Atoms Multi-e- Including Spin
The orbital approximation: Each e- occupies its own orbital
(r1 , r2 , r3 , ...) = (r1 ) (r2 ) (r3 )...
Individual orbitals: ~ Hydrogenic orbitals Modified by: nuclear charge & e- repulsions Electron spin: s = ms = + () or - () State of e- described by 4 quantum numbers For electron "1":
z
(1) n , l , ml , ms
Each independent e- wave function spatial and spin part One e- in 1s orbital with "up" spin
(1)1, 0, 0, +1/2 = 1, 0, 0 (r , , ) = 1s
spatial
spin
Two e-s in a 1s orbital, opposite spins
= 1s (1) 1s ( 2) (1) (2)
But this is not actually an acceptable wave function...
1
Pauli Exclusion Principle
Simple form: No two e-s can have the same set of four quantum numbers
n , l , ml , ms
No more than two e-s can occupy an orbital If two e-s do occupy an orbital spins must be paired
Full form: The total wave function must be antisymmetric under the interchange of two identical fermions What this means: The total wave function must change sign if the labels of two identical e-s are swapped.
(1,2) = - (2,1) acceptable wave function
Potential full wave functions for He atom:
= 1s (1) 1s (2) (1) (2) = 1s (1) 1s ( 2) (1) (2)
= 1s (1) 1s ( 2) (1) (2) = 1s (1) 1s ( 2) (1) (2)
Try linear combinations:
None of these are acceptable
+ = 1s (1) 1s (2) - = 1s (1) 1s (2)
1 [ (1) (2) + (1) (2)] 2 1 [ (1) ( 2) - (1) (2)] 2
Still unacceptable
Acceptable
= Full wave function for the ground state of He
2
The Aufbau (building up) Principle
Aufbau principle = method for determine e- configuration of the ground state of multielectron atoms and ions Treat e-s as if they were independent. (orbital approximation) Each e- described by the same set of 4 quantum numbers that is used to describe H atom states: n, l, ml, ms Order of energy levels: Can differ from H atom Due to e- e- repulsions = shielding Generally, order given by:
Electrons are arranged in the lowest possible energy levels consistent with Pauli the Principle But look out for exceptions: Cr = [Ar]4s13d5 (Half-filled d subshell lower in energy than 4s) 3
Spin-Orbit Coupling
In addition to main lines, spectra of atoms show extra splittings Due to interaction of spin and orbital magnetic moments Electrons in atoms have both: orbital angular momentum spin angular momentum
l = 0, 1, 2, ... ml (lz) = 0, 1, 2, ... l
which interact by spin-orbit coupling:
s= ms (sz) = r r r j = l + s vectors angular Total
j = l
Selection rules:
momentum
n = any integer,
l = 1,
j = 0, 1
Without spin-orbit coupling
With spin-orbit coupling
s
p
d
n 3d 3
S1 2
s
p
d
D5 2
3s
3p
P3 2
P1 2
P3 2 P1 2
D3 2
2s
2p
2
S1 2
Fine structure splitting
1s
1
S1 2
4
Spin-Orbit Coupling Single Electron
Interaction of orbital and spin magnetic moments Vector sum of spin and orbital angular momentum
SOC Depends on size and relative orientation of vectors Total angular momentum described by quantum numbers:
j and mj
For 1-electron system:
s= j=l
Spin-orbit coupling energy
ESOC = (A2/2)[j(j+1) - l(l+1) - s(s+1)] (J) A = spin-orbit coupling constant Etotal = Eorbital(n,l,ml) + ESOC.(l,s,j)
Fine-structure splittings SOC Depends strongly on Z (...