Chemistry Reference
In-Depth Information
polyelectrolytes moving in the background of an ion gas. In these Monte Carlo
simulations, the PB equation was solved for each test conformation, and the
solution produced the free energy for the conformation. The PB equation
was solved with an FD-MG algorithm (and compared with more standard
solution techniques). The most efficient method presented in Ref. 148 was
the MG algorithm coupled with a configurational-bias Monte Carlo
(CBMC) procedure for simulating the flexible chains. In that approach, a lin-
earized form of the PB potential (the Debye-H ¨ ckel potential),
e
k
r
r
v
DH
Þ
kT
¼
ð
r
Z
2
l
½
60
B
was employed to guide the generation of a new trial chain in the CBMC pro-
cess. In Eq. [60],
Z
is the charge,
l
B
is a physical parameter termed the Bjer-
rum length (which is about 7
˚
is the inverse of the Debye-
H ¨ ckel screening length. Then, following chain growth, the full PB equation
was solved to yield a correction to the free energy beyond the Debye-H ¨ ckel
level. That is, the simulation moves on the free energy surface determined by
the nonlinear PB level of theory. Tens of thousands of numerical solutions of
the PB equation were required. Simulations were performed for a range of
polyelectrolyte charge densities both above and below the Manning condensa-
tion limit. This kind of simulation is similar in spirit to the Born-Oppenheimer
ab initio simulation methods discussed above in the electronic structure sec-
tion; in those methods the nuclei move based on forces determined by the
ground-state electronic surface (Hellman-Feynman forces).
182
Here the
mobile ion distribution is assumed to relax quickly for a given polymer con-
figuration, and the free energy of the configuration is computed from that
equilibrium distribution. These ideas have also been applied to simulations
of large-scale colloid systems and proteins.
222-225
In addition, the accuracy
of the PB approximation for ions near DNA has been addressed by Ponomar-
ev, Thayer, and Beveridge.
226
While real-space grid methods have been applied to continuum dielectric
problems for some time, their utilization for computations of electrostatic
forces in molecular-level simulations is rather new. Long-ranged forces such
as the Coulomb potential present difficult challenges for molecular simulation,
and those components of molecular dynamics codes typically consume a large
fraction of the computational time. Significant efforts have been directed at
improving the speed and scaling of electrostatics calculations.
The Coulomb potential in simulations of periodic systems is typically
computed with the Ewald method.
227
This clever idea from 1921 partitions
the problem into a locally screened potential and a long-range part that can
be handled with a momentum space sum. The method formally scales as
N
3=2
if an optimal width parameter is used for the often-used Gaussian screen-
ing functions. The locally screened part of the potential decays rapidly with
in water), and
k
