Download down - Display Materials Lab.

Ch 11. Quantum States for
Many-Electron Atoms and
Atomic Spectroscopy (A.S.)
• States of many - e- atoms are grouped into terms and levels
• A.S. is useful for obtaining information on the levels of atoms and
→
→
understanding the coupling of S and l
• A. spectroscopies are widely used in analytical chemistry
• Laser and excited atoms are of interest.
MS310 Quantum Physical Chemistry
11.1 Good quantum numbers, terms, levels,
and states
What about the quantum number of many-electron atom?
In H atom, n, l, ml, and ms are used.
Each operator lˆ 2 , lˆ , ŝ 2 , and ŝ commute with the hamiltonian.
z
z
→ independent of time
→ n, l, ml, and ms : Good quantum number!
However, in many-electron atoms, these numbers are not good
quantum number!
Case of Z<40, we can separate the angular momentum and spin
momentum.
→ define total orbital momentum vector L and total spin
momentum vector S
L   l i , S   s i , | L | L( L  1), | S | S ( S  1)
i
MS310 Quantum Physical Chemistry
We define the scalar ML and MS
M L   l zi , M S   szi
i
i
Next, we define L̂2 , L̂z , Ŝ 2 , and Ŝ z
Lˆ z   lˆz ,i , Lˆ2  (  lˆi ) 2 , Sˆ z   sˆ z ,i , Sˆ 2  (  sˆi ) 2
i
i
i
i
Good quantum number in many-electron
atoms : L, S, ML, MS
MS310 Quantum Physical Chemistry
Although this configuration is very useful, angular momentum and
spin momentum interact in real atom when L>0, S>0
→ spin-orbit coupling (a magnetic interaction)
If spin-orbit coupling occurs, operator L̂2 , L̂ , Ŝ 2, and Ŝ don’t
z
z
commute with hamiltonian.
2
However, the operators Ĵ and Ĵ z commute with hamiltonian.
Total angular momentum J is defined by
J  LS
In this case, the only good quantum numbers are J and MJ, the
projection of J on the z axis.
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
When no electron-electron repulsion : electron ‘configuration’
Take electron-electron repulsion : Term(group of states that has the
same L and S)
When Z>40, effect of spin-orbit coupling increase, and good
quantum numbers are J and MJ : Level(groups of 2J+1 states),
energy depends on J.
If external magnetic field applied, each state split(same J, different
MJ and energy depends on both J and MJ)
ex) Carbon atom configuration 1s22s22p2 : 3 terms, 5 levels, and 15
states
MS310 Quantum Physical Chemistry
11.2 The energy of a configuration depends on
both orbital and spin angular momentum
Consider the He, 1s12s1 configuration
2 electrons are both l=0(s orbital) : |L|=0
Magnitude of spin angular momentum vector :
| s | s( s  1)
Vector s has a 2s+1=2 orientations.
In this case, there are 2 spins can only.
parallel (α(1)α(2), β(1)β(2)) and antiparallel (α(1)β(2), β(1)α(2))
MS310 Quantum Physical Chemistry
Calculate the MS value.
twice MS = ms1+ms2=0, each MS = ms1+ms2=1 and MS = ms1+ms2=-1
We know S ≥ |MS|
→ S=1 when |MS|=1, MS=±1
MS takes -S to S : S=1 group include MS = 1,0,-1 → triplet
When S=0, there are only MS = 0 → singlet
Singlet and triplet : associated with paired and unpaired electrons
Singlet and triplet wavefunction is given by
1
1
[1s(1)2 s( 2)  2 s(1)1s( 2)]
[ (1) ( 2)   (1) ( 2)]
2
2
 (1) ( 2)
1
 triplet 
[1s(1)2 s( 2)  2 s(1)1s( 2)]{
 (1) ( 2)
}
2
1
[ (1) ( 2)   (1) ( 2)]
2
 singlet 
MS310 Quantum Physical Chemistry
Vector model of the singlet and triplet states
MS310 Quantum Physical Chemistry
We approximate the potential is spherically symmetry.
However, if l>0, probability distribution is not spherically symmetrical.
