Image Methods for Electrostatic Potentials

Wei CaiDepartment of Mathematics and StatisticsUniversity of North Carolina at Charlotte

Charlotte, NC 28223-0001, USA

Abstract

Image method is a classical method for finding electrostatic potentialsin dielectric or conducting objects with special geometries. The imagemethod provides simple and analytical formulas to represent the reactionfield on a charge in an object as a result of the polarization effect from thesurrounding material. In this paper, we will provide an overview of theclassical results of image methods for electrostatic potentials in simple ge-ometries, and recent developments in constructing image approximationsfor reaction fields in a dielectric sphere.

1 Introduction

Electrostatics plays a major role in many physical problems such as electro-magnetic wave propagation, structure and folding and stability of proteins.In the case of biomolecule simulations, the long range electrostatic interac-tions strongly depend on the solvent environment surrounding the biomoleculeunder study. When modeling a biological system numerically, it has beenchallenging, however, to account for such environment in a manner that iscomputationally efficient and physically accurate at the same time. As such,theoretical modeling of electrostatic interactions has been and remains an im-portant subject of theoretical and computational studies of biomolecules.

One of the fundamental problems in electrostatics is to find the poten-tial distribution of a single charge in a heterogeneous medium, for instance,a charge embedded in a cavity of one dielectric constant surrounded/adjacentto another medium with a different dielectric constant. A classic example forthis setting is to find the solvation energy of a water molecule in the infinitesolvent environment, which was first studied by Born in [3] with the Born sol-vation energy formula. More recently, in efforts to speed up the computationalmodeling of the solvation of macromolecules, hybrid models of representation

1

[31, 23, 22] of the biomolecule and solvent is being studied to take advantageof the accuracy of explicit atomic models [21, 24, 34] of the biomolecules andwater shells on one hand, and the speed of implicit models [10, 2] using contin-uum dielectric representations for the infinite extent of solvent away from thebiomolecules on the other hand. In the hybrid model, a spherical or ellipticcavity is usually used to enclose the biomolecule and some solvent molecules.The key physical quantity for molecular dynamics simulation of the moleculesinside the cavity is to calculate the reaction field of charges from the infiniteextent of continuum dielectric solvent (pure water or ionic solvent).

Electric charges within the cavity will polarize the surrounding solventmedium, which in turn makes a contribution, called the reaction field, to theelectric potential within the cavity. The electric potential inside the cavity isexpressed as Φ=ΦS + ΦRF where ΦS is the potential from the source chargeinside the cavity, and ΦRF is the reaction field. Fast and accurate calculationof such a reaction field has a far-reaching impact on computational simulationsfor chemical and biological systems involving electrostatic interactions withina solvent. In the case of a spherical cavity, the reaction field can be calculatedusing the classical Kirkwood series expansion [19, 20]. Although in theoryany desired degree of accuracy can be obtained using the series expansion, itsconvergence rate is slow near the cavity boundary.

For some cavity geometries, classical image methods [26] provide simpleand analytic ways to compute the reaction fields. As computational model-ing of biological systems has gained increasing prominence for the study ofproteins and membrane, research on image methods for dielectric materialshave attracted more attentions. For this reason, several image charge ap-proaches have been proposed in which the reaction field for a dielectric sphereis represented in terms of the potential of a single image charge, including theFriedman image approximation [13] and the Abagyan-Totrov modified imageapproximation [1]. However, by using only one image charge these meth-ods were limited in accuracy. Recently, a high-order accurate approximationusing multiple image charges was proposed [4, 8, 9] which was found to per-form about 20-30 times faster than the Kirkwood expansion in typical highaccuracy calculations. Moreover, combined with the fast multipole methods[16, 15], the multiple image approximation has the potential to calculate elec-trostatic interactions among N charges inside the spherical cavity in O(N)operations.

In this paper, we will provide an overview of the traditional results ofimage methods for electrostatic potentials in simple geometries, and the recentdevelopment alluded above in constructing image approximations for reactionfields of a dielectric sphere. For image methods applied to multiple objects

2

such as cylinders and spheres, please refer to [36, 18, 38, 6, 5].

