WEBVTT
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NICOLA MARZARI:
OK, good morning.
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And welcome to lecture 6.
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We're still working on
electronic structure methods.
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And today in particular, we'll
finish up our introduction
00:00:14.560 --> 00:00:18.040
of the Hartree-Fock methods,
if you want the cornerstone
00:00:18.040 --> 00:00:20.110
of quantum chemistry.
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Really developed
in the late '20s,
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we have Douglas Hartree
here from the University
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of Cambridge and Fock, I
think, from the University
00:00:28.780 --> 00:00:30.670
of St. Petersburg.
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And then we'll also go into
Density-Functional theory that,
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in many ways, is a much
more recent approach.
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The theory itself was
developed in the '60s
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by Walter Kohn in collaboration
with Pierre Hohenberg
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and in collaboration
with Lu Sham.
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But really, I would say
it's only in the mid '70s
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and then really in
the mid '80s that it
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started to take off as
actually a practical approach
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to study the electronic
structure, especially
00:00:59.920 --> 00:01:03.520
of solids, and has become very
popular, popular to the point
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that the Nobel Prize
for chemistry in '98
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was given to Walter Kohn for
DFT and John Pople, a quantum
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chemist, for the development
of quantum chemistry methods.
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Let me remind you before we go
on into the lecture of the two
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main conclusions of last class.
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The first one was the
appropriate matrix formulation
00:01:34.830 --> 00:01:36.900
of the Schrodinger
equation that is something
00:01:36.900 --> 00:01:39.060
that is very powerful
and very useful
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when we actually need
to solve a differential
00:01:41.820 --> 00:01:43.920
equation on a computer.
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And the hypothesis there was
that we had chosen a basis set
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that I indicate here with phi.
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That would be the basis set in
which I would expand my ground
00:01:54.960 --> 00:01:56.070
stateway function.
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And so once the
basis set is chosen--
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it could be sines and cosines
with different wavelengths.
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It could be localized wave
functions like Gaussians.
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It could be just points
on a discrete grid.
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But once my basis set is
chosen, and once I know what is
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my electronic structure
problem-- that is,
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once I know what is the
potential in my Hamiltonian
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operator--
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this here, these integrals,
are just numbers.
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So solving the Schrodinger
equation-- that is,
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finding the key E eigenvalues
and the corresponding
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eigenvectors--
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is nothing else than solving
this linear algebra problem
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where, again, these coefficients
here form a vector that really
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gives me what are the terms
in the expansion of my wave
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function, in terms
of orthonormal basis.
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And so linear algebra problems,
we have a matrix here.
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And it will have, if it's
an order n matrix, n values
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for which the determinant is 0.
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And those are the
eigenvalues for which
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we can find the solution.
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So this is important not
only in electronic structure,
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but in a lot of applied
computational approaches.
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And the other very
useful principle here
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that I have listed is the
variational principle.
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That is, again, it is possible
to define a functional--
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that is, if you
want an algorithm--
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that takes as input
a generic function
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and gives us output, a number.
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And the functional
that we use is
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the functional that
I've written here
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with the integral expectation
value of the Hamiltonian
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on an arbitrary wave
function divided by basically
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a normalization term.
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And it can be proven--
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and it was given
to us an exercise--
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that this quantity
is always greater
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or equal than the ground state.
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And this is very
powerful because, if we
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have no idea on what the
solution for our ground state
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problem is, well, we
can just try out a few.
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And our best
solution-- although,
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we will never know if it's
the exact one or not--
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will be the one that gives
the lowest expectation value
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for this term.
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And here is an example
on how we could actually
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go on and try to solve a very
simple problem that is finding
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the eigenstates and eigenvalues
for a hydrogen atom using
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the variational principle.
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And instead of trying actually
different wave functions
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at a time, what we can do
that is very meaningful
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is choose a parametric
format for the wave function.
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So you see, I'm writing
here a generic wave function
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that decays as an exponential
as a function of the r distance
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from the nucleus from the
center of the electron.
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So if you wanted
this parameter here,
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alpha determines how steep
the decay of this wave
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function here.
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So by writing it
in this way, I'm
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not considering any more
just one way function.
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But I am considering an
entire family of functions
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with different decays.
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And so what I can do is try
this family of wave functions.
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And so what I would do
is stick this expression
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in this expectation value.
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And then actually,
the constant, C,
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is what we call a
normalization constant.
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It always cancels out, you
see, from the integral below
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and the integral above.
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This C squared
would just go away.
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And so overall, the variational
principle and expectation value
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is a parametric
function of alpha.
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And the optimal alpha
will be the alpha
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that gives me the minimum
value for this number.
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Again, we can't go
below the ground state
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for this expectation value.
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So the lower that we
can go, the better.
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And if we use this particular
choice of wave function--
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and this is, again, a very
simple analysis problem
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that you could actually
work out by yourself.
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I've actually written here all
the terms of the normalization
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integral.
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What is its value as a function
of this parameter once you
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do it properly in
spherical coordinates,
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considering that this is
really the radial distance,
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and this is the expectation
value of the kinetic energy,
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and this is the expectation
value of-- you see minus 1
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over r.
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That is the potential
for the electron
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in the field of a proton.
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And if you minimize this
expression-- very easy--
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what you obtain is
alpha equal to 1.
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So the optimal solution
is then alpha equal to 1.
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It turns out that, by solving
directly the differential
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equation, the
Schrodinger equation,
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we actually know that this
exponential of minus r
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is actually the exact solution.
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And so what happens,
in this case,
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is that this parametric
family of wave functions
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is the right form.
00:06:57.300 --> 00:07:01.020
That is, it contains actually
the exact the ground state
00:07:01.020 --> 00:07:03.300
among all its possible forms.
00:07:03.300 --> 00:07:05.190
And that exact ground
state is reached
00:07:05.190 --> 00:07:06.750
when alpha is equal to 1.
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And the variational principle
will give us alpha equal to 1
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without having any
need of solving
00:07:14.850 --> 00:07:18.660
the very complex radial
Schrodinger equation that I've
00:07:18.660 --> 00:07:23.250
shown at a certain point
in the class on Tuesday
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where we really need to solve
for the spherical harmonics
00:07:26.280 --> 00:07:29.880
and find the Legendre
polynomial, Laguerre terms
00:07:29.880 --> 00:07:32.200
in the radial part,
and so on and so forth.
00:07:32.200 --> 00:07:33.870
So this becomes very simple.
00:07:33.870 --> 00:07:36.330
And for this reason, the
variational principle
00:07:36.330 --> 00:07:37.830
is very powerful.
00:07:37.830 --> 00:07:41.370
And we'll actually see
how Hartree and Fock use
00:07:41.370 --> 00:07:46.710
that to find out a way to tackle
the problem of many electrons
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interacting because, up to now,
we have really only thought
00:07:50.550 --> 00:07:54.240
of the problem of one
electron in a potential.
00:07:54.240 --> 00:07:57.990
But the problem of many
interacting electrons
00:07:57.990 --> 00:08:01.330
increases its
complexity very quickly.
00:08:01.330 --> 00:08:05.940
So I'm just showing here what
would be, say, the Schrodinger
00:08:05.940 --> 00:08:09.690
equation for a two-electron
atom, say something typically
00:08:09.690 --> 00:08:13.460
like the atom of helium.
00:08:13.460 --> 00:08:18.950
You have a nucleus that
has really two protons.
00:08:18.950 --> 00:08:25.270
And then you have two
electrons around the nucleus,
00:08:25.270 --> 00:08:26.540
going around.
00:08:26.540 --> 00:08:29.470
And so what is the Hamiltonian,
the quantum mechanical
00:08:29.470 --> 00:08:31.340
Hamiltonian for this problem?
00:08:31.340 --> 00:08:35.840
Well, what we have is the
kinetic energy terms here.
00:08:35.840 --> 00:08:40.600
So we have a Laplacian that is
a second derivative in space
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for each one of the coordinates.
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So you see this is
the wave function.
00:08:45.310 --> 00:08:48.460
The wave function
is an amplitude
00:08:48.460 --> 00:08:52.660
that is a function of the
combined position of r1
00:08:52.660 --> 00:08:55.780
describing the first
electron and r2 describing
00:08:55.780 --> 00:08:57.010
the second electron.
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So you have these two
second derivatives.
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And then there is
the term that deals
00:09:04.750 --> 00:09:09.310
with the attraction
between electron 1,
00:09:09.310 --> 00:09:12.310
say this electron
here, and the nucleus.
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We have an attraction term
here, Coulombic attraction,
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that goes basically as 2/r or as
Z/r if we think of the nucleus
00:09:21.430 --> 00:09:24.610
as having charge Z. And so
there is attraction here
00:09:24.610 --> 00:09:26.770
of the first electron
to the nucleus
00:09:26.770 --> 00:09:30.610
and then attraction of the
second electron to the nucleus.
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And then the last term is
actually a repulsive term--
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so there should be
actually a plus sign here--
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between the two electrons.
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And so you see it depends--
00:09:42.220 --> 00:09:45.130
again, Coulombic comparison
between the electrons--
00:09:45.130 --> 00:09:50.800
on the distance between r1 and
r2, the instantaneous position.
00:09:50.800 --> 00:09:53.830
So you see, this is how
the Hamiltonian becomes,
00:09:53.830 --> 00:09:58.120
in the case of an atom that has
two electrons and everything
00:09:58.120 --> 00:10:03.340
here, the final generalization
still for one atom
00:10:03.340 --> 00:10:04.810
but with many electrons.
00:10:04.810 --> 00:10:06.520
So in the case where,
again, you have
00:10:06.520 --> 00:10:10.750
a nucleus with many
electrons orbiting around,
00:10:10.750 --> 00:10:13.540
what you will have
is a wave function
00:10:13.540 --> 00:10:21.530
that is a combined amplitude of
n different spatial variables.
00:10:21.530 --> 00:10:27.340
So if we have, say, an atom
like iron that has 26 electrons,
00:10:27.340 --> 00:10:31.630
well, that wave function
is a combined amplitude
00:10:31.630 --> 00:10:34.360
of 78 coordinates.
00:10:34.360 --> 00:10:36.460
And the Hamiltonian
operator, again,
00:10:36.460 --> 00:10:40.180
is our second derivative
for each electron in there
00:10:40.180 --> 00:10:43.960
as an attractive term
for each electron that
00:10:43.960 --> 00:10:49.390
is attracted to the nucleus that
now has a charge Z equal to 26.
00:10:49.390 --> 00:10:54.490
And then each electron-- so this
sum i goes over each electron.
00:10:54.490 --> 00:10:59.140
Each electron will be
repelling each other electron.
00:10:59.140 --> 00:11:03.120
So there is an overall
repulsive term.
00:11:03.120 --> 00:11:05.310
And we actually call
this term here--
00:11:05.310 --> 00:11:06.900
this is the difficult term.
00:11:06.900 --> 00:11:10.470
We call this term
here a two-body term
00:11:10.470 --> 00:11:14.310
because it depends on the
simultaneous possession of two
00:11:14.310 --> 00:11:15.300
electrons.
00:11:15.300 --> 00:11:17.530
Well, if you want,
the first the terms
00:11:17.530 --> 00:11:21.450
here are called one-body
terms because they act only
00:11:21.450 --> 00:11:23.850
on one electron
at a time, quantum
00:11:23.850 --> 00:11:27.000
kinetic energy and attraction.
00:11:27.000 --> 00:11:31.830
And truly, this
Schrodinger equation
00:11:31.830 --> 00:11:35.790
becomes overly complex
already by the time
00:11:35.790 --> 00:11:37.330
when we have two electrons.
00:11:37.330 --> 00:11:41.760
So we can't solve analytically
even the helium atom, OK?
00:11:41.760 --> 00:11:44.342
We can solve everything
about the hydrogen atom.
00:11:44.342 --> 00:11:46.050
And we can solve the
Schrodinger equation
00:11:46.050 --> 00:11:47.970
for a number of very
simple problems.
00:11:47.970 --> 00:11:52.600
But already, for the helium
atom, we can't do that.
00:11:52.600 --> 00:11:55.890
And so in the '40s
and '50s, people
00:11:55.890 --> 00:11:58.230
developed variational
approaches.
00:11:58.230 --> 00:12:01.330
They were making an uncertain
hypothesis on this wave
00:12:01.330 --> 00:12:04.410
function, expanding
this wave function
00:12:04.410 --> 00:12:08.790
in a series of other functions,
depending parametrically
00:12:08.790 --> 00:12:10.110
on a number of coefficients--
00:12:10.110 --> 00:12:13.740
2, 3, 5, 8, 200 coefficients.
00:12:13.740 --> 00:12:17.670
And the more coefficients
there were, the more flexible
00:12:17.670 --> 00:12:19.350
that wave function was.
00:12:19.350 --> 00:12:23.910
And so systematically, by adding
more and more flexibility,
00:12:23.910 --> 00:12:26.370
you could go lower and lower.
00:12:26.370 --> 00:12:28.860
And hopefully, you
were converging.
00:12:28.860 --> 00:12:31.380
And say for something
like the helium atom,
00:12:31.380 --> 00:12:37.350
well, the total energy of two
electrons in the helium atom
00:12:37.350 --> 00:12:43.950
is something like
5.8, 5.8 Rydbergs.
00:12:43.950 --> 00:12:47.190
And you could get
very, very close,
00:12:47.190 --> 00:12:51.300
down to a fraction of
hundreds of electron volts
00:12:51.300 --> 00:12:54.000
just by 3 or 5 parameters.
00:12:54.000 --> 00:12:56.430
But then as you go to
more and more electrons,
00:12:56.430 --> 00:13:00.720
the problem really has a
complexity that explodes.
