Eigenvalues equation

The exchange matrix, K, is just the rate, k, times the unit matrix. In block fonn, the full matrix for two sites is given in the eigenvalue equation, (B2.4.38). 

The value of detennines how much computer time and memory is needed to solve the -dimensional Sj HjjCj= E Cj secular problem in the Cl and MCSCF metiiods. Solution of tliese matrix eigenvalue equations requires computer time that scales as (if few eigenvalues are computed) to A, (if most eigenvalues are... 

This is equivalent to finding the lowest eigenvalue 1. (which is always negative and approaches zero at convergence) of the generalized eigenvalue equation... 

As has been shown previously [243], both sets can be described by eigenvalue equations, but for the set 2 it is more direct to work with projectors Pr taking the values 1 or 0. Let us consider a class of functions/(x), describing the state of the system or a process, such that (for reasons rooted in physics)/(x) should vanish for X D (i.e., for supp/(x) = D, where D can be an arbifiary domain and x represents a set of variables). If Pro(x) is the projector onto the domain D, which equals 1 for x G D and 0 for x D, then all functions having this state property obey an equation of restriction [244] ... 

With either of these diabatic reference Hamiltonians, the LHSFs satisfy the eigenvalue equation... 

This is one way of finding eigenvalues. All atoniie and moleeular energy levels are eigenvalues of a speeial eigenvalue equation ealled the Schwedinger equation. 

This equation is an eigenvalue equation for the energy or Hamiltonian operator its eigenvalues provide the energy levels of the system... 

The first of these equations is ealled the time-independent Sehrodinger equation it is a so-ealled eigenvalue equation in whieh one is asked to find funetions that yield a eonstant multiple of themselves when aeted on by the Hamiltonian operator. Sueh funetions are ealled eigenflinetions of H and the eorresponding eonstants are ealled eigenvalues of H. 

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. 

In summary, to solve an eigenvalue equation, we first solve the determinantal equation ... 

The eigenveetors v(k) are found by plugging the above values for )ik into the basie eigenvalue equation... 

Note that even after requiring normalization, there is still an indeterminaney in the sign of v(3). The eigenvalue equation, as we reeall, only speeifies a direetion in spaee. The sense or sign is not determined. We ean ehoose either sign we prefer. 

VIII. Hermitian Matrices and The Turnover Rule The eigenvalue equation ... 

Suppose that the function g(x) obeys some operator equation (e.g., an eigenvalue equation) such as... 

Remember that ai is the representation of g(x) in the fi basis. So the operator eigenvalue equation is equivalent to the matrix eigenvalue problem if the functions fi form a complete set. 

Consider in more detail the calculation of the determinant in (3.70) for the potential (3.66). The eigenvalue equation... 

Consider a more general eigenvalue equation without imposing the periodic boundary condition,... 

Note also that Equation 3 is an eigenvalue equation an equation in which an operator acting on a function produces a multiple of the function itself as its result, having the general form ... 

You will see shortly that an exact solution of the electronic Schrodinger equation is impossible, because of the electron-electron repulsion term g(ri, r2). What we have to do is investigate approximate solutions based on chemical intuition, and then refine these models, typically using the variation principle, until we attain the required accuracy. This means in particular that any approximate solution will not satisfy the electronic Schrodinger equation, and we will not be able to calculate the energy from an eigenvalue equation. First of all, let s see why the problem is so difficult. 

There are several ways in which we can proceed with the derivation of the HP equations. The traditional one is to look for an eigenvalue equation for the HP orbitals... 

Roothaan actually solved the problem by allowing the LCAO coefficients to vary, subject to the LCAO orbitals remaining orthonormal. He showed that the LCAO coefficients are given from the following matrix eigenvalue equation ... 

By analogy with solid-state studies. Slater had the idea of writing the atomic Hartree-Fock eigenvalue equation... 

In the context of the HF-LCAO model, we seek a solution of the matrix eigenvalue equation... 

The equations may be simplified by choosing a unitary transformation (Chapter 13) which makes the matrix of Lagrange multipliers diagonal, i.e. Ay 0 and A This special set of molecular orbitals (f> ) are called canonical MOs, and they transform eq. (3.40) mto a set of pseudo-eigenvalue equations. 

The Hartree-Fock equations form a set of pseudo-eigenvalue equations, as the Fock operator depends on all the occupied MOs (via the Coulomb and Exchange operators, eqs. (3.36) and (3.33)). A specific Fock orbital can only be determined if all the other occupied orbitals are known, and iterative methods must therefore be employed for determining the orbitals. A set of functions which is a solution to eq. (3.41) are called Self-Consistent Field (SCF) orbitals. 

We may again chose a unitary transfonnation which makes tlie matrix of the Lagrange multiplier diagonal, producing a set of canonical Kohn-Sham (KS) orbitals. The resulting pseudo-eigenvalue equations are known as the Kohn-Sham equations. 

That relation follows from the fact that the operator X2 + Y2 has positive eigenvalues. To see this, write the eigenvalue equations for X, Y, and X2 + Y2 ... 

It is clear from the preceding section that, until inequality (2.30) holds, the moments satisfy Eq. (2.24). Using this fact in Eq. (2.36) and the eigenvalue equation... 

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