5
QUANTUM MECHANICS
AND ATOMIC STRUCTURE
CHAPTER
5.1 The Hydrogen Atom
5.2 Shell Model for Many-Electron Atoms
5.3 Aufbau Principle and Electron Configurations
5.4 Shells and the Periodic Table:
Photoelectron Spectroscopy
5.5 Periodic Properties and Electronic Structure
General Chemistry I
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Colors of
Fireworks
from atomic emission
red from Sr
orange from Ca
yellow from Na
green from Ba
blue from Cu
General Chemistry I
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5.1 THE HYDROGEN ATOM
- The hydrogen atom is the simplest example of a one-electron atom
or ion (i.e. H, He+, Li2+, …).
- For solution of the Schrödinger equation,
Cartesian coordinates, x, y, z
spherical coordinates, r, q, f
to express angular orientation.
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Energy Levels
 For a hydrogen atom, V(r) = Coulomb potential energy
 Solutions of the Schrödinger equation
E = En =
_
Z 2e4me
8 0 n h
2 2 2
n = 1, 2, 3,.....
1 rydberg = 2.18×10-18 J
 Principal quantum number n indexes the individual
energy levels.
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Coordinate Systems
V(x, y, z) – can be expressed in various coordinate systems.
Spherical polar coordinates (r, θ, Ф) vs.
Cartesian coordinates (x, y, z)
 h 2   2

2
2 



V
(
x
,
y
,
z
)
 2  2
 ( x, y, z )  E ( x, y, z )
2
2 
8

m

x

y

z




For Coulomb potential, V, convenient to express
in SPC (V = -e2/r); Cartesian, V = -e2/[x2 + y2 +z2]1/2
- Cartesian Coordinates
 h 2   2

2
2 
e2
 2  2  2  2 
 ( x, y, z )  E ( x, y, z )
2
2
2
8

m

x

y

z
(
x

y

z
)




- Spherical Polar Coordinates (SPC)
 h 2  1   2  
1
 

 2  2 r
 2
 sin q
q
 8 m  r r  r  r sin q q 
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 2  e 2 


 ( x, y, z )  E ( x, y, z )
 2 2
2 
 r sin q f  r 
5
Quantization of the Angular Momentum
any integral value from 0 to n-1
value of l
0
1
2
3
orbital type
s
p
d
f
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orbitals
6
For every value of n, n2 sets of quantum numbers
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Energy level diagram for
the H atom
These sets of state are said to be degenerate.
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Boundary Conditions
Boundary Conditions yield quantum numbers!
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Wave Functions
radial part
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angular part: spherical harmonics
10
- Spherical Polar Coordinates and the Wave Function
 h 2  1   2  
1
 

 2  2 r
 2
 sin q
q
 8 m  r r  r  r sin q q 
1
 2  e 2 


 ( x, y, z )  E ( x, y, z )
 2 2
2 
 r sin q f  r 
  h 2  1   2    e 2

 1
 

 E  ( x, y, z )   2
 2  2 r
 
 sin q
q
 r sin q q 
 8 m  r r  r   r

1
2 

 ( x, y , z )  0
 2 2
2
 r sin q f 
  h 2    2    e 2 r 2
 1  

2
r


Er

(
x
,
y
,
z
)

sin
q
 2  


 sin q q 
r
q


 8 m  r  r  

1 2 

 ( x, y , z )  0
 2
2
 sin q f 
Depends only on r
Depends only on q, f
- Schrödinger wave function
→ factored into radial (R) and angular (Y) functions
 (r,q ,f )  R(r )Y (q ,f )
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Orbitals
EXAMPLE 5.1
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Specification of the Wave Functions: Orbitals

Schrödinger’s Wave Function
i.e.) n = 3 (only one orbit in the Bohr- de Broglie model)
32 (nine) different ways the electron can vibrate to form
a standing wave → 9 degenerate quantum states

