On the algorithm and theory side, the advent of semilocal and especially nonlocal descriptions of exchange and correlation effects within the density functional theory (DFT) now allows describing bonding in molecules and solids with a sufficient accuracy to compare quantitatively to experiments for many classes of systems. It is therefore becoming possible to develop a semiquantitative description of a large number of systems and processes relevant to PV development. As such, we briefly review DFT and its various extensions to include nonlocal terms that are important for long-range dispersion interactions that dominate many self-assembly processes, molecular surface adsorption processes, solution processes, and biological and polymeric materials. Additionally, as computational simulation of many physicochemical processes, especially those related to fabrication of materials, often requires using direct molecular dynamics (MD) simulations or methods that allow treatment of systems under extreme environments, we also review new quantum dynamics methodologies. We note that considerable progress is still needed in designing efficient electronic structure algorithms that can accurately treat excited state processes and that can speed up the required evaluation of energy and forces (see “Prospects on Improving Computational Approaches” section). Some relatively short-term hope may arise from new computer architectures and programming paradigms such as general programming of graphical processing units (GPGPU).
Electronic Structure: DFT and DFT-Based Methods
DFT proposes to solve electronic structure problems using the electron charge density, ρ(r), as the fundamental variable,54 which is formally based on the Hohenberg and Kohn theorems.55 In practice, DFT is applied using the Kohn–Sham (KS) ansatz, using a mean-field approach.56 The KS method represents the density as a linear combination of the inner products of independent spin-orbital functions and the energy as a functional of ρ(r) as
where the first and second terms are the kinetic energy of independent particles, T0[ρ], and the Coulomb interaction energy, and J[ρ] = ½ ∫∫dr′drρ(r′)ρ(r)/|r′ − r|. The term υext(r) is the external potential generated by the nuclei and experienced by the electrons, and VNN is the nuclear repulsion energy for a fixed nuclear configuration. In eq 1, the contribution Exc[ρ] is the exchange-correlation energy, which includes the electron exchange interaction as well as the many-body contributions to the kinetic and electron–electron repulsion potentials (i.e., part of Vee[ρ]) that are not included in T0[ρ] or J[ρ], that is, Exc[ρ] = Vee[ρ] + T[ρ] − J[ρ] − To[ρ]. The explicit expression of Exc remains unknown; however, there are many approaches that have shown satisfactory results using standard approximations such as general gradient approximations and hybrid functionals. Such approaches are grouped according to their treatment of the density into “generations” or “ladder's rungs.”57 The most common are based on the local density approximation or generalized gradient approximation (GGA).58 Although some functionals have shown impressive results, those are not always transferable for every problem and can often fail for the description of long-range interactions and excited states. The origins of these difficulties are attributed to the incorrect cancellation of electron self-interaction,59 incorrect treatment of dynamic correlation, and so forth. There are many strategies to avoid these problems, some of which involve the inclusion of explicit terms from wavefunction theories (hybrid functionals), treatments with optimized effective potentials, an adjustment to the asymptotic correction exchange correlation potentials, and the addition of empirical energy terms.
The pervasiveness of hydrogen bonding and London forces among biological60–62 and supramolecular63–65 entities has motivated persistent efforts to develop reliable computational methodologies for noncovalent interactions.66, 67 Extensive work has established the importance of high levels of electron correlation, as embodied by coupled cluster through perturbative triples, CCSD(T),68 for the proper characterization of dispersion attractions. Yet, achieving this “gold standard” of chemical accuracy69 presents a formal cost of O(N7), where N is proportional to system size. Meanwhile, the efficient DFT [formally, O(N3) − O(N4), or O(N5) for double hybrids) can falter for even qualitative descriptions.70–73 The introduction of the DFT + dispersion (DFT-D) and exchange-hole dipole moment (XDM)74–79 methods and the crafting of several long-range functionals80–89 have attempted to improve the quantum chemical techniques available for nonbonded systems by introducing an largely empirical dispersion term. These are computationally efficient and promising approaches that have emerged from a very active field that strives to improve performance for classes of noncovalent interactions governed by disparate intermolecular forces and spanning broad-length scales.
