Dr M. Mehrdad University of Guilan, Department of Chemistry,

Rasht, Iran

MO theory considers the electrons in molecules to occupy MOs that are formed by linear combinations (addition and subtraction) of all the atomic orbitals on all the atoms in the structure.

In MOT, electrons are not confined to an individual atom plus the bonding region with another atom. Instead, electrons are contained in MOs that are highly delocalized – spread across the entire molecule.

MOT is based on the Schrödinger equation. HΨ = EΨ H: Hamiltonian operator Ψ: wavefunction describing an orbital E: the energy of an electron in a particular orbital

obtain Ψ and this equation Ψ = Σciφi (linear combinations of all the atomic orbitals) ci = coefficient φi = atomic orbital

To construct group MO’s, one needs to understand how to combine AO’s properly.

This procedure is called qualitative molecular orbital theory(QMOT) 2

Rules of QMOT

1. Consider valence orbitals only (e.g., for Carbon, 2s, 2px, 2py and 2pz) 2. Form completely delocalized MO’s as linear combinations of s and p AO’s. Remember, combination of n AO’s gives n MO’s 3. MO’s must be either symmetric or antisymmetric with respect to the symmetry operations of the molecule. 4. Compose MO’s for structures of higher symmetry and then produce MOs for related, but less symmetric systems by systematic distortion of the MOs for higher symmetry. For example, for the CH2

system, start with linear HCH (D∞h) then bend the system (C2v). 5. Molecules with similar molecular structures, e.g., CH3 and NH3, have qualitatively similar MO’s, the major difference being the number of valence electrons that occupy the common MO system. 6. The total energy of the system is the sum of the MO energies of the individual valence electrons. ( occupied MO’s) 7. If the two highest energy MO’s of a given symmetry derive primarily from different kinds of AO’s (e.g., s and p), then mix the two MO’s to form hybrid orbitals. For example, for the AH2

system (p.3), mix C and E orbitals to form hybrid C’ and E’. 3

8. When two orbitals interact, the lower energy orbital is stabilized and the higher energy orbital is destabilized. The out-of-phase or antibonding interaction between the two starting orbitals always raises the energy more than the corresponding in-phase or bonding interaction lowers the energy..

(energy of stabilization, estab, is always smaller

that energy of destabilization, edestab. Thus,

4electron-2center interaction is always

repulsive. )

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9. When two orbitals interact, the lower energy orbital mixes into itself the higher energy one in a bonding way, whereas the higher energy orbital mixes into itself the lower energy one in an antibonding way.

(If orbitals of different energy interact

(b), the one of lower energy, B, will

contribute more in binding orbital; the

one of higher energy, A, will contribute

more in antibonding orbital.)

b)

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10. The smaller the initial energy gap between the two interacting orbitals, the

stronger the mixing interaction.

11. The larger the overlap between interacting orbitals, the larger the

interaction. (σ-bonds are stronger than π-bonds.)

12. The more electronegative elements have lower energy AO’s.

13. A change in the geometry of a molecule will produce a large change in the

energy of a particular MO if the geometry change results in changes of AO

overlap that are large.

14. The AO coefficients are large in high energy MO’s with many nodes or

complicated nodal surfaces.

15. Energies of orbitals of the same symmetry classification cannot cross each

other. Instead such orbitals mix and diverge.

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example

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Symmetry Elements E: Identity operation

عنصر یکسانیCn: Proper rotation

محور چرخشی

CH3

H

H

H3C

C2

H3C

H

H

CH3

H

H

H

H

H

H

H

H

H

H

H

HC6

H

H

H

H

H

H

C2

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Symmetry Elements

i: Inversion

عمل وارونگی W

Cl

Cl Cl

Cl

Cl

Cl

i

WCl

Cl Cl

Cl

Cl

Cl

sh: Horizontal Mirror Plane

صفحه آیینه ای افقی

sv: Vertical Mirror Plane

صفحه آیینه ای عمودی

C4

H

H

H

H

H

H

H

H

sh H

H

H

H

H

H

H

H

Br Br

HH

C2

sv

Br Br

HH

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Symmetry Elements Sn: Improper rotation: combination Cn and sh

عمل چرخشی انعکاسی S2 is equivalent to inversion (i)

Me

Me

Me

MeC2

Me

Me

Me

Mesh

Me

Me

Me

Me

S2

center of symmetry

H1 H2

H4H3C4

H4H3

H1 H2

sh

H3 H4

H1H2

S4

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Symmetry Groups

• Groups with no proper rotation axis

– C1: Only E (i.e. no symmetry elements)

– Cs: E and s

– Ci: E and i

– Sn: E, Sn (S1 = Cs; S2 = Ci)

• Groups with one proper rotation axis

– Cn: E, Cn only

– Cnv: E, Cn, and n sv (linear unsymmetrical molecules are C∞v)

