Lanczos model

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

The Lanczos method has been widely applied to the dynamics in Hubbard and Heisenberg model Hamiltonians[39]. The spectral intensity for an operator O is given by... 

Once the Lanczos coupling constants an,pn have been computed, we could construct the Green function TZ(u) = c0(< 0 ul — UJ cho), which is defined in Eq. (49). If in the formula (49) for 7Z(u), we use the Lanczos representation for the matrix Ml - U based on Eq. (60) within the infinite chain model, we shall have ... 

The memory function formalism leads to several advantages, both from a formal point of view and from a practical point of view. It makes transparent the relationship between the recursion method, the moment method, and the Lanczos metfiod on the one hand and the projective methods of nonequiUbrium statistical mechanics on the other. Also the ad hoc use of Padd iqiproximants of type [n/n +1], often adopted in the literature without true justification, now appears natural, since the approximants of the J-frac-tion (3.48) encountered in continued fraction expansions of autocorrelation functions are just of the type [n/n +1]. The mathematical apparatus of continued fractions can be profitably used to investigate properties of Green s functions and to embody in the formalism the physical information pertinent to specific models. Last but not least, the memory function formaUsm provides a new and simple PD algorithm to relate moments to continued fraction parameters. 

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. ... 

The non-adiabatic quantum simulation procedures we employ have been well described previously in the literature, so we describe them only briefly here. The model system consists of 200 classical SPC flexible water molecules," and one quantum mechanical electron interacting with the water molecules via a pseudopotential. 2 The equations of motion were integrated using the Verlet algorithm with a 1 fs time step in the microcanonical ensemble, and the adiabatic eigenstates at each time step were calculated with an iterative and block Lanczos scheme. Periodic boundary conditions were employed using a cubic simulation box of side 18.17A (water density 0.997 g/ml). 

The spectral Lanczos decomposition method is designed for effective calculation of the matrix functions. This method has found a useful application in the solution of linear equations in electromagnetic modeling (Druskin and Knizhnerman, 1994 Druskin et ah, 1999)). Let us consider, for example, the matrix equation (12.35). The formal solution of this equation has the form... 

Druskin, V., Knizhnerman, L., and P. Lee, 1999, New spectral Lanczos decomposition method for induction modeling in arbitrary 3D geometry Geophysics, 64, 701-706. 

F. Becca, A. Parola, S. Sorella (2000) Ground-state properties of the Hubbard model by Lanczos diagonalizations. Phys. Rev. B 61, pp. R16287-R16290 R. W. Hall (2002) An adaptive, kink-based approach to path integral calculations. J. Chem. Phys. 116, pp. 1-7... 

The dynamic response functions of finite interacting systems have most commonly been obtained from an explicit computation of the eigenstates of the Hamiltonian and the matrix elements of the appropriate operators in the basis of these eigenstates [115]. This has been a widely used method particularly in the computation of the dynamic NLO coefficients of molecular systems and is known as the sum-over-states (SOS) method. In the case of model Hamiltonians, the technique that has been widely exploited to study dynamics is the Lanczos method [116]. The spectral intensity corresponding to an operator O is given by ... 

The CEO computation of electronic structure starts with molecular geometry, optimized using standard quantum chemical methods, or obtained from experimental X-ray diffraction or NMR data. For excited-state calculations, we usually use the INDO/S semiempirical Hamiltonian model (Section IIA) generated by the ZINDO code " however, other model Hamiltonians may be employed as well. The next step is to calculate the Hartree— Fock (HE) ground state density matrix. This density matrix and the Hamiltonian are the Input Into the CEO calculation. Solving the TDHE equation of motion (Appendix A) Involves the diagonalization of the Liouville operator (Section IIB) which is efficiently performed using Kiylov-space techniques e.g., IDSMA (Appendix C), Lanczos (Appendix D), or... 

The combination of the linear-couphng model with the Lanczos algorithm allows for an extremely efficient computation of even very complex vibronic spectra. 

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