Variational principle, Hylleraas

The derived formulae are rarely employed in practise, because we only very rarely have at our disposal all the necessary solutions of eq. (5.16). The eigenfunctions of the operator appeared as a consequence of using them as the complete set of functions (e.g., in expanding There are, however, some numerical methods 

This proves the Hylleraas variational principle. The last equality follows from the first-order perturbational equation, and the last inequality from the fact that is assumed to be the lowest eigenvalue of (see the variational principle). What is the minimal value of the functional under consideration Let us insert X = We obtain 

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Hylleraas variational principle (p. 246) perturbation (p. 240) perturhational method (p. 240) perturbed system (p. 240)... 

Hylleraas variational principle (p. 209) Hylleraas equation (p. 210) asymptotic convergence (p. 210)... 

The precise connection with finite dimensional matrix formulas obtains simply from Lowdin s inner and outer projections [21, 22], see more below, or equivalently from the corresponding Hylleraas-Lippmann-Schwinger-type variational principles [24, 25]. For instance, if we restrict our operator representations to an n-dimensional linear manifold (orthonormal for simplicity) defined by... 

Explicitly correlated wave function fheory [14] is anofher imporfanf approach in quantum chemistry. One introduces inter-electron distances together with the nuclear-electron distances and set up some presumably accurate wave function and applies the variation principle. The Hylleraas wave function reported in 1929 [15] was the first of this theory and gave accurate results for the helium atom. Many important studies have been published since then even when we limit ourselves to the helium atom [16-28]. They clarified the natures and important aspects of very accurate wave functions. However, the explicitly correlated wave function theory has not been very popularly used in the studies of chemical problems in comparison with the Hartree-Fock and electron correlation approach. One reason was that it was generally difficult to formulate very accurate wave functions of general molecules with intuitions alone and another reason was that this approach was rather computationally demanding. 

Still, there was a discrepancy of0.12 eV that continued to bother Hylleraas when he returned to Oslo in 1928. Hylleraas considered this discrepancy a serious problem. On the one hand, attaching great importance to the variational principle, he was happy to see that his calculations approached the experimental number from the right side. On the other hand, he could not see how further extensions of the Cl expansion could improve the situation much, hinting at a limit of 24.49 eV [3] - still... 

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