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Approximation by Perturbation Method:

The eigenstates of complex systems, such as atoms and molecules, cannot be solved for precisely using the Schrödinger Equation alone. Electron-electron repulsion and electromagnetic field interactions have a significant effect on the energy eigenvalues of a system and cannot be ignored. The perturbation method is one method used to contend with such. It assumes the system in question is being perturbed by an electromagentic disturbance.

This method treats the hamiltonian, the eigenfunction, and the energy eigenvalue of the system each as a power series:

eq.1

eq.2

eq.3

The (0) indicates the unperturbed form, (1) indicates the first-order perturbed form. Other terms are negelected for simplicity's sake. The letter l is a fictitious parameter used only to keep track of various terms and is equal to one. Inserting eqs. 1, 2, and 3 into the Schrödinger Equaiton:

eq.4

Multiplying through:

eq.5

Terms containing l ² are ignored, again for simplicity. For more accurate calculations, several more terms would be included to arrive at a better approximation. Organizing terms of l :

eq.6

The coefficients of each power of l equal zero separately. Demonstrating each:

eq.7

And:

eq.8

To continue further, one treats the perturbed wave function as a linear combination of the unperturbed wave function:

eq.9

Implementing this in eq.8 to determine E⁽¹⁾:

eq.10

Knowing that the unperturbed wave function is an eigenfunction of the unperturbed hamiltonian:

eq.11

Canceling the first terms on each side:

eq.12

Multiplying through by < Y |:

eq.13

The first-order correction to the energy of the system therefore can be determined from the unperturbed eigenfunction, pending one knows the form of the perturbed hamiltonian. For example, the energy correction caused by electron-electron replusion in a ground-state helium atom can be represented by the perturbed hamiltonian which appears as:

eq.14

The variable r₁₂ is the 'distance' between the two electrons. Knowing that the helium atom wavefunction is the product of the two hydrogen-like wavefunctions associated with each electron, eq. 13 can be rewritten to determine the first-order energy correction:

eq.15

The perturbation method is only accurate when the perturbation in energy is small relative to the zero-order energy state of the system. Such is not the case in the helium atom, where the electron-electron replusion is comparable to the attraction induced by the nucleus. Therefore, this method is not typically used for the helium atom. But being 94.6% accurate with respect to the most accurate calculation made of -79.0 eV while using first-order corrections (-74.8 eV), it is certainly better than nothing (Mortimer 480). The helium atom example was shown to clarify eq. 13. No numerical second order corrections have yet been made when dealing with systems containing coulomb potentials due to the difficult nature of expressing the second-order perturbed hamiltonian.