Excited States of the Helium Atom

Excited States of the Helium Atom.—The variation method can be applied to the lowest triplet state of helium as well as to the lowest singlet state, inasmuch as (neglecting 

Numerous investigations by Hylleraas and others2 have shown that wave mechanics can be applied in the treatment of other states of the helium atom. We shall not discuss further the rather complicated calculations. 

If both electrons occupy the same space orbital, the wave function for an excited state is similar to that of the ground state. For the configuration (2s )  

The probability density for two electrons and the electric charge density are analogous to those of the ground state. 

If the electrons occupy different space orbitals there are four possible states for each pair of space orbitals. For the (ls)(25) configuration we can write four antisymmetric wave functions that are products of a space factor and a spin factor  

The number of states can be determined by counting the number of ways of arranging two spins up-up, down-down, up-down, and down-up. As shown in Eq. (18.4-2), the up-down and down-up arrangements combine in two ways, corresponding to symmetric and antisymmetric spin factors. 

The probability density for finding the two electrons irrespective of spins is obtained by integrating the square of the wave function over the spin coordinates. Consider the state corresponding to in Eq. (18.4-2a). The probability of finding electron 1 in the volume element fr and electron 2 in the volume element (fir2 irrespective of spins is 

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Note that in the present case the matrix elements depend on the final density p . Moreover, because this density is obtained from the transformed wavefunction, they also depend on the expansion coefficients. For this reason, Eq. (177) must be solved iteratively. Such a procedure has been applied - in a sample calculation - to the 2 S excited state of the helium atom. The upper-bound character of the energy corresponding to the energy functional for the transformed wavefunction [ p( r,- ) with respect to the exact energy is guaranteed by... 

Figure 8.2 Lower excited states of the helium atom.

When we combine the spatial and spin wavefunctions, the overall wavefunction must be antisymmetric with respect to exchange of electrons. It is therefore only admissible to combine a syrrunetric spatial part with an antisymmetric spin part, or an antisymmetric spatial part with a symmetric spin part. The following functional forms are therefore permissible functional forms for the wavefunctions of the ground and first few excited states of the helium atom ... 

Pretend that electrons are bosons with zero spin. Describe how the excited states of the helium atom would differ from the actual excited states in the orbital approximation. 

Excited States of the Helium Atom. Degenerate Perturbation Theory... 

The perturbation method as described in the previous section does not apply if several wave functions correspond to the same zero-order energy (the degenerate case). For example, the zero-order orbital energies of the 2s and 2p hydrogen-like orbitals are all equal, so that all of the states of the (1 X2 ) and (U)(2/ ) helium configurations have the same energy in zero order. A version of the perturbation method has been developed to handle this case. We will describe this method only briefly and present some results for some excited states of the helium atom. There is additional information in Appendix G. 

Use the separation theorem to show how the discussion of the ground and first excited states of the helium atom (p. 15) could be modified in order to obtain upper bounds on the energies, still using orbital products as expansion functions. 

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