Exchange-correlation functionals developed for covalent systems are often successful in treating hydrogen bonding and other electrostatically dominated noncovalent interactions that act over short-length scales (<2 Å).90 In contrast, the dispersion attraction, which arises from correlated motions of electrons, is primarily prominent over longer distances [∼2–5 Å (medium range) and >5 Å (long range)], and none of the typical components of a functional, the local electron density (ρσ), its gradient (▿ρσ, present in semilocal GGA functionals), or its kinetic energy (τσ, present in nonlocal meta-GGA functionals), is fully capable of acting over a suitable span of length scales. The original introduction of hybrid functionals included long-range behavior (nonlocality) through Hartree–Fock exchange; however, there are still issues with local correlation and, therefore, many such functionals are also unable to correctly describe the R−6 asymptotic distance dependence of dispersion forces. A number or research groups have developed successive families of functionals of parameterized hybrid meta-GGAs for main-group thermochemistry and noncovalent interactions.82, 84 By including systems with noncovalent interactions into the fitting set and incorporating double the amount of nonlocal exchange, the functionals strive to capture medium-range electron correlation, for reasonable results up to ∼5 Å separation of complexes.83, 84 This area of research and development continues to introduce improved functionals that are providing very encouraging results.91
The challenges to adapt efficient scaling wavefunction and DFT methods for treatment of dispersion have been the subject of several recent reviews in the literature.84, 92–96 Among the empirical treatments developed are dispersion-corrected atom-centered potentials97 and DFT-D.80, 98–100 For common implementations of the former, a pseudo-potential optimized to replicate reference interaction energies is placed over each atom, thereby incorporating medium-range correlation but not capturing the correct asymptotic form, whereas for the latter, a damped atom–atom dispersion term is added to the total DFT energy, thereby accurately encoding the long-range attraction, yet necessitating a carefully tuned damping function to evenhandedly address double counting of the correlation energy across all system configurations and types of interaction. The exchange-hole dipole moment (XDM) is constructed similarly to DFT-D as a correction to the base electronic energy but, by modeling the instantaneous dipole that arises between an electron and its exchange hole, it generates dispersion coefficients that incorporate the effect of the chemical environment. There are also efforts to include medium-range dispersion in conventional semilocal DFT or to incorporate correlation components from wavefunction theory.86–89 Both approaches can often address long-range effects through an additional DFT-D-like term. A more complex task is the development of explicitly nonlocal correlation functionals from first principles. Currently at the forefront are vdW-DF101 and VV09,102 which show very good promise but do not necessarily exceed the accuracy of some of the more empirical methods.
The local or semilocal character of conventional density functionals necessarily leads to underestimate the long-range correlation interactions that capture attractive van der Waals forces. Initially proposed by Wu and Yang98 for ab initio DFT after its first introduction into the tight binding DFT approach by Elstner et al.103 and further developed by Grimme,80, 99, 100 the DFT + dispersion (DFT-D) method appends to the base functional a scaled, damped, and fitted leading term to the well-known dispersion energy series,
Many calculations follow the DFT-D2 variant, where the correction takes the explicit form:
The dispersion coefficients, C, are obtained from the geometric mean of tabulated elemental values, are summed over interatomic distances, Rij, modulated by a damping function, fdamp(Rij), that gradually activates the dispersion correction over a distance characterized by the sum of the two atomic vdW radii, whereas an overall scaling term, s6, is optimized to be unique to each DFT exchange functional. The scaling factors used are sB2PLYP6 = 0.55, sPBE06 = 0.6, sPBE6 = 0.75, sB9706 = 0.7564, sBP866 = 1.05, sB3LYP6 = 1.05, and sB976 = 1.25. Grimme et al.100 recently presented a refined method, DFT-D3, that incorporates an additional R−8 term in the dispersion series and adjusts the C combination formula and damping function. The individual atomic C are interpolated from several reference values based on coordination numbers extracted from the molecular structure, rather than assigned solely by atomic identity as in DFT-D2, and thereby incorporate some awareness of the chemical environment into an otherwise largely heuristic correction. The −D3 dispersion has the following form:
where sr and s8 are the customary nonunity parameters fitted for individual functionals. These parameters are tabulated in ref.100.
A strategy to place dispersion corrections on a somewhat less-empirical footing than DFT-D is the “semiclassical” exchange-hole dipole moment model first formulated in 2005 by Becke and Johnson74 (later placed on firmer theoretical foundations76–78). This approach is based on the concept that the nonzero dipole moment of an exchange hole104 can induce an instantaneous dipole moment and, thus, dispersion interactions between nonoverlapping systems. The method uses first-principles calculation of the dispersion coefficients for the classical dispersion force and forms a fully self-consistent and first-principles treatment of dispersion effects (the form of the dispersion interaction is not based on the first-principles calculation, but the coefficients are based on this calculation).
