1
Electronic Structure of Atoms
Chemical properties of atoms (and molecules) are
given by their electronic structure.
We need to know:
• electron energy
• spatial distribution of electrons
Knowledge about electronic structure of atoms was
obtained by studies of radiation emitted by excited
atoms (from ground state to excited state by adding
energy – thermal, electrical - spark, arc)
2
Electromagnetic Radiation
c = 2.998 108 m s−1 speed of light
James C. Maxwell
(1831-1879)
Heinrich Hertz
(1857 - 1894)
3
Wavelength, frequency, wavenumber,
amplitude
ν λ = c
c = 2.998 108 m s−1
ΰ = 1/λ [cm−1]
4
Electromagnetic Radiation
Wavelength, λ [m]
380 nm 780 nm
5
Spectrum
Character of light:
• Wave (interference) Huygens, Young
• Corpuscular (linear rays, reflection) Newton
6
Spectrum
Sun spectrum: He, Fe, Mg,...
Absorption spectrum
Emission spectrum
Continuous spectrum
7
Line Spectra of Elements
Emission spectrum
Absorption spectrum
8
Emission Line Spectra of Elements
Cu
Zn
Wavelength, nm
H
He
Li
9
Quantized Energy
Planck constant h = 6.626 10−34 J s
ΔE = n h ν = n h c / λ
Max Planck
(1858 - 1947)
NP in Physics 1918
1900 Energy of radiation with wavelength λ could be
absorbed or emitted only in discrete amount = quantum
Quantum of light = photon
E1
E2
E1
E2
E2 -E1 = h ν
Ground state
Excited state
Energy
10
Black Body Radiation
Black Body = perfectly absorbs all incoming radiation,
perfectly emits all wavelengths
Atoms = oscillators
Quantized Energy E = h ν
Max Planck derived
Energy emitted at wavelength λ
is only a function of temperature
UV catastrophe
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
1
2
5
2
kT
hc
e
hc
P
λ
λ
λ
π
11
Black Body Radiation
Stefan-Boltzmann Law
Energy emitted from a unit area
per time
T
konst
=maxλ
Wien’s Law
4
TP ×= σ
12
Photoelectric Effect
photon
Cathode
alkali metal
1887 Heinrich Hertz
1898 J. J. Thomson
Observations
• electrons are emitted from the metal
surface upon irradiation by UV (alkali
metals by visible light)
• minimum ν, photons of lower energy
cannot eject electrons
• kinetic energy of photoelectrons
depends on ν, increases with higher
light energy, but not dependent on its
intensity
13
Photoelectric Effect
Below ν0 no emission
No matter how intense the light is!
Kinetic energy of
photoelectrons
kinetic energy of
photoelectrons
depends on ν,
increases with higher
light energy, but not
dependent on its
intensity
hν0 = work function
14
Photoelectric Effect
Φ = photoelectron flux
KE =
Kinetic
energy
hν0 = work function
I = UV light
Intensity
minimum ν0
Incr. with ν
Does not depend on I
Incr. with IDoes not
depend on ν
15
Photoelectric Effect
1905
Albert Einstein
(1879-1955)
NP in Physics 1921
Particle character of electromagnetic radiation
Light = photons
Photon energy E = h ν
Ejected electron energy Ekin = ½ mv2
h ν = Ei + ½ mv2
Ekin = h (ν – ν0)
ν0 = metal characteristic
h = Planck constant
Ei = hν0 = work function
16
Photoelectric Effect
h ν0
h ν0
h ν
h ν
Ekin = h (ν – ν0)
h ν = Ei + ½ mv2
Ei = hν0
work function
Photon energy
E = h ν
Ejected electron energy
Ekin = ½ mv2
17
Hydrogen Emission Spectrum
Line spectrum of light emitted by H atoms
Lines have constant wavelengths
18
Hydrogen Emission Spectrum
m → n
⎟
⎠
⎞
⎜
⎝
⎛
−= ∞ 22
111
mn
R
λ
Balmer series in visible range (1855)
19
Rydberg Equation
Experimental result from spectral mearurements
(visible, infrared, ultraviolet regions)
Rydberg constant, R∞ = 109678 cm−1
n, m = integers,
n = 2, m = 3, 4, 5, 6,.... Balmer series in visible range (1st in 1855)
Rydberg equation holds only for H spectrum
⎟
⎠
⎞
⎜
⎝
⎛
−= ∞ 22
111
mn
R
λ
20
Spectral Series
⎟
⎠
⎞
⎜
⎝
⎛
−= ∞ 22
111
mn
R
λ
n = 1, m = 2, 3,.... Lyman
n = 2, m = 3, 4,.... Balmer
n = 3, m = 4, 5,.... Paschen
n = 4, m = 5, 6,.... Bracket
n = 5, m = 6, 7,.... Pfund
21
Bohr’s Model of Atom
Fcoul Fcf
r
v
Niels Bohr
(1885 - 1962)
NP in Physics 1922
