Diagonalization discrete-basis

The bound-state type of approach that was followed by Hylleraas—albeit fraught with a couple of erroneous conclusions—eventually evolved in the 1960s into methods aiming at the calculation and identification of resonance states from the behavior of roots of diagonalized energy matrices constructed from discrete basis sets, as a function of a scaling parameter, or size of discrete basis sets, or size of the artificial box inside which the continuous spectrum is artificially discretized. 

In one way or another, the discrete-basis diagonalization methods of the 1960s attempt to provide an understanding of the behavior of roots of Hamiltonian matrices, in view of the picture of quasi-localization of resonance eigenfunctions. For example. Pels and Hazi [50] write ... 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

In the case of a finite system described by a finite basis set the spectrum of G E) and WIE) are discrete andG (E) has isolated real poles (31,99). As a result, the solution for the propagator consists in the diagonalization of the WfE) matrix... 

A numerical solution of the Schrodinger equation in Eq. [1] often starts with the discretization of the wave function. Discretization is necessary because it converts the differential equation to a matrix form, which can then be readily handled by a digital computer. This process is typically done using a set of basis functions in a chosen coordinate system. As discussed extensively in the literature,5,9-11 the proper choice of the coordinate system and the basis functions is vital in minimizing the size of the problem and in providing a physically relevant interpretation of the solution. However, this important topic is out of the scope of this review and we will only discuss some related issues in the context of recursive diagonalization. Interested readers are referred to other excellent reviews on this topic.5,9,10... 

Discrete Variable Representation Basis Obtained by Simultaneous Diagonalization. 

To consider evolutionary optimization as a sequence of discrete selection steps requires some explanation. In fact, it is justified only on the basis of a hierarchical order of the rate terms, in which the off-diagonal terms usually are much smaller than the diagonal ones, decreasing in binomial or Poisson-ian progression with increasing Hamming distance between template and (erroneous) replica. As a consequence, the deterministic order around an... 

In the procedure (Appendix 9B) to evaluate the lineshape (9.40) we use the representation defined by the states 1/) that diagonalize the Hamiltonian in the ( 5), /> ) subspace. Of course any basis can be used for a mathematical analysis. It was important and useful to state the physical problem in terms of the zero-order states 5) and ]/) because an important attribute of the model was that in the latter representation the ground state g) is coupled by the radiation field only to the state 5), which therefore has the status of a doorway state. This state is also referred to as a resonance state, a name used for the spectral feature associated with an underlying picture of a discrete zero-order state embedded in and coupled to a continuous manifold of such states. 

The reduced density matrix can be converted into a discrete representation that involves sums over all the Slater determinants, MOs, and basis functions. This matrix will in general have many off-diagonal elements. The matrix is Hermi-tian therefore it can be diagonalized. The orbitals that result from the diagonal reduced-density matrix are called natural orbitals, and the diagonal elements are the occupation numbers for these orbitals. The natural orbitals are orthonormal molecular orbitals having maximal occupancy. 

In this context, the idea of discrete numerical basis sets, introduced by Sa-lomonson and Oster (129) for the bound-state problem and combined with the complex-rotation method by Lindroth (30), is very interesting. One-particle basis functions are defined on a discrete grid inside a spherical box containing the system under cosideration. The functions are evaluated by diagonalizing the discretized one-particle complex-rotated Hamiltonian. Such basis sets are then used to compute autoionizing state parameters by means of bound-state methods (30,31,66). 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

In general, when O is not 4 q, the discrete representation of the one-matrix in the basis of HF spin orbitals is not diagonal. However, since y is Hermitian, it is possible to define an orthonormal basis i ,-, related to (y,) by a unitary transformation, in which the matrix representation of the one-matrix is diagonal. The elements of the orthonormal set in which y is diagonal are called natural spin orbitals. To make the above explicit, we start with the relation between two orthonormal bases tjt and y ] (see Eqs. (1.63) and (1.65))... 

Lastly we note that the width of the well identified bound states is zero. But if we diagonalize a hamiltonian with a Coulombic tail, where sits an infinity of discrete but loosely bound Rydberg states, we will reproduce in our projected calculation one discrete state which averages these states, and is surely also an admixture of continuum states. Typically cross sections to these states are small. If one is really interested in cross sections to them an adjunct basis, P, may be used which connects them to P through the potential matrix elements. A t-matrix expression which has the desired state as the final entry, but P%, as the approximate state wave function, can be used. For example, if charge transfer is responsible for a small amount of flux loss from the target nucleus it is not necessary to use a two-centered basis the procedure described can be used, e.g.,... 

We note that Tg, Eq. (3.11) cannot be simply written in the form of Eq. (3.4). In such a case the general prescription is to introduce a suitable orthogonal basis set that diagonalizes the operator under study, or is to enable efficient evaluation of its operation (using the so-called discrete variable representation [240]). In the former approach, even a wave function spatially confined (such as would be the case when the angular coordinate represents molecular vibration) would be a coherent sum of a potentially large number of spatially global basis functions. Here we outline a spatially local method devised by Dateo and Metiu [106] based on the Fourier transform. 

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