many-electron atoms, part ii hund’s rules selection ... 8- atomic... · ©, 2017 uwe burghaus,...
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©, 2017 Uwe Burghaus, Fargo, ND, USA
• Many-electron atoms, part II • Hund’s rules • Selection rules
• Hyperfine structure • Stark effect
Atomic spectroscopy (part V)
©, 2017 Uwe Burghaus, Fargo, ND, USA
• Atomic spectroscopy (part I) • Absorption spectroscopy • Bohr model • QM of H atom (review)
• Atomic spectroscopy (part II) • Visualization of wave functions
• Atomic spectroscopy (part III) • Angular momentum (details)
• Orbital angular momentum • Spin
• Spin-Orbit coupling • Zeeman effect
• Multi-electron systems (part I)
• Multi-electron systems (part II) - today
PChem – Quantum mechanics
Pauli exclusion principle
Q.M. version Different electrons cannot be distinguished by any known experiment. Electrons are indistinguishable. We cannot distinguis a ms = +1/2 electron from a ms = -1/2 electron All (electronic) wave functions must be antisymmetric (under the interchange of any two electrons).
We can distinguish particles by tracing their path. However, in Q.M. the Heisenberg uncertainty principle prevents that.
Classical mechanics
General chemistry version Each state can hold only two electrons with opposite spin. No two electrons in an atom can have the same four quantum numbers n, l,ml,ms
Quantum mechanics
PChem – Quantum mechanics
Orbital approximation
• Orbital approximation: using one-electron wave functions to describe multi-electron systems
• Each electron in a many-electron system occupies its own one-electron function (called an orbital).
Examples hydrogen atoms: orbitals are the well-known solutions to the Schrödinger equation (1s, 2s, 2p orbitals, etc.). many-electron molecule: e.g. linear combinations of hydrogen wave functions.
PChem – Quantum mechanics
Multi-electron Schrödinger eq.
PChem – Quantum mechanics
Singlet / Triplet states – wave functions
12
2 1 1 2[ ( ) ( ) ( ) ( )]α β α β−
antisymmetric
S=0; MS = 0
symmetric α α
α β α β
β β
( ) ( )
[ ( ) ( ) ( ) ( )]
( ) ( )
1 2 112
2 1 1 2 0
1 2 1
M
M
M
S
S
S
=
+ =
= −
z z z z
ms = +1/2 ms = +1/2 ms = +1/2
ms = -1/2 ms = -1/2
ms = +1/2
ms = -1/2 ms = -1/2
S=0 S=1
vector model of spins
for two electron system
• Physics of Atoms and Molecules, B.H. Bransden, C.J. Joachain, Wiley Press
chapter 5
• Haken, Wolf Chapter 15, 20
Hund’s rules • Engel/Reid Ch. 21.8/21.9
• Spectra of Atoms and Molecules, 3rd Ed., Peter F. Bernath, Oxford University Press, p. 135
Read • Spectra of Atoms and Molecules, 3rd Ed., Peter F.
Bernath, Oxford University Press, Chapter 5.5 to 5.9
PChem 476
Aufbau principle
Building up principle
PChem – Quantum mechanics
What is a term symbol?
n 2S+1 L J
principal quantum number
(defines the energy)
multiplicity (number of possible
different wave functions)
L+S
angular momentum L=0 s state L=1 p state L=2 d state
If we neglect spin-orbit coupling the total energy is independent of MS and ML
Term: same L and S but different ML and MS
PChem – Quantum mechanics
Object: Predicts lowest energy term of a configuration.
Advantage: One can determine lowest energy term without knowing all the other terms.
Friedrich Hund, Göttingen, in the 1920s [ from Wikipedia ]
PChem – Quantum mechanics
Hund’s rules:
Frederick Hund (1896-1997)
Friedrich Hermann Hund (1896 – 1997) was a German physicist known for his work on atoms and molecules. Hund worked with Schrödinger, Dirac, Heisenberg, Max Born. He was Born's assistant, working on quantum interpretation of band spectra of diatomic molecules. • He published more than 250 papers and essays. • Hund discovered the tunnel effect • He was 101 years old when he died http://en.wikipedia.org/wiki/Friedrich_Hund
PChem – Quantum mechanics
Hund’s rules
1) Hund’s maximum multiplicity rule: An atom in its ground state adopts a configuration with the greatest number of unpaired electrons. Or For a set of terms arising from a given electron configuration, the lowest-lying term is generally the one with the maximum spin multiplicity.