→ there are different repulsive interaction depending on ml values.
→ repulsive interaction between electrons : depends on l and s.
Only ‘partially’ filled subshells contribute to L and S.
How can one calculate it?
If spin-orbit coupling is neglected : total energy independent from ML
and MS.
→ group of different quantum state : same L and S value, different
ML and MS values.(it means degeneracy)
Group of states : ‘term’
L and S values for the term : 2S+1L, L=0,1,2,3… : symbol S,P,D,F…
MS310 Quantum Physical Chemistry
Degeneracy : 2L+1 for L value, 2S+1 for S value
→ (2L+1)(2S+1) : degeneracy of a term, 2S+1 : multiplicity
If filled subshell or shell, M L 
→ degeneracy=1(only the 1S)
m
i
li
 0, M S   m si  0
i
Term symbol : independent from principal quantum number
C : 1s22s22p2 ,Si : 1s22s22p63s23p2 : same set of terms
How are terms generated for a given configuration?
→ consider the ‘not-filled’ subshells(filled subshells doesn’t
contribute the term)
Possible values of L and S : Clebsch-Gordon series
For 2-electron case, allowed L values are given by l1+l2, l1+l2-1, …,
|l1-l2| and allowed S values are s1+s2 and s1-s2
MS310 Quantum Physical Chemistry
The different ways in which 2 electrons can be placed in p orbital
is shown.
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
Relative energy of different terms : the Hund’s rule
Rule 1 : The lowest energy term is that which has the greatest
spin multiplicity. For example, the 3P term of an np2
configuration is lower in energy than the 1D and 1S terms.
Rule 2 : For terms that have the same spin multiplicity, the term
with the greatest orbital angular momentum lies lowest in
energy. For example, the 1D term of an np2 configuration is
lower in energy than the 1S term.
Hund’s rules imply that the energetic consequences of e- - erepulsion are greater for spin than for orbital angular momentum.
MS310 Quantum Physical Chemistry
11.3 Spin-orbit coupling breaks up a term
into levels
Until now, we said the all states in a term have the same energy.
However, in real case, spin-orbit coupling occurs and terms are
split into closely spaced levels.
See the total angular momentum vector J
Magnitude of J can take the L+S, L+S-1, …, |L-S|
For example, 3P term has J=2,1,0.
→ 5 states with 3P2, 3 states with 3P1, 1 state with 3P0
Therefore, total states are 9.
Nomenclature : 2S+1LJ
2J+1 states have different MJ values associated with each J values.
Generally, there are (2L+1)(2S+1) states in 2S+1LJ
Coupling : add L•S term into total energy operator
MS310 Quantum Physical Chemistry
Taking spin-orbit coupling, it gives Hund’s third rule
Rule 3 : The order in energy of levels in a term is given by the
following :
If the unfilled subshell is exactly or more than half fill, the
level with the highest J value has the lowest energy.
If the unfilled subshell is less than half fill, the level with the
lowest J value has the lowest energy.