Electrons move around nucleus
in circular orbits,
equilibrium of centrifugal and
Coulombic forces
FO = FC
1913
Z
2
0
22
4 r
Ze
r
mv
πε
=
22
Bohr’s Model of Atom
E = Ekin + Epot = 1/2 m v2 − Z e2 / 4 π e0 r = − Z e2 / 8 π e0 r
Electrons moves on allowed orbits with certain definite E and r
On allowed orbits do not emit energy = stacionary states
Lowest energy state = the most stable = ground state
Higher states = excited states
Quantized change of energy state E2 − E1 = hν
Spectrum line
2
0
22
4 r
Ze
r
mv
πε
= 2
0
2
4 mv
Ze
r
πε
=
23
Bohr’s Model of Atom
Bohr’s postulates: electron angular momentum is an integer
multiple of Planck quantum (h/2π)
n = quantum number
plug in from m v2 = Z e2 / 4 π e0 r
for n = 1 and Z = 1
a0 = e0 h2 / π m e2
a0 = 0.529 Å Bohr radius of H atom
hn
h
nmvr ==
π2
Z
a
nr 02
=
nh
Ze
v
0
2
2ε
=
Orbit radius
Speed of electron
24
Bohr’s Model of Atom
E = Ekin + Epot = 1/2 m v2 − Z e2 / 4 π e0 r
E0 (= m e4 / 8 e0
2 h2) = 2.18 10 −18 J
(1 eV = 1.6 10 −19 J)
E0 = 13.6 eV
Ionisation potential
of H atom
2
2
0
n
Z
EEn −=
Quantized energy
Energy of
an electron at
level n
Energy of electron
25
Bohr’s Model of Atom
The stronger is an
electron bound to
nucleus, the lower is
its energy
(more negative)
E = 0 Energy of electron
26
Ionisation Energy
Atomic number, Z
Energy for removing a bound electron
27
Bohr’s Model of Atom
Energy difference between two levels
E2 − E1 = (− E0 Z2 / n2
2) − (− E0 Z2 / n1
2)
ΔE = h ν = h c / λ
⎟
⎠
⎞
⎜
⎝
⎛
−= 2232
0
4
11
8
1
mnch
me
ελ
Equation is identical to Rydberg’s !!!
2
2
22
0
4
2
2
0
8 n
Z
h
me
n
Z
EEn
ε
−=−=
Energy of
an electron at
level n
28
Spectral Series of H Atom
⎟
⎠
⎞
⎜
⎝
⎛
−= ∞ 22
111
mn
R
λ
n = 1, m = 2, 3,.... Lyman
n = 2, m = 3, 4,.... Balmer
n = 3, m = 4, 5,.... Paschen
29
Limitations of Bohr’s Model
• Simple and easy to understand
• Explained lines in the H spectrum
• Explained quantization of energy in atoms
• Cannot be used for multielectron atom spectra
• Only for atoms of “hydrogen-type”
(nucleus = Zn+, only one electron)
Fundamentally flawed model
Overcome by quantum-mechanic model
30
Wave-like Character of Light
diffraction, interference, refraction, polarisation
Christian Huygens
Augustin J. Fresnel
Thomas Young
James C. Maxwell
Heinrich Hertz
31
Particle-like Character of Light
Black body radiation, photoelectric effect, line spectra,
maximum wavelength of X-rays, Compton scattering
Albert Einstein
Max Planck
Wilhelm K. Roentgen
Henry Moseley
Niels Bohr
Arthur Compton
32
Particle-like Character of Light
Electromagnetic radiation = wave
E = h ν
Electromagnetic radiation = particles – photons
Compton scattering 1922
Photon’s mass mf
E = h ν = h c / λ
E = mf c2
Arthur H. Compton
(1892 - 1962)
NP in Physics 1927
c
h
mf
λ
=
33
Compton Scattering
Photons scattered on core electrons,
no change in energy
wavelength
the scattering of monochromatic Xrays
from electrons in a carbon target,
scattered x-rays with a longer
wavelength than those incident upon
the target, the shift of the wavelength
increases scattering angle
N = number of photons
Photons scattered on outer
electrons, energy
transferred, wavelength
increases
34
Dual Character of Light
λ - incident x-ray photon
wavelength
λ’ - scattered x-ray photon
wavelength, longer than incident
one
the shift of the wavelength
increases scattering angle θ
( )θλλ cos1' 2
−=−
cm
h
e
35
Wave-like Character of Electrons
Louis de Broglie
(1892 - 1987)
NP in Physics 1929
1923 de Broglie
Electron has a wavelength
Planck + Einstein
E = h ν = h v / λ E = m v2
particle
v = speed of electron
mv = p = momentum of electron
wave
Wavelength λ
mv
h
=λ
36
Scattering of Electrons on Ni Crystal
1927
C. J. Davisson
(1881-1958)
L. Germer
G. P. Thomson
(1892-1975)
NP in Physics 1937
E = e V = ½ m v2
Experimental evidence of wave
character of electrons. Particles
would scatter evenly in all directions.