2 Method of Images for Simple Geometries

Let Ω be the region where the electrostatic potential Φ(r), due to the presenceof a source charge q located at rs inside Ω, will be found and the potential willsatisfy the Poisson equation

∇ · (ε(r)∇Φ(r)) = −4πqδ(|r− rs|), r ∈ Ω (1)

where the potential will satisfy a homogeneous boundary conditions if Ωc is aperfect conductor

Φ(r) = 0, r ∈ ∂Ω (2)

or the continuity condition if both Ω and Ωc (the exterior of Ω) are dielectricmaterial,

Φ(r+) = Φ(r−), (3)

εo∂Φ(r+)

∂n= εi

∂Φ(r−)∂n

where n is the external normal to the boundary, and εi is the dielectric constantinside Ω and εo the dielectric constant outside Ω (namely in Ωc) and + and −denotes the limit taking from outside and inside Ω, respectively. Meanwhile,the potential Φ(r) is assumed to decay to zero when r →∞.

In general, the solution to (1) has to be solved with numerical approx-imation, and analytical forms of the solutions are only available for simplegeometries. The solution for the total potential Φ(r) is usually decomposedinto two parts - one part for the potential due to the source charge at rs andthe second part the reaction field which represents the polarization effect ofthe material in Ωc, namely,

Φ(r) =q

4πεi|r− rs||+ ΦRF(r). (4)

For selected geometries, the method of images provide simple and ana-lytical solutions to the reaction field ΦRF(r). In this section, we will reviewsome classic image solutions to simple geometries of conducting bodies anddielectrics.

For simplicity of presentation, we will introduce the following notations

κ =εoεi

, Γ =εo − εiεo + εi

=κ− 1κ + 1

, Υ =2εo

εo + εi(5)

3

and the potential at r due to a point charge q located in rs in a homogeneousfree space with dielectric constant ε is denoted by

U [q, rs, ε] = U [q, rs, ε](r) = − q

2πεln(|r− rs|) r = (x, y) ∈ R2 , (6)

orV [q, rs, ε] = V [q, rs, ε](r) =

q

4πε|r− rs| r = (x, y, z) ∈ R3 (7)

with the argument r omitted in most cases when no confusion occurs.

2.1 Potential of a point charge in the presence of a conducting3-D half space

Consider a point charge located at rs = (0, 0, d) along the z-axis above aconducting plane which is assumed to be grounded (at zero potential). So, wehave Ω = r =(x, y, z), z ≥ 0 and the homogeneous boundary condition (2)is assumed. The solution to (1) is given by (4). And, the reaction field inthis case can in fact be represented by the field of an image charge q′ = −qwhich is located below the conducting plane z = 0 at the mirror image locationrim = (0, 0,−d) and the effect of this image potential (reaction potential) isto satisfy the required zero potential boundary conditions (2). So, we have

ΦRF(r) = − q

4πεi|r− rim| = V [−q, rim, εi]. (8)

2.2 Potential of a point charge in the presence of a dielectric3-D half space

Again, we have a point charge located at rs = (0, 0, d) above a dielectric halfspace and the potential in Ω will have to satisfy the continuity conditions (3)on the interface z = 0. Again the potential is given by (4). The reaction fieldcan be represented by an image charge, specifically, for z > 0, we have

ΦRF(r) = V [q′, rim, εi] (9)

whererim = (0, 0,−d), q′ = −Γq .

Meanwhile, the total potential in the lower half plane can also be representedby a second image locating at the source point in the upper half plane -consistent with the non-singular feature of the potential there, i.e., for z < 0,

Φ(r) = V [q′′, rs, εo] (10)

whereq′′ =

2εo(εo + εi)

q.

4

2.3 Potential of a line charge and a dielectric cylinder

In this case, the problem is a 2-D problem for the cross section of the cylinder,let Ω = r = (r, θ), r > a denote the exterior of the cylinder of radius a whichcontains a source charge q at location rs = (rs, θs = 0), rs > a in the polarcoordinates.

The potential inside Ω is the superposition of the potential from the sourcecharge and two images at 0 and rim = (a2

rs, 0), [37] i.e.

Φ(r) = U [q, rs, εi] + U [−q, 0, εi] + U [q′, rim, εi], | r| > a (11)

whereq′ = −Γq.

while the potential outside Ω (inside the cylinder) is given by a second imageq′′ at location rs

Φ(r) = U [q′′, rs, εo], | r| < a (12)

where q′′ = Υq.

