Transcript Lecture 3

Chemistry 445.
Lecture 3.
Molecular Orbital Theory
We start by reminding ourselves of the shapes and signs of the
wavefunction on the atomic orbitals. Below are the s and three p
orbitals, showing boundary surfaces (H&S Fig. 1.9)
Note: Pink color indicates sign of wavefunction opposite to that of
the white part of the orbital.
The five atomic d-orbitals
Orbitals and Quantum numbers:
The solution to the Schrödinger waveequation leads to a set of wavefunctions that
yields 4 types of quantum numbers instead
of the single quantum number yielded by the
Bohr model.
These are:
1) The principal quantum number, n, which
has values of 1,2,3,… This corresponds to
the quantum number n in the Bohr model of
the atom.
Quantum numbers (contd.)
2) The Azimuthal quantum number l, which has values
of 0 to (n-1) for each value of n. The different values of l
correspond to orbital types as follows:
l
=
0
1
2
3
letter used
=
s
p
d
f
3) The magnetic quantum number ml, can have values
of –l through 0 to +l for each value of l.
Value of l
0
1
2
3
possible values of ml
0
-1,0,+1
-2,-1,0,+1,+2
-3,-2,-1,0,+1,+2,+3
4) The spin quantum number (ms). This can
have values of +½ or –½. This means that for
each value of ml there are two values of ms. It
is this that leads to the occupation of each
orbital by two electrons of opposite spin, i.e.
with ms = +½ or –½.
These quantum numbers lead to the shells
(different values of n) and subshells
(different values of l) that lead to our modern
understanding of chemistry. The number of
orbitals in each sub-shell (1 for s, 3 for p, 5
for d, and 7 for f sub-shells) is determined by
ml, and ms determines that only two
electrons of opposite spin can occupy each
orbital.
Representation of orbitals:
Schrödinger’s model:
z
y
x
s-orbital
p-orbital
(1 of 3)
d-orbital
(1 of 5)
f-orbital
(1 of 7)
The H-atom compared to many-electron
atoms:
4p
n=4
n=4
4s
4p
4d
4f
4s
n=3
3s
3p
3d
3s
n=1
1s
H-atom. All subshells within
same shell have the same energy.
2p
n=2
energy
energy
2p
3p
n=3
n=2
2s
3d
2s
n=1
1s
Many-electron atom. Subshells
within same shell have different
energies.
The essence of MO theory is that overlap of two orbitals always occurs in
two ways. In one (bottom), the two 1s orbitals shown here overlapping have
the same sign of the wavefunction, and so a net overlap occurs. This
produces a lower energy bonding orbital. In the upper case, the two orbitals
are of opposite sign, and so no net overlap occurs. This produces a higher
energy anti-bonding orbital.
higher energy
anti-bonding orbital
Sign of wavefunction is opposite
+
1s
σ*1s
1s
+
σ1s
1s
1s
sign of wavefunction is the same
lower energy
bonding orbital
Drawing up a Molecular Orbital (MO) diagram for H2
arrow represents
electron in 1s orbital
energy
1s atomic
orbital of H
atom
1s atomic
orbital of H
atom
energy level
of 1s orbital
of H-atom
Drawing up a Molecular Orbital (MO) diagram for H2
σ*1s anti-bonding
molecular
orbital in H2
molecule
energy
1s atomic
orbital of H
atom
1s atomic
orbital of H
atom
σ1s bonding
molecular
orbital in H2
molecule
These are the molecular
orbitals of the H2 molecule
Molecular Orbital (MO) diagram for H2 molecule (bond
order = 1)
σ* 1s
asterisk denotes
anti-bonding orbital
arrow = electron
1s atomic
orbital of H
atom
σ 1s
atom
1s atomic
orbital of H
Some observations on MO diagrams:
A single bond consists
of a shared pair of
electrons (Lewis). In
MO theory Bond
Order (BO) =
(No. of e’s in
bonding levels –
no. of e’s in antibonding levels)/2
the two arrows are
opposite in direction
indicating a pair of spin-paired
electrons of opposite spin
BO for H2 = (2-0)/2 = 1
in labeling the molecular
orbitals, the type of
overlap is specified
(σ or π), and the atomic
orbitals involved indicated.
because of the Pauli exclusion
Principle each orbital can contain
a maximum of two electrons, which
must be of opposite spin
Some more observations on MO diagrams:
In MO theory the reason molecules form is
because the bonding orbitals formed are lower
in energy than the atomic orbitals, and the
electrons are lowered in energy by this amount.
The greater the
drop in energy
the stronger the
bond. For the H2
molecule the
drop is 218 kJ/mol
so the enthalpy of
dissociation of the
H2 molecule is
436 kJ/mole
Even more observations on MO diagrams:
Photon of
Energy = hv
Electron excited
to anti-bonding level
BO = (1-1)/2 = 0
for excited state
MO diagrams show how a photon of energy = hv = the difference in energy
between two MO’s, can cause an electron to be excited to the higher
energy level MO. In this excited state the bond order = zero and so the
H2 molecule can photo-dissociate. Whether the transition can occur is also
determined by the parity of the orbitals (g or u) – see later.
Identification of bonding and non-bonding molecular
orbitals.
A bonding MO has no nodal plane between the two atoms forming
the bond, i.e. the electron density does not go to zero at a node. An
anti-bonding MO has a nodal plane where electron-density = zero:
nodal plane
σ(1s) bonding orbital
π(2p) bonding orbital
σ*(1s) anti-bonding orbital
π*(2p) anti-bonding orbital