Vibrational states stationary

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

To simulate the vibrational progression, we obtain the Franck—Condon factors using the two-dimensional array method in ref 64. We consider 1 vibrational quantum v = 0) from the EC stationary point and 21 vibrational quanta (z/ = 0, 1,. .., 20) from the GC stationary point. The Franck— Condon factors are then calculated for every permutation up to 21 quanta over the vibrational modes. It is necessary in order to get all Franck—Condon factors of the EC stationary point with respect to each three alg vibrational state (Figure 6) of the GC to sum to one. One obtains a qualitative agreement between the calculated and the experimental emission profiles (Figure... 

Electronic transition between stationary states consists in the transfer of a photon by the Wheeler-Feynmann handshake mechanism which implies the photon to exist between the radial surfaces of the two vibrating states before emission or absorption, exactly as envisaged in Schrodinger s beat model for electron transition. 

In general, a minimum of the energy surface corresponds to a set of stationary vibrational states of the molecular system. The position of the energy minimum is commonly called the equilibrium geometry Re. Analogously, we denote the expectation values for the molecular geometries in the vibrational states 0,1,2,... by R0, Rx, R2, etc. In most cases Ro is very close to Re. There are also exceptions to this correspondence which are important in the theory of intermolecular forces. We distinguish several cases ... 

Some potential wells are too shallow to sustain stationary vibrational states. The best known example of this kind is the dimer of helium He2 (Fig. 1). 

The simplest way to combine electronic stnicture calculations with nuclear dynamics is to use harmonic analysis to estimate both vibrational averaging effects on physico-chemical observables and reaction rates in terms of conventional transition state theory, possibly extended to incorporate tunneling corrections. This requires, at least, the knowledge of the structures, energetics, and harmonic force fields of the relevant stationary points (i.e. energy minima and first order saddle points connecting pairs of minima). Small anq)litude vibrations around stationary points are expressed in terms of normal modes Q, which are linearly related to cartesian coordinates x... 

If the system has two minima of dipolar type equivalent in energy, as in the case of, e.g., the ammonia molecule, then instead of each vibrational state in the minimum, two stationary states divided by an energy interval 2 A arise owing to tunneling (inversion splitting). We constrain ourselves to the consideration of a(T) for the lowest inversion states only. In this approximation... 

The validity of the l -centroid approximation is based on a stationary phase argument (see Section 5.1.1) (Tellinghuisen 1984). For two vibrational states,... 

Because the initial vibrational state for absorption spectra often is v = 0, the vibrational nonstationary state typically produced initially is an only slightly distorted Gaussian wavepacket centered at R"g. Conservation of momentum requires that this approximately minimum-uncertainty wavepacket be launched at the turning point on the upper surface, R e = R"g, which lies vertically above R"g. [This is a consequence of the stationary phase condition, see Sections 5.1.1 and 7.6 and Tellinghuisen s (1984) discussion of the classical Franck-Condon... 

It is instructive to compute the time correlation function in the simple case that the ground and excited state potentials are harmonic but differ in their equilibrium position and frequency. This is particularly simple if the initial vibrational state is the ground state (or, in general, a coherent state (52)) so that its wave function is a Gaussian. We shall also use the Condon approximation where the transition dipole is taken to be a constant, independent of the nuclear separation, but explicit analytical results are possible even without this approximation. A quick derivation which uses the properties of coherent states (52) is as follows. The initial state on the upper approximation is, in the Condon approximation, a coherent state, i /,(0)) = a). The value of the parameter a is determined by the initial conditions which, if we start from a stationary state, are that there is no mean momentum and that the mean displacement (x) is the difference in the equilibrium position of the two potentials. In general, using m and o> to denote the mass and the vibrational frequency... 

It is the wave function which renders stationary the functional and not the nuclei (external potential) which determine the wave function. This is a feature of the present approach. This is due to the assumption that the stationary wave function defines not only a stationary geometry but also an equivalence class where all operations respecting the symmetry properties yield values ofp for which the inequality hold EK(pok) < E.(p) for systems having bound vibrational states. 

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