37
Bragg Equation
X-rays
Electrons
de Broglie
wavelength
of electron λ
38
Electron as a Standing Wave
Electron = wave
de Broglie
Standing wave on a circle
of radius r
n λ = 2 π r
Combined equations
This is Bohr’s postulate !
mv
h
=λ
mvr
h
n =
π2
39
Heisenberg Uncertainity Principle
1927
The more precisely the position (x) is determined,
the less precisely the momentum (p = m v) is known
in this instant, and vice versa.
h = 6.626 10−34 J s
Electron in H atom in ground state
v = 2.18 106 m s−1
error 1%, Δv = 104 m s−1
Δx = 0.7 10−7 m = 70 nm
a0 = 0.053 nm
Not possible to find precisely the position of an
electron in an atom
Werner Heisenberg
(1901 - 1976)
NP in Physics 1932
2
h
≥ΔΔ px
40
Heisenberg Uncertainity Principle
The product of the uncertainty in an energy
measurement (ΔE) and the uncertainty in the time
interval of the measurement (Δt) equals h/2π or more.
h = 6.626 10−34 J s
2
h
≥ΔΔ tE
41
Heisenberg Uncertainity Principle
Energy of electrons is know very precisely from
emission spectra
Position of an electron cannot be measured precisely
Circular orbits with defined radii = nonsense
State of an electron has to be described by quantum
mechanics
a0 = 0.053 nm – the most probable radius of electron
42
Ĥ Ψ = E Ψ
Schrödinger Equation
Erwin Schrödinger
(1887 - 1961)
NP in Physics 1933
1926 Schrödinger equation = postulate
∂2 Ψ ∂2 Ψ ∂2 Ψ 8π2m
∂ x2 ∂ y2 ∂ z2 h2
+ ++ (E −V) Ψ = 0
Ĥ = Hamilton operator of total energy (E),
Kinetic and potential (V) energy
43
Schrödinger Equation
Ĥ Ψ = E Ψ
44
Schrödinger Equation
Second-order partial differential equation
Exact solution ONLY for H and one-electron systems (He+, Li2+,....)
Approximate solutions for many-electron atoms (He,...) and molecules
The solution of differential equation are pairs (E, Ψ ):
• proper wave functions (Eigenfunctions) Ψ
orbitals | Ψ |2 – space distribution of e
• proper values of electron energy in orbitals (Eigenvalues) E
To one value of E could belong several wave functions (degenerate)
Ĥ Ψ = E Ψ
45
Wavefunctions
Ψ(x,y,z) – solution of a stationary Schrödinger eq.
Only certain states of electron are allowed - Ψ(x,y,z)
Ψ is a complex function of coordinates x, y, z, has no
physical meaning, positive and negative values
| Ψ |2 – probability density of electron position
Ψ depends on integers – quantum numbers
46
Born Interpretation of Wavefunction
Ψ(x,y,z) solution of a stationary Schrödinger eq.,
(Ψ no physical meaning)
| Ψ |2 dV probability of finding electron in volume dV
at position r
(dV= dx dy dz)
Max Born
(1882 - 1970)
NP in Physics 1954
dV
47
“I think I can safely say that nobody understands Quantum Mechanics”