Download Topic II. Chapter 9

Topic II Quantum chemistry
Chapter 9
Atomic structure and spectra
Optical spectrometer for emission spectra
q : the diffraction angle.
d : is the slit spacing of
the diffraction grating.
n : an integer number.
l : the wavelength of the
diffracted light.
Emission spectrum of H atom
EM radiation of discrete frequencies emitted by the energetically excited
atoms as they discard energy and return to the lower energy states.
Balmer series (n1 = 2)
in visible light
Lyman series (n1 = 1)
in ultraviolet region
Paschen series (n1 = 3)
in infrared region
Rydberg formula
 1
 1
1 
~    RH  2  2 ,
c l
 n1 n2 
n1  1, 2, 3, 4 , 5,   
n2  n1  1, n1  2, n1  3,  
 : Frequency (Hz)
l : Wavelength (cm)
~: Wavenumber (cm-1)
RH = 109677 cm-1
Rydberg constant for H atom
1 cm-1 = 3 × 1010 s-1
Hydrogenic atoms
• Hydrogenic atom (H) or ions
( He+, Li+2,..) : single electron
• In this kind of atom or ions,
there are only two particles,
one nucleus of mass mn with a
positive charge of Ze and one
electron of mass me with
charge -e.
• The Coulombic potential
energy between the nucleus
and the electron
 Ze 2
V (r ) 
40 r
• The Schrodinger’s equation of
the hydrogenic atom
2 2 


  (r )  V (r )(r )  E(r )
2
μ
mn me
mn  me
The reduced mass
• The wave function is subject to
the boundary conditions:
As r  ,
 (r ) and  (r )  0
• The Schrodinger’s equation of
the hydrogenic atom can be
solved exactly.
Separation of variables in spherical polar coordinates
In spherical polar coordinates
Normalization


0
2 2 
1
  2
 2 2
r
r r r
1 2
1 

2
 

sin
q
sin 2 q 2 sin q q
q

2
0
0
dr  dq d r 2 sin θ|ψ(r,θ,φ)|2  1
(r , q, )  R(r )Y (q, )
2
R(r) : Radial wavefunction
Y(q,) : Angular wavefunction
 2  d 2 R(r ) 2 dR(r )  l (l  1) 2
 
 

R(r )  V (r ) R(r )  ER(r )
2
2
2  dr
r dr 
2r

2Y (q, )  l (l  1)Y (q, )
Boundary conditions
Y (q, )  Y (q,   2)

0
R(r ) and R(r )  0,
as r  .
Normalization for R(r)
and Y(q,)


0
dr r 2 | R(r ) |2  1
2
dq sin θ  d | Y (q, ) |2  1
0
Wave function of 1s orbital (l = 0)
 n,l ,m (r ,q ,  )  Rn,l (r )Yl ,m (q ,  )
l
As  approaches to the electron
mass me, a ≈ a0 the Bohr radius.
l
The radial wavefunction depends on quantum
number n and l.
The angular wavefunction depends on
quantum number l and ml.
E1s
Wavefunction for 1s orbital of H atom (Z=1)
Z
R1s (r )  2
a

3/ 2



 exp   Zr 

 a 



1
Y0,0 (q, ) 
41/ 2
1 Z
| 1 s ( r ) | 2  
  a
40 2
a
e 2
3



 exp   2Zr 

 a 



40  2
a0 
 52.9177pm
2
me e
Z 2 2
Z 2e 4


2
2a
32 2  02  2
Wave function of 2S orbital (l = 0)
1  Z 
R2 s (r ) 
2  a 
1
Y0, 0 (q, ) 
41/ 2
1  Z
2
|  2 s (r ) | 
8  a
3/ 2
 Zr
 Zr 

1

exp


 2a
 2a 





2
E2 s
Z 
  
2
 2  2a
e 4
Z
  
2 2 2
 2  32  0 
2
3
  Zr  2


 1   exp   Zr 
  2a 
 a 



2
Radial wavefunctions: Rn,l (r ) .
Properties of s orbitals
• Spherical symmetry (independent of angle)
• Wave function at r = 0 is not zero.
• The radial wave function has n-1 nodes.
Radial distribution function of s orbital
Probability density is defined as the
probability for finding electron within
per unit volume.
probabilit y density  | (r ) |2
Radial distribution function P(r) is the
probability density for finding electron
at a distance r from the nucleus within
a spherical shell per unit thickness.
P(r )  4r 2 | (r ) |2
For 1s orbital
Electron densities of 1s and 2s orbitals
 2r 
4r 2

