Eigenvalue equations function

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] ... 

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. 

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 ... 

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. 

The adiabatic electronic potential energy surfaces (a function of both nuclear geometry and electric field) are obtained by solving the following electronic eigenvalue equation... 

Instead of using repeated solution of a suitable eigenvalue equation to optimize the orbitals, as in conventional forms of SCF theory, we have found it more convenient to optimize by a gradient method based on direct evaluation of the ener functional (4), ortho normalization being restored after every parameter variation. Although many iterations are required, the energy evaluation is extremely rapid, the process is very stable, and any constraints on the parameters (e.g. due to spatial symmetry or choice of some type of localization) are very easily imposed. It is also a simple matter to optimize with respect to non-linear parameters such as orbital exponents. 

This looks like an eigenvalue equation, but is not, since instead of regenerating fa, a sum of functions 4>j is obtained. For the complete set of molecular orbitals the equation can be written in matrix notation as... 

In recent years density-functional methods32 have made it possible to obtain orbitals that mimic correlated natural orbitals directly from one-electron eigenvalue equations such as Eq. (1.13a), bypassing the calculation of multi-configurational MP or Cl wavefunctions. These methods are based on a modified Kohn-Sham33 form (Tks) of the one-electron effective Hamiltonian in Eq. (1.13a), differing from the HF operator (1.13b) by inclusion of a correlation potential (as well as other possible modifications of (Fee(av))-... 

To calculate G (2.12), in addition to the various matrices and vectors we have described, we need the weights wm derived from the coefficients a of the wave function in solution the latter are obtained by solving the appropriate eigenvalue equation, discussed in the next Section. 

The functions Xav and Xbw which describe the nuclear motions in the initial and final states, are defined more precisely as belonging to the basis sets XavJ and Xbw> constituted by the solutions of the eigenvalue equations ... 

As one can see, for each set of parameters Z, k, L and d, one can select a in such a way that Eqs. (21) and (22) are fulfilled. Then, the resulting expectation values correspond to the variational minima and are equal to the appropriate exact eigenvalues of either Dirac or Schrodinger (or rather Levy-Leblond) Hamiltonian. However the corresponding functions do not fulfil the pertinent eigenvalue equations they are not eigenfunctions of these Hamiltonians. This example demonstrates that the value of the variational energy cannot be taken as a measure of the quality of the wavefunction, unless the appropriate relation between the components of the wavefunction is fulfilled [2]. 

The first of these equations is called the time-independent Schrodinger equation it is a so-called eigenvalue equation in which one is asked to find functions that yield a constant multiple of themselves when acted on by the Hamiltonian operator. Such functions are called eigenfunctions of H and the corresponding constants are called eigenvalues of H. 

If the functional applied does not include any exact exchange then (A-B) will be a diagonal matrix that only depends on the orbital eigenvalues. The working equation then becomes a single standard eigenvalue equation ... 

Fix an eigenvalue E. Suppose we have a solution to the eigenvalue equation for the Schrodinger operator in the given form. I.e, suppose we have a function a 1 and a spherical harmonic function such that... 

The valence electron wave function p can be determined from the one-electron self-consistent field eigenvalue equation... 

The coefficients a, (3, y and5 are obtained by imposing the Bloch conditions with periodicity A,p, the continuity conditions of the wave function and its derivative at L/2, and finally by normalization in the surface unit. The solution of the eigenvalue equation for E gives the electronic energy dispersion for the n-th subband with energy... 

We want an eigenvalue equation because (cf. Section 4.3.4) we hope to be able to use the matrix form of a series of such equations to invoke matrix diagonalization to get eigenvalues and eigenvectors. Equation (5.35) is not quite an eigenvalue equation, because it is not of the form operation on function = k x function, but rather operation on function = sum of (k x functions). However, by transforming the molecular orbitals to a new set the equation can be put in eigenvalue form (with a caveat, as we shall see). Equation 5.35 represents a system of equations... 

The Hartree-Fock equations (5.47) (in matrix form Eqs. 5.44 and 5.46) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction i//, to generate an energy value ,-, times i/q. Pseudoeigenvalue because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the Hartree-Fock equations F depends on i// because (Eq. 5.36) the operator contains J and K, which in turn depend (Eqs. 5.29 and 5.30) on i//. Each of the equations in the set (5.47) is for a single electron ( electron 1 is indicated, but any ordinal number could be used), so the Hartree-Fock operator F is a one-electron operator, and each spatial molecular orbital i// is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the, full description of each of these electrons requires a spin function 7 or jl (Section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital i// do not have all four quantum numbers the same (for an atomic Is orbital, for example, one electron has quantum numbers n= 1, / = 0, m = 0 and s = 1/2, while the other has n= l,l = 0,m = 0 and s = —1/2), and so the Pauli exclusion principle is not violated. 

The standard strategy for solving the KS eigenvalue equations, like that for solving the HF equations, which they resemble, is to expand the KS orbitals in terms of basis functions r/j (with m functions in the set) ... 

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