Hamiltonian, second-quantized general form

In order to be able to write out all the terms of the direct Cl equations explicitly, the Hamiltonian operator is needed in a form where the integrals appear. This is done using the language of second quantization, which has been reviewed in the mathematical lectures. Since, in the MR-CI method, we will generally work with spin-adapted configurations a particularly useful form of the Hamiltonian is obtained in terms of the generators of the unitary group. The Hamiltonian in terms of these operators is written,... 

This representation among others removes one more inconsistency in quantum chemistry one generally deals with the systems of constant composition i.e. of the fixed number of electrons. The expression eq. (1.178) allows one to express the matrix elements of an electronic Hamiltonian without the necessity to go in a subspace with number of electrons different from the considered number N which is implied by the second quantization formalism of the Fermi creation and annihilation operators and on the other hand allows to keep the general form independent explicitly neither on the above number of electrons nor on the total spin which are both condensed in the matrix form of the generators E specific for the Young pattern T for which they are calculated. 

This general notation is deceptively simple. The bra is an excited determinant. There is an equation for each excited determinant, and each level of excitation leads to a different type of equation. Furthermore, the equations are all coupled, and they are non-linear in the amplitudes. However, they may be formulated in a quasilinear manner [27], and they have been solved for a wide range of CC schemes. The operator HN is the Hamiltonian written in second-quantized form minus the energy of the reference determinant, i.e. HN = H— < 0 /7 0 >. The subscript C restricts the operator product of HN and eT to connected terms. Once the CC equations have been solved, the CC correlation energy can be calculated from... 

The general Hamiltonian of a molecule interacting with an external field in second quantization form reads "... 

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