Diagonal method

Beck M H and Meyer H D 1998 Extracting accurate bound-state spectra from approximate wave packet propagation using the filter-diagonalization method J. Chem. Phys. 109 3730... 

Chen R Q and Guo H 1996 A general and efficient filter diagonalization method without time propagation J. Chem. Phys. 105 1311... 

Narevicius E, Neuhauser D, Korsch H J and Moiseyev M 1997 Resonances from short time complex-scaled cross- correlation probability amplitudes by the filter-diagonalization method Chem. Phys. Lett. 276 250... 

Assuming that the diabatic space can be truncated to the same size as the adiabatic space, Eqs. (64) and (65) clearly define the relationship between the two representations, and methods have been developed to obtain the tians-formation matrices directly. These include the line integral method of Baer [53,54] and the block diagonalization method of Pacher et al. [179]. Failure of the truncation assumption, however, leads to possibly important nonremovable derivative couplings remaining in the diabatic basis [55,182]. 

The Jacobi method is probably the simplest diagonalization method that is well adapted to computers. It is limited to real symmetric matrices, but that is the only kind we will get by the formula for generating simple Huckel molecular orbital method (HMO) matrices just described. A rotation matrix is defined, for example. 

The Newton-Raphson block diagonal method is a second order optimizer. It calculates both the first and second derivatives of potential energy with respect to Cartesian coordinates. These derivatives provide information about both the slope and curvature of the potential energy surface. Unlike a full Newton-Raph son method, the block diagonal algorithm calculates the second derivative matrix for one atom at a time, avoiding the second derivatives with respect to two atoms. 

As illustrated above, even quite small systems at the CISD level results in millions of CSFs. The variational problem is to extract one or possibly a few of the lowest eigenvalues and -veetors of a matrix the size of millions squared. This cannot be done by standard diagonalization methods where all the eigenvalues are found. There are, however, iterative methods for extraeting one, or a few, eigenvalues and -veetors of a large matrix. The Cl problem eq. (4.6) may be written as... 

As indicated by the Kronecker deltas in the above equation, the resulting Hamiltonian matrix is extremely sparse and its action onto a vector can be readily computed one term at a time.12,13 This property becomes very important for recursive diagonalization methods, which rely on matrix-vector multiplication ... 

As the modification of the original Hamiltonian matrix is involved in the diagonalization methods discussed above, we denote such approaches as direct diagonalization to distinguish them from the recursive ones discussed below. The direct diagonalization process is illustrated in Figure 1. 

In recent years, state-of-the-art recursive diagonalization methods have been applied to bound-states problems for LiCN,152 H20,12,117,239-241 CH2,242 HCN,i3,80,105-107,241 40,67,164,243-245 246 H0C1,247,248 N02,76,249-253... 

Eigenvlaues and Eigenvectors with an Implicitly Restarted and Filter Diagonalization Method. 

Residual Filter Diagonalization Method for Interior Eigenvalues of Physical Systems. 

Diagonalization Method to Calculate Expectation Values of Operators Non-Commutative to the Hamiltonian. Vibrational Assignment of HOC1. 

Diagonalization Method for Quantum Eigenenergy Calculation or for Spectral Analysis of Time Signals. 

State Spectra from Approximate Wave Packet Propagation Using the Filter-Diagonalization Method. 

Quasienergies from Short Time Cross-Correlation Probability Amplitudes by the Filter-Diagonalization Method. 

Resonance States by a Time-Independent Filter Diagonalization Method for Large Hamiltonian Matrices. 

Reference Deconvolution, Phase Correction and Line Listing of NMR Spectra by the ID Filter Diagonalization Method. 

Filter Diagonalization Method for Spectral Analysis of Noisy Data with Nonlocalized Features. 

Filter Diagonalization Methods with the Lanczos Method for Calculating Vibrational Energy Levels. 

Time-Dependent Solutions. In the time-dependent case, the diffusion equations given by Eq. 6.23 are coupled. However, they can be uncoupled by again using the diagonalization method. 

The variational condition determining the coefficients cJt is cubic in the unknowns, but iterative techniques permit these coefficients to be determined by repeated use of matrix diagonalization methods. Under most conditions it is possible to choose an iterative process facilitating convergence there is much RHF experience, and inordinate difficulties are not usually experienced. Because of the occupancy assumptions, it is possible without loss of generality to take the RHF spatial orbitals as orthogonal, and this is an important feature simplifying the calculations. 

The general methods of computing dynamic NMR spectra which have so-far been reported may be divided into two groups. One involves the point by point approach while the other is based on the diagonalization method. 

Contrary to the point by point approach the diagonalization method consists of the generation of an entire lineshape function in one step. (13, 14, 57-60) Time-consuming calculations are carried out only once. The resulting set of complex numbers can be used for a simple calculation of the lineshape (absorption and dispersion modes) at any desired point on the frequency axis. Thus, the complex matrix from equation (147) can be diagonalized by a similarity transformation using an co-independent complex matrix W ... 

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