Mass-polarization term

In this way the mass polarization term may be removed from the Hamiltonian. However, the resulting electronic wave functions which are obtained are then dependent upon the nuclear masses as well as the nuclear charges and such wave functions are an inconvenient basis from which to investigate nonadiabatic processes. 

From the present calculations, the expectation value of the operator r 2 may provide a direct physical picture about the thermodynamic stability and dissociation of Hj-like molecules. As shown in Fig. 16, there is a vertical jump of the mean value ru at Xc. We note that there are similarities and differences between helium-like atoms and Hj-like molecules. In Section V.A of heliumlike systems, based on an infinite mass assumption, we show that the electron at the critical point leaves the atom with zero kinetic energy in a first-order phase transition. This limit corresponds to the ionization of an electron as the nuclear charge varies. For the Hj-like molecules, the two protons move in an electronic potential with a mass-polarization term. They move apart as X approaches its critical point and the system approaches its dissociation limit through a first-order phase transition. 

The observation of two different branches leads us to investigate the similarity between the molecule-like systems of the right branch, k > Km, such as the Ps , and the atom-like systems of the left branch, k < Km, such as ppd. The parameter k measures the strength of the mass polarization term, which is due to the motion of the two identical particles with respect to the third particle. The mass polarization term is then a measure of the momentum correlation of the two identical particles with respect to the third particle. If k Km, as in the case of a molecule such as Hj, then the light particle with mass m and charge q... 

The reduced-mass and mass-polarization terms arise on transforming the many-electron plus nucleus Hamiltonian to center of mass coordinates. 

For high precision calculations, and especially for the isotope shift, it is necessary to include also the motion of the nucleus in the center-of-mass (CM) frame. A transformation to CM plus relative coordinates yields the additional — fx./M)V V2 mass polarization term in the modified Hamiltonian... 

Here He is the electronic Hamiltonian operator and Hmp is called the mass-polarization (Mtot is the total mass of all the nuclei). The mass-polarization term arises because it is not possible to rigorously separate the centre of mass motion from the internal motion for a system with more than two particles. We note that He only depends on the nuclear positions (via Vne and Vm, see eq. (3.23)), but not on their momenta. 

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