Application of the Ritz method

Oflie products (I ) (S, l ) maybe used to produce linear combinations that are automati- 

Each of the eigenfunctions will correspond to some J, Mj and to a certain parity. The problem is solved. 

The dipole moment in the above formula takes into account that the charge distribution in the C... AB system depends on the nuclear configuration, i.e. on R, 

R and r, e.g., the atom C may have a net charge and the AB molecule may change its dipole moment when rotating. 

Heijmen et al. carried out accurate calculations of the hypersurlace V for a few atom-diatomic molecules, and then using the method described above the Schrodinger equation is solved for the nuclear motion. Fig. 73 gives a comparison of theory and experiments for the CS 0 complex with the He atom.s All the lines follow from the electric-dipole-allowed transitions [those for which the sum of the integrals in the formula I J — J ) is not equal to zero], each line is associated with a transition (/ , I , j ) (/, V, /). 

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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. 

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 ... 

The same set of equations ( secular equations ) is obtained after using the Ritz method (Chapter 5). This pertains to almost all applications. For complex functions the equations are only slightly more complicated. 

The most interesting application for our purposes is to construct MOs by the linear combination of atomic orbitals (LCAO) method, where the variable parameters are the coefficients of the linear combination of some basic orbitals y 9 (Ritz method). It can be shown that, in this case, the best orbitals are obtained by solving the eigenvalue equation for matrix H ... 

Presently, the widely used post-Hartree-Fock approaches to the correlation problem in molecular electronic structure calculations are basically of two kinds, namely, those of variational and those of perturbative nature. The former are typified by various configuration interaction (Cl) or shell-model methods, and employ the linear Ansatz for the wave function in the spirit of Ritz variation principle (c/, e.g. Ref. [21]). However, since the dimension of the Cl problem rapidly increases with increasing size of the system and size of the atomic orbital (AO) basis set employed (see, e.g. the so-called Paldus-Weyl dimension formula [22,23]), one has to rely in actual applications on truncated Cl expansions (referred to as a limited Cl), despite the fact that these expansions are slowly convergent, even when based on the optimal natural orbitals (NOs). Unfortunately, such limited Cl expansions (usually truncated at the doubly excited level relative to the IPM reference, resulting in the CISD method) are unable to properly describe the so-called dynamic correlation, which requires that higher than doubly excited configurations be taken into account. Moreover, the energies obtained with the limited Cl method are not size-extensive. 

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