1 Introduction to Computational Chemistry

1.1 What Do We Want to Know?

Chemistry is about understanding and predicting how matter behaves. What structure will a molecule adopt? Will a reaction proceed? What colour will a compound absorb? How tightly will a drug bind to its target? These are hard questions. Molecules are quantum mechanical systems, and exact solutions to the equations governing their behaviour exist only for the simplest cases. Experiments can answer many questions, but they are often slow, expensive, or impractical — we cannot easily measure the structure of a transition state or track individual atoms as they move. Computational chemistry offers another route. By combining physical theory with numerical methods, we can calculate molecular properties, simulate how systems evolve, and predict behaviour before ever entering the laboratory. Different questions require different methods, and a central skill in computational chemistry is matching the question you want to answer to an appropriate computational approach.

1.1.1 Questions About Structure and Energetics

Many fundamental questions in chemistry reduce to questions about energy. Which arrangement of atoms is most stable? Will this reaction proceed spontaneously? How difficult is it to cross a reaction barrier? The most stable structure corresponds to an energy minimum. Whether a reaction is thermodynamically favourable depends on the energy difference between reactants and products. The difficulty of a reaction—how fast it proceeds—relates to the energy barrier that must be overcome to convert reactants to products. To answer such questions computationally, we use methods that can calculate or estimate molecular energies as a function of atomic positions.

1.1.2 Questions About Dynamics

Other questions are inherently about motion and change. How does a protein flex? How does a lithium-ion move through a battery material? How do molecules move through membranes or filters? For these, we generate trajectories that track how atomic positions evolve over time. Trajectories reveal mechanisms and timescales — from femtosecond bond vibrations to the slow diffusion of atoms through solids. Where questions about structure ask what state is stable, dynamics asks how systems move and change.

1.1.3 Questions About Ensemble Properties

Thermodynamic quantities like pressure, free energy, and entropy describe bulk systems, not individual molecules in single configurations. Even questions about a single molecule, like “what conformations does it populate?”, require considering the distribution of structures the molecule explores, not just the global energy minimum. Answering these questions requires sampling many configurations and computing averages.

1.1.4 Questions About Electronic Character

Some questions cannot be answered by knowing molecular energies alone. What wavelength of light will a molecule absorb? Is a transition metal complex paramagnetic or diamagnetic? Where on a molecule is an electrophilic attack most likely to occur? These questions concern the electronic structure of molecules—the arrangement and behaviour of electrons within the molecular framework. A molecule’s colour depends on the energy gaps between electronic states. Magnetic properties depend on the presence and coupling of unpaired electron spins. The regioselectivity of a reaction—which site on a molecule reacts—depends on the distribution of electron density in the frontier orbitals. Answering these questions requires methods that explicitly describe electrons and their quantum mechanical behaviour, not just the total energy of a given nuclear configuration.

1.1.5 Two Classes of Questions

Within these four groups of questions, there is a natural division. Questions about structure, dynamics, and ensemble properties all depend on knowing how molecular energy varies with atomic positions—the potential energy surface. Questions about electronic character require information about the electrons themselves.

1.2 Two Domains of Computational Chemistry

1.2.1 The Potential Energy Surface

For questions about structure, dynamics, and ensemble properties, a central concept is the potential energy surface (or PES). The PES describes how the energy of a molecular system varies as a function of atomic positions. For a diatomic molecule, this is simply a curve showing energy versus bond length. For larger molecules, the PES becomes a high-dimensional surface—impossible to visualise directly, but conceptually similar.

The topology of the PES encodes much of chemistry. Energy minima correspond to stable molecular structures—reactants, products, and stable intermediates. First-order saddle points, which are minima in all directions except one, correspond to transition states—the highest-energy points along minimum-energy reaction pathways. The depth of a minimum indicates relative thermodynamic stability; the height of a barrier is related to how fast or slow a transformation is.

1.2.2 Domain 1: Working with the PES

The first domain of computational chemistry comprises methods for working with the PES—approximating it efficiently and extracting chemical information from it.

Where does the PES come from? The PES can be obtained in several ways. The most rigorous approach calculates energies directly from quantum mechanics—solving (approximately) the electronic Schrödinger equation for each nuclear configuration. This provides accurate energies but is computationally expensive, limiting such calculations to relatively small systems or few configurations.

A more economical approach approximates the PES using classical force fields, also called molecular mechanics. Force fields represent the energy as a sum of simple analytical functions. The parameters in these functions are fitted to reproduce experimental data or quantum mechanical calculations. Force fields sacrifice accuracy for speed, enabling calculations on large biomolecules, polymers, or condensed phases that would be inaccessible with quantum methods.

An emerging approach uses machine-learned interatomic potentials (MLIPs), which are trained on quantum mechanical data to model the PES with near-quantum accuracy at a fraction of the cost. This is an exciting, rapidly developing field, but it sits outside the scope of this course.

