Quantum Theory and Molecular Energy
Going to consider:
1) The energy of individual molecules
2) The Classical (Newtonian) equations for this energy (1st year Physics)
3) The Quantum equations for this energy
4) The Schrödinger Equation which is the source of these expressions
5) The interpretation of the wavefunction
6) The solution of the Schrödinger Equation for some simple cases
Not going to consider:
1) Why quantum mechanics had to be introduced in the early 20th century.
2) The philosophical meaning of a wave mechanical description of matter
3) Some of the absolutely fascinating mathematical procedures needed to solve some
of the most important quantum chemical problems.
Strongly recommend that anyone finding this interesting take PY2002 as an optional
module in 3rd year. The theoretical/philosophical background will be covered there.
The energy equations considered in this course are the ones used in CM2007 and
CM3003 to explain the energies of spectroscopic transitions and wavefunctions are used
to explain the intensity of transitions.
The topics covered here are developed in CM3109/CM4113 to show the connection
between the energy of individual molecules and thermodynamics; the basics of molecular
bonding; the theory of molecular structure calculations – calculating geometries,
properties and reactivity from Classical and Quantum approaches.
Molecular Energy
Consider a diatomic molecule, A – B, as an example. It consists of 2 nuclei, masses, mA
and mB, chemically bonded by electrons.
Consider the energy of the nuclei separately from the electrons (a reasonably good
approximation). The electrons are replaced (represented) by a ‘spring’ and the two nuclei
are separated by the bond length, R.
mB
mA
R
The bond length ‘vibrates’ about an average value Re.
This is a good model of the reality.
Classical (Newtonian) physics shows that the energy of two mass points (the nuclei)
joined in this manner and moving in 3 dimensions can be separated into three
components.
Translational Energy.
The motion of the whole molecule through space. The energy is Kinetic energy.
Etranslation = m v2 / 2 = p2 /2m
v
m = mA + mB
p=mv
velocity
total mass
momentum
Note: v2 = vx2 + vy2 + vz2 therefore there are actually 3 separate components of the
translational energy:
Etranslation = m (vx2 + vy2 + vz2 )/ 2
= m vx2/ 2 + m vy2/ 2 + m vz2/ 2
= px2 /2m + py2 /2m + pz2 /2m
Rotational Energy.
Rotation of the molecule about its centre of mass. The energy is the Kinetic energy of
rotation. The bond length of the molecule is fixed at its average value, Re.
x
z
y
Rotation about y axis shown. There is no rotation
about z axis.
centre of
mass
Erotation = I ω2 /2 = μ Re2 ω2 /2 for rotation about an axis
I = μ Re2
μ = mA mB / (mA + mB)
momentum of inertia
reduced mass
There are actually two components of rotational energy, rotation about the X and Y axes
but they are identical axes, therefore ωx2 = ωy2 = ω2
Erotation = μ Re2 ωx2 /2 + μ Re2 ωy2 /2 = I ω2
Vibrational Energy.
To write an energy expression for the vibration of the bond length, have to assume a
model for this motion. The common (and easiest) model is Hooke’s Law or the ‘Simple
Harmonic Oscillator’. This states that as the bond gets longer (or shorter) than Re there is
a force restoring it to Re.
The energy consists of the Kinetic energy of vibration and Potential energy of the
‘spring’.
This results in an energy expression:
Evibration = μ vR2/2 + (k/2)(R – Re)2
vR
k
velocity of vibrational motion along the bond length
the force constant – the larger k the stronger the ‘spring’
Enulear = Etranslation + Erotation + Evibration
The six individual components (3 trans, 2 rot, 1 vib) outlined above correspond exactly to
the energy equation obtained by writing:
z
E = kinetic energy of A + kinetic energy of B
+potential energy of AB
mB, xB, yB zB
mA, xA , yA, zA
= mAvxA2 /2 + mAvyA2/2 + mAvzA2/2 + mAvxB2/2 + mAvyB2/2
+ mAvzB2/2 + (k/2)(R – Re)2
x
y
It is just a matter of changing from 6 coordinates
(vxA, vyA, vzA, vxB, vyB, vzB) to 6 coordinates
(x, y, z) of centre of mass, (x, y) of rotation and R of
vibration.