– Cnh: E, Cn, and sh

• Dihedral Groups: Groups with n C2 axes to Cn

– Dn: E, Cn, and n C2 axes to Cn

– Dnh: E, Cn, n C2 axes, and sh (linear symmetrical molecules are D∞h)

– Dnd: E, Cn, n C2 axes, and n sv

• Cubic Groups: Groups with more than one Cn (n ≥ 3)

– Td: symmetry of a regular tetrahedron: 4 C3

– Oh: symmetry of a regular octagon: 6 C4

– Ih: symmetry of a regular icosahedron: 12 C5

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F

CCl Br

H

C1

H

Br BrBr

HH

Cs

HH

FCl

FCl

Ci

CH3

CH3

C2

NH H

H

C3v

H F

F H

C2h

D6h

H

CH H

H

Td

WC

C C

C

C

C

O

O

O

O

O

O

Oh

Ih

D2

S4

Br

Br Br

Br

HH

H

H

H

H HH

HH

D3d

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O C S

C•v∞

O C O

D•h∞

Symmetry Decision Tree

Linear?

Find

principal

axes

C∞v or

D ∞h

More than one

Cn (n ≥ 3)

Cubic

T, O, I

Cs, Ci

or C1

Cn is the

principal

axis?

nC2

to Cn?

S2n colinear

w/ Cn?

n vertical

mirror

planes

S2n

sh? sv? Dnd

Dnh Dn

sh? sv? Cn

Cnv

Cnh

Yes

No

Yes

Yes

Yes

Yes

Yes

Yes

Yes None

No

No

No

No No

No

No

Physical Chemistry, Joseph H.

Noggle, 2nd ed., Scott Foresman

& Co, Glenview, IL, 1996, pg 840.

Yes

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To illustrate the procedures of qualitative MO theory, we will “build” the MO’s of planar CH3. We choose planar CH3 because it is more symmetrical.

We will be using: - three H 1s orbitals, - one C 2s orbital, - and three C 2p orbitals.

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1. First, mix the carbon 2s orbital with the three H AO’s in-phase to

produce orbital A

which is symmetric with respect to the C3-axis of symmetry of the molecule. [Focus on the low-lying, bonding MO’s, the orbitals of which are mixed in-phase (bonding).] Using out-of-phase mixing, one gets the high energy, antibonding

orbital E.

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2. Next, use the carbon p-orbitals. The pz AO cannot mix with any of the H orbitals, since they all lie on the nodal plane of this orbital.

We thus have an MO that is simply an atomic p-orbital, D.

The px and py AO’s can mix with the 1s orbitals of the H atoms to give

favorable interaction patterns, as seen in MO’s B and C.

B and C are degenerate – of the same energy. 17

Orbitals A, B, C and D are the group orbitals for planar CH3.

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MOs for Planar AH3

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A diagram that follows orbital energies as a function of angular

distortions is termed a Walsh diagram.

Pyramidalization lowers the energy of orbital A slightly (slight H-H

interaction);

it raises the energies of B and C more because of the loss of overlap

between the p orbitals and the hydrogen orbitals.

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The biggest impact, however, is on orbital D.

This orbital is non-bonding when planar, but becomes increasingly bonding upon pyramidalization.

Orbitals A-C are strongly C-H bonding, whether planar or pyramidal.

In a VB model, we would want three C-H bonds, each with two electrons, for

a total of six C-H bonding electrons.

The two models agree. With QMOT, we still have three C-H bonds, described

by three occupied MOs that are strongly C-H bonding.

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Consistent with Rule 7, 7. If the two highest energy MO’s of a given symmetry derive primarily from different kinds of AO’s then mix the two MO’s to form hybrid orbitals.

MO’s D and E, having the same symmetry, but one based on a carbon p

orbital and the other using a 2s orbital, leads to mixing of these two

orbitals to form D’ and E’.

D’ now looks more like a lone pair orbital, and it resembles an spi hybrid

at carbon.

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Group orbitals – a collection of partially delocalized orbitals that is consistently associated with a functional group or similar collection of atoms in a molecule.

A, B, C, D (or D’) are the Group orbitals of the methyl group, and we can

use these orbitals to model the bonding in any molecules that contain the methyl group.

1. Low-lying C-H bonding orbitals derived from carbon 2s orbitals and of s-symmetry are termed s(CH3) orbitals. 2. The C-H bonding orbitals that are derived from carbon 2p orbitals, of -symmetry, are termed (CH3) orbitals. They are a degenerate pair. 3. The other orbital of s-symmetry, that derived from the carbon pz AO, points away from the H’s and is termed the s(out) orbital.

http://www.chem.ox.ac.uk/vrchemistry/orbitals/html/page11.htm 25

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