Ritz variation method

MacDonald J K L 1933 Successive approximations by the Rayleigh-Ritz variation method Phys. Rev4Z 830-3... 

Schwartz, H. M., Phys. Rev. 103, 110, "Ground state solution of the non-relativistic equation for He." More rapid convergence in the Ritz variational method by inclusion of half-integral powers in the Hylleraas function. 

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

As an alternative to the Ritz variational method, a variational perturbation method based on a perturbation expansion can be utilized. Here, the Hamiltonian is separated into an unperturbed Hamiltonian and a perturbing potential,... 

In quantum calculations, the Rayleigh-Ritz variational method is widely used to approximate the solution of the Schrodinger equation [86], To obtain exact results, one should expand the exact wave function in a complete basis set... 

Simulation of confinement by penetrable boxes represents a more realistic physical model. A very simple approach was proposed by Marin and Cruz [18], where they used the Rayleigh-Ritz variational method via a trial wave function for the ground state, which consists of two piecewise functions, one for the inner region (r < ro), and the other for the outer region (r > ro). The trial wave function is defined as follows ... 

We obtain the same equation if in the Hylieiaas functional [Eq. (5.28)], the variational functitm x is expanded as a linear combination [Eq. (5.29)], and then vary dj in a similar way to that of the Ritz variational method described on p. 238. 

We should fix the problem in some way. Let us focus on a particular unperturbed state The slightest perturbation applied would force us to consider the Ritz variational method to determine the resulting state. We can choose all functions as possible expansion functions ... 

The first thing to do is to adapt the wave function of the unperturhed system to the perturhation. To this end, we wiU use the Ritz variational method with the expansion functions m = l,2.g ... 

In practice, the Ritz variational method is used most often. One of the technical problems to be solved is the size of the basis set. Enormous progress in computation and software development now facilitates investigations that 20 years ago were absolutely beyond the imagination. The world record in quantum chemistry means a few billion expansion functions. To accomplish this, quantum chemists have had to invent some powerful methods of applied mathematics. 

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,... 

Kotos and Wohuewicz applied the Ritz variational method (see Chapter 5) to the hydrogen molecule with the following trial function ... 

What about electronic correlation in excited electronic states Not much is known for excited states in general. In our case of Eq. (10.23), the Ritz variational method would give two solutions. One would be of lower energy corresponding to /r < 0 (this solution has been approximated by us using the perturbational approach). The second solution (the excited electronic state) will be of the form xj/exc = in such a simple two-state model, the coefficient k can be found just... 

Slater determinants are usually constructed from molecular spinorbitals. If, instead, we use atomic spinorbitals and the Ritz variational method (Slater determinants as the expansion functions), we would get the most general formulation of the valence bond (VB) method. The beginning of VB theory goes back to papers by Heisenbeig, the first application was made by Heitler and London, and later theory was generalized by Hurley, Lennard-Jones, and Pople. The essence of the VB method can be explained by an example. Let us take the hydrogen molecule with atomic spinorbitals of type liaO and Vst (abbreviated as aa and b ) centered at two nuclei. Let us construct from them several (non-normalized) Slater determinants, for instance ... 

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