Biorthogonal sets

A subtly different procedure must be followed if the matrix M is non-Hermitian the main difference is that we must now operate with M and M, which are no longer equal. We have thus to consider the biorthogonal sets of vectors ... 

We do this first directly, then by the moment method, and finally by using projective techniques and biorthogonal sets. 

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,... 

Note that the formal superposition, Eq. (1.11), reproduces a physical attribute, yielding the present derivation of special relativity a tangible conception outside a purely abstract understanding. Another important observation, associated with the biorthogonal setting of the system, entails that the analysis shows that the formulation turns out to be nonstatistical. We notice moreover that the description for a zero rest mass particle (photon) corresponds to a degenerate singularity of the equations since... 

Owing to the presence of the antihermitian part of the eigenfunctions n> are not orthogonal, and they must be combined with conjugate functions [Pg.326]

Keywords Overlap Orthogonalization Biorthogonal sets Multiconfiguration perturbation theory Multistate... 

In addition to these second-order corrections to the RPA matrices there are three new matrices due to the operators. In the following, we present explicit expressions for them in terms of spatial orbitals (f>p and for two spin-free operators Pa and using a biorthogonal set of double excitation operators qq (Bak et ai, 2000). 

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,... 

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],... 

Similar remarks apply to practical computational schemes, such as CR-CC(2,3), based on the biorthogonal MMCC theory, as defined by Eqs. (30) and (39). In this case, to avoid the calculation of the entire set of moments of CC equations for a given... 

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 ... 

The solutions, Eq. (1.10), correspond each to a root of the characteristic equation Eqs. (1.6-1.8). Although the general setting of the complex symmetric forms ensures biorthogonality, the eigenvectors for % (and similarly for T) obtain simply as... 

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... 

Sometimes real, symmetric wavelets or wavelets with more vanishing moments and at the same time with smaller support than corresponding orthogonal wavelets or simply more regular", in closed form defined wavelets are required. One way to obtain them is to construct two sets of biorthogonal wavelets r/r and its dual x). One of these two biorthogonal wavelets is consequently used to decompose the signal and the second one to reconstruct it. The numbers M,N of... 

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