Use it, we can determine the lowest energy level in same term
Lowest energy of np2 configuration : 3P0 level
Lowest energy of np4 configuration : 3P2 level, it describes O
MS310 Quantum Physical Chemistry
Level diagram of Carbon(ground state)
np2 configuration : 3P0 level is lowest level
MS310 Quantum Physical Chemistry
Ex) excited configuration of C : 1s2 2s2 2p1 3d1
l1  1, l 2  2, s1  s2 
3
L  3,2, 1 
F


1
S  1, 0 
F
3
D
1
D
1
2
3
P
1
P
1) L  3, S  1  3 F
J  ( L  S ),  , L  S  ( 3  1),  , 2  4, 3, 2
L  3, S  0  1 F
 3 F4 , 3 F3 , 3 F2
J  ( 3  0),  , 3  0  3
 1F3
2) L  2, S  1  3 D
J  ( L  S ),  , L  S  ( 2  1),  , 1  3, 2, 1
L  2, S  0  1 D
3) L  1, S  1  3 P
L  1, S  0  1 P
 3 D 3 , 3 D 2 , 3 D1
J  ( 2  0),  , 2  0  2
 1D2
J  (1  1),  , 1  1  2, 1, 0
J  (1  0),  , 1  0  1
 3 P2 , 3 P1 , 3 P0
 1 P1
Degeneracy : total 60 states
3d electron : 5 different ml, 2 different ms : total 10 combination
2p electron : 3 different ml, 2 different ms : total 6 combination
→ 6x10=60 quantum states in 1s22s22p13d1
MS310 Quantum Physical Chemistry
11.4 The essentials of atomic spectroscopy
Spectroscopy : see the ‘transition’
What transitions are allowed? ‘selection rule’
Selection rule : obtained by dipole approximation(8.4)
Very useful although forbidden transition in the dipole
approximation can occur in higher level theory
Dipole selection rule
∆n=±1 for vibration
∆J=±1 for rotation(J:rotational quantum number)
In atomic level : consider the spin-orbit coupling
: ∆l=±1, ∆L=0,±1, ∆J=0,±1 and ∆S=0
(J:total angular momentum, L+S)
How use it? Transition of Cs → ‘atomic clock’(frequency of
transition : 9192631770 s-1)
MS310 Quantum Physical Chemistry
me e 4
Energy level of H atom : E n  
8 02 h2 n 2
Absorption frequency is given by
4
m
e
1
1
1
1
~
e
  2 3 ( 2  2 )  RH ( 2  2 )
8 0 h c ninitial n final
ninitial n final
RH : Rydberg constant, 109677.581 cm-1
ninitial=1 : Lyman series
ninitial=2 : Balmer series
ninitial=3 : Paschen series
ninitial=4 : Brackett series
ninitial=5 : Pfund series
MS310 Quantum Physical Chemistry
More general display : Grotrian diagram
Case of He atom
Solid line : allowed, dashed line : forbidden transition
MS310 Quantum Physical Chemistry
11.5 Analytical techniques based on atomic
spectroscopy
Example : detect toxic metal using the atomic emission and
atomic absorption spectroscopy
MS310 Quantum Physical Chemistry
Sample : very small droplet(1-10μm)
Heated zone : electrically heated graphite furnace or plasma arc
source → convert the state to excite states
Atomic emission spectroscopy
Light emitted by excited-state atoms → transitions back down
to the ground state : dispersed into its component wavelengths
by a monochromator
Intensity : proportional to # of excited-state atoms : character of
‘atom’
nupper/ nlower : 6x10-4 for 3000K Na
Use photomultiplier, spectral transition for nupper/ nlower < 10-10
It used for detect the 589.0nm and 589.6nm Na emission
MS310 Quantum Physical Chemistry
Atomic absorption spectroscopy
Difference : light pass through the heated zone, absorption
occurs from lower state to excited states and detected
→ we see the ‘absorption’
Sensitivity : 10-4 μg/ml for Mg, 10-2 μg/ml for Pt
MS310 Quantum Physical Chemistry
11.6 The Doppler effect
Doppler effect : shift of frequency
MS310 Quantum Physical Chemistry
Shifted frequency is given by
  0
vz
c
vz
1
c
1
1
In non-relativistic region, formula is more simple.    0
vz
1
c
In real case, ‘distribution’ of speed : follows the MaxwellBoltzmann distribution.
→ all velocity directions : ‘equally’ distributed
→ large range and <vz>=0
Therefore, there are no shift but ‘broadening’ occurs.
: Doppler broadening
MS310 Quantum Physical Chemistry
11.7 The He-Ne laser
Selection rule : ∆l=±1 for electron, ∆L=0 or ±1 for atom
Photon-assisted transition (see 8.2)
- Absorption : photon induces a transition to higher level
- Spontaneous emission : excited state relaxes to lower level
- Stimulated emission : photon induces a transition from
excited state to lower level
System is described by
B12  ( ) N1  B21 ( ) N 2  A21 N 2
Use blackbody spectral density function, we can obtain
A21 16 2 3
B12  B21 ,

B21
c3
MS310 Quantum Physical Chemistry
If stimulated emission dominant : N2>N1 : population inversion
Key of laser : stable population inversion
1 to 4 : external source(electric
field)
4 to 3 : relaxation(spontaneous
emission)
2 to 1 : similar to 4 to 3
(spontaneous emission)
Lasing transition : 3 to 2
How can make it?
