Harmonic Ansatz

The relations derived up to this point are not sufficient to solve the problem of the dispersion of the optical branch for small k values. The retardation effect must also be taken into account. Because of the finite velocity of electromagnetic waves, the forces at a certain point of time and space in a crystal are determined by the states of the whole crystal at earlier times. A precise description of the dispersion effect therefore requires the introduction of Maxwell s equations. With a harmonic ansatz for P and P which is analogous to Eq. (II.15) they lead to the relation... 

The last equation is a variation principle for the coupled cluster quasi-energy and wavefunction within oscillating harmonic external fields. If one inserts a perturbation and Fourier expansion as ansatz for the cluster amplitudes... 

Note that with the simplified potential (14.61) our problem becomes mathematically similarto that of a harmonic oscillator, albeit with a negative force constant. Because of its linear character we may anticipate that a linear transformation on the variables X and V can lead to a separation of variables. With this in mind we follow Kramers by making the ansatz that Eq. (14.62) may be satisfied by a function f of one linear combination of x and v, that is, we seek a solution of the form... 

Figure C3.5.6 compares the result of this ansatz to the numerical result from the Wiener-Khintchine theorem. They agree well and the ansatz exhibits the expected exponential energy-gap law (VER rate decreases exponentially with Q). The ansatz was used to determine the VER rate with no quantum correction Q = 1), with the Bader-Beme harmonic correction [61] and with a correction based [M, M] on EgelstafPs method [62]. The Egelstaff corrected results were within a factor of five of experiment, whereas other eorrections were off by orders of magnitude. This ealeulation represents the present state of the art in computing VER rates in such difficult systems, inasmuch as the authors used only a model potential and no adjustable parameters. However the ansatz proeedure is clearly not extendible to polyatomic molecules or to diatomic molecules in polyatomic solvents.

The GED approach uses a mixed QC ansatz to construct a separable model for the electronuclear problem. As it corresponds to the case of a set of identical fermions, the electronic part is fully quantized in the GED scheme. The PCB is endowed with a mass distribution allowing for a correspondence with the external potential created by nuclei. The harmonic oscillator model of nuclear dynamics then follows as a quantum extension of the latter model, one where mass fluctuations are described by normal modes related... 

The orthonormality restriction for the spinors, Eq. (8.108), may be split into two parts in the case of atoms. The product ansatz for the spinor automatically yields orthonormal angular parts (coupled spherical harmonics cf. chapter 9). But these do not contain information about the principal quantum numbers in the composite indices i and j. For this reason, the restriction to orthonormal spinors results in the orthonormality restriction for radial functions... 

While for the rotational states in the first instance only the equilibrium distance between the nuclei is important, for the vibrational states the matter is not that simple, even if the potential law between the two atoms is assumed to be known. The classical picture of a harmonic or anharmonic oscillator is insufficient, since with these approximations the potential at infinity is completely wrong and quantum-mechanically the full curve is of importance. The ansatz used by Pues which... 

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