Solid State Chemistry Chapter 3 Atomic Structure and Spectra.

Solid State Chemistry

Chapter 3Atomic Structure and Spectra

AGENDA

The structure and spectra of Hydrogenic atoms

The structure of many-Electron atoms1. Pauli’s principle 2. Penetration and shielding3. Building up principle

The spectra of complex atoms1. Quantum defects2. Singlet and triplets3. Spin-orbit coupling4. The total angular momentum5. Term symbols and selection rules

The Structure of Many-electron Atoms

)()(,, 2121 rrrr

21 :atom Helium

electronsother theall of presence by the modified chargesnuclear toingcorrespondbut

orbitals, hydrogenic theresembling as orbitals individual The

s

The Orbital ApproximationJustification 10.5

The Pauli Principle

paired. bemust spinsr then theiorbital, oneoccupy do twoif

and, orbitalgiven any occupy may electrons than twomore No

:principleexclusion Pauli

Pauli principle:

When the labels of any two identical fermions are exchanged, the total wavefunction changes sign. When the labels of any two identical bosons are exchanged, the total wavefunction retains the same sign

(2,1) = -(1,2) for two electrons

Penetration and Shielding

orbitals d lower than lie orbitals p and shell,

given a of orbitals pn energy thain lower liegenerally orbitals s

ZZeff

:constant shielding chargenuclear shielded a sexperienceelectron

:chargenuclear Effective

Penetration and Shielding

electron p a than shielding less sexperienceelectron s

:electron p a than shellinner through

n penetratiogreater a haselectron sA

state ground itsin atoman of shell

outermost in the electrons the:electrons Valence

21 :atom Li 12 ss

fdps

The Building-up (Aufbou) Principle

orbitals 3din repulsionselectron -electron Strong -

ncorrelatioSpin -

electrons unpaired ofnumber

greatest heion with tconfigurat a

adopts state ground itsin atomAn -

:rulety multiplici maximum sHund'

themof oneany occupyingdoubly

before subshellgiven a of

orbitalsdifferent occupy Electrons -

6s 5p 4d 5s 4p 3d 4s 3p 3s 2p 2s s1

Ionization Energies and Electron Affinities

2

1

energy, ionization second The -

energy, ionizationfirst The -

I

I

Ionization Energies and Electron Affinities

atom phase-gas a to

attacheselectron an when

releasedenergy the:Eaffinity Electron -

2

5)(

ionization ofenthalpy Standard -

ea

RTITHion

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2

5)(

gain,electron ofenthalpy Standard

.exothermic

is attachmentelectron that implies 0E

ioneg

eg

eg

ea

XHXH

RTETH

H

ea

Self-consistent Field Orbitals

i

iji

i r

e

r

ZeV

0

2

0

2

42

1

4

)1()1()correction exchange()1(electrons)other ()1()1( 22222 ppppp EVVH

• The first term on the left is the contribution of the kinetic energy and the attraction of the electron to the nucleus, just as in a hydrogenic atom

• The second takes into account the potential energy of the electron of interest due to the electrons in the other occupied orbitals

• The third term takes into account the spin correlation effects.

The Spectra of Complex Atoms

Quantum defects and ionization limits

2

2

2

~

validis variationn1

thatdiffuse so are that states excited some :state Rydberg

)(~

quantity empiricalpurely a : defect, Quantum

pyspectrosco atomic :energies ionization ofion Determinat

n

R

hc

Iv

n

hcRE

Singlet and Triplet States

singlet

rulety multiplici maxinum sHund'

)2()1()2()1()21()2 ,1_( 21

)2()1( ,)2()1()2()1()21()2 ,1( ),2()1( 21

ncorrelatioSpin :differenceenergy oforigin The

state.singlet t theenergy thain lower lies state triplet The

The Spectrum of Atomic Helium

states triplet andsinglet between ns transitioradiative No -

1 :ionconfigurat excitedOnly - 11nls

Spin-Orbit Coupling

momentumangular orbital thefrom arising field magnetic with the

moment magneticspin ofn interactio the:couplingorbit -Spin

The Total Angular Momentum

low is momentumangular total theopposed, are momentaangular twowhen the

high; is momentumangular totaltheparallel,nealy are momentaangular orbital andspin When the

opposed) are (when they 2

1

direction) same in the are momentaangular two(when the 2

1

lj

lj

Fine Structure

) Z(asnumber atomic with

sharply increases coupling the-

chargenuclear on the depends

couplingorbit -spin theofStrength

)1()1()1(2

1

A constant, couplingorbit -Spin

4

,,

sslljjhcAE jsl

Term Symbols and Selection Rules

J. number, quantum momentumangular totalthe

of value theis )Pin 2

3 theexample,(for

symbol termon thesubscript right The 3.

term. theofty multiplici the

gives )Pin 2 theexample,(for

symbol termin thet superscripleft The 2.

L. number, quantum

momentumangular orbital total theindicates

examples) in the Dor P example,(for letter The 1.

232

232

Total Orbital Angular Momentum

Total angular momentum quantum number L

Angular momentum =

{L(L + 1)}1/2h

L = l1+l2, l1+l2-1….,|l1-l2|

Clebsch-Gordan series

L: 0,1,2,3,4,… (S,P,D,F,G…)

Example:

d2 electron

Multiplicity

Total spin angular momentum quantum number S

Spin angular momentum =

{S(S + 1)}1/2h

S = s1+s2, s1+s2-1….,|s1-s2|

Multiplicity: 2S + 1

Example:

Two unpaired electrons

Total Angular Momentum

Total angular momentum quantum number J

J = j where j = l + ½, |l – ½|

Example:

[Ne]3s1

[Ne]3p1

Russell-Saunders coupling:

If spin-orbit coupling is weak, then it is effective only when all the orbital momenta are operating cooperatively

J = L + S, L + S – 1,….., |L – S|

Example:

[Ne]2p13p1Selection rules

S = 0 L = 0,±1 l = ±1 J = 0,±1

Short Summary

The Structures and spectra of many electron atoms

1. The Pauli principle

2. Penetration and shielding

3. Singlet and triplet states

4. Spin-orbit coupling

5. Term symbols and selection rules

HW#2: 10.3d, 10.7d, Exe: 10.4b, 10.6a, 10.8a, 10.12b, 10.18b, 10.19a