‘optical resonator’
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
Condition of constructive interference : nλ = n(c/ν) = 2d
Next constructive condition : n → n+1
Difference of frequency : ∆ν = c/2d, bandwidth of cavity
# of nodes : determined by 2 factors
1) frequency of resonator modes
2) width in frequency of the stimulated emission transition
Width of transition : given by the Doppler broadening(by the
thermal motion of gas-phase atoms or molecules)
MS310 Quantum Physical Chemistry
a) resonator transition : depends on the Doppler linewidth
b) through the threshold : only 2 peaks survive
MS310 Quantum Physical Chemistry
How can He-Ne laser act?
1) He 1s2 configuration(1S term) → 1s2s configuration(1S and
3S term) by the electric field : ‘pumping transition’
2) by the collision, energy of He transfer to Ne(not obey the
selection rule) : 1S to 2p55s, 3S to 2p54s
3) by the lasing transition(stimulated emission), these states
go to 2p54p and 2p53p : 632.8nm
4) by the spontaneous emission, 2p53p goes to 2p53s
5) by the coalitional deactivation, 2p53s goes to 2p6 (ground
state)
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
11.8 Laser isotope separation
Speed of gas : proportional to M-1/2
→ it used for separation of 235U,
material of nuclear bomb
Potential difference occurs the laser
isotope separation. Why?
→ real nuclear potential is not a
coulomb potential : energy difference
(calculated by the perturbation theory)
Difference of IE of U : 2 x 10-3%
It can negligible when l>0 : effective
potential is repulsive potential
MS310 Quantum Physical Chemistry
11.9 Auger electron and X-ray photoelectron
spectroscopies
Application of spectroscopy : analysis of gas-phase and surface
Character of these 2 spectroscopies : ejection of electron, measure
the electron energy
Electron eject to the material, it through the vacuum and outside the
vacuum, electron collide to other materials and loss the energy.
→ energy of atomic level when electron within the ‘inelastic mean
free path’
(mean free path : average length of atom and molecule can move
without the collision)
Inelastic mean free path
2 atomic layer when 40eV
10 atomic layer when 1000eV
MS310 Quantum Physical Chemistry
Auger electron spectroscopy(AES)
: apply the X-ray and measure the low-energy electron
Principle of AES
1) Inject large energy(electron of X-ray photon)
2) An electron is ejected from a low-lying level
3) Hole of core electron filled through the relaxation from a higher
level electron
4) By energy conservation, third electron eject from the
higher level
Electron beam is focused to a spot size on the order of 10-100 nm
→ can make a map of elemental distribution at the solid surface
with very high lateral resolution
MS310 Quantum Physical Chemistry
Schematic diagram of AES
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
X-ray photoelectron spectroscopy(XPS)
: apply the X-ray and measure the high-energy electron
By the energy conservation, Ekinetic = hν - Ebinding
MS310 Quantum Physical Chemistry
We can see the chemical shift in the XPS.
Case of CF3COOCH2CH3
High electronegativity of F
→ net electron withdrawal to F in the
CF3 group
→ electron in C are deshielding and
binding energy increase
: large, positive chemical shift
Similarly, C in COO group are large
deshielding, too.
C in CH2 group : next to O by the single
bond : small chemical shift
C in CH3 group : no electron withdrawal
by the any atoms : no chemical shift
MS310 Quantum Physical Chemistry
Surface sensitivity of XPS : Fe film on a crystalline MgO surface
By the X-ray, 2p electron eject to the Fe surface.
Spin angular momentum s
→ coupled with orbital angular momentum l.
