Question: Why is Y2 directly related to spectroscopic observations. Why is Y2 and not Y used to predict observables?

Answer:
From the purely formal point of view, the wavefunction of a system is a mathematical function that contains all the information that is required to describe all of the measurable properties of the system by performing the appropriate mathematical operations on the wavefunction. Wavefunctions are obtained by solving the Schrödinger equation of the system with appropriate boundary conditions. Each state of a system can be described by a unique wavefunction, usually denoted as Y.

The wavefunction of a particle, such as an electron or a photon, is a mathematical formula which is a solution of the Schrördinger equation for a set of assumed boundary conditions. In principle if the boundary conditions are correct, then the wavefunction allows the prediction, or more precisely the computation of all of the measurable properties of the particle because quantum mechanics teaches how to extract such predictions if knowledge of the wavefunction is available. Thus, the application of quantum mechanisms to study molecules boils down to the solution of the appropriate Schrördinger equation to produce the appropriate wavefunction and then using the wavefunction and the rules of quantum mechanics to compute observables of interest.

In systems of photochemical interest the wavefunction of the orbiting electrons, the vibrating nuclei and the precessing spins are of greatest interest. The mathematical details of wavefunctions and their use to compute observables will not be dealt with in this text, but each wavefunction will be translated into a geometric form that is readily visualizable and which will allow an intuitive and qualitative computation of observable properties. Thus, the wavefunction of the electrons will be represented by the well known orbitals of organic molecules, the wavefunctions of vibrating nuclei will be represented by the less well known graphical features of the wavefunctions and the wavefunctions of the electron spin will be represented by vectors. Each of these representation is admittedly approximate, but powerful in obtained rapid and remarkably consistent predictions concerning molecular properties such as energies, transition rates, reaction paths, etc.

The “physical” interpretation of the wavefunction, Y was suggested by Born. He used the analogy to the classical interpretation of electromagnetic radiation in which the square of the amplitude of an electromagnetic wave is interpreted as the intensity of the wave. In quantum terms the intensity is identified as the number of electrons. Born then postulated that since the square of the amplitude is proportional to the number of photons, it is also proportionally to the probability of finding a photon. Since the square of the classical wavefunction Yc is the square of its amplitude, it followed that the square of the quantum mechanical wavefunction for a particle (photon or electron) can be viewed as a density or probability of finding the particle.

The student may be perplexed by the fact that the square of the wavefunction, Y2, rather than the wavefunction, Y, is “more physically real”. Why use Y at all? Answers to this question may vary, but the most practical reason is that the laws of quantum mechanism are very conveniently expressed in terms of Y rather than its square. At the level of discussing electrons and photons, the concept of Y is very useful. At the level of measurement of the properties of electrons and photons, the concept of Y2 is what is required for agreement with experiment.

Question: Does the Born-Oppenheimer approximation break down when there is a First order Hamiltonian?

This question is better phrased, “when does the Born-Oppenheimer approximation, which is a good Zero order approximation, break down” or “what are the first order corrections that make the Born-Oppenheimer approximation a better approximation”. Since all of quantum mechanical systems except for the hydrogen atom must be dealt with at some level of approximation, called the Zero order approximation, there is always room for improvement in a first order correction which produces a First order approximation. The Zero order approximation is arbitrary, but should always be such that any corrections to the system are relative small in terms of energy.

Not back to the question of when does the Born-Oppenheimer approximation break down? The general answer must be whenever the electrons cannot follow the nuclear motion instantaneously. This will happen when the nuclei are moving too fast for some reason or the electrons cannot move fast enough for some reason. When either of these situations occurs the electron state gets “mixed up” and cannot figure out how to reorganize itself smoothly to keep the lowest energy. This is the perfect situation for a “transition” between state. We say the wavefunction of the electron becomes “mixed”.

Wavefunctions mix best when there is a small or negligible energy difference between the states mixing, when there is significant overlap between the states, when the states have the same overall symmetry and when the mixing perturbation takes full advantage of the overlap and symmetry of the situation. When Zero order energy surfaces representing the wavefunction come close in energy for a given nuclear geometry, these situations are the best for state mixing and for the breakdown of the Born-Oppenheimer approximation. We’ll be dealing with such situations throughout the course.

Question: When discussing electron exchange why don’t we explicitly consider the energy change due to spin?

Answer: In Chapter 2 we have artificially broken down the “true” but unknowable wavefunction Y into separate and non-interacting electron (y), vibrartional () and spin () parts. The calculations of the exchange energy are dealing only with the y or electronic part. In this Zero order approximation there is no spin to deal with. We can always deal with spin as a first or second order perturbation. In the case of the exchange energies between S1 and T1 we see that the order of magnitude of the values of J, the exchange integral, are of the order of kcal/mole. Spin interactions between electrons possess energies of the order of 10-3 kcal/mole or less, clearly a small value compare to the value of J for excited states.
On the other hand we shall see that when we are dealing with radical pairs and diradicals, the value of J may be very small because of extremely small overlap (this idea shows up in the last part of Chapter 2). In these cases the Zero order approximation may have to include the energy due to spin effects and other magnetic effects. See for example, Section 2.37.

Question: Does an electron in interstellar space have spin?

Answer: Absolutely. Charge, mass and spin are inherent properties of electrons (at non-relativistic seeds) and cannot be removed by simply pulling them out of an orbital. However, the orbital angular momentum of an electron obviously depends on what orbit an electron is in and is equal to zero when the electron is in a s-orbital or in interstellar space. See section 2.32.