Ch 11. Quantum States for Many-Electron Atoms and Atomic Spectroscopy (A.S.) MS310 Quantum Physical...

Ch 11. Quantum States for Ch 11. Quantum States for Many-Electron Atoms and Many-Electron Atoms and

Atomic Spectroscopy (A.S.)Atomic Spectroscopy (A.S.)

MS310 Quantum Physical Chemistry

• 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 and

• A. spectroscopies are widely used in analytical chemistry

• Laser and excited atoms are of interest.

→

S→

l

11.1 Good quantum numbers, terms, levels,11.1 Good quantum numbers, terms, levels, and statesand states


What about the quantum number of many-electron atom?In H atom, n, l, ml, and ms are used.Each operator , , , and commute with the hamiltonian. → 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

2l zl2s zs

)1(||,)1(||,s,l SSSLLLSL ii

i


We define the scalar ML and MS

i

ziSi

ziL sMlM ,

Next, we define , , , and 2L zL2S zS

22,

22, )ˆ(ˆ,ˆˆ,)ˆ(ˆ,ˆˆ

ii

iizz

ii

iizz sSsSlLlL

Good quantum number in many-electron atoms : L, S, ML, MS


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 , , , and don’t commute with hamiltonian.However, the operators and commute with hamiltonian.

Total angular momentum J is defined by

In this case, the only good quantum numbers are J and MJ, the projection of J on the z axis.

2L zL2S zS

2J zJ

SLJ


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

11.2 The energy of a configuration depends onboth orbital and spin angular momentum


Consider the He, 1s12s1 configuration2 electrons are both l=0(s orbital) : |L|=0Magnitude of spin angular momentum vector :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))

)1(|s| ss


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 → tripletWhen S=0, there are only MS = 0 → singletSinglet and triplet : associated with paired and unpaired electrons

Singlet and triplet wavefunction is given by

}

)]2()1()2()1([2

1)2()1(

)2()1(

)]{2(1)1(2)2(2)1(1[2

1

)]2()1()2()1([2

1)]2(1)1(2)2(2)1(1[

2

1

ssss

ssss

triplet

singlet


Vector model of the singlet and triplet states


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…


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, → degeneracy=1(only the 1S)

Term symbol : independent from principal quantum numberC : 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 seriesFor 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

0,0 i

siSi

liL mMmM


The different ways in which 2 electrons can be placed in p orbital is shown.


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.

Relative energy of different terms : the Hund’s rule

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- - e-

repulsion are greater for spin than for orbital angular momentum.

11.3 Spin-orbit coupling breaks up a term 11.3 Spin-orbit coupling breaks up a term into levelsinto 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 JMagnitude 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


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 levelLowest energy of np4 configuration : 3P2 level, it describes O


Level diagram of Carbon(ground state)np2 configuration : 3P0 level is lowest level

Ex) excited configuration of C : 1s2 2s2 2p1 3d1

PDF

PDF

0,1

1,2,3

,2,1

111

333

21

2121

S

L

ssll

111

03

13

233

211

13

23

33

3

311

23

33

43

3

P101,),01(P0,1

P,P,P0,1,211,),11(P1,1 3)

D202,),02(D0,2

D,D,D1,2,31,),12(,),(

D1,2 2)

F303,),03(F0,3

F,F,F2,3,42,),13(,),(

F1,3 1)

JSL

JSL

JSL

SLSLJ

SL

JSL

SLSLJ

SL

Degeneracy : total 60 states3d electron : 5 different ml, 2 different ms : total 10 combination2p electron : 3 different ml, 2 different ms : total 6 combination → 6x10=60 quantum states in 1s22s22p13d1MS310 Quantum Physical

Chemistry


11.4 The essentials of atomic spectroscopy11.4 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)


Energy level of H atom :

Absorption frequency is given by

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

2220

4

8 nh

emE e

n

)11

()11

(8

~222232

0

4

finalinitialH

finalinitial

e

nnR

nnch

em


More general display : Grotrian diagram

Case of He atomSolid line : allowed, dashed line : forbidden transition


11.5 Analytical techniques based on atomic 11.5 Analytical techniques based on atomic spectroscopyspectroscopy

Example : detect toxic metal using the atomic emission and atomic absorption spectroscopy


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


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

11.6 The Doppler effect11.6 The Doppler effect


Doppler effect : shift of frequency


Shifted frequency is given by

c

vc

v

z

z

1

1

0

In non-relativistic region, formula is more simple.

