Rayleigh quotient Dirac

However, there is a stationary variation principle of precisely the type employed in the quantum chemical linear variation method. In the derivation of the Roothaan equations based on finite basis set expansions of Schrodinger wavefimctions, one insists only that the Rayleigh quotient be stationary with respect to the variational parameters, and then assumes that the variational principle guarantees an absolute minimum. In the corresponding linear equations based on the Dirac equation, the stationary condition is imposed, but no further assumption is made about the nature of the stationary point. 

The developments of this section show that for energy solutions in the domain of interest to us, the Rayleigh quotient is bounded below, and there is therefore no danger of variational collapse when solving the Dirac equation in a kinetically balanced finite basis. For the Dirac-Hartree-Fock equations, the only addition is the electron-electron interaction, which is positive and therefore will not contribute to a variational collapse. 

What is the relation between the lORA energies and the ZORA and Dirac energies There is a correspondence at =0 and we expect that the correspondence continues in the vicinity of this point. Unlike the ZORA equation, we cannot perform a scaling to obtain a relation with the Dirac ESC equation, and therefore we cannot obtain a direct relation with the Dirac eigenvalues. What we can do is to make use of the Rayleigh quotient for (18.37) to obtain a relation between the ZORA and lORA eigenvalues, since ZORA and lORA have the same Hamiltonian but a different metric. For an arbitrary wave function r] . 

With the expression for the ZORA energy in terms of the Dirac energy from (18.9), the Rayleigh quotient is... 

The fact that the Rayleigh quotient for the ZORA wave fiinction yields the Dirac eigenvalue logically leads into another, even simpler, approach than lORA to the improvement of ZORA, and one that preceded lORA the scaled ZORA method (van Lenthe et al. 1994). It is the renormalization terms in the lORA Hamiltonian that present the main difficulty. But for any given function, renormalization may be achieved by a simple scaling. We therefore make the approximations... 

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