Diagonalization procedure

As has been shown in the foregoing subsection, the stationary states (r) and F (r) are solutions of the full Hamiltonian H with given energies E, and Eu. This means their Hamilton matrix is diagonal (the subscript I is used now to indicate all such stationary states characterized by Roman numbers I, II, III.)  

It is also possible to derive this result from a different starting point, without explicitly taking into account the time dependences. In this approach one starts with the ansatz 

If equ. (7.90a) is presented in matrix form (Hik) using column vectors (ak), one gets the equivalent representation 

The last equation shows that the unknown coefficients are those solutions which diagonalize the Hamilton energy matrix Hik. Hence, they can be found by a diagonalization procedure where for non-trivial solutions the condition 

In order to obtain the eigenvectors, restrictions from the requirement of orthonormality for the wavefunctions 4yr) have to be also incorporated. This will be demonstrated for the case of a two-state system defined in equ. (7.86). Here these restrictions lead to the additional conditions 

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Nevertheless, equation (A 1.1.145) fonns the basis for the approximate diagonalization procedure provided by perturbation theory. To proceed, the exact ground-state eigenvalue and correspondmg eigenvector are written as the sums... 

Very similarly, higher-order processes can be shown to yield a size-consistent redistribution of the intensity of shake-up states among themselves, via multiple 2h-lp/2h-lp interactions. Any restriction on this balance will therefore yield a size-inconsistent description of correlation bands, which will tend to vanish with increasing system size (11). A nice example is provided here, with the necessary introduction of a lower limit on pole strengths in the block-Davidson diagonalization procedure. 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

The cycle matrix of Table XXII is a tabulation of mechanism (43) with p = 0, a = 0, and t = 0, and the row vector (51) consists of the coefficients in (43) with = 0, x = 0, and ij/ = 0. Any three independent cycles could have been chosen to generate Table XXII and any mechanism for the overall reaction could have been chosen to establish the row vector (45). The choices we made are arbitrary and depend on the diagonalization procedure used to find the matrix of Table XXI, which is far from unique. The important point is that the list of direct mechanisms we are looking for is unique and independent of how the above choices are made. 

If the basis set used is finite and incomplete, solution of the secular equation yields approximate, rather than exact, eigenvalues. An example is the linear variation method note that (2.78) and (1.190) have the same form, except that (1.190) uses an incomplete basis set. An important application of the linear variation method is the Hartree-Fock-Roothaan secular equation (1.298) here, basis AOs centered on different nuclei are nonorthogonal. Ab initio and semiempirical SCF methods use matrix-diagonalization procedures to solve the Roothaan equations. 

The direct Cl equations are obtained by combining the normal Cl equations (3.3) with an iterative diagonalization procedure. (The same direct Cl equations can also be obtained within a perturbation theory approach). Since diagonalization procedures have been described in another set of lectures we will here only repeat the most essential results. The simplest iterative procedure is obtained by moving everything but the diagonal terms in the Cl equations over to the right hand side and assume that this side of the equations can be obtained from the Cl vector of the previous iteration C. ... 

These relations will be used in the next section for an example of a more extended diagonalization procedure see Section 5.3.2. 

The orthonormalization conditions reduce the number of independent variation variables as compared to this estimate, but do not reduce so to say the number of numbers to be calculated throughout the diagonalization procedure. 

The solution of Eq. (4.4) can be obtained with a diagonalization procedure which gives for the eigenvalues X,. the equation... 

Equation [73] has the same form as the equations of motion for molecules with constrained internal coordinates, and we already know that such equations can be solved effectively using the SHAKE algorithm4 ° Equations [72] and [73] play a key role in the Car-Parrinello method and enable one to run the dynamics for both ionic and electronic degrees of freedom in parallel. With carefully chosen effective mass p and a small time step, the electronic state adjusts itself instanteously to the nuclear configuration (Born-Oppenheimer principle), and, therefore, the atomic dynamics is computed along the system s Born-Oppenheimer surface. Note that there is no need to carry out the costly matrix-diagonalization procedure for performing electronic structure calculations. 

In this case the diffusion equations are already uncoupled so we do not need to use the diagonalization procedure discussed above. The solution to the set of uncoupled linear ordinary differential equations (Eq. 5.3.1) is obtained as... 

We consider a classical equilibrium system of independent harmonic oscillators whose positions and velocities are denoted XjVy = iy, respectively. In fact, dealing with normal modes implies that we have gone through the linearization and diagonalization procedure described in Section 4.2.1. In this procedure it is convenient to work in mass-normalized coordinates, in particular when the problem involves different particle masses. This would lead to mass weighted position and... 

With multidimensional problems, the grid basis set is more suitable for matrix diagonalization procedures, which yield a desired number of the lowest eigenvalues. Extension of the procedure to three and more dimensions is straightforward. 

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