Lecture-III

Schrödinger equation can be solved exactly for H atom but can be extended for 1-electron system, like: He+, Li2+ , Be3+ etc…

But for He, there are three centers (2 e and 1 nucleus) gives more complicated picture where famous three body problem of physics should be taken into consideration.To get real solution, approximation of wave function is necessary, that can be done by using self-consistent field method (HartreeFock Method)

Four Quantum NumbersEach electron in an atom can be described uniquely by the four quantum numbers.

Three rules involving quantum numbers • Pauli Exclusion Principle • Aufbau Principle • Hund's Rule

Pauli exclusion principle

no two electrons in an atom can have the same four quantum numbers. ∴The maximum number of electrons in any orbital is two. The maximum number of electrons in a shell (or subshell) is 2x the number of orbitals in the shell (or subshell).ms = +1/2 or –1/2 (up or down)

Writing electronic configurations

Electronic configurations: a method of describing the orbital arrangement of electrons in an atom.

Atomic orbital diagram: pictorially represents electronic configurations.

Hund’s rule For degenerate orbitals, lowest energy is obtained when spin is maximized. this means…

1. Electrons will fill the subshellorbitals, one at a time, until each orbital has one electron.

2. All electrons will have the same spin (either up or down, or either +1/2 or –1/2)

3. Only then will electrons be paired.

Electron Filling Pattern1s2s 2p3s 3p 3d4s 4p 4d 4f5s 5p 5d 5f 5g6s 6p 6d 6f 6g 6h7s 7p

1s 2s 2p 3s 3p 4s 3d 4p 5s

4d 5p 6s 4f 5d 6p 7s

1s1

1s22s1

1s22s22p63s1

1s22s22p63s23p64s1

1s22s22p63s23p64s23d104p65s1

1s22s22p63s23p64s23d104p65s24d10

5p66s1

1s22s22p63s23p64s23d104p65s24d105p66s24f145d106p67s1

H1

Li3

Na11K19

Rb37Cs55Fr87

He2

Ne10

Ar18

Kr36

Xe54

Rn86

1s2

1s22s22p6

1s22s22p63s23p6

1s22s22p63s23p64s23d104p6

1s22s22p63s23p64s23d104p65s24d105p6

1s22s22p63s23p64s23d104p65s24d10

5p66s24f145d106p6

Alkali metals all end in s1

Alkaline earth metals all end in s2

really should include He, but it fits better later.He has the properties of the noble gases.

s2s1 S- block

Transition Metals -d block

d1 d2 d3s1d5 d5 d6 d7 d8

s1d10 d10

The P-block p1 p2 p3 p4 p5 p6

F - blockinner transition elements

f1 f5f2 f3 f4 f6 f7 f8 f9 f10 f11f12 f14f13

Each row (or period) is the energy level for s and p orbitals.

1

2

3

4

5

6

7

d orbitals fill up after previous energy level, so first d is 3d even though it’s in row 4.

1

2

3

4

5

6

7

3d

f orbitals start filling at 4f

1

2

3

4

5

6

7 4f

5f

Writing electron configurations the easy way

Yes there is a shorthandYes there is a shorthand

Electron Configurations repeatThe shape of the periodic table is a representation of this repetition.When we get to the end of the column the outermost energy level is full.This is the basis for our shorthand.

The ShorthandWrite symbol of the noble gas before the element, in [ ].Then, the rest of the electrons.Aluminum’s full configuration:

1s22s22p63s23p1

previous noble gas Ne is: 1s22s22p6

so, Al is: [Ne] 3s23p1

More examplesGe = 1s22s22p63s23p64s23d104p2

• Thus, Ge = [Ar] 4s23d104p2

Hf = 1s22s22p63s23p64s23d104p65s2

4d105p66s24f145d2

• Thus, Hf = [Xe]6s24f145d2

The Shorthand Again

Sn- 50 electrons The noble gas before it is Kr

[ Kr ]

Takes care of 36Next 5s2

5s2

Then 4d10

4d10Finally 5p2

5p2

The theory underlying Hund’s rule of maximum multiplicity

1. Minimization of electron-electron repulsion- There is less repulsion between electrons in

different orbitals (different regions in space)

Electrons in different orbitals feel a greater Z*, thus they are more stable

The theory underlying Hund’s rule of maximum multiplicity

2. Maximization of exchange energy stabilization- This is a quantum mechanical effect that causes systems with electrons of the same energy and spinto be more stable.

