5.4 Concurrent multiscale methods
The concurrent approaches couple two or more methods and execute them
simultaneously with continuous information transfer across scales in
contrast to the minimal coupling methods which attempt to do the
opposite. In this class of methods, the behavior at each scale depends
strongly on the phenomena at other scales. A successful algorithm in the
concurrent method implements a smooth coupling between the scales. In
concurrent simulations, often, two distinct domains with different
scales are linked together through a buffer or overlap region called the
handshake region [18].
5.4.1 Quantum mechanics molecular mechanics (QM/MM) Simulations:
An example of concurrent include mixed quantum mechanics/ molecular
mechanics (QM/MM) methods combining MD using the empirical force-field
approach with electronic structure methods [19, 20, 107] to produce
a concurrent multiscale method [76, 108-119]. In the QM/MM
simulations, the system is sub-divided into two sub-regions, the quantum
mechanical sub-region (QM region) where the reactive events take place,
and the molecular mechanical sub-region (which provides the complete
environment around the reactive chemistry) [109, 111]. Since
electronic structure methods are limited by the number of atoms they can
handle (typically 50-500), the QM sub-region is restricted to a small
number of atoms of the total system. For example, in an enzymatic
system, the quantum region can consist of Mg2+ ions,
water molecules within 3 Ã… of the Mg2+ ions, parts of
the substrate molecules, and the catalytic amino acid residues
(such
as aspartic acids). The remaining protein and solvent molecules are
treated classically using the regular classical force-field.
In QM/MM simulations, wave function optimizations are typically
performed in the quantum (or QM) sub-region of the system using an
electronic structure method such as density functional theory (DFT)
[20]. In this step, the electrostatic coupling between the QM and
the MM sub-regions is accounted for: i.e., the charges in the MM
sub-region are allowed to polarize the electronic wave functions in the
QM sub-region. The forces in the quantum sub-region are calculated using
DFT on-the-fly, assuming that the system moves on the Born-Oppenheimer
surface [111, 120]. That is, we assume a clear timescale of
separation between the electronic and nuclear degrees of freedom and the
electronic degrees of freedom are in their ground state around the
instantaneous configurations of the nuclei. The forces on the classical
region are calculated using a classical force-field. Besides, a mixed
Hamiltonian (energy function) accounts for the interaction of the
classical and the quantum sub-regions. For example, since the QM/MM
boundary often cuts across covalent bonds, one can use a link atom
procedure [114] to satisfy the valences of broken bonds in the QM
sub-region. Also, bonded terms and electrostatic terms between the atoms
of the QM region and those of the classical region are typically
included [112].
From a practitioner’s stand-point, QM/MM methods are implemented based
on existing interfaces between the electronic structure and the
molecular dynamics programs; one example implementation is between
GAMESS-UK [121] (an ab-initio electronic structure prediction
package) and CHARMM [25]. The model system can then be subjected to
the usual energy minimization and constant temperature equilibration
runs at the desired temperature using the regular integration procedures
in operation for pure MM systems; it is customary to carry out QM/MM
dynamics runs (typically limited to 10-100 ps because of the
computationally intensive electronic structure calculations) using a
standard 1 fs time step of integration. The main advantage of the QM/MM
simulations is that one can follow reactive events and dissect reaction
mechanisms in the active site while considering the explicit coupling to
the extended region. In practice, sufficient experience and care is
needed in the choices of the QM sub-region, and the many alternative
choices of system sizes, as well as the link-atom schemes, need to be
compared to ensure convergence and accuracy of results [112]. The
shorter length of the dynamics runs in the QM/MM simulations (ps)
relative to the MM MD simulations (ns) implies that sufficiently
high-resolution structures are usually necessary for setting up such
runs as the simulations only explore a limited conformational space
available to the system. Another challenge is an accurate and reliable
representation of the mixed QM/MM interaction terms [115]. These
challenges are currently being overcome by the suitable design of
next-generation methods for electronic structure and molecular mechanics
simulations [51, 122]. Other examples of concurrent methods linking
electronic structure and or molecular mechanics scales include Car
Parrinello molecular dynamics (CPMD) [123, 124] and mixed molecular
mechanics/ coarse-grained (MM/CG) [77, 125].
5.4.2 Linking atomistic and continuum models : In several
applications involving solving continuum equations in fluid and solid
mechanics, there is a need to treat a small domain at finer (often
molecular or particle-based resolution) to avoid sharp fronts or even
singularities. In such cases linking atomistic and continuum domains
using bridging algorithms are necessary. A class of algorithms that
realize this challenging integration have been reviewed in [18]:
examples include the quasicontinuum approach, finite-element/ atomistic
method, bridging scale method, and the Schwartz inequality method
[126, 127], which all employ domain decomposition bridging by
performing molecular scale modeling in one (typically a small domain)
and integrating it with continuum modeling in an adjoining (larger)
domain, such that certain constraints (boundary conditions) are
satisfied self-consistently at the boundary separating the two domains.
Such approaches are useful for treating various problems involving
contact lines.