Orbitals
replacement of Bohr’s orbit for 3D Schrödinger’s wave function
- Radial part, Rnℓ(r)
Laguerre polynomial (rn-1) x e-r/(na0 )
- Angular part, Yℓm(θ,Ф)
Legendre function (sin θ & cos θ) x eimФ
* a0 ← Bohr’s radius
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No radial node
n=1
Spherical
Symmetry:
no angular
nodes
n = 2 1 radial node
n = 3 2 radial nodes
n = 2 No radial node
1 angular
node
n = 3 1 radial node
2 angular
nodes
n = 3 No radial node
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Sizes and Shapes of Orbitals
 Graphical representations of the orbitals
1) Slicing up 3D space into various 2D and 1D regions and
examining the value of wave function at each point.
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2) Looking only at the radial behavior (“vertical slice”)
R100(r)
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s orbitals
 s orbital:
constant Y
All s orbitals are
spherically symmetric.
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- 1s (n = 1, ℓ = 0, m = 0)  R10(r) and Y00(θ,Ф)
a function of r only
* spherically symmetrical
* exponentially decaying
* no nodes
- 2s (n = 2, ℓ = 0, m = 0)  R20(r) and Y00(θ,Ф)
zero at r = 2a0 = 1.06Å
nodal sphere or radial node
[r<2ao: Ψ>0, positive] [r>2ao: Ψ<0, negative]
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2s
3s
isosurface
1s
A surface of points
with the same value
of wave functions
wave function
plot
radial
probability
density
(or function)
plot
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Radial Distribution Functions: RDFs
Prob(r) = r2[R(r)]2
2

f q
0
 r 2 sin q dq df
2
0
 4 r 2   4 r 2 R (r )Y (q , f )
2
 4 r 2 R (r ) Y (q ,f )
2
 4 r R (r ) 1/ 2 
2
2
2
2
 r 2 R (r )
2
2
┃Ψ┃2 x 4πr2 dr
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Boundary Surfaces
- No clear boundary of an atom
- Defining the size of atom as the extent
of a “balloon skin” inside which 90%
of the probability density of the electron
is contained.
1s: 1.41 Å, 2s: 4.83 Å, 3s: 10.29 Å
An ns orbital has n-1 radial nodes.
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p Orbitals

2p0 orbital : R21 Y10 (n = 2, ℓ =1, m = 0)
2
· Ф = 0 → cylindrical symmetry about the z-axis
· R21(r) → r/a0 no radial nodes except at the origin
· cos θ → angular node at θ = 90o, x-y nodal plane
· r cos θ → z-axis 2p0 → labeled as 2pz
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2p+1 and 2p-1 orbitals: R21Y11, R21Y1-1 (n = 2, ℓ = 1, m = ±1)
· Y11(θ,Ф) → e±iФ = cosФ ± i sinФ ← Euler’s formula was used
· taking linear combinations of these gave two real orbitals,
called 2px and 2py
- px and py differ from pz only in the angular
factors (orientations).
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Wave function and RDF plots for 2p and 3p orbitals
2p
3p
r2Rnl2
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d Orbitals
 d-orbitals
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Orbital Shapes and Sizes: General Notes
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Stern-Gerlach Experiment: experimental evidence
for the existence of electron spin
Stern and Gerlach (1926): Ground state Na atoms were passed through a
strong inhomogeneous magnetic field.
· All spin magnetisms in inner orbitals
→ cancelled, except an electron in 3s
orbital.
· Y (θ,Ф)(3s) → no rotation
· The intrinsic magnetism of the 3s electron
→ only possible to respond to the external
field.
· 3s electrons → line up with their north
poles either along or against the main
north pole of the magnet.
The lack of a “straight-through” beam
with the magnet → clear evidence of twovalued electron’s magnetism.
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Electron Spin
 ms, spin magnetic quantum number
- An electron has two spin states,
as ↑(up) and ↓(down), or a and b.
- the values of ms, only +1/2 and -1/2
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5.2 SHELL MODEL FOR MANY-ELECTRON
ATOMS
- In many-electron atoms, Coulomb potential energy equals the sum of
nucleus-electron attractions and electron-electron repulsions.
- No exact solutions of Schrödinger equation
- In a helium atom,
r1 = the distance of
electron 1 from
the nucleus
General Chemistry I
r2 = the distance of
electron 2 from
the nucleus
r12 = the distance
between the
two electrons
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r1≡ (r1,θ1,Ф1)
interelectron distance
r12 ≡┃r1 – r2┃
r2≡ (r2,θ2,Ф2)
Interparticle distance in He
Repulsion between electrons:
This new cross-term makes the
Schrodinger equation impossible
to solve exactly and causes the
Bohr model to fail for He and
other atoms
Screening of one electron
by another
General Chemistry I
(r1,q1,f1, r2 ,q2 ,f2 )  (r1, r2 )
34
Hartree Orbitals
 Self-consistent field (SCF) orbital approximation
method by Hartree, Fock, Slater, and others
- all electrons were treated as independent, meaning that we
neglect the repulsion term.
- Building up an effective nuclear charge Zeffe
(r1,q1,f1, r2 ,q2 ,f2 )  (r1, r2 )   a (r1 ) b (r2 )
i.e.) for He,
Zeff(He) = 1.6875 < 2
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 Three simplifying assumptions by Hartree
The wave function becomes a product of these one-electron orbitals.
3. The effective field is spherically symmetric; that is, it has no angular
dependence.
The equations for the unknown effective field and the unknown
one-electron orbitals must be solved by iteration until a selfconsistent solution is obtained.
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E.g. for the ground state of He,
 1  a0  3/ 2  Zr / a  1  a0  3/ 2  Zr / a 
 (r1 , r2 )   a ( r1 ) b ( r2 )  
e 1 0 
e 2 0




