Bi-orthogonal sets

One can say that operators are related to a bi-orthogonal set with respect to Xfc. Given a set of nonorthogonal functions %k, the bi-orthogonal (or reciprocal) functions are defined by... 

All in all, the bi-orthogonal dilated electron propagator offers a simple extension of the real electron propagator technique and with the incorporation of higher order decouplings like the E3, E ADC(3) etc. and suitably large and flexible basis sets should offer same power and effectiveness in the treatment of metastable anions and cations as done by its real counterpart for stable bound systems. 

In u-MCPT version step 1 is missing, reciprocal vectors to the set 0> and A) s are directly constructed, giving (0 and (A. The spectral form of the zero-order Hamiltonian is nonsymmetric in both formulations, due to the use of bi-orthogonal vector sets ... 

Before proceeding to applications, let us discuss another important aspect of averaged second-order energies Eqs. (15) and (17). Both formulae are of multipartitioning nature the zero-order operator varies with principal determinant AT). This affects not only the bi-orthogonal vector set, but also the zero-order excitation... 

The set ket and bra wavefunctions k and 71k satisfy the property of bi-orthogonality many-body systems... 

The two sets 0, and 0j are often said to be bi-orthogonal relationship between them is easily seen to be... 

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