Hamiltonian vibrational corrections

Atieh et al. have calculated spin-Hamiltonian parameters (chemical shifts and spin-spin couplings) for serine (Fig. 8c), taking into account solvent and conformer effects, and zero-point vibrational correction. The authors performed the calculations at the B3LYP/6-311-I--I-G theory level. 

PhD thesis for the quantization of electron-vibrational Hamiltonian. Now a revised version will be presented for the complete electron-hypervibrational Hamiltonian which corrects the original version which totally failed during comparison tests with the results of the Born-Handy ansatz. 

The theoretical method, as developed before, concerns a molecule whose nuclei are fixed in a given geometry and whose wavefimctions are the eigenfunctions of the electronic Hamiltonian. Actually, the molecular structure is vibrating and rotating and the electric field is acting on the vibration itself. Thus, in a companion work, we have evaluated the vibronic corrections (5) in order to correct and to compare our results with experimental values. 

That effective hamiltonian according to formula 29, with neglect of W"(R), appears to be the most comprehensive and practical currently available for spectral reduction when one seeks to take into account all three principal extramechanical terms, namely radial functions for rotational and vibrational g factors and adiabatic corrections. The form of this effective hamiltonian differs slightly from that used by van Vleck [9], who failed to recognise a connection between the electronic contribution to the rotational g factor and rotational nonadiabatic terms [150,56]. There exists nevertheless a clear evolution from the advance in van Vleck s [9] elaboration of Dunham s [5] innovative derivation of vibration-rotational energies into the present effective hamiltonian in formula 29 through the work of Herman [60,66]. The notation g for two radial functions pertaining to extra-mechanical effects in formula 29 alludes to that connection between... 

In the present simulation we apply a simple model consisting of two vibrational modes in the relevant system. The first one will contribute to the Stokes shift as well as to the Herzberg-Teller correction, while the second one only to the Stokes shift. Next, we will assume a mutual coupling of the modes in the bilinear form Q Qi for the excited state. This type of coupling is usually omitted in literature. The Hamiltonian 77s is written as... 

The first term of (3.289) represents a translational Stark effect. A molecule with a permanent dipole moment experiences a moving magnetic field as an electric field and hence shows an interaction the term could equally well be interpreted as a Zeeman effect. The second term represents the nuclear rotation and vibration Zeeman interactions we shall deal with this more fully below. The fourth term gives the interaction of the field with the orbital motion of the electrons and its small polarisation correction. The other terms are probably not important but are retained to preserve the gauge invariance of the Hamiltonian. For an ionic species (q 0) we have the additional translational term... 

We use the operator on the left-hand side of equation (7.169) as the zeroth-order vibrational Hamiltonian. The remaining terms in the effective electronic Hamiltonian, given for example in equations (7.124) and (7.137), are treated as perturbations. In a similar vein to the electronic problem, we consider only first- and second-order corrections as given in equations (7.68) and (7.69) to produce an effective Hamiltonian 3Q, which is confined to act within a single vibronic state rj, v) only. Once again, the condition for the validity of this approximation is that the perturbation matrix elements should be small compared with the vibrational intervals. It will therefore tend to fail for loosely bound states with low vibrational frequencies. 

We have seen that the dependence of Bn(R) on the vibrational coordinate causes a mixing of the vibrational level of interest with neighbouring levels. This mixing results in centrifugal distortion corrections to all the various parameters Xn(R) in the perturbation Hamiltonian 3C when combined in a cross term. The operator has the same form as in the original term, for example, (2/3) /6T 0(S, S) for the spin spin dipolar term, multiplied by N2. The coefficient which qualifies this term has the general form... 

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