Molecular Models

2.2. Molecular Models#

2.2.1. Classical Atomistic Models#

  • One Atom is treated as 1 particle

  • Interactions with in a molecule, i.e, Intramolecular forces are bonded interactions, angles and dihedrals (or torsional angles) as illustrated in Fig. 2.1.

Visualization of bonds, angles, and dihedrals in a molecule

Fig. 2.1 Visualization of bonds, angles, and dihedrals in a molecule with four atoms.#

2.2.1.1. Non-bonded interactions#

  • Van der Waals (dispersion forces): induced dipoles

    \( u(r) \sim \frac{1}{r^6} \quad \quad \text{(where } r \text{ = distance between particles)} \)

  • Repulsion (electron cloud overlap)

    \(u(r) \sim e^{-r/l}\), which is slow/expensive to evaluate

  • Combine Van der Walls and short ranged repulsion, one gets the Lennard-Jones Potential (short-ranged):

    \( u(r) = 4\varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^6 \right] \)

  • Coulomb electrostatics (long-ranged):

    \( u(r) = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q_1 q_2}{r} \), where \(\varepsilon_r = 2.61\) is the relative permeability.

Note

The Lennard-Jones potential (as shown in Fig. Fig. 2.2) is incredibly popular for a whole range of models and is frequently used across fields. It can be used to model simple model liquid systems, predict phase diagrams, and much more.

Sketch of the Lennard-Jones potential

Fig. 2.2 Lennard-Jones potential \(U(r)\) as function of distance \(r\). Here, \(\sigma\) corresponds to a particle size, and \(-\varepsilon\) is the interaction energy at the minimum of the potential, which is located at \(r=2^{1/6}\sigma\).#

It is helpful to plot these non-bonded interaction potentials with python and matplotlib, vary the parameters to familiarize ourselves with the shapes and the effects of each parameter.

2.2.1.2. Bonded, Angle, and Dihedral interactions#

2.2.2. Total potential energy#

The total potential engery is the sum of all one-body, two-body, and higher-body interactions. One-body interactions are for example caused by external fields or forces. Two body interactions are interactions as described above.

\( U = \sum_i u(r_i) + \sum_i \sum_{j>i} u(r_i,r_j) + \sum_i \sum_{j>i} \sum_{k>j} u(r_i,r_j,r_k)+ \ldots \)

2.2.3. Two-Body Approximation#

Three-body (and higher) forces are expensive to compute (combinatorics!). Usually, we truncate at two-body interactions and use an effective potential (Fig. 2.3), that reproduces relevant properties.

Other forces (e.g., polar forces) are often neglected.

Effective pair potential that incoperates some approximation of three-body effects.

Fig. 2.3 “Effective” pair potential that incoperates some approximation of three-body effects.#

Atomistic force fields often consists of two-body non-bonded interactions, bonds, angles and dihedrals.

2.2.3.1. Additional References#

2.2.4. Two Complementary Approaches to Designing Models#

  • Use experimental data, invent a model to mimic the problem (semi-empirical approach). But: one cannot reliably extrapolate the model away from the empirical data.

  • Use foundation of physics (Maxwell, Boltzmann, and Schrödinger) and numerically solve the mathematical problem, determine the properties (first-principles/ab initio methods)

2.2.5. Coarse-Grained Models#

  • Atomistic resolution may be impractical for large systems.

Example: Nanoparticles in Water

1 water molecule ≈ 3 Å, 1 nanoparticle ≈ 10 nm results in 1 NP diameter ≈ 30 water molecules. If we compare volume \(V_{np}=\frac{4\pi}{3}(5\text{nm})^3\), then we need \(\sim 37,000\) water molecules to fill the same space as one nanoparticle. If, additionally, we are interested in dilute situations, the vast majority of the system is comprised of the solvent.

In these cases, we study models of the particles we care about with effective interactions that coarsen out the pieces of the system we do not care about, i.e the solvent, as illustrated in Fig. 2.4.

Illustration of coarse-graining by removing solvent.

Fig. 2.4 Illustration of coarse-graining by removing solvent. The top shows so-called explicit solvent, e.g. water molecules, and the bottom panel shows the same system where the solvent is replaced with a homogenous “background” that is treated implicitly by modifing the pair interactions. This is called a implicit solvent model.#

Coarse-graining removes degrees of freedom and is therefore less accurate, and often considered a empirical approach.

  • How to get these effective interactions:

    • Force matching

    • Relative entropy minimization

    • Inverse Boltzmann iteration

2.2.5.1. Simple Examples for coarse-grained Models#

Even though a coarse-grained or simplified model lost some amout of detail and degrees of freedom, they often offer a lot of insights.

