Biorthogonal operators

It is possible to arrive at the results of Section 1.9.2 in a different way. We note that all complications due to nonorthogonality arise from the anticommutator (1.9.12). We therefore introduce a transformed set of annihilation operators ap that satisfy the anticommutation relation 

To satisfy this relation, the annihilation operators are chosen as 

That these operators indeed satisfy the anticommutation relation (1.9.22) can be verified by substitution and use of (1.9.12). The operators a pOQ behave just like excitation operators in an orthonormal basis  

A general one-electron operator can now be expanded in the excitation operators OpOQ 

Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge University Press, 1967. P. A. M. Dirac, The Principles of Quantum Mechanics, 4th edn, Oxford University Press, 1958. 

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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)... 

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 ... 

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 ... 

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. ... 

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,... 

As discussed earlier, the present interpretation of the truth table can be obtained from conventional representations with the use of a non-positive definite metric A All = —A22=1 Ai2=Ai2=0. In this picture, we can use conventional brake nomenclature, while for another selection of A, leading e.g. to a complex symmetric choice, it would require complex symmetric realisations. In both cases, the formulation is biorthogonal. With this realisation, we can make an identification between Eqs. (1.63) and (1.66), making the replacement q = /c(r), where q is related to the probability function/operator of the simple proposition Q = P. Hence, we realise a probabilistic origin combined with the nonclassical, self-referential character of gravitational interactions. Note also the analogy between the formulations, i.e. that the result of a classical measurement, i.e. the truth or... 

The discrete biorthogonal wavelet transform is also computationally very efficient, requiring only 0( ) operations, where n is the number of processed data. The most used filters are certainly the 9/7 filters which are by default in the JPEG 2000 norm. 

In the case of the one-dimensional multigroup diffusion operator, Q, it is shown that the spectral subspaces generated by the generalized eigenfunctions of Q are complete, and biorthogonal expansions in these generalized... 

This operator is considered as the Hamiltonian for the bond i. Apart from overlap effects reflected by the biorthogonal integral list, it describes an isolated bond. The next three terms of the total Hamiltonian in Eq. (16.32) describe interbond interactions. The expressions of these interaction Hamiltonians can be obtained by the partitioning of the summation labels according to Eq. (16.31). The pairwise interaction operator H after simple algebraic manipulations can be given in the following compact form ... 

The zeroth-order Hamiltonian H is not Hermitian, since the creation and annihilation operators are not the adjoints of each other. Owing to this non-Hermiticity various eigenstates of H do not form an orthonormal set. This is the same situation we were faced with in the intermolecular PT using the biorthogonal formulation (cf. Sect. 15), thus we have to apply here the same formulae for energy corrections, Eqs. (15.47). Application of these PT formulas... 

In one version we collect pairs of matrices S 2 into matrices and express the Hamiltonian in terms of the one- and two-electron integrals over the origmal overlapping basis orbitals and of the biorthogonal creation and annihilation operators... 

Comparison with Eq. (21) indicates that in overlapping basis the elements of the spin-dependent first- and second-order density matrix can be obtained as expectation values of operator strings constructed from biorthogonal creation and annihilation operators ... 

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). 

To sum up we note that the present level of formulation does not distinguish between classical- and quantum mechanics. A further characteristic reveals biorthogonality implying that the coefficients c,- are not to be associated with a probability interpretation, since they obey the rule Cj + c = 1. As emphasized, the operators in Eq. 1.63 are principally non-self-adjoint and non-normal and hence they might not commute with each other as well as their own adjoint. The order appearing in the resulting operator relations therefore has to be respected. 

Since the orhitals are nonorthogonal, we have introduced annihilation operators 5 that are biorthogonal to the creation operators in the sense that the anticommutation relations... 

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... 

Since the triples operator (14.4.11) is redundant, we cannot set up a projection basis (/I that is biorthogonal to the linear combination of CSFs ix) in (14.4.13). We shall simply assume that the constitute a linearly independent basis for the space spanned by the linearly dependent vectors /u.3 but we shall not specify their detailed form. 

T, and (4) Tf. Alternative (1) is said to be fully covariant, (2) is fully cowtravariant, and the other two are mixed representations. In principle, one is free to formulate physical laws and quantum chemical equations in any of these alternative representations, because the results are independent of the choice of representation. Furthermore, by applying the metric tensors, one may convert between all of these alternatives. It turns out, however, that it is convenient to use representations (3) or (4), which are sometimes called the natural representation. In this notation, every ket is considered to be a covariant tensor, and every bra is contravariant, which is advantageous as a result of the condition of biorthogonality in the natural representation, one obtains equations that are formally identical to those in an orthogonal basis, and operator equations may be translated directly into tensor equations in this natural representation. On the contrary, in fully co- or contravariant equations, one has to take the metric into account in many places, leading to formally more difficult equations. 

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