Intermolecular Forces:Electrostatics

•“Dielectrics

•Different classical electrostatic interactions

Coulomb’s Law•Like charges repel, unlike charges attract•Force is directly along a line joining the two charges

ke = 8.988109 Nm2/C2

1 22

ee

k q qF

r

q1 q2

r

1 22

0

ˆ4e

q qF r

r

0 = 8.85410-12 C2/ (N●m2)•This can change when not in

vacuum

Dielectric

If we place a charge in a media with orientable/polarizable dipoles, the charge will be

“solvated” by the dipoles

The dielectric constant tells us about how electric fields are weakened due to mobility of dipoles.

Dielectric constants depend on mobility, size and polarizability of dipoles

Not readily defined in a heterogenous flexible medium!

Dielectric

In a complex material, poisson’s equation

In a homogenous material, a scale factor

Multiple charges

e i

i

k qV

r

q1

q2

q3

r1 r2

r3

We can handle multiple charges by considering each on explicitly, or by a multipole expansion

Multipole expansion (qualitatively)

When outside the charge distribution, consider a set of charges as being a decomposition of a

monopole, a dipole { and higher order terms}

The monopole term is the net charge at the center of the charges {often zero}

The dipole moment has its positive head at the center of the positive changes, and its negative tail

at the center of the negative charges

Multipole expansion The multipole expansion expands a potential in a

complete set of functions:

00

(cos )4

i

ii

qP

r

The significance is that we can study the different poles one by one, tounderstand any charge distribution

Where might we have a significant dipole moment?

Where might we have a significant quadrapole moment?

Charge-Charge Interaction

r

1 22

04p

q qE

r

0 = 8.85410-12 C2/ (N●m2)

When might we have charge-charge interactions?

Charge-Dipole Interaction

-

20

cos

4p

pqE

r

+

p

EppEU~~cos

+

12

0

ˆ4

qE r

r

��������������

What is ?

Dipole-Dipole Interaction

-21

304p

p p KE

r

+

-

+

Since we have two different vectors, there are two angles, and so the angular component

becomes complicated (see page20)

The angular component is interesting when one has restricted motion, but otherwise only

the radial component is essential

When might restricted motion by interesting?

Why is the angular component not interesting when one hasunrestricted motion?

Npole-Mpole Interaction

1

1p m n

Er

In general, when there are different “poles” interacting, the interaction energy has a r-dependence that increases with increasing

order of the pole.

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}

Why are we only considering dipoles

much?

Induced DipolesWhen 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

Induced Dipoles2

0 0

0 0 2

E E EE p dE E dE

2

0 4 202 (4 )

qE

r

•When the field is due to a charge

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

20

0 6 202 (4 )

pE

r

Proportionality constant depends on

geometry if fixed; 2 if thermal motion

Induced Dipoles

1 26

Er

Precise calculation requires high-quality QM

calc; form from radiation and matter

Included as part of vdW interactions

Thermal Averaging: ion-dipole

exp( / ) exp( cos / )B BE k T pE k T

•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:

coth( / ) /( )B Bp p pE k T k T E

Thermal Averaging: Results

2 2 / 3 BE p E k T

2 / 3 Bp p E k T

•In the high T approximation:

•What is the high T approximation?

•This means that the mean Energy is

•When is the high T approximation realistic?

•This means that the mean Energy is:

2 2 2 40/((4 ) 3 )BE p q k Tr

Thermal Averaging: Dipole-Dipole•In the high T approximation:

2 2 2 61 2 02 /((4 ) 3 )BE p p k Tr

•Note the range!

Why don’t I consider thermal motion with charge-charge interactions?

Hierarchy

1 2q q

r

2

qp

r

1 23

p p

r

2

4

q

r

20

6

p

r

21 2 0

6

p

r

2 2

4

q p

Tr

2 21 2

6

p p

Tr

1 2q q

r

Fixed Thermal

Ion-ion

Charge-dipole

Dipole-dipole

Charge-molecule

Dipole-molecule

Induced dipole-induced dipole