Molecular Electronic Wave Functions


Dr. John R. VanWazer, a Senior Scientist at the Central Research Department of the Monsanto Company, was at that time, the world's foremost phosphorus chemist. He and his associates had collected the nuclear magnetic resonance (NMR) chemical shifts of thousands of molecules containing the P31 atom. Yet, they were not able to explain the values found in the experimental NMR chemical shift data using classical techniques.

So, in 1964, Dr. VanWazer hired me, as a theoretical physicist, to work with him and his immediate group. This research was intended for me to work out an explanation for the experimental NMR chemical shift values using proper quantum mechanical ideas and techniques.

When this work started, I had time to consider many aspects of the properties of the matter. This led to my new ideas in particle physics. I never published this work,

The Five Moiety Model of the Universe

This was because I found out very quickly that one lepton, the electron, e-, and the photon were the only two fundamental particles that mattered in chemistry. I quickly moved on to what is presented, below.

The nuclear or particle physicist entire universe as a huge number of nuclei scattered throughout all of space. Surrounding these nuclei are enough electrons and neutrinos, collectively called leptons, to keep everything in balance physical balance. These individuals are concerned with the inner structure of the nucleus, giving little concern to the surrounding leptons.

The molecular or atomic physicist see the universe as being composed of entities which are points in space (highly localized) that have the attributes of mass, charge, and spin. These are the nuclei. These are surrounded by a huge number of leptons and photons.

The molecular physicists are concerned with the properties and dynamics of the leptons, only. So, considering only a single molecule, for the purposes of most of chemistry, each nucleus within a molecule can be considered to be at rest with respect to the other nuclei in the molecule. Surrounding these nuclei are the electrons, in sufficient numbers to keep everything in balance. It is believed by the molecular physicists that the structure, properties and dynamics of the electrons determine the mechanisms of chemistry.

Classical Mechanics

First, I will give a short review to establish the context of what is to follow.

In classical mechanics we deal with forces that act on masses that exist at points. Also, classical mechanics deals with masses and charges (particles) that that have positions, velocities and accelerations. In short, we are dealing with the motion of particles.

The quest in classical mechanics is to derive a set of differential equations of motion, which are given in terms of the masses, velocities and accelerations of the particles. Classical mechanics gives us the techniques to derive the differential equations of motion in very elegant ways.

The differential equations that are given for this system are generally equations for all possible motions of the particles. In order to arrive at a single solution for a particular system that we have in mind, we must give boundary conditions or conditions (values of the positions, velocities and accelerations) that occur in our system at specified times. From this set of conditions, our solution is correct for the system under study.

In the past (before the time of quantum mechanics) schemes were developed to derive the equations of motion in terms of the kinetic and potential energies of the particles in a system. In the system under consideration, there may be a huge number of particles in the system. Each particle had a well defined position and velocity (or momentum). It was found (by Lagrange) that if one were could define a function, T, which is the sum of all of the kinetic energies of the particles in the system (for each particle the kinetic energy is one half times the mass of the particle times the square of its velocity). Also, Lagrange defined a scalar function, V, which is the sum of all of the potential energies of the particles. Then, a function, L, could be defined by the simple relationship L = T - V. We call this function, the Lagrangian. In terms of this function we can state all of the equations of motion in a conservative system by an equation which we call Lagrange's Equations:

$$\begin{align} \frac{ d }{ dt} \frac{\partial L}{\partial \dot{q}^m} - \frac{\partial L}{\partial q^m} = 0 \\ \end{align} $$

When I first studied classical mechanics, I was struck by the beauty and simplicity of being able to describe all of the equations of motion for a system with a huge number of particles - by one equation. This dramatic simplification of the mathematical description of a system is what I was inspired to do in my work with quantum mechanical electronic wave functions.