2.4 Potential of a line charge and a conducting cylinder

As a limiting case of (11), we can find the potential outside the cylindercentered at origin due to a line charge placed at (rs, θs = 0), rs > a as

Φ(r) = U [q, rs, εi] + U [q′, rim, εi], | r| > a (13)

where q′ = −q, rim = (a2

rs, 0).

2.5 Potential of a line charge in a 3-D wedge of angle π/m withconducting boundary

Let Ω = (r, θ),− π2m ≤ θ ≤ − π

2m , r > 0 be a dielectric wedge with dielectricconstant εi with a conducting boundary and a line charge is placed at (rs, θs)− π

2m < θs < − π2m , rs > 0. The potential inside the wedge Ω can be found [37]

by m charges, including the source charge and (m − 1) image charges, withcharge q and, m image charges with charge −q, namely,

Φ(r) =m∑

i=1

U [q, (rs, θs + i2π

m), εi] +

m∑

i=1

U [−q, (rs,−θs + i2π

m), εi] (14)

5

2.6 Potential of a point charge in a 3-D dielectric wedge

Nikoskinen and Lindel [29] proposed an ingenious image method to find thepotential due to a point charge outside a dielectric wedge W = r =(ρ, ϕ, z)|−Θ < ϕ < Θ of dielectric constant εo in cylindrical coordinates. So in thiscase, our domain Ω = W c = r =(ρ, ϕ, z)|Θ < ϕ < 2π−Θ which will have adielectric constant εi. The potential due to a charge q located at rs=(ρs, ϕs, zs)∈ Ω can be expressed using discrete images and surface image density at bothreal physical and complex locations as follows.

The potential inside Ω (outside the wedge)

Φ(r) = V [q, (ρs, ϕs, zs), εi] +N∑

n=1

V [q(−Γ)n, (ρs, αen, zs), εi]

+M∑

m=1

V [q(−Γ)m, (ρs, βem, zs), εi], (15)

+

i∞∫

−i∞V [qe(ζ), (ρs, ζ, zs), εi]dζ (ρ, ϕ, z) ∈ Ω

where the first term is from the source charge and the second summation isfrom N = integer[ 2π−ϕs

2(π−Θ) ] discrete image charges of q(−Γ)n located at

(ρs, αen, zs), αe

n = (−1)n[2n(π −Θ) + ϕs − 2π]

and M = integer[ ϕs

2(π−Θ) ] discrete image charges of q(−Γ)m located at

(ρs, βen, zs), βe

n = (−1)m[ϕs − 2m(π −Θ)]

and a line image of density qe(ζ) distributed along the imaginary location(ρs, ζ, zs), ζ ∈ (−i∞, i∞) and

qe(ζ) = qe(ip) = − iq

2π

∞∫

−∞[qee(t) + qeo(t)]e−itpdt

where

qeo(t) =−Γ sinh(2tΘ) sinh[t(π − ϕs)]sinh(tπ) + Γ sinh[t(π − 2Θ)]

−N∑

n=1

(−Γ)n sinh(tαen)−

M∑

m=1

(−Γ)m sinh(tβem)

qee(t) =−Γ sinh(2tΘ) cosh[t(π − ϕs)]sinh(tπ)− Γ sinh[t(π − 2Θ)]

−N∑

n=1

(−Γ)n cosh(tαen)−

M∑

m=1

(−Γ)m cosh(tβem).

6

Similarly, the potential outside Ω (inside the wedge)

Φ(r) = Υ(V [q, (ρs, ϕs, zs), εo] +N∑

n=1

V [qΓn, (ρs, αin, zs), εo]

+M∑

m=1

V [qΓm, (ρs, βim, zs), εo] (16)

+

π+i∞∫

π−i∞V [qi(ζ), (ρs, ζ, zs), εo]dζ).

