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

2. 11.1 Good quantum numbers, terms, levels, and states and states MS310 Quantum Physical Chemistry What about the quantum number of many-electron atom? In H atom, n, l, m l, and m s are used. Each operator,,, and commute with the hamiltonian. → independent of time → n, l, m l, and m s : 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

3. MS310 Quantum Physical Chemistry We define the scalar M L and M S Next, we define,,, and Good quantum number in many-electron atoms : L, S, M L, M S

4. 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,,, 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 M J, the projection of J on the z axis.

5. MS310 Quantum Physical Chemistry

6. 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 M J : Level(groups of 2J+1 states), energy depends on J. If external magnetic field applied, each state split(same J, different M J and energy depends on both J and M J ) ex) Carbon atom configuration 1s 2 2s 2 2p 2 : 3 terms, 5 levels, and 15 states

7. 11.2 The energy of a configuration depends on both orbital and spin angular momentum MS310 Quantum Physical Chemistry Consider the He, 1s 1 2s 1 configuration 2 electrons are both l=0(s orbital) : |L|=0 Magnitude 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))

8. MS310 Quantum Physical Chemistry Calculate the M S value. twice M S = m s1 +m s2 =0, each M S = m s1 +m s2 =1 and M S = m s1 +m s2 =-1 We know S ≥ |M S | → S=1 when |M S |=1, M S =±1 M S takes -S to S : S=1 group include M S = 1,0,-1 → triplet When S=0, there are only M S = 0 → singlet Singlet and triplet : associated with paired and unpaired electrons Singlet and triplet wavefunction is given by

9. MS310 Quantum Physical Chemistry Vector model of the singlet and triplet states

10. 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 m l 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 M L and M S. → group of different quantum state : same L and S value, different M L and M S values.(it means degeneracy) Group of states : ‘term’ L and S values for the term : 2S+1 L, L=0,1,2,3… : symbol S,P,D,F…

11. 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, → degeneracy=1(only the 1 S) Term symbol : independent from principal quantum number C : 1s 2 2s 2 2p 2,Si : 1s 2 2s 2 2p 6 3s 2 3p 2 : 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 l 1 +l 2, l 1 +l 2 -1, …, |l 1 -l 2 | and allowed S values are s 1 +s 2 and s 1 -s 2

12. MS310 Quantum Physical Chemistry The different ways in which 2 electrons can be placed in p orbital is shown.

13. MS310 Quantum Physical Chemistry

18. Rule 1 : The lowest energy term is that which has the greatest spin multiplicity. For example, the 3 P term of an np 2 configuration is lower in energy than the 1 D and 1 S 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 1 D term of an np 2 configuration is lower in energy than the 1 S term. Hund’s rules imply that the energetic consequences of e - ­ - ­e - repulsion are greater for spin than for orbital angular momentum.

19. 11.3 Spin-orbit coupling breaks up a term into levels MS310 Quantum Physical Chemistry 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, 3 P term has J=2,1,0. → 5 states with 3 P 2, 3 states with 3 P 1, 1 state with 3 P 0 Therefore, total states are 9. Nomenclature : 2S+1 L J 2J+1 states have different M J values associated with each J values. Generally, there are (2L+1)(2S+1) states in 2S+1 L J Coupling : add LS term into total energy operator

20. 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 np 2 configuration : 3 P 0 level Lowest energy of np 4 configuration : 3 P 2 level, it describes O

21. MS310 Quantum Physical Chemistry Level diagram of Carbon(ground state) np 2 configuration : 3 P 0 level is lowest level

22. Ex) excited configuration of C : 1s 2 2s 2 2p 1 3d 1 Degeneracy : total 60 states 3d electron : 5 different m l, 2 different m s : total 10 combination 2p electron : 3 different m l, 2 different m s : total 6 combination → 6x10=60 quantum states in 1s 2 2s 2 2p 1 3d 1 MS310 Quantum Physical Chemistry

23. 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 )

24. MS310 Quantum Physical Chemistry Energy level of H atom : Absorption frequency is given by R H : Rydberg constant, 109677.581 cm-1 n initial =1 : Lyman series n initial =2 : Balmer series n initial =3 : Paschen series n initial =4 : Brackett series n initial =5 : Pfund series

25. MS310 Quantum Physical Chemistry More general display : Grotrian diagram Case of He atom Solid line : allowed, dashed line : forbidden transition

