods are presented for the study of collisions between a linear, symmetric triatomic molecule and an atom by three‐dimensional quasiclassical trajectory calculations. Application is made to the investigation of translational to rotational and translational to vibrational energy transfer in the systems CO2–Kr, CO2–Ar, and CO2–Ne. Potential‐energy surfaces based on spectroscopic and molecular beam scattering data are used. In most of the calculations, the CO2 molecule is initially in the quantum mechanical zero‐point vibrational state and in a rotational state picked from a Boltzmann distribution at 300°K. The energy transfer processes are investigated for translational energies ranging from 0.1 to 10 eV. Translational to rotational energy transfer is found to be the major process for CO2–rare gas collisions at these energies. Below 1 eV there is very little translational to vibrational energy transfer. The effects of changes in the internal energy of the molecule, in the masses of the collidants, and in the potential‐energy parameters are studied in an attempt to gain understanding of the energy transfer processes.

The mean square amplitudes of internal coordinates in a harmonically vibrating molecule are developed to second order in a perturbation theory treatment where the unperturbed problem corresponds to uncoupled coordinate oscillators and the perturbation arises from coupling terms in the kinetic and potential energies. Exact and perturbation theory calculations are compared.

The single‐resonance damped oscillator model for molecular polarizability is combined with the Lorentz model for the local field in order to obtain expressions describing the dielectric constant and absorption coefficient of pure condensed phases. These expressions are functions of three parameters: the transition dipole matrix element, the molecular resonance frequency, and the lifetime. It is found that condensed phases exhibit a macroscopic resonance frequency different from the molecular resonance frequency, and equations are supplied whereby one may calculate this shift. The shape of the condensed phase absorption coefficient is found to be unsymmetrical about the resonance frequency for intense bands; a method is described in which one exploits this lack of symmetry in order to find the transition dipole matrix element. Attenuated total reflection data for four liquid phase infrared bands are analyzed in terms of the absorption model.

Reduced isotopic partition function ratios (s/sub 2//s/sub 1/)f are employed in the calculation of isotope effects on thermodynamic equilibrium constrants. The quadratic force constants of the molecular force field are needed to evaluate (s/sub 2//s/sub 1/)f. Often these force constants are directly deduced from observed fundamentals in vibrational spectra and the (s/sub 2//s/sub 1/)f values so obtained are labeled (ANHARM). In a theroretically more valid procedure that is more difficult, one corrects observed fundamentals for anharmonicity on the basis of observed overtone and combination bands and then deduces force constants from these observed harmonic frequencies. The (s/sub 2//s/sub 1/)f values obtained from these force constants are labeled (HARM). (HARM) values and (ANHARM) values are evaluated and the isotope effects calculated with these values are discussed. It is concluded that the consistent use of (ANHARM) values in such calculations is a valid procedure.