00:13:00.720 --> 00:13:03.970
And in all of this,
I want to remind you,
00:13:03.970 --> 00:13:07.710
I'm not using any more the
international system of units.
00:13:07.710 --> 00:13:10.650
So I'm not using
meters and seconds.
00:13:10.650 --> 00:13:13.290
But I'm using what are
called atomic units.
00:13:13.290 --> 00:13:17.160
And there is a handout on your
Stellar web page reminding
00:13:17.160 --> 00:13:21.930
that the unit of energy is
what we call the Hartree that
00:13:21.930 --> 00:13:26.880
corresponds to another set
of unit that are often used.
00:13:26.880 --> 00:13:30.510
Say a Hartree
corresponds to Rydberg.
00:13:30.510 --> 00:13:33.750
And the Rydberg is
13.6 electron volts.
00:13:33.750 --> 00:13:37.020
And electron volt is the
energy of an electron that
00:13:37.020 --> 00:13:39.570
feels a potential
difference of 1 volt.
00:13:39.570 --> 00:13:42.090
And 1 electron volt
is often considered
00:13:42.090 --> 00:13:45.210
23.5 kilocalories per mole.
00:13:45.210 --> 00:13:47.230
In the electronic
structure literature,
00:13:47.230 --> 00:13:49.020
you'll see all these numbers.
00:13:49.020 --> 00:13:53.790
Often, chemists use the
kilocalorie per mole unit.
00:13:53.790 --> 00:13:56.910
Physicists tend to use
the electron volts.
00:13:56.910 --> 00:14:00.990
And again, I remind you that
the average kinetic energy
00:14:00.990 --> 00:14:05.850
of an atom at room temperature
is 0.04 electron volts.
00:14:05.850 --> 00:14:10.920
And the binding of water
dimer is 0.29 electron volts.
00:14:10.920 --> 00:14:13.380
So the electron volts
is really the order
00:14:13.380 --> 00:14:20.350
of magnitude of the binding
of weakly attached molecules.
00:14:23.600 --> 00:14:24.290
OK.
00:14:24.290 --> 00:14:26.340
So now, the problem--
00:14:26.340 --> 00:14:29.120
and that's where the Hartree
and Hartree-Fock solution become
00:14:29.120 --> 00:14:30.560
important--
00:14:30.560 --> 00:14:33.680
is to deal with a
realistic system.
00:14:33.680 --> 00:14:36.380
That is, we need to
deal with molecules
00:14:36.380 --> 00:14:38.390
or we need to deal with solids.
00:14:38.390 --> 00:14:40.130
So we need to
deal, in principle,
00:14:40.130 --> 00:14:45.180
with a Schrodinger equation in
which we have a lot of nuclei.
00:14:45.180 --> 00:14:50.210
I'm denoting here with capital
R the position of the nuclei.
00:14:50.210 --> 00:14:51.920
Again, if you want
to find out what
00:14:51.920 --> 00:14:54.680
is the structure of benzene,
if you want to find out
00:14:54.680 --> 00:14:56.660
what is the electron
structure of silicon,
00:14:56.660 --> 00:14:59.510
you need to keep in mind
that, say for benzene, you
00:14:59.510 --> 00:15:02.990
will have six Coulombic
attractive centers that
00:15:02.990 --> 00:15:06.320
are the six carbon
nuclei in a ring
00:15:06.320 --> 00:15:10.320
and also the six
1/r protons around.
00:15:10.320 --> 00:15:14.570
So the position of
those nuclei determines
00:15:14.570 --> 00:15:18.870
what is the potential
acting on your electrons.
00:15:18.870 --> 00:15:21.500
So in principle, the
Schrodinger equation
00:15:21.500 --> 00:15:24.590
will be something like
I've written here.
00:15:24.590 --> 00:15:27.680
But we can make this
fundamental simplification
00:15:27.680 --> 00:15:31.580
that will always be the case
for the classes that follow.
00:15:31.580 --> 00:15:35.540
That is, we never treat the
nuclei as quantum particles,
00:15:35.540 --> 00:15:36.140
OK?
00:15:36.140 --> 00:15:40.300
In principle, also the
nuclei are quantum particles.
00:15:40.300 --> 00:15:42.290
So they would have
their own wavelength.
00:15:42.290 --> 00:15:46.230
They come in as variables
into the wave function.
00:15:46.230 --> 00:15:48.890
And you would need to
calculate, say, things
00:15:48.890 --> 00:15:51.500
like the quantum kinetic
energy of the nucleus.
00:15:51.500 --> 00:15:54.230
That is the Laplacian,
the second derivative
00:15:54.230 --> 00:15:56.000
of the overall wave
function with respect
00:15:56.000 --> 00:15:57.620
to the nuclear coordinates.
00:15:57.620 --> 00:16:00.470
In practice, remember
the Bernoulli relation.
00:16:00.470 --> 00:16:05.590
The nuclei are so heavy that
their wavelength is very, very
00:16:05.590 --> 00:16:06.090
small.
00:16:06.090 --> 00:16:09.650
So we can truly treat them
as classical particles.
00:16:09.650 --> 00:16:13.790
And the electrons are so
much faster than the nuclei
00:16:13.790 --> 00:16:18.350
that even if the nuclei move,
the electrons can always,
00:16:18.350 --> 00:16:21.800
much faster than the
nuclei, follow this movement
00:16:21.800 --> 00:16:25.820
and reorganize themselves as
to be getting the lowest energy
00:16:25.820 --> 00:16:26.990
state possible.
00:16:26.990 --> 00:16:30.230
Basically, you have to imagine
this picture of the molecule
00:16:30.230 --> 00:16:32.420
vibrating at room temperature.
00:16:32.420 --> 00:16:35.690
And the nucleus then will
have a kinetic energy.
00:16:35.690 --> 00:16:38.100
All the mass of the
molecule is in the nuclei.
00:16:38.100 --> 00:16:40.730
So all the effects of
temperature, if you want,
00:16:40.730 --> 00:16:42.680
are in the vibration
of the nuclei.
00:16:42.680 --> 00:16:46.520
But these vibrations are
very slow from the point
00:16:46.520 --> 00:16:48.150
of view of the electrons.
00:16:48.150 --> 00:16:50.840
So the electrons see that
the nucleus is slowly moving
00:16:50.840 --> 00:16:55.310
and rearrange themselves as
to being in the ground state
00:16:55.310 --> 00:16:58.640
for that instantaneous
configuration of the nuclei.
00:16:58.640 --> 00:17:01.610
Obviously, if the nucleus
starts moving very, very
00:17:01.610 --> 00:17:06.329
fast by any chance, well, then
the electrons can't do this.
00:17:06.329 --> 00:17:08.130
They can't follow any more.
00:17:08.130 --> 00:17:10.099
And so if you
want, you can start
00:17:10.099 --> 00:17:12.140
having electronic excitation.
00:17:12.140 --> 00:17:14.780
That is the electrons
are not anymore
00:17:14.780 --> 00:17:17.960
on the lowest energy state
possible for that given
00:17:17.960 --> 00:17:19.640
configuration.
00:17:19.640 --> 00:17:22.339
That doesn't really
happen, especially
00:17:22.339 --> 00:17:25.220
if we are just considering
a molecule a solid at room
00:17:25.220 --> 00:17:26.210
temperature.
00:17:26.210 --> 00:17:30.650
But it could happen, say, if we
have an external potential that
00:17:30.650 --> 00:17:32.240
changes very fast.
00:17:32.240 --> 00:17:35.780
If we shine a laser
light on a molecule,
00:17:35.780 --> 00:17:39.590
then light is nothing less
than a electromagnetic field.
00:17:39.590 --> 00:17:42.830
And laser light will
typically have a frequency
00:17:42.830 --> 00:17:46.190
that is fast and comparable to
the frequency of the electrons.
00:17:46.190 --> 00:17:49.205
And so all these adiabatic
approximations break down.
00:17:49.205 --> 00:17:52.340
And lo and behold, we can
actually excite with the laser
00:17:52.340 --> 00:17:54.950
the electron in a
higher energy state.
00:17:54.950 --> 00:17:56.960
But for what you are
seeing in this class,
00:17:56.960 --> 00:18:01.670
we'll always think of the
electrons in a ground state.
00:18:01.670 --> 00:18:05.060
That this is what people
call often the adiabatic
00:18:05.060 --> 00:18:08.480
or the Born-Oppenheimer
approximation.
00:18:08.480 --> 00:18:14.330
These two terms, in most cases,
are used in the same way.
00:18:14.330 --> 00:18:20.480
Although, chemists tend to make
a subtle distinction about what
00:18:20.480 --> 00:18:23.940
adiabatic means and
Born-Oppenheimer means.
00:18:23.940 --> 00:18:26.810
And adiabatic really
refers to the coupling
00:18:26.810 --> 00:18:30.440
between the different potential
surfaces for the electrons,
00:18:30.440 --> 00:18:32.630
depending on the
velocity of the nuclei.
00:18:32.630 --> 00:18:35.270
And Born-Oppenheimer
implies that there
00:18:35.270 --> 00:18:37.580
is no influence of the
ionic motion of one
00:18:37.580 --> 00:18:39.050
single electronic surface.
00:18:39.050 --> 00:18:40.940
But I mean, this is
fairly technical.
00:18:40.940 --> 00:18:45.020
Just remember that sometimes
these two terms actually mean
00:18:45.020 --> 00:18:47.930
something very different
and very specific.
00:18:47.930 --> 00:18:48.650
OK.
00:18:48.650 --> 00:18:53.960
So this has now become our
most general expression
00:18:53.960 --> 00:19:00.920
for the Hamiltonian and also for
the energy of a set of nuclei
00:19:00.920 --> 00:19:02.150
and a set of electrons.
00:19:02.150 --> 00:19:06.050
That is really our picture
of a molecule or a solid.
00:19:06.050 --> 00:19:12.130
And again, we will have
the nuclei generating
00:19:12.130 --> 00:19:16.120
Coulombic attractive potentials
in every position where
00:19:16.120 --> 00:19:16.970
they are.
00:19:16.970 --> 00:19:19.720
So there is what we call
an electron nucleus term.
00:19:19.720 --> 00:19:22.300
That is an attractive
term in which
00:19:22.300 --> 00:19:26.020
we have a sum over
each and every electron
00:19:26.020 --> 00:19:30.070
because each and every electron
feels the potential of all
00:19:30.070 --> 00:19:31.180
the nuclei.
00:19:31.180 --> 00:19:38.060
And the sum of all the nuclei
refers to a sum of Z/r terms.
00:19:38.060 --> 00:19:41.050
So in whole space, wherever
there is a nucleus,
00:19:41.050 --> 00:19:43.420
there is a 1/r term.
00:19:43.420 --> 00:19:50.020
And this sum over nuclei
is the overall potential
00:19:50.020 --> 00:19:51.550
for the electronic system.
00:19:51.550 --> 00:19:55.000
And each electron
feels this potential.
00:19:55.000 --> 00:19:57.760
And this is the fundamental
attractive term.
00:19:57.760 --> 00:20:00.460
So electrons are
attracted to the nuclei.
00:20:00.460 --> 00:20:03.830
But also, electrons
repel each other.
00:20:03.830 --> 00:20:05.560
And this is the other term.
00:20:05.560 --> 00:20:08.620
So you see each
and every electron
00:20:08.620 --> 00:20:11.920
has a charge 1 in
atomic units and has
00:20:11.920 --> 00:20:15.700
a Coulombic repulsion with
each and every other electron.
00:20:15.700 --> 00:20:17.800
And then, of course,
for each electron,
00:20:17.800 --> 00:20:21.820
we have the quantum
kinetic energy here.
00:20:21.820 --> 00:20:25.840
And so I listed here in the
Hamiltonian all these terms.
00:20:25.840 --> 00:20:29.950
We have the quantum
kinetic energy here.
00:20:29.950 --> 00:20:33.260
We have the
electron-electron repulsion.
00:20:33.260 --> 00:20:37.000
And we have the
electron-nucleus attraction.
00:20:37.000 --> 00:20:39.160
And then there is
a last term that
00:20:39.160 --> 00:20:41.380
is truly a classical
term because it
00:20:41.380 --> 00:20:43.460
involves only the nuclei.
00:20:43.460 --> 00:20:45.940
And so it's a
repulsive term that is
00:20:45.940 --> 00:20:48.100
the nucleus-nucleus repulsion.
00:20:48.100 --> 00:20:52.150
So if you want to think for a
moment of a hydrogen molecule,
00:20:52.150 --> 00:20:57.520
say, what you would there is
a nucleus, another nucleus,
00:20:57.520 --> 00:21:02.140
and then a wave function of
all the electrons around.
00:21:02.140 --> 00:21:06.910
And there is a classic term that
is the electrostatic repulsion
00:21:06.910 --> 00:21:08.630
between the two nuclei.
00:21:08.630 --> 00:21:14.260
And there is a quantum term of
repulsion between the electrons
00:21:14.260 --> 00:21:16.720
and an attraction between
the electron and the nuclei.
00:21:16.720 --> 00:21:20.950
And basically, all these
electrostatic terms--
00:21:20.950 --> 00:21:22.570
nucleus-nucleus,
electron-nucleus,
00:21:22.570 --> 00:21:26.770
and electron-electron--
almost balance themselves.
00:21:26.770 --> 00:21:29.620
Each and every
one is very large.
00:21:29.620 --> 00:21:32.620
But then the combination
of these three almost
00:21:32.620 --> 00:21:34.720
cancels itself out.
