Biorthonormal sets

Appendix A. Construction of Dual Complex Conjugate Biorthonormal Sets... 

As noted in Section 13.6.1, we would like the EOM-CC excited states (13.6.2) to be orthonormal. Orthonormality in the usual sense presents problems of the sort discussed in Section 13.1.4. Indeed, even the calculation of the norm of the ground state (CC CC> is cumbersome since the excitation operators in T do not commute with the de-exdtation operators in 7. We solve this problem by resorting to biorthogonality, expanding the bra states in a set of configurations that, toother wddi the ket states (13.6.2), constitute a biorthonormal set Adc )ting the notation... 

To illustrate the use of the biorthogonal basis, let us consider the norm of the CCSD state. Expanding in a full biorthonormal set of basis functions, we find that the squared norm of the... 

A set C having this property is said to be complex conjugate biorthonormal. 

It is further evident that it would greatly simplify the overlap problem if one could introduce dual bases

linearly independent set tj) = (/> , 4>2,..., [Pg.97]

As in the EOMXCC theory, the left-hand operators Cl, Eq. (234), are more exponential than the EOMCC operators Lr. Together with the right-hand eigenstates 7 1 ), the left-hand states ( Cl form a biorthogonal set, which can be made biorthonormal,... 

We now introduce a biorthonormal complete set of functions defined in the q subspace... 

We may now use the method of Appendix A for the case of a biorthonormal discrete set of eigenfunctions. We then have that... 

Vector set c biorthonormal to c . j is obtained by the transformation C = CLpo, fulfilling... 

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