Amplitude matrix definition

The amplitude of electron tunneling transfer with simultaneous change of the vibrations in the case of the non-adiabatic asymptotics (56) may be found, if to substitute the expression (56) instead of F ) to the definition (9) and the acceptor wave function in the matrix element (9) should be its total expression... 

The 5-matrix is unitary and symmetric, while the T-matrix is symmetric. This particular definition of the T-matrix reduces for scattering by a central potential to the phase-shift factor in the scattering amplitude,... 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

The discussion in the Introduction led to the convincing assumption that the strain-dependent behavior of filled rubbers is due to the break-down of filler networks within the rubber matrix. This conviction will be enhanced in the following sections. However, in contrast to this mechanism, sometimes alternative models have been proposed. Gui et al. theorized that the strain amplitude effect was due to deformation, flow and alignment of the rubber molecules attached to the filler particle [41 ]. Another concept has been developed by Smith [42]. He has indicated that a shell of hard rubber (bound rubber) of definite thickness surrounds the filler and the non-linearity in dynamic mechanical behavior is related to the desorption and reabsorption of the hard absorbed shell around the carbon black. In a similar way, recently Maier and Goritz suggested a Langmuir-type polymer chain adsorption on the filler surface to explain the Payne-effect [43]. 

Analogous to the present use of complex time to construct amplitudes which refer to a definite energy is the more familiar use of complex energy to describe time dependence. The most common example of this is when considering the decay of a prepared state.60 The probability amplitude that the system has not decayed from its initial state < > is a diagonal matrix element... 

Clearly, the best correlation pattern complying exactly with the expected Lorentzian form is obtained in the case of the AvAF amplitudes in connection with a comparison of frequencies with adiabatic internal frequencies (Og. Adiabatic internal modes, the amplitude definition of Eq. (64) and the force constant matrix f as a suitable metric for comparison provide the right ingredients for a physically well-founded CNM analysis. 

Within the density-matrix formalism (Vol. 1, Sect. 2.9) the coherent techniques measure the off-diagonal elements pab of the density matrix, called the coherences, while incoherent spectroscopy only yields information about the diagonal elements, representing the time-dependent population densities. The off-diagonal elements describe the atomic dipoles induced by the radiation field, which oscillate at the field frequency radiation sources with the field amplitude Ak(r, t). Under coherent excitation the dipoles oscillate with definite phase relations, and the phase-sensitive superposition of the radiation amplitudes Ak results in measurable interference phenomena (quantum beats, photon echoes, free induction decay, etc.). 

The same quantities (6.28) are identified with hole-particle amplitudes which are just equal to matrix elements of hole-particle excitation operators Ct By definition, Ck generates the superposition of k-excited configurations of the corresponding order k (for more detail see [39,42]). Within the customary hole-paiticle formalism, the first k indices O]... in Cai...ay, are related to states of particles which are excited above a sea of occupied states, whereas the second k indices i. .. 4 (occupied orbitals) are related to the possible hole states in the same sea. This well-known interpretation is also suitable for designing correlation indices. To this end, let us consider the normalization condition which is, evidently. 

Since the internal amplitude is by definition zero, the term with denominator and all disconnected terms of the right-hand-side vanish [4,5] and the matrix element is obtained as right-hand-side of the amplitude update equation for s ... 

The large amplitude of the continuum resonance states is a direct result of the non Hermitian properties of Hamiltonian (i.e. the resonance eigenfunctions which are associated with complex eigenvalues are not in the Hermitian domain of the molecular Hamiltonian). Let us explain this point in some more detail. As was mentioned above Moiseyev and Priedland [7] have proved that if two N x N real symmetric matrices H and H2 do not commute, there exists at least one value of parameter A = such that matrix H + XH2 possesses incomplete spectrum. That is at A A there are at least two specific eigenstates i and j for which ]imx j ei — ej) =0 and also lim ( i j) — 0- Since and ipj are orthogonal (within the general inner product definition i.e., i/ il i/ j) = = 0 not in the... 

Given the definition of S matrix operator, S = S2 S2+, the probability amplitude to scatter from an initial state i of the reactant arrangement a to a final state / of the product arrangement fi is written as the matrix element of the S operator [153],... 

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