Nuclear momenta

We can make further approximations to simplify the NRF of the Hamiltonian presented in equation (75) for non-dynamical properties. For such properties, we can freeze the nuclear movements and study only the electronic problem. This is commonly known as the clamped nuclei approximation, and it usually is quite good because of the fact that the nuclei of a molecule are about 1836 times more massive than the electrons, so we can usually think of the nuclei moving slowly in the average field of the electrons, which are able to adapt almost instantaneously to the nuclear motion. Invocation of the clamped nuclei approximation to equation (75) causes all the nuclear contributions which involve the nuclear momentum operator to vanish and the others to become constants (nuclear repulsion, etc.). These constant terms will only shift the total energy of the system. The remaining terms in the Hamiltonian are electronic terms and nuclear-electronic interaction contributions which do not involve the nuclear momentum operator. 

To consider the nature of this approximation one should notice that the nuclear kinetic energy operator acts both on the electronic and the nuclear parts of the BO wavefunction. Hence, the deviations from the adiabatic approximation will be measured by the matrix elements of the nuclear kinetic energy, T(Q), and of the nuclear momentum. The approximate adiabatic wavefunctions have the following off-diagonal matrix elements between different vibronic states ... 

Equation (15) becomes exact in the e —> 0 limit, and can be made more explicit by evaluating the kinetic term by inserting a complete set of states in the nuclear momentum representation to obtain, for example. 

In the beta-decay allowed approximation, we neglect the variation of the lepton wave-functions over the nuclear volume and the nuclear momentum (this is equivalent to neglecting all total lepton orbital angular momenta L > 0). The total angular momentum carried off by the leptons is their total spin i.e. 5 = 1 or 0, since each lepton has When the lepton spins in the final state are antiparallel, se+s = stot = 0 the process is the Fermi transition with Vector coupling constant g = Cv (e.g. a pure Fermi decay 140(J r = 0+) —>14 N(JJ = 0+)). When the final state lepton spins are parallel, se + sv = stot = 1> the process is... 

To do so we replace the nuclear momentum operator with its classical counterpart Pk such that... 

T can be further simplified by eliminating the nuclear momentum coupling operator by using the condition... 

Here the square of the vibrational matrix element is written as a double integral in respect to q i and q, the latter of which represents the same nuclear coordinate. The K, (t) corresponds to the single-mode generating function that involves the nuclear momentum operator for the promoting mode q ... 

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