Insertion of Eq. (56) in Eq. (54) results in ⟨α1α2|ˆHI|β1β2⟩=12[⟨α1α2|ˆv|β1β2⟩−⟨α1α2|ˆv|β2β1⟩−⟨α2α1|ˆv|β1β2⟩+⟨α2α1|ˆv|β2β1⟩]=⟨α1α2|ˆv|β1β2⟩−⟨α1α2|ˆv|β2β1⟩=⟨α1α2|ˆv|β1β2⟩AS.