qi qj 1 1 qi

’ 1’

4πµ0 Uel = . (13.102)

µr rij 2 µr Ri

i<j i

6 To mention a few modi¬cations of the expression (13.105) to compute the e¬ective “distance”

GB

fij : Hawkins et al. (1996) add another parameter to the term in square brackets; Onufriev et

GB

al. (2004) modify the 1/µr to exp(’κfij )/µr ) to take care of an ionic strength in the solvent;

Schaefer and Karplus (1996) retain a dielectric constant in the macromolecule.

13.8 Multipole expansion 353

Here, the ¬rst term is the electrostatic energy of a distribution of charges in

a medium (see (13.44) on page 340) and the second term is the sum of the

Born energies of the charges (see (13.60) on page 343). After subtracting

the direct vacuum Coulomb energy we obtain

qi qj

1 1

4πµ0 ”Gel = ’ 1’ , (13.103)

GB

2 µr fij

i j

with

GB

fij = rij for i = j,

= Ri for i = j. (13.104)

GB

Still et al. (1990) propose the following form for fij that includes the

case of the dilute liquid of charges-in-cavities, but has a much larger range

of validity:

1/2

2

rij

= rij + Ri Rj exp ’

GB 2

fij . (13.105)

4Ri Rj

The “e¬ective Born radii” Ri are to be treated as parameters that depend

on the shape of the explicitly treated system and the positions of the charges

therein. They are determined by comparison with Poisson“Boltzmann cal-

culations on a grid, by free energy perturbation (or integration) calculations

of charging a molecule in explicit solvent or from experimental solvation free

energies. The e¬ective Born radius is related to the distance to the surface

as we can see from the following example.

Consider a charge q situated in a spherical cavity at a distance d from the

surface. This is equivalent to the situation treated in Section 13.7.4, see Fig.

13.2, with s = a ’ d. The solvation free energy in the image approximation

(13.97) then becomes

1 (µr ’ 1) q2

4πµ0 UR = ’ , (13.106)

2 (µr + 1) 2d(1 ’ d/2a)

which is nearly equivalent to the GB equation with an e¬ective Born radius

of 2d(1’d/2a). This equals twice the distance to the surface of the sphere for

small distances, reducing to once the distance when the charge approaches

the center.

13.8 Multipole expansion

Consider two groups of charges, A and B, with qi at r i , i ∈ A and qj at

r j , j ∈ B (see Fig. 13.3). Each group has a de¬ned central coordinate r A

354 Electromagnetism

qi r

i

rA

qj r

j

A rB

B

Figure 13.3 Two interacting non-overlapping groups of charges

and r B , and the groups are non-overlapping. The groups may, for example,

represent di¬erent atoms or molecules. For the time being the medium

is taken as vacuum (µ = µ0 ) and the charges are ¬xed; i.e., there is no

polarizability. It is our purpose to treat the interaction energy of the two

distributions in terms of a multipole expansion. But ¬rst we must clearly

de¬ne what the interaction energy is.

The total energy of the two groups of charges is, according to (13.42), and

omitting the self-energy terms in the potential:

1 1

U= i∈A qi φ(r i ) + j∈B qi φ(r j ). (13.107)

2 2

Here φ(r i ) is the sum of φA (r i ) due to all other charges in A, and φB (r i )

due to all charges in B. Furthermore, UA = 1 i∈A qi φA (r i ) is the internal

2

electrostatic energy of group A (and, similarly, UB ). The total electrostatic

energy equals

U = UA + UB + UAB , (13.108)

with UAB being the interaction energy between group A and group B:

1 1

B A

UAB = i∈A qi φ (r i ) + j∈B qj φ (r j ). (13.109)

2 2

Both terms in UAB are equal, which can be easily veri¬ed by inserting

qj

1

, i∈A

φB (r i ) =

4πµ0 rij

j∈B

and

1 qi

, i∈B

φA (r j ) =

4πµ0 rij

i∈A

13.8 Multipole expansion 355

into (13.109). Therefore the interaction energy UAB can also be written as

qi φB (r i ),

UAB = (13.110)

i∈A

with

qj

1

φB (r) = . (13.111)

|r ’ r j |

4πµ0

j∈B

Thus we consider B as the source for the potential acting on A. By omitting

the factor 1/2, (13.110) represents the total interaction energy. It would be

incorrect to add any interaction of charges in B with potentials produced by

A.

We now proceed to write the interaction UAB in terms of a multipole

expansion. This can be done in two ways:

(i) The potential φB (r) in (13.110) is expanded in a Taylor series around

the center r A , involving derivatives of the potential at r A .

(ii) The source terms qj (rj ) in (13.111) are expanded in a Taylor series

around the center r B .

Both methods lead to nearly equivalent multipole de¬nitions, with subtle

di¬erences that we subsequently discuss. Combined they result in a descrip-

tion of the interaction between two charge clouds as a sum of interactions

between multipoles. The expansions are only convergent when riA < rAB ,

resp. rjB < rAB , which is ful¬lled when the charge distributions do not