The many-electron Hamiltonian is expressed as

where the Mulliken notation is used for the two-electron integrals. The Fock matrix then has the elements

with an element of the 1-matrix, and the single-particle density operator . The tilde denotes operators relative the Hartree-Fock spin orbitals, *i.e.*,

with and .

The average value of the Hamiltonian can be expressed in a number of ways

Note that

with

Thus,

because

and because the trace of a ...

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