Quantum Mechanics

When distances become very small, it is not possible to state the position and velocity of a particle with infinite precision. We must work with the uncertainty in the knowledge of the position of a particle and the uncertainty in the knowledge of the velocity (or momentum, which equals the product of the velocity times the mass of the particle.) These uncertainties in the values are related by the famous Heisenberg Uncertainty Principle. This states that the product of the two uncertainties must be greater than or equal to a number, Plank's Constant, h.

So, instead of giving the position (or momentum), of a particle, we must deal with statistical distribution functions. We call these wave functions. These functions allow us to calculate the probability that a particle will be found at the fixed position that we would have like to have used in classical treatments.

This 'cloud-like' function has given students a great trouble in visualizing the particle. So, I would like to offer a simple problem - one with a purpose - to indicate how we might visualize a wave function as if they were localized at a point. Hence, I present the Golf Ball Problem.

The Golf Ball Problem

So, the quest is to find (electronic) wave functions for the electrons in a subject molecule. From these wave functions we can calculate many of the quantities that are of interest.

In quantum mechanics, we do not have the luxury of saying that any particle is located at a point. Rather, all discussion is given in terms of distribution functions that may be visualized in the same way that we did with our mass density function in the problem with the golf ball being attracted to the center of the earth.

Any particle under investigation is described by a distribution function (a wave function.) This wave function may be visualized as the entire distribution function being localized to a point. This will dramatically simplify visualization of the atoms by most people.

That is, we can take the cloud like distribution functions of quantum mechanics and, if they are suitably well behaved, we can imagine that a distribution function is all concentrated at a point.

A quantum mechanic might say that we all know that we must deal with wave functions; we may not deal with particles in the classical sense. A casual observer can now say that now we can look at these quantum mechanical wave functions as if the particle did indeed exist, at a point.

Atomic or Molecular Physics

For many years, I have striven to develop techniques to delineate precise quantum-mechanical molecular electronic wave functions for polyatomic molecules, in terms of bond angles and bond distances between the nuclei in a molecule. I would like to try to describe what I have done to accomplish this end. Furthermore, I would like to offer my opinion as to what all of this means.

I will describe what I have done in development of techniques for defining what I have called localized orbitals, which are (almost) precise quantum mechanical wave functions. These localized orbitals are distribution functions which can be visualized readily by a high school student to show what is 'going on' within a molecule. Hopefully, this will give insight into the mechanisms of chemistry.

But, since the localized orbitals are not quite accurate enough for precise quantum mechanical calculations, I will describe the Delta Transformation, that allows us to take localized orbitals and convert them into Equivalent Orbitals which are properly anti-symmetrized wave functions, accurate enough for rather precise quantum mechanical calculations.

Electronic Molecular Wave Functions - Localized Orbitals and the Delta Transformation

This reference is an excellent discussion of the subjects of Localized Orbitals, Equivalent Orbitals and the Delta Transformation:

Localized Orbitals

What has been presented is a method for conjuring distribution functions for the electrons in a molecule. These are transformed into quantum mechanical electronic wave function by a very simple (looking) equation, the Delta Transformation.

All of this process is what I had hoped to develop.

What All of This Implies

We now have a technique for calculating the energy (and other properties) of atoms and molecules, with great accuracy.

Let us now consider two atoms (or molecules), A and B. It is our intention to allow A to collide and join with B to form a new molecule AB. We have the ability to define the electronic wave function for AB using the techniques described above. Furthermore, we have the ability to calculate the wave function for the extended molecule AB` where the distance between A and B which is greater than we find in the final molecule, AB.

The energy of the intermediate molecule, AB`, is greater than that of AB. If we were to find physical conditions to lower the energy of AB`, we have found how to catalyze the chemical reaction. For each of these finds, we have a new patent, and a new commercial chemical process.

This has been the goal of the theoretical work that I have done in this area. With modern computers, we have the ability to carry out these calculations. This was impossible at the time that I started this work.

The work to bring these ideas to fruition lies ahead.