The first term in (16) is from an image charge q located at (ρs, ϕs, zs) and thesecond summation is from N = integer[π−ϕs

2Θ ] discrete image charges of qΓn

located at(ρs, α

en, zs), αi

n = π + (−1)n[ϕs − π + 2nΘ]

and M = integer[ϕs−π2Θ ] discrete image charges of qΓm located at

(ρs, βin, zs), βe

n = π + (−1)m[ϕs − π − 2mΘ]

and a line image of density qe(ζ) distributed along the imaginary location(ρs, ζ, zs), ζ ∈ (−i∞, i∞) and

qi(ζ) = qi(π − ip) = − iq

2π

∞∫

−∞[qie(t) + qio(t)]e−itpdt

where

qio(t) =sinh[t(π − ϕs)]

sinh(tπ) + Γ sinh[t(π − 2Θ)]− sinh[t(π − ϕs)]

−N∑

n=1

Γn sinh[t(π − αin)]−

M∑

m=1

Γm sinh[t(π − βim)],

qie(t) =cosh[t(π − ϕs)]

sinh(tπ)− Γ sinh[t(π − 2Θ)]− cosh[t(π − ϕs)]

−N∑

n=1

Γn cosh[t(π − αin)]−

M∑

m=1

Γm cosh[t(π − βim)].

7

2.7 Potential of a point charge in the presence of a conductingsphere

For a charge q outside a conducting sphere, i..e Ω = r, |r| > a and the po-tential in Ω will be given as the sum of the primary field at rs = (rs, θs, φs)in the spherical coordinates and the potential of an image charge −q at theKelvin image (1885) which is the inversion point rk = (a2

rs, θs, φs) with respect

to the sphere,

Φ(r) = V [q, rs, εi] + V [−q, rk, εi], r ∈ Ω. (17)

2.8 Potential of a point charge in the presence of a dielectricsphere

In this case, we consider the sphere Ω = r, |r| ≤ a with a charge q locatedat rs = (rs, θs, φs) ∈ Ω, the potential inside and outside the sphere can beapproximated by those of a point charge at the Kevin image location and aline image charges with a power law distribution density along a ray extendingfrom the Kelvin image point rk = (a2

rs, θs, φs) to infinity (for the reaction field

inside the sphere) or along a ray between the origin and the Kevin point (for re-action field outside the sphere if the source charge is outside the sphere). Thisrepresentation has been discovered by several authors independently. How-ever, it is still not well known in the electrostatic research communities. Theearliest such a result was obtained by C. Neumann [28] in 1883, and, then byFinkelstein [11] in 1977, and later rediscovered in the 1990s independently byLindell and Norris [25, 30]. The result shows that the reaction field inside thesphere

ΦRF(r) =qk

4πεi|r− rk| +∫ ∞

rk

q′(x)4πεi|r− x|dx (18)

= V [qk, rk, εi] +∫ ∞

rk

V [q′(x),x, εi]dx, r ∈ Ω.

where x =xrsrs

, rk = a2

rs

qk = −Γa

rsq, q′(x) =

εi(εi − εo)(εi + εo)2

q

a

(rk

x

)εo/(εi+εo), rk ≤ x.

8

Meanwhile, the total potential outside the sphere is given by a point imagecharge at rs and a line charge distributing along the ray from origin to rs, i.e.

Φ(r) =q′′s

4πεo |r− rs| +∫ rs

0

q′′(x)4πεo |r− x|dx

= V [q′′s , rs, εo] +∫ rs

0V [q′′(x),x, εo]dx, (19)

where

q′′s = Υq, q′′(x) = − q

rs

ΓΥ2

(x

rs

)−εi/(εi+εo)

, 0 ≤ x ≤ rs.

3 The Discrete Image Approximations for ReactionFields in a Dielectric Sphere

In this section, we will present the recent development of constructing discreteimage approximations to the reaction field for a dielectric sphere Ω = r, |r| <awith dielectric constant εi while the dielectric constant outside the sphere isassumed to be εo . The starting point of most of the results is the classicalseries expansion due to Kirkwood (Kirkwood series expansion) [19] for the re-action fields. The recent results on multiple image approximations are basedon the elegant line image formula in (18) through appropriate Gauss integra-tion. Also, extension to the Poisson-Boltzmann equation is also included whenthe medium outside the sphere has an ionic density.

