Second quantization of the Born-Oppenheimer Hamiltonian

The Born-Oppenheimer Hamiltonian describes the electronic degrees of freedom. A convenient representation of fermion Hamiltonians is by second quantization. As this representation is widely used in this book, we give a brief discussion of it here. A good discussion may be found in (Landau and Lifshitz 1977) or (Surjan 

In Dirac notation we may represent a single-particle electronic state as the ket f). Suppose that the single-particle states f) form an orthonormal basis. The projection of i) onto the coordinate representation, r), (where r) is an eigenstate of the position operator, f) gives the single-particle wave function (or 

It is often convenient to regard f) and cj i r) as different, but essentially equivalent 

Then we may define the creation operator, cj, such that it creates an electron in the orbital 4 i r). Formally, 

Since electrons carry spin we also need to define the creation operator c, which creates an electron with spin a in the spin-orbital, 

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