Biorthogonal basis

Bader analysis, 667 balance, kinetic, 132 band, conduction, 533-534 band gap, 537 band structure, 523, 527 band, valence, 537, 610 bandwidth, 532 barrier as shell opening, 948 barrier of dissociation, 801 barriers of reaction, 948 basis, biorthogonal, 513 basis set, atomic, 428, 431,el37... 

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). 

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... 

Since the complex scaling of the exponents of the primitive basis set will lead to a complex primitive basic set u and hence to the loss of biorthogonality central to our constructions, the Moiseyev-Corcoran approach has been adopted by us /44-46/ and Donnelly /21,47-50/ in the construction of the molecular dilated electron propagator. 

The first-order state /i) is therefore obtained simply by applying Fq to the zeroth-order state, with the caution of rejecting /o)(/ol Fq j/g), because the contribution coming from the state /q) is already includ in the basis set. We could go on that way, as this is just how Mori proceeds in his celebrated papers. However, to avoid the Hermitian assumption made by Mori, we build up a biorthogonal basis set. This means that the state /i) has to be associated with the corresponding left state (we assume (/q = (/o )... 

After building up this biorthogonal basis set, we would be led naturally to expand over it the operator F, so as to introduce the following kind of time evolution ... 

The extension of the recursion method to non-Hermitian operators possessing real eigenvalues has been carried out by introducing an appropriate biorthogonal basis set in close analogy with the unsymmetric Lanczos procedure. Non-Hermitian operators with real eigenvalues are encountered, for instance, in the chemical pseudopotential theory. Notice that the two-sided recursion method in formulation (3.18) is also valid for relaxation operators, as previously discussed. 

It is easily seen by inspection that the biorthogonal basis set definition (3.55) cmnddes with the definifion (3.18) ven in the discussion of the Lanczos method. We recall that the dynamics of operators (liouville equations) or probabilities (Fokker-Planck equations) have a mathematical structure similar to Eq. (3.29) and can thus be treated with the same techniques (see, e.g., Chapter 1) once an appropriate generalization of a scalar product is performed. For instance, this same formalism has been successfully adopted to model phonon thermal baths and to include, in principle, anharmonicity effects in the interesting aspects of lattice dynamics and atom-solid collisions. ... 

It appears clear from Chapters I, III, and IV that the Mori theory is the major theoretical tool behind the algorithm illustrated in Section II, which derives the expansion parameters X, and from the moments s . This theory also affords us with a second straightforward way of determining these parameters that of deriving them directly from the biorthogonal basis set of states fi) and j/-) (Eqs. 2.15). As discussed at length in Chapter IV, this is an especially stable way of building up X, and A. The Lanczos method fol-... 

For the sake of simplicity, we shall assume in this review that the basis orbitals x form an orthonormal set. Generalization to the nonorthogonal case can be done in a straightforward manner using the biorthogonal technique [127, 128, 129, 109, 113, 126],... 

One of the major findings is that a single pair of biorthogonalized orbitals forms a physical basis for describing the tunneling transfer. How different is this orbital firom a the canonical HOMO In a naive Koopmans picture, the tunneling orbital should be HOMO. To investigate this question we have performed a series of calculations on Heme a of cytochrome c oxidase. Fig. 

In general, we will have to assume that the 5 and 5° spaces are not orthogonal. This means that there does not exist a vector in the 5-space which is orthogonal to all of the vectors of the 5°-space (22). In addition, the states )° constitute a nonorthonormal basis set for the model space, 5°. From a physical point of view, it is important to have a one-to-one correspondence between the exact eigenvectors, 1 ), and the vectors )°. However, another basis set, denoted w)°, n = 1,, biorthogonal to the previous one w)°, n = 1, N has to be defined and used in Bloch s formulation. These vectors satisfy the following equations ... 

Let us now eonstruet the so-called biorthogonal basis b, b2,hz with respeet to the basis vectors Oi. 02. 3 of the primitive lattiee, i.e., the vectors that satisfy the biorthogonality relations ... 

Let us now construct the so called biorthogonal basis bi,b2,b3 with respect to the basis vectors a, a2, of the primitive lattice, i.e. the vectors that satisfy the... 

Bloch function (p. 435) symmetry orbital (p. 435) biorthogonal basis (p. 436) inverse lattice (p. 436)... 

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