Classical semi-empirical force fields ChE210D Today's lecture: approximate descriptions of interatomic interactions suitable for descriptions of many organic, inorganic, and other non-reacting systems. The classical picture The classical approximation The extensive computational demands of electronic structure calculations mean that their application to even modest-sized molecular systems is quite limited. Fortunately, to good approximation, we don’t need to solve the Schrodinger equation for many systems to accurate- ly reproduce their properties. Instead, we can use a classical description, which ignores the motions of the electrons and describes the time-evolution of the nuclear positions alone. A classical approach uses a force field or classical potential energy function that approximates the quantum ground-state potential energy surface due to electronic structure and internuclear interactions, as a function of the positions of the nuclei. Classical descriptions work very well under the following conditions: • the Born-Oppenheimer approximation is valid • the electronic structure is not of interest • the temperature is modest (not too low) • there is no bond breaking or forming • electrons are highly localized (metals and pi-bonded systems are delocalized) Basic features In the classical approximation, we describe a system by the positions and momenta of all of the atomic nuclei: 푁 퐫 = (푥1, 푦1, 푧1, 푥2, … , 푦푁, 푧푁) 푁 퐩 = (푝푥,1, 푝푦,1, 푝푧,1, 푝푥,2, … , 푝푦,푁, 푝푧,푁) Even though technically we deal with nuclei, we can think of the fundamental particle as an atom. Unlike quantum uncertainty, each atom has a definite position 퐫 and momentum 퐩. © M. S. Shell 2019 1/24 last modified 9/29/2019 Alternatively, we could consider the velocity 퐯 instead of the momentum, since the two simply differ by a constant mass factor. We will actually use the momentum more often, since there are several reasons why it is a more natural in statistical mechanics. A microstate is just one “configuration” of the system. In a classical system, one microstate is characterized by a list of the 3푁 positions 퐫N and 3푁 momenta 퐩푁, for a total of 6푁 pieces of 푁 푁 information. For a microstate 푚 we might use the notation (퐩푚, 퐫푚) to indicate specific values of these variables. For any microstate, we can calculate the total, potential, and kinetic energies. The potential energy function depends on the positions 푈(퐫푁) and the kinetic energy function depends on the momenta 퐾(퐩푁). The Hamiltonian of a classical system is the function that gives the energy of a microstate: 푁 푁 푁 푁 퐻(퐩 , 퐫 ) = 퐾(퐩 ) + 푈(퐫 ) The kinetic energy term simply follows | |2 푁 퐩푖 퐾(퐩 ) = ∑ 2푚푖 푖 Interactions between atoms are described by a potential energy function that depends on the positions but not the momenta of all of the atoms, 푈(퐫1, 퐫2, … ). Fundamentally, the potential energy function approximates ground state energy that would be obtained by solving the electronic structure using Schrodinger’s equation, for fixed nuclear positions. The time evolution of the system is deterministic and described by Newton’s equations: 2 푑 퐫푖 푚 = 퐅 for all atoms 푖 푖 푑푡2 푖 Or alternatively, 푑퐩푖 푑푈 = − (퐫1, 퐫2, … ) 푑푡 푑퐫푖 Here, the LHS gives the time rate of change of momentum of atom 푖 and the RHS gives the force on atom 푖 which, by definition, is the derivative of the potential energy function with respect to atom 푖’s position. The total energy is constant during a system’s time evolution according to Newton’s laws. This is a statement of the conservation of energy. It is readily shown: © M. S. Shell 2019 2/24 last modified 9/29/2019 2 푑퐻 푑푈 푑퐫푖 푑 퐩푖 = ∑ + ∑ ( ) 푑푡 푑퐫푖 푑푡 푑푡 2푚푖 푑푈 푑퐫푖 퐩푖 푑퐩푖 = ∑ ( + ) combining the sums 푑퐫푖 푑푡 푚푖 푑푡 푑푈 푑퐫푖 푑퐫푖 푑퐩푖 푑퐫푖 = ∑ ( + ) using the fact that 퐩푖 = 푚푖 푑퐫푖 푑푡 푑푡 푑푡 푑푡 푑퐫푖 푑푈 푑퐩푖 = ∑ ( + ) 푑푡 푑퐫푖 푑푡 푑퐫푖 푑퐩푖 푑푈 = ∑ × 0 = 0 using Newton's law = − 푑푡 푑푡 푑퐫푖 The atomic force field The most important aspect of the classical description is the potential energy function 푈(퐫푁). This function takes in a set of positions of all of the atoms and returns an energy due to the interatomic interactions. This energy approximates that which one would obtain by solving the Schrodinger equation for all of the electrons in the multi-atom system, for given fixed positions of the nuclei. The approximation stems from examining the various modes by which atoms can interact according to the Schrodinger equation, and patching simple, often first-order theoreti- cal expressions for these together. 