Differentiability eigenvalue

The last section discussed angular momentum from the viewpoint of solving the differential eigenvalue equations that result from expressing L2... 

Bridges, T.J. and Morris, P. (1984). Differential eigenvalue problems in which the parameters appear nonlinearly, J. Comp. Phys. 55 437-460. 

This reduces the Schrodinger equation to = 4/. To solve the Schrodinger equation it is necessary to find values of E and functions 4/ such that, when the wavefunction is operated upon by the Hamiltonian, it returns the wavefunction multiplied by the energy. The Schrodinger equation falls into the category of equations known as partial differential eigenvalue equations in which an operator acts on a function (the eigenfunction) and returns the... 

From the definitions (3.316) and (3.318) of the two Fock operators / and /, we can see that the two integro-differential eigenvalue equations (3.312) and (3.313) are coupled and cannot be solved independently. That is,/ depends on the occupied orbitals, through Jj, and depends on the occupied a orbitals, through JJ. The two equations must thus be solved by a simultaneous iterative process. 

There is a linear algebra equivalent of the differential eigenvalue equation. Instead of a differential operator acting on a function, a square matrix multiplies a column vector, and this is equal to a constant, the eigenvalue, times the vector ... 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

Hence, in order to contract extended BO approximated equations for an N-state coupled BO system that takes into account the non-adiabatic coupling terms, we have to solve N uncoupled differential equations, all related to the electronic ground state but with different eigenvalues of the non-adiabatic coupling matrix. These uncoupled equations can yield meaningful physical... 

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. 

This technique is not always successful, but can be used for the type of cases illustrated now. Consider a two-dimensional differential equation that is of second order and of the eigenvalue type ... 

The net result is that we now have two first-order differential equations of the eigenvalue form ... 

This set of ordinaiy differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. If Euler s method is used for integration, the time step is hmited by... 

To avoid numerical differentiation (which is inherently unstable) one uses the fact that an eigenvalue can be expressed as Ai = v Tvf where are the corresponding normalized left and right eigenvectors. Differentiation of the eigenvalue with respect to any parameter is then equivalent to the differentiation of the transfer matrix, and one finds... 

Frost and Pearson treated Scheme XV by the eigenvalue method, and we have solved it by the method of Laplace transforms in the preceding subsection. The differential rate equations are... 

When the piston is assumed to be measured from the LGSs (top eurve), the corresponding mode is singular because one cannot measure the contribution of each layer to the total piston. This case is not realistic, since no wavefront sensor measures the piston. When the tilts are measured from the LGSs (case of the polychromatic LGS), the odd piston is not measured again. The even piston is no longer available. And the two odd tilt modes are not also, because whereas the tilt is measured, the differential tilt between the two DMs is not one does not know where the tilt forms. Thus there are 4 zero eigenvalues. 

When is an eigenvalue of r(.B),. E is a pole. The corresponding operator, r(JS), is nonlocal and energy-dependent. In its exact limit, it incorporates all relaxation and differential correlation corrections to canonical orbital energies. A normalized DO is determined by an eigenvector of T Epou) according to... 

In the unrestricted Hartree-Fock method, a single-determinant wave function is used with different molecular orbitals for a and jS spins, and the eigenvalue problem is solved with separate F and F matrices. With the zero differential overlap approximation, the F matrix elements (25) become... 

In order to obtain nonzero spin densities even on hydrogen atoms in tt radicals, one has to take the one-center exchange repulsion integrals into account in the eigenvalue problem. In other words, a less rough approximation than the complete neglect of differential overlap (CNDO) is required. This implies that in the CNDO/2 approach also, o and n radicals have to be treated separately (98). 

Before giving further motivations, we would like to recall the basic aspects concerned with the elementary problem of determining eigenfunctions and eigenvalues for the differential equation... 

In the next step, which is numerically the most demanding, the differential equations (3) are solved. Two possible strategies using a variational expansion of the single particle wave functions, /., are described below. After the eigenvalues and eigenfunctions have been found, a new ("output") charge density can be... 

To conclude this section on two-compartment models we note that the hybrid constants a and p in the exponential function are eigenvalues of the matrix of coefficients of the system of linear differential equations ... 

The benefit is now that the HF equations are turned from complicated integro-differential equations into pseudo-eigenvalue equations for the unknown expansion coefficients c. 

The differential equation for X(x) is exactly of the form given by (4.13) for a one-dimensional harmonic oscillator. Thus, the eigenvalues Ex are given by equation (4.30)... 

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