Helium atom perturbation treatment

The value of the polarizability a of an atom or molecule can be calculated by evaluating the second-order Stark effect energy — %aF2 by the methods of perturbation theory or by other approximate methods. A discussion of the hydrogen atom has been given in Sections 27a and 27e (and Problem 26-1). The helium atom has been treated by various investigators by the variation method, and an extensive approximate treatment of many-electron atoms and ions based on the use of screening constants (Sec. 33a) has also been given.3 We shall discuss the variational treatments of the helium atom in detail. 

PERTURBATION TREATMENT OF THE HELIUM-ATOM GROUND STATE... 

Section 9,3 Perturbation Treatment of the Helium-Atom Ground State 253... 

We now reconsider the helium atom from the standpoint of electron spin and the Pauli principle. In the perturbation treatment of helium in Section 9.3, we found the zeroth-order wave function for the ground state to be 15(1)15(2). To take spin into account, we must multiply this spatial function by a spin eigenfunction. We therefore consider the possible spin eigenfunctions for two electrons. We shall use the notation a(l)a(2) to indicate a state where electron 1 has spin up and electron 2 has spin up a(l) stands for a(Wji). Since each electron has two possible spin states, we have at first sight the four possible spin functions ... 

The Hartree SCF Method. Because of the interelectronic repulsion terms the Schrodinger equation for an atom is not separable. Recalling the perturbation treatment of helium (Section 9.3), we can obtain a zeroth-order wave function by neglecting these repulsions. The Schrddinger equation would then separate into n one-electron hydrogenlike equations. The zeroth-order wave function would be a product of n hydrogenlike (one-electron) orbitals ... 

In quantum mechanics the uncertainty principle tells us that we cannot follow the exact path taken by a microscopic particle. If the microscopic particles of the system all have different masses or charges or spins, we can use one of these properties to distinguish the particles from one another. But if they are all identical, then the one way we had in classical mechanics of distinguishing them, namely by specifying their paths, is lost in quantum mechanics because of the uncertainty principle. Therefore, the wave function of a system of interacting identical particles must not distinguish among the particles. For example, in the perturbation treatment of the helium-atom excited states in Chapter 9, we saw that the function li(l )2i(2), which says that electron 1 is in the li orbital and electron 2 is in the 2s orbital, was not a correct zeroth-order wave function. 

In the perturbation method the Hamiltonian is written as + H, where corresponds to a Schrodinger equation that can be solved. The perturbation term H is arbitrarily multiplied by a fictitious parameter k, so that A. = 1 corresponds to the actual case. The method is based on representations of energy eigenvalues and energy eigenfunctions as power series in A. and approximation of the series by partial sums. The method can be applied to excited states. In the helium atom treatment the electron-electron repulsive potential energy was treated as the perturbation term in the Hamiltonian operator. 

Suppose we take the interelectronic repulsions in the li atom as a perturbation on the remaining terms in the Hamiltonian. By the same steps used in the treatment of helium, the unperturbed wave functions are products of three hydrogenlike functions. For the ground state,... 

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