Helium Atom Variational Principle
a) How many electrons and protons are in the helium atom?
2 electrons, 2 protons
b) What is the screening effect?
Extra electrons effectively “screen out” some of the charge from the protons
in the nucleus. For the specific example of helium, each electron orbiting the
nucleus will not feel the total charge of two protons, but will feel an effective
charge less than 2e because of the other electron present.
e2
~2
521 + 522 − 4π
The Hamiltonian for the Helium atom is H = − 2m
0
2
r1
+
c) Explain the origin (or meaning) of each term in H.
~2
− 2m
521 + 522 : This term corresponds to the kinetic energy of the two electrons.
e2
2
2
− 4π
+
r1
r2 : This is the potential energy of the nucleus on the first and
0
second electrons
respectively.
e2
4π0
1
−
−
r1 −→
r2 |
|→
: This is the interaction effect of the two electrons.
−
−
Make a drawing showing the vectors →
r1 and →
r2 .
r1
e-
r2
r2
e-
r1
2e+
1
2
r2
1
− →
.
−
r1 −→
r2 |
|−
3
Z
−Z(r1 +r2 )/a
, the
d) When using the trial wave function ψ (r1 , r2 ) = πa
3e
2
expectation value of the Hamiltonian is hHi = 2Z − 4Z(Z − 2) − 45 Z E1 .
Find the optimal value for Z.
To optimize Z, we will take the derivative with respect to Z of hHi, set it to
zero, then solve for Z.
d hHi
5
27
= 4Z − 8Z + 8 −
E1 = −4Z +
E1 = 0
dZ
4
4
⇒Z=
27
.
16
What is the meaning of Z?
Z is the effective charge the electrons feel from the nucleus, which will be decreased from 2 because of the screening effect.
e) Estimate the upper bound for the ground state of Helium.
We can estimate this upper bound by plugging our optimized Z back into the
Hamiltonian.
27
27 27
5
hHi = 2
−4
− 2 − Z E1
16
16 16
4
"
= −2
27
16
2
+4
27
16
2 #
E1 =
272
729
E1 =
E1
16 ∗ 8
128
∼
= 5.695E1 = −77.452eV
How far off the experimental value EHe ?
The difference is -77.452 eV + 78.985 eV = 1.533 eV
Pretty close.
2