Zero-point Energy Contributions Research Articles

Theoretical analysis of kinetic isotope effects. Deuterium isotope effects in the reaction of methyl with methane

A detailed analysis of the primary and secondary deuterium isotope effects in the reaction of CH/sub 3/ with methane has been done using the MINDO/3 method. The analysis is performed by partitioning the total isotope effects into its rotational, translational, vibrational excitation, and zero-point contributions. By using the symmetry properties of the reactants and transition state, the role of the new vibrational modes created in the transition state is discussed. It is shown that they cannot be neglected in the discussion of the secondary isotope effects. In contrast, they have a very small role in the primary isotope effects. The main contribution to the primary effects is due to the changes in the zero-point energy contribution of the vibrational modes of the reactants.

On the contribution of intramolecular zero point energy to the equation of state of solid H2

Experimental evidence shows that the internal zero-point energy of the H2 molecule exhibits a relatively strong pressure dependence in the solid as well as changing considerably upon condensation. We show that these effects contribute about 6% to the total sublimation energy and to the pressure in the solid state. Methods to modify the ab initio isolated pair potential to account for these environmental effects are discussed.

The lattice energy of magnesium hydride

The lattice energy of MgH 2 has been calculated using a Born-Mayer model. The result is U 0 = −2906.5 kJ mol −1 which may be compared with the Born-Haber cycle value of −2721.3 kJ mol −1. This difference indicates that there is an appriciable covalent contribution to the bonding. The importance of the zero-point energy contribution to the lattice energy in the case of the light metal hydrides is discussed.

Model calculations of primary hydrogen isotope effects

Model calculations of primary hydrogen isotope effects for four and five-centre transition states, taking account of bending vibrations and proton tunnelling, are reported. The results reinforce those based on consideration of stretching vibrations alone, and suggest that both zero-point energy and tunnelling contributions are at a maximum for a symmetrical transition state. For proton transfer to halide ions, large isotope effects persist for product-like transition states owing to the low frequency of hydrogen bending vibrations. Swain's relation between deuterium and tritium isotope effects is well obeyed. It seems better to consider the reaction co-ordinate mode for an atom transfer reaction as deriving from a translation rather than from a vibration in the reactants.

The relation between (elastic) and (calorimetric) nearT= 0 K

The first-order quantum corrections to the elastic frequencies (zero-point energy) and to the specific heat (vibrational anharmonicity) of a Bravais crystal near T = 0 are considered. It is explicitly shown that these corrections maintain the equality of Θ0c and Θ0el to the same order in λ = /α0(ML0/NA)1/2, which is an appropriate ordering parameter. The effect on the elastic frequencies of the zero-point energy, which is a first-order quantum correction to Φ0 (the static lattice energy), is determined by applying perturbation theory, first to the secular equation of elasticity theory, and then to the secular equation of Born-von Karman theory. In the former case the term proportional to the zero-point energy contribution to the elastic constants is chosen as the perturbation. A symbolic expression is obtained for the effect of zero-point energy on the elastic frequencies. In the latter case all the strain-dependent terms in an expansion of the dynamical matrix about the undeformed lattice sites are chosen as the perturbation. An explicit expression is obtained for the contribution of the zero-point energy to the elastic constants and the equation of state.

On the Strain Dependence of the Vibrational Frequency Distributions of Simple Lattices

Expressions are derived for the first even moments ω2 and ω4 of the normal frequency distribution of a rigid system of particles interacting with central forces. These are used to obtain the strain dependence of ω2 and ω4 for any cubic lattice whose atoms all occupy centres of symmetry, up to the second order in the homogeneous strain parameters. By a different method the strain dependence of ω6 is derived for the special case of a face-centred cubic lattice with a potential (r) between nearest neighbours, under the conditions of static equilibrium. The results are applied first to a face-centred cubic lattice with a 6-12 potential between all neighbours, and used to derive values of ωD(n) = [1/3(n + 3) ωn]1/n for n = 2 and n = 4. In earlier approximations the strain dependence of ln ωD(n) has been assumed not to vary with n, but the present results show that this is invalid for shear strain, and hence for calculating the elastic constants c44 and c11 -c12. Neglect of the interaction between all except nearest neighbours only slightly alters the strain dependence of ln ωD(n), and calculations with a more general potential (r) are therefore made only with nearest neighbour interaction. Strong variation with n (indicating strong distortion of the shape of the frequency distribution under strain) is found to be always present for shear strain unless the cubic term in the pair potential is small. Results for n = 1 and n = 0, required for calculations of thermoelastic properties at T = 0 and at very high temperatures respectively, can be estimated by extrapolation from n = 6, 4 and 2. In particular, the contribution of the zero-point energy to the elastic constants can be determined to within about 10%, provided that the dominant forces are central forces between nearest neighbours.

On a Non-Linear Differential Equation for the Zero-point Energies of the Rare Gas Solids

Domb and Salter (3) have postulated that the Debye characteristic temperature for the rare gas solids should be given by where C involves atomic constants and the atomic mass of the rare gas, r is the mean interatomic distance and Φ(r) is the interatomic potential. This formula provides a good approximation for the heavier rare gas solids but breaks down for the lighter inert elements. On the basis of Born's argument (1) that the dynamic rather than the static equilibrium positions should be considered, Domb (2) suggested that (1) might be replaced by , where the additional term is effectively the zero-point energy contribution to the lattice equilibrium. (R is the gas constant.) From experimental measurements of the Gruneisen parameter γ Domb obtained numerical values which, when substituted in the right-hand side of (2), yielded values of vD that appeared to compare favourably with direct experiments. More recently Hooton (6) considered the problem of lattice dynamics with anharmonic forces (which are the dominant factor in the lighter rare gas solids) and developed a theoretical argument which also led to Domb's equation (2).