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Derivation of the Hamiltonian in terms of
Unitary Group Generators

March 1994

We begin with the second-quantized form of the one- and two-electron operators (see Szabo and Ostlund [1], p. 95),

 (1)

 (2)

where the sums run over all spin orbitals . Thus the Hamiltonian is

 (3)

Now integrate over spin, assuming that spatial orbitals are constrained to be identical for and spins. A sum over all 2K spin orbitals can be split up into two sums, one over K orbitals with spin, and one over K orbitals with spin. Symbolically, this is

 (4)

The one-electron part of the Hamiltonian becomes

 (5)

After integrating over spin, this becomes

 (6)

The two-electron term can be expanded similarly to give

 (7)

Now we make use of the anticommutation relation

 (8)

and we swap the order of and , introducing a minus sign. This yields

 (9)

Now we use the anticommutation relation between a creation and an annihilation operator, which is

 (10)

This relation allows us to swap the aq and in each term, to give
 = + (11)

Now we observe that and can both be written , and also that and are both 0. This simplifies our equation to
 = - (12)

Now we introduce the unitary group generators, which we write as [2]

 (13)

and the Hamiltonian becomes

 (14)

This is the Hamiltonian in terms of the unitary group generators [3].

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C. David Sherrill
2000-04-18