Molecular Dynamics Monte Carlo and Molecular Mechanics

The molecular dynamics, Monte Carlo, and molecular mechanics methods are well reviewed and proven modeling methods and we only discuss our particular implementations. These methods require the specification of molecular potential energy functions to describe the various many-body interactions common in molecular systems. Since our interest here is to understand the structure and morphology in solution and in dry single molecule nanoparticles, we have elected to use potential functions with a proven record of accurate prediction of structure and electronic spectra for conjugated organic molecules. These potentials are harmonic or Morse oscillators for the bond stretching and angle bending terms (both in- and out-of-plane), truncated Fourier series for the torsion interactions (regular dihedral and improper), and Lennard-Jones 6-12 plus Coulomb potentials for the nonbonded interactions. A number of standard force fields fall into this category, such as the MM2, MM3, MM4, Dreiding, UFF, MMFF, CHARMM, AMBER, GROMOS, TRIPOS, and OPLS models [30]. In the present study, we have used parameters defined within the MM3 model as this particular parameterization has proven to give very accurate results for structural optimizations of many conjugated organic molecules [31]. Of particular importance is the capability of the MM3 model to account for intermolecular interactions of the p-electron densities through the dependence of the stretching and torsion terms on iterative self-consistent field (SCF) evaluations for the relevant p-conjugated bonds. The overall reliability of this model for structural calculations has continually been demonstrated for numerous aromatic compounds (benzene, biphenyl, annulene) [32] and conjugated systems (t-stilbene and even multiple oligomers of PPV) [33]. The structures were also verified by comparing the results to those obtained from identical simulations using calculations with no assumed potentials by semiempirical and ab initio quantum mechanics.

Monte Carlo methods used in macromolecular science generally begin by constructing a Markov chain generated by the Metropolis algorithm (i.e., sampling of states according to their thermal importance: Boltzmann distribution for the ensemble under consideration, usually the canonical ensemble) [34]. As the chain length of a simulated system becomes longer, it quickly becomes necessary to introduce a series of biased moves in which additional information about the system is incorporated into the Monte Carlo selection process in such a way as to maintain detailed balance. The most commonly used biased sampling techniques are the continuum and concerted rotation moves. These modern algorithms or slightly modified versions can efficiently generate dense fluid polymer systems for chain lengths of 30-100 monomers. Longer polymer chain lengths pose additional convergence problems and often require the use of other types of biased moves, in particular the double bridging moves [35]. In the present study, we are primarily interested in understanding the morphology and molecular structure of polymeric molecules composed of PPV-based molecules in dilute solution. The applicability of the Monte Carlo methods to the current situation is somewhat different than in dense fluids but the biased moves developed for that regime are still valid. Addition of solvent through continuum models (discussed below) or explicit atoms can also be easily implemented. One of our primary interests in using the Monte Carlo method is to ensure adequate equilibration of longer chain polymers, i.e., those with hundreds of monomers. Molecular mechanics and MD methods can also be used but generally require considerably longer times to equilibrate. For shorter chain lengths, however, we use these methods to obtain temporal data on the dynamical processes of chain self-organization.

Molecular mechanics methods use the laws of classical physics to predict structures and properties of molecules by optimizing the positions of atoms based on the energy derived from an empirical force field describing the interactions between all nuclei (electrons are not treated explicitly) [36]. As such, molecular mechanics can determine the equilibrium geometry in a much more computationally efficient manner than ab initio quantum chemistry methods yet, the results for many systems are often comparable. However, as molecular mechanics treats molecular systems as an array of atoms governed by a set of potential energy functions, the model nature of this approach should always be noted.

MD simulations essentially consist of integrating Hamilton's equations of motion over small time steps. Although these equations are valid for any set of conjugate positions and momenta, Cartesian coordinates greatly simplifies the kinetic energy term. In our MD simulations, the integrations of the equations of motion are carried out in Cartesian coordinates, thus giving an exact definition of the kinetic energy and coupling, and the classical equations of motion are formulated using our geometric statement function approach, which reduces the number of mathematical operations required by a factor of ~60 over many traditional approaches. These coupled first-order ordinary differential equations are solved using novel symplectic integrators developed in our laboratory that conserve the volume of phase space and robustly allow integration for virtually any timescale.

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