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Operators in second quantization

Insertion of Eq. (56) in Eq. (54) results in α1α2|ˆHI|β1β2=12[α1α2|ˆv|β1β2α1α2|ˆv|β2β1α2α1|ˆv|β1β2+α2α1|ˆv|β2β1]=α1α2|ˆv|β1β2α1α2|ˆv|β2β1=α1α2|ˆv|β1β2AS.