c

vz1

10

In real case, ‘distribution’ of speed : follows the Maxwell-Boltzmann distribution. → all velocity directions : ‘equally’ distributed → large range and <vz>=0

Therefore, there are no shift but ‘broadening’ occurs. : Doppler broadening

11.7 The He-Ne laser11.7 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

221221112 )()( NANBNB System is described by

Use blackbody spectral density function, we can obtain

3

32

21

212112

16,

cB

ABB


If stimulated emission dominant : N2>N1 : population inversionKey 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’


Condition of constructive interference : nλ = n(c/ν) = 2dNext constructive condition : n → n+1Difference 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)


a) resonator transition : depends on the Doppler linewidth b) through the threshold : only 2 peaks survive


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)


11.8 Laser isotope separation11.8 Laser isotope separationSpeed 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

11.9 Auger electron and X-ray photoelectron11.9 Auger electron and X-ray photoelectron spectroscopiesspectroscopies


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


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 level3) Hole of core electron filled through the relaxation from a higher level electron4) 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


Schematic diagram of AES


X-ray photoelectron spectroscopy(XPS) : apply the X-ray and measure the high-energy electron

By the energy conservation, Ekinetic = hν - Ebinding


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


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

21

2

12

12

32

)2(

)2(

2/1

2/3

pI

pI

Therefore, different ratio between Fe(II) and Fe(III)


11.10 Selective chemistry of excited state11.10 Selective chemistry of excited state : O(: O(33P) and O(P) and O(11D)D)

Interaction of sunlight with molecules in the atmosphere : set of chemical reactions

Oxygen : major species in atmosphere reaction dynamic equilibrium of oxygen and ozone.

23

23

32

2

2OOO

OOhO

MOMOO

OOhO

M : other molecule(oxygen or nitrogen)


Less than 315nm, dissociation of oxygen occurs as

)()( 132 DOPOO

1D term : 190kJ/mol of excess energy than 3P termIt is used to overcome an activation barrier to reaction.

molkJQOHOHOHDO

molkJQOHOHOHPO

/120,)(

/70,)(

21

23

1D to 3P transition : ∆S=0 → forbidden : 1D are long-lived species and it depleted by the reaction.

Because of the excess energy of 1D species, it can make the reactive hydroxyl and methyl radical.

341

21

)(

)(

CHOHCHDO

OHOHOHDO

11.11 Configurations with paired and unpaired 11.11 Configurations with paired and unpaired electron spins differ in energyelectron spins differ in energy


Consider the first excited state of He : 1s12s1

Schrödinger equation is given by

Singlet wavefunction is given by(ignore the spin part)

)2,1()2,1()4

ˆˆ(120

2

21 singletE

r

eHH

)]2(1)1(2)2(2)1(1[2

1sssssinglet

21120

2

21 )]2(1)1(2)2(2)1(1)[4

ˆˆ)](2(1)1(2)2(2)1(1[2

1

ddssssr

eHHssssEsinglet

002

2

21 2)2,1(ˆ)2,1(

an

eEddH nsn

Use (H-like orbitals)

21120

2

21 )]2(1)1(2)2(2)1(1)[4

)](2(1)1(2)2(2)1(1[2

1

ddssssr

essssEEE sssinglet

Therefore, integral becomes to

Write the integral as the Esinglet = E1s + E2s + J12 + K12

21120

2

12212

12

2

0

2

12 )]1(2)2(1[)1

)](2(2)1(1[8

,)]2(2[)1

()]1(1[8

ddssr

sse

Kddsr

se

J


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 stateJ12 : coulomb integral, K12 : exchange integral

Consider electron 2 approaches to electron 1Singlet :

Triplet :

Therefore, triplet wavefunction has a greater degree of electron correlation than singlet wavefunction → unpaired spin has lower energy than paired spin.

)1(2)1(12)]1(1)1(2)1(2)1(1[2

1)]2(1)1(2)2(2)1(1[

2

1ssssssssss

0)]1(1)1(2)1(2)1(1[2

1)]2(1)1(2)2(2)1(1[

2

1 ssssssss


- 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 :

Summary Summary

Ek=hνincident-Ebinding

Ch 11. Quantum States for Many-Electron Atoms and Atomic Spectroscopy (A.S.) MS310 Quantum Physical...

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Transcript of Ch 11. Quantum States for Many-Electron Atoms and Atomic Spectroscopy (A.S.) MS310 Quantum Physical...