Introduction to molecular structure – Part I

Contents:

atomic orbitals - generalities models used in developing the molecular structuremolecular orbital theoryapplication for hydrogen-ion molecule diatomic moleculesexamples of molecular orbital diagrams

Mathematical description of electrons

Schrödinger equation which for one electron system has the form:

0)(8

2

2

2

2

2

2

2

2

VEh

m

zyx

The solution of this equation are themselves mathematical formulas, but not differential equations. They describe the electrons as waves can be represented graphically. These graphs are 3D pictures that show electron density and are called orbitals or electron clouds.

e

Atomic orbitals

Waves added togehter can occur in two ways:

When waves interact, they don't reflect off each other--they combine. If the amplitudes of the waves are both positive or both negative, the combined wave will have a larger amplitude.

• destructive interference•constructive interference

If the waves have opposite amplitudes, the resulting wave will have a smaller amplitude.

first-wave

second-wave

sum of the waves

Types of atomic orbitals

1s orbital + -- -

+ +

• atomic orbitals: s, p, d, f,g

For example, in a simple lowest-energy state of hydrogen atom, the electrons are most likely to be found within a sphere around the nucleus of an atom. In a higher energy state, the shapes become lobes and rings, due to the interaction of the quantum effects between the different atomic particles

z

x y

+

z

x y

+

2s orbital

Each p orbital has a node. The orbitals lobe are labeled (+) and (-) which referes to the signs of the wave function .

2p orbitals

Again to Shrödinger equation ....

Possible to solve for one electron system, but not possible for molecules containing two or more electrons ...need for approximation

Principals models of molecular structure

1. Molecular orbital theory

2. Valence bond theory

Formation of chemical bonding

EH where represents the many-electron wave-function

Valence Bond Theory

• Explains the structures of covalently bonded molecules

‘how’ bonding occurs

• Principles of VB Theory: Bonds form from overlapping atomic orbitals and electron pairs are

shared between two atoms Lone pairs of electrons are localized on one atom

Molecular Orbital (MO) Theory

• was developed in 1920’s• explains the distributions and energy of electrons in molecules• useful for describing properties of compounds

– Bond energies, electron cloud distribution and magnetic properties

• basic principles of MO Theory– Atomic orbitals combine to form molecular orbitals– Molecular orbitals have different energies depending on overlaping type :

• Bonding orbitals (lower energy than corresponding atomic orbitals AO)• Nonbonding orbitals (same energy as corresponding AO)• Antibonding orbitals (higher energy than corresponding AO)

Note: Both theories are limited case, but Chapter 9 treat a more exact theory

Molecular orbital theory - hydrogen molecule–ion H2+

1s 1s

1 bond

atomic orbitals atomic orbitals

molecular orbitals

and

2 antibond

molecular orbitals

Hydrogen atom has only 1 electron situated on the s orbital

An application: the hydrogen molecule-ion H2+

• the simplest system: two nuclei and one electron

• the equation which characterize this system is given by:

R

e

r

e

r

e

mH

BAe 0

2

0

2

0

22

2

4442

e

A B

rA rB

R

kinetic energy of the electron

attraction of the electron e

by A

attraction of the electron

e by B

repulsion between A and B

Coordinates used to specify the hamiltonian

for the hydrogen molecule-ion

the set of solution lead to construction of the molecular potential energy

the molecular potential energy curves vary with internuclear distance, R.

.... and the equation set of solution

Bonding: the electronic density transfer into the internuclear region: lowering of energy

Antibonding: the electronic density undergo to a small expansion around nuclei increasing the energy

Bonding and antibonding of atomic orbitals molecular orbital formation

Atomic orbitals of the H2+ are added in the same way as the waves. The in-phase

addition of two 1s orbitals will form a molecular orbital with electron density between two nuclei bonding orbital. The out-of-phase addition of the same type of orbitals will produce a molecular orbital with no electron density between two nucleii, called node antibonding orbital.

Bonding and antibonding of atomic orbitals for H2+ - interpretation

2 antibonding orbital

1 bonding orbital

.

In a bonding orbital, the nuclei are attracted to the accumulation of electron density in the inter nuclear region.

In an antibonding orbital, the nuclei are attracted to an accumulation of electron density outside the internuclear region

Mathematical system in bond formation

Theoretically, molecular orbitals can be formed by overlapping the wave function of the atomic orbitals. This overlaping procedure is constructed by forming linear combination of atomic orbitals, LCAO. Adition of the atomic orbitals give the bonding MO:

BBAAB

BBAAA

cc

cc

A, B denote molecular orbitals

A, B wave function for the atomic orbital of atom

having the nucleus in A, B respectively;

cA, cB electrons coeficients which show what part of the electron was involved in the bond formation

bonding

antibonding

e

A B

rA rB

R

Atkins notations

Liniar combination of atomic orbitals LCAO

For species with many atoms, LCAO will have the form:

r

rrc

In principle, we should use an infinite basis set for a precise recreation of the molecular orbital, but in practice only a finite basis set is used. In Atkins is assume that the members of the basis set are real and each one is normalize to 1.