 Z
   Z 




 1  a  3/ 2
 1  a

Z
r
/
a
eff 1
0


 0  e
 0
   Z eff 
   Z eff


Z eff 3  Zeff r1 / a0
 Z eff r2 / a0

e
e
 a 03





3/ 2




1s2

1s orbital
General Chemistry I
e
 Z eff r2 / a0
occupancy
1s orbital
37
The ground state of Ar, Z = 18
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sum of the radial probability density functions
of all the occupied orbitals.
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Shielding Effects
- Energy-level diagrams for many-electron atoms
1) The degeneracy of the p, d, and f orbitals is removed, due to
the difference of Zeff from the Coulomb field.
2) Energy values are distinctly shifted from the values of corresponding
H orbitals due to the strong attraction by nuclei with Z > 1.
- Each electron attracted by the nucleus, and repelled by the
other electrons.
→ shielded from the full nuclear attraction by the other electrons.
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- effective nuclear charge, Zeff e < Ze
- Hartree orbital energy
(rydbergs)
E.g. For Ar (Z =18), Zeff(1) ~ 16, Zeff(2) ~ 8, Zeff(3) ~ 2.5
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Energy level diagram for many-electron atoms
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 Penetration:
s-electron – very close to the nucleus
highly penetrates through the inner shell.
p-electron – penetrates less than an s-orbital
effectively shielded from the nucleus
In a many-electron atom, because of the effects of
penetration and shielding, the order of energies
of orbitals in a given shell is s < p < d < f.
Order of orbital shielding (for fixed value of n):
Zeff(s) > Zeff(p) > Zeff(d) > Zeff(f)
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5.3 AUFBAU PRINCIPLE AND
ELECTRON CONFIGURATIONS
The Building-up (Aufbau) Principle is a set of rules that
allows us to construct ground state electron configurations
of the elements.
1.
2.
3.
Assume electrons ‘occupy’ orbitals in such a way as
to minimize the total energy (lowest energy first).
Assume a maximum of two electrons can ‘occupy’ an
orbital and these must have opposite spins. (Pauli’s
Exclusion Principle: no two electrons in an atom can
have the same set of quantum numbers).
Assume electrons occupy unoccupied degenerate
subshell orbitals first, and with parallel spins (Hund’s
rule).
In building-up electron configurations, in order of energy,
subshell energy overlap must be taken into account (see
later).
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Electron configurations of first ten elements (H to Ne)
Electron configuration of an atom is a list of all its
occupied orbitals, with the numbers of electrons that
each one contains.
Standard notation
Box and arrow notation
(showing electron spins)
 H, 1s1
 He, 1s2
 Li, 1s22s1 or [He]2s1
Valence electron
Core electrons
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 Be, 1s22s2 or [He]2s2
 B, 1s22s22p1 or [He]2s22p1
 C, 1s22s22p2 or [He]2s22p2
→
1s22s22px12py1
General Chemistry I
parallel spins
by Hund’s rule
46
- Elements in Period 2 (from Li to Ne), the valence shell with n = 2.
- p-block elements: N to Ne, filling of p orbitals
s-block elements: H to Be, filling of s orbitals
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 Magnetic properties as a test of electronic configurations
- paramagnetic: a substance attracted into a magnetic field
with one or more unpaired electrons
- diamagnetic: a substance pushed out of a magnetic field
with all electrons paired
Examples:
paramagnetic: H, Li, B, C
diamagnetic: He, Be
C
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 n = 3: Na, [He]2s22p63s1 or [Ne]3s1 to Ar, [Ne]3s23p6
 n = 4: from Sc (scandium, Z = 21) to Zn (zinc, Z = 30)
the next 10 electrons enter the 3d-orbitals (d-block elements)
 The (n+l) rule
Order of filling subshells in neutral atoms is
determined by filling those with the lowest
values of (n+l) first. Subshells in a group with
the same value of (n+l) are filled in order
of increasing n, due to orbital screening.
order: 1s < 2s < 2p < 3s < 3p < 4s < 3d
< 4p < 5s < 4d < …
 n = 5: 5s-electrons followed by the 4delectrons
n = 6: Ce (cerium, [Xe]4f15d16s2)
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Alternative Pictorial Representation of Orbital
Energy Order
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Anomalous Configurations
Exceptions to the Aufbau principle
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Filling of electron subshells in relation
to the periodic table