PChem – Quantum mechanics
Hund’s 1st rule - EXAMPLE
px py pz
obeys Hund’s rule
px py pz
does not obey Hund’s rule
1) Hund’s maximum multiplicity rule: An atom in its ground state adopts a configuration with the greatest number of unpaired electrons. Or For a set of terms arising from a given electron configuration, the lowest-lying term is generally the one with the maximum spin multiplicity.
p4 configuration: [↑↓] [↑] [↑] rather than [↑↓] [↑] [↓] or [↑↓] [↑↓][ ]
PChem 476
PChem – Quantum mechanics
Hund’s rules
1) Hund’s maximum multiplicity rule: An atom in its ground state adopts a configuration with the greatest number of unpaired electrons. Or For a set of terms arising from a given electron configuration, the lowest-lying term is generally the one with the maximum spin multiplicity.
2) For terms that have the same spin multiplicity, the term with the greatest orbital angular momentum lies lowest in energy. Example (2nd rule): E(1D) < E(1S) Why? 1D: L=2 < 1S: L=0
PChem – Quantum mechanics
Hund’s rules
1) Hund’s maximum multiplicity rule: An atom in its ground state adopts a configuration with the greatest number of unpaired electrons. Or For a set of terms arising from a given electron configuration, the lowest-lying term is generally the one with the maximum spin multiplicity.
2) For terms that have the same spin multiplicity, the term with the greatest orbital angular momentum lies lowest in energy.
PChem – Quantum mechanics
Hund’s 2nd rule – simple explanation
2) For terms that have the same spin multiplicity, the term with the greatest orbital angular momentum lies lowest in energy.
• What is the most likely configuration? • Emin for smallest electron-electron repulsion
• Electrons repel less if they stay out of each others way • Electrons travel in same direction clockwise/counter clockwise L = max
Lowes Energy Term
3P
5D
PChem 476
Procedure for finding lowest energy term of a configuration
from Engel/Reid p. 478, example 21.7 similar example, see Bernath, p. 136
discussion
PChem 476
Procedure for finding lowest energy term of a configuration
discussion
PChem 476
Procedure for finding lowest energy term of a configuration
discussion
PChem 476
Procedure for finding lowest energy term of a configuration
discussion
PChem 476
Spin-Orbit coupling (e.g. even in undergrad books Engel/Reid Ch. 21.9)
No external fields: All states with the same J value have the same energy
PChem 476
Spin-Orbit coupling (e.g. even in undergrad books Engel/Reid Ch. 21.9)
No external fields: All states with the same J value have the same energy
PChem 476
PChem – Quantum mechanics
Hund’s rules
1) Hund’s maximum multiplicity rule: An atom in its ground state adopts a configuration with the greatest number of unpaired electrons. Or For a set of terms arising from a given electron configuration, the lowest-lying term is generally the one with the maximum spin multiplicity.
2) For terms that have the same spin multiplicity, the term with the greatest orbital angular momentum lies lowest in energy.
3) If the unfilled subshell is exactly or more than half full, the level with the highest J value less than half full, the level with the lowest J values has the lowest energy
PChem – Quantum mechanics
Hund’s 3rd rule example
3) If the unfilled subshell is exactly or more than half full, the level with the highest J value less than half full, the level with the lowest J values has the lowest energy
PChem – Quantum mechanics
Hund’s 3rd rule 2nd example
Spin-Orbit coupling
No external fields: All states with the same J value have the same energy
With an external (magnetic) field: In an external magnetic field states with the same J but different MJ have different energies. (Zeeman effect)
PChem 476
MJ
PChem 476
MJ
PChem 476
MJ
Explanation from Levine “Quantum Chemistry” Traditional explanation incorrect …?
PChem 476
PChem 476
E1
E2
E1
E2
photon photon
absorption spontaneous emission
E1
E2
photon
stimulated emission
photon
photon
A21 B21 B12
N1
N2
])([)(dt
dN21212121
1 ABNBN ++−= νρνρ
21
12
1
2
NN
BB
>
νννσ d
hcB )(
12 = 2213
3
21 || µνc
A ≈
This is a complicated story if considered in detail.
