Oscillator quantum energy

With the assumption of hannonic oscillators, the molecule s quantum energy levels are... 

When the electron transfer process is coupled to classical reorientation modes and to only one harmonic oscillator whose energy quantum h( is high enough for only the ground vibrational level to be populated, the expression of the electron transfer rate is given by [4, 9] ... 

Note that if j = 1, (9.12) is formally identical with the classical expression (9.7) the classical multiple oscillator model, which will be discussed in Section 9.2, is even more closely analogous to (9.12). However, the interpretations of the terms in the quantum and classical expressions are quite different. Classically, o30 is the resonance frequency of the simple harmonic oscillator quantum mechanically 03 is the energy difference (divided by h) between the initial or ground state / and excited state j. Classically, y is a damping factor such as that caused by drag on an object moving in a viscous fluid quantum mechanically, y/... 

Quantum-chemical calculations for pyrylium including one, two, or three water molecules using DFT and 6-31 + G(d,p) basis set revealed that the aromaticity (estimated by harmonic oscillator stabilization energy, HOSE natural resonance theory, NRT harmonic oscillator model of aromaticity, HOMA and nucleus-independent chemical shifts, NICS) is not influenced by water molecules [82],... 

Fig. 4.1.2 Harmonic oscillator with the energy E = p2/(2m) + (1/2)kq2 (which is the equation for an ellipse in the (q,p)-space). In the quasi-classical trajectory approach, E is chosen as one of the quantum energies, and all points on the ellipse may be chosen as initial conditions in a calculation, i.e., corresponding to all phases a [0, 27r].

The quantum-mechanical description of a polyatomic system may be extrapolated from the treatment of the diatomic molecule. Starting again with the harmonic oscillator, the energy levels for the entire system (E) can be given in terms of the characteristic frequencies (v,) and quantum numbers ( ,) of a series of independent harmonic oscillators ... 

For an isolated molecule in the rigid rotor, harmonic oscillator approximation, the (quantum) energy states are sufficiently regular to allow an explicit construction of the partition function, as discussed in Chapter 12. For a collection of many particles the... 

Here again, therefore, we obtain for our term scheme an equidistant succession of energy levels, as in Bohr s theory. The sole difference lies in the fact that the whole term diagram of quantum mechanics is displaced relative to that of Bohr s theory by half a quantum of energy. Although this difference does not manifest itself in the spectrum, it plays a part in statistical problems. In any case it is important to note that the linear harmonic oscillator possesses energy hv in. the lowest state, the so-called zem-jpoint energy. 

The oscillation frequencies in the case of HC1 are at 5=2877 and 5=5657 (in wave numbers, i.e. number of waves per cm.). The corresponding bands appear in the case of absorption at the ordinary temperature. They correspond, therefore, to a change in the oscillation quantum number for which the initial state has so little energy that it is present to a considerable degree at the ordinary temperature that, however, can only be the oscillation state n2- -0. We assign, therefore, to the two bands observed the two transitions... 

Identifying as the Morse oscillator quantum number n we obtain for the energy... 

Any neutral atom or molecule can be schematized in terms of a positive centre surrounded by a cloud of negative electrification. If the positive centre is displaced from its equilibrium position, it will oscillate with a frequency v. According to the quantum theory, even in the lowest possible energy state, an oscillator possesses energy hvQ. Thus the atom or molecule wiU, from this cause alone, always... 

Fig. 4.16. A graphic representation of the 2-D hamonic oscillator wave function (isolinesj. Panels a) through ( i) show the wave functions labeled by a pair of oscillation quantum numbers (ui, V2). The higher the energy, the la ger the number of node planes. A reader acquainted with the wave functions of the hydrogen atom will easily recognize a striking resemblance between these figures and the orbitals.

---