Theoretical chemistry

If it were not for the too long wavelengths of visible light, we could see through a microscope the chemical building blocks of all matter: atoms. At a magnification of twenty five million times, the smallest atom, hydrogen, would still seem to be about only one millimeter in size. Although, the word atom comes from the Greek word atomos, which means unbreakable, atoms are actually separable into a relatively small core with a positive charge and a cloud of negatively charged electrons, surrounding the core. In classical mechanics, a moving body with an electric charge (such as an electron) loses kinetic energy by radiating an electro-magnetic field. For an atom, this would imply that an electron flying around the core would be doomed to spiral down into a crash landing onto the core. That this apparently does not happen in reality is a quantum mechanical effect: the remarkable behavior of the electron around the core is in a way similar to a standing wave in a guitar string. The wave pattern of nodes and anti-nodes for an electron is called an orbital; it reflects the space in which the electron moves. According to quantum mechanical laws, one orbital can contain up to two electrons with opposite values for the quantum number $s$, called electron spin. The orbitals for an atom are classified in shells, with a principle quantum number $n$, and the orbitals in one shell can again be divided into sub-shells, each with a so-called angular momentum quantum numbers $l$ and a magnetic quantum number $m_l$. From this buildup of the electrons, each with its unique set of the four quantum numbers $\{n,l,m_l,s\}$, i.e. from the electronic structure, the properties of the different atoms can be understood. The development of accurate quantum mechanical models and computer programs to calculate and understand the electronic structure and its derivable physical properties is one of the most important achievements in theoretical chemistry, for which John A. Pople and Walter Kohn were rewarded with the Nobel prize for chemistry in 1998.

In general, electrons prefer to group together as much as possible into completely filled orbitals, sub-shells and shells, because such configurations have the lowest energy. Consider for example, the first shell ($n=1$), which has only one sub-shell, with only one orbital, so that it can contain just two electrons. For a hydrogen atom, which has only one electron, the first shell is thus half-filled. It can, however, lower its energy by sharing its electron with another hydrogen atom. Together, the atoms form a molecule by shaping a new (molecular) orbital, completely filled by the two electrons. The electrons shared by the atoms form a chemical bond. The energy release by this bond formation is significant; a 60 Watt light bulb could burn for one hour from the energy that would come free if we would have only one gram of hydrogen atoms transforming into hydrogen molecules. On the other hand, in helium, which has two electrons, the first shell is already completely filled so that it cannot further lower its energy by forming chemical bonds. Helium, and the other inert gasses (neon, argon, krypton, xenon and radon) are the only atoms with completely filled shells, which are therefore found as such in nature. All other atoms, are usually only found in nature to be chemically bound to other atoms, forming molecules, metals or crystals.

The relative stability of a molecule is given by its energy of formation (also: minus the atomization energy), which is the energy that would come free if the molecule is formed from bare atoms. In a chemical reaction, the atoms in one or more molecules (the reactants) regroup to form one or more new molecules (the products). The reaction energy is simply the difference between the energies of formation of the products and reactants. At room temperature, most chemical reactants are activated, i.e. one first has to inject energy into the reactants in order to start the chemical reaction, even if the reaction energy is negative. The existence of such energy barriers is actually very fortunate, and if they would suddenly disappear, this theses, for instance, would instantaneously react explosively with oxygen in the air to form carbon dioxide and water vapor, before you could reach to the scientific juicy parts -- and that would have been even the least of your problems. For most chemical reactions between molecules in the gas-phase, the reaction rate constant, $k$, depends exponentially on the absolute temperature, $T$, as reflected by the well-known Arrhenius equation:

$$k=A \cdot e^{-E_a/RT}$$ (1)

Here, $R$ is the universal gas constant, $E_a$ is the activation energy and A is a pre-exponential factor whose dependence on the temperature is usually neglected. Within collision theory, this empirical equation is rationalized as the product of the (Boltzmann) fraction of collisions between the molecules with enough energy to react ( $\propto e^{-E_a/RT}$) and $A$, which is related to the total number of collisions in the gas per second and the reactive cross section (i.e. the probability that the reactants have the required orientation to react). Alternatively, the activation energy, $E_a$, can be associated to the energetically most unfavorable geometric configuration, the so called transition state, which the reactants have to surpass when they transform into products. Now, the Boltzmann factor reflects the probability to find the system in the transition state, and A is a frequency factor connected to the decay of the transition state into products. The latter interpretation justifies Arrhenius-type reaction rates of reactions in solution, in which case the reactants do not actually collide at high velocities. Equation 1.1 predicts that storing meat in a refrigerator will slow down rotting processes and heating up an egg speeds up the chemical reaction that makes it solid. We can also understand that the addition of an extra chemical that stabilizes the transition state (i.e. a catalyst), thus lowering the activation energy, will speed up the chemical reaction. And finally, we see that the reaction rate of a specific reaction in solution will change when this reaction is performed in the gas phase or in another solvent, because the parameters $A$ and $E_a$ are dependent on the reaction environment.

Of course, we want to answer the ultimate question about life, the universe and everything from first principles by computer simulation[1]. For gas phase reactions, one can nowadays calculate the structure of the reacting molecules along a reaction path together with the reaction and activation energies with only a few mouseclicks. Entropic contributions (in equation 1.1 concealed in $A$) can be estimated with a sum over states after calculating the molecular vibrations. For chemical reactions in solution however, computer simulation is a bit more involved. For instance, embedding the reactants by a few spheres of solvent molecules would hardly mimic the liquid after optimizing the geometry and also the calculation of the vibrations would become horribly cumbersome. Instead, one needs to sample over many solvent configurations, either stochastically within a Monte Carlo simulation or deterministic using molecular dynamics. The advantage of the latter is that it can also be used to study the reaction dynamics. And last but not least, the molecular dynamics trajectories can by visualized in illustrative movies, offering a realistic impression of the motions of the chemical building blocks which we can never see in reality, not even with the best microscope. Two examples of such visualizations, are found in the bottom corner of this thesis by rapidly flipping the pages, showing two chemical reactions in water (see also the caption on the last pages of this thesis).