MO Theory

H2+ and H2 solns

Solutions to Hydrogen Molecule Ion

2, E2 = -10.16 eV (for H2 )

1, E1 = 1.37 eV (for H2)

Solutions to Hydrogen MoleculeMOs created from combinations of p-orbitals

pzA - pzB

pxA + pxB, pyA + pyB pxA - pxB, pyA - pyB

pzA + pzB

Solutions to Hydrogen Moleculepx+ px OR py + py

px - px OR py - py

pz - pz

pz + pz

Parity

Gerade = symmetric

with inversion

Ungerade = antisymmetric with inversion

represents center of inversion

inversion

inversion

MO Energy Level Diagram for Homonuclear Diatomics

*

*

*

lone atom lone atom

1s 1s

2s 2s

2p 2p

Molecular Term Symbols

• ML = (over all e-) • identifies “z-component” of angular momentum

of an e-

• Symbols used to id

| | 0 1 2 3 4

Molecular Term Symbols

• Angular momentum about “z-axis” for all electrons is LM

= |ML|

Symbol used to id

0 1 2 3 4

Molecular Term Symbols

• Symbol is 2S + 1 g/u

• 2S + 1 is multiplicity as already used for atomic term symbols

• g or u identifies overall parity– To determine overall parity, make use of multiplication

of symmetric and antisymmetric functions• If the term is a term, a right superscript of + or

– is added to indicate whether the wavefunction is symmetric or antisymmetric with respect to reflection through a plane containing the two nuclei

Molecular Term Symbols

Remember sigma orbs:

Remember pi orbs:

From s orbs From pz orbs

From px orbsFrom py orbs

Molecular Term Symbols

Remember sigma-star orbs:

Remember pi-star orbs:

From s orbitals From p orbitals

From px orbitals From py orbitals

Spectroscopy – Selection Rules

= 0, +1, -1

S = 0

note = Ms

= 0

note refers to spin-orbit coupling and

= 0, +1, -1

Molecular Term Symbols

• Molecular Orbitals not always so “clear-cut”

• Remember how orbitals change energy as go across PT– Can affect MO energy pattern too

MO Energy Level Diagram for Homonuclear Diatomics

Atkins, Fig 14.30

As you move to the right on PT, 2s and 2p energy gap increases. Early, in the period, then, this permits mixing of 2s and 2pz orbitals.

Essentially LCAOs involving four orbitals are made. The sigma orbitals that we thought of as being made by the 2s orbitals are lowered in E while the sigma orbitals that we thought of as being made by the 2pz orbitals are raised in E.

MO Energy Level Diagram for Homonuclear Diatomics (N2 and “before”)

*

*

*

lone atom lone atom

1s 1s

2s 2s

2p 2pUse this diagram for N2 and earlier in PT

Taking a look at

heteronuclear diatomic molecules

Taking a look at

heteronuclear diatomic molecules

MOs of HF

E = -0.491 au

Unoccupied, E = -0.124 eV

Occupied, E = -0.3523 au

E = -1.086 au

MOs of HF

H atom F atomH – F molecule

1s

1s2s

2p

Computational Chemistry

• Considering complexity of the calculations we’ve been doing, certainly, using computers to do these calcs should be useful Computational Chemistry

• For polyatomic molecules can make LCAOs MO = cii

i constitute basis set (computational forms of atomic orbitals)

– Use variation theory to find ci

– To find structure of molecule, must move nuclei and find MOs find structure with lowest overall energy

Computational Chemistry

• May “solve” for MOs using ab initio or semi-empirical methods– Semi-empirical methods: empirical parameters

substituted for some “integrals” to save time in calculations

– Ab initio methods: supposedly make no assumptions• NOTE: computational chemistry may determine Energy

and some other properties without using quantum chemistry– Such calculations are referred to as molecular

mechanics calculations

Valence Bond Theory

• H2

• Initial approx is = 1sA(1) 1sB(2)– But, is this a symm or antisymm wavefxn?

