Multigraphs in Chemistry

Today we are going to talk about multigraphs and apply them to chemistry.  A multigraph is just a graph in which multiple edges and loops are allowed.  Multiple edges are the name for having more than one edge between two vertices, and loops are edges both of whose ends are at the same vertex.  It turns out we can once again represent Lewis structures like this.  A lone pair is a loop, a double bond is two edges, and a triple bond is three edges.

Above is a multigraph, courtesy of the great Wikipedia:

Note that you have the red multiple edges and the blue loops.

So what are the properties of a mutigraph we are concerned with?  Once again we have vertices, edges, and degree.  The vertices are the gray dots in the above diagram, and the edges are the black, red, and blue in the diagram.  Degree is defined to be the number of edges adjacent to a vertex, where loops are counted twice, one for each end.  In this way the sum of all degrees is still twice the number of vertices.

This is nice.  For starters we can represent any Lewis structure much more completely with this convention.  Graph theoretic properties once again have interesting meanings.  Every edge still represents an electron pair.  The degree of a vertex still represents the number of electrons in its valence shell after bonding, where loops are counted twice for a vertex.  Formal charge has an interesting representation now.  It is the difference between the number of electrons present on the individual atom and the degree in the accompanying graph.

Pictures might be an excellent aid, but it’s better to construct lewis structures yourself!  Draw the Lewis dot structure of your favorite compound, say glucose.  Go on now.  Do it on paper.  Start by making a graph with the atoms as vertices.  Now, turn the lone pairs into loops, so for every lone pair, drawn an edge from that lone pair’s atom to itself.  Finally, draw an edge between two atoms with a single bond, draw two between two with a double bond, and draw three between two with a triple bond.  There you go.

Now, once again these multigraphs can be applied to calculate the formula of certain compounds!  Let’s start with hydrocarbons because they are simple.  Say we have an acyclic hydrocarbon with 5 carbons and 8 hydrogens.  What bonds can it have?

Note that this is an alkane so there are no cycles.  There are also no lone pairs because carbon is not very electronegative and instead makes 4 bonds.  The carbons in total have degree 20 because they need to be adjacent to 4 edges (electron pairs).  Since 8 of these are taken up by carbon-carbon sigma bonds (there are 4, because of our tree with 4 edges, but each is counted twice for each of the carbons), and 8 by C-H bonds, C-C pi bonds are counted a total of 4 times, which means that there are 2 of them.  So either there is a triple bond or two double bounds.

Let’s do another example.  What does carbon monoxide look like?  First, we can draw the simple graph for it, which is a carbon connected to an oxygen.  The edges in the simple graph represent the sigma bonds that do not hold lone pairs, and onto them we will draw the extra multiple edges which represent pi bonds.  Clearly 5 edges need to be drawn in some capacity, since we need 10 total valence electrons.  4 are for the carbon, and 6 for the oxygen.  The only way to do this is to make two extra pi bonds between C and O and then give each of C,O a lone pair.  So we have a graph on two vertices connected by a triple edge, each of which has a loop attached to it.

So we have seen that the idea of interpreting Lewis structures as simple graphs can be extended to interpreting them as multiple graphs.  Once again, this allows us to mathematically capture the structure of many compounds and rationalize their structures to some extent using degree arguments.  Once again, there is still nothing about the actual shape of the molecules involved here, but that is covered very nicely in our previous posts on Group Theory.  Stay tuned for next time!


Quantum Chemistry and the Born-Oppenheimer Approximation

 As mentioned in our previous blog post, the goal of this post and later posts is to eventually get into the nitty gritty mathematics of quantum chemistry. Before we can begin, we must discuss what exactly quantum chemistry is used for, and the many methods that it involves. Quantum chemistry, as the name implies, is the application of quantum mechanics to atoms and subatomic particles in order to provide extremely accurate models of what’s exactly happening. Molecules are small enough that classical mechanics doesn’t always provide a good description. Therefore, quantum mechanics must be used, and by a well-known postulate of quantum mechanics, quantum mechanical models provide all knowable information about a system.The concepts of orbitals and the probability of electrons discussed in the previous post are a direct result of quantum mechanics, and hence many people accept that quantum mechanics is the reason chemistry can exist today in its rigorous form.  This blog post will introduce some of the mathematics used in quantum chemistry and give an overview of the many methods that are used to approximate the behaviors of systems such as the hydrogen atom, and many other elements and molecules.