Total angular momentum j with 2 possible values
J = L+S, L+S-1, …, |L-S|
→ j = 1+1/2 = 3/2 and j = 1-1/2 = ½
Ratio of photoemission signal : ratio of degeneracy
3
2

1
I ( 2 p3 / 2 )
2

2
1
I ( 2 p1 / 2 )
2  1
2
Therefore, different ratio between Fe(II) and Fe(III)
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
11.10 Selective chemistry of excited state
: O(3P) and O(1D)
Interaction of sunlight with molecules in the atmosphere : set of
chemical reactions
Oxygen : major species in atmosphere reaction
dynamic equilibrium of oxygen and ozone.
O2  h  O   O 
O   O2  M  O3  M
O3  h  O   O2
O   O3  2O2
M : other molecule(oxygen or nitrogen)
MS310 Quantum Physical Chemistry
Less than 315nm, dissociation of oxygen occurs as
O2  O  ( 3 P )  O  (1 D)
1D
term : 190kJ/mol of excess energy than 3P term
It is used to overcome an activation barrier to reaction.
O  ( 3 P )  H 2O  OH  OH , Q  70kJ / mol
O  (1 D)  H 2O  OH  OH , Q  120kJ / mol
to 3P transition : ∆S=0 → forbidden : 1D are long-lived species
and it depleted by the reaction.
1D
Because of the excess energy of 1D species, it can make the
reactive hydroxyl and methyl radical.
O  (1 D )  H 2O  OH  OH
O  (1 D )  CH 4  OH  CH 3
MS310 Quantum Physical Chemistry
11.11 Configurations with paired and unpaired
electron spins differ in energy
Consider the first excited state of He : 1s12s1 2
e
Schrödinger equation is given by ( Hˆ 1  Hˆ 2 
) (1,2)  E singlet (1,2)
4 0 r12
1


[1s(1)2 s( 2)  2 s(1)1s( 2)]
Singlet wavefunction is given by singlet
2
(ignore the spin part)
2
1
e
E singlet   [1s(1)2 s( 2)  2 s(1)1s( 2)]( Hˆ 1  Hˆ 2 
)[1s(1)2 s( 2)  2 s(1)1s( 2)]d 1d 2
2
4 0 r12
e2
ˆ
Use  (1,2) H n (1,2)d 1d 2  E ns   2
(H-like orbitals)
2n  0a0
Therefore, integral becomes to
E singlet
1
e2
 E1 s  E2 s   [1s(1)2 s( 2)  2 s(1)1s( 2)](
)[1s(1)2 s( 2)  2 s(1)1s( 2)]d 1d 2
2
4 0 r12
Write the integral as the Esinglet = E1s + E2s + J12 + K12
e2
1
e2
2
J 12 
[1s(1)] ( )[2 s( 2)] d 1d 2 , K 12 

8 0
r12
8 0
2
 [1s(1)2s(2)](
MS310 Quantum Physical Chemistry
1
)[1s( 2)2 s(1)]d 1d 2
r12
In triplet state, result changes to Etriplet = E1s + E2s + J12 – K12
It gives the important result.
1) absence the electron repulsion : Etotal = E1s + E2s
2) including the coulomb repulsion, singlet and triplet state
separately, change of the value of energy J12 + K12 and J12 – K12
3) J12 >0, K12>0 : triplet state must be lower energy than
singlet state
J12 : coulomb integral, K12 : exchange integral
Consider electron 2 approaches to electron 1
Singlet : 1 [1s(1)2 s( 2)  2 s(1)1s( 2)]  1 [1s(1)2 s(1)  2 s(1)1s(1)]  2 1s(1)2 s(1)
Triplet :
2
2
1
1
[1s(1)2 s( 2)  2 s(1)1s( 2)] 
[1s(1)2 s(1)  2 s(1)1s(1)]  0
2
2
Therefore, triplet wavefunction has a greater degree of
electron correlation than singlet wavefunction → unpaired
spin has lower energy than paired spin.
MS310 Quantum Physical Chemistry
Summary
- Study the atomic emission spectroscopy and
atomic absorption spectroscopy
- Laser : population inversion(stable excited
state)
- Auger spectroscopy : third electron is ejected
through intermediate process from incident
high energy
- X-ray photoemission spectroscopy : Ek=hνincident-Ebinding
MS310 Quantum Physical Chemistry