Helium Ground State:
Variational Method +
Closing Part: Doubly- Excited States
Here we will perform the variational method for ground state of
helium, based on an idea of
effective charge determining the "screened" nuclear charge. The idea
is that we make
a model, where we assume that the electrons still behave as
independent particles, but each
electron weakens a bit the attraction between the other one and the
nucleus, i.e. the "cloud" of
one electron partly "screens" the nucleus, as seen by the other
electron.
To do this, we first evaluate the "perturbation theory as before. We
use the expressions for
hydrogen-like situation, but build in
(using the virial theorem see this: http://en.wikipedia.org/wiki/Virial_theorem,
also http://math.ucr.edu/home/baez/virial.html)
and use "scaling relations" - the terms depend on Z squared, but the
operations do not.
Thus something must come from the wavefunction-dependence on the Z - or the effective one, which we
call z
1_evaluate_H_for_Z_and_for_effective_z_virial_trick.png
1_evaluate_H_for_Z_and_for_effective_z_virial_trick.png
Thus the E(z) has been evaluated. But how to find the "correct"
screening?
We can prove simply a theorem: when using any wavefunction, the
expectation value in this arbitrary
state can never be lower than the expectation value evaluated for
the true ground state.
The proof is very simple.
Thus the task is to find a function giving the MINIMUM.
Since we have established the E(z), we simply find the "correct" z
by setting the derivative to zero
(no, it will not give us the maximum, there is no maximum value
....)
2_G.S.minimizes_energy_exp.val_Variational_find_min.png
2_G.S.minimizes_energy_exp.val_Variational_find_min.png
The result is: z = Z - 5/16,
or roughly z=Z-0.4
This is not the best possible value, but best for the model.
Better approximations.
One thing most important: remove the "independent particlesW
One method is configuration mixing - we shall discuss it
again later
Other methods - variational with lifting the product wafenction
approximation
Hylleraas method, Pekeris method
3_variational_res_and_Hylleraas.png
3_variational_res_and_Hylleraas.png
Technical points on evaluating the repulsion.
The 6-dim integral must be evaluated.
General method - use the so-called multipole expansion - very
general
4_Evaluating_repulsion_multipoles_Y_LM.png
4_Evaluating_repulsion_multipoles_Y_LM.png
Spherical harmonics - as a special case of orthogonal polynomials
Here we use a schematic from 2011 - all the steps are illustrated
4a_2011_-evaluate-repulsion.png
2011 copy
4a_2011_-evaluate-repulsion.png
2011 copy
4b_2011_-evaluate-repulsion-1-term.png
2011 copy
4b_2011_-evaluate-repulsion-1-term.png
2011 copy
Here is an illustration we did this time: first that for ground
state (and other s-states) there is
only the "monopole" contribution.
Then the double integral over the two radial coordinates is shown
5_evaluating_repulsion_spherical_harmonics.png
5_evaluating_repulsion_spherical_harmonics.png
the double integral over the two radial coordinates is shown
6_evaluating_repulsion_radial_integrals.png
6_evaluating_repulsion_radial_integrals.png
6b_2011_-evaluate-repulsion-result.png
2011 copy
6b_2011_-evaluate-repulsion-result.png
2011 copy
Doubly excited states of helium.
These are very interesting states: they look like bound states, but
the repulsion is so large
that the energy is sufficient to free one of the electrons.
This is illustrated by the energy diagram.
Such states are then unstable. Similar states appear also in ions.
An atomic system (or in other cases, an ion) brought into such state
will then decay after some time. One electron can leave, there is
enough energy for that.
This type of states is called autoionizing states.
If this state is a result of a scattering process - such states are
called resonances.
Decay of some ions via this mechanism is also called Auger effect -
or non-radiative de-excitation
(we shall mention this in connection with the radiation by excited
atoms towards the end of this course)
a1-He-spectra.png 2011
copy
a1-He-spectra.png 2011
copy