00:21:34.720 --> 00:21:39.340
And that's why actually the
binding energy of a molecule
00:21:39.340 --> 00:21:42.830
is much, much smaller than
any of the energy, say,
00:21:42.830 --> 00:21:46.150
of two charges at that
distance repelling each other
00:21:46.150 --> 00:21:50.800
or even just the energy
of a core electron
00:21:50.800 --> 00:21:52.210
very close to its nucleus.
00:21:52.210 --> 00:21:54.670
And that's why also electronic
structure calculations are
00:21:54.670 --> 00:21:58.240
very delicate because
what you need to find out
00:21:58.240 --> 00:22:04.390
is a total energy that is
the combination of terms that
00:22:04.390 --> 00:22:08.960
largely cancel each other out.
00:22:08.960 --> 00:22:11.500
And so you need to be
very accurate in order
00:22:11.500 --> 00:22:15.340
to actually decide if something
like a hydrogen molecule
00:22:15.340 --> 00:22:20.880
binds together or breaks apart.
00:22:20.880 --> 00:22:21.510
OK.
00:22:21.510 --> 00:22:25.740
Now, this is truly
a problem of greater
00:22:25.740 --> 00:22:27.570
computational complexity.
00:22:27.570 --> 00:22:30.220
And as I said over
and over again,
00:22:30.220 --> 00:22:34.140
we can't really deal,
even computationally,
00:22:34.140 --> 00:22:38.820
with an object that has all the
information content of a wave
00:22:38.820 --> 00:22:39.730
function.
00:22:39.730 --> 00:22:43.200
So let me actually
go through this
00:22:43.200 --> 00:22:47.160
into the next slide, in which
I've written out explicitly
00:22:47.160 --> 00:22:49.680
the example of the iron atom.
00:22:49.680 --> 00:22:51.930
So that has, again,
26 electrons.
00:22:51.930 --> 00:22:53.700
So the electromagnetic
wave function
00:22:53.700 --> 00:22:57.240
will in itself
have 78 variables.
00:22:57.240 --> 00:22:59.250
And if you think
about how many numbers
00:22:59.250 --> 00:23:03.600
do we need to store this object
with any kind of precision,
00:23:03.600 --> 00:23:09.600
well, suppose that we even limit
ourselves to a very, very core
00:23:09.600 --> 00:23:15.510
sampling, only say 10 values
around either nucleus.
00:23:15.510 --> 00:23:18.060
Well, even to give
this wave function,
00:23:18.060 --> 00:23:21.580
to give this amplitude with
this very core sampling,
00:23:21.580 --> 00:23:24.090
we would need 10
to the 78 numbers.
00:23:24.090 --> 00:23:27.690
So basically, there is no
way we can numerically deal
00:23:27.690 --> 00:23:30.280
with the complexity
of the wave function.
00:23:30.280 --> 00:23:34.590
And this is where the power
of the variational principle
00:23:34.590 --> 00:23:38.640
and the ideas of
Hartree-Fock came together.
00:23:38.640 --> 00:23:44.250
And we'll first discuss
the first idea of Hartree
00:23:44.250 --> 00:23:46.200
in dealing with this problem.
00:23:46.200 --> 00:23:49.770
Remember, what we have is a
set of interacting electrons.
00:23:49.770 --> 00:23:54.612
And I like to compare each
one of you to an electron.
00:23:54.612 --> 00:23:56.070
So what you need
to think, and what
00:23:56.070 --> 00:23:58.005
the complexity of
this problem is,
00:23:58.005 --> 00:24:02.430
is that at every instant
in time each one of you
00:24:02.430 --> 00:24:07.810
is interacting or actually
repelling with each one else.
00:24:07.810 --> 00:24:10.650
So this is the complexity
of a many-body problem.
00:24:10.650 --> 00:24:13.740
In order to understand
what's happening,
00:24:13.740 --> 00:24:18.390
each thing needs to know
what everyone else is doing.
00:24:18.390 --> 00:24:21.930
And Hartree
introduced the concept
00:24:21.930 --> 00:24:25.110
of independent particles
and effective potential.
00:24:25.110 --> 00:24:27.480
This is something
that comes over not
00:24:27.480 --> 00:24:29.940
only in electronic
structure, but it comes over
00:24:29.940 --> 00:24:32.310
in a lot of problems
where we actually
00:24:32.310 --> 00:24:37.980
need to deal with a very large
number of interacting elements
00:24:37.980 --> 00:24:40.290
and interacting particles.
00:24:40.290 --> 00:24:45.900
And the general idea is that
we can approximate and try
00:24:45.900 --> 00:24:50.130
to solve this problem
by not considering what
00:24:50.130 --> 00:24:53.130
each electron instantly does.
00:24:53.130 --> 00:24:56.550
But we can solve the
problem by treating
00:24:56.550 --> 00:25:01.620
what one electron would
do in a field that,
00:25:01.620 --> 00:25:06.420
on average, represents what
all the other electrons would
00:25:06.420 --> 00:25:07.470
be doing.
00:25:07.470 --> 00:25:10.990
So if we want to think,
say, what I would be doing--
00:25:10.990 --> 00:25:14.430
I shouldn't try to find the
solution that instantaneously
00:25:14.430 --> 00:25:17.550
knows about what each and
everyone else is doing.
00:25:17.550 --> 00:25:22.080
But I could try to find a
solution for myself interacting
00:25:22.080 --> 00:25:26.220
electrostatically with the
average charge distribution
00:25:26.220 --> 00:25:28.090
that everyone else does.
00:25:28.090 --> 00:25:30.330
So instead of
having to know what
00:25:30.330 --> 00:25:32.670
is the instantaneous
position of all
00:25:32.670 --> 00:25:34.710
the other interacting
electrons, I
00:25:34.710 --> 00:25:38.130
could make an approximation
that I could actually really
00:25:38.130 --> 00:25:41.490
just try to deal with
the way I interact
00:25:41.490 --> 00:25:44.980
with an average distribution
of everyone else.
00:25:44.980 --> 00:25:48.600
So this is, if you want,
the concept of mean field
00:25:48.600 --> 00:25:50.640
or effective potential.
00:25:50.640 --> 00:25:53.850
We are averaging over
all the variables.
00:25:53.850 --> 00:25:56.700
And there is actually
a mathematical way
00:25:56.700 --> 00:26:00.390
to do this that I'll
introduce in this moment.
00:26:00.390 --> 00:26:04.230
But if you want, the Hartree
solution really leads to this.
00:26:04.230 --> 00:26:06.330
It leads to a
Schrodinger equation
00:26:06.330 --> 00:26:09.120
in which we are actually
trying to solve the problem
00:26:09.120 --> 00:26:12.030
of a single electron at a time.
00:26:12.030 --> 00:26:16.410
But that electron feels the
average electrostatic charge
00:26:16.410 --> 00:26:20.220
distribution of all
the other electrons.
00:26:20.220 --> 00:26:24.360
And one can actually
obtain this directly
00:26:24.360 --> 00:26:26.520
from the variational principle.
00:26:26.520 --> 00:26:31.770
That is, one can make an
answer for the wave function.
00:26:31.770 --> 00:26:36.630
What is written here is this
most generic wave function.
00:26:36.630 --> 00:26:39.420
And one can make what
turns out to be actually
00:26:39.420 --> 00:26:42.120
a fairly severe approximation.
00:26:42.120 --> 00:26:46.980
That is, we can say that this
many-body wave function can
00:26:46.980 --> 00:26:53.027
actually be written as a product
of single particle orbitals.
00:26:53.027 --> 00:26:55.110
So you see, what's happening,
when we are actually
00:26:55.110 --> 00:27:01.380
making this hypothesis, is
that varying r1 will change
00:27:01.380 --> 00:27:03.420
the amplitude of
the wave function
00:27:03.420 --> 00:27:06.930
independently from what
happens to r2 and rn.
00:27:06.930 --> 00:27:10.180
These have all become
independent variable.
00:27:10.180 --> 00:27:14.970
There is no combined
effect what r2 and r1 are
00:27:14.970 --> 00:27:17.430
doing-- so your any
couple, any triplet,
00:27:17.430 --> 00:27:19.270
and so on and so forth.
00:27:19.270 --> 00:27:22.530
So if you want, if we fix
all the other variables,
00:27:22.530 --> 00:27:26.790
we can independently look at
what each one of these orbitals
00:27:26.790 --> 00:27:27.730
is doing.
00:27:27.730 --> 00:27:29.670
And you can think
just what would
00:27:29.670 --> 00:27:31.330
come from a Taylor expansion.
00:27:31.330 --> 00:27:32.490
This is an approximation.
00:27:32.490 --> 00:27:35.610
It's not a true solution, OK?
00:27:35.610 --> 00:27:39.300
Suppose that we were
dealing with two variables.
00:27:39.300 --> 00:27:42.820
Well, I mean, something like--
00:27:42.820 --> 00:27:45.790
think of a generic wave
function that could be written
00:27:45.790 --> 00:27:53.320
as, say, the exponential of the
square root of r1 plus r2 that
00:27:53.320 --> 00:27:58.310
can't be decoupled in the
product of two wave functions.
00:27:58.310 --> 00:28:01.660
So the product of two
single-particle wave functions
00:28:01.660 --> 00:28:04.240
is something simple
that doesn't capture
00:28:04.240 --> 00:28:07.600
the complexity of all the
possible two-body wave
00:28:07.600 --> 00:28:08.360
functions.
00:28:08.360 --> 00:28:11.300
And so in this, it's
an approximation.
00:28:11.300 --> 00:28:14.000
And so Hartree made
this approximation
00:28:14.000 --> 00:28:17.260
and then asked
himself, what happens
00:28:17.260 --> 00:28:21.400
if I actually throw
this approximation
00:28:21.400 --> 00:28:23.980
in the variational principle?
00:28:23.980 --> 00:28:26.560
Now, the difference
is that, instead
00:28:26.560 --> 00:28:31.360
of having just a function that
varies parametrically, now what
00:28:31.360 --> 00:28:33.550
we can really vary
in our valuational
00:28:33.550 --> 00:28:40.460
principle are the shapes of all
these single-particle orbitals.
00:28:40.460 --> 00:28:43.660
So what we are
asking ourselves is
00:28:43.660 --> 00:28:45.970
a function of this
former throwing
00:28:45.970 --> 00:28:48.760
into the variational
principle will give me
00:28:48.760 --> 00:28:53.840
a condition that each of these
orbitals need to satisfy.
00:28:53.840 --> 00:28:56.890
So if you want, you really
need to now calculate
00:28:56.890 --> 00:29:01.060
with functional analysis what
are the differential equations
00:29:01.060 --> 00:29:04.960
that each of these
orbitals, phi 1 to phi n,
00:29:04.960 --> 00:29:08.770
needs to satisfy so that
the overall expectation
00:29:08.770 --> 00:29:13.240
value of the energy is minimum
for a wave function written
00:29:13.240 --> 00:29:17.050
in this restricted
class of product
00:29:17.050 --> 00:29:20.080
of single-particle orbitals.
00:29:20.080 --> 00:29:22.730
And so when you
actually work out
00:29:22.730 --> 00:29:27.520
the fairly complex functional
analysis of this problem--
00:29:27.520 --> 00:29:30.130
that's actually described,
if you're interested,
00:29:30.130 --> 00:29:31.750
in one of the
references that we have
00:29:31.750 --> 00:29:34.390
given you, the Bransden
and Joachain book
00:29:34.390 --> 00:29:37.570
on the physics of
atoms and molecules--
00:29:37.570 --> 00:29:40.900
what you obtain for
the specific case,
00:29:40.900 --> 00:29:46.180
again of a Hamiltonian operator
in which the potential is given
00:29:46.180 --> 00:29:50.470
by a linear combination of
attractive Coulombic potential,
00:29:50.470 --> 00:29:52.660
is a set of equations.
00:29:52.660 --> 00:29:55.630
That is, what you
obtain is a new set
00:29:55.630 --> 00:29:57.820
of differential equations.
00:29:57.820 --> 00:30:01.990
Instead of having, if you want,
one single Schrodinger equation
00:30:01.990 --> 00:30:04.510
for a many-body wave function--
00:30:04.510 --> 00:30:07.420
what you obtain if you are
dealing with n particles
00:30:07.420 --> 00:30:13.090
is n different differential
equations, each one
00:30:13.090 --> 00:30:17.710
being a differential equation
for only a single particle wave
00:30:17.710 --> 00:30:19.060
function.
00:30:19.060 --> 00:30:21.190
This is still fairly complex.
00:30:21.190 --> 00:30:23.890
But the complexity
of this has gone
00:30:23.890 --> 00:30:28.300
from being a complexity of a
wave function of 78 variables
00:30:28.300 --> 00:30:31.990
to, say, the complexity
of 26 equations
00:30:31.990 --> 00:30:34.360
in three variables
each, if we are
00:30:34.360 --> 00:30:36.340
working in three dimensions.
00:30:36.340 --> 00:30:38.680
And actualyl, the
form of this equation
00:30:38.680 --> 00:30:40.900
is very intriguing
because this really
00:30:40.900 --> 00:30:45.400
looks like a Schrodinger
equation for one electron.
00:30:45.400 --> 00:30:49.120
You see, here is the
quantum kinetic energy
00:30:49.120 --> 00:30:51.190
term for this electron.
00:30:51.190 --> 00:30:53.950
And then there is
the interaction
00:30:53.950 --> 00:30:57.820
between this single electron
with the Coulombic distribution
00:30:57.820 --> 00:30:59.140
of nuclei.
00:30:59.140 --> 00:31:00.640
And then there is a term--
00:31:00.640 --> 00:31:05.650
you see, this appropriately
is called the Hartree term--
00:31:05.650 --> 00:31:13.070
in which this electron, i, is
actually feeling a Coulombic
00:31:13.070 --> 00:31:15.890
repulsion-- this is
the Coulombic term--
00:31:15.890 --> 00:31:19.160
of a charge density
distribution.
