Hilbert matrix

Our test problem will involve the Hilbert matrix of order b, defined by... 

For exercise find cond(H ) of the Hilbert matrix 1- defined by (1.69). 

Use Gaussian elimination and back-substitution to solve the linear system of Eq. (A.1-2), with m = n = 3. Oij = l/ i+j — 1), and bj = 1,0.0. The coefficient matrix A given here is called the Hilbert matrix, it arises in a nonrecommended brute-force construction of orthogonal polynomials from monomials N. 

Now it will be shown that the Hilbert matrix norm and the euclidean vector norm satisfy the four requirements as well as the compatibility relationship. To show that the first condition is satisfied, let A be any real nonsingular, nonzero matrix (A 0). Then there exists a vector X such that X , = 1 and such that the euclidean norm of the vector AX is a maximum and nonzero. Moreover, the particular vector X that satisfies these conditions is the eigenvector Xt corresponding to the largest eigenvalue That is... 

By assuming the Hilbert space of dimension N, one can easily establish the relation between coupling matrices and by considering tbe (i/)tb matrix element of V ... 

The symbol M represents the masses of the nuclei in the molecule, which for simplicity are taken to be equal. The symbol is the Kionecker delta. The tensor notation is used in this section and the summation convention is assumed for all repeated indexes not placed in parentheses. In Eq. (91) the NACT appears (this being a matrix in the electronic Hilbert space, whose components are denoted by labels k, m, and a vector with respect to the b component of the nuclear coordinate R). It is given by an integral over the electron coordinates... 

Projecting the nuclear solutions Xt( ) oti the Hilbert space of the electronic states (r, R) and working in the projected Hilbert space of the nuclear coordinates R. The equation of motion (the nuclear Schrddinger equation) is shown in Eq. (91) and the Lagrangean in Eq. (96). In either expression, the terms with represent couplings between the nuclear wave functions X (K) and X (R). that is, (virtual) transitions (or admixtures) between the nuclear states. (These may represent transitions also for the electronic states, which would get expressed in finite electionic lifetimes.) The expression for the transition matrix is not elementaiy, since the coupling terms are of a derivative type. 

In Section IV.A, the adiabatic-to-diabatic transformation matrix as well as the diabatic potentials were derived for the relevant sub-space without running into theoretical conflicts. In other words, the conditions in Eqs. (10) led to a.finite sub-Hilbert space which, for all practical purposes, behaves like a full (infinite) Hilbert space. However it is inconceivable that such strict conditions as presented in Eq. (10) are fulfilled for real molecular systems. Thus the question is to what extent the results of the present approach, namely, the adiabatic-to-diabatic transformation matrix, the curl equation, and first and foremost, the diabatic potentials, are affected if the conditions in Eq. (10) are replaced by more realistic ones This subject will be treated next. 

This concludes our derivation regarding the adiabatic-to-diabatic tiansforma-tion matrix for a finite N. The same applies for an infinite Hilbert space (but finite M) if the coupling to the higher -states decays fast enough. 

In our introductory remarks, we said that this section would be devoted to model systems. Nevertheless it is important to emphasize that although this case is treated within a group of model systems this model stands for the general case of a two-state sub-Hilbert space. Moreover, this is the only case for which we can show, analytically, for a nonmodel system, that the restrictions on the D matrix indeed lead to a quantization of the relevant non-adiabatic coupling term. 

We intend to show that an adiabatic-to-diabatic transformation matrix based on the non-adiabatic coupling matrix can be used not only for reaching the diabatic fi amework but also as a mean to determine the minimum size of a sub-Hilbert space, namely, the minimal M value that still guarantees a valid diabatization. 

For example one forms, within a two-dimensional (2D) sub-Hilbert space, a 2x2 diabatic potential matrix, which is not single valued. This implies that the 2D transformation matrix yields an invalid diabatization and therefore the required dimension of the transformation matrix has to be at least three. The same applies to the size of the sub-Hilbert space, which also has to be at least three. In this section, we intend to discuss this type of problems. It also leads us to term the conditions for reaching the minimal relevant sub-Hilbert space as the necessary conditions for diabatization. ... 

In Section IV, we introduced the topological matrix D [see Eq. (38)] and showed that for a sub-Hilbert space this matrix is diagonal with (-1-1) and (—1) terms a feature that was defined as quantization of the non-adiabatic coupling matrix. If the present three-state system forms a sub-Hilbert space the resulting D matrix has to be a diagonal matrix as just mentioned. From Eq. (38) it is noticed that the D matrix is calculated along contours, F, that surround conical intersections. Our task in this section is to calculate the D matrix and we do this, again, for circular contours. 

We restrict ourselves to finite-dimensional Hilbert spaces, making H a Her-mitian matrix. We denote the eigenvalues of H q) by Efc(g) and consider the spectral decomposition... 

Linear algebraic problem, 53 Linear displacement operator, 392 Linear manifolds in Hilbert space, 429 Linear momentum operator, 392 Linear operators in Hilbert space, 431 Linear programming, 252,261 diet problem, 294 dual problem, 304 evaluation of methods, 302 in matrix notation, simplex method, 292... 

Where Njk is the elements of the Hilbert-Noda transformation matrix given by... 

As stated in the introduction, we present the derivation of an extended BO approximate equation for a Hilbert space of arbitary dimensions, for a situation where all the surfaces including the ground-state surface, have a degeneracy along a single line (e.g., a conical intersection) with the excited states. In a two-state problem, this kind of derivation can be done with an arbitary t matrix. On the contrary, such derivation for an N > 2 dimensional case has been performed with some limits to the elements of the r matrix. Hence, in this sence the present derivation is not general but hoped that with some additional assumptions it will be applicable for more general cases. 

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