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.