00:31:19.160 --> 00:31:22.640
Remember that if we take
this square model of a wave
00:31:22.640 --> 00:31:26.090
function, we obtain the
probability of finding
00:31:26.090 --> 00:31:27.470
an electron somewhere.
00:31:27.470 --> 00:31:29.660
And the charge density is
nothing less than that.
00:31:29.660 --> 00:31:32.720
It's the probability of
finding an electron somewhere.
00:31:32.720 --> 00:31:35.600
And so you see, what we
have is, for the electron
00:31:35.600 --> 00:31:41.240
that we have denoted as i, the
interaction between the wave
00:31:41.240 --> 00:31:46.850
function and a potential, as
usual a potential, in which we
00:31:46.850 --> 00:31:49.580
have the Coulombic term.
00:31:49.580 --> 00:31:52.040
And the repulsion, the
Coulombic repulsion
00:31:52.040 --> 00:31:55.580
is between the electron,
i, and the charged density
00:31:55.580 --> 00:32:00.170
distribution of each and
every other electron, j.
00:32:00.170 --> 00:32:04.070
So you see, this sum goes
over all the other electrons.
00:32:04.070 --> 00:32:06.350
So my many-body
Schrodinger equation
00:32:06.350 --> 00:32:10.670
has become a series
of equations that we
00:32:10.670 --> 00:32:13.700
call single-particle equations,
a differential equation
00:32:13.700 --> 00:32:15.680
for each and every electron.
00:32:15.680 --> 00:32:18.020
And those differential
equations,
00:32:18.020 --> 00:32:20.360
as I said, have been
obtained formally
00:32:20.360 --> 00:32:25.130
just by applying this answer
to the variational principle,
00:32:25.130 --> 00:32:28.400
are in the form of a
Schrodinger-like equation
00:32:28.400 --> 00:32:31.670
with a kinetic energy,
an attractive potential.
00:32:31.670 --> 00:32:35.720
And now, we have only a
mean field interaction
00:32:35.720 --> 00:32:38.870
between the electrons
because electron i
00:32:38.870 --> 00:32:42.410
doesn't instantaneously
need to know what each
00:32:42.410 --> 00:32:44.180
and every other electron does.
00:32:44.180 --> 00:32:49.250
But it actually only interacts
with the average charge density
00:32:49.250 --> 00:32:53.690
distribution that is given
by the square model of j.
00:32:53.690 --> 00:32:54.560
OK.
00:32:54.560 --> 00:32:56.510
So this is a great
simplification.
00:32:56.510 --> 00:33:00.950
And it actually allowed some
of the first calculations,
00:33:00.950 --> 00:33:02.390
say, on atoms.
00:33:02.390 --> 00:33:05.180
Actually, this was
developed in the late '20s.
00:33:05.180 --> 00:33:08.450
And very quickly, it was
realized what was wrong.
00:33:08.450 --> 00:33:09.800
And we'll see that in a moment.
00:33:09.800 --> 00:33:15.080
That is the lack of
correlation because there
00:33:15.080 --> 00:33:18.380
is a specific role that
electrons are fermions.
00:33:18.380 --> 00:33:20.270
But really, this
was the first time
00:33:20.270 --> 00:33:24.710
in which we had a workable
differential equation
00:33:24.710 --> 00:33:26.750
for our many-body system.
00:33:26.750 --> 00:33:31.790
And the most
important conclusion
00:33:31.790 --> 00:33:34.790
of this, and where
the complexity
00:33:34.790 --> 00:33:37.520
of the many-body
problem comes back in,
00:33:37.520 --> 00:33:41.840
is that this new operator,
this new overall Hartree
00:33:41.840 --> 00:33:44.510
operator acting on
the wave function,
00:33:44.510 --> 00:33:48.440
has become, as we
say, self-consistent.
00:33:48.440 --> 00:33:52.610
That is, the operator
itself depends
00:33:52.610 --> 00:33:54.940
on what the other electrons do.
00:33:54.940 --> 00:33:58.610
So it depends on
what the solution
00:33:58.610 --> 00:34:02.390
to all the other
differential equations are.
00:34:02.390 --> 00:34:06.440
So if you want to know
what electron i does
00:34:06.440 --> 00:34:08.900
in the mean field of
all the other electrons,
00:34:08.900 --> 00:34:12.350
you need to know what is
the wave function of each
00:34:12.350 --> 00:34:14.510
and every other electron.
00:34:14.510 --> 00:34:17.030
But in order to solve
the wave function of each
00:34:17.030 --> 00:34:18.590
and every other
electron, you will
00:34:18.590 --> 00:34:22.620
need to know also what
electron i is doing.
00:34:22.620 --> 00:34:25.070
And so really, this
is very different.
00:34:25.070 --> 00:34:29.239
The Hamiltonian operator
here is not anymore given
00:34:29.239 --> 00:34:31.820
at the beginning of the
problem, but actually
00:34:31.820 --> 00:34:35.719
needs to be found
because the operator--
00:34:35.719 --> 00:34:38.239
the action on one
electron depends
00:34:38.239 --> 00:34:40.022
on what the other electrons do.
00:34:40.022 --> 00:34:42.230
And in order to find out
what the other electrons do,
00:34:42.230 --> 00:34:43.790
we need to solve the set.
00:34:43.790 --> 00:34:45.170
So this set of--
00:34:45.170 --> 00:34:46.730
say, the case of iron again--
00:34:46.730 --> 00:34:53.159
26 differential equations needs
to be solved simultaneously,
00:34:53.159 --> 00:34:53.659
OK?
00:34:53.659 --> 00:34:57.440
So we actually solve
it iteratively.
00:34:57.440 --> 00:35:02.510
We start with a guess for what
the wave function would be.
00:35:02.510 --> 00:35:04.970
And we try to find
a combined solution.
00:35:04.970 --> 00:35:08.330
I'll describe in a moment
what the algorithm is.
00:35:08.330 --> 00:35:11.480
So this is how the many-body
complexity comes in.
00:35:11.480 --> 00:35:15.170
That is, we need to solve a
differential equation for which
00:35:15.170 --> 00:35:17.960
we don't even know,
at the beginning, what
00:35:17.960 --> 00:35:24.150
is the operator acting on our
single-particle wave function.
00:35:24.150 --> 00:35:27.410
And so the concept
of self-consistency
00:35:27.410 --> 00:35:30.440
and of iterative
solution will basically
00:35:30.440 --> 00:35:33.950
be always present in all
the electronic structure
00:35:33.950 --> 00:35:37.520
approaches that we are
going to see in this class.
00:35:37.520 --> 00:35:41.120
And it's actually very
simple to figure out
00:35:41.120 --> 00:35:43.880
what could be an
actual algorithm
00:35:43.880 --> 00:35:46.160
to try to get to the solution.
00:35:46.160 --> 00:35:50.300
That is, we need to start with
an arbitrary guess for all
00:35:50.300 --> 00:35:53.360
those 26 orbitals.
00:35:53.360 --> 00:35:56.810
And once we have
that arbitrary guess,
00:35:56.810 --> 00:36:00.260
we can construct the charge
density of every electron.
00:36:00.260 --> 00:36:04.700
And so we can construct what
would be the operator acting
00:36:04.700 --> 00:36:07.610
on each one of the phi i.
00:36:07.610 --> 00:36:14.000
That is, we can
construct this term here.
00:36:14.000 --> 00:36:18.050
Once now we know what actually
our differential equation is,
00:36:18.050 --> 00:36:19.890
we can solve it.
00:36:19.890 --> 00:36:24.140
So we find what are the ground
states of each of those 26
00:36:24.140 --> 00:36:25.800
differential equations.
00:36:25.800 --> 00:36:28.280
And now, with those
ground states,
00:36:28.280 --> 00:36:30.890
we can construct again
a charge density of each
00:36:30.890 --> 00:36:32.420
and every single electron.
00:36:32.420 --> 00:36:33.710
We put it together.
00:36:33.710 --> 00:36:36.110
We have a new Hartree operator.
00:36:36.110 --> 00:36:40.130
And we can solve these
differential equations again.
00:36:40.130 --> 00:36:44.690
And we keep iterating
this until what
00:36:44.690 --> 00:36:48.680
we obtain is a Hartree operator
in each differential equation
00:36:48.680 --> 00:36:50.550
that doesn't change anymore.
00:36:50.550 --> 00:36:55.190
And so we have now a set of
self-consistent orbitals.
00:36:55.190 --> 00:36:58.940
In most cases, you actually
don't get to convergence.
00:36:58.940 --> 00:37:02.510
So a lot of the
algorithmic advances
00:37:02.510 --> 00:37:04.610
that has been done
in the 20th century
00:37:04.610 --> 00:37:06.800
is actually to do
with this problem.
00:37:06.800 --> 00:37:10.310
That is, we need to find
ways to get this procedure
00:37:10.310 --> 00:37:13.850
to converge to an actually
self-consistent point.
00:37:13.850 --> 00:37:16.920
But the concept is all here.
00:37:16.920 --> 00:37:19.400
And if you want,
the simplest thing
00:37:19.400 --> 00:37:22.940
that you can do to
make sure that you're
00:37:22.940 --> 00:37:25.220
going to iterate
to self-consistence
00:37:25.220 --> 00:37:28.200
is to move very slowly.
00:37:28.200 --> 00:37:31.490
That is, whenever you
have a set of solutions,
00:37:31.490 --> 00:37:34.910
you don't want to construct
your Hartree operator,
00:37:34.910 --> 00:37:37.670
this new charge density,
with the solution.
00:37:37.670 --> 00:37:42.620
But you want just to modify a
little bit your previous charge
00:37:42.620 --> 00:37:46.550
densities to go in the direction
of the new charge density
00:37:46.550 --> 00:37:48.050
that you were calculating.
00:37:48.050 --> 00:37:53.600
So you somehow try to minimize
the change in the operators
00:37:53.600 --> 00:37:55.650
from one iteration to the other.
00:37:55.650 --> 00:38:00.800
And that tends to be actually
fairly functional in a lot
00:38:00.800 --> 00:38:02.840
of problems.
00:38:02.840 --> 00:38:06.080
Now, Hartree was a very
interesting character,
00:38:06.080 --> 00:38:10.340
again growing up in Cambridge
at the turn of the century.
00:38:10.340 --> 00:38:13.460
He became an expert in
differential equations
00:38:13.460 --> 00:38:15.920
during the First World War
because basically people
00:38:15.920 --> 00:38:20.480
had the problem of sending
cannonballs across the lines.
00:38:20.480 --> 00:38:23.530
So there was a lot of
differential equations.
00:38:23.530 --> 00:38:29.840
And so mid '20s, he
developed this general idea
00:38:29.840 --> 00:38:32.420
of the Hartree equation
that really what we
00:38:32.420 --> 00:38:35.210
call coupled
integral differential
00:38:35.210 --> 00:38:37.220
equation because
there are derivatives
00:38:37.220 --> 00:38:38.750
and there are integrals.
00:38:38.750 --> 00:38:41.390
And so what you need
to do is now solve
00:38:41.390 --> 00:38:44.840
this equation that can't
really be solved analytically.
00:38:44.840 --> 00:38:47.180
And luckily, that
was the time in which
00:38:47.180 --> 00:38:50.930
people were developing, if
you want, the first computers.
00:38:50.930 --> 00:38:53.750
No electronic in
there-- so computing
00:38:53.750 --> 00:38:55.580
machine, but mechanical.
00:38:55.580 --> 00:38:58.520
And one of the first
computers was at MIT.
00:38:58.520 --> 00:39:01.970
This is actually a picture
from the MIT archives.
00:39:01.970 --> 00:39:05.420
There was an electrical
engineer named Vannevar Bush,
00:39:05.420 --> 00:39:08.780
to which Building 13 the
Bush Building, is named
00:39:08.780 --> 00:39:14.780
that developed one of the
first mechanical differential
00:39:14.780 --> 00:39:16.400
equation solvers.
00:39:16.400 --> 00:39:19.580
And so Hartree came at
the end of the '20s to MIT
00:39:19.580 --> 00:39:21.680
to actually solve
the Hartree equation,
00:39:21.680 --> 00:39:25.760
I guess, in some building
here exactly on that machine,
00:39:25.760 --> 00:39:29.360
and then went back to Cambridge.
00:39:29.360 --> 00:39:33.850
And so this, I thought,
was an interesting note.
00:39:33.850 --> 00:39:36.010
So what do we obtain
when we actually
00:39:36.010 --> 00:39:38.570
solve the Hartree equation?
00:39:38.570 --> 00:39:41.950
Well, the most
fundamental concept
00:39:41.950 --> 00:39:45.700
is that we are losing
some information
00:39:45.700 --> 00:39:48.670
on what's happening
instantaneously
00:39:48.670 --> 00:39:50.770
to all the electrons.
00:39:50.770 --> 00:39:53.170
This is what we call,
generically speaking,
00:39:53.170 --> 00:39:54.310
a correlation.
00:39:54.310 --> 00:39:57.863
And I'll give you the technical
definition in a few slides.
00:39:57.863 --> 00:39:59.530
But basically, this
is what's happening.
00:39:59.530 --> 00:40:03.070
Let's consider the case
of the helium atom.
00:40:03.070 --> 00:40:06.010
That is the simplest
case in which you
00:40:06.010 --> 00:40:08.000
have more than one electron.
00:40:08.000 --> 00:40:10.150
So what you have
is two electrons.
