posite charges at a distance d in the x-direction in a cubic cell with an edge

of length a. The Coulomb force is depicted in Fig. 6.4 as a function of d/a.

When d/a is not small, the periodicity artefact is considerable, amounting

to a complete cancelation of the force at d = a/2 and even a sign reversal.

Artifacts of periodicity are avoided if modi¬ed interaction potentials are

used that vanish for distances larger than half the smallest length of the unit

cell. In this way only the nearest image interactions occur. Of course, this

involves a modi¬cation of the potential that causes its own artifacts, and

needs careful evaluation. Care should also be taken with the handling of cut-

o¬s and the long-range parts in the potential function, as described on page

159: sudden cut-o¬s cause additional noise and erroneous behavior; smooth

146 Molecular dynamics

Figure 6.3 Construction of a molecular-shaped triclinic box. The molecule (top

left) is expanded with a shell of size equal to half the minimum distance required

between atoms of images (top right). Subsequently these shapes are translated into

a close-packed arrangement (middle left). Middle right: the unit cell depicted with

one molecule including its shel l of solvent. Bottom left: the unit cell as simulated

(solvent not shown); bottom right: reconstructed molecules. Figures reproduced by

permission of Tsjerk Wassenaar, University of Groningen (Wassenaar, 2006). See

also Bekker et al., 2004.

cuto¬s strongly modify the interaction. There is no good solution to avoid

periodicity artifacts completely. The best strategy is to use consistent forces

and potentials by inclusion of complete lattice sums, but combine this with

6.2 Boundary conditions of the system 147

100

75

50

single pair

← force lattice sum

25

0 0

’25 ’5

energy ’

’50 ’10

single pair

lattice sum

’75 ’15

’100 ’20

0.2 0.4 0.6 0.8 1

Distance (fraction of box size)

Figure 6.4 The Coulomb energy (black) and the force (grey) between an isolated

positive and a negative unit charge at a distance d (solid curves) is compared with

the energy and force between the same pair in a cubic periodic box (dashed curves).

studying the behavior of the system as a function of box size. In favorable

cases it may also be possible to ¬nd analytical (or numerical) corrections to

the e¬ects of either periodicity or modi¬cations of the interaction potentials

(see the discussion on electrostatic continuum corrections on page 168).

148 Molecular dynamics

6.2.2 Continuum boundary conditions

Other boundary conditions may be used. They will involve some kind of

re¬‚ecting wall, often taken to be spherical for simplicity. The character of

the problem may require other geometries, e.g., a ¬‚at wall in the case of

molecules adsorbed on a surface. Interactions with the environment out-

side the “wall” should represent in the simplest case a potential of mean

force given the con¬guration of atomic positions in the explicit system. In

fact the system has a reduced number of degrees of freedom: all degrees of

freedom outside the boundary are not speci¬cally considered. The situation

is identical to the reduced system description, treated in Chapter 8. The

omitted degrees of freedom give rise to a combination of systematic, fric-

tional and stochastic forces. Most boundary treatments take only care of

the systematic forces, which are derivatives of the potential of mean force.

If done correctly, the thermodynamic accuracy is maintained, but erroneous

dynamic boundary e¬ects may persist.

For the potential of mean force a simple approximation must be found.

Since the most important interaction with the environment is of electro-

static nature, the electric ¬eld inside the system should be modi¬ed with

the in¬‚uence exerted by an environment treated as a continuum dielectric,

and “ if appropriate “ conducting, material. This requires solution of the

Poisson equation (see Chapter 13) or “ if ions are present “ the Poisson“

Boltzmann equation. While for general geometries numerical solutions us-

ing either ¬nite-di¬erence or boundary-element methods are required, for

a spherical geometry the solutions are much simpler. They can be either

described by adding a ¬eld expressed in spherical harmonics, or by using

the method of image charges (see Chapter 13).

Long-range contributions other than Coulomb interactions involve the r’6

dispersion interaction. Its potential of mean force can be evaluated from

the average composition of the environmental material, assuming a homoge-

neous distribution outside the boundary. Since the dispersion interaction is

always negative, its contribution from outside the boundary is not negligible,

despite its fast decay with distance.

Atoms close to the boundary will feel modi¬ed interactions and thus de-

viate in behavior from the atoms that are far removed from the boundary.

Thus there are non-negligible boundary e¬ects, and the outer shell of atoms

must not be included in the statistical analysis of the system™s behavior.

6.3 Force ¬eld descriptions 149

6.2.3 Restrained-shell boundary conditions

A boundary method that has found applications in hydrated proteins is the

incorporation of a shell of restrained molecules, usually taken to be spherical,

between the system and the outer boundary with a continuum. One starts

with a ¬nal snapshot from a full, equilibrated simulation, preferably in a

larger periodic box. One then de¬nes a spherical shell in which the atoms are

given an additional restraining potential (such as a harmonic potential with

respect to the position in the snapshot), with a force constant depending

on the position in the shell, continuously changing from zero at the inner

border of the shell to a large value at the outer border. Outside the shell a

continuum potential may be added, as described above. This boundary-shell

method avoids the insertion of a re¬‚ecting wall, gives smooth transitions at

the two boundaries, and is easy to implement. One should realize, however,

that molecules that would otherwise di¬use are now made rigid, and the

response is “frozen in.” One should allow as much motion as possible, e.g.,

restrain only the centers of mass of solvent molecules like water, leaving

rotational freedom that allows a proper dielectric response. This and most

other boundary methods do not allow elastic response of the environment

and could produce adverse building up of local pressure (positive or negative)

that cannot relax due to the rigidity of the boundary condition.

Examples of spherical boundary conditions are the SCAAS (surface-con-

strained all-atom solvent) model of King and Warshel (1989), which imposes

harmonic position and orientation restraints in a surface shell and treats the

shell by stochastic Brownian dynamics, and the somewhat more complex

boundary model of Essex and Jorgensen (1995). In general one may ques-

tion the e¬ciency of boundary methods that are sophisticated enough to

yield reliable results (implying a rather extensive water shell around the so-

lute, especially for large hydrated (macro)molecules), compared to periodic

systems with e¬cient shapes, as discussed above.

6.3 Force ¬eld descriptions

The fact that there are many force ¬elds in use, often developed along dif-

ferent routes, based on di¬erent principles, using di¬erent data, specialized

for di¬erent applications and yielding di¬erent results, is a warning that

the theory behind force ¬elds is not in a good shape. Ideally a force ¬eld