Recently, one of the present authors helped perform a systematic study of techniques for treating noncovalent interactions within the computationally efficient DFT framework through comparison to benchmark-quality evaluations of binding strength compiled for molecular complexes of diverse size and nature.105 Through comparison with available CCSD(T)/CBS benchmarks, the performance of DFT methods was evaluated both overall and for hydrogen-bonded, mixed-influence, and dispersion-bound subsets. Complete evaluations of test sets were conducted with aug-cc-pVDZ and aug-cc-pVTZ, and closer analysis with families of Dunning double-ζ and triple-ζ basis sets have permitted appraisal of these bases as well as the utility of the counterpoise correction. The efficiency and accuracy of functionals deliberately crafted to encompass long-range forces, a posteriori DFT + dispersion corrections (DFT-D2 and DFT-D3), and exchange-hole dipole moment (XDM) theory was assessed against a large collection of 469 energy points. For most results, M05-2X, B97-D3, and B970-D2 yield superior values in conjunction with aug-cc-pVDZ, for a mean absolute deviation of 0.41–0.49 kcal/mol, and B3LYP-D3, B97-D3, ωB97X-D, and B2PLYP-D3 dominate with aug-cc-pVTZ, affording, together with XYG3/6-311+G(3df,2p), a mean absolute deviation of 0.33–0.38 kcal/mol. These results provide a benchmark for many of the different possible approaches for including dispersion into DFT and highlight the fact that indeed chemical accuracy can be achieved for a nonbiased set of test systems. This is important as it allows the computational efficiency of DFT to be maintained while enabling accurate calculation for a much broader class of molecular materials. However, one should be cautious and note that for anisotropic systems with a high polarizabilty, nonadditive cross-polarization effects can yield a substantially altered van der Waals attraction strength that may cause inaccuracies, and thus, care is still required when using such methods.106, 107 More exact approaches such as the “vdW Density Functional”108–110 may offer an improved general capability for the latter case; however, this is still an active research area.
An extension to the standard DFT approach, time-dependent DFT (TDDFT), uses the correspondence between a time-dependent external potential and the time-dependent one-particle density to enable computation of excited states properties. However, as with standard DFT, difficulties with the current exchange-correlation functionals still lead to problems with electron self-interaction, which means that TDDFT will generally overestimate wavefunction delocalization. These errors tend to shift the vertical excitation energies (absorption energy) when compared with experiment, and as such a quantitative comparison to experiment is unlikely. A recent review presents an overview of the current concerns for polymeric materials.111
Quantum Dynamics-Based Methods
A complete description of molecular systems requires solving the full time-dependent Schrödinger equation for the constituent electrons and nuclei. This task is currently out of reach of state-of-the-art computational approaches and resources. To simplify the problem, it is usually assumed that the motion of electrons (fast) and nuclei (slow) can be separated and that Born-Oppenheimer approximation is valid. As a consequence, the time-dependent equations are replaced by (a) classical Newton's equations for nuclei and (b) time-independent equation for electrons. The resulting method is referred to as Born-Oppenheimer MD (BOMD)112–115 or its extended Lagrangian variants116, 117 in which nuclei move classically on the ground-state potential energy surface (or on its approximation) obtained on-the-fly self-consistently (or are propagated classically) from the stationary electronic structure.
The high computational demand of the electronic structure calculation renders BOMD only practical in implementations based on an independent particle model (Hartree–Fock, tight-binding, and density functional theories). In these methods, all nuclei move on the potential energy surface generated by the electronic structure, and the total “electronic” energy of the system can be expressed as a sum over occupied orbitals
in which the electronic structure is represented by the density matrix P. VNN represents the interaction between nuclei and does not depend on the electronic structure. The second, so-called one-electron term, Hcore, depends linearly on P and represents the kinetic energy of electrons and interaction of electrons with nuclei. The last term, G(P), describes the electron–electron repulsion and depends quadratically on P. The last equation can be used regardless of whether one solves a time-independent electronic structure (BOMD) problem or quantum-dynamical time-dependent Schrödinger equation. It is orbital free (but not basis set free) and is more general. The choice of molecular orbital is arbitrary as the electronic structure represented by density matrix P may be arbitrarily described by many different choices of molecular orbitals.