00:40:10.150 --> 00:40:13.930
What does the Hartree
equation tell us about--
00:40:13.930 --> 00:40:16.180
or what do the Hartree
equations tell us
00:40:16.180 --> 00:40:18.020
about these two electrons?
00:40:18.020 --> 00:40:20.560
Well, we have two equations.
00:40:20.560 --> 00:40:23.950
And in each of
them, one electron
00:40:23.950 --> 00:40:28.900
is going to feel an average
electrostatic repulsion
00:40:28.900 --> 00:40:30.910
from the other electron.
00:40:30.910 --> 00:40:34.210
So what we have is that
this right electron here
00:40:34.210 --> 00:40:37.120
is attracted to
the nucleus and is
00:40:37.120 --> 00:40:45.430
repelled by a spherically
symmetrical average charge
00:40:45.430 --> 00:40:46.720
distribution.
00:40:46.720 --> 00:40:48.140
So this is what's happening.
00:40:48.140 --> 00:40:50.950
This is what Hartree tells us.
00:40:50.950 --> 00:40:57.510
But in reality,
electrons instantaneously
00:40:57.510 --> 00:41:02.610
try to keep themselves
as far apart as possible
00:41:02.610 --> 00:41:04.840
because electrons
repel each other.
00:41:04.840 --> 00:41:10.260
So in a simplified way, you
could think of electron 1
00:41:10.260 --> 00:41:17.340
and electron 2 trying to orbit
the nucleus as much as possible
00:41:17.340 --> 00:41:20.100
in a position of phases.
00:41:20.100 --> 00:41:23.730
So the true two
interacting electrons
00:41:23.730 --> 00:41:28.650
try to be as far as possible
at every moment in time
00:41:28.650 --> 00:41:34.860
during their revolution
around the helium nucleus.
00:41:34.860 --> 00:41:37.620
But this instantaneous
correlation--
00:41:37.620 --> 00:41:41.610
that is, the fact that
the wave functions tries
00:41:41.610 --> 00:41:46.530
to keep the electron as far
away as possible-- is lost
00:41:46.530 --> 00:41:50.760
in the Hartree equations
because what we do is we
00:41:50.760 --> 00:41:54.600
are really having one
electron interacting
00:41:54.600 --> 00:41:58.800
with the average position
of the other electron.
00:41:58.800 --> 00:42:02.070
And so in the Hartree
equation, there
00:42:02.070 --> 00:42:07.140
are a lot of terms that have to
do with our initial electron,
00:42:07.140 --> 00:42:12.030
red, being too close to
the green charge density
00:42:12.030 --> 00:42:13.600
distribution.
00:42:13.600 --> 00:42:16.710
So if you want, the
wave function-- that is,
00:42:16.710 --> 00:42:19.560
the overall solution of
the Hartree equation--
00:42:19.560 --> 00:42:24.180
tends to have too much
electrostatic repulsion
00:42:24.180 --> 00:42:26.490
between electron
1 and electron 2.
00:42:26.490 --> 00:42:31.890
And that's why ultimately the
energy of this Hartree function
00:42:31.890 --> 00:42:34.380
is higher than
the true solution.
00:42:34.380 --> 00:42:37.890
This is what is missing in
the Hartree equation, the fact
00:42:37.890 --> 00:42:40.800
that what you want is
a lot of correlation.
00:42:40.800 --> 00:42:44.610
That is, electrons want to
keep each other apart as
00:42:44.610 --> 00:42:45.840
much as possible.
00:42:45.840 --> 00:42:50.520
But that, if you want, is really
an instantaneous solution.
00:42:50.520 --> 00:42:53.280
It's what people call
dynamical correlation.
00:42:53.280 --> 00:42:56.170
Electrons want to keep
apart from each other.
00:42:56.170 --> 00:42:57.900
But if you start
looking at a mean field
00:42:57.900 --> 00:43:00.390
solution in which
only one interacts
00:43:00.390 --> 00:43:02.640
with the average
charge density, you
00:43:02.640 --> 00:43:05.610
have lost the
possibility of having
00:43:05.610 --> 00:43:10.350
this instantaneous
non-symmetric distribution.
00:43:10.350 --> 00:43:14.050
And so in general, this is
what we call correlation.
00:43:14.050 --> 00:43:19.330
And this is what is missing
in the Hartree picture.
00:43:19.330 --> 00:43:22.780
There is another set of very
fundamental concepts that
00:43:22.780 --> 00:43:26.350
I'll describe in a moment
that the wave function--
00:43:26.350 --> 00:43:29.620
the answer for the
Hartree wave function--
00:43:29.620 --> 00:43:34.360
doesn't satisfy a fundamental
rule for wave functions.
00:43:34.360 --> 00:43:36.280
We say it's not anti-symmetric.
00:43:36.280 --> 00:43:38.845
And I'll show you in
a moment what it is.
00:43:38.845 --> 00:43:41.590
That is, if you want,
it doesn't respect
00:43:41.590 --> 00:43:46.750
a fundamental constraint
on the shape of functions.
00:43:46.750 --> 00:43:48.610
And so that's, if
you want, an error.
00:43:48.610 --> 00:43:52.570
And that's another source of
error in our final estimate
00:43:52.570 --> 00:43:54.100
of the energy.
00:43:54.100 --> 00:43:57.850
And in particular,
what it doesn't do--
00:43:57.850 --> 00:44:02.170
it does not remove what is the
accidental degeneracy, the fact
00:44:02.170 --> 00:44:06.100
that there is the same
energy for electrons that
00:44:06.100 --> 00:44:10.510
have the same principle quantum
number n and the same angular
00:44:10.510 --> 00:44:13.180
momentum number l.
00:44:13.180 --> 00:44:15.940
But really, what
is most important
00:44:15.940 --> 00:44:19.930
is this lack of a
physical constraint
00:44:19.930 --> 00:44:22.880
and this lack of correlation.
00:44:22.880 --> 00:44:27.020
Very soon, I mean
probably the same year
00:44:27.020 --> 00:44:31.250
or a year later, Hartree,
and independently Fock,
00:44:31.250 --> 00:44:35.210
realized that one could
actually find a better solution
00:44:35.210 --> 00:44:38.960
to the problem
satisfying one, again,
00:44:38.960 --> 00:44:42.410
of the fundamental
rules of nature that
00:44:42.410 --> 00:44:46.550
had been discovered in the
'20s during the development
00:44:46.550 --> 00:44:48.740
of quantum mechanics.
00:44:48.740 --> 00:44:52.040
And so one of these
rules was what
00:44:52.040 --> 00:44:56.700
is called the
spin-statistic correlation.
00:44:56.700 --> 00:44:59.510
And this is really very general.
00:44:59.510 --> 00:45:03.260
First of all,
there is a division
00:45:03.260 --> 00:45:06.020
in elementary
particles that says
00:45:06.020 --> 00:45:09.350
that all elementary
particles can be called
00:45:09.350 --> 00:45:12.320
either fermions or bosons.
00:45:12.320 --> 00:45:15.290
And so things like electrons
are actually fermions.
00:45:15.290 --> 00:45:17.780
They have a half-integer spin.
00:45:17.780 --> 00:45:20.930
But there is a
fundamental difference
00:45:20.930 --> 00:45:23.660
between fermions and bosons.
00:45:23.660 --> 00:45:31.070
And in particular, they satisfy
different statistical rules
00:45:31.070 --> 00:45:34.670
for an ensemble of many
interacting electrons
00:45:34.670 --> 00:45:36.440
or for many bosons.
00:45:36.440 --> 00:45:38.360
And this rule-- so
this is nothing else
00:45:38.360 --> 00:45:41.990
that-- again, another rule
like in classical mechanics--
00:45:41.990 --> 00:45:43.790
you have Newton's
equation of motion.
00:45:43.790 --> 00:45:46.550
In quantum mechanics,
you have a rule
00:45:46.550 --> 00:45:51.800
that wave functions that
described fermions-- that
00:45:51.800 --> 00:45:56.030
is, a wave function
that describes electrons
00:45:56.030 --> 00:45:58.490
needs to have this
overall shape.
00:45:58.490 --> 00:46:05.720
That is, it needs to change sign
when we invert two variables.
00:46:05.720 --> 00:46:09.980
So we have this general
form for the wave function.
00:46:09.980 --> 00:46:12.770
We exchange two variables.
00:46:12.770 --> 00:46:18.080
And what needs to happen
is that the wave function
00:46:18.080 --> 00:46:19.910
needs to change sign.
00:46:19.910 --> 00:46:21.620
And this has something to do--
00:46:21.620 --> 00:46:23.480
this can actually
be demonstrated.
00:46:23.480 --> 00:46:25.930
But it's what people call
quantum field theory.
00:46:25.930 --> 00:46:27.680
So it's an advanced concept.
00:46:27.680 --> 00:46:29.480
But it's a very simple rule.
00:46:29.480 --> 00:46:32.315
And it's a very simple symmetry
of the wave functioning.
00:46:32.315 --> 00:46:36.170
In the same way, if you
want, you have in a crystal,
00:46:36.170 --> 00:46:39.500
you have physical
symmetries for what
00:46:39.500 --> 00:46:41.210
could be some of
your properties,
00:46:41.210 --> 00:46:44.540
like an elastic tensor or a
piezoelectric coefficient.
00:46:44.540 --> 00:46:47.960
Well, what you have is a
fundamental symmetry for a wave
00:46:47.960 --> 00:46:49.730
function describing electrons.
00:46:49.730 --> 00:46:53.540
It needs to change sign when
you invert two coordinates.
00:46:53.540 --> 00:47:05.150
And so what the Hartree
solution didn't have
00:47:05.150 --> 00:47:08.510
was exactly this
anti-symmetry requirement.
00:47:08.510 --> 00:47:12.500
Remember, the Hartree
solution was just the product
00:47:12.500 --> 00:47:15.560
of single-particle orbitals.
00:47:15.560 --> 00:47:18.710
But Hartree very quickly
realized that you can actually
00:47:18.710 --> 00:47:22.430
satisfy this
symmetry requirement
00:47:22.430 --> 00:47:27.350
if instead of taking just
the product of n orbitals,
00:47:27.350 --> 00:47:33.170
you take the sum of the
product of n orbitals
00:47:33.170 --> 00:47:37.470
where you interchange
the variables in all
00:47:37.470 --> 00:47:38.940
the possible ways--
00:47:38.940 --> 00:47:42.330
putting a plus or
minus sign in front,
00:47:42.330 --> 00:47:45.300
depending on how much
interchanges you had.
00:47:45.300 --> 00:47:48.420
And I think it's very simple
to think of this problem if you
00:47:48.420 --> 00:47:50.910
have, say, only two electrons.
00:47:50.910 --> 00:47:53.020
So you have only two orbitals.
00:47:53.020 --> 00:47:59.190
And so we could call these two
orbital, say, alpha and beta.
00:47:59.190 --> 00:48:04.170
And so what we would have
is the Hartree solution
00:48:04.170 --> 00:48:09.270
that is the product of the alpha
orbital function of the first r
00:48:09.270 --> 00:48:11.700
variable and the
beta orbital function
00:48:11.700 --> 00:48:13.420
of the second variable.
00:48:13.420 --> 00:48:17.760
And this doesn't change sign
if we exchange 1 with 2.
00:48:17.760 --> 00:48:19.260
It becomes a different function.
00:48:19.260 --> 00:48:23.430
It doesn't become the same
function with the sign changed.
00:48:23.430 --> 00:48:26.970
But you can see that what we can
do, without increasing really
00:48:26.970 --> 00:48:29.670
the complexity of the problem--
that is, still dealing
00:48:29.670 --> 00:48:33.120
with just the need of describing
two orbitals, what we could do
00:48:33.120 --> 00:48:36.840
is take as an
ansatz for the wave
00:48:36.840 --> 00:48:40.560
function describing two
electrons something that
00:48:40.560 --> 00:48:51.090
is actually alpha 1 B2
minus B1 alpha 2, OK?
00:48:51.090 --> 00:48:55.740
So we have still two orbitals
that we need to figure out.
00:48:55.740 --> 00:48:58.110
We need to figure out
what is the shape of alpha
00:48:58.110 --> 00:49:00.090
and what is the shape of beta.
00:49:00.090 --> 00:49:02.820
But now, we are
using as an ansatz
00:49:02.820 --> 00:49:05.220
for this two-electron
wave function
00:49:05.220 --> 00:49:08.790
something that
actually changes sign
00:49:08.790 --> 00:49:12.120
when you exchange 1 with 2.
00:49:12.120 --> 00:49:16.410
So this is the trial
wave function of Hartree.
00:49:16.410 --> 00:49:19.740
And this is the
trial wave function
00:49:19.740 --> 00:49:22.920
of what we call the Hartree-Fock
method that is basically
00:49:22.920 --> 00:49:27.630
a trial wave function that
has built-in anti-symmetry
00:49:27.630 --> 00:49:31.410
constraint for
exchange of particles.
00:49:31.410 --> 00:49:36.090
And this can be generalized
to the case of n particles.
00:49:36.090 --> 00:49:41.220
And really, what we
call a sum of n terms
00:49:41.220 --> 00:49:43.230
with all the possible
permutations,
00:49:43.230 --> 00:49:45.810
with all the possible
signs, is nothing less
00:49:45.810 --> 00:49:47.160
than a determinant.
00:49:47.160 --> 00:49:49.620
If you think at what
a determinant is,
00:49:49.620 --> 00:49:54.880
well, this determinant is
going to have one element.
00:49:54.880 --> 00:49:57.930
It's going to have a sum
of terms that each of them
00:49:57.930 --> 00:50:01.770
are rated as one term
from each column.
