Energy zero-point vibration

Relative Energies, Zero Point Vibrational Energies... 

The isomers of C2H2 are 2Ol(s), acetylene and 2O2(s), vinylidene. Both have singlet ground states. A third isomer was reported to have appeared in a theoretical calculation, but, it turned out to have been a computational artifact [8]. Table 17.1 presents relative self consistent field (SCF) energies, zero point vibrational energies (ZPVE), and dipole moments of the isomers. 

Relative SCF Energies, Zero Point Vibrational Energies (ZPVE), and Dipole Moments of 201 (s) and 2O2(s)... 

In Table 17.12 we show relative SCF energies, zero point vibrational energies, and dipole moments of the open molecules. The quantum chemical treatments of the three open but branched constructs did not generate additional molecules. 

Relative SCF Energy, Zero Point Vibrational Energy (ZPVE), and Dipole Moment of the Open C5H2 Molecules... 

We will consider only two open molecules of C10H2, lOOl(s) and 10O2(s). Table 17.38 presents their relative energies, zero point vibrational energies and dipole moments. 

ZPE ZPVE zero point energy zero point vibrational energies ... 

Here tSEehc, tSZPVE, Er and A5 are the diiferences in electronic energy, zero-point vibrational energy, thermal energy and entropy between the products and reactants, respectively. To calculate these thermodynamic parameters, the basic thermochemical quantity is the partition function q(F,7), which is the corresponding quantity of the total partition function Q in volume V. The total partition function Q is the product of partition functions for translation, rotation, vibration and electronic degrees of freedom. 

Zero energy is the energy which a species would have at absolute zero in the absence of zero-point vibrational energy. 

There is a stack of rotational levels, with term values such as those given by Equation (5.19), associated with not only the zero-point vibrational level but also all the other vibrational levels shown, for example, in Figure 1.13. However, the Boltzmann equation (Equation 2.11), together with the vibrational energy level expression (Equation 1.69), gives the ratio of the population of the wth vibrational level to Nq, that of the zero-point level, as... 

More commonly, the resonant two-photon process in Figure 9.50(c) is employed. This necessitates the use of two lasers, one at a fixed wavenumber Vj and the other at a wavenumber V2 which is tunable. The first photon takes the molecule, which, again, is usually in a supersonic jet, to the zero-point vibrational level of an excited electronic state M. The wavenumber of the second photon is tuned across the M to band system while, in principle, the photoelectrons with zero kinetic energy are detected. In practice, however, this technique cannot easily distinguish between electrons which have zero kinetic energy (zero velocity) and those having almost zero kinetic energy, say about 0.1 meV... 

Figure 6-19. Bond dissociation curve showing the different zero-point vibrational energies of isotopic species R-H and R-D.

At low temperatures nearly all bonds will be in their lowest vibrational level, n = 0, and will, therefore, possess the zero-point vibrational energy, Eq = hvl2. Presuming the molecule behaves as a simple harmonic oscillator, the vibrational frequency is given by... 

To compute zero-point vibration and thermal energy corrections to total energies as well as other thermodynamic quantities of interest such and the enthalpy and entropy of the system. 

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. 

The inequality of the C—H bonds in the radical cation implies that all C—H bonds do not have the same force constants. In a simplistic approximation, the zero-point vibrational energy (ZPVE) of a C—H stretching vibration will be proportional to (k/mn), where k is the force constant of the C—H bond and j// is the mass of the hydrogen nucleus. The effect on the ZPVE of replacing one proton by a deuteron will hence depend on the deuteration site, such that the ZPVE will be lowered more if the deuteron occupies a site with a larger fcrce constant, i.e. a shorter bond. This, in general, means a site with low unpaired spin density. 

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 ... 

Here qi2D-vib refers to the electronic ground state of the transition state and q -vih to the vibrational ground state of the transition state. We have combined the t vo zero-point vibrations vith AE into an effective activation energy A act- We shall later explain how this important quantity can be measured. 

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