INTRODUCTION I have decided to do a project on how to ...

When using the perturbation method I will assume thate 2 / r12is small enough <strong>to</strong> beconsidered as a minor modification or perturbation of the opera<strong>to</strong>r0 ?H , an opera<strong>to</strong>r forwhich we may directly calculate the eigenfunctions0ψ . I will further hope that theperturbation of the opera<strong>to</strong>r0ψ will not be <strong>to</strong>o different from ψ (which cannot bedirectly evaluated)Thus when evaluating the ground state energy of a helium a<strong>to</strong>m in a given state, I willcombine the use of the correct hamil<strong>to</strong>nion opera<strong>to</strong>r H ? = H ? 0+ e 2 / r12with an incorrect0wave functionψ for that state which is regarded <strong>to</strong> be fairly close <strong>to</strong> what the correct0wave function should be. The nonrepulsive opera<strong>to</strong>r?H is said <strong>to</strong> be perturbed <strong>to</strong> the firs<strong>to</strong>rder by terme 2 / r12, and the present method of approximation is called a first orderperturbation method. The previous approximation method in which I ignored electron0repulsion al<strong>to</strong>gether is called a zero order perturbation method, since I used?H directlyas the opera<strong>to</strong>r. This means that I added no perturbation terms <strong>to</strong> the nonrepulsive0hamil<strong>to</strong>nion that I used. It is important <strong>to</strong> mention that for a given state,ψ is not aneigenfunction of H ? and therefore the operation of H ? 0onψ will not yield an exactenergy eigenvalue. Rather, because of the perturbation of energy due <strong>to</strong> the repulsionterm, the value of E obtained by the operation H ? 0onψ for a given state will depend onthe relative positions of the two electrons so that an approximate average value of E ~must be calculated through use of the mean value postulate as~E =∞∫ψ−∞∞∫− ∞*0ψH?ψ*0ψ00dτdτ=∞∫− ∞ψ*0( H?0 2 0ψ + e ψ / r )∞∫− ∞ψ*0ψ0dτ12dτSubstituting of equation (29) in<strong>to</strong> equation (33) yields~E =∞∫−∞but since for any given state,ψ*00E ψ0dτ+∞∫−∞ψ∞∫−∞*00E is constantψψ*002( e / r )dτ12ψ0dτ(33)(34)8