Atomic force and virial theorems in the presence of external fields

The statement of the atomic action principle which enables one to calculate the properties of an atom in a molecule in the presence of an electromagnetic field is (compare eqn (8.145) for the field-free case) 

The Hamiltonian appearing in eqn (8.225) for a system of electrons, charge = — e, and nuclei with general charge -t- Z e is 

Ehrenfest s first relation, the definition of the velocity and its associated momentum, is obtained using the commutator of H and the position vector r. This commutator, using the above Hamiltonian and multiplied by m, does indeed yield k as the momentum of an electron in an electromagnetic field 

The evaluation of the right-hand side of eqn (8.228) for F = mi and equating the result to the right-hand side of eqn (8.225) then yields an expression for the atomic average of the momentum a in the presence of a varying electromagnetic field 

The atomic statement of the Ehrenfest force law is obtained by setting the generator F equal to the momentum a. The commutator of the many-particle Hamiltonian eqn (8.226) and a can be expressed as 

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ACTION PRINCIPLE FOR A QUANTUM SUBSYSTEM 8.5 8.5.3 Atomic force and virial theorems in the presence of external fields... 

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