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All atoms in a given period have the same type of core with the same n.
All atoms in a given group have analogous valence electron configurations
that differ only in the value of n.
Group 18/VIII
Group 1/IA
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
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5.4 SHELLS AND THE PERIODIC TABLE:
PHOTOELECTRON SPECTROSCOPY
 A shell is defined precisely as a set of orbitals that have the same
principal quantum number.
 The photoelectron spectroscopy (PES) principle
determining the energy level of each orbital by measuring the
ionization energy required to remove each electron from the atom
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Basic
design
of a
PE
spectrometer
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- Koopmans’ approximation,
- with the frozen orbital
approximation:
the orbital energies are the same
in the ion, despite the loss of an
electron.
E.g. for Ne with 1s2 2s2 2p6
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5.28 Photoelectron spectroscopic studies of silicon atoms excited by X-rays of wavelength
9.890 x 10-10 m show four peaks in which the electrons have speeds (all x 107 m/s)
of (1) 2.097, (2) 2.093, (3) 2.014, and (4) 1.971.
(a) Calculate the ionization energy of the electrons corresponding to each peak.
(b) Assign each peak to an orbital of the silicon atom.
1eV = 1.6022 x 10-19 J; c = 2.99792 x 108 m/s; h = 6.62607 x 10-34 J s;
me = 9.10938 x 10-31 kg
Solutions
(a) E(X-rays) = h = hc/  =
(6.62607 x 10-34 J s)(2.99792 x 108 m/s)
9.890 x 10-10 m
= 2.0085 x 10-16 J = (2.0085 x 10-16 J)
1 eV
1.6022 x 10-19 J
= 1253.6 eV
Using the PES equation, IE = h - KE = h - 1/2mev2,
1 eV
(9.10938 x 10-31 kg)(2.097 x 107 m/s)2
_
IE(1) = 1253.6 eV
1.6022 x 10-19 J
2
= 3.5 eV
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IE(2) =1253.6 eV
_ (9.10938 x 10-31 kg)(2.093 x 107 m/s)2
2
1 eV
1.6022 x 10-19 J
= 8.3 eV
IE(3) =1253.6 eV
_ (9.10938 x 10-31 kg)(2.014 x 107 m/s)2
2
1 eV
1.6022 x 10-19 J
= 100.5 eV
IE(4) =1253.6 eV
_ (9.10938 x 10-31 kg)(1.971 x 107 m/s)2
2
1 eV
1.6022 x 10-19 J
= 149.2 eV
(b) The ground-state electron configuration of silicon is 1s22s22p63s23p2. Peak 1
corresponds to removal of silicon’s 3p electrons, which are its least tightly bound
electrons.
Peaks 2, 3, and 4 correspond to removal of Si’s 3s, 2p, and 2s electrons
respectively. The 1s electron does not give a peak. It is probably at an energy
lower than –1253.6 eV and inaccessible with the X-radiation used in this
experiment.
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(Fig. 5.25)
Energies of
atomic
subshells,
determined
by PES
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5.5 PERIODIC PROPERTIES AND
ELECTRONIC STRUCTURE
 Atomic radius: defined as half the distance between the centers of
neighboring atoms
 Ionic radius: its share of the distance between neighboring
ions in an ionic solid
 General trends
- r decreases from left to right across
a period
(effective nuclear charge increases)
- r increases from top to bottom down
a group
(change in n and size of valence shell)
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- The rate of increase changes
considerably.
i.e. Li+, Na+, to K+
substantial change
to Rb+, Cs+
small change due to
filling d-orbitals
 Lanthanide contraction
filling of the 4f orbitals
(poor shielders)  radii
of 3rd row D-block elements
similar to those of 2nd row
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 Molar volumes (cm3 mol-1) of atoms in the solid phase
= size of the atoms + geometry of the bonding
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Periodic Trends in Ionization Energies
 From left to right, generally increase in IE1 due to the increase of Zeff
 From top to bottom, generally decrease in IE1 due to the increase of n
 From He to Li, a large reduction in IE1
2s e- further than 1s e-, and 2s e- sees
a net +1 charge
 From Be to B, slight reduction in IE1
fifth e- in a higher energy 2p orbital
 From N to O, slight reduction in IE1
2 e- in the same 2p orbital leading to
greater repulsion
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Electron Affinity
- The periodic trends in EA parallel those in IE1 with one unit lower shift
e.g. F to F-, large EA because of its closed-shell configuration
(kJ mol-1)
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10 Problem Sets
For Chapter 5,
6, 18, 22, 28, 32, 38, 44, 46, 48, 56
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