Diagonal set

A triangular matrix is a matrix all of whose elements above or below the main diagonal (set of elements an,. . . , a j) are zero. 

The NOESY spectrum has a diagonal set of peaks (open circles) as well as peaks which are off the diagonal (filled circles). The off-diagonal peaks occur at positions where a proton on the Fi axis is close in space to a one on the F2 axis. In the schematic spectrum on the right, the off-diagonal signals show that Ha must be located near Hd and Hb must be located near He. 

Part (h) From the discussion of the Jacobi method (Sec. 2.8), it would seem that Gauss-Seidel would be the best method of solution for a predominantly diagonal set. However, because calculations in MATLAB are based on matrices, the Jacobi method in matrix form is considerably faster tlian the Gauss-Seidel method in formula form in the MATLAB workspace. 

Jacobi Iterative method for solution of predominantly diagonal sets of simultaneous linear algebraic equations. 

It is important to realize that while the uppennost diagonal elements of these matrices are numbers, the other diagonal element is a matrix of dimension N. Specifically, these are the matrix representations of Hq and Fin the basis q which consists of all the original set, apart from i.e. 

The off-diagonal elements in this representation of h and v are the zero vector of lengtii (for h) and matrix elements which couple the zeroth-order ground-state eigenfunction members of the set q (for v) ... 

Having done this we solve the Scln-ddinger equation for the molecule by diagonalizing the Hamiltonian matrix in a complete set of known basis fiinctions. We choose the basis functions so that they transfonn according to the irreducible representations of the synnnetry group. 

The Hamiltonian matrix will be block diagonal in this basis set. There will be one block for each irreducible representation of the synnnetry group. 

Since the vibrational eigenstates of the ground electronic state constitute an orthonomial basis set, tire off-diagonal matrix elements in equation (B 1.3.14) will vanish unless the ground state electronic polarizability depends on nuclear coordinates. (This is the Raman analogue of the requirement in infrared spectroscopy that, to observe a transition, the electronic dipole moment in the ground electronic state must properly vary with nuclear displacements from... 

Iterative approaches, including time-dependent methods, are especially successfiil for very large-scale calculations because they generally involve the action of a very localized operator (the Hamiltonian) on a fiinction defined on a grid. The effort increases relatively mildly with the problem size, since it is proportional to the number of points used to describe the wavefiinction (and not to the cube of the number of basis sets, as is the case for methods involving matrix diagonalization). Present computational power allows calculations... 

To see physically the problem of motion of wavepackets in a non-diagonal diabatic potential, we plot in figure B3.4.17 a set of two adiabatic potentials and their diabatic counterparts for a ID problem, for example, vibrations in a diatom (as in metal-metal complexes). As figure B3.4.17 shows, if a wavepacket is started away from the crossing point, it would slide towards this crossing point (where where it would... 

Let us express the displacement coordinates as linear combinations of a set of new coordinates y >q= Uy then AE = y U HUy. U can be an arbitrary non-singular matrix, and thus can be chosen to diagonalize the synmietric matrix H U HU = A, where the diagonal matrix A contains the (real) eigenvalues of H. In this fomi, the energy change from the stationary point is simply AF. = t Uj A 7- h is clear now that a sufBcient... 

A better set of parameters can be chosen as the above-diagonal elements of an antisynnnetric matrix K, used... 

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