Hamiltonian, second-quantized nonrelativistic

Combining the results of Sections 1.4.1 and 1.4.2, we may now construct the full second-quantization representation of the electronic Hamiltonian operator in the Bom-Oppenheimer approximation. Although not strictly needed for the development of the second-quantization theory in this chapter, we present the detailed form of this operator as an example of the construction of operators in second quantization. In the absence of external fields, the second-quantization nonrelativistic and spin-free molecular electronic Hamiltonian is given by... 

The formulation of the relativistic CASPT2 method is almost the same as the nonrelativistic CASPT2 in the second quantized form. In this section, firstly we express the relativistic Hamiltonian in the second quantized form, and then, we give a summary of the CASPT2 method [11, 12],... 

Ad (i) The nonrelativistic Born-Oppenheimer many-body Hamiltonian projected to a given basis set can be most conveniently specified by the usual second quantized form. For convenience, the underlying basis set is assumed to be orthonormalized MBPT calculations are usually performed in the MO basis which meets this criterion. 

Here E and E are the exact energies of the two individual molecules A and B when they are isolated, while E" is the exact energy of the supersystem (molecular complex, for example). Theoretically, these quantities can be obtained from the exact solution of the Schrodinger equation for the corresponding systems. (We remain within the nonrelativistic Born-Oppenheimer model.) This requires the definition of the Hamiltonians H", H and H" , and one feels challenged to handle these Hamiltonians in a common (e.g., perturbational) scheme. This point is not at all trivial especially if approximate model Hamiltonians are used. In what follows we shall consider this issue emphasizing the points where the second quantized approach can help to clarify the situation. 

In second quantization, the electronic Hamiltonian operator is expressed as a linear combination of strings of creation and annihilation operators. The following form is appropriate for a spin-free, nonrelativistic electronic system ... 

We are now in a position to write up the second-quantization representation of the nonrelativistic and spin-free molecular electronic Hamiltonian in the orbital basis ... 

---