         Note that for all non-relativistic atoms– meaning atoms that don’t need to be “corrected” using Einstein’s Special Relativity– the time independent Schrodinger equation, as mentioned in the previous blog post, is all that is needed to determine all knowable information about the system.  These atoms usually include smaller elements,  specifically those from the 2nd transition row and above, since these are less massive and therefore have less electrons and interactions between the subatomic molecules. However, a huge problem with solving the Schrodinger equation as it has 162 variables that need to be solved in order to determine the necessary information. These 162 variables describe the spatial coordinate of the electrons and the nuclei, essentially the location of these subatomic particles in a certain Hilbert Space. As a result, the Born-Oppenheimer approximation was created in order to solve this problem and it essentially splits the Schrodinger equation into two separate solves. The process is to solve each Schrodinger equation separately for the electrons and the nuclei.

This is represented by the equation, 

          The  is essentially the solution to to the electronic Schrodinger equation, producing the particular wave function that’s dependent on electrons only. The reason why this particular wave function is so useful is because by taking its partial derivatives, one can gain information about Dipole moments, polarizability, molecular structures, and even spectra: electronic, photoelectron, vibrational, rotaional, NMR, etc. If you forget, the wave equation is essentially the superposition of a bunch of states, and can be represented as:

where C_i’s are the constants and the The kets , specify the different quantum “alternatives” available – a particular quantum state. The partial derivatives are taken with respect to each ket, so the partial derivatives of a wave function are simply,

where f(a) = , and X1, X2, …, XN, are simply the kets . Note that “a” in this case is simply the parameter of the function which are the electrons if the wave is the electronic wave function. The way partials are taken is that every variable is held constant except for an individual variable such as X1. So the first partial derivative with respect to X1 in the figure would simply be C1, since all the other kets or variables become 0 as they’re “constants”, and the derivative of the first ket itself is just 1. Thus, the remaining value would be C1.

          The brief idea of how the electronic wave function is found is that the nuclei spatial coordinates are fixed in a certain way, most often in an equilibrium configuration. Of course, on the other hand, the nuclear Schrodinger equation is solved by fixing the spatial coordinates of the electrons at hand, most often in the equilibrium configuration. The reason why the Born-Oppenheimer approximation is so successful is because of the high ratio between the nuclear and electronic masses. Even though the Born-Oppenheimer approximation greatly reduces the amount of work that is needed to be done to solve the total Schrodinger equation, the new Schrodinger equations are still extremely difficult to solve. As a result, various methods have been created by notable scientists in order to correctly and systematically solve these equations. We will describe the two most notable ones, the Hartree-Fock method, and the many-body Pertubation theory.

          In computational chemistry, the Hartree-Fock method approximates the electronic wave function of the interaction of a large quantity of particles  using the Slater Determinant. In other words, the method is based on the assumption that electrons only feel an average charge distribution due to other electrons. However it is important to keep in mind that such is only an approximation. Orbitals are only a proposed theory; they do not exist in reality. Due to the fact that electrons travel in random motion, their charge distribution is random too. Nevertheless, in this case, it is safe to assume the average distribution for all electrons.

Figure Above: The slater determinant.  is the Hartree-Fock wave function, and  refers to the molecular orbital i.

       Another method of higher precision is the electron perturbation theory. This is also known as the Moller – Plessent Perturbation Theory. Based on the Hartree-Fock wave function, this method treats electrons as small perturbation. Although this is more accurate, the cost is significantly higher. Scientists are making an effort to reduce the cost. The perturbation theory is basically the Hartree – Fock determinant plus additional correction factors that account for different orbitals. It has the following general form:

      Now, one may ask is how exactly to solve the various Schrodinger equations, as they seem like a bunch of Greek symbols. Unfortunately, given the scope of these series of blog posts, we cannot teach one how to solve the Schrodinger equation for large molecules, but for the next couple of posts, we will lead up to the solving of the Schrodinger equation for the hydrogen atom.