n + µr (n + 1) a2

4πµ0 a

n=0

Here Pn (cos θ) are the Legendre polynomials (see page 361). Because of the

axial symmetry of the problem, the result does not depend on the azimuthal

angle φ of the observation point. This equation is exact, but is hard to

350 Electromagnetism

evaluate. Now, by expanding the ¬rst term in the sum:

n+1 1

(1 + c + c2 + · · ·),

= (13.91)

n + µr (n + 1) µr + 1

where

1

c= , (13.92)

(µr + 1)(n + 1)

one obtains an expansion of ¦R (r, θ):

(0) (1) (2)

¦R = ¦R + ¦R + ¦R + · · · . (13.93)

The exact expression for the terms in this expansion is

∞ n

(µr ’ 1) a q r

(k)

=’ k

¦R (r, θ) (n + 1) Pn (cos θ).

(µr + 1)k+1 s 4πµ0 (a2 /s) a2 /s

n=0

(13.94)

(0)

The zero-order term ¦R is just the Legendre-polynomial expansion (for

r < a2 /s) of the potential at (r, θ) due to a charge qim at a position a2 /s on

the z-axis:

’1/2

2

a2 a2

qim

(0)

+ r ’ 2r cos θ

2

¦R (r, cos θ) =

4πµ0 s s

qim

= , (13.95)

4πµ0 |r ’ r im |

with

(µr ’ 1) a

qim = ’ q. (13.96)

(µr + 1) s

(1) (0)

The ¬rst-order term ¦R is considerably smaller than ¦R /(µr + 1), which

(0)

is also true for the ratio of subsequent terms, so that ¦R is a good approxi-

mation to the exact reaction potential when µr 1.

So, in conclusion: the reaction potential of a source charge q at position

(s, 0) within a sphere (vacuum, µ0 ) of radius a is well approximated by the

Coulomb potential (in vacuum) of an image charge qim (13.96) located at

position r im = (a2 /s, 0) outside the sphere on the same axis as the source.

The interaction free energy of the source q at distance s from the center

(s < a) with its own reaction potential in the image approximation is

µr ’ 1 q2 a2

1 (0) 1

UR (s) = q¦R (s) = ’ . (13.97)

a2 ’ s2

2 2 µr + 1 4πµ0 a

13.7 Quasi-stationary electrostatics 351

We can compare this result with the exact Born free energy of solvation.

For a charge in the center (s = 0), we obtain

µr ’ 1 q2

1

UR (0) = ’ , (13.98)

2 µr + 1 4πµ0 a

while the Born energy, according to (13.60), equals

µr ’ 1 q2

1

=’

UBorn . (13.99)

2 µr 4πµ0 a

The di¬erence is due to the neglect of higher order contributions to the

reaction potential and is negligible for large µr . We can also compare (13.97)

with the exact result in the other limit: s ’ a (d a) approaches the

case of a charge q at a distance d = a ’ s from a planar surface. The

source is on the vacuum side; on the other side is a medium (µr ). The

exact result for the reaction potential is the potential of an image charge

qim = ’q(µr ’ 1)/(µr + 1) at the mirror position at a distance d from the

plane in the medium (see Exercise 13.3). In this limit, (13.97) simpli¬es to

µr ’ 1 1 q2

1

UR (d) = ’ , (13.100)

2 µr + 1 4πµ0 2d

which is exactly the free energy of the charge in the reaction potential of its

mirror image.

When the sphere contains many charges, the reaction potentials of all

charges add up in a linear fashion. Thus all charges interact not only with

all other charges, but also with the reaction potential of itself (i.e., with its

own image) and with the reaction potentials (i.e., the images) of all other

charges.

13.7.5 The generalized Born solvation model

When charges are embedded in an irregular environment, e.g. in a macro-

molecule that itself is solvated in a polar solvent, the electrical contribution

to the free energy of solvation (i.e., the interaction energy of the charges with

the reaction potential due to the polar environment) is hard to compute. The

“standard” approach requires a time-consuming numerical solution of the

Poisson (or linearized Poisson“Boltzmann) equation. In simulations with

implicit solvent, the extra computational e¬ort destroys the advantage of

omitting the solvent and explicit solvent representation is often preferred.

Therefore, there is a pressing need for approximate solutions that can be

rapidly evaluated, for simulations of (macro)molecules in an implicit solvent

352 Electromagnetism

(see Section 7.7 on page 234). The generalized Born solvation model was

invented to do just that. The original introduction by Still et al. (1990) was

followed by several re¬nements and adaptations,6 especially for application

to macromolecules and proteins (Onufriev et al., 2004).

The general problem is how to compute forces and energies for simulations

of explicit (macro)molecules in an implicit polar solvent. The direct inter-

actions between the explicit particles in the macromolecule (described by

the “vacuum energy” Vvac ) must be augmented by the solvation free energy

”Gsolv between these particles and the solvent. The total solvation free

energy consists of an electrostatic term ”Gel and a surface term ”Gsurf .

Consider the following sequence of processes:

(i) start with the system in vacuum;

(ii) remove all charges, i.e., remove the direct Coulomb interactions;

(iii) solvate the uncharged system, i.e., add the surface free energy;

(iv) add all charges back in the presence of the solvent.

The total potential of mean force is

mf

Vtot = Vvac + ”Gsurf + ”Gel . (13.101)

The surface free energy is usually taken to be proportional to the solvent

accessible surface area, with a proportionality constant derived from the

experimental free energy of solvation of small molecules. Onufriev et al.

(2004) quote a value of 0.005 kcal/mol par ˚2 = 2.1 kJ/mol per nm2 ,

A

but this value may be di¬erentiated depending on the particular atom type.

The free energy of adding all charges back in the presence of the solvent is

the total electrostatic interaction in the polarizable medium. From this the

direct Coulomb interaction (in vacuum) should be subtracted in order to

obtain ”Gel , because the direct Coulomb interaction has been removed in

step (ii) above.

Now consider the total electrostatic interaction for a dilute system of

charges qi , each in a (small) spherical vacuum cavity with radius Ri centered

at position r i , embedded in a medium with relative dielectric constant µr

(“dilute liquid of charges-in-cavities”). When all distances r ij are large

compared to all radii Ri , the total electrostatic energy is