푈 typically has two main components, energies due to bonded and nonbonded interactions. These arise as different interpretations of the solution of the Schrodinger equation for the electron clouds; it turns out that our notion of bonded interactions naturally manifest as solutions to the wavefunctions when atoms approach at close range. Bonded interactions Bond stretching An accurate description of bond stretching that well-describes quantum-mechanical solutions is the empirical Morse potential: © M. S. Shell 2019 3/24 last modified 9/29/2019 2 −푎(푑−푑0) 푢(푑) = 퐷푒[1 − 푒 ] where 푑 is the length of the bond,푎 a constant, 푑0 the equilibrium bond length, and 퐷푒 the well depth minimum. However, this form is rarely used. It requires three parameters per bond and is somewhat expensive to compute in simulation due to the exponential term. Since the energy scales of bond stretching are so high, bonds rarely deviate significantly from the equilibrium bond length. Thus we can use a second-order Taylor expansion around the energy minimum: 2 푢(푑) = 푎(푑 − 푑0) where 푎 is a different constant from above. Bond angle bending This potential accounts for deviations from the preferred hybridization geometry (e.g., sp3). Again a common form is a second-order Taylor expansion about the energy minimum: 2 푢(휃) = 푏(휃 − 휃0) where 휃 is the bond angle between three atoms and 푏 and 휃0 are constants. Bond torsions/dihedrals fig © M. S. Shell 2019 4/24 last modified 9/29/2019 These interactions occur among four atoms and account for energies of rotations along bonds. Unlike bonding and angle terms, these degrees of freedom are “soft” in that torsional energies are often not so high as to only allow small deviations from an equilibrium structure. Generic torsional interactions can be approximated by a cosine expansion, 푁 푛 푢(휔) = ∑ 푐푛 cos(휔) 푛=0 where 휔 is torsional angle, 푛 is a summation index, 푐푛 are summation coefficients, and 푁 is the number of terms in the expansion. In practice, an alternative but ultimately equivalent expres- sion is often used: 푁 푢(휔) = ∑ 푐푛[1 + cos(푛휔 − 훾)] 푛=0 where the 푐푛 are different from above and the parameter 훾 is an additional offset parameter. This form has the advantage of converging faster for a given number of terms in the expansion. In practice, generally only one or two terms is used. Nonbonded interactions The nonbonded interactions apply to any atoms that are not closely bonded (generally with a bond order greater than 3 or 4), either within the same molecule or between two different molecules. These interactions are described using a pairwise decomposition of the energy. Formally, we can decompose the potential energy function into interactions involving single atoms, pairs of atoms, triples of atoms, and so on and so forth: 푁 푁 푁 푁 푁 푁 푁 푈(퐫 ) = ∑ 푢1(퐫푖) + ∑ ∑ 푢2(퐫푖, 퐫푗) + ∑ ∑ ∑ 푢3(퐫푖, 퐫푗, 퐫푘) + ⋯ 푖=1 푖=1 푗=푖+1 푖=1 푗=푖+1 푘=푗+1 Typically, by symmetry, the two- and three-body terms do not depend on the absolute posi- tions of the molecules but on the relative positions: 푁 푁 푁 푁 푁 푁 푁 푈(퐫 ) = ∑ 푢1(퐫푖) + ∑ ∑ 푢2(푟푖푗) + ∑ ∑ ∑ 푢3(푟푖푗, 푟푖푘, 푟푗푘) + ⋯ 푖=1 푖=1 푗=푖+1 푖=1 푗=푖+1 푘=푗+1 We see that multi-body effects are first included at the level of the pair potential. We would like to truncate this expansion beyond that point for the following reason: the two- and three- body interactions require a loop over atom pairs and triplets, respectively, and thus their © M. S. Shell 2019 5/24 last modified 9/29/2019 relative computational expense goes as 푁2 and 푁3. The cubic scaling might make our simula- tions prohibitive for modest system sizes. By truncating the higher order interactions, we typically neglect important effects as these make nonnegligible contributions to the total energy. Instead, we approximate the higher- order energies by renormalizing the pair interactions. At this point, we have an effective pair potential rather than the one which would result from this systematic expansion: 푁 푁 푁 푁 푈(퐫 ) ≈ ∑ 푢1(퐫푖) + ∑ ∑ 푢eff(푟푖푗) 푖=1 푖=1 푗=푖+1 Electrostatics In the classical approximation, atoms can have a net charge, which may be a partial or formal charge. Here, the charges are typically determined so as to reproduce the same electrostatic potential that would be given by the true electronic structure and electron density distribution. Atoms with partial charges interact through Coulomb’s law, 푞푖푞푗 푢(푟푖푗) = 4휋휖0푟푖푗 for the two atoms 푖 and 푗 separated by distance 푟푖푗. The partial charges are given by 푞푖 and 푞푗, and 휖0 is the free space permittivity. Note that electrostatic interactions are long-ranged and thus require special treatment, which we will discuss in more detail in Simulations of Bulk Phases. van der Waals attractions Correlations between the instantaneous electron densities surrounding two atoms gives rise to an attractive energy.