Chapter 10. Molecular structure

Chapter 10. Molecular structure

-The Born-Oppenheimer approximation -Valence bond theory (VB theory)

shared e- pair

-Molecular orbital theory (MO theory)

extension of atomic orbital

-Molecular orbitals for polyatomic systems

The Born-Oppenheimer approximation [1]

• assuming that nuclei is stationary (it is heavier and slower than electron)

• solve Shrodinger eqn for e- alone

• nuclei in H

₂

move 1 pm while e- speeds 1000 pm

•Molecular potential E curve (E_kof the stationary nuclei is 0) Equilibrium bond length

D_e, D₀=D_e-hv/2

•A molecular potential curve

•Eq. bond length~E min.

Valence bond theory

:

pairing of the e-s and the accumulation of e- density in the internuclear region (from pairing)

10.1 Homonuclear diatomic molecules

The hydrogen molecules

more stable

) ( )

(

_{1 1 2}

1

r r

B

A H S

H S





 

Valence bond wavefunction

cylindrical symmetry around internuclear axis

zero orbital angular momentum around internuclear axis

σ bond

• Opposed spin (pairing) -> spatial wavefunction total VB for 2 electrons

interchanging the label 1 and 2,

Due to Pauli principle,

Homonuclear diatomic molecules

ex. N₂

conventionally,

2 1 1 1

2 2s p_x2p_y2p_z z-axis

overlap of 2P_x, 2P_y orbitals: bond

(orbital angular momentum about the internuclear axis)



Polyatomic molecules

2 2 1 1

2 2s p_x2p_y2p_z ex. H₂O

for O,

How about NH₃?

σbond: spin pairing of e- with cylindrical symmetry

bond: approximate symmetry



2 1 1

2 2s p_x2p_y For C in CH₄,

Although energy was required to promote the e-, it is more than recovered by the promoted atom’s ability to form 4 bonds

1 1 1 1

2 2s p_x2p_y2p_z

only two bonds?

four bonds!

Hybridization ex. CH

₄

Four sp

³

hybrid orbitals

σ bond

Directional character:

constructive interference between s and p

increased bond strength

three σ bonds

contribution of s and p:

for h₁, h₂and h₃~ 1:2 (square of coefficients)

10.11

3^rd2p: not included in the hybridization bond -> locks the planar arrangement



for ethyne, CHCH sp hybrid orbitals

σ bond σ bond

remaining 2p orbitals two bonds 

10.13

Hybridization of N atomic orbitals:

N hybrid orbitals

ex. sp

³

d

²

hybridization -> 6 equivalent hybrid orbitals octahedron… SF

₆

Molecular orbital theory

10.3 The hydrogen molecule-ion

10.4 Homonuclear diatomic molecules

10.5 Heteronuclear diatomic molecules

The hydrogen molecule-ion

For H

₂⁺

,

The Schrodinger eqn can be solved under Born-Oppenheimer approximation.

But, it cannot be extended to polyatomic system.

Linear combination of atomic orbitals (LCAO-MO)

For H₂⁺,

A denotes 

_H1sA

, and B denotes 

_H1sB

Normalization~

10.14 10.15

Not independent

How to draw?

Ex. 10.1

Bonding orbitals

probability density if e-s were confined to the atomic orbital A

probability density if e-s were confined to the atomic orbital B

•An extra contribution to the density

•probability density in the internuclear region

•e- accumulation in regions

where atomic orbitals overlab and interfere constructively

1

: bonding orbital ex. 1





10.16

e- accumulation when R is suitably large

10.18

1 1

H s H s

E

_

 E  E

_

 E

The antibonding orbital is more antibonding than the bonding orbital is bonding

10.19

Antibonding orbitals

Destructive interference

internuclear nodal plane exist!

*

*

: bonding orbital ex. 2

 anti



if occupied, reduce cohesion

and raise E compared to E

10.20

10.21

10.22

10.24

The structure of diatomic molecules

H₂

He cannot form diatomic molecule.

