Chemistry Reference
In-Depth Information
distance from a given nucleus, so that part of the calculation scales linearly
with system size. An alternative to the momentum-space sum is to discretize
the continuous Gaussian functions on a grid and solve the long-ranged
part with either an FFT
228
or real-space MG method.
229
This approach results
in
N
log
N
(FFT) or
N
(MG) scaling and is called the particle-mesh-Ewald
(PME) method. It is now a standard for large-scale biomolecular simulations,
and the method has been extended to include fixed and induced dipolar
interactions.
230
It should be emphasized that the proper handling of long-ranged electro-
statics has been shown to be crucial for accurate free energy calculations in
molecular simulations. Errors of many kilo calories/mole can occur if the elec-
trostatic potential is artificially truncated. In addition, if proper self-energy
corrections are included, accurate free energies for charged systems can be
obtained with remarkably few solvent (water) molecules in the simulation
box.
227,231,232
In fact, it makes more sense physically in modeling ion free ener-
gies to calculate those free energies for a charged system, since neutralization
with an ion of the opposite charge necessarily creates a high-concentration
environment. These issues have not been widely recognized until recently.
Finally, we list some interesting developments related to alternative
forms of electrostatics calculations. Pask and Sterne
233
pointed out that real-
space electrostatic calculations for periodic systems require no information
from outside the central box. Rather, we only need the charge density within
the box and the appropriate boundary conditions to obtain the electrostatic
potential for the infinite system. These ideas were used earlier in an initial
MG effort to compute Madelung constants in crystals.
160
So long as charge
balance exists inside the box, the computed potential is stable and yields an
accurate total electrostatic energy. Thus, questions about conditional summa-
tion of the 1
r
potential to obtain the physical electrostatic energy are unne-
cessary. This has also been noted in the context of Ewald methods for the
simulation of liquids (See Chapter 5 in Ref. 227).
Thompson and Ayers
234
presented an interesting new technique for sol-
ving the Poisson equation ''inside the box.'' The electrostatic potential is
expanded in a set of sine functions, which leads to simple analytic forms for
the Coulomb energy for systems of electrons in a box. The method is closely
related to FFT methods and scales linearly with system size. Both finite and
periodic systems were considered. Juselius and Sundholm
235
developed a FE
method for electrostatics that employs Lagrange interpolating polynomials
as the basis. Rather than utilizing an iterative approach (as in the MG meth-
od), innovative procedures for direct numerical integration on the grid result
in
N
3=2
computational complexity. A parallel implementation was shown to
scale linearly with system size since the matrix multiplications in the innermost
loop are independent.
In the spirit of real-space methods, where all of the function updates in
solving the Poisson equation are near-local in space, Maggs and Rossetto
236
=