26. MS310 Quantum Physical Chemistry 11.5 Analytical techniques based on atomic spectroscopy Example : detect toxic metal using the atomic emission and atomic absorption spectroscopy

27. 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’ n upper / n lower : 6x10 -4 for 3000K Na Use photomultiplier, spectral transition for n upper / n lower < 10 -10 It used for detect the 589.0nm and 589.6nm Na emission

28. 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

29. 11.6 The Doppler effect MS310 Quantum Physical Chemistry Doppler effect : shift of frequency

30. MS310 Quantum Physical Chemistry Shifted frequency is given by In non-relativistic region, formula is more simple. In real case, ‘distribution’ of speed : follows the Maxwell- Boltzmann distribution. → all velocity directions : ‘equally’ distributed → large range and =0 Therefore, there are no shift but ‘broadening’ occurs. : Doppler broadening

31. 11.7 The He-Ne laser MS310 Quantum Physical Chemistry 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 Use blackbody spectral density function, we can obtain

32. MS310 Quantum Physical Chemistry If stimulated emission dominant : N 2 >N 1 : 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’

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35. 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)

36. MS310 Quantum Physical Chemistry a) resonator transition : depends on the Doppler linewidth b) through the threshold : only 2 peaks survive

37. MS310 Quantum Physical Chemistry How can He-Ne laser act? 1) He 1s 2 configuration( 1 S term) → 1s2s configuration( 1 S and 3 S term) by the electric field : ‘pumping transition’ 2) by the collision, energy of He transfer to Ne(not obey the selection rule) : 1 S to 2p 5 5s, 3 S to 2p 5 4s 3) by the lasing transition(stimulated emission), these states go to 2p 5 4p and 2p 5 3p : 632.8nm 4) by the spontaneous emission, 2p 5 3p goes to 2p 5 3s 5) by the coalitional deactivation, 2p 5 3s goes to 2p 6 (ground state)

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39. 11.8 Laser isotope separation Speed of gas : proportional to M -1/2 → it used for separation of 235 U, 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

40. 11.9 Auger electron and X-ray photoelectron spectroscopies spectroscopies MS310 Quantum Physical Chemistry 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

41. 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

42. MS310 Quantum Physical Chemistry Schematic diagram of AES

43. MS310 Quantum Physical Chemistry

44. X-ray photoelectron spectroscopy(XPS) : apply the X-ray and measure the high-energy electron By the energy conservation, E kinetic = hν - E binding

45. MS310 Quantum Physical Chemistry We can see the chemical shift in the XPS. Case of CF 3 COOCH 2 CH 3 High electronegativity of F → net electron withdrawal to F in the CF 3 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 CH 2 group : next to O by the single bond : small chemical shift C in CH 3 group : no electron withdrawal by the any atoms : no chemical shift

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

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48. 11.10 Selective chemistry of excited state : O( 3 P) and O( 1 D) : O( 3 P) and O( 1 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. M : other molecule(oxygen or nitrogen)

49. MS310 Quantum Physical Chemistry Less than 315nm, dissociation of oxygen occurs as 1 D term : 190kJ/mol of excess energy than 3 P term It is used to overcome an activation barrier to reaction. 1 D to 3 P transition : ∆S=0 → forbidden : 1 D are long-lived species and it depleted by the reaction. Because of the excess energy of 1 D species, it can make the reactive hydroxyl and methyl radical.

50. 11.11 Configurations with paired and unpaired electron spins differ in energy MS310 Quantum Physical Chemistry Consider the first excited state of He : 1s 1 2s 1 Schrödinger equation is given by Singlet wavefunction is given by (ignore the spin part) Use (H-like orbitals) Therefore, integral becomes to Write the integral as the E singlet = E 1s + E 2s + J 12 + K 12

51. MS310 Quantum Physical Chemistry In triplet state, result changes to E triplet = E 1s + E 2s + J 12 – K 12 It gives the important result. 1) absence the electron repulsion : E total = E 1s + E 2s 2) including the coulomb repulsion, singlet and triplet state separately, change of the value of energy J 12 + K 12 and J 12 – K 12 3) J 12 >0, K 12 >0 : triplet state must be lower energy than singlet state J 12 : coulomb integral, K 12 : exchange integral Consider electron 2 approaches to electron 1 Singlet : Triplet : Therefore, triplet wavefunction has a greater degree of electron correlation than singlet wavefunction → unpaired spin has lower energy than paired spin.

52. MS310 Quantum Physical Chemistry - 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 E k =hν incident -E binding