In the Born Oppenheimer case, the electronic structure is determined through minimization of the energy according to the variational principle. This is usually achieved by diagonalization of the total matrix Hamiltonian [F = Hcore + G(P)]. Alternatives to diagonalization methods exist (e.g., a Fermi operator expansion or direct density matrix minimization) and are described elsewhere (see refs.118–120 and references therein). Once the electronic structure and corresponding nuclear forces are known, the molecular motion can be determined and related properties can be described.
Alternatively to time-independent solutions of the electronic structure problem in dynamics, one can integrate a time-dependent Schrödinger equation along with Newton equations of motion for the nuclei. Another option includes quantum nuclear effects into dynamics for selected nuclei. Other possible approaches include quantum wavepacket dynamics, path integral, and Bohmian trajectory approaches. Notably, quantum dynamical treatment plays an especially important role in proton and electron transfer processes.118–120
Classical Mechanics-Based Approaches
Although many of the intrinsic processes governing OPV operation are quantum mechanical in nature (e.g., exciton generation/recombination, charge carrier mobility, and charge transfer), structural and dynamical processes of long-chain molecules typically used in OPVs are better described within classical or statistical mechanics. In this case, particle-based methods such as classical MD, Monte Carlo (MC), molecular mechanics (MM), and various coarse-grained versions can be of great utility. These are well-tested methods with an established track record of success within the field of polymer physics, and therefore, we only briefly introduce them (e.g., see refs.121 and122 for more details).
For molecular-based materials, these methods require the specification of potential energy functions (usually known as force fields) to describe the various relevant many-body interactions. As one of the main interests here is to understand the structure and bulk morphology in solution-cast thin films, the use of potential functions with a proven record of accurate prediction of structure and vibrational spectra for conjugated organic molecules are required. Such potentials are typically composed of harmonic or Morse oscillators for the bond-stretching and angle-bending terms (both in plane 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. Several standard force fields fall into this category, such as the MM2, MM3, MM4, Dreiding, UFF, MMFF, CHARMM, AMBER, GROMOS, TRIPOS, and OPLS models.123
MD simulations (for details see refs.124–126) essentially consist of numerically integrating Hamilton's, Newton's, or Lagrange's equations of motion using small integration time steps. Although these equations are valid for any set of conjugate positions and momenta, the use of Cartesian coordinates greatly simplifies the kinetic energy term. The resulting coupled first-order, ordinary differential equations can be solved using numerical integrators.127 Despite the simplicity of MD methods, the simulation of millions of atoms over time-scales up to several seconds is not possible without substantial reduction of the number of equations of motion, a feat typically achieved by course-graining groups of atoms into one element (see “Course-Grained Models” section). This approach not only reduces the number of equations, but when proper separation of the time scales for motion can be achieved, it also allows the use of considerably larger integration time steps and thus helps to bridge the large gap between simulation and experimental time scales.
Molecular mechanics (MM) methods (for details see ref.128) apply 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). As such, MM 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, at least qualitatively. However, as MM 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.
Course-grained models (CGMs) consist of replacing an atomistic description of a molecule with a lower resolution model that averages away some of the fine details of the interactions. Numerous CGMs (differing in the metric used to obtain the course-grained potential, such as the fitting to the forces or to structural features, and in the definition of the course grain) have been developed to investigate the longer time- and length-scale dynamics that are critical to many long-chain molecular processes, such as polymer, lipid membranes, and proteins. Coarse graining can also refer to the removal of certain degrees of freedom (e.g., vibrational modes between two atoms) by freezing the bonds, bends, or torsional degrees of freedom, but more typically it implies that two or more atoms are collapsed into a single particle representation (the so-called united atom model was one of the first popular CGMs). Fundamentally, the level to which a system may be coarse grained is bound by the accuracy in the dynamics and structural properties desired from a simulation. We refer to several review articles for more details.129–131
FT representations can be effectively obtained from CGMs and offer an advantage of significantly reducing the number of differential equations for the particles to a diffusion equation that can be numerically solved using a self-consistent algorithm.132–134 Another advantage of the FT approach is that it ensures thermodynamic equilibrium of the final morphology and thus provides access to modeling much larger length and time scales. Such improvement is difficult to obtain with the particle-based approaches (atomistic or course grained) due to the existence of a very high-dimensional potential energy landscape plagued with multiple local minima. The field theoretic approach can also incorporate realistic polymer models that are often difficult for macroscopic, continuum simulations. In fact, the numerical self-consistent FT (SCFT) for dense polymer melts has been proven to be highly successful in describing the complex morphologies in multiblock copolymers and even for nanoparticles in polymers.132, 133, 135, 136 SCFT simulations for a dense melt of copolymers in bulk use chemical specificity of blocks through the value of the Flory parameter χ, which controls the strength of the chain segregation that drives the formation of microphase segregated morphologies. The course-grained Hamiltonian of a dense melt of n polymer chains containing distinct monomers A and B can be written as follows:
The first term in eq 8 gives the free-energy contribution from the so-called Gaussian thread model. The position vector of a bead's s contour steps along the α-th chain is given by the space curve . The second term is the free-energy contribution in a model where dissimilar monomers in contact have a higher free energy than like monomers in contact. The tendency for chemically dissimilar monomers to segregate is described by the χ value and is a function of the specific chemical species A and B as well as temperature. The model given by eq 8 corresponds to a linear AB diblock copolymer. However, it can be easily generalized to a larger number of blocks to include architectural asymmetry such as branched copolymer architectures, surface/substrate corrugation, and hybrids with nanoparticles.
