Zero-point motion

Table 2 gives our calculated results for the equilibrium volume Vq, bulk modulus Bq, and enthalpy of formation AH. Theoretical results refer to T=0, uncorrected for zero point motion, whereas experimental values refer to room temperature. Note that the extensive quantities AH and Vq arc reported per atom in the present paper, i.e., divided by the total number of atoms. As well known the LDA underestimates the volume. Comparing the bulk modulus for T3 and D8s we see that the addition of Si to pure Ti has a large (26 %) effect on the bulk modulus, indicating that p electrons of Si have a strong effect on the bonding in this system. 

This expression is exact within our original approximation, where we have neglected relativistic effects of the electrons and the zero-point motions of the nuclei. The physical interpretation is simple the first term represents the repulsive Coulomb potential between the nuclei, the second the kinetic energy of the electronic cloud, the third the attractive Coulomb potential between the electrons and the nuclei, and the last term the repulsive Coulomb potential between the electrons. 

There are two general cases of dipole-dipole forces those between molecules in which the distribution of electronic charge is centrosymmetric and those in which it is not. In the first case, there are no permanent electrical dipoles, whereas there is a permanent dipole if the charge distribution is non-centro-symmetric. When permanent dipoles are not present, there are nevertheless fluctuating dipoles as a result of atomic vibrations. These are always present because of zero-point motion. At temperatures greater than 0°K, thermal energy further excites the molecular vibrational modes which create fluctuating electric dipoles. 

Electronic properties generally do not depend on mass or lifetime therefore the adiabatic total-energy surfaces and also the electronic structure of muonium should be very similar to that of hydrogen. However, its dynamical behavior (zero-point motion, vibrational frequencies, diffusion,. ..) may differ from that of H because of the difference in mass. Most of the results discussed in this chapter will be applicable to both hydrogen and muonium (although for convenience I will usually refer to hydrogen). Dynamical features that may be distinct for the hydrogen vs. muonium cases will be discussed in Parts VI and VII, respectively. 

One study (DeLeo et al., 1988 Fowler et al., 1989) has found that neutral H at the B site in Si has a tendency to preferentially bind to one of the two Si neighbors, leading to an asymmetric configuration, with Si—H distances of 1.48 A and 1.77 A respectively. This tendency was interpreted in terms of a pseudo-Jahn-Teller distortion. However, the potential barrier that leads to the asymmetric position is so low (< 0.2 eV) that it can readily be surmounted by zero-point motion of the proton. Experimental observation of such an asymmetry is therefore unlikely, except maybe through an isotope shift measurement in an infrared experiment (DeLeo et al., 1988). None of the other theoretical approaches has produced this type of asymmetry. 

The effect of zero-point motion on the hyperfine constant has been the subject of some controversy. Manninen and Meier (1982) argued that motional averaging (over all positions sampled in the vibrational ground... 

Hazony Y (1966) Effect of zero-point motion on the Mossbauer spectra of K[Pg.99]

The atoms in a crystal are vibrating with amplitudes determined by the force constants of the crystal s normal modes. This motion can never be frozen out because of the persistence of zero-point motion, and it has important consequences for the scattering intensities. 

The quantity we have calculated here is appropriately compared to De, the bond energy from the bottom of the curve. This differs from the experimental bond energy. Do, by the amount of energy due to the zero point motion of the vibration. There is no vibration in our system, since the nuclei are infinitely massive. We use the theoretical result for comparison, since it is today considered more accurate than experimental numbers. 

The enthalpy obviously depends on temperature. However, even at T=0 the enthalpy of an intermediate may differ substantially from the electronic binding energy due to the zero-point motion in vibrational degrees of freedom. 

Because the absolute value of Uuncor measured in a crystal depends on the amplitude of the zero point motion and includes contributions from static disorder, it is more appropriate to compare how (A ) and Uuncor vary with temperature. Differentiating eqn (9.7) with respect to temperature gives eqn (9.9) ... 

The electronic charge distribution of H2 is inversion symmetric and, therefore, H2 is necessarily non-polar. The HD molecule possesses a nearly identical electronic cloud. Nevertheless, HD does feature a (weak) permanent dipole moment. It is of a non-adiabatic nature and arises from the fact that the zero-point motion of the proton takes place with a greater amplitude than that of the deuteron. As a consequence, the side of the proton is slightly more positively charged than that of the deuteron if the HD molecule is in the vibrational ground state. 

It is interesting to note that the vibrational model of the nucleus predicts that each nucleus will be continuously undergoing zero-point motion in all of its modes. This zero-point motion of a quantum mechanical harmonic oscillator is a formal consequence of the Heisenberg uncertainty principle and can also be seen in the fact that the lowest energy state, N = 0, has the finite energy of h to/2. 

Energy of Zero-Point-Motion is calculated for each nucleus rather than estimated or ignored. (3) Elaborate shape definitions are replaced by a matching procedure where the fragment interaction has the correct asymptotic form. (4) Microscopically calculated mass paramater functions are employed in two-dimensional action integrals. Mass asymmetry as well as charge asymmetry are fully taken into account. 

In terms of quantum mechanics, a system with zero energy is impossible. A quantum system must possess a minimum energy of Ev. This postulate is due to the irrepressible zero-point motion imposed on microscopic systems by the uncertainty principle and by quantization. Thus, the classical concept of nuclei in space and associated motion is replaced by the concept of a nuclear or vibrational... 

Suppose that a molecule starts off on an excited surface ij/ and makes a trajectory from A to B on the ij/ surface during its zero-point motion. Classically,... 

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