00:50:01.770 --> 00:50:04.650
And we are taking all the
possible permutations.
00:50:04.650 --> 00:50:08.640
And we are taking a plus
sign or a minus sign
00:50:08.640 --> 00:50:11.280
in the sum of all
these terms, depending
00:50:11.280 --> 00:50:13.080
on how many
permutations they are.
00:50:13.080 --> 00:50:16.800
This is just the linear algebra
definition of a determinant.
00:50:16.800 --> 00:50:21.930
And you see, again, a
determinant of two functions
00:50:21.930 --> 00:50:24.570
is-- that's what I've
written here in green,
00:50:24.570 --> 00:50:26.010
right here in the corner.
00:50:26.010 --> 00:50:34.890
If we are asking what
alpha 1 determinant beta
00:50:34.890 --> 00:50:45.360
1 beta 2 and alpha and alpha--
00:50:45.360 --> 00:50:49.600
so beta 1 and alpha 2.
00:50:54.380 --> 00:51:00.030
So this would be the specific
expression of this determinant
00:51:00.030 --> 00:51:01.960
for the case of two particles.
00:51:01.960 --> 00:51:04.500
And if you just solved
that determinant,
00:51:04.500 --> 00:51:06.610
you have only these two terms.
00:51:06.610 --> 00:51:10.110
So we haven't increased, in
going into the Hartree-Fock
00:51:10.110 --> 00:51:12.360
method, the complexity
of the problem
00:51:12.360 --> 00:51:17.190
that we need to solve because we
still need to find n functions,
00:51:17.190 --> 00:51:19.200
where n is the
number of electrons.
00:51:19.200 --> 00:51:21.810
And we need to find out the
appropriate differential
00:51:21.810 --> 00:51:25.380
equation that descend from
the variational principle
00:51:25.380 --> 00:51:29.940
once we stick this determinant
into the variational principle.
00:51:29.940 --> 00:51:33.780
And again, it's not very
complex functional analysis.
00:51:33.780 --> 00:51:36.990
And the Bransden Joachain
describes that in detail.
00:51:36.990 --> 00:51:41.370
But with this new solution,
with this ansatz, what
00:51:41.370 --> 00:51:44.400
we find is a new set of
differential equations
00:51:44.400 --> 00:51:48.120
that look a lot like
the Hartree equation
00:51:48.120 --> 00:51:49.380
that we had written before--
00:51:49.380 --> 00:51:51.790
I'll go back to
this in a moment--
00:51:51.790 --> 00:51:54.250
but have an additional term.
00:51:54.250 --> 00:51:59.620
So what we find are, again,
single-particle equations.
00:51:59.620 --> 00:52:02.730
So we have an integral
differential equation
00:52:02.730 --> 00:52:08.880
for each single-particle orbital
lambda that is written in red.
00:52:08.880 --> 00:52:13.440
And now, what we have is
a set of additional terms.
00:52:13.440 --> 00:52:17.100
So we still have the
quantum kinetic energy
00:52:17.100 --> 00:52:18.450
for that electron.
00:52:18.450 --> 00:52:20.220
And we still have
the interaction
00:52:20.220 --> 00:52:22.740
between that electron
and the collection
00:52:22.740 --> 00:52:26.140
of attractive
Coulombic potentials.
00:52:26.140 --> 00:52:30.130
We still have the Hartree
electrostatic term
00:52:30.130 --> 00:52:34.600
in which the electron
lambda interacts
00:52:34.600 --> 00:52:38.230
with the charge density
phi star mu times phi
00:52:38.230 --> 00:52:42.670
mu of every other electron mu.
00:52:42.670 --> 00:52:45.530
And I'll come to what goes
into this sum in a moment.
00:52:45.530 --> 00:52:47.500
But basically, we
have electron lambda
00:52:47.500 --> 00:52:50.560
interacting with the charge
density of electron mu.
00:52:50.560 --> 00:52:53.770
And obviously, it's a Coulombic
electrostatic repulsion.
00:52:53.770 --> 00:52:55.960
So there is a 1/r term.
00:52:55.960 --> 00:53:01.510
But now, there is a new
term with a minus sign
00:53:01.510 --> 00:53:06.430
that comes out only from the
anti-symmetry requirement.
00:53:06.430 --> 00:53:09.430
And that is what is
called the exchange term
00:53:09.430 --> 00:53:13.880
and is the new player in
the Hartree-Fock equation.
00:53:13.880 --> 00:53:16.690
And it's a little bit
exotic because, if you want,
00:53:16.690 --> 00:53:19.960
now we don't have any
more an operator that
00:53:19.960 --> 00:53:24.100
could be, say, a local operator,
a charge density distribution
00:53:24.100 --> 00:53:25.720
acting on an orbital.
00:53:25.720 --> 00:53:28.880
But it's really become
a known local operator
00:53:28.880 --> 00:53:33.060
because the orbital
on which I'm acting
00:53:33.060 --> 00:53:37.150
has gone inside
the orbital sign.
00:53:37.150 --> 00:53:42.220
So this new term here that
we call the exchange term is
00:53:42.220 --> 00:53:47.290
a purely quantum mechanical
term that comes out exclusively
00:53:47.290 --> 00:53:50.980
by the anti symmetry
requirements on what
00:53:50.980 --> 00:53:53.860
happens to a fermionic
wave function
00:53:53.860 --> 00:53:57.410
when we invert coefficients.
00:53:57.410 --> 00:54:01.340
And there is another fundamental
distinction between the Hartree
00:54:01.340 --> 00:54:05.120
and the Hartree-Fock equations.
00:54:05.120 --> 00:54:09.590
This sum here-- you
see this time over mu--
00:54:09.590 --> 00:54:13.220
in the Hartree
equation was running
00:54:13.220 --> 00:54:16.850
over all the
electrons but the one
00:54:16.850 --> 00:54:19.280
that we were considering, OK?
00:54:19.280 --> 00:54:23.390
And now instead, this
sum mu is running
00:54:23.390 --> 00:54:26.880
on all the possible electrons.
00:54:26.880 --> 00:54:29.660
So what you have
in this term here
00:54:29.660 --> 00:54:33.830
is what is technically
called self-interaction.
00:54:33.830 --> 00:54:35.870
Suppose that for
a moment you were
00:54:35.870 --> 00:54:42.050
trying to solve the hydrogen
atom with just one electron.
00:54:42.050 --> 00:54:45.230
Well, what happens in
the Hartree equation
00:54:45.230 --> 00:54:47.000
for the hydrogen atom--
00:54:47.000 --> 00:54:49.700
there is obviously
no exchange term.
00:54:49.700 --> 00:54:52.970
And there would be
also no Hartree term
00:54:52.970 --> 00:54:55.790
because, in the Hartree equation
that you've seen before,
00:54:55.790 --> 00:54:58.580
what you have is a sum over
all the other electrons.
00:54:58.580 --> 00:55:00.300
But there are no
other electrons.
00:55:00.300 --> 00:55:01.670
So this term is not there.
00:55:01.670 --> 00:55:04.160
And what you
trivially recover is
00:55:04.160 --> 00:55:08.450
the single-particle Schrodinger
equation for the hydrogen atom.
00:55:08.450 --> 00:55:11.390
In the Hartree-Fock
equations, now even
00:55:11.390 --> 00:55:14.480
for the hydrogen
atom, what you have
00:55:14.480 --> 00:55:16.880
is that now you have this term.
00:55:16.880 --> 00:55:21.650
You have actually an
unphysical self-interaction.
00:55:21.650 --> 00:55:27.335
That is, you have electron
1 interacting with itself.
00:55:27.335 --> 00:55:29.210
And that's really not
interacting with itself
00:55:29.210 --> 00:55:32.030
in a local way.
00:55:32.030 --> 00:55:36.290
But it's interacting with
its own charge distribution.
00:55:36.290 --> 00:55:39.140
This is actually a sort
of unphysical thing.
00:55:39.140 --> 00:55:45.320
But what happens exclusively
in the Hartree-Fock formulation
00:55:45.320 --> 00:55:47.480
is that there is a second term.
00:55:47.480 --> 00:55:49.700
There is the exchange term.
00:55:49.700 --> 00:55:53.270
And the exchange term-- you
see, there is a minus sign--
00:55:53.270 --> 00:55:57.950
cancels for the hydrogen atom
the self-interaction term
00:55:57.950 --> 00:55:59.240
exactly.
00:55:59.240 --> 00:56:03.170
If you just think-- if you
have only one electron,
00:56:03.170 --> 00:56:05.780
you have that there
is only one phi.
00:56:05.780 --> 00:56:06.980
So the lambda goes away.
00:56:06.980 --> 00:56:08.130
The mu goes away.
00:56:08.130 --> 00:56:10.670
So here, the electron
is interacting
00:56:10.670 --> 00:56:12.780
with its charge distribution.
00:56:12.780 --> 00:56:14.360
But then this term here--
00:56:14.360 --> 00:56:17.600
just remove all the sum
and remove mu and lambda--
00:56:17.600 --> 00:56:20.480
is canceled out thanks
to the minus sign
00:56:20.480 --> 00:56:22.400
by the exchange term.
00:56:22.400 --> 00:56:26.510
So Hartree-Fock
formalism is actually
00:56:26.510 --> 00:56:30.110
what we call self-interaction
corrected, OK?
00:56:30.110 --> 00:56:33.230
An electron, even if it's
a mean field picture,
00:56:33.230 --> 00:56:35.330
doesn't interact with itself.
00:56:35.330 --> 00:56:39.170
And that's actually a very
beautiful symmetry property
00:56:39.170 --> 00:56:42.170
that other approaches like
density functional theory
00:56:42.170 --> 00:56:43.790
do not satisfy.
00:56:43.790 --> 00:56:48.120
And truly, a lot of the
problems that come out
00:56:48.120 --> 00:56:51.560
in density functional theory
that otherwise perform really
00:56:51.560 --> 00:56:55.590
well have to do with this
self-interaction problem.
00:56:55.590 --> 00:56:58.970
And so those problems
are very significant,
00:56:58.970 --> 00:57:02.480
if you think for a moment,
of problems like dissociation
00:57:02.480 --> 00:57:05.090
of a molecule in atoms.
00:57:05.090 --> 00:57:07.610
So when you really
consider how--
00:57:07.610 --> 00:57:09.290
and we'll discuss
this in detail--
00:57:09.290 --> 00:57:14.660
how the energy changes along a
process in which the electron
00:57:14.660 --> 00:57:17.870
needs to localize itself
from a shared bonds
00:57:17.870 --> 00:57:21.380
to a localized bond and that
self interaction problem
00:57:21.380 --> 00:57:24.980
kills density function
theory and would actually not
00:57:24.980 --> 00:57:28.130
be present to begin
with in Hartree-Fock.
00:57:28.130 --> 00:57:31.730
So this is actually
very important.
00:57:31.730 --> 00:57:32.900
There is another thing.
00:57:32.900 --> 00:57:35.750
Actually, the
Hartree-Fock equation
00:57:35.750 --> 00:57:41.090
has a beauty in the fact
that the operator acting
00:57:41.090 --> 00:57:44.120
on the single-particle
orbitals does not
00:57:44.120 --> 00:57:50.160
depend on which orbital you
are looking at because here we
00:57:50.160 --> 00:57:52.660
have a sum over all the states.
00:57:52.660 --> 00:57:56.790
So this does not depend on which
electron you are looking at.
00:57:56.790 --> 00:57:58.920
That's different from
the Hartree equation.
00:57:58.920 --> 00:58:00.330
In the Hartree
equation, there is
00:58:00.330 --> 00:58:03.150
a Hartree term
where this sum would
00:58:03.150 --> 00:58:05.880
exclude the electron itself.
00:58:05.880 --> 00:58:07.980
So the Hartree
equations are actually
00:58:07.980 --> 00:58:13.740
more complex to solve because
the operator changes depending
00:58:13.740 --> 00:58:16.350
on which electron
it's acting on,
00:58:16.350 --> 00:58:21.420
while the Hartree-Fock
equation has the same operator.
00:58:21.420 --> 00:58:23.490
And so what we really
need to find out,
00:58:23.490 --> 00:58:26.550
if we are solving
the iron atom, what
00:58:26.550 --> 00:58:31.560
are the 26 lowest
energy solutions
00:58:31.560 --> 00:58:34.320
for all the electrons in here.
00:58:34.320 --> 00:58:37.300
So that's a very good thing.
00:58:37.300 --> 00:58:39.480
But there are terms--
00:58:39.480 --> 00:58:42.120
there are integrals
that are actually very
00:58:42.120 --> 00:58:44.820
expensive still to calculate.
00:58:44.820 --> 00:58:50.220
And ultimately, it's terms like
this that give us the scaling
00:58:50.220 --> 00:58:51.660
cost of the equation.
00:58:51.660 --> 00:58:55.110
Often, what we saw in such
a problem, we want to know
00:58:55.110 --> 00:58:56.535
what is it scaling cost.
00:58:56.535 --> 00:58:59.070
That is, how much
our calculation
00:58:59.070 --> 00:59:01.740
becomes more expensive,
say, if we double
00:59:01.740 --> 00:59:04.050
the size of the system
because that basically
00:59:04.050 --> 00:59:06.660
tells us how large we can go.
00:59:06.660 --> 00:59:10.020
And because of
these integrals that
00:59:10.020 --> 00:59:13.320
involve three orbitals and the
fact that then to calculate
00:59:13.320 --> 00:59:16.620
an energy of one more
sum, really the scaling
00:59:16.620 --> 00:59:20.550
cost of the Hartree-Fock
equations, in principle,
00:59:20.550 --> 00:59:22.690
goes as the fourth power.
