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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:.

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Annoucements

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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

Multipole expansion

The multipole expansion expands a potential in a complete set of functions:

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

0= 8.85410-12 C2/ (N●m2)

When might we have charge-charge interactions?

### Charge-Dipole Interaction

+

q

-

p

+

What is U in this configuration?

What is q?

### Dipole-Dipole Interaction

Since we have two different vectors, there are two angles, and so the angular component becomes complicated (see pages 19-20)

+

-

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 r-dependence that increases with increasing order of the pole.

Why are we only considering dipoles much?

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

• Integrate to determine the mean value of p in the direction of the field:

### Thermal Averaging: Results

• In the high T approximation:

• What is the high T approximation?

• When is the high T approximation realistic?

• This means that the mean Energy is

• This means that the mean Energy is:

### Thermal Averaging: Dipole-Dipole

• In the high T approximation:

• 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

First approximation: with the polarizability being the coefficient

What difficulties might there be with this approx?

### Induced Dipoles

• When the field is due to a charge

• When the molecule has a scalar polarizability, and there is a dipole:

Proportionality constant depends on

geometry if fixed; 2 if thermal motion

### Induced Dipoles

Precise calculation requires high-quality QM calc; form from radiation and matter

Included as part of vdW interactions

### Hierarchy

Fixed

Thermal

Ion-ion

Charge-dipole

Dipole-dipole

Charge-molecule

Dipole-molecule

Induced dipole-induced dipole