The representation of one- and two-electron operators

Expectation values correspond to observables and should therefore be independent of the representation given to the operators and the states. Since expectation values may be expressed as sums of matrix elements of operators, we require the matrix element of a second-quantization operator between two ON vectors to be equal to its counterpart in first quantization. An operator in the Fock space can thus be constructed by requiring its matrix elements between ON vectors to be equal to the corresponding matrix elements between Slater determinants of the first-quantization operator. 

Before proceeding to determine the form of the operators in second quantization, we recall that the matrix elements between Slater determinants depend on the spatial form of the spin orbitals. Since the ON vectors are independent of the spatial form of spin orbitals, we conclude that the second-quantization f )eratws - in contrast to their first-quantization counterparts - must depend on the spatial form of the spin orbitals. 

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