1
Hydrogen Atom
The simplest system: p + e
Schroedinger equation can be
solved exactly
Spherical symmetry
Potential energy between p + e
r
e
V
0
2
4πε
−=
2
Polar Coordinates
Ψ(x,y,z) → Ψ(r,θ, φ) x = ?
y = ?
z = r cos q
Take advantage of spherical symmetry
3
Radial and Angular Part
Ψn, l, m (r,θ, φ) = N × Rn, l (r) × χl, m(θ, φ)
Separation of variables
Rn, l (r) = radial part of the wave function, depends only on
distance from a nucleus - r
χl, m(θ, φ) = angular (angles) part of the wave function,
depends only on direction - θ, φ
N = normalisation constant
In order to be ∫| Ψ |2 dV = +1
Normalisation condition - electron is definitely somewhere
with probability = 1
4
Quantum Numbers
Principal quantum number n, (1 to ∞)
Angular momentum quantum number l, (0 to n −1)
l = 0 (s), 1 (p), 2 (d), 3 (f), 4 (g), 5 (h), ........
Magnetic quantum number ml, (+ l, .....0, ..... −l)
For each l there is (2l + 1) values of ml
Magnetic spin quantum number ms (±½)
Rn, l (r) depends on quantum numbers n and l
χl, m(θ, φ) depends on quantum numbers l and ml
5
Wave Functions of H atom
• solution of Schrödinger
equation
• complex function of coordinates
x, y, z or better r, φ, θ
• no physical meaning
• positive and negative values
• | Ψ |2 probability density of
finding electron e
6
Radial Part of the Wave Function of H Atom
1 (p)
1 (p)
0 (s)
l
±1
0
0
ml
2 (Z/2a0) 3/2 (1 − Zr/2a0) exp(− Zr/2a0)2 (L)
2/√3 (Z/2a0) 3/2 (Zr/2a0) exp(− Zr/2a0)2 (L)
2 (Z/a0) 3/2 exp(− Zr/a0)1 (K)
Rn, l (r)n
7
Electron Energy in H-type Atoms
μ = reduced mass of nucleus-electron
e = elementary charge, ε0 = permitivity of vacuum
Z – the higher a nucleus charge the stronger is an
electron bound, the lower energy has, one-electron ions
(He+, Li2+,....)
n – the higher a principal number the less stable e is
Corresponds to Bohr’s eq.!!
2
2
22
0
4
8 n
Z
h
eN
E A
n
ε
μ
−=
2
2
0
n
Z
EEn −=
8
Electron Energy in H-type Atoms
E1 = −13.6 eV
(13.6 eV = 1 Ry)
Energy depend ONLY on n
E2 = ?
2
2
22
0
4
8 n
Z
h
eN
E A
n
ε
μ
−=
9
Principal Quantum Number n
Gives the levels energy
Higher n has higher energy less
stable
n same as in the Bohr’s model
Attains values 1 to ∞
For each n there is n2 of
degenerate levels
Σ (2l + 1) = n2
l = 0
l = n − 1
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Orbital Angular Momentum
L = Orbital angular momentum (vector)
L = m × v × r = p × r
( )1+= llL h
Describes movements of
electrons in orbitals
L
11
Angular Momentum Quantum Number l
l orbital
0 s
1 p
2 d
3 f
4 g
5 h
6 i
7 j
8 k
L = Orbital angular momentum
L = m × v × r
Type of orbital, (0 to n −1)
these orbitals are not filled by electrons
in atoms in ground state
( )1+= llL h
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Magnetic Quantum Number ml
l orbital ml
0 s 0
1 p 1, 0, −1
2 d 2, 1, 0, −1, −2
3 f 3, 2, 1, 0, −1, −2, −3
4 g
5 h
6 i
π2
h
mmL llz == h
these orbitals are not
filled by electrons in
atoms in ground state
13
Quantization of Orbital Angular Momentum
( )1+= llL h
π2
h
mmL llz == h
14
1sn = 1
543210l =
hgfdps
2p2sn = 2
n = 6
n = 5
n = 4
n = 3
6s
5s
4s
3s
6h6g6f6d6p
5g5f5d5p
4f4d4p
3d3p
For each n there is n2 of degenerate levels
15
Magnetic Spin Quantum Number ms
Stern-Gerlach experiment
S = h/2π [s (s +1)]½
s = ½
SZ = ms h/2π
S = spin
momentum
vacuum
Inhomogeneous magnetic field
Furnace with Ag
16
Magnetic Spin Quantum Number ms
S = h/2π [s (s +1)]½
s = ½
SZ = ms h/2π
ms = ±½
17
Ψ = Wave Functions
Ψ = solution of Schrödinger
equation
| Ψ |2 = probability density of e
| Ψ |2 dV = probability density
of finding electron e in volume
dV = distribution of electron
density
1
s
18
Probability Density
Polar coordinates
Rn, l (r) radial function
dV = 4πr2 dr (spherical layer of thickness dr)
Radial distribution function
P = 4πr2 | Ψ |2 dr = 4πr2 R2
n, l (r) dr
P = probability density
of finding electron e
in volume
of spherical layer
of thickness dr
In a distance r
19
Wave Functions
Probability
density
Radial
distribution
function
Orbital
Sign change
Zero values
20
Orbitals
Position of electrons cannot be established – Heisenberg’s
principle – only probability
Radial function – probability of finding e in a direction away
from nucleus (to r = ∞) and number of nodes = zero values
of radial distribution function
Angular function = shape of orbitals (number of nodal
planes)
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s - Orbitals
Rn, l (r) = Radial function, depends on r only
χl, m(θ, φ) = angular function, is a constant for sorbitals
