Second Quantized Form of the Born-Oppenheimer Hamiltonian

4 The Second Quantized Form of the Born-Oppenheimer Hamiltonian 

The Born-Oppenheimer electronic Hamiltonian introduced in Eq. (1.2) consists of two parts the first one is a sum of one-electron operators, the second one is a sum of two-electron operators  

The one-electron part contains the kinetic energy and the nuclear-electron attraction, while the two-electron part is due to the electron-electron repulsion. The corresponding matrix elements, h y and [ iv Xa], have been introduced by Eqs. (1.9)-(1.11). 

In Sect. 4.1 we have learned how to convert the one-electron part into the second quantized form [cf. Eq. (4.27)]. Similiary, the representation of the two-electron part is given by Eq. (4.38) of Sect. 4.2. Thus, the Hamiltonian in the second quantized form can be written down immediately as  

the Greek labels refer to an aribtrary orthonormalized basis set. Note that the labels a and X appear in a reverse order in the operator string as compared to that in the integral list. This is not a misprint, but a consequence of the eliminated negative sign in Eq. (4.37). 

---