Electro-nuclear separation theory

Let us first define the general problem, namely, to find solutions to the Schrodinger equation for the global system 

Q) is the Hamiltonian of the interacting electrons and nuclei. The configuration variables of ne electrons, r=(r1.rnc), and nN nuclei, Q=(Qi.QnN) are defined with respect to a coordinate frame fixed in the laboratory. The nuclei are identified by their nuclear charges, Z=(eZ.eZ N), where e is the electron charge, and the mass vector (or block diagonal 3nNx3nN matrix) M=(m,.m ). 

In general, H(r,Q) is invariant to a translation of origin of the laboratory frame, not to be mixed up with the space fixed frame which moves with the center-of-mass commonly used [1]. Let us select a particular (arbitrary for the time being) origin with real space coordinates u. Applying the corresponding translation operator H(r,Q) changes into  

The molecular Hamiltonian, Hm(T), in the fixed frame with origin u, is given 

The configuration coordinates of electrons (p) and nuclei (R) in the new frame are related to the laboratory one by rk = u + pk, Qk. = u + Rk, symbolically written as r =u+p, Q=u+R, and T=(p,R). Ke represents the electrons kinetic energy operators Vee (p), VeN(p, R) and Vnn(R) are the standard Coulomb interaction potentials they are invariant to origin translation. The vector u is just a vector in real space R3. Kn is the kinetic energy operator of the nuclei, and in this work the electronic Hamiltonian He(r Z) includes all Coulomb interactions. This Hamiltonian would represent a general electronic system submitted to arbitrary sources of external Coulomb potential. 

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