- The more exchanges possible, the more stable the electron configuration of the subshell

For an s-orbital (subshell), the spins must be different, so no exchanges are possible

Two electrons of the same spin, one exchange is possible:

For a p subshell, there are different orbitals of the same energy and exchanges are possible.

Two electrons of opposite spin, no exchange is possible:

One exchange

Initial arrangement

Three electrons of same spin, three exchanges are possible:

Initial arrangement

One exchange

Second exchange

Third exchange

The exchange energy explains why half-filled subshellsare unusually stable.e.g. the electron configuration of Cr: [Ar]4s1 3d5 instead of [Ar]4s2 3d4

Πc

Πe

pairingenergyen

ergy

Πc = Coulombic energy(destabilizing)

Πe = exchange energy(stabilizing)

Hypothetical arrangement

N

O

3 Πe

1 Πc + 3 Πe

The IE anomaly at nitrogen and oxygen C

1 Πe

Exceptions to Electron FillingThere are two exceptions to the electron filling pattern in the first 40 elements.• Chromium• Copper

ChromiumCr24

1s2 2s2 2p6 3s2 3p6 4s1 3d5

Why might this exception occur?

24 electrons

What is the expected (incorrect) electronicconfiguration for chromium?

CopperCu29

1s2 2s2 2p6 3s2 3p6 4s1 3d10

Why might this configuration be more stablethan 4s2 3d9?

29 electrons

What is the expected (incorrect) electronicconfiguration for copper?

Spectroscopists use the word termto describe the energies involved in

an electronic transition.

Term Symbols

1s

2shν

−=ν −22

21

1

n1

n1cm737,109

Note: this calculation only worksfor hydrogen with one electron!

Term symbols are an abbreviated descriptionof the energy, angular momentum and spin

of an atom in a particular state.

Term Symbols

d-d electronic transitionsare responsible for

the color of metal ions.hν

Term SymbolsAngular Momentum and Spin State

orbitals have differentangular momentum values-2 -1 0 +1 +2

spin state depends on thenumber of unpaired electronseach spin is -1/2 or +1/2

Term SymbolsMultiplicity (number of spectral lines)

MSspin angularmomentum

2S +1

netelectronspin (S)

01/21

3/22

5/2

123456

singletdoublettripletquartetquintetsextet

net electronsin samedirection

012345

Obeying the Pauli Exclusion Principle,but allowing the d-electrons of a vanadium(III)

ion to occupy any other legal patterns,how many different patterns could exist and

what would be the magnetic propertiesof each pattern?

Term Symbols

V3+: 1s2 2s2 2p6 3s2 3p6 3d2

Some examples

ML = - 4Ms = 0

ML = - 1Ms = 0

ML = - 2Ms = -1

ML = 0Ms = 0

ML = +1Ms = +1

Microstates

How many microstates are possible?

10 × 91 × 2= 45 microstates

Ms

ML 0-1-2-3-4

+1+2+3+4

+10-1

2211

211

54321

4321

2211

211 3F

“triplet F”

Ms

ML 0-1-2-3-4

+1+2+3+4

+10-1

11

143211

3211

11

1

3F3P

“triplet P”

Ms

ML 0-1-2-3-4

+1+2+3+4

+10-1

32211

2211

3F3P1G

“singlet G”

Ms

ML 0-1-2-3-4

+1+2+3+4

+10-1

211

11

3F3P1G1D

Ms

ML 0-1-2-3-4

+1+2+3+4

+10-1

211

11

3F3P1G1D