P ( r )  3 exp  


a
 a 
Radial wave function of 2p orbital (l = 1)
1  Z
R2 p (r ) 
24  a
Y1, 0 (q, ) 




3/ 2
 Zr
Zr
exp  
 2a
a





3
cos q
4
1  Z
2
|  2 p (r ) | 
32  a
5



 r cos q2 exp   Zr 

 a 



2
E2 p
2
Z

 
  
2
2
2

a
 
4
Z

e
 
  
2 2 2
 2  32  0 
2
Angular wave functions
Wave functions of 2p orbitals (l=1)
 2,1,0 (r , q, )  R2,1 (r )Y1,0 (q, )
1/ 2
 Z 

 
3 
 24a0 
 Zr   3 
Zr
   
exp  
a0
 2a0   4 
1/ 2
cos q
1/ 2
 Z 
 Zr 

 

 
r
cos
q
exp
5 
 32a0 
 2a0 
 2 p   2,1,0
 2,1, 1 (r , q, )  R2,1 (r )Y1, 1 (q, )
1/ 2
 Z3 

 
3 
24
a
0 

z
 Zr   3 
Zr
   
exp  
a0
2
a
 8 
0 

1/ 2
sin qe i
 2p
x
 2p
y
1/ 2
 Z5 
 Zr 
 i




 
r
exp

sin
q
e
5 
 2a 
0 
 64a0 

1
 2,1,1  2,1,1 

2
i
 2,1,1  2,1,1 

2
Properties of p orbitals
• The three orbitals are denoted as px, py and pz.
• All p orbitals have a symmetric double–lobed
shape separated by a node plane.
• Magnitude of angular momentum: L  2
• Angular momentum along z axis:
  for ml  1

L z   0 for ml  0
  for m  1
l

Wave functions of d orbitals (l=2)
 3, 2,m (r ,q ,  )  R3, 2 (r )Y2,m (q ,  )
l
l
 d  xyf (r )
xy
 d  yzf (r )
yz
 d  zxf (r )
zx