3.1 The Kirkwood series expansion of reaction field for a di-electric sphere

For a charge q inside a dielectric sphere Ω and the potential is the solutionof (1) and (3) for Ω, which is given by (4) and the reaction field ΦRF(r) atan observation point r=(r, θ, φ) inside the sphere can be expressed in terms ofthe Legendre polynomials of cos θ [19], namely,

ΦRF(r) =q(εi − εo)

4πεia

∞∑

n=0

(n + 1

εin + εo(n + 1)

) (rrs

a2

)nPn(cos θ), (20)

where Pn(x), n = 0, 1, 2, · · · , are the Legendre polynomials.The expansion in (20) is termed as the Kirkwood expansion, which has a

fast convergence when r is away from the sphere boundary. In the case thatthe point charge is close to the boundary of the sphere, when calculating thereaction field at a point also close to the boundary, the convergence rate by the

9

Kirkwood series expansion is slow due to rrs/a2 ≈ 1, requiring a great numberof terms in the series expansion to achieve good accuracy in the reaction field.

3.2 The Kirkwood image approximation

Assume that εi < εo. Then expanding the term (n + 1)/(εin + εo(n + 1)) in(20) in terms of εin/εo(n + 1) < 1 yields

n + 1εin + εo(n + 1)

=1εo

[1− εi

εo

n

n + 1+

(εiεo

)2 (n

n + 1

)2

− · · ·]

,

which enables us to write the reaction field given in (20) as

ΦRF(r) = B(0)(r) + B(1)(r) + B(2)(r) + · · · , (21)

where for k=0, 1, 2, · · · , we have

B(k)(r) = (−1)k (εi − εo)4πεiεo

a

rs

q

(a2/rs)

(εiεo

)k ∞∑

n=0

(n

n + 1

)k (r

a2/rs

)n

Pn(cos θ).

In particular, the first term in (21) is

B(0)(r) =(εi − εo)4πεiεo

a

rs

q

(a2/rs)

∞∑

n=0

(r

a2/rs

)n

Pn(cos θ), (22)

which is exactly the Legendre polynomial expansion of the Coulomb potentialat the point r inside the sphere due to a point charge of strength qK outsidethe sphere at the conventional Kelvin image point rk=(rk, 0, 0) [27], namely,

B(0)(r) =qK

4πεi|r− rk| = V [qK , rk, εi],

where

qK =εi − εo

εo

a

rsq, rk =

a2

rs.

The Kirkwood image approximation to the reaction field is then defined as

ΦRF(r) ≈ ΦK(r) = B(0)(r). (23)

Now let us consider the second term in (21) which can be written as

B(1)(r) =− (εi − εo)4πεiεo

a

rs

q

(a2/rs)

(εiεo

) ∞∑

n=0

(r

a2/rs

)n

Pn(cos θ),

+(εi − εo)4πεiεo

a

rs

q

(a2/rs)

(εiεo

) ∞∑

n=0

(1

n + 1

)(r

a2/rs

)n

Pn(cos θ). (24)

10

Similarly, the first series in (24) is exactly the Legendre polynomial expansionof the Coulomb potential at the point r inside the sphere due to a point chargeof strength q′

Koutside the sphere at the Kelvin image point rk, where

q′K

= −εi(εi − εo)ε2o

a

rsq.

On the other hand, using the integral identity

1n + 1

= rn+1k

∫ ∞

rk

1xn+2

dx, (25)

which is valid for all n ≥ 0, the second series in (24) can be written as

∫ ∞

rk

[qK(x)4πεix

∞∑

n=0

( r

x

)nPn(cos θ)

]dx, (26)

whereqK(x) =

εi(εi − εo)ε2o

q

a

(rk

x

), rk ≤ x.

Note that qK(x) can be regarded as the density function of a continuous linecharge extending along the radial direction from the Kelvin image point rk

to infinity. Also, the integrand in (26) is exactly the Legendre polynomialexpansion of the Coulomb potential at the point r inside the sphere due to acharge of strength qK(x) outside the sphere at the point x=(x, 0, 0). Hence weget

B(1)(r) =q′K

4πεi|r− rk| +∫ ∞

rk

qK(x)4πεi|r− x|dx.

The Kirkwood image approximation (23) can then be improved by includ-ing B(1)(r) as a correction potential, and the resulting image approximationis referred to by us as the improved Kirkwood image approximation, namely,

ΦRF(r) ≈ ΦIK(r) = B(0)(r) + B(1)(r) =qIK

4πεi|r− rk| +∫ ∞

rk

qK(x)4πεi|r− x|dx

(27)

= V [qIK , rk, εi] +∫ ∞

rk

V [qK(x),x, εi]dx,

where

qIK = qK + q′K

= −(εi − εo)2

ε2o

a

rsq.