  • Hard spheres: \( u(r) = \begin{cases} \infty & r < d \\ 0 & r \geq d \end{cases} \)

  • Lennard-Jones spheres for liquids

  • Bead-spring polymers:

    • FENE (Finite Extensible Nonlinear Elastic) Spring model

2.2.5.2. Additonal References#

2.2.6. Force Fields#

  • Parametrized, self-consistent models fit to thermodynamic or structural data.

  • Designed for specific systems:

    • OPLS: hydrocarbons

    • SPC/E, TIP4P/2005, OPC, …: water

    • AMBER, CHARMM: biomolecules

    • TraPPE: phase equilibrium

    • Martini: coarse-grained lipids and biomolecules in water

  • Machine Learning (ML) models trained on quantum mechanics data are becoming popular.

  • You can create or refit your own force field, however validation is crucial.

  • How to get the parameters and functional forms for these force fields:

    • experimental information to parameterize model parameters

    • ab-initio calculations from Density Functional Theory (DFT) or other quantum methods

2.2.7. Reduced Units#

  • SI units are too small and often inconvinient for atomic systems.

  • Use reduced units where quantities are of order 1. This also helps with numerical stability and readability of simulation code.

2.2.7.1. Unit Systems#

Traditional unit systems can be looked at as follows:

Quantity

SI

CGS

Imperial

length

m

cm

ft

mass

kg

g

lb

time

s

s

s

Other units follow from the base units defined abve in the table.

Example: Unit Consistency

Energy: \( E = \frac{[\text{mass}] [\text{length}]^2}{[\text{time}]^2} = \frac{m l^2 }{ t^2} \rightarrow J = \frac{kg \cdot m^2}{s^2}\) ✅

  • For reduced units, choose \( l, m, E \) based on the problem:

    • \(l = \) length = diameter of atom (LJ \(\sigma\), Å)

    • \( m = \) mass = mass of atom (10s amu)

    • \( E = \) energy = interaction energy or thermal energy at reference temperature (\(k_B T\)) (LJ \(\varepsilon\))

This then leads to derived units from your base reduced units:

Derived unit

Relation to base units

area

\([\mathrm{length}]^2\)

volume

\([\mathrm{length}]^3\)

time

\([\mathrm{energy}]^{-1/2} \cdot [\mathrm{length}] \cdot [\mathrm{mass}]^{1/2}\)

velocity

\([\mathrm{energy}]^{1/2} \cdot [\mathrm{mass}]^{-1/2}\)

force

\([\mathrm{energy}] \cdot [\mathrm{length}]^{-1}\)

pressure

\([\mathrm{energy}] \cdot [\mathrm{length}]^{-3}\)

charge

\(`\left(4 \pi \epsilon_{0} \cdot [\mathrm{energy}] \cdot [\mathrm{length}] \right)^{1/2}\)

Here, \(\epsilon_{0}\) is permittivity of free space.

Note that for example time, often called \(\tau\), is a derived unit in reduced unit systems, where it is a base unit in traditional unit systems. We can do that because we can choose \( l, m, E \) sutiable for the system of interest.

2.2.7.2. GROMACS Units#

Quantity

Unit

Length

\(l\)

nm

Mass

\(m\)

amu (g/mol)

Energy

\(\varepsilon\)

kJ/mol

Charge

\(q\)

e (\(=1.602 \cdot 10^{-19}\) coulombs)

Quantity

Derived Unit

Time

\(\tau = \sqrt{\frac{m \cdot \sigma^2 }{ \varepsilon}}\)

ps

Velocity

\(l/\tau\)

nm/ps

Force

\(\varepsilon/l\)

kJ/mol/nm

Pressure

\(\varepsilon/l^3\)

kJ/mol/nm³

Viscosity

\(\varepsilon \tau/l^3\)

kJ/mol·ps/nm³

Diffusivity

\(l^2/\tau\)

nm²/ps

  • Temperature is either in its own units or derived via Boltzmann constant: \( T = \frac{E}{k_B} \)

Example: Time Units

Time in reduced units: \( \tau = \sqrt{\frac{[\text{mass}] [\text{length}]^2 }{ [\text{energy}]}} \rightarrow \tau = \sqrt{\frac{m \cdot \sigma^2 }{ \varepsilon}}\).

Time in GROMACS units: \(\tau = \sqrt{(10^{-3}kg/mol\cdot (10^{-9}m)^2)/(10^3 J/mol)} = 10^{-12}s\) = ps. ✅

2.2.7.3. LAMMPS Units#

LAMMPS offers many different unit systems to use.

Warning

Simulation software may use inconsistent units (e.g., LAMMPS “real” units). Always convert carefully.

Arbitrary units are useful for generalizing results to new problems →Corresponding states principle.

2.2.8. Resources#

  • Allen & Tildesley, “Simulations of liquids” Appendix B - Reduced Units

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