Welcome back everyone. I'm Charles Clark.
And in this part, we're going to look at the application of the Gaussian functions
that you derived at the end of the last segment and their value for a wide range
of practical calculations using something called the Variational Theorem.
Now, I must say that I always carry a Gaussian with me wherever I go.
this one, as very simple to remember. it's the ground state of a harmonic
oscillator, and you can use it in a variety of applications for estimating
ground state energies of other systems. That's an exercise that you conduct in
the assignment this week. So, let's proceed.
[SOUND] So, here is the three-dimensional version of the way function that you
derived the end of last segment. And here's the one I prefer to remember
that has the, it, it expresses the the length scale, the harmonic oscillator,
through this parameter d which is the square root of h bar over m omega.
so d has to have the units of length. And now, have a quick InVideo quiz just
to refresh your appreciation of that. Well, the answer to that inline quiz was
a of course sort of apparent right here and the messages, the way function has
dimensional units. it's the units of the inverse square root
of length times the number of dimensions. So, in one dimension, it's 1 over the
root length. in three dimensions, it's 1 over length
to the three halves power. [SOUND] Gaussians have become a major
workhorse of actually of chemistry rather than physics.
And I think it's it seems plausible to think that much more work using quantum
mechanics is done by chemists rather than by physicists.
So, for example the Noble Prize in Chemistry in 1998 was awarded to this
man, John Pople for his development of computational methods in quantum
chemistry. And what he did in particular, was lead
the development, the very large scale program development effort for using
Gaussians to compute molecular structure. And if you if you look at Google Scholar
and look at the just search on Gaussian orbitals, seems that the top two entries
that I got. this one here is co-authored by Pople,
7307 citations and next to it, a paper that reports Gaussian basis.
And so there's choices of particular values of Gaussian orbitals that are
optimized for doing calculations of of molecular structure and light atoms.
And this it's been cited 14,000, over 14,000 times.
Now, I don't know about you, but where I live that's a large number.
For example, if you look at the most cited paper in Physics, using the same
Google search engine, it's this paper by Steven Weinberg.
Also Nobel laureate in Physics cited a little bit over 10,000 times.
That's a very large number, largest number in Physics.
but as you can see the the work in chemistry is it generates a lot of effort
in terms of number citations. That's not a statement of its absolute
importance, but it does indicate the the effect that this development has had in
the field of science as a whole. Why has there been such a great attention
paid to these Gaussian orbitals. Well, it has to do with the use of very
important theorem of quantum mechanics, the Variational Theorem, which states
that for any Hamiltonian and for any function that we choose, so this is not
necessary an eigenfunction, it's just some arbitrary function.
I should say, this is a course in physics, not in mathematics.
So, there's some qualifications, so I say here.
But, let's just let those be debated in mathematician study group.
for any Hamiltonian and for any function that's normalized, the expectation value
of the Hamiltonian over that function is greater than or equal to the ground-state
energy. [SOUND].
That is the exact, it's greater than or equal to the exact minimum energy.
And so this means that if you have computational capabilities and a rich an
intelligent way of choosing lots and lots of trial functions you can, you can make
start to make estimates, of the ground-state energy by continuing
refinement of approximations, to get you to the lower value.
Now, for those of you who are, sitting peacefully and undisturbed right now, if
you haven't yet proven any theorem in quantum mechanics, this would be a good
one to prove. So, for such people, you might consider
pausing and thinking about how you would prove this.
I'm going to go, run through the proof. It's just a few lines on the next slide.
And I'll just if you want a suggestion this is a proof which is quite
straightforward, if you expand the, the trial wave function psi in a sequence of
eigenfunctions of the Hamiltonian Age. So, I'll pause while you think and then
well, if you pause, I'm going to continue but you might pause and, and think about
how that, how to prove like that might look.
[SOUND] Well, here is the, here is the proof of the Variation of Theorem.
So, this is the statement of theorem up to this point and now what we do is just
use a familiar expansion of the arbitrary weigh function psi, I wish I should have
put a, made this a cat form. In terms of the I, in terms of a, well it
could be any orthonormal basis, but in particular we'll choose as our basis the
functions that are eigenfunctions, the Hamiltonian.
Okay. So we expand psi in the eigenfunctions of
Hamiltonian H. And now, we take the expectation value of
psi H psi. That's the thing that we're supposed to
be computing. And so now, we just, we just put this
form of psi into the the the Ha-, the Hamiltonian.
So, we're just, this part is the, the cat side of that and it's the, the
[INAUDIBLE] side here and then we just sandwich the Hamiltonian.
And between are these are all very routine mechanical procedures.
Now the critical point here is that the Hamiltonian is diagonal in the
eigenfunc-, in its owneigenfunctions. So, this matrix element here reduces just
to the energy. And we just, we just take its a chronicle
delta MN but just replace it by En. And so, now we have a we have oh, you
know, a weighted sum of the energies and Well the, so the product, the product of
these two terms here is a positive number because they're complex conjugates.
I guess I should have emphasized that. So, this in equality implies, that is if
we replace E, this sum here is greater than the sum where instead of En, we have
the smallest value of all the [UNKNOWN] values, which is E0.
And then, then this, then, then there's this identity here, this, this series
collapses. So, I hope this, you can think about this
intuitively. You're taking your, you're projecting the
arbitrary trial function onto all of these, the eigenstates of the system.
And you're taking the weighted average energy and that's gotta be greater than
the least energy. That's a basic idea.
I hope this was clear. Well, why should anyone really care about
this? And the reason has to do with a
remarkable result derived by Max Born and Robert Oppenheimer, which sort of laid
the grounds for practical and accurate calculation of materials properties.
And basically the Born-Oppenheimer approach shows that the electrons are
really the glue that holds everything together.
And if you want to find the [INAUDIBLE] equilibrium configuration of atoms in a
solid or the positions of nuclei in a molecule, what you're going to do is
minimize the total energy of the system that will be, that will be the ground
state of you know, a very complicated material system.
And in, in most of the cases the cases of most of the materials that you're
familiar with, this approach can be implemented by solving the Schrodinger
equation for the ground-state of the system at some fixed value of the, of the
nuclear coordinate. So, for a molecule, we'd say, you take
the nucle-, nuclei of some arbitrary distance R, solve the total energy, and
then, as a function of R, you'll get an, you'll get an E of R, which, which
displays a minimum typically. And that, that minimum then, defines the
equilibrium, the equilibrium bond length in the molecule.
So, generalizing this to a complicated system [INAUDIBLE] it's computationally
intensive, of course, but it's an approach that's had great success.
And indeed in 1998, the Nobel Prize in chemistry was shared by Walter Kohn, who
developed a so, density functional techniques that, that greatly improved
the computational speed of of such methods.
So, this the the ability of quantum chemistry and condensed theoretical,
condensed matter Physics to understand and discover new materials, was
recognized by the Nobel Prize in the 1998.