Lanczos states

Mapping from the Krylov-SchrOdinger to the Lanczos States 195... 

The alternative recursion (57) involves the monic Lanczos states ] ) Here, the term monic" serves to indicate that for any given integer n, the highest state ) in the finite sum, which defines the vector jr ), always has an overall multiplying coefficient equal to unity similarly to a monic polynomial [2], The Lanczos states ) are orthogonal, but unnormalized, as opposed to orthonormalized Lanczos states ( linear combinations of powers of the operator U(r) acting on the initial state 0)- Therefore, due to the relation 4> ) = U (r) 0) from Eq. (36), the vectors tjrn) and [ ) are certain sums of Schrodinger states ). ... 

In Section 5, we have introduced the finite-dimensional Lanczos subspace CM cH spanned by the Lanczos state vectors via ... 

Matrix elements of the evolution operator in terms of the Lanczos states... 

Besides the signal points c , the key ingredients of spectral analysis are the Lanczos coupling constants an,f . They are defined in terms of the elements of the basis set [ fn) of the Lanczos state vectors, as stated in Eq. (54). The common matrix element from Eq. (54) is of the following type ... 

This duality enables switching from the work with the Lanczos state vectors fn) to the analysis with the Lanczos polynomials Q (m). A change from one representation to the other is readily accomplished along the lines indicated in this section, together with the basic relations from Sections 11 and 12, in particular, the definition (142) of the inner product in the Lanczos space CK, the completeness (163) and orthogonality (166) of the polynomial basis Qn,k -... 

Further, regarding ijrn), it is also possible to have the explicit Lanczos algorithm by deriving the expression that holds the whole result with no recourse to recurrence relations. For example, applying the explicit Lanczos polynomial operator Q (U) from Eq. (177 to tq) = 0) will generate the wave packet IV n) according to ijrn) = Q (U) V o) as in Eq. (91). Therefore, the final result is the following expression for the explicit Lanczos states Vr ) ... 

Hence, as expected, the general Lanczos state tjrn) is indeed a linear combination of the Schrodinger states r). The availability of the explicit formula (181) for any state vector t/r ) also permits deriving the explicit expressions for the coupling constants an,fin. Thus, inserting Eq. (181) into Eq. (54), we have ... 

MAPPING FROM THE KRYLOV-SCHRODINGERTOTHE LANCZOS STATES... 

Hence, the inverse mapping T ) —>- ir ) between the Schrodinger and Lanczos states is carried out by means of the matrix /i of modified moments... 

Lanczos states and by analyzing the expansion coefficients on the molecular basis functions, we have removed the truncation errors of our previous works [3,4], we have discussed the convergence of the spectrum with respect to the dimensions of the molecular and Lanczos bases, we have calculated 1304 B2... 

The calculation of the molecular eigenstates with the MVCM model, necessary in traditional time-independent methods, can prove to be very cumbersome or even unfeasible. However, time-independent effective solutions, practicable for reduced-dimensionality models (in practice when the number of relevant normal coordinates is less than 10), may be obtained by taking advantage of the Lanczos iterative tridiagonalization of the Hamiltonian matrix [130]. The Lanczos algorithm proves to be very suitable for the computation of low-resolution spectra however, its effectiveness is better highlighted in a time-dependent framework. In fact, it can be easily realized that Lanczos states are only sequentially coupled, and it is therefore clear that only a limited number of states is necessary to describe short-time dynamics since the latter is the only relevant information for low-resolution spectra (see Chapter 10). 

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