### Helium ground state

```Helium ground state
H total
2
2
2
− 2 2
2
e
2
e
e
=
∇1 + ∇ 22 ) −
−
+
(
2m
4πε 0 r1 4πε 0 r2 4πε 0 r1 − r2
Approximate antisymmetrized wave function (neglecting electronelectron interactions)
Ψ (r1 , r2 ) =
Energy neglecting
e-e interactions
Approximate ground state
evaluated with the total
Hamiltonian
ψ 1s (r1 )ψ 1s (r2 )
2
( ↑↓ − ↓↑ )
−13.6 Z 2
E = 2×
= − 108.8 eV
2
n
E=
Ψ H total Ψ
Ψ Ψ
= −74.83 eV
Helium excited states
One electron in 1s and one in 2s,
E=
He
Ψ H red
Ψ
Ψ Ψ
13.6* 22 13.6* 22
=−
−
= −68 eV
2
2
1
2
The antisymmetric solution Ψ = 0 for r1 = r2 .
Helium excited states
E = hf = hc/λ
Names refer to
approximate
solutions
http://physics.nist.gov/PhysRefData/Handbook/Tables/heliumtable5.htm
Exchange (Austauschwechselwirking)
1
ψ A ( r1 , r2 ) =
(φ1 (r1 )φ2 (r2 ) − φ1 (r2 )φ2 (r1 ) )
2
1
ψ A H ψ A = [ φ1 (r1 )φ2 (r2 ) H φ1 (r1 )φ2 (r2 ) − φ1 (r1 )φ2 (r2 ) H φ1 (r2 )φ2 (r1 )
2
− φ1 (r2 )φ2 (r1 ) H φ1 (r1 )φ2 (r2 ) + φ1 (r2 )φ2 ( r1 ) H φ1 (r2 )φ2 (r1 ) ]
1
ψ S ( r1 , r2 ) =
(φ1 (r1 )φ2 (r2 ) + φ1 (r2 )φ2 (r1 ) )
2
1
ψ S H ψ S = [ φ1 (r1 )φ2 (r2 ) H φ1 (r1 )φ2 (r2 ) + φ1 (r1 )φ2 (r2 ) H φ1 (r2 )φ2 (r1 )
2
+ φ1 (r2 )φ2 (r1 ) H φ1 (r1 )φ2 (r2 ) + φ1 (r2 )φ2 (r1 ) H φ1 (r2 )φ2 (r1 ) ]
The difference in energy between the ψA and ψS is twice the exchange energy.
Exchange
The exchange energy can only be defined when you speak of multi-electron
wavefunctions. It is the difference in energy between the symmetric solution
and the antisymmetric solution. There is only a difference when the electronelectron term is included. Coulomb repulsion determines the exchange energy.
In ferromagnets, the antisymmetric state has a lower energy. Thus the state
with parallel spins has lower energy.
In antiferromagnets, the symmetric state has a lower energy. Neighboring
spins are antiparallel.
Many electrons
Consider a gold atom (79 electrons)
2
⎛ d2
∂ψ
=− ⎜ 2
i
∂t
2m ⎝ dx1
3 x 79 = 237 terms
Ψ( x1, y1, z1,
d2 ⎞
79e2
+ 2 ⎟Ψ −
Ψ
dz79 ⎠
4πε 0rj
79 terms
+
e2
4πε 0rij
Ψ
79 x 78 = 3081 terms
2
x79 , y79 , z79 , t ) is a complex function in 237 dimensions
Ψ(r1, rN )
2
is the joint probability of finding an electron
at position r1, r2, ... rN.
Numerical solution of the Schrödinger
equation for one electron
∂Ψ −
e
2
=
∇ Ψ−
Ψ
i
∂t
2m
4πε 0 r
2
2
z
Discretize Ψ to solve
numerically. For one
electron ~ 106 elements are
needed.
y
x
Numerical solution for many electrons
For a numerical solution, divide Hilbert space along each axis
into 100 divisions.
z1
y1
100237 = 10474
x1
y2
x2
There are 1068 atoms in the
Milky Way galaxy
There are ~ 1080 atoms in the
observable universe
Intractable problem
We know the equation that
has to be solved. We know
how to solve it but we don't
have the computer resources
A Matlab that will calculate the
time evolution of an n-electron
atom
Quantum computation
Sometimes it is possible to map one intractable problem onto
another.
If you map an intractable problem onto a system of interacting
electrons and then measure the energy levels of the electron
system, you can find solutions to the intractable problem.
