Fock Space and Occupation Number Vectors

Up to this point we have tailored the second-quantization formalism in close connection to the independent-particle picture introduced before. However, the formalism can be generalized in an even more abstract fashion. For this we introduce so-called occupation number vectors, which are state vectors in Fock space. Fock space is a mathematical concept that allows us to treat variable particle numbers (although this is hardly exploited in quantum chemistry see for an exception the Fock-space coupled-cluster approach mentioned in section 8.9). Accordingly, it represents loosely speaking all Hilbert spaces for different but fixed particle numbers and can therefore be formally written as a direct sum of N-electron Hilbert spaces. 

The occupation number vector can be written as a sequence of O s and I s according to the occupation of a particular one-electron state 

In particular, we may generate any reference state from the vacuum state, 

What has been said so far assumed exactly known one-particle states. These are, however, not known in quantum chemistry the orbitals are only approximations and, hence, a Cl-like expansion of the state results (compare section 8.5). However, the formalism can still be elegantly employed in chemistry [284,352] as a bookkeeping scheme that also takes care of the Slater-Condon rules as demonstrated above. Consequently, the total state is then to 

Finally, in expectation values sequences of annihilation and creation operators stemming from the second-quantized Hamiltonian and from the states in bra and ket of the full bra-ket must be evaluated for which rules such as Wick s theorem, which implements the anticommutation relations of operator pairs to obtain a relation to normal ordered operator products, can be beneficial [65,353]. 

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