Dynamics on the PES. Given a PES (whether from quantum calculations, force fields, or machine learning models), molecular dynamics (MD) simulations allow us to simulate how atoms and molecules move over time. Starting from initial positions and velocities, MD integrates Newton’s equations of motion to generate a trajectory—a sequence of configurations showing how the system moves across the energy landscape. From such trajectories, we can study molecular vibrations, diffusion, and conformational changes. Because a trajectory provides a sequence of different configurations, we can also average over MD trajectories to obtain estimates of thermodynamic properties.

Sampling the PES. Another way to obtain thermodynamic averages is to sample configurations directly using random numbers. In Markov chain Monte Carlo (MCMC), we generate configurations according to their Boltzmann probabilities, without following physical trajectories. Using random numbers for sampling can also be used to model dynamic processes. Kinetic Monte Carlo (kMC) generates stochastic trajectories by sampling events rather than configurations, selecting which transition occurs and when based on rate constants. This is useful for modelling dynamics in cases where MD is impractical, such as processes occurring over very long timescales.

1.2.3 Domain 2: First-Principles Calculation of Chemical Properties

The second domain comprises methods that calculate chemical properties directly from quantum mechanics—the electronic structure methods. These methods serve two distinct purposes.

Accurate energies. The energy calculated at each nuclear configuration defines a quantum mechanical PES, and these calculations underpin the other approaches we have discussed—force field parameters are fitted to quantum mechanical data, and machine-learned potentials are trained on databases of quantum mechanical energies.

Electronic properties. Electronic structure methods also provide access to properties that depend on how electrons are arranged or how they respond to external effects like light or magnetic fields. The questions from the previous section—about colour, magnetism, and reactivity—all require information contained in the electronic wave function or electron density, not just the total energy.

1.3 How This Course Fits Together

The methods introduced above form the content of this course. The four main sections—Molecular Mechanics and Molecular Dynamics, Monte Carlo Methods, Electronic Structure Theory, and Drug Discovery—correspond to different aspects of the two domains above. Understanding how these sections relate helps make sense of the course as a whole.

1.3.1 MM/MD and MC: Working with the PES

The first two sections focus on Domain 1: given a potential energy surface, how do we extract chemical information from it?

The MM/MD section introduces force fields as computationally tractable approximations to the PES, suitable for large systems and long timescales. With a force field in hand, molecular dynamics simulations reveal how systems evolve—how molecules vibrate, how proteins flex, how solutes diffuse through solvents. As we noted earlier, MD trajectories also provide a route to thermodynamic averages by sampling configurations over time.

The Monte Carlo section introduces an alternative approach to sampling. Metropolis Monte Carlo generates configurations according to their Boltzmann weights without following trajectories, enabling direct calculation of thermodynamic averages. Kinetic Monte Carlo uses random sampling to model rare events, connecting microscopic rate constants to observable kinetics.

1.3.2 Electronic Structure: First-Principles Methods

The third section covers Domain 2: calculating chemical properties from first principles using electronic structure theory.

This section serves two purposes. First, it explains how we can calculate accurate energies that characterise the PES. Force fields and machine learning models are parameterised against quantum mechanical calculations, and the accuracy of any energy-based calculation ultimately traces back to electronic structure theory. Second, it addresses the questions about electronic character raised earlier—spectra, magnetism, reactivity—which require explicit treatment of electrons, not just energies.

1.3.3 Drug Discovery: Computational Chemistry in Action

The final section illustrates how computational chemistry methods can be used and combined to address problems in a specific application domain.

Predicting whether a small molecule will bind to a protein target draws on both domains of computational chemistry. Docking algorithms search for low-energy binding poses on the PES of the protein-ligand complex. Molecular dynamics simulations reveal binding site flexibility. Free energy calculations use statistical sampling to predict binding affinities. Where key interactions require more accurate treatment than classical force fields provide, we can use hybrid methods: these combine quantum mechanical methods for modelling the binding site with classical force fields for the surrounding protein.

This section also introduces data-driven approaches—quantitative structure-activity relationships (QSAR) and pharmacophore models—that predict activity from molecular structure without explicitly modelling the physics.

1.4 Summary

Computational chemistry provides methods for understanding and predicting molecular behaviour. The questions we ask fall into broad classes: those concerning the potential energy surface (structure, dynamics, ensemble properties), and those requiring explicit treatment of electrons (accurate energies and electronic properties such as spectra and magnetism).

These classes map onto two domains of computational methods. Domain 1 focuses on working with the PES—constructing it and extracting information through dynamics or statistical sampling. Domain 2 focuses on calculating chemical properties from first principles—obtaining accurate energies from quantum mechanics and computing electronic properties that depend on the wave function, not just the total energy.

This course covers both domains. Force fields are intuitive (atoms connected by springs, charges interacting), and molecular dynamics uses Newton’s laws, familiar from physics. Monte Carlo uses random numbers rather than equations of motion, but works with the same energy landscapes. Electronic structure theory uses quantum mechanics to calculate the PES itself and to access properties that depend on electrons—spectra, magnetism, reactivity. Drug discovery draws on all of these approaches, illustrating how they combine to address real chemical problems.

Throughout the course, a central skill is learning to match questions to methods—and recognising when different methods need to be combined. No single approach answers every question, and knowing what each method can and cannot do is essential for applying computational chemistry effectively.