Transcript Lecture #3 09/02/04

Annoucements
Office Hours:
Dr. Salsbury 4-6pm Mon, 4-5 Tues, by appt
{occasionally have to cancel; such as next
Monday 08/08/04}
Dr. Fetrow 2:30-3:30 Thursday
Questions, concerns comments?
Multipole expansion
The multipole expansion expands a potential in a
complete set of functions:


Pi (cos  )  

4 0 i 0
r
q

i
The significance is that we can study the different poles one by one, to
understand any charge distribution
Where might we have a significant dipole moment?
Where might we have a significant quadrapole moment?
Charge-Charge Interaction
r
q1q2
Ep 
2
4 0 r
0 = 8.85410-12 C2/ (N●m2)
When might we have charge-charge interactions?
Charge-Dipole Interaction
+

-
~
~
U   pE cos    p  E
p
+
E
 pq cos 
Ep 
4 0 r 2
q1
4 0 r
2
rˆ
What is U in this configuration?
What is ?
Dipole-Dipole Interaction
Since we have two different vectors, there are
two angles, and so the angular component
becomes complicated (see pages 19-20)
+
+
-
Ep 
p1 p 2 K
4 0 r 3
When is this a minimum?
The angular component is interesting when
one has restricted motion, but otherwise only
the radial component is essential
Why is the angular component not interesting when one has
unrestricted motion?
When might restricted motion by interesting?
Npole-Mpole Interaction
In general, when there are different “poles”
interacting, the interaction energy has a rdependence that increases with increasing
order of the pole.
Ep 
1
r
m  n 1
The decreasing range of the electrostatics is
why higher order poles are less important,
especially in biomolecules, where they many
charges and dipoles {and quadrupoles around}
Thermal Averaging: ion-dipole
•Recall: At nonzero finite temperature, thermal energy can result in the
population of multiple states inside an ensemble
•What does this mean?
•We have to consider the statistical weight of each possible orientation
exp( E / kBT )  exp( pE cos  / kBT )
•Integrate to determine the mean value of p in the direction of the field:
p  p coth( pE / kBT )  kBT /( E)
Thermal Averaging: Results
•In the high T approximation:
•What is the high T approximation?
p  p2 E / 3kBT
•When is the high T approximation realistic?
•This means that the mean Energy is
E   p2 E 2 / 3kBT
•This means that the mean Energy is:
E   p q /((4 0 ) 3kBTr )
2
2
2
4
Thermal Averaging: Dipole-Dipole
•In the high T approximation:
E  2 p12 p22 /((4 0 )2 3kBTr 6 )
•Note the range!
Why don’t I consider thermal motion with charge-charge interactions?
Induced Dipoles
When a molecule is placed in an external field,
the electron distribution is distorted
For example: when a molecule is placed in
water, the electric fields from the water
molecules will change the electron
distributions
pE
First approximation: with the polarizability
being the coefficient
What difficulties might there be with this
approx?
Induced Dipoles
E
E
E2
E    p  dE   0  E  dE   0
2
0
0
•When the field is due to a charge
q2
E   0 4
2r (4 0 ) 2
•When the molecule has a scalar polarizability, and there is a dipole:
p0 2
E   0 6
2r (4 0 ) 2
Proportionality constant depends on
geometry if fixed; 2 if thermal motion
Induced Dipoles
E
1 2
r
6
Precise calculation requires high-quality QM
calc; form from radiation and matter
Included as part of vdW interactions
Hierarchy
Ion-ion
Charge-dipole
Dipole-dipole
Charge-molecule
Dipole-molecule
Fixed
Thermal
q1q2
r
qp
 2
r
q1q2
r
q2 p2

Tr 4
p12 p2 2

Tr 6
p1 p2
 3
r2
q
 4
r

 p 20
r6
Induced dipole-induced dipole

1 2 p 20
r6