1 2
 d 2 2  x  y 2 f (r )
x y
2
3 2 2
d 2 
3z  r f (r )
z
2
• Magnitude of angular momentum:
•Angular momentum along z axis:
L z  2, , 0,  ,  2
L  6
Energy levels of hydrogenic atoms
Z2 ~
En   2 hcRN ,
n
e 4
~
hcRN 
32 2  02  2
n  1, 2 , 3,   
Rydberg constant
me e 4
 ~
~
R 
RN 
me
8c 02 h3
n: Principle quantum number ~ ~
For H atom, Z=1 and mn=mp, RN  R .
Observation of electron spin
Stern-Gerlach experiment (1921)
The magnet is the source of an
inhomogeneous field.
A beam of silver atoms is shot
through the magnetic field.
In classical physics, the result is expected
to be a continuous spectrum, since the
orientations of the electron spin can take
all angles.
In experiment, the observed outcome
splits into two beams.
Electron spin
• Based on the theory of relativity, W. Pauli was the first to suggest a fourth
quantum number assigned to the electron to explain the Stern-Gerlach
experiment.
• In 1925, Goudsmit and Uhlenbeck proposed that the electron must have an
intrinsic angular momentum.
• The relativistic quantum theory proposed by Dirac in 1928 showed that the
intrinsic spin of the electron required a fourth quantum number as a
consequence of the theory of relativity.
• An intrinsic angular momentum that every electrons possess for all time is
described by spin quantum number s =1/2, with s fixed at this single
value.
• Electron spin is a purely quantum mechanical phenomenon and there is no
counterpart in classical mechanics.
• The spin of an electron can be in one of two states, which are distinguished
by the spin magnetic quantum number ms, with ms=1/2 for spin up (↑)
and ms = -1/2 for spin down (↓).
A classical representation for the two
allowed spin states of an electron
Spin angular momentum
S  s( s  1)
2
2
The magnitude of the angular momentum in each case
is 3  2 , but the direction of each spin is either spin up for a
electrons or spin down for b electrons.
Other particles in nature also have intrinsic spin
Proton and neutron: s = 1/2 (Fermions: Particles with a
half integer spin)
Photon: s = 1 (Bosons: Particles with an integer spin)
Quantum number
• For atoms in 3-dimensonal space, the electronic states should be specified
by three quantum numbers.
• The three quantum numbers come from the boundary conditions on
wavefunctions.
a. The wavefunctions must decay to zero as they extend to infinity.
b. The wavefunctions must match as we encircle the equator.
c. The wavefunctions must match as we encircle the poles.
Three quantum numbers: n, l, ml
n : Principle quantum number
n =1, 2, 3,∙∙∙
l : Orbital quantum number
l = 0,1,2, ∙∙∙, n-1.
ml : Magnetic quantum number
ml = l, l-1, l-2, ∙∙∙, -l
For the hydrogenic atoms , the energy levels depend only on the principle quantum
number n.
Shell structures of atoms
For a given n, there are n
allowed l values.
For a given l, there are
2l+1 allowed ml values.
n = 1, 2, 3, 4, ∙∙∙
K, L, M, N, ∙∙∙
l = 0, 1, 2, 3,∙∙∙
s, p, d, f,∙∙∙
Spectral transition and selection rules
As an electron makes a transition
from state i to the state j, the
wave function of the electron
changes from i(r) to j(r).
Not all transitions between all available orbitals of H atom are
possible, but only those that conserve angular momentum.
Since a photon has angular momentum of one unit, the selection
rules for one photon process, in which one photon is absorbed
or emitted, are
l  1
ml  0,  1
Allowed transitions are experimentally observable.
Forbidden transitions are not observable or
weakly observable in experiments.
Transitions between the energy levels of H atom
• Allowed transitions
2p → 1s (l = 1)
3p → 1s (l = 1)
3d → 2p (l = 1)
• Forbidden transitions
2s → 1s (l = 0)
3d → 1s (l = 2)
Emission lines and ionization energy
• As an electron jumps from an
energy level with quantum number
n2 to a lower energy level with
quantum number n1, the energy loss
of the electron is
E  En2  En1
~
~
 hcRH hcRH
  2  2
n1
 n2




• The loss energy of the electron is
carried away by a radiated photon of
wavenumber ~ so that E  hc~ .
E ~  1
1 
~

 RH  2  2 
hc
 n1 n2 
Ionization energy I is the minimum
energy to remove an electron
completely from an atom.
An electron corresponding to
completely remove from an atom is
the state with n = ∞.
For H atom, the ionization energy I
is the energy required to raise the
electron from the ground state with
n = 1 to the state with n = ∞.
~
I  hcRH  13.6 eV
Observation of magnetic quantum number: Zeeman effect
The splitting of spectral lines caused by an external
magnetic field is called the Zeeman effect.
If the magnetic field is inhomogeneous,
the atom will experience a force as
passing through the magnetic field.
Many-electron atoms
Coulomb interactions in an atom with N electrons
 

 Ze
e
V (r1 , r2 ,..., rN )  
 
i 1 40 ri
i  j ,i  j 40 rij
N
2
Nucleus-electron
interactions
2
2
 



n
ˆ
H  
 V (r1 , r2 ,..., rN )
i 1 2me
N
K.E. of electrons
N
2
Electron-electron
interactions
ri : Distance between the nucleus
and the i-th electron
rij: Distance between the i-th and
j-th electrons
The mass of nucleus is much heavy, so the motions of electrons are relative to the nucleus.
No exact solution for the Schrodinger’s equation of atoms with electrons more than one.
The orbital approximation
• Neglect the electron-electron
interaction first. So, the electrons in
an atom are independent of one
another.
• The wavefunction for the noninteracting electrons is a product of
the wavefunctions of each individual
electron in a hydrogenic atom. This
is the orbital approximation.
• The orbitals of each individual
electron can be thought as the
hydrogenic orbitals, but with a
modified nuclear charge Zeff due to
the presence of all other electrons in
the atom.
Wavefunction of many electrons
 (1,2,, N )   (1) (2)  ( N )
i: hydrogenic wavefunction of
the i-th electron
He atom in the ground state
1s 2 (1,2)  1s (1)1s (2)
3/ 2
1  Z eff