• So, make LCs– = 1sA(1) 1sB(2) + 1sB(1) 1sA(2)

– = 1sA(1) 1sB(2) - 1sB(1) 1sA(2)

• In this case, turns out that + is lower E

Valence Bond Theory

• Ground state wavefunction would bebond = [1sA(1) 1sB(2) + 1sB(1) 1sA(2)][(1)(2) – (2)(1)]

• 2 electrons in overlapping orbitals – with spins paired

Remember CH4

• If try to make combinations of the valence s of C with s of H, will be different type of wavefxn, hence diff’t kind of bond than when make combination of a p of C with an s of H

• DON’T see any diff in bonding of 4 H’s– Make LCs of valence orbitals on central atom– Call these LCs hybrid orbitals– Use these hybrid orbitals to make sigma bonds with H– Atomic orbitals NOT used to make sigma bonds used

to make pi bonds (Huckel method for conjugated)

Hybrid Orbitals

• Valence s and p orbitals on C hybrids1 = a12s + a22px + a32py + a42pz

= b12s + b22px + b32py + b42pz

= c12s + c22px + c32py + c42pz

= d12s + d22px + d32py + d42pz

• Consider ethyne– Only two hybrids

1 = s + pz and y2 = s – pz

– Leftover px and py on one C overlap with px and py on other C

Simplification to MO Approach

Huckel Approach

Symmetry of Molecules

Determining Point Groups

Special Group?

No

Cn

YesC∞v , D∞h , Td , Oh , Ih , Th

No

h

Yes

S2n or S2n and i only, collinearwith highest order Cn

YesSn

Cs

Yes

YesCiNo

C1

No

nC2 perpendicular to Cn

Yes h

DnhYesNo

n dNoDn

No

h

YesCnh

Non v

CnvYes

NoCn

DndYes

iNo

C2v Character TableC2v E C2 v(xz) v

’(yz)

A1 1 1 1 1

A2 1 1 -1 -1

B1 1 -1 1 -1

B2 1 -1 -1 1

Now go practice!!!

Applying Symmetry to MOs

Water

MOs of Water

HOMO-4

Looks like s orbital on O, nbo

E = -18.6035 au

a1

MOs of Water

HOMO-3 from two viewpoints

Looks like s orbital on O with constructive interference with 1 - bo E = -0.9127 au

a1

MOs of Water

HOMO-1

Looks like combination of p on O along C2 with constructive interference with bo (close to nbo)

E = -0.3356 au

HOMO-2

Looks like combination of p on O (perp to C2, but in plane of molecule) with constructive interference with bo

E = -0.4778 au

a1b2

MOs of Water

HOMO from two viewpoints

Looks like p orbital on O, perpendicular to plane of molecule - nbo E = -0.2603 au

b1

MOs of Water

LUMO

Looks like combination of p on O along C2 with destructive interference with abo

E = -0.0059 au

LUMO +1

Looks like combination of p on O (perp to C2, but in plane of molecule) with destructive interference with abo

E = 0.0828 au

a1b2

Filling Pattern for Water

1a1 (nbo)

2a1 (bo)

1b2 (bo)

3a1 (bo/nbo)

1b1 (nbo)

4a1 (abo)

2b2 (abo)

Molecular Spectroscopy

• Molecule has a number of motions– Translational, vibrational, rotational, electronic

• Sum them to get total energy of molecule• Changes may occur in any of these

modes through absorption or emission of energy– Vibrational: IR– Rotational: Microwave– Electronic: UV-Vis

CHP 16, 17, 18 of text

Statistical Mechanics

• Quantum gives you possible energy levels (states)– In a real sample, not all molecules in the same energy

level• With statistics and total energy, can predict (on

average) how many molecules in each state– Dynamic Equilibrium– Role of Temperature

• Can predict macroscopic properties/behavior– Heat capacity, pressure, etc.

CHP 19, 20 of text