LNPhysiqueAtomique2016
Towards bigger atoms
The central field approximation (CFA)
• To start with, we still ignore the spin-orbit interaction
2
• The Schrödinger equation
for the spatial part :
3
N ✓
X
1 2
4
ri
2
1=1
⁃
⁃
⁃
Z
ri
◆
N
X
1 5
+
(~r1 , ~r2 , . . . , ~rN ) = E (~r1 , ~r2 , . . . , ~rN )
r
ij
j>i
3N-dimensional differential equation
Not separable
The 1/rij -term is too large for a very accurate
perturbation treatment
Effective potential
• A large part of the 1/rij -term will be radial
• On an individual valence electron, the other electrons
will act like an almost spherical screening of the nuclear
charge
• The effective radial part of the total potential, felt by
one electron:
Z
VCF (r) =
+ S(r)
r
⁃ with S (r) being the screening potential from the
(N-1) other electrons
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LNPhysiqueAtomique2016
N
X
1
• The term S (r) will include all the radial part of
r
j>i ij
• The angular part of the mutual interaction term, we
will treat as a perturbation
Form of VCF
• Asymptotically, when ri ! 1 :
) rij ⇡ ri
N
X1 1
Z
Z
VCF (r) ⇡
+
=
ri
r
j=1 i
N +1
ri
⁃
for a neutral atom, Z = N :
1
VCF (r) ⇡
ri
• Asymptotically, when ri ! 0 :
) rij ⇡ rj
*N 1 +
X 1
Z
VCF (r) ⇡
+
⇡
ri
r
j=1 j
Z
ri
• In between the limits, an electron will feel an effective
Z, between 1 and Z
Ze↵ (r)
VCF (r) =
ri
2
LNPhysiqueAtomique2016
• Usually, we can only guess VCF , or calculate it
numerically
• Nevertheless, even without knowing the exact form of
VCF and ψ0 , we can understand a lot of atomic
structure
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LNPhysiqueAtomique2016
Perturbative treatment
• We now treat the reminder of the total Hamiltonian,
N
X
1
the angular part of
, as a perturbation; Hres
r
ij
j>i
H = HCF + Hres
◆ X
◆ X
N ✓
N ✓
N
X
1 2
Z
1
H=
rri +
+
2
ri
r
i=1
i=1
j>i ij
◆ X
N ✓
N
X
Z
(all)
VCF (r) =
+
S(r)
r
i=1
i=1
HCF
◆
N ✓
N
X
X
1 2
(all)
=
rri + VCF (ri ) =
Hi
2
i=1
i=1
Hres
N
X✓
= H HCF
◆ X
◆ X
◆
N ✓
N
N ✓
X
1 2
Z
1
1 2
=
rri +
+
rri
2
r
r
2
i
i=1
i=1
j>i ij
i=1
◆
✓
◆
✓
N
N
N
N
X 1
X
X
X
Z
Z
+
S(ri )
=
r
r
r
i
ij
i
j>i
i=1
i=1
i=1
N
X
1
=
r
j>i ij
N
X
S(ri )
i=1
4
(all)
VCF (ri )
LNPhysiqueAtomique2016
zero-order wave functions, ψCF
• Schrödinger equation:
N 
X
1 2
HCF CF =
rri + VCF (ri )
2
i=1
CF
= ECF
CF
• This is a separable equation :
r1 ) u2 (~r2 ) . . . uN (~rN )
CF = u1 (~
• This is N separate equations, of the type :

1 2
r + VCF (r) unlml (~r) = Enl unlml (~r)
2 r
⁃ where
unlml (~r) = Rnl (r) Ylml (✓, ')
• The solutions will be similar to the hydrogenic ones
n = 1, 2, 3, . . .
l = 0, 1, . . . , n 1
m = l, l + 1, . . . , l
• The total (zero-order) energy :
N
X
ECF =
Eni li
i=1
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LNPhysiqueAtomique2016
Electron configurations, Orbitals
• The individual one-electron wav functions will be a bit
different from hydrogenic ones
• But the potential is central, and the will be close to the
hydrogenic
⁃ Logical to use the hydrogenic notation
• Possible solutions :
u1s , u2s , u2p , u3s , u3p , . . .
• We say the electrons “occupy the orbitals” :
1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, . . .
The Pauli principle
• Two electrons may not be in the same state:
• the set of quantum numbers, ( n , l , ml , ms ) has to be
unique for every electron
• For one combination of ( n , l , ml ) , there may be two
electrons ( ms=+½ , ms=-½ )
• For one particular orbit ( n , l ), there may be
2 ( 2l + 1 ) electrons
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LNPhysiqueAtomique2016
l = 0 ; “s-orbital” ; 2 electrons
l = 1 ; “p-orbital” ; 6 electrons
l = 2 ; “d-orbital” ; 10 electrons
l = 3 ; “f-orbital” ; 14 electrons
……
• For the ground sate, the electrons will gradually fill up
the lowest energy orbitals
• Energy order (with lowest first) :
1s
2s
2p
3s
3p
4s
3d
4p
5s
4d
5p
6s
4f
5d
6p
7s
5f
6d
7p
8s
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LNPhysiqueAtomique2016
The periodic system
• We gradually “build up” the atom
⁃ “the aufbau-principle”
⁃ (“règle de Klechkowski”)
• Electronic configuration of the ground states of the
atoms:
1
2
3
4
5
6
7
8
9
10
11
H
He
Li
Be
B
C
N
O
F
Ne
Na
: 1s
: 1s2
(full)
: 1s2 2s
: 1s2 2s2
(full)
: 1s2 2s2 2p
: 1s2 2s2 2p2
: 1s2 2s2 2p3
: 1s2 2s2 2p4
: 1s2 2s2 2p5
: 1s2 2s2 2p6
(full)
: 1s2 2s2 2p6 3s = [Ne] 3s
…………
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LNPhysiqueAtomique2016
1
2
H
hydrogen
1s
3
Li
lithium
2s
11
Na
sodium
3s
19
K
4
5
Be
2p
12
boron
13
Mg
magnesium
3s2
20
Ca
21
Sc
22
23
Ti
24
V
25
Cr
Mn
26
Fe
27
28
Co
29
Ni
30
Cu
Zn
31
32
vanadium
3d3 4s2
chromium
3d5 4s
manganese
3d5 4s2
iron
3d6 4s2
cobalt
3d7 4s2
nickel
3d8 4s2
copper
3d10 4s
zinc
3d10 4s2
4p
37
38
39
40
41
42
43
44
45
46
47
48
49
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
strontium
5s2
yttrium
4d 5s2
zirconium
4d2 5s2
niobium
4d4 5s
4d5 5s
technetium
4d5 5s2
ruthenium
4d7 5s1
rhodium
4d8 5s
palladium
4d10
silver
4d10 5s
cadmium
4d10 5s2
5p
55
56
57-71
72
73
74
75
76
77
78
79
80
81
lanthanides
hafnium
5d2 6s2
tantalum
5d3 6s2
tungsten
5d4 6s2
rhenium
5d5 6s2
osmium
5d6 6s2
iridium
5d7 6s2
platinum
5d9 6s
104
105
Db
106
Sg
107
Bh
108
Hs
109
Mt
110
Cs
cesium
6s
87
Ba
barium
6s2
Fr
francium
7s
88
Ra
radium
89-103
actinides
7s2
Hf
Rf
rutherfordium
6d2 7s2
57
La
58
Ta
dubnium
6d3 7s2
Ce
59
lanthanum
5d 6s2
cerium
4f 5d 6s2
4f3
89
90
91
Ac
actinium
6d 7s2
Th
thorium
6d2 7s2
W
seaborgium
6d4 7s2
60
Pr
praseodymium
6s2
Pa
protactinium
5f2 6d 7s2
Re
bohrium
6d5 7s2
Nd
61
Os
hassium
6d6 7s2
Pm
62
Ir
meitnerium
6d7 7s2
Sm
63
Pt
Ds
Eu
112
Cn
113
Uut
Gd
65
promethium
4f5 6s2
samarium
4f6 6s2
europium
4f7 6s2
gadolinium
4f7 5d 6s2
4f9
92
93
94
95
96
97
U
Np
neptunium
5f4 6d 7s2
Pu
plutonium
5f6 7s2
Am
americium
5f7 7s2
9
Tl
111
Rg
neodymium
4f4 6s2
uranium
5f3 6d 7s2
Hg
thallium
6p
roentgenium
Cm
curium
5f7 6d 7s2
copernicium
6d10 7s2
Tb
terbium
6s2
Bk
berkelium
5f9 7s2
66
Ge
ununtrium
7p
Dy
67
10
F
fluorine
2p5
2p6
15
16
17
18
P
S
Cl
phosphorus
3p3
sulphur
3p4
chlorine
3p5
3p6
33
34
35
36
As
Se
Br
Ne
neon
Ar
argon
Kr
selenium
4p4
bromine
4p5
krypton
4p6
50
51
52
53
54
Sn
tin
82
Pb
lead
6p2
114
Fl
flerovium
7p2
Ho
holmium
4f11 6s2
98
99
Cf
9
O
oxygen
2p4
arsenic
4p3
dysprosium
4f10 6s2
californium
5f10 7s2
8
N
nitrogen
2p3
germanium
4p2
5p2
mercury
5d10 6s2
6d9 7s2
64
In
Si
silicon
indium
5d10
darmstadtium
6d8 7s2
Au
gold
6s
Ga
gallium
rubidium
5s
molybdenum
14
3p2
titanium
3d2 4s2
Zr
Al
aluminium
3p
scandium
3d 4s2
Y
7
C
carbon
2p2
calcium
4s2
Sr
6
B
beryllium
2s2
potassium
4s
Rb
He
helium
1s2
Es
einsteinium
5f11 7s2
68
Sb
Te
I
Xe
antimony
5p3
tellurium
5p4
iodine
5p5
xenon
5p6
83
84
85
86
Bi
Po
At
Rn
bismuth
6p3
polonium
6p4
astatine
6p5
6p6
115
Uup
116
Lv
117
Uus
118
Uuo
ununpentium
7p3
Er
erbium
69
livermorium
7p4
Tm
70
7p5
Yb
71
ununoctium
7p6
Lu
4f12 6s2
4f13 6s2
ytterbium
4f14 6s2
lutetium
4f14 5d 6s2
100
Fm
101
Md
102
No
103
Lr
fermium
5f12 7s2
thulium
ununseptium
radon
mendelevium
5f13 7s2
nobelium
5f14 7s2
lawrencium
5f14 7s2 7p
LNPhysiqueAtomique2016
• Chemical properties are given by the number of
valence electrons (outermost orbital)
⁃ alkalis
⁃ alkaline earths
⁃ ……
⁃ metals
⁃ ……
⁃ halogens
⁃ rare gases
• Optical properties are also given by the valence
electrons
• Inner orbital are typically only accessible with x-rays
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