Lanczos functions

The natures of the Lanczos functions and various properties of Stleltjes orbitals are best demonstrated by specific example. In Figure 3 are shown the first ten radial Lanczos functions for the ls->kp spectrum of a hydrogen atom (18). These are obtained from solution of Equations (4) for j=l to 10 employing the appropriate hydrogenic Hamiltonian in the operator A(H) and the Is ground state orbital in the test function. In this case. Equation (4b) can be solved for the v. in terms of Laguerre functions with constant expo-... 

Figure 3. Radial Lanczos functions of Equations (4) for Is-Hcp excitation and Ionization of atomic hydrogen (18) The abscissa spans 24 a, the ordinate 0.07 a.u.

Abstract. We present novel time integration schemes for Newtonian dynamics whose fastest oscillations are nearly harmonic, for constrained Newtonian dynamics including the Car-Parrinello equations of ab initio molecular dynamics, and for mixed quantum-classical molecular dynamics. The methods attain favorable properties by using matrix-function vector products which are computed via Lanczos method. This permits to take longer time steps than in standard integrators. 

A block Lanczos algorithm (where one starts with more than one vector) has been used to calculate the first 120 normal modes of citrate synthase [4]. In this calculation no apparent use was made of symmetry, but it appears that to save memory a short cutoff of 7.5 A was used to create a sparse matrix. The results suggested some overlap between the low frequency normal modes and functional modes detennined from the two X-ray conformers. 

A commonly used approach for computing the transition amplitudes is to approximate the propagator in the Krylov subspace, in a similar spirit to the time-dependent wave packet approach.7 For example, the Lanczos-based QMR has been used for U(H) = (E — H)-1 when calculating S-matrix elements from an initial channel (%m )-93 97 The transition amplitudes to all final channels (Xm) can be computed from the cross-correlation functions, namely their overlaps with the recurring vectors. Since the initial vector is given by xmo only a column of the S-matrix can be obtained from a single Lanczos recursion. 

In this chapter, we also discussed several schemes that allow for the computation of scalar observables without explicit construction and storage of the eigenvectors. This is important not only numerically for minimizing the core memory requirement but also conceptually because such a strategy is reminiscent of the experimental measurement, which almost never measures the wave function explicitly. Both the Lanczos and the Chebyshev recursion-based methods for this purpose have been developed and applied to both bound-state and scattering problems by various groups. 

Resonance Energies, Widths, and Wave Functions Using a Lanczos Method in Real Arithmetic. 

Contracted Basis Functions with a Lanczos Eigensolver for Computing Vibrational Spectra of Molecules with Four or More Atoms. 

Matrix Spectroscopy The Preconditioned Green-Function Block Lanczos Algorithm. 

Strategies for Spectral Analysis in Dissipative Systems Filter Diagonalization in the Lanczos Representation and Harmonic Inversion of the Chebychev Order Domain Autocorrelation Function. 

Constraining Basis Function Indices When Using the Lanczos Algorithm to Calculate Vibrational Energy Levels. 

Every Lanczos iteration with P, however, requires two operations of the Green s function onto a vector,... 

We have again utilized the Lanczos algorithm to compute the B strength function for the l3°Cd(0+) l30In(1 + ) Gamow-Teller transitions. Low-... 

Fig. 2. Calculated GT-strength function for the 130Cd ground-state (ON decays to the low-lying 1+ (T=16) states in 130In as a sequence of the number of Lanczos iterations. The abscissa is the excitation energy in 130In. For convenience of drawing, the minimum width is chosen to be 100 keV.

The recursion (84) can be extended to operators and matrices. This is done by using the Cayley-Hamilton theorem [2], which states that for a given analytic scalar function/(m), the expression for its operator counterpart/(U) is obtained via replacement of u by U as in Eq. (6). In this way, we can introduce the Lanczos operator and matrix polynomials defined by the following recursions ... 

This is the orthogonality relation of the two Lanczos polynomials Q (m) and Qm(u) with the weight function, which is the residue dk [48]. We recall that the sequence Q = (Q ( z-) coincides with the set of eigenvectors of the Jacobi matrix (60). 

The orthogonal characteristic polynomials or eigenpolynomials Qn(u) play one of the central roles in spectral analysis since they form a basis due to the completeness relation (163). They can be computed either via the Lanczos recursion (84) or from the power series representation (114). The latter method generates the expansion coefficients q , -r through the recursion (117). Alternatively, these coefficients can be deduced from the Lanczos recursion (97) for the rth derivative Q /r(0) since we have qni r = (l/r )Q r(0) as in Eq. (122). The polynomial set Qn(u) is the basis comprised of scalar functions in the Lanczos vector space C from Eq. (135). In Eq. (135), the definition (142) of the inner product implies that the polynomials Qn(u) and Qm(u) are orthogonal to each other (for n= m) with respect to the complex weight function dk, as per (166). The completeness (163) of the set Q (u) enables expansion of every function f(u) e C in a series in terms of the... 

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