Biorthonormality

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]

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

In these equations, H is the matrix representation, while Rk and Lk are vectors, Lk being a row vector and Rk being a column vector. The left- and right-hand eigenvectors can be normalized so that they share a biorthonormal relationship [24] ... 

Biorthonormality of the left-hand and right-hand eigenvectors may be enforced such that... 

Note that the biorthonormality of the left- and right-hand states does not imply orthonormality of the left- or right-hand states among themselves, for example,... 

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... 

Multiplying (11.9.12) from the left by a determinant in the biorthonormal basis j, we anive at the following equation for the Cl corrections ... 

For CAS and RAS expansions, the matrix elements between nonorthogonal states are conveniently obtained using the procedure of Section 11.9. For each pair of states, we determine a biorthonormal basis - see Section 1.9.3. The overlap and Hamiltonian matrix elements of this basis are obtained by transforming the integrals and Cl vectors as described in Sections 1.9.3 and 11.9, respectively. A subsequent diagonalization of the resulting small Hamiltonian matrix (suitably weighted by... 

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... 

For projection, the important point is the existence of a biorthonormal basis that satisfies (13.7.54) and (13.7.55) its explicit representation is not required to carry out the projection. 

The bra states are easily normalized so as to form a biorthonormal basis by a simple scaling of the doubly excited states with a — b and i = j ... 

However, since the bra states are used only for projection of the coupled-cluster equations (which is zero for the optimized wave function), normalization is unimportant. In fact, use of the biorthogonal basis (13.7.59) rather than the biorthonormal basis (13.7.60) simplifies our algebraic manipulations considerably. 

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... 

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