00:59:22.690 --> 00:59:25.470
So if I'm starting a
system with two electrons,
00:59:25.470 --> 00:59:28.530
and then I want to study a
system with four electrons,
00:59:28.530 --> 00:59:31.590
well, the cost has
gone up 16 times.
00:59:31.590 --> 00:59:37.630
And a fourth power, like any
power, kills you very rapidly.
00:59:37.630 --> 00:59:41.280
So that's why we can't study
a molecule like benzene
00:59:41.280 --> 00:59:42.390
with Hartree-Fock.
00:59:42.390 --> 00:59:45.000
But we can't study
really something
00:59:45.000 --> 00:59:48.300
like DNA with Hartree-Fock
or really none
00:59:48.300 --> 00:59:50.700
of the standard electronic
structure methods.
00:59:50.700 --> 00:59:53.970
And a lot of effort that
goes into developing
00:59:53.970 --> 00:59:57.600
a linear scaling
methods, that is methods
00:59:57.600 --> 01:00:00.510
in which the computational
cost of your calculation
01:00:00.510 --> 01:00:03.600
doubles if you double
the number of electrons.
01:00:03.600 --> 01:00:07.620
And at the end, nature
is linear scaling
01:00:07.620 --> 01:00:09.900
because really you can
imagine that the wave
01:00:09.900 --> 01:00:12.750
function of the electrons
here doesn't have anything
01:00:12.750 --> 01:00:15.360
to do with the wave function
of the electrons there.
01:00:15.360 --> 01:00:19.290
So there is really,
when you go far away, no
01:00:19.290 --> 01:00:22.590
exchange of information
between wave functions
01:00:22.590 --> 01:00:24.490
in different parts of space.
01:00:24.490 --> 01:00:28.200
And so there is ultimately
a linear scaling nature.
01:00:28.200 --> 01:00:32.920
But our algorithms, in
general, are not yet there.
01:00:32.920 --> 01:00:38.460
And we'll discuss a little bit
in some of the later classes
01:00:38.460 --> 01:00:39.660
how to solve this problem.
01:00:42.470 --> 01:00:49.710
Before going on, I wanted to
show one set of very simple
01:00:49.710 --> 01:00:54.840
conclusions actually of having
a wave function with the proper
01:00:54.840 --> 01:00:57.330
symmetry-- that is, having
a wave function written
01:00:57.330 --> 01:00:59.490
as a Slater determinant--
01:00:59.490 --> 01:01:04.350
because that form
gives us automatically
01:01:04.350 --> 01:01:06.120
what is called the
Pauli principle.
01:01:06.120 --> 01:01:08.340
If you remember what
the Pauli principle is,
01:01:08.340 --> 01:01:12.510
it's that you can't have two
fermions-- two electrons,
01:01:12.510 --> 01:01:16.060
in particular-- in the
same quantum state.
01:01:16.060 --> 01:01:22.200
So you can't have two electrons
having, say, the same orbital.
01:01:22.200 --> 01:01:25.590
And that's obvious
because, in a determinant,
01:01:25.590 --> 01:01:28.590
two electrons having
the same orbital
01:01:28.590 --> 01:01:33.640
would mean that two columns in
the determinant are identical.
01:01:33.640 --> 01:01:36.930
And when two columns in a
determinant are identical,
01:01:36.930 --> 01:01:40.170
the determinant
linearly dependent.
01:01:40.170 --> 01:01:43.290
And so the solution is 0.
01:01:43.290 --> 01:01:46.350
So a lot of good things
came out Hartree-Fock.
01:01:46.350 --> 01:01:49.680
In particular, one could
start solving atoms.
01:01:49.680 --> 01:01:53.140
And one would recover, say,
the shell structure of atoms.
01:01:53.140 --> 01:01:56.280
So if you would obtain
the Hartree-Fock solution
01:01:56.280 --> 01:01:58.710
for something like
an argon atom,
01:01:58.710 --> 01:02:02.650
and then say plot the overall
charge density of the system,
01:02:02.650 --> 01:02:08.710
well, it would start
to look like this as we
01:02:08.710 --> 01:02:11.860
move from the center outwards.
01:02:11.860 --> 01:02:14.760
So it would clearly
show the fundamentals
01:02:14.760 --> 01:02:16.630
of the periodic table
nature of things.
01:02:16.630 --> 01:02:20.170
That is, it would
show a 1s shell.
01:02:20.170 --> 01:02:23.920
And then it would show
a 2s and a 2p shell.
01:02:23.920 --> 01:02:26.772
And this is something that
some of the other approaches,
01:02:26.772 --> 01:02:28.480
like the Thomas Fermi
approach that we'll
01:02:28.480 --> 01:02:31.210
see in a moment that were
being developed at the time,
01:02:31.210 --> 01:02:32.620
didn't have.
01:02:32.620 --> 01:02:35.170
In general, Hartree-Fock
is very good
01:02:35.170 --> 01:02:39.130
to describe atomic properties.
01:02:39.130 --> 01:02:45.940
And what is most important is
a well-defined approximation
01:02:45.940 --> 01:02:48.070
in the variational principle.
01:02:48.070 --> 01:02:50.740
Remember, one of the
fundamental powers
01:02:50.740 --> 01:02:53.050
of the variational
principle is that if we
01:02:53.050 --> 01:02:57.280
make our wave function
ansatz, our [INAUDIBLE] wave
01:02:57.280 --> 01:03:01.580
function more and more flexible,
we become better and better.
01:03:01.580 --> 01:03:05.800
So Hartree-Fock is
a certain hypothesis
01:03:05.800 --> 01:03:07.780
and gives us certain energies.
01:03:07.780 --> 01:03:11.560
But what we can do is
make our wave functions
01:03:11.560 --> 01:03:13.510
more and more flexible--
01:03:13.510 --> 01:03:17.740
write them not only just
as a single determinant,
01:03:17.740 --> 01:03:21.530
but a single determinant
plus something else.
01:03:21.530 --> 01:03:25.150
And the solution that we'll find
will be computationally more
01:03:25.150 --> 01:03:28.430
expensive to find, but
is going to be better.
01:03:28.430 --> 01:03:34.150
So Hartree-Fock, in principle,
can be improved indefinitely.
01:03:34.150 --> 01:03:37.420
That is something very
powerful conceptually.
01:03:37.420 --> 01:03:40.000
It's practically very
complex because those
01:03:40.000 --> 01:03:42.110
costs that we are
scaling already
01:03:42.110 --> 01:03:44.570
in a simple Hartree-Fock,
like the fourth power,
01:03:44.570 --> 01:03:46.390
keep going up.
01:03:46.390 --> 01:03:49.210
What you will see in
this differential theory
01:03:49.210 --> 01:03:51.820
is that besides
being a theory that
01:03:51.820 --> 01:03:54.460
tends to give more
accurate results
01:03:54.460 --> 01:03:56.440
for a lot of
physical properties,
01:03:56.440 --> 01:03:58.750
it's something that also
scales a bit better.
01:03:58.750 --> 01:04:02.380
It scales as the third
power of the size.
01:04:02.380 --> 01:04:04.570
But it's a theory
that can't really be
01:04:04.570 --> 01:04:07.390
improved in any systematic way.
01:04:07.390 --> 01:04:10.840
One can find ingenious
ways to make it better.
01:04:10.840 --> 01:04:15.160
But there isn't a brute
force improvement strategy
01:04:15.160 --> 01:04:19.310
like there is in Hartree-Fock.
01:04:19.310 --> 01:04:23.060
The Hartree-Fock operator
included the last term
01:04:23.060 --> 01:04:25.760
that we have called
the exchange term.
01:04:25.760 --> 01:04:29.450
And so for every possible atom,
for every possible molecule,
01:04:29.450 --> 01:04:33.800
for every system, there is
a well-defined Hartree-Fock
01:04:33.800 --> 01:04:34.610
energy.
01:04:34.610 --> 01:04:38.600
And this Hartree-Fock energy is
going to be good, or very good,
01:04:38.600 --> 01:04:39.830
or sort of so-so.
01:04:39.830 --> 01:04:44.360
But it's always going to be
higher than the true ground
01:04:44.360 --> 01:04:46.490
state energy of our system.
01:04:46.490 --> 01:04:48.830
And actually, what
is technically
01:04:48.830 --> 01:04:52.760
called the correlation
energy is the difference
01:04:52.760 --> 01:04:55.790
between the true energy of your
system and the Hartree-Fock
01:04:55.790 --> 01:04:56.880
energy.
01:04:56.880 --> 01:04:59.210
So when people talk
about correlation energy,
01:04:59.210 --> 01:05:03.830
they refer to all the
energy that is not captured
01:05:03.830 --> 01:05:06.260
by a Hartree-Fock approach.
01:05:06.260 --> 01:05:09.500
And in that sense, it's
a well-defined quantity.
01:05:09.500 --> 01:05:12.860
Although, it involves a
generic term correlation
01:05:12.860 --> 01:05:14.240
that can mean a lot of things.
01:05:14.240 --> 01:05:16.175
And I'll show you
a few examples.
01:05:20.700 --> 01:05:23.880
There is one more
thing that we can do.
01:05:23.880 --> 01:05:27.600
We have never discussed
up to now spin.
01:05:27.600 --> 01:05:31.170
But in reality, an
electron is described
01:05:31.170 --> 01:05:35.080
by a wave function that
doesn't have only space parts.
01:05:35.080 --> 01:05:37.200
So in order to
describe an electron,
01:05:37.200 --> 01:05:40.815
we don't only describe what
the distribution of its wave
01:05:40.815 --> 01:05:43.530
function is in
space, but we also
01:05:43.530 --> 01:05:47.700
specify what is the
spin of the electron.
01:05:47.700 --> 01:05:51.030
And that has to do
basically with spin
01:05:51.030 --> 01:05:55.470
being an operator that can be
simultaneously diagonalized
01:05:55.470 --> 01:05:57.180
with a set of--
01:05:57.180 --> 01:05:58.830
well, it becomes complex.
01:05:58.830 --> 01:06:00.570
But it's another quantity.
01:06:00.570 --> 01:06:02.280
You can think of it as a color.
01:06:02.280 --> 01:06:04.770
We need to specify if our
electron is red or blue.
01:06:04.770 --> 01:06:06.990
Or in particular,
we need to specify
01:06:06.990 --> 01:06:10.830
what is its spin, what is
its projection with respect
01:06:10.830 --> 01:06:12.310
to an axis.
01:06:12.310 --> 01:06:14.635
And so in this,
you need to think
01:06:14.635 --> 01:06:17.040
of a wave function
of an electron
01:06:17.040 --> 01:06:20.580
not having only a
spatial distribution.
01:06:20.580 --> 01:06:25.000
But it has another property
besides the spatial variables
01:06:25.000 --> 01:06:26.700
that is called the spin.
01:06:26.700 --> 01:06:30.930
And you can make a
sort of approximation--
01:06:30.930 --> 01:06:35.650
that is what is called a
restricted Hartree-Fock
01:06:35.650 --> 01:06:36.510
scheme--
01:06:36.510 --> 01:06:41.550
in which you decide
that an electron of spin
01:06:41.550 --> 01:06:44.730
up and an electron
of spin down will
01:06:44.730 --> 01:06:48.360
have the same spatial
part, so the same wave
01:06:48.360 --> 01:06:49.980
function in space.
01:06:49.980 --> 01:06:52.920
And their wave
function differs only
01:06:52.920 --> 01:06:55.620
because you describe an
electron with spin up
01:06:55.620 --> 01:06:57.480
and an electron with spin down.
01:06:57.480 --> 01:07:00.630
Again, this corresponds to
the classical periodic table
01:07:00.630 --> 01:07:01.300
picture.
01:07:01.300 --> 01:07:03.420
You are constructing
the periodic table.
01:07:03.420 --> 01:07:07.650
You go, say, from hydrogen--
one electron in the 1s level--
01:07:07.650 --> 01:07:12.210
to helium-- one electron in
the 1s level with spin up
01:07:12.210 --> 01:07:16.110
and another electron in the
same level with the spin down.
01:07:16.110 --> 01:07:19.440
Actually, if you think,
the periodic table
01:07:19.440 --> 01:07:22.320
itself is not a truth.
01:07:22.320 --> 01:07:27.390
It's just a Hartree-Fock
picture of electrons, OK?
01:07:27.390 --> 01:07:31.590
In principle, you shouldn't
be able to talk about
01:07:31.590 --> 01:07:33.750
single-particle quantities--
01:07:33.750 --> 01:07:37.950
1s, 2s quantities-- because,
in reality, if you have iron,
01:07:37.950 --> 01:07:40.290
you have a many-body
wave function
01:07:40.290 --> 01:07:45.220
that is an overall function
of all the electrons.
01:07:45.220 --> 01:07:49.200
It's only when you enter
into a Hartree-Fock picture
01:07:49.200 --> 01:07:53.730
that you can have a well-defined
concept as a single orbital
01:07:53.730 --> 01:07:57.120
for an electron and what's
the energy for that electron.
01:07:57.120 --> 01:07:59.700
So if you want, what we
think of this beautiful thing
01:07:59.700 --> 01:08:03.370
as the periodic table is nothing
else than the Hartree-Fock
01:08:03.370 --> 01:08:06.000
solution of the atoms.
01:08:06.000 --> 01:08:08.490
And again, we can
make the approximation
01:08:08.490 --> 01:08:13.740
in which we fill up every
orbital, every spatial part
01:08:13.740 --> 01:08:16.170
with two electrons
with the same spin.
01:08:16.170 --> 01:08:18.899
That tends to be a
very good approximation
01:08:18.899 --> 01:08:23.220
for a lot of problems, say
a lot of bound systems.
