Mass polarization

Here Hg is the electronic Hamilton operator and H p is called the mass-polarization (Mtot is the total mass of all the nuclei and the sum is over all electrons). We note that He depends only on the nuclear positions (via Vne and Vnn, see eq. (3.23)) and not on their momenta. 

The electronic wave function has now been removed from the first two terms while the curly bracket contains tenns which couple different electronic states. The first two of these are the first- and second-order non-adiabatic coupling elements, respectively, vhile the last is the mass polarization. The non-adiabatic coupling elements are important for systems involving more than one electronic surface, such as photochemical reactions. 

Neglecting the mass polarization and reintroducing the kinetic energy operator gives... 

In the Bom-Oppenheimer picture the nuclei move on a potential energy surface (PES) which is a solution to the electronic Schrodinger equation. The PES is independent of the nuclear masses (i.e. it is the same for isotopic molecules), this is not the case when working in the adiabatic approximation since the diagonal correction (and mass polarization) depends on the nuclear masses. Solution of (3.16) for the nuclear wave function leads to energy levels for molecular vibrations (Section 13.1) and rotations, which in turn are the fundamentals for many forms of spectroscopy, such as IR, Raman, microwave etc. 

Wilets, L., and Cherry, I. J., Phys. Rev. 103, 112, Lower bound to the ground state energy and mass polarization in helium-like atoms. ... 

The conventional differential cross-section1,2 refers to the cross-section per solid angle in the center-of-mass polar coordinate. The desired doubly differential cross-section is then related to the measured quantity by the simple equation... 

Technically, the time-independent Schrodinger equation (2) is solved for clamped nuclei. The Hamiltonian is broken into its electronic part, He, including the nuclear Coulomb repulsion energy, and the nuclear Hamiltonian HN. At this level, mass polarization effects are usually neglected. The wave function is therefore factorized as usual (r,X)= vP(r X)g(X). Formally, the electronic wave function d lnX) and total electronic energy, E(X), are obtained after solving the equation for each value of X ... 

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. 

Other two-electron operators are the mass-polarization and the spin-orbit coupling operator. A two-electron operator gives non-vanishing matrix elements between two Slater determinants if the determinants contain at least two electrons and if they differ in the occupation of at most two pairs of electrons. The second quantization representation of a two-electron operator must thus have the structure... 

Before the individual parts of this function are discussed, the energy eigenvalue will be considered. The ground state energy g of the helium atom is just the energy value for double-ionization which can be determined accurately by several different kinds of experiments. Before the experimental value can be compared with the calculated one, some small corrections (for the reduced mass effect, mass polarization, relativistic effects, Lamb shift) are necessary which, for simplicity, are... 

Table 6. 1/n expansion coefficients bi for the Bethe logarithms of helium. The coefficients di and da give the finite mass correction due to mass polarization effects on the wave function. See Eqs. (16) and (17)... 

Table 2 also shows the QED contributions to helium-like resonance lines in vanadium as determined by Drake. The QED contributions are also expressed as a proportion of the relevant transition in ppm. The level at which our measurements test these contributions is between 5.7% and 8%. The theoretical QED contributions include mass polarization and nuclear size effects but these contribute less than 1% to the total. If the QED contributions to the 21 states are assumed to be correct, then the Is QED contribution is measured to 6%. 

The Hamiltonian in eqn.(10) can be viewed as representing the internal motions of a three-particle system or as the total energy of a two-particle system with fictitious masses mi, m2 and charges 91, q2 interacting with a charge 90 at the origin together with their mutual Coulomb interaction and a momentum dependent mass polarization potential . 

Drake[56] has computed a total mass polarization correction given as... 

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