Dirac Equation Generalized for Two Bound-State Electrons

Dirac Equation Generalized for Two Bound-State Electrons 

According to the Dirac equation for an electron in external electromagnetic fields, Eq. (5.121), we should make the following ansatz for the two-electron system including the external potential energy, Vnuc, of resting nuclei. 

Two problems need to be addressed next. First, what to do with the two different time derivatives and, second, how are the scalar and vector potentials of the electrons to be chosen. With respect to the former issue, we adopt the single absolute time frame, ti,t2 t of nonrelativistic theory. This reduces the two time derivatives to one with respect to the new absolute time t. 

This product ansatz is, of course, also not exactly known as we shall discuss later in this chapter. In chemistry, explicit and sufficiently accurate many-electron calculations are the aim. The solution functions tpi and ip2 of Eqs. (8.2) and (8.3) are not known because the structure of the potentials is too complicated even in the case of only two electrons — as we shall see — to allow for an analytic solution. We will see in the later sections 8.5 - 8.7 how to deal with this problem by invoking a mean-field approach. 

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