11

Furthermore, by evaluating the integral in (27) explicitly, a compact ana-lytical form of the improved Kirkwood image approximation can be obtainedas

ΦIK(r) =qIK

4πεi|r− rk| +

εi − εo4πε2

o

q

aξln

(√1− 2 cos θξ + ξ2 + ξ − cos θ

1− cos θ

),

if cos θ 6= 1,

−εi − εo4πε2

o

q

aξln(1− ξ), if cos θ = 1,

where ξ=rrs/a2.

3.3 The Friedman image approximation

Alternatively, expanding the term (n + 1)/(εin + εo(n + 1)) in (20) in terms ofεi/((εi + εo)(n + 1)) < 1 results in

n + 1εin + εo(n + 1)

=1

εi + εo

[1 +

εiεi + εo

1n + 1

+(

εiεi + εo

)2 (1

n + 1

)2

+ · · ·]

,

(28)which enables us to write the reaction field given in (20) as

ΦRF(r) = R(0)(r) + R(1)(r) + R(2)(r) + · · · , (29)

where for k=0, 1, 2, · · · , we have

R(k)(r) =(εi − εo)εk

i

4πεi(εi + εo)k+1

a

rs

q

(a2/rs)

∞∑

n=0

(1

n + 1

)k (r

a2/rs

)n

Pn(cos θ). (30)

In particular, the first term in (29) is

R(0)(r) =εi − εo

4πεi(εi + εo)a

rs

q

(a2/rs)

∞∑

n=0

(r

a2/rs

)n

Pn(cos θ),

which is the Legendre polynomial expansion of the Coulomb potential at thepoint r inside the sphere due to a point charge of strength qF outside thesphere at the Kelvin image point rk, namely,

R(0)(r) =qF

4πεi|r− rk| = V [qF , rk, εi],

whereqF = −Γ

a

rsq.

12

The Friedman image approximation to the reaction field [13] is thus definedas

ΦRF(r) ≈ ΦF(r) = R(0)(r). (31)

Next, let us consider the second term in (29) which is

R(1)(r) =(εi − εo)εi

4πεi(εi + εo)2a

rs

q

(a2/rs)

∞∑

n=0

(1

n + 1

)(r

a2/rs

)n

Pn(cos θ).

Using the integral identity (25) again, R(1)(r) can be regarded as the potentialof a continuous line charge extending along the radial direction from the Kelvinimage point rk to infinity with the charge density function given by

qF(x) =εi(εi − εo)(εi + εo)2

q

a

(rk

x

), rk ≤ x,

namely,

R(1)(r) =∫ ∞

rk

qF(x)4πεi|r− x|dx =

∫ ∞

rk

V [qF(x),x, εi]dx. (32)

Note that the line charge qF(x) is a constant multiple of the line charge qK(x).Also, integrating (32) leads to a compact analytical form for R(1)(r) as

R(1)(r) =

εi − εo4π(εi + εo)2

q

aξln

(√1− 2 cos θξ + ξ2 + ξ − cos θ

1− cos θ

), if cos θ 6= 1,

− εi − εo4π(εi + εo)2

q

aξln(1− ξ), if cos θ = 1.

Consequently, the Friedman image approximation can be improved by includ-ing R(1)(r) as a correction potential, and the resulting image approximationis referred to by us as the improved Friedman image approximation, namely,

ΦRF(r) ≈ ΦIF(r) = R(0)(r) + R(1)(r). (33)

The Friedman image approximation has been applied in many areas in-cluding molecular dynamics or Monte Carlo simulations [39, 40, 33]. From(28) and (30), we can see that the accuracy of the improved Friedman imageapproximation is limited to the second order of the dielectric constant contrast,

i.e.(

εiεi+εo

)2. On the other hand, it is clear from (30) that the approximation

holds even when the source charge rs approaches to the sphere boundary.

13

3.4 The Abagyan-Totrov image approximation

The Friedman image approximation (31) or (33) provides insufficient accuracy.In particular, when rs tends to zero so that the point charge is located in thecenter of the sphere, the reaction field energy based on the Friedman imageapproximation (31) does not reproduce the Born formula [1].

