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.