Complex atoms and the Periodic System of the elements
Non-central forces due to
electron repulsion
Central field approximation
Æelectronic orbitals
Ælift degeneracy of l
E nl = −
R(hc )
(n − δ l )2
Æ “Aufbau” principle
W. Ubachs – Lectures MNW-4
Complex atoms and the central field approximation
The potential energy in a multi-electron atom:
Z
Ze 2
V =−
4πε 0 i =1
∑
1
1
e2
+
r
r r
ri 4πε 0 i < j ri − r j
attraction
to nucleus
∑
repulsion between
electrons
V is a non-central potential
CFA:
⎛ r⎞
ri ⎟
The overall effect of V ⎜
⎜
⎟
⎝ i ⎠
∑
is centrally directed toward the nucleus
Vc (r )
−
Schrödinger equation
[
]
h2 2
−
∇1 + ∇ 22 + ⋅ ⋅ ⋅ + ∇ 2Z Ψ + VΨ = EΨ
2m
W. Ubachs – Lectures MNW-4
e2
4πε 0 r
for r → ∞
Then V can be written in terms of an
“effective screened nuclear charge”
Vcfa (r ) = −
The wave function depends on
3Z spatial coordinates
r r
r
Ψ (r1, r2 ,⋅ ⋅ ⋅, rZ )
Ze 2
for r → 0
−
4πε 0 r
Note:
Z eff (r )e 2
4πε 0 r
Vatom = Vcfa (r ) + VeeNC
'
Separation of variables in the central field approximation
Product wave function
r r
r
r
r
r
Ψ (r1, r2 ,⋅ ⋅ ⋅, rZ ) = ψ 1 (r1 )ψ 2 (r2 ) ⋅ ⋅ ⋅ψ Z (rZ )
1. Angular part of the wave function
is the same, hence angular functions
Υli mi (θ i , φi )
Potential and eigenenergy
V=
Z
∑
Vc (ri )
E=
i =1
Z
∑
Ei
i =1
Insertion of trial yields a set of
equations:
2. Radial part of the wave function
⎡ Z eff (r ) h 2l(l + 1) ⎤
h2 d
rRnl + ⎢
−
+
⎥ Rnl = Enl Rnl
2
2mr dr
2mr ⎥⎦
⎢⎣ r
Energy Enl depends on l
r
r
h2 2 r
−
∇ i Ψ (ri ) + Vc (ri )Ψ (ri ) = Ei Ψ (ri )
2m
The potential function is not Coulombic
i.e. is not 1/r but
−
Z eff (r )
W. Ubachs – Lectures MNW-4
r
Energy Enl does not depend on m
Wave functions for single “orbital”
r
ψ i (ri ) = Rnl (ri )Υli mi (θ i , φi ) ↑, ↓
Screening in the central field approximation
High l
Low l
For low l values (and same n)
electron comes closer to the nucleus
More Coulomb attraction
More binding energy
Lower l states Æ lower energy
W. Ubachs – Lectures MNW-4
Li
Lowering of low
W. Ubachs – Lectures MNW-4
l quantum states as an effect of screening
Screening and the quantum defect
Levels described with:
Enl = −
RNa
(n − δ l )2
With quantum defects:
δs = 1.35
δp = 0.86
δd = 0.01
δf = 0.00
W. Ubachs – Lectures MNW-4
Aufbau principle for multi-electron atoms
Eigenfunctions in multi-electron atom
r r
r
r
r
r
Ψ (r1, r2 ,⋅ ⋅ ⋅, rZ ) = ψ 1 (r1 )ψ 2 (r2 ) ⋅ ⋅ ⋅ψ Z (rZ )
1) Electrons fill the one-electron orbitals
r
ψ i (ri ) = Rnl (ri )Υli mi (θ i , φi ) ↑, ↓
into a “configuration”:
3) For filled shells
∑
r
ml i = 0 ⇒ Ltot = 0
shell
∑
r
sl i = 0 ⇒ Stot = 0
shell
r
∏ψ i (ri )
i
Degeneracy 2n2 structures
the Periodic System
2) Pauli principle dictates: single
occupancy
Note:
this is about ground states of the atoms
W. Ubachs – Lectures MNW-4
Ground state orbital configurations for multi-electron atoms
Noble gases
He configuration (1s ) ; closed shell
2
2
2
6
Ne conf. (1s ) (2 s ) (2 p ) ; closed shell
Ar conf.
(1s )2 (2s )2 (2 p )6 (3s )2 (3 p )6; closed shell
Alkali metals
Li
Na
K
one open shell electron
(1s )2 (2s )
(1s )2 (2s )2 (2 p)6 (3s ) one open shell electron
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s ) ....
Earth alkali metals
Mg
Ca
(1s )2 (2s )2 (2 p)6 (3s )2 closed
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )2
W. Ubachs – Lectures MNW-4
Binding energies of the
one-electron orbitals
vary with Z:
Screening effects
Irregularities and degeneracies
Transition metals; effect of 3d orbitals:
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )2
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )2 (3d )
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )2 (3d )2
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )2 (3d )3
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )(3d )5
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )2 (3d )5
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )2 (3d )6
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )2 (3d )7
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )2 (3d )8
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )(3d )10
(1s )2 (2s )2 (2 p)6 (3s )2 (3 p )6 (4s )2 (3d )10
W. Ubachs – Lectures MNW-4
ground term
1S
0
2D
3/2
3F
2
4F
3/2
7S
3
6S
5/2
5D
Competition in allocation
of electrons
4
4F
9/2
3F
4
2S
Near degeneracy
of 3d and 4s orbitals
1/2
1S
0
Coupling of the
Angular momenta
The periodic system of the elements
W. Ubachs – Lectures MNW-4
Ionization Potentials vary over the periodic structures
-Shell closing for the noble gases;
-Alkali metals
outer electron least binding energy
W. Ubachs – Lectures MNW-4