1s (1) 
  a0



1  Z eff

1s (2) 
  a0



3/ 2
 Z eff r1 

exp  
 a0 
 Z eff r2 

exp  
a0 

Pauli exclusion principle
• In the ground state of He atom, one
electron is in the spin-up sate and
the other electron is in the spindown state. The two electrons are
said to be in a pair , denoted as  ,
and have net spin angular
momentum zero.
• Each electron state is specified
with four quantum numbers: (n, l,
ml and ms).
•
•
•
Pauli exclusion principle for electrons
(identical fermions):
No more than two electrons may
occupy any given orbital, and if two
electrons do occupy one orbital, then
their spins must be paired (), with
one spin up and one spin down.
Thus, no two electronic states have the
same quantum number.
Pauli principle:
When the labels of two identical
fermions are exchanged, the total
wavefunction should change sign.
When the labels of two identical
bosons are exchanged, the sign of the
total wavefunction is the same.
(1, 2)  (2, 1)
+: Boson
- : Fermion
Singlet and triplet states of two electrons
Two spin wavefunctions for a single electron:
• a wavefunction for electron in the spin-up state
• b wavefunction for electron in the spin-down state
s1  12 , s2  12
Four spin states for two electrons:
The four spin states are the singlet state (S = 0) and
the triplet state (S = 1).
S: total spin angular momentum
  
