Biorthogonality

Stanton JF, Bartlett RJ (1993) The equation of motion coupled-cluster method - a systematic biorthogonal approach to molecular-excitation energies, transition-probabilities, and excited-state properties. J Chem Phys 98 7029... 

The biorthogonality and completeness relations presented above do not uniquely define the reciprocal basis vectors and mi a list of (3N) scalar components is required to specify the 3N components of these 3N reciprocal basis vectors, but only (3N) —fK equations involving the reciprocal vectors are provided by Eqs. (2.186-2.188), leaving/K more unknowns than equations. The source of the resulting arbitrariness may be understood by decomposing the reciprocal vectors into soft and hard components. The/ soft components of the / b vectors are completely determined by the equations of Eq. (2.186). Similarly, the hard components of the m vectors are determined by Eq. (2.187). These two restrictions leave undetermined both the fK hard components of the / b vectors and the Kf soft components of the K m vectors. Equation (2.188) provides another fK equations, but still leaves fK more equations than unknowns. Equation (2.189) does not involve the reciprocal vectors, and so is irrelevant for this purpose. We show below that a choice of reciprocal basis vectors may be uniquely specified by specifying arbitrary expressions for either the hard components of the b vectors or the soft components of m vectors (but not both). 

Biorthogonality conditions (2.186-2.189) and completeness relation (2.190) are equivalent to the biorthogonality and completeness relations (2.5) and (2.6) obeyed by partial derivatives in the full space, if we identify... 

Let us now describe this graphical construction in more mathematical terms. Although the reference axes R ) are generally nonorthogonal, one can always construct the associated set of conjugate vectors R ) that are biorthogonal to the R ), namely,... 

The biorthogonality relations (11.15) make clear the far-reaching symmetry between intensive vectors R ) and their conjugates R ) in the geometrical formalism. The formal symmetry is also seen in relations of the form... 

The biorthogonality property (12.4) then allows one to easily evaluate scalar products of each excess intensity RK) with basis axis vectors R ) or R ). The scalar product of extensive Xj) with (12.76) gives... 

The operators Sj are said to be biorthogonal to the operators a , since they... 

As already mentioned the derivation above leaves the interpretation, classical or quantum to the eye of the beholder. The second remark concerns biorthogonality, which implies that the coefficients c, will not be associated with a probability interpretation since we have the rule c + c = 1. The operators, in Eqs. (65)-(68), are in general non-selfadjoint and nonnormal (do not commute with its own adjoint), hence the order between them must be respected. We finally note that the general kets in Eq. (68) depend on energy and momenta, whereas in the conjugate problem, to be introduced below, they rely on time and position. Introducing well-known operator identifications, (h = 2nh is Planck s constant and V the gradient operator)... 

J. J. W. McDouall, in Valence Bond Theory, D. L. Cooper, Ed., Elsevier, Amsterdam, The Netherlands, 2002, pp. 227-260. The Biorthogonal Valence Bond Method. 

Other multiconfiguration VB methods have also been devised, like the biorthogonal valence bond method of McDouall (35,36) or the spin-free approach of McWeeny (37). For an overview of these methods, the reader is advised to consult a recent review (1). 

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