First-Principles-Based Multiscale, Multiparadigm Molecular Mechanics and Dynamics Methods for Describing Complex Chemical Processes - Multiscale Molecular Methods in Applied Chemistry - page 12

Fig. 3 Conventional (Dreiding) valence interatomic potentials. Sub-indices 0 indicate equilib-
rium values, k constants are related to force constants for vibrational frequencies, c constants are
related to an energy barriers, and n refers to periodicity

flowing within a dielectric medium ( 1 in a vacuum but larger values are used

for various media), expressed conventionally as

e ¼

C 0 X
i

;

Q i Q j
e

U Coulomb ¼

R ij SR ij ;

R on ; R off (10)

> j

where C 0 corresponds to a unit conversion scalar (e.g., for energy in kcal/mol,
distances in ˚ , and charge in electron units, C 0 ¼ 332.0637), Q i,j to the pairwise

point charges, R ij to the interparticle distance, and S to a cutoff function.
One additional term included in Dreiding accounts for weak hydrogen bonded
interactions, as a mixture of 3-body angles (between an H atom, and H donor and
acceptor atoms) and non-bonded terms (between donor and acceptor atoms), and is
given by

E HB ð R ; q AHD Þ¼ E b ð R Þ E a cos q AHD ð ð Þ Þ: (11)

The most time-consuming aspect of MD simulations for large systems corre-
sponds to the calculation of long-range non-bond interactions, (7) and (8), which
decrease slowly with R . This scales as O ( N 2 ) for an N particle system (e.g., a
protein with 600 residues would have ~6,000 atoms requiring ~18 million terms to
be evaluated every time step). One way to reduce this cost is to allow the long-range
terms to be cut off smoothly after a threshold value ( S function in (9) and (10)).
Alternatively, our Cell Multipole Method (CMM) [ 40 ] (and the Reduced CMM
[ 41 ]) enable linear scaling, reducing the computational cost while retaining accu-
racy over large-scale systems.