Based on the Friedman image approximation (31), Abagyan and Totrovproposed a modified image approximation which is more accurate and lesscomputationally intensive than the improved Friedman image approximation(33). Instead of an exact expression for R(1)(r), a position-independent cor-rection potential Rcorr is added to the Friedman image approximation (31) sothat for the particular case of a charge in the center of the sphere one gets theexact solution. The position-independent correction potential Rcorr is definedas

Rcorr = −Γq

4πaεo.

The Abagyan-Totrov modified image approximation to the reaction field isthen defined as

ΦRF(r) ≈ ΦAT(r) = R(0)(r) + Rcorr. (34)

The reaction field energy based on the Kirkwood image approximation(23), however, reproduces the Born formula when rs tends to zero. This impliesthat, when the source is located around the center of the sphere, the Kirkwoodimage approximation (23) should perform better than the Friedman imageapproximation (31). On the other hand, one can show that ΦK(r)=ΦAT(r) as rs

tends to zero, indicating that for relatively small values of rs, the Kirkwood andthe Abagyan-Totrov image approximations are comparable in terms of theiraccuracy. Applications of the Abagyan-Totrov modified image approximationcan be found in [1, 32, 17].

3.5 The high-order accurate multiple image approximation

In essence, the image approximations to the reaction field discussed in theprevious subsections all employ a single image charge to represent the re-action field with limited accuracy. Recently, a high-order accurate multipleimage (MI) approximation to the reaction field was proposed [4, 8, 9] basedon (18)-(19). The high-order accurate MI approximation to the reaction fieldis obtained by representing the continuous line charge q′(x) in (18) with dis-crete charges constructed through an appropriate numerical quadrature [4].More precisely, without losing any generality, let sm, wm,m=1, 2, · · · ,M , bethe Gauss quadrature points and weights on the interval [−1, 1], which can beobtained with the program ORTHPOL [14]. Then, the numerical quadrature

14

for approximating the integral in (18) is

∫ ∞

rk

q′(x)4πεi|r− x|dx ≈

M∑

m=1

qm

4πεi|r− xm| =M∑

m=1

V [qm,xm, εi],

where xm=(xm, 0, 0), and for m=1, 2, · · · ,M ,

qm = −Γεi

2εo

wmxm

aq, xm = rk

(2

1− sm

)(εi+εo)/εo

.

Accordingly, the high-order accurate multiple image approximation to thereaction field is defined as

ΦRF(r) ≈ ΦMI(r) =qMI

4πεi|r− rk| +M∑

m=1

qm

4πεi|r− xm| (35)

= V [qMI , rk, εi] +M∑

m=1

V [qm,xm, εi].

where qMI = Υq. Similar multiple image approximation for the potentialoutside the sphere can be obtained by the same quadrature discretization of(19) [4]. Due to the linear nature of the electrostatic potential from multiplecharges, the multiple image approximation (35) to the reaction field can beused for each physical charge individually and then superimposed to producethe total reaction fields of multiple charges.

4 Image Charges for an Aqueous Solvent

In many biological application, we need to consider the ionicity of the solventfor the material outside the sphere. In the Debye-Huckel theory [7], the mobileion concentration in the ionic solvent is given by a Boltzmann distributionin the mean field approximation. For a solvent of weak ionic strength, thelinearized Poisson-Boltzmann equation [12]

∇2Φ(r)− λ2Φ(r) = 0, |r| > a (36)

can be used to approximate the screened Coulomb potential in the solvent andKirkwood expansion can also be found for the solution of (36). Here, λ is theinverse Debye screening length defined by

λ2 =8πNAe2ρA

1000εokBTcs, (37)

15

where NA is Avogadro’s number, ρA is the solvent density, e is the protoniccharge (4.803×10−10esu), kB is the Boltzmann constant, T is the absolutetemperature, and cs is the ionic concentration measured in molar units. From(37), we see that the inverse Debye screening length λ is proportional to thesquare root of the ionic concentration cs. In particular, for 1:1 electrolytes(monovalent:monovalent salts like NaCl), λ ≈ 0.33

√csA

−1at room tempera-ture (25o), with εo=78.5 and A=10−10m [35].