The optimum value of the coefficients are found by applying the variation principle, which means to solve the secular equation:

fomation of basis set

r

rsrsr ESHc 0)(

See section 6.10 from Atkins.

Hrs – matrix element of the hamiltonian

Srs – overlap matrix element

The energies for LCAO-MO:

S

kj

R

eEE

S

kj

R

eEE

s

s

14

14''

0

2

1

''

0

2

1

The energy expression are derived in Atkins, chapter 8.3

Positive integral which is attributed to the total Coloumbic potential energy arising from the interaction of electron clound around A with nucelus in B and B with nucelus in A, respectively

Interaction of the overlap electronic density with nucleus A

A B

j’

A B

k’

A B

The molecular orbital energy level diagram of H+

2 in LCAO approximation

E-

E+

2

1

molecular orbitals

atomic orbitals

En

erg

y

E1s E1s

AB

Diagram asymmetry: an antibonding orbital is more antibonding than a bonding orbital is bonding

The bonding orbital is of lower energy than its atomic orbitals and the antibonding orbital is of higher energy

Cal. with LCAO: R=2.5a0 E=170kJ/mol

Exp: R=2.0a0 ; E=255kJ/mol

How we will use this theory?

1. we used the atomic orbitals from the hydrogen molecule-ion to construct

the wave –functions for many electrons atoms, by adding them one by one into the

calculations

2. we will use the orbitals of the hydrogen molecule-ion to build the MO for diatomic

molecules

Diatomic molecules – angular momentum

• the spatial symmetries of atomic orbitals and the number of each symmetry type are determined by the angular momentum of the electron.

•the angular momentum vector for molecule case will lie along the bond axis. The quantum number in this case is denoted by and it is analogous to the quantum number m in the atomic case. The possible values for are: 0, 1, 2, 3... correspondingly, the allowed values of the angular momentum about the internuclear axis are 0, ±1 (h/2p), ±2(h/2p), etc., or in general, ±l(h/2p). Thus when is different from zero, each energy level is doubly degenerate corresponding to the two possible directions for the component along the bond axis.

   The molecular orbitals are labelled according to the values of the quantum number l . When = 0, they are called s orbitals; when = 1, p orbitals; when   = 2, d orbitals, etc.

the molecule as a whole rotates in space and the nuclei contribute to the total angular momentum of the system. The nuclei and the electrons of a diatomic molecule can rotate around both axes which are perpendicular to the bond axis

Overlaping of 2p orbitals

2px, 2py, 2pz: the direction of orbitals along the axes.

1. Overlaping of 2px orbitals:

Constructive interference from the 2px orbitals

Destructive interference for the 2px orbitals

2. Overlaping of 2py /2pz orbitals

Constructive interference from the 2py orbitals

Destructive interference from the 2py orbitals

Basic principle of molecular orbital formation:

-Linear combination of atomic orbitals that have the same symetry species, i.e.span the same irreducible representation, within the molecular point group. As was shown in chapter 5.16 ( see M.H. Lecture), only orbitals of the same symetry species may have a nonzero overlap (S0) and hence contribute to the bonding

Diatomic molecules

Criteria for selecting the correct bonding

Group theory provides techniques for selecting the atomic orbitals that may contribute to bonding, but others types of arguments must be used to decide whether these orbitals do in fact contribute and to what extent two important criteria:

1. Atomic orbitals must be neither too diffuse or too compact (their overlaping is too diffuse to have a high significance)

2. The energies of the orbitals should be similar.

Filling of Orbitals

The valence eletctrons of the atomic orbitals are used to fill in the molecular orbitals. The rules of filling in the MO are the same as filling in the AO

Diatomic molecules – periodic table where only s and p orbitals are important

1. Homonuclear molecules – the same atomic species:ex: Li2, O2, N2...

2. Heteronuclear molecules – different atomic species: ex: CO, HF...

HOMO and LUMO are acronyms for Highest Occupied Molecular Orbitals and Lowest Unoccupied Molecular Orbitals respectively.

The energy level difference of the two (HOMO-LUMO) can (sometimes) serve as a measure of the excitability of the molecule: the smaller the energy, the easier it will be excited.

The HOMO and LUMO state

AO AOMO

2s 2s2s

The MO diagram for Li2E

nerg

y

2p 2p

*2s

2p

2p

*2p

*2p

Ene

rgy

The MO diagram for O2

2p

2p

2p

*2p

*2p

AO AOMO

2s 2s2s

*2s

2p

12 valence eletrons2 valence electrons

HOMO

LUMO

LUMO

HOMO

MO for CO molecule

C has 4 valence electrons O has 6 valence electrons

Carbon AO

MO

2s2s

2s

*2s

2p2p

2p

*2p

*2p

2p

Oxygen AO

HOMO

LUMOThe CO molecule has 10 valence

eletrons

..... To be continued by Staffan in two weeks