01:08:23.220 --> 01:08:27.630
And we'll see the case of the
hydrogen molecule in a moment.
01:08:27.630 --> 01:08:32.399
But you could actually make your
wave function more flexible,
01:08:32.399 --> 01:08:36.630
saying that say orbitals
don't need to be paired.
01:08:36.630 --> 01:08:39.270
That is, an electron with a spin
up and an electron with a spin
01:08:39.270 --> 01:08:41.500
down, even if they are
very close in energy,
01:08:41.500 --> 01:08:46.380
can have two wave functions in
which the space part differs.
01:08:46.380 --> 01:08:55.830
And this is really an ansatz
that contains this in itself.
01:08:55.830 --> 01:08:59.300
So an unrestricted
Hartree-Fock solution
01:08:59.300 --> 01:09:01.880
will always give
you a lower energy
01:09:01.880 --> 01:09:04.100
than a restricted solution.
01:09:04.100 --> 01:09:06.740
And we'll see in a
moment an example.
01:09:06.740 --> 01:09:10.340
And this is the case of the
dissociation of a hydrogen
01:09:10.340 --> 01:09:11.609
molecule.
01:09:11.609 --> 01:09:17.899
So when we go back and try to
understand what is the bonding
01:09:17.899 --> 01:09:20.029
between, say, two
hydrogen atoms--
01:09:20.029 --> 01:09:22.250
and we had seen in one
of the first lectures,
01:09:22.250 --> 01:09:25.160
we discussed about potentials--
01:09:25.160 --> 01:09:29.600
that is, what is the energy
of a system as a function
01:09:29.600 --> 01:09:31.479
of the nuclear distances.
01:09:31.479 --> 01:09:32.960
And this is what
we are doing here.
01:09:32.960 --> 01:09:35.569
We are trying to look at
the energy of the system
01:09:35.569 --> 01:09:38.720
as a function of the
hydrogen-hydrogen distance.
01:09:38.720 --> 01:09:41.240
And there will be an equilibrium
distance that corresponds
01:09:41.240 --> 01:09:42.979
to the minimum of the energy.
01:09:42.979 --> 01:09:45.500
And this is what classical
potential tried to do.
01:09:45.500 --> 01:09:47.960
They tried to replicate
what is the energy
01:09:47.960 --> 01:09:50.420
of a system as a function
of the nuclear coordinates.
01:09:50.420 --> 01:09:52.850
And they tend to do very
well, as Professor Ceder has
01:09:52.850 --> 01:09:58.020
told you, closer to the region
where they have been fitted.
01:09:58.020 --> 01:10:00.650
If we have created a
potential around here,
01:10:00.650 --> 01:10:03.140
we tend to be able to
reproduce things very well.
01:10:03.140 --> 01:10:05.730
Obviously, it's very
easy to even find out
01:10:05.730 --> 01:10:08.420
the potential that
reproduced all these curves.
01:10:08.420 --> 01:10:11.750
But when you start to have more
than two atoms interacting,
01:10:11.750 --> 01:10:14.960
there are all these problems of
bond-breaking and bond-forming
01:10:14.960 --> 01:10:18.500
that can't really be given
by classical potential
01:10:18.500 --> 01:10:22.670
and can be given by quantum
mechanical calculations.
01:10:22.670 --> 01:10:25.730
And so in principle, we
have an ideal solution,
01:10:25.730 --> 01:10:27.620
that is if we were able.
01:10:27.620 --> 01:10:30.110
And nowadays, with
numerical accuracy,
01:10:30.110 --> 01:10:33.290
we have basically been able
to solve almost perfectly
01:10:33.290 --> 01:10:33.930
this problem.
01:10:33.930 --> 01:10:36.860
We are able to find out
what is the total energy
01:10:36.860 --> 01:10:41.060
of the system as a function
of the nuclear coordinates.
01:10:41.060 --> 01:10:43.280
And then for this
specific problem,
01:10:43.280 --> 01:10:46.880
the Coulombic potentials
at different distances,
01:10:46.880 --> 01:10:50.210
we can find a
Hartree-Fock solution.
01:10:50.210 --> 01:10:53.450
And with only two
electrons, we can
01:10:53.450 --> 01:10:55.730
find a restricted
Hartree-Fock solution
01:10:55.730 --> 01:10:58.820
in which we say, well,
these two electrons are
01:10:58.820 --> 01:11:03.530
going to have the same orbital
part in the wave function.
01:11:03.530 --> 01:11:08.000
They just differ in having a
spin up and spin down term.
01:11:08.000 --> 01:11:10.130
And that makes them orthogonal.
01:11:10.130 --> 01:11:13.970
So certain things are going to
happen in the exchange term.
01:11:13.970 --> 01:11:19.010
But then we plotted this energy
as a function of the distance.
01:11:19.010 --> 01:11:21.590
And this is what we have.
01:11:21.590 --> 01:11:24.850
And then we can
release this condition.
01:11:24.850 --> 01:11:27.940
We can say these
two electrons don't
01:11:27.940 --> 01:11:32.000
need to have the same orbital
part for the wave functions.
01:11:32.000 --> 01:11:33.970
They can have different parts.
01:11:33.970 --> 01:11:36.310
And we can do that calculation.
01:11:36.310 --> 01:11:39.730
And what we obtain is the
unrestricted Hartree-Fock
01:11:39.730 --> 01:11:41.620
solution.
01:11:41.620 --> 01:11:45.820
And you see two fundamental
things coming out from here.
01:11:45.820 --> 01:11:50.250
First of all is that all
these Hartree-Fock approaches
01:11:50.250 --> 01:11:52.080
give you an energy
that is obviously
01:11:52.080 --> 01:11:54.540
larger than the exact energy.
01:11:54.540 --> 01:11:56.280
It's the variational principle.
01:11:56.280 --> 01:11:58.560
We can get lower and lower.
01:11:58.560 --> 01:12:01.170
But we will never be able--
and that's very good--
01:12:01.170 --> 01:12:04.350
to go below the true energy.
01:12:04.350 --> 01:12:07.380
So the more flexible we
make our wave function
01:12:07.380 --> 01:12:10.570
with both Hartree-Fock
methods, the more
01:12:10.570 --> 01:12:14.010
we'll be able to recover
this last electron
01:12:14.010 --> 01:12:16.350
volt of correlation energy.
01:12:16.350 --> 01:12:19.530
So this is all where
our effort is going.
01:12:19.530 --> 01:12:21.780
But you see,
Hartree-Fock is already
01:12:21.780 --> 01:12:26.520
doing extremely well in giving
us the equilibrium distance.
01:12:26.520 --> 01:12:28.770
I mean, this is the Hartree-Fock
equilibrium distance.
01:12:28.770 --> 01:12:30.920
And this is the exact
equilibrium distance.
01:12:30.920 --> 01:12:33.550
So it's doing a good job.
01:12:33.550 --> 01:12:37.290
What restricted Hartree-Fock
is not doing well
01:12:37.290 --> 01:12:40.770
is giving us the
dissociation energy.
01:12:40.770 --> 01:12:46.470
So restricted Hartree-Fock works
very well around the minimum.
01:12:46.470 --> 01:12:48.330
Well, you really
should have this sort
01:12:48.330 --> 01:12:50.400
of physical picture
of your ground
01:12:50.400 --> 01:12:54.960
state being given by a bonding
combination of 1s orbitals.
01:12:54.960 --> 01:12:57.870
Really, this is what the
covalent bond for a hydrogen
01:12:57.870 --> 01:13:02.820
molecule is, is the two
1s orbitals covalently
01:13:02.820 --> 01:13:04.330
overlapping.
01:13:04.330 --> 01:13:07.170
And so restricted Hartree-Fock
does very well here.
01:13:07.170 --> 01:13:11.430
And it's basically identical
to unrestricted Hartree-Fock.
01:13:11.430 --> 01:13:16.290
But formally, unrestricted will
always be lower than restricted
01:13:16.290 --> 01:13:20.910
because it contains the
restricted solution because,
01:13:20.910 --> 01:13:24.690
in order to have a
restricted solution,
01:13:24.690 --> 01:13:27.780
you just need to have the
orbital part for the two
01:13:27.780 --> 01:13:29.190
electrons to be identical.
01:13:29.190 --> 01:13:31.570
But because it can
also be different,
01:13:31.570 --> 01:13:35.430
it will always be lower than
the restricted Hartree-Fock.
01:13:35.430 --> 01:13:38.850
And you see, when we
break this system apart--
01:13:38.850 --> 01:13:45.720
when we want to go from a bound
hydrogen molecule to two atoms,
01:13:45.720 --> 01:13:49.380
the restricted Hartree-Fock
is doing very poorly.
01:13:49.380 --> 01:13:52.590
You'll see, it'll
give us an energy
01:13:52.590 --> 01:13:56.800
that is a very poor predictor
of the dissociation energy.
01:13:56.800 --> 01:13:59.880
The dissociation energy-- the
true disassociation energy
01:13:59.880 --> 01:14:02.610
of the system is the
distance between the minimum
01:14:02.610 --> 01:14:04.428
here and the 0 value.
01:14:04.428 --> 01:14:05.970
That this is, it's
the energy that we
01:14:05.970 --> 01:14:08.640
need to spend to break
apart the molecule.
01:14:08.640 --> 01:14:12.520
Unrestricted Hartree-Fock
will do very well.
01:14:12.520 --> 01:14:15.570
I mean, obviously, it goes
to 0 in this scale when
01:14:15.570 --> 01:14:16.560
we are far apart.
01:14:16.560 --> 01:14:19.110
And so we have this 1
electron volt error.
01:14:19.110 --> 01:14:22.110
But the restricted Hartree-Fock
is doing very poorly.
01:14:22.110 --> 01:14:23.010
And why is that?
01:14:23.010 --> 01:14:25.680
Well, basically because
the restricted Hartree-Fock
01:14:25.680 --> 01:14:30.600
is really doubly occupying
the same spatial part
01:14:30.600 --> 01:14:33.460
of the same bonding combination.
01:14:33.460 --> 01:14:35.670
And that's good when
the system is bound.
01:14:35.670 --> 01:14:37.650
But when you break
it apart, it's
01:14:37.650 --> 01:14:41.670
very poor because what
you really want is--
01:14:41.670 --> 01:14:47.190
in your solution, you want to
mix in another determinant that
01:14:47.190 --> 01:14:50.640
is given by the
anti-bonding combination
01:14:50.640 --> 01:14:54.340
because if you think
of the bonding state
01:14:54.340 --> 01:14:58.950
as always a pile-up of
charge in between the atoms.
01:14:58.950 --> 01:15:01.110
But when the molecule
disassociates,
01:15:01.110 --> 01:15:05.070
you really want to have a
solution that has zero charge
01:15:05.070 --> 01:15:06.840
density between the atoms.
01:15:06.840 --> 01:15:10.740
And that looks much more like
the anti-bonding combination
01:15:10.740 --> 01:15:12.880
of 1s orbitals.
01:15:12.880 --> 01:15:14.310
And so the restricted
Hartree-Fock
01:15:14.310 --> 01:15:15.880
doesn't have this freedom.
01:15:15.880 --> 01:15:17.790
So it does it very poorly.
01:15:17.790 --> 01:15:21.240
And unrestricted has
the freedom of having
01:15:21.240 --> 01:15:24.750
two orbitals that are different
for the two electrons.
01:15:24.750 --> 01:15:28.950
And so it just puts one orbital
on one atom and another orbital
01:15:28.950 --> 01:15:32.970
in another atom instead of
having a single orbital doubly
01:15:32.970 --> 01:15:34.440
occupied.
01:15:34.440 --> 01:15:38.710
And so unrestricted Hartree-Fock
is going to be much better,
01:15:38.710 --> 01:15:44.100
especially for problems like
the bond-breaking reactions
01:15:44.100 --> 01:15:51.530
or for problems in which
you have isolated spins,
01:15:51.530 --> 01:15:53.450
so you have atoms--
01:15:53.450 --> 01:15:58.735
you have single electrons
that are not paired.
01:15:58.735 --> 01:16:00.360
There are actually
a number of theorems
01:16:00.360 --> 01:16:03.000
that can be derived from
the Hartree-Fock equation.
01:16:03.000 --> 01:16:04.410
I won't dwell on them.
01:16:04.410 --> 01:16:08.220
They are generically
called Koopmans' theorems.
01:16:08.220 --> 01:16:11.100
And they have to do
with calculations, say,
01:16:11.100 --> 01:16:14.010
of quantities like
the ionization energy
01:16:14.010 --> 01:16:15.510
or the electron affinity.
01:16:15.510 --> 01:16:16.890
What is the initiation energy?
01:16:16.890 --> 01:16:20.250
It's the energy that you
need to spend to remove
01:16:20.250 --> 01:16:22.140
an electron from an atom.
01:16:22.140 --> 01:16:24.960
Or the electron
affinity is the energy
01:16:24.960 --> 01:16:28.620
that you gain when an
electron captures--
01:16:28.620 --> 01:16:31.930
sorry, when an atom
captures an extra electron.
01:16:31.930 --> 01:16:34.740
So how do you calculate them,
say, in an electronic structure
01:16:34.740 --> 01:16:35.580
calculation?
01:16:35.580 --> 01:16:37.150
Say the ionization energy?
01:16:37.150 --> 01:16:40.020
Well, it'll just be given
by the difference, say
01:16:40.020 --> 01:16:42.900
for the case of iron
atoms, of the Hartree-Fock
01:16:42.900 --> 01:16:45.960
solution with 26 electrons
and the Hartree-Fock
01:16:45.960 --> 01:16:48.210
solution with 25 electrons.
01:16