Right-hand eigenvectors

Here, /j and rj are the l" left- and the J right-hand eigenvectors of the non-Hermitian Hamiltonian H. The operator is represented on the space spanned by the manifold created by the excitations out of a Hartree-Fock reference determinant, including the null excitation (the reference function). When we calculate the transition probability between a ground state g) and an excited state ]e), we need to evaluate and The reference function is a right-... 

Similarly we may define a right-hand eigenvector B satisfying... 

Each eigenvalue X, has associated with it a column, or right-hand eigenvector r, and a row, or left-hand eigenvector s . The X are given by the roots of the equation PX — XI = 0 (I is the unit matrix). 

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] ... 

Herein the vectors are the unit eigenvectors of the Cartesian coordinate system and 1 the right-hand eigenvectors of the value matrix W as discussed in Appendix 4. Formally the transformed differential equation is of the same general type as (11.12) ... 

The leading term of 1, which does not appear in T (cf. Eq. [29]), is required in order that the left- and right-hand eigenvectors have unit overlap with one another. Unlike the cluster operators, f , the operators act to the left on Therefore, it is convenient to define them as de-excitation operators (or, equivalently, as bra-state excitation operators),... 

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

Table 1 Left- and right-hand eigenvectors of zero-order Hamiltonians in p-MCPT and u-MCPT...

The eigenvalue analysis of (3.12) is performed using the HQR algorithm. The left- and right-hand eigenvectors of the original problem are then restored in the form... 

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