Pauli exclusion principle and Hund’s rule

Bond order

bond length, bond strength

number of e- in bonding orbital

number of e- in antibonding orbital

Period 2 diatomic molecules

All orbitals of the appropriate symmetry contribute to a molecular orbital.

ex. σ orbital: LC of all orbitals that have cylindrical symmetry

general form of σ orbital

general form

(2s and 2p have distinct different E)

10.26

orbitals



x y

* *

2p , 2p orbitals :

, orbital (degenerate) and , orbital (degenerate)

x y

x y

 

 

10.27

10.28

The overlap integral

Fig. 10.29

10.30

10.31

assumption: 2s and 2p contributes different MO

10.33

10.34

Figure 10.35

Figure 10.37

Figure 10.38

Heteronuclear diatomic molecules

ex. CO or HCl polar bond:

e- distribution in the covalent bond between the atoms is not evenly shared





H-F 

partial negative charge partial positive charge

-Polar bonds

nonpolar bond

2 2

A B

c  c

pure ionic bond: one coeff. 0 ex) A+B-: c_A=0, c_B=1

for HF, σ bond in HF, mainly F2p

10.39

Pauling electronegativity

most electronegative

least electronegative greater the difference in electronegativities, more polar

Mulliken electronegativity scale

-The variation principle

discussing bond polarity

method of getting the coefficients

Methodology of the variation principle:

if an arbitrary wavefunction is used to calculate the E, the calculated E is never less than the true E.

(we need to calculate the wavefunction until the E is lowest.)

secular equations

Coulomb integral: E of e- (_Afor A, or _Bfor B) negative.

resonance integral: 0 if the orbitals do not overlap at eq bond length, it is negative.

in LC is real, but, not normalized



differentiation

Coulomb integral resonance integral

wrt c_Aor c_B

secular determinant,

-Two simple cases [1]

1. homonuclear diatomic molecules

(when two atoms are the same, _A = _B = )

bonding antibonding

-Two simple cases [2]

2. heteronuclear diatomic molecules with S=0

then,

(when E difference is larger than the resonance integral)

The orbitals are almost pure A and pure B…

Calculating the molecular orbitals of HF

Molecular orbitals for polyatomic systems:

MO spreading over entire molecules general form of MO:

diatomic molecules: linear

polyatomic molecules: not necessarily linear

10.6 The Huckel approximation 10.7 Computational chemistry

10.8 Prediction of molecular properties

The Huckel approximation

-Ethene and frontier orbitals

orbitals as LCAOs of the C2p



C2p orbitals on atom A

-for Ethene

3. all remaining resonance integrals are equal (to β)

1. all diagonal elements: α-E

2. off-diagonal elements between neighboring atoms: β. 3. all other elements: 0

+: bonding conformations - : antibonding

β: negative

1  2

excitation E: 2 lβl LUMO (lowest unoccupied molecular orbital)

-The matrix formulation of the Huckel method

overlap matrix For a two-atom system,

note) N orbitals (here two) -> N eigenvalues (E) and N column vectors (c)

We need to N eqns.

In Huckel approximation, overlap matrix, S, is unit matrix.

Introducing two matrices for simplicity,

Then,

-Butadiene and -electron binding energy

• LCAO-Mos for butadiene,

• total -electron bindingenergy, E_

butadiene ethene

More internuclear node, higher E

Figure 10.43

-Benzene and aromatic stability

-bond formation energy: 8β

aromatic stability:

1. strong σ bonds due to regular hexagon

(σframe work is relaxed and without strain)

10.44

10.45

Extended Huckel theory

-The matrix formulation of the theory

diagonal component: 

off-diagonal component: (for HT, β or 0) difference betw. HT and EHT:

1. EHT considers σ and  2. overlap integral

-Population analysis

Overlap population Mulliken population analysis atomic population

Still EHT cannot predict the correct 3-d structures.

-The Hartree-Fock eqns

-Semiempirical and ab initio method -Density functional theory

-The Hartree-Fock eqns

Hartree-Fock eqns

Fock operator

Hartree-Fock eqns

Roothaan eqns

-Semiempirical and ab initio method

10.46

-Density functional theory

exchange correlation E

local density approximation

-Electron density and the electrostatic potential surfaces

-Thermodynamic and spectroscopic properties

Structure 10.5

Structure 10.6

Structure 10.8 Structure 10.9