In [8, 9], we have developed various image approximations in the order ofthe small parameter u=λa, for instance, a fourth order approximation is givenas follows.

ΦRF(r) =qK

4πεi|r− rK |+

∫ ∞

rK

qL1(x)4πεi|r− x|dx +

∫ ∞

rK

qL2(x)4πεi

(1

|r− x| −1x

)dx

+ ΦC1 + ΦC2(r) + ΦC3(r) + O(u4), (38)

= V [qK , rk, εi] +∫ ∞

rK

V [qL1(x),x, εi]dx

+∫ ∞

rK

(V [qL2(x),x, εi](r)− V [qL2(x),x, εi](0))dx

+ ΦC1 + ΦC2(r) + ΦC3(r) + O(u4)

where ΦC1 is a constant, position-independent correction potential defined as

ΦC1 =q

4πεia

(C0(u) + Γ− δ1

σ1

), (39)

and on the other hand, ΦC2(r) is a position-dependent correction potentialgiven by

ΦC2(r) =q

4πεia

(C1(u) + Γ− δ1

1 + σ1− δ2

1− σ2

)r

rK

cos θ. (40)

the potential ΦC3(r) is defined as

ΦC3(r) =q

4πεia

(C2(u) + Γ− δ1

2 + σ1− δ2

2− σ2

)(r

rK

)2

P2(cos θ). (41)

Here, P2(cos θ)=(3 cos2 θ − 1)/2. Also, denoting σ0 = εoεi+εo

, δ0 = εi(εi−εo)(εi+εo)2

δ1 =α2σ1 − α1

σ1 + σ2, σ1 =

√β2

2 − 4β1 + β2

2,

δ2 =α2σ2 + α1

σ1 + σ2, σ2 =

√β2

2 − 4β1 − β2

2.

16

α1 = −Γ(

a1

a3− b1

b3

), β1 =

b1

b3,

α2 = −Γ(

a2

a3− b2

b3

), β2 =

b2

b3.

a1 = −(εi + εo)u2 − 2(εi − εo), a2 = (εi − εo)(2− u2), a3 = 4(εi − εo),

b1 = −εo(2− u2), b2 = −(εi + εo)u2 − 2(εi − εo), b3 = 4(εi + εo).

And,qK = −Γ

a

rS

q,

qL1(x) =δ1q

a

(x

rK

)−σ1

, rK ≤ x. (42)

qL2(x) =δ2q

a

(x

rK

)σ2

, rK ≤ x. (43)

Discrete image can be constructed from (38) [9] given as

Φ(r) ≈ 14πεi

(qK

|r− rK |+

M∑

m=1

qL1m

|r− xm| +M∑

m=1

qL2m

|r− x′m|

)

+ ΦC1 + ΦC2(r) + ΦC3(r) (44)

= V [qK , rk, εi] +M∑

m=1

V [qL1m ,xm, εi] +

M∑

m=1

V [qL1m ,x′m, εi]

+ ΦC1 + ΦC2(r) + ΦC3(r)

qL1m = 2−τσcτδ1ωm

(xm

rK

)σc−σ1 xm

aq, (45)

and

qL2m = 2−τσcτδ2ωm

(xm

rK

)σc+σ2 xm

aq. (46)

Here, sm, ωm,m=1, 2, · · · ,M , are the Jacobi-Gauss or Jacobi-Gauss-Radaupoints and weights based on the Jacobi polynomial P (α,β)(x)- on the interval[−1, 1] with α=τσ− 1 and β=0, with τ > 0. The parameter σc > 0 is tunablefor optimal computational efficiency. For example, depending on the value ofu=λa, either of the two natural choices σc=σ1 and σc=1 − σ2 could performbetter than the other.

17

5 Conclusions

In this paper, we provide a review of classical image methods for calculatingelectrostatic potentials for several cases of simple geometries and also recentdevelopments of images for a dielectric sphere with immediate applications incomputer simulations of bimolecular and other material systems.

Acknowledgments

The author thanks the support of the National Institutes of Health (grantnumber: NIGMS 1R01GM083600-01) for the work reported in this paper.Also, the author thanks for the many discussions with Dr. S. Z. Deng, DonaldJacobs, and Andriy Baumketner during the writing of this paper.

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