Zero translational energy

If A+ and B are formed in their ground states and if these species and the electron have zero translational energies, then the standard enthalpy of reaction 4.10 at T = 0 is equal to the appearance energy of A+ at T = 0, Ao(A+/AB). It becomes obvious from this definition that when reporting a value for an appearance energy, it is essential to state the parent molecule (indicated by /AB). Otherwise, we cannot identify the remaining species in the net reaction 4.10. 

The appearance energy of a cation A can be defined as the standard enthalpy of reaction (16), where a molecule AB in the gas phase at T= 0 K is ionized and excited to a state AB by means of an electron or a photon, followed by decomposition into the fragments A" and B, provided that A and B are in their ground states and all the species, including the electron, have zero translational energies. 

The use of the spectral resolution of the identity in this form is not fully justified. A sudden cut in the electrie field may leave the molecule with a non-zero translational energy. However, in the above Spectral resolution one has the stationary states computed in the centre-of-mass coordinate system, and therefore translation is not taken into account. 

Electron affinity of a species M is the minimum energy required for the process M M + e where M and M are in their ground rotational, vibrational and electronic states and the electron has zero translational energy. 

From equation (A3.9.2). we can see that at low E, the acceleration into the well dominates the energy loss, that is, 5 does not reduce to zero with decreasing E. Below a critical translational energy, given by... 

In order to predict the energy of a system at some higher temperature, a thermal energy correction must be added to the total energy, which includes the effects of molecular translation, rotation and vibration at the specified temperature and pressure. Note that the thermal energy includes the zero-point energy automatically do not add both of them to an energy value. 

Note that we don t need to run any calculations on H at aU. Its electronic energy is 0 since it has no electrons, and its only other non-zero energy term is the translational energy term AE, which is equal to RT = 0.889 kcal moT. ... 

Note that at zero Kelvin, the molecule has a zero-point vibrational energy of ( hv) and translational energy equal to (3/i2/8 i/,2). V - Uu is the energy we add above these amounts to get to a temperature T. 

MMl represents the mass and moment-of-inertia term that arises from the translational and rotational partition functions EXG, which may be approximated to unity at low temperatures, arises from excitation of vibrations, and finally ZPE is the vibrational zero-point-energy term. The relation between these terms and the isotopic enthalpy and entropy differences may be written... 

The molecule has jRT from translational energy, RT from the term pV, RT from the two rotational degrees of freedom, and then the zero-point vibrational energy. The atom has only contributions from translational energy and the PV term ... 

Therefore, Eact = RT, which shows that even when threshold energy E() is zero, the Eact has a non-zero value. In case of line of center model, a collision is reactive only if the component of the relative translational energy along the line joining the centre of mass of the two molecules exceeds E0. In case of line of centers of model, the rate constant is given by... 

The proviso T = 0 signifies that AB is in its electronic, vibrational, and rotational ground states and has no translational energy. The word isolated indicates the perfect gas model. The minimum energy condition ensures that AB+ is also in its electronic, vibrational, and rotational ground states and the translational energies of AB+ and e are both zero it also indicates that the products in reaction 4.1 do not interact, that is, they also conform with the perfect gas model. 

W1/W2 theory and their variants would appear to represent a valuable addition to the computational chemist s toolbox, both for applications that require high-accuracy energetics for small molecules and as a potential source of parameterization data for more approximate methods. The extra cost of W2 theory (compared to W1 theory) does appear to translate into better results for heats of formation and electron affinities, but does not appear to be justified for ionization potentials and proton affinities, for which the W1 approach yields basically converged results. Explicit calculation of anharmonic zero-point energies (as opposed to scaling of harmonic ones) does lead to a further improvement in the quality of W2 heats of formation at the W1 level, the improvement is not sufficiently noticeable to justify the extra expense and difficulty. 

Neither of these vibrations corresponds to stretching vibrations of AH or BH. The antisymmetric vibrational mode represents translational motion in the transition state and has an imaginary force constant. The symmetric transition-state vibration has a real force constant but the vibration may or may not involve motion of the central H(D) atom2,12 13. If the motion is truly symmetric, the central atom will be motionless in the vibration and the frequency of the vibration will not depend on the mass of this atom, i.e. the vibrational frequency will be the same for both isotopically substituted transition states. It is apparent that under such circumstances there will be no zero-point energy difference... 

Hase s trajectory value for the association rate constant, /cp of 1.04 cm- s maybe used in conjunction with the above Langevin value of the collisional stabilization rate constant to yield a unimolecular dissociation rate constant of 3.75 x 10 ° s and a lifetime of 27 ps. In each case, these values are in excellent agreement with the order of magnitude of lifetimes predicted by Hase s calculations for cr/CHjCl collisions at relative translational energies of 1 kcal mor , rotational temperatures of 300 K, and vibrational energies equal to the zero-point energy of the system. 

A AZPE = AZPEii — AZPEd AZPEt) corresponds to the terms for the reactions of monodeuteriated aldehydes. Terms defined by IE = MMl x EXC x EXP (IE is the Isotopic exchange equilibrium, MMl is the mass moment of inertia term representing the rotational and translational partition function ratios, EXC is the vibrational excitation term and EXP is the exponential zero point energy). 

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