Zero-point vibration

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

Figure 5.4 Part of the microwave spectmm of cyanodiacetylene. (The many satellite transitions in each group are due to the molecule being in not only the zero-point vibrational state but also a multitude of excited vibrational states.) (Reproduced, with permission, from Alexander, A. J., Kroto, H. W. and Walton, D. R. M., J. Mol. Spectrosc., 62, 175, 1967)...

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... 

Table 5.3 Values of bond length in the Tq, and in the equilibrium configuration, r, zero-point vibrational state, e, for N2...

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... 

The temperature dependences of k, calculated by Hancock et al. [1989], are given in fig. 48. The crossover temperature equals 25-30 K. The weak increase of k T) with decreasing temperature below is an artefact caused by extending the gas-phase theory prefactor to low temperatures without taking into account the zero-point vibrations of the H atom in the crystal. For the same reason the values of the constants differ by 1-2 orders of magnitude from the experimental ones. 

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. 

Basis-set sensitivity is important, as it was for the calculation of electric dipole moments. Several experimental values in the literamre refer to measurements corrected neither for zero-point vibrations nor for centrifugal effects. 

They take no account of the zero point vibrations. So for example, in the case of a diatomic molecule the dissociation constants De and Dq are related by... 

Zero point vibration, 235 Ziegler type catalyst, 162 Zinc-tin, alloy (Zn6Sn5), calculation of thermodynamic quantities, 136... 

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. 

Since the force constant [A of Eq. (9-90)] is independent of the isotope, it is easy to show that the ratio of the zero-point vibrational frequencies is... 

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

Combining [52] the zero point vibrational corrections of Ref. [51] with the CCSD results obtained in the t-aug-cc-pVTZ and d-aug-cc-pVQZ basis sets we obtained the estimates for the ZPV corrected 70, A and B coefficients listed in Table 4. An experimental estimate for 70, A, and B has been derived by Shelton by fitting the results of ESHG measurements to the expresssion = To(l + A0JI2 + This... 

The pointwise given MCSCF results for zero point vibrational corrections for the ESHG hyperpolarizability were added to the CCSD results obtained from the dispersion coefficients and than fitted to a fourth-order polynomial in... 

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. 

Fig. 5.18 Four in-plane Fe modes, predicted on the basis of B3LYP calculations, contributing to the pair of experimental features at 312 and 333 cm in Fe(TPP)(NO). For ease of visualization, each arrow is 100(ot,/tofj,) longer than the zero-point vibrational amplitude of atom j. Color scheme as in Fig. 5.15 (taken from [101])...

Conroy and Perlow [235] have measured the Debye-Waller factor for W in the sodium tungsten bronze Nao.gWOs. They derived a value of/= 0.18 0.01 which corresponds to a zero-point vibrational amplitude of R = 0.044 A. This amplitude is small as compared to that of beryllium atoms in metallic beryllium (0.098 A) or to that of carbon atoms in diamond (0.064 A). The authors conclude that atoms substituting tungsten in bronze may well be expected to have a high recoilless fraction. 

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