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 BornOppenheimer approximation
break down when there is a First order Hamiltonian?
This question is better phrased, “when does the BornOppenheimer
approximation, which is a good Zero order approximation, break down”
or “what are the first order corrections that make the BornOppenheimer
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 BornOppenheimer 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 BornOppenheimer 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 noninteracting 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 103 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 nonrelativistic 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 sorbital or in interstellar space. See section
2.32.
