Ritz variational procedure

We will use the Ritz variational method (see Chapter 5, p. 238) to solve the Schrodinger equation. What should we propose as the expansion functions It is usually recommended that we proceed systematically and choose first a complete set of functions depending on R, then a complete set depending on R, and finally a complete set that depends on the f variables. Next, one may create the complete set depending on all five variables (these functions wUl be used in the Ritz variational procedure) by taking all possible products of the three functions depending on R, R,... 

An alternative approach, utilizing the same wavefunction representation, was used by Handy (1996) in implementing a Rayleigh-Ritz variational procedure with respect to the fi dependence of the a -coefficients. 

Here N+1 is the number of unit cells, m the number of basis functions in it, a the elementray translation, Xs js) the s-th atomic orbital (AO) centered at the s-th atom (s S) in the j-th unit cell (this atom has the position vector " jq) and n is the band index. After a Ritz variational procedure one obtains a generalized eigenvalue equation... 

Instead of applying tail cancellation as in Sect.2.1 where we derived the KKR-ASA equations, one may use the linear combination of muffin-tin orbitals (5.27) directly in a variational procedure. This has the advantages that it leads to an eigenvalue problem and that it is possible to include non-muffin-tin perturbations to the potential. According to the Rayleigh-Ritz variational principle, one varies y to make the energy functional stationary, i.e. 

Wheeler and collaborators [3], in the context of nuclear physics, showed at that time that the limit in the variational procedure potential itself was not reached. Indeed, the Rayleigh-Ritz (RR) variational scheme teaches us how to obtain the best value for a parameter in a trial function, i.e., exponents of Slater (STO) or Gaussian (GTO) type orbital, Roothaan or linear combination of atomic orbitals (LCAO) expansion coefficients and Cl coefficients. Instead, the generator coordinate method (GCM) introduces the Hill-Wheeler (HW) equation, an integral transform algorithm capable, in principle, to find the best functional form for a given trial function. We present the GCM and the HW equation in Section 2. 

An important observation about the EMM approach is that it is manifestly a scale-translation (affine map) invariant variational procedure, unlike other approaches, such as Rayleigh-Ritz. At each order, the variation with respect to the Cj s is actually optimizing over all possible affine map transforms of the polynomial sampling function. 

The Hamiltonian (equation 1) and the crystal orbital (equation 3) (which is a linear combination of Bloch functions because we have more than one basis function centered in one unit cell) leads through Ritz s variational procedure to the generalized eigenvalue equation... 

In the present paper we suggest a simple technique for calculating the electronic structure in the presence of a SDW, based on the variational Ritz procedure. 

The variational Ritz procedure reduces the problem of solving (1) to solving the generalized eigenvalue problem ... 

Structural analysis, initially developed on an intuitive basis, later became identified with variational calculus, in which the Ritz procedure is used to minimize a functional derived mathematically or arrived at directly from physical principles. By substituting the final solutions into the variational statement of the problem and minimizing the latter, the FEM equations are obtained. Example 15.2 gives a very simple demonstration of this procedure. 

The Ritz procedure is a special case of the variational method, in which the parameters c enter linearly 4>(x c) = c l, where I, - are some known basis functions that form (or more exactly, in principle form) the complete set of functions in the Hilhert space. This formalism leads to a set of homogeneous linear equations to solve (secular equations), from which we find approximations to the ground- and excited-state energies and wave functions. 

The major difficulty with non-Hermitian matrices is that in general, no variational principle exists for their eigenvalues, and therefore the Ritz procedure is not applicable. In addition, the Lanczos recursion eq D1, which is based on the Hermiticity of // does not yield an orthonormal set of vectors Wm when... 

The Ritz procedure is a special case of the variational method, in which the parameters c enter d> linearly d>(x c) = where are some known basis func-... 

It has been shown that many differential equations that originate from the physical sciences have equivalent variational formulations." This is the basis for the well-known Rayleigh-Ritz procedure which in turn forms the basis for the finite element methods. 

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