S  s1  s2
S  0 or 1
• Singlet state (S = 0)
- (1,2)  21/ 2 a(1)b(2)  b(1)a(2)
• The wavefunction changes sign, as the
labels of the two electrons are
exchanged.
• Triplet state (S = 1)
a(1)a(2)
  (1,2)  21/ 2 a(1)b(2)  b(1)a(2) 
b(1)b(2)
• The wavefunction is the same sign, as
the labels of the two electrons are
exchanged,
Electronic wavefunction of He atom
To be satisfied with the Pauli exclusion principle, the total wavefunction of He
atom should be antisymmetric, as exchanging the labels of the two electrons.
• In the ground state,
the two electrons are both in the
1s state, and their spins are in the
singlet state in pair.
The total wavefunction can be
expressed as a Slater determinant.
(1,2)  1s (1)1s (2) (1,2)
1 1s (1)a(1) 1s (2)a(2)
(1,2) 
2 1s (1)b(1) 1s (2)b(2)
 1s (1)1s (2)  (1,2)
Shell structures of many-electron atoms
• Electronic ground state of
H atom: 1s1 (n=1, l=0, ml=0,
ms=1/2)
• Electronic ground state of
He atom: 1s2
(n=1, l=0, ml=0, ms=1/2;
n=1, l=0, ml=0, ms=-1/2)
• There are only two
electrons in K shell (n=1)
and the two electrons form
a closed K shell.
• Li atom: three electrons
Two in K shell (n=1) and
one in L shell (n=2).
Electronic ground state
of Li atom:
1s2, 2s1 or 1s2 2p1 ?
Does the third electron in
Li go to the 2s orbital or
the 2p orbital?
Penetration and shielding
In many-electron atoms, 2s and 2p orbitals (and all orbitals
with the same principle quantum number) are not degenerate
because of the shielding and penetration effects.
In electrostatics, for an electron at distance r0 from
the nucleus, the Coulomb interactions only come
from those electrons with distance r < r0.
The shielding constant is different for s and p electrons,
because they have different radial distributions.
The effect of the negative charge with r <
r0 is to lower the charge of the nucleus to
an effective charge Zeff e.
Zeff =Z –  : The effective atomic number
 : The shielding constant due to the
electrons inside the sphere of radius r0
• Comparison between s orbital and p orbital in a shell
• The wavefunction of s orbital has spherical symmetry and is not zero at r =0.
• The s electron is more close to the nucleus than the p electron of the same
shell, because the closeness of the innermost peak of the s orbital to the
nucleus.
• An s electron has a greater penetration through inner shell and experience
less shielding than a p electron, so the shielding constant of a s electron is
smaller than that of a p electron.
• For C atom (Z=6),  ≈ 2.78 for 2s orbital but  ≈ 2.86 for 2p orbital.
An s electron has lower energy than a p electron of the same shell.
The order in energy of the orbitals is s < p < d < f .
Electronic ground state of Li atom: 1s2, 2s1 or [He] 2S1
Core electrons: the electrons form a close shell
Valance electrons: the electrons in the ourmost shell of an atom
Only the valance electrons are responsible for the chemical bond.
Building-up principle
• Rule I: The order of occupation of orbitals
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 5d,… (The periodic table)
The order of occupation is approximately the order of energies of the
individual orbital.
•
Rule II: According to Pauli exclusion principle, each
orbital may accommodate up to two electrons, for spin
up and spin down.
• Examples for electronic configuration of ground state
Be atom: 1s2, 2s2
B atom: 1s2, 2s2, 2p1
C atom: 1s2, 2s2, 2px2 or 1s2, 2s2, 2px1, 2py1 ?
The two 2p electrons are in the same p orbital in a spin pair or they are in
different p orbitals and both spin up?
If the two electrons in the same p orbital, they have the same wavefunction
and the repulsive energy between them increases.
If the two electrons in different p orbitals, they are separated apart due to
different wavefunctions so their repulsive energy is relatively lower.
• Rule III: Electrons occupy different orbitals of a given
subshell before doubly occupying any one of them.
C (Z=6) atom: [He], 2s2, 2px1, 2py1
N (Z=7) atom: [He], 2s2, 2px1, 2py1, 2pz1
O (Z=8) atom: [He], 2s2, 2px2, 2py1, 2pz1
The two electrons in the same p orbital are in a spin pair (  ) .
The single electrons in py and pz orbitals are both spin-up or
they are one spin-up and one spin-down?
2py1(↑), 2pz1(↑) or 2py1(↑), 2pz1(↓) ?
Quantum mechanics requires that electrons in different orbitals
with parallel spins should stay well apart in space, so that the two
electrons have a lower repulsive energy.
• Rule IV (Hund’s rule): In the ground state, an atom adopts a
configuration with the great number of unpaired electrons.
F (Z=9) atom: [He], 2s2, 2px2, 2py2, 2pz1
Ne (Z=10) atom: [He], 2s2, 2px2, 2py2, 2pz2 (Closed L shell for n=2)
Na (Z=11) atom: [Ne], 3s1
Ar (Z=18) atom: [Ne], 3s2, 3px2, 3py2, 3pz2 (Closed M shell for n=3)
Occupation of d orbitals
K (Z=19) atom: [Ar], 4s1
Ca (Z=20) atom: [Ar], 4s2
Sc (Z=21) atom: [Ar], 4s2, 3d1 or 4s1, 3d2 ?
1. The energies of 3d orbitals are always lower than the energy of 4s orbital.
2. The most probable distance of a 3d electron from the nucleus is less than
that of a 4s electron. So, two d electrons repel each other more strongly
than two 4s electrons.
Cr (Z=24) atom: [Ar], 3d5, 4s1
Cu (Z=29) atom: [Ar], 3d10, 4s1
Configurations of cations and anions
• Cations: by removing electrons from a neutral atom or ion
Fe (Z=26) atom : [Ar] 3d6, 4s2
The 3d orbitals in Fe are lower in energy than the 4s orbital.
Fe+3 ion: [Ar] 3d5
• Anions: by adding electrons to a neutral atoms
O atom: [He], 2s2, 2px2, 2py1, 2pz1
O-2 ion: [He], 2s2, 2px2, 2py2, 2pz2
Periodic trends in atomic properties
Two properties of atoms
I.Atomic radius
II.Ionization energy
Both are correlated with
the effective nucleus
charge of an atom
Variation of atomic radius through the periodic table
I. Atomic radius
• Determining the number of
chemical bonds that the atom
can form
• Determining the chemical
properties of the element
Two general rules for determining atomic radius from the periodic table
I. Atomic radii decrease from left to right across a period
II. Atomic radii increase down in each group
Exception:
The second rule is applied for the first five periods, but not for period six.
First ionization energies of elements
The minimum energy necessary to remove an electron from the atom
• In a period, the increase in ionization energy is due to the higher
effective nucleus charges.
• The drop between Be and B is because of the 2p electron is less strongly
bound than the 2s electron.
• The drop between N and O is due to the electron-electron repulsion
between the spin paired electrons in O atom.
Exercises
• 9A.2(b), 9A.3(b), 9A.6(a), 9A.7(a), 9A.9(a)
• 9B.1(a)
• 9C.2(a), 9C.3(a), 9C.4(a)