2112 BB =
i
f I = Intensity of transition
2*fi || M τψµψ dI ji∫=∝
transition dipole moment
https://en.wikipedia.org/wiki/Selection_rule
Symmetry-forbidden transitions µ: odd for dipole transitions :must have different symmetry (product will then be even) jiψψ
Spin-forbidden transitions ∆S=0 singlet triplet forbidden
mathematically complex topic
Quantum mechanics
Particle in box - example Po
tent
ial e
nerg
y
X-coordinate
∞ ∞
wall wall
x = 0 x = L
Rules for quantum number n = 1, 2, 3, …
n = 0 excluded ! n < 0 not required
2
22
8mLhnEn =Energy quantization
3,1 ±±=∆↔n
ug
2*2*fi |)(| || M τψψτψµψ dexdI jiji ∫∫ −==∝
µ : gerage / even symmetry
0|)(| 2* ≠−∫ τψψ dex ji If integrand even
gguu =⊗⊗
guug =⊗⊗
Plausible explanation
Chang, p. 520
1±=∆l
Particle in box – H atom (one electron selection rules)
...3,2,1,01,0
1
±±±=∆±=∆
±=∆
nml
Similar symmetry considerations can be used
principal quantum number
angular momentum quantum number
magnetic quantum number
n = 1, 2, 3, … l = 0, 1, 2, 3, …, n-1 ml = 0, ±1, ± 2, ± 3, …, ± l
ms = +1/2, -1/2 spin orientation quantum number
See e.g. P.W. Atkins Quanta for a “complete” list
∆J = 0, ± 1 but J=0 → 0
Atoms
∆L = 0, ± 1 but L=0 → 0 ∆l = ± 1 g→u, u → g
(electric dipole transitions)
(magnetic dipole transitions)
(electric quadrupole transitions)
Molecules * * *
https://en.wikipedia.org/wiki/Selection_rule
PChem 476
Spin orbit coupling / fine structure
https://en.wikipedia.org/wiki/Spin%E2%80%93orbit_interaction
Rather complex topic Mathematically involved when done in detail
Plausible explanation (quasi classical) • The electron (with orbital angular moment l) “loping the nucleus” generates a current • That current generates a magnetic field, Bl ∼ l • The magnetic moment of the spin, µs, interacts with Bl • Different orientations of µs generate different interaction energies, Vls = - µsBl ∼LS • For a one electron system: duplet peaks • One can also say that the magnetic moments interact; µl and µs
Experimental result • Spectra of alkali atoms do show peak splitting, doublets • Systems with one valence electron • Except s-states these do not show doublets
21
jm =
21
jm −=
z
21
21
43
43
23
jm =
23
jm −=
23
23
p-electron, example • Orbital angular momentum: l=1 • Spin angular moment: s=1/2 j=3/2 and j=1/2
All states with the same j have the same energy
l-s coupling / spin-orbit coupling
Schematics for alkali atoms (treaded at one-electron states including spin-orbit coupling)
Coupling of nuclear spin, I, and J (electronic spin)
F = J + I
Bernath, p. 147
same term same energy
same j same energy
Zeeman effect
Feinstructure
Bransden, p. 242
PChem 476
Zeeman effect
The effect: Splitting of spectral lines due to an external magnetic field
00 BgmBV jjjm j−== µ
lz mL =
Selection rules ∆mj = 0, +-1
(mj = j, j-1, …, -j)
Line splitting: Splitting is described by orientation quantization
• Normal Zeeman effect (l only) • Anomalous Zeeman effect (j only) • Paschen-Back effect (B large)
Pieter Zeeman
Zeeman effect
The effect: Splitting of spectral lines due to an external magnetic field
Pieter Zeeman
Stark effect
The effect: Splitting of spectral lines due to an external electric field
Johannes Stark (German physicist, 1874 – 1957) Nobel Prize in Physics (1919)
Haken, Wolf, p. 228
Zeeman effect Stark effect
What is the difference?
Atomic spectroscopy (part I) Absorption spectroscopy Bohr model QM of H atom (review)
Atomic spectroscopy (part II) Visualization of wave functions
Atomic spectroscopy (part III) Angular momentum (details)
Orbital angular momentum Spin
Spin-Orbit coupling Zeeman effect
Many-electron atoms, part I Pauli Principle Singlet vs. triplet Term symbols LS vs. jj coupling
Many-electron atoms, part II Hund’s rules Selection rules Hyperfine structure Stark effect
What did we do in this class segment?
• xxx
• Spectra of Atoms and Molecules, 3rd Ed., Peter F.
Bernath, Oxford University Press, Chapter 6 – rotational spectroscopy