The coarse-grained Hamiltonian is used to form the partition function of a dense melt of n copolymer chains. The monomer density operators may be recast as continuous monomer density fields by application of a Hubbard-Stratonovich transformation that introduces chemical potential fields w(r) conjugate to the monomer density fields.137 The partition function has the following form:
where ρA and ρB are the density fields and ωA and ωB are the conjugate chemical potential fields that are introduced through the FT transformations. The p field is introduced through the constraint of overall monomer incompressibility. The integrals are over all possible realizations of the fields with F given by
where Q is the single-chain partition function and is a functional of the chemical potential fields and a function of the chain length N. In the mean-field approximation, the extremum of F is assumed to make the largest contribution to the overall sum. In that case, the resulting system of equations can be solved self-consistently using an iterative algorithm to solve for the ρ(r) and ω(r) fields that satisfy the SCFT mean-field equations.
Monte Carlo Methods
MC methods (for details see ref.138) 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). 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 MC selection process in such a way as to maintain a detailed balance. The most commonly used biased sampling techniques are the continuum configuration bias and concerted rotation moves.139 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.140, 141 Addition of solvent via continuum models (implicit solvation models)142 or explicit atoms can also be implemented. A primary interest in using the MC method is to ensure adequate equilibration of longer chain polymers, that is, those with hundreds of monomers. MM and MD methods can also be used but generally require considerably longer times to equilibrate, as they often spend long times in local minima.
A useful extension of the MC method is to introduce a stochastic time variable into the simulation to allow modeling of the rates of specified processes. This approach is referred to as kinetic MC (KMC) and is often used to examine aspects of crystal-growth dynamics143 and to estimate charge transport.144 The method requires input from other computational or experimental methods to provide the energetics for the processes; however, it has been used to obtain critical information for a number of relevant cases. For example, Liu et al.145 used KMC to examine the growth of a C60 phase on graphite. Extensions of KMC to treat nucleation and growth in solutions can also be achieved.146
Multiscale methods generally involve using two or more of the above computational approaches in order to treat a broader set of processes that may span multiple length and time scales (e.g., see refs.147–149). Typically, the so-called multiscale modeling is actually more of a multistep approach (a loose integration), whereby a needed parameter/information at one length or time scale is obtained using an appropriate computational approach that is subsequently used to start another type of simulation for a different scale. For example, aligned nanorod inclusions in polymeric materials were recently studied using this type of approach.150 First, the self-assembly of a mixture of A-coated and B-coated rods in an AB phase-separating blend was examined with dissipative particle dynamics151 simulations. This showed that the steric repulsion between ligands causes the coated rods to preferentially align end to end within the minority phase of the binary blend. The information was then used to formulate a coarse-grained approach, which combined a Cahn–Hilliard152 model for the polymer blend with a Brownian dynamics153, 154 simulation for the rods. Interestingly, the authors found that a small volume fraction of B rods in the majority B phase promotes the percolation of A-like rods within the A phase so that the percolation threshold for the A rods is significantly lowered. However, if the number of B nanorods in the B phase exceeds a particular volume fraction, the B particles inhibit the percolation of the A rods. The output from the morphological studies was then used as input to the lattice spring model155 to characterize the mechanical behavior of the composite revealing that nanorods oriented along the tensile direction contribute to the enhancement of the macroscopic mechanical properties.