Residual vibration energy

Note from (9) that a molecule has some vibrational energy (a half quantum ) even if its vibrational quantum number, v, is zero. This residual vibrational energy (which presumably persists even at 0° K) is called zero-point energy. It is connected with the uncertainty in the relative positions of the atoms in low energy molecules as dictated by the Heisenberg principle (p. 7). 

Zero-point energy The zero-point energy is the residual vibrational energy of a harmonic oscillator at the lowest vibrational state. It arises from the fact that the position of a particle is uncertain and therefore its momentum and hence its kinetic energy cannot be exactly zero. 

The term A h° T) is the linear combination, weighted by the stoichiometric numbers, of the residual vibrational energies of each of the substances. 

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

A given molecule has 3n-6 normal modes of vibration (3n-5 if linear), where n is the number of atoms. Each mode i has a characteristic vibrational frequency Vj and a residual energy even at absolute zero temperature. The total zero-point vibrational energy is thus ... 

Table 13.6 Differences in vibrational energy levels (v = 0-5) and residual 20) of two PECs with different gauge origins (perpendicular magnetic field, FC orders...

We can see that this relation is the product of two terms the preexponential term and the variation in internal energy due to residual vibrations at the temperature of 0 K. 

In this equation, D is the difference between the origin of energies and the minimum value of the curve. De is the residual energy, which is the vibration energy at absolute zero hV(/2, a-q is the inter-atom distance for the minimum energy, which is the equilibrium distance of the molecule. Constant a depends on the light speed c molecule reduced mass p. (p is related to the atomic masses by the equations in [10.5]) Plank s constant h and value x as defined by [10.5]. Constant a is written ... 

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