Many-electron atoms, part II Hund’s rules Selection ... 8- atomic... · ©, 2017 Uwe Burghaus, Fargo, ND, USA • Many-electron atoms, part II • Hund’s rules • Selection rules

©, 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