Real-energy method

As regards the autoionization widths. Table 4.1 contains the CESE results and the results of two previous theoretical ones that were obtained at some level of approximation using real-energy methods [125, 126]. The experimental ones are the ones quoted in Ref. [125]. 

Variational the energies predicted by a method ought to be an upper bound to the real energy resulting from the exact solution of the SchrPdinger equation. 

The impossibility of x being equal to about 1 V, as suggested by Kamieifki, " " has been demonstrated by Frumkin on the basis of a discussion of the real energies of hydration. Estimates from the variation in the solution surface potential with electrolyte molarity have yielded the value of +0.025 0.010 V.21 For methanol, the same method results in a value of -0.09 V.146 Later the authors of that investigation stated that both estimated values should be understood as the lower limits of surface potentials of water and methanol. "... 

The application of the Chebyshev recursion to complex-symmetric problems is more restricted because Chebyshev polynomials may diverge outside the real axis. Nevertheless, eigenvalues of a complex-symmetric matrix that are close to the real energy axis can be obtained using the FD method based on the damped Chebyshev recursion.155,215 For broad and even overlapping resonances, it has been shown that the use of multiple cross-correlation functions may be beneficial.216... 

Tab. 14.3 A list of some of the major properties which are determined for real materials using the standard model and employing the pseudopotential total energy method.

In most cases, except those in earlier comparative studies between the real-photon method and the dipole-simulation method, the absolute cross-section values obtained by both methods agree with each other [27]. Comparison of obtained cross-section values between the two methods were discussed in detail [27, 2, and references therein] and summarized in conclusion [5]. It should be noted, at least briefly, that it is essentially difficult to accurately obtain the absolute values of photoabsorption cross sections (u) in the dipole-simulation experiments, and it is necessary to use indirect ways in obtaining those values as the application of the TKR sum rule, Eq. (3), to the relative values of the cross sections obtained partly with theoretical assumptions. Moreover, in some cases, in relatively earlier dipole-simulation experiments, particularly of corrosive molecules upon their electron optics with poorer energy resolutions, serious discrepancy from the real-photon experiments was clearly pointed out in the obtained absolute values of photoabsorption cross sections [5,20,25-28]. 

Within the real-space method, the kinetic energy operator is expressed by the finite-difference scheme. Here, we derive the matrix elements for the kinetic energy operator of one dimension in the first-order finite difference. By the Taylor expansion of a wavefunction i/r (/) at the grid point Z we obtain the equations,... 

S. Fitzwater and H. A. Scheraga, Proc. Natl. Acad. Sci. USA, 79, 2133 (1982). Combined-Information Protein Structure Refinement Potential Energy-Constrained Real-Space Method for Refinement with Limited Diffraction Data. 

In the case of the DV-Xa method, an average charge density is used which may not lead to the real energy minimum. On the other hand, this charge density contains the contribution of all electrons, even that one which is used to calculate the interaction with the charge cloud, so the simple functional proposed by Slater... 

As / tends to zero the averaging procedure corresponds to evaluating the T-matrix element just above the branch cut on the real energy axis. It is numerically possible to choose such a large basis that the width I can be reduced to a value for which each averaged T-matrix element is independent of I. Such a calculation, however, involves enormous computational labour. The method has not yet been implemented up to the angular-momentum values required for a realistic calculation. 

Eigenfunction expansions as used in Ref. 168 are not accurate near the critical point. Instead, we developed a shooting point method in order to make a direct numerical integration of Eq. (110) with the condition Eq. (112). Real energies (bound and virtual) were found by bisection methods, and for complex energies it was necessary to combine the Newton-Raphson and grid methods. 

In conclusion it may be said that the real energies of a delocalized and/or solvated electron in solution can be computed only if the Volta potential difference in the electrode-solution system is known. The methods based on the estimation of surface potentials enable the chemical, or ideal energies to be evaluated. 

What is the position and width of an investigated resonance In order to answer this fundamental question, one can use complex coordinates within computational methods that had been originally developed for bound states. The real part of a complex energy, e, that one obtains for the resonance, constitutes its position, i.e., the actual real energy. The imaginary part gives the width of the resonance, P = —2/m(e), so it determines the lifetime of the state, T = p. 

The total width of the resonance is directly given by the resonance complex energy. In the case where many channels of autodetachment are open, the question of partial widths for the decay into individual channels arises. This always requires analysis of the wave fimction. The problem of obtaining partial widths from complex coordinate computation has been discussed by Noro and Taylor (39) and Bcicic and Simons (40), and recently by Moiseyev (10). However, these considerations do not seem to have found a practical application. Interchannel coupling for a real, multichannel, multielectron problem has been solved in a practical way within the CESE method by Nicolaides and Mercouris (41). According to this theory the partial widths, 7, and partial shifts to the real energy, Sj, are computed to all orders via the simple formula... 

Let me repeat the above argument, using the hydrogenic Hamiltonian with complex coordinates, which is the hallmark of the CCR method. According to mathematical analysis [107, 108], the poles of the hydrogenic H(rd ) are real and correspond to the discrete spectrum below threshold. As 6 increases from zero, these poles must remain at their initial positions on the real energy axis. 

For example, this form is in harmony with the superposition of energy states in Eq. (2), whose coefficients have been obtained formally by Fano [29]. Although, for the solution of particular problems involving unstable states, we have implemented, in conjunction with the methods of the SSA, the real-energy, Hermitian, Cl in the continuum formalism that characterizes Fano s theory, e.g.. Refs. [78, 82-87] and Chapter 6, in this chapter I focused on the theory and the nonperturbative method of solution of the complex eigenvalue Schrodinger equation (CESE), Eq. (27). 

The contribution of fhe open channels is fhen faken info account by appropriate methods that employ different function spaces, symbolized here by X s (as = asymptotic) (Eq. (1)). These methods are based either on K - matrix theory and numerically computed scattering orbitals for atoms and diatomics (real energy-dependent) (see [17, 29, 79-87]), or via diagonalization of fhe non-Hermifian mafrices of fhe CESE-SSA for field-free or field-induced resonances. 

The aforementioned approach to the TIMEP of ground or low-lying excited states that is based on the use of a single basis set characterized for many years the conventional quantum chemistry methods. These are the methods which obtain the wavefunctions and real energies either by direct diagonalization of huge Hamiltonian matrices or by incorporating the... 

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