Many-electron systems
In such a quantum system, the repeated interactions
between particles create quantum correlations, or
entanglement. As a consequence, the wave function of
the system is a complicated object holding a large
amount of information, which usually makes analytical
calculations impractical. In fact, many-body theoretical
physics ranks among the most computationally intensive
fields of science.
http://en.wikipedia.org/wiki/Many-body_problem
The Central Dilemma of Solid State Physics
The Schrödinger explains everything but can explain nothing.
From a fundamental point of view it is impossible to describe
electrons in a metal correctly - Ashcroft and Mermin
Neglect the e-e interactions
Consider a gold atom (79 electrons)
⎛ d2
− ⎜ 2
2m ⎝ dx1
2
3 x 79 = 237 terms
d2 ⎞
79e2
+ 2 ⎟Ψ −
Ψ
dz79 ⎠
4πε 0rj
79 terms
+
e2
4πε 0rij
Ψ
= EΨ
79 x 78 = 3081 terms
2
Out of desperation: We simplify the model for a solid until the
Schrödinger can be solved. If the 3081 electron - electron terms
are neglected, the equation can be solved exactly and the total
wave function is a product of atomic orbitals.
Antisymmetrized product wave functions
Ψ (r1 , r2 ,
, r79 ) = φ179s ↑ (r1 ), φ179s ↓ (r2 ),… , φ679s ↑ (r79 )
The standard first approximation for the many-electron
wave-function of any atom is an antisymmetrized
product of atomic orbitals.
1s2 2s22p63s23p63d104s24p64d104f145s25p65d106s1
Electron configurations
Filling of electron shells
Ni: 3d84s2
Cu: 3d104s1
Why isn't Ni 3d94s1 or 3d10?
You can evaluate the energy of any electron configuration.
Ψ (r1 , r2 ,
, r28 ) = φ128s ↑ (r1 ), φ128s ↓ (r2 ),… , φ328d ↑ (r27 )φ428s ↑ (r29 )
E=
Ψ H Ψ
Ψ Ψ
Lithium
The antisymmetric 3 electron wavefunction can be written
φ1Lis ↑ (r1 ) φ1Lis ↑ (r2 ) φ1Lis ↑ (r3 )
1 Li
Li
Li
Li
Ψ(r1, r2 , r3 ) = φ1s ↑,φ1s ↓,φ2s ↑ =
φ1s ↓ (r1 ) φ1Lis ↓ (r2 ) φ1Lis ↓ (r3 )
3! Li
φ2s ↑ (r1 ) φ2Lis ↑ (r2 ) φ2Lis ↑ (r3 )
There are two possible configurations
φ ↑,φ ↓,φ ↑
Li
1s
Li
1s
Li
2s
and
φ1Lis ↑,φ1Lis ↓,φ2Lis ↓
Construct the Hamiltonian matrix to see which has the lowest energy
Berylium
The antisymmetric 4 electron wave function can be written,
Ψ(r1, r2 , r3 , r4 ) = φ1Bes ↑,φ1Bes ↓,φ2Bes ↑,φ2Bes ↓
φ1Bes ↑ (r1 )
Be
1 φ1s ↓ (r1 )
=
4! φ2Bes ↑ (r1 )
φ2Bes ↓ (r1 )
φ1Bes ↑ (r2 )
φ1Bes ↓ (r2 )
φ2Bes ↑ (r2 )
φ2Bes ↓ (r2 )
φ1Bes ↑ (r3 )
φ1Bes ↓ (r3 )
φ2Bes ↑ (r3 )
φ2Bes ↓ (r3 )
φ1Bes ↑ (r4 )
φ1Bes ↓ (r4 )
φ2Bes ↑ (r4 )
φ2Bes ↓ (r4 )
Gold
The antisymmetric 79 electron wave function can be written,
Ψ (r1 , r2 ,
, r79 ) = φ179s ↑ (r1 ), φ179s ↓ ( r2 ),… , φ679s ↑ (r79 )
The determinant of an N×N matrix has N! terms.
79! = 8.95 × 10116
(more than the atoms in the observable universe)
matrix gets too big.
Pauli exclusion
The sign of the wave function must change when two electrons are exchanged
1 ψ1 (r1 ) ψ 2 (r1 )
1
Ψ(r1, r2 ) =
=
(ψ1(r1 )ψ 2 (r2 ) −ψ1(r2 )ψ 2 (r1))
2 ψ1 (r2 ) ψ 2 (r2 )
2
If two columns are exchanged, the wave function changes sign.
If two columns are the same, the determinant is zero.
The Pauli exclusion principle only holds in the noninteracting electron
approximation when the many electron wave function can be written as
a product of single electron wave functions, only one electron can
occupy each single electron state.
```