(l = 0) = SPHERICAL SHAPE
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Atomic Orbital 1s
Rn, l (r)
n = 1, l = 0
Wave Function 1s
23
Radial Distribution Function
Rn, l (r) = Radial function of H atom
4πr2 R2
n, l (r) = Radial distribution function
rmax = the most probable radius
for 1s
rmax = a0 Bohr’s radius
4πr2 R2
n, l (r)
24
4πr2R2
n,l(r)=Radialdistributionfunction
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Nodes
Number of nodes = n − l −1
• Wavefunction changes sign
• Radial distribution function has zero value
27
Effect of Z Radial Function s
With increasing nucleus charge the
maximum of radial distribution function
approaches closer to the nucleus
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
0
3
0
ln, exp2(r)R
a
Zr
a
Z
Radial distribution function 1s
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4πr2 (Rnl)2
29
Angular Wave Function
Angular wave function gives the shape of orbitals
The same for all values of n
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p - Orbitals
x
y
z
pz
py
px
n = 2, l = 1, m = 1,0,−1
31
n = 2, l = 1, m = 0 n = 3, l = 1, m = 0
2p - orbitals 3p - orbitals
32
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2p - orbitals 3p - orbitals
Wave Function = Radial × Angular
+
−
+
+
−
−
34
Angular Wave Function of d-Orbitals
35
d - Orbitals
dZ2
dX2-Y2
dXY
dXZ
dYZ
36
d - Orbitals
x
y
z
dZ2 x
y
z
dYZ
x
y
dX2-Y2
x
y
dXY
37
f - Orbitals
38
Nodes
Spherical nodes = n − l −1
for s, p, d, f,....
Radial wave function
Nodal planes
Angular wave function:
Orbital no.
s 0
p 1
d 2
f 3
. .
. .
Only s-orbitals
have non-zero
value of wave
function at the
nucleus
39
Energy of H-Atom Orbitals
Energetically degenerate levels
n
2
2
22
0
4
8 n
Z
h
eN
E A
n
ε
μ
−=
Energy depends only on n
40
Energy Levels in Many-Electron Atoms
No degeneration
Energies depend on n and l
41
Energy Levels in Many-Electron Atoms
More stable orbital
has a lower energy
Madelung’s Rule
(up to Ca)
1. Lower for (n + l)
2. When n + l same
lower n
3p 4s
4p 3d
42
Many-Electron Atoms
Penetration and Screeneing
2s and 2p penetrate 1s
2s penetrate more than 2p
E(2s) < E(2p)
but maxima r(2s) > r(2p)
1s
2p 2s
43
Relative Energies of s, p, d Orbitals
E(3s) < E(3p) < E(3d)
r(3s) > r(3p) > r(3d)
44
Slater’s Orbitals
Orbitals for many-electron atoms - approximate
• orbitals (wave functions) of hydrogen type
• angular part: same as for H
• radial part:
R (r) = N r n*−1 exp(− Z* r/n*)
Z* = A charge acting on an electron
= Nucleus charge (Z+) – charge of other electrons
n* = effective quant. number (for K, L, M = n)
Ei = − N (Z*i /ni) N = 1313 kJ mol −1
45
Efective Nucleus Charge
Z* = Z − σ
σ = screening constant, sum for all electrons
(1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p)(5d)(5f)...
Slater’s rules
e on the right does not screen, no contribution to σ
Within a group screens 0.35 (1s only 0.30)
n − 1 (s,p) screens 0.85
n − 2 and lower screens 1.00
If an electron is in d or f, all on the left screens 1.00
46
Efective Nucleus Charge
Z* = Efective Nucleus Charge
Z* = Z − σ
A charge acting on an electron
= Nucleus charge (Z+) – charge of other electrons
K (1s)2(2s,2p)8(3s,3p)8(3d)1
σ(3d) = 0 x (0.35) + 8 x 1.00 + 10 x 1.00 = 18
Z* = 19 − 18 = 1
K (1s)2(2s,2p)8(3s,3p)8 (4s)1
σ(4s) = 0 x (0.35) + 8 x 0.85 + 10 x 1.00 = 16.8
Z* = 19 − 16.8 = 2.2
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0
2
4
6
8
10
12
14
16
H He Li Be Be C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
Efective Nucleus Charge
Efective charge
Z* He (1s)2
σ(1s) = 1 x (0.30) = 0.30
Z* = 2 − 0.30 = 1.70
48
Efective charge
Z*
1s electrons are not screened
Other
electrons
are
screened
49
Radius of maximum
electron density
r(2s) > r(2p)
r(3s) ~ r(3p)
50
Energies of 2s and 2p Orbitals
Closer for light elements
51
Electron Configurations of Ground State
Atoms
Aufbau Principle:
Electron levels are filled by electrons
in the order of increasing energy, to
maintain the lowest atom energy
Pauli Principle:
Two electrons cannot have all 4
quantum numbers the same
Hund’s Rule:
In degenerate orbitals, the state with
maximum unpaired electrons is the
most stable
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Electron Configurations of C
54
Electron Configurations of Valence Shell
(Ne)
55
Orbital
energy
Placing electrons in
orbitals can change
order of energy
levels
Starting at Sc,
3d orbitals have
lower energy than 4s
56
4s
57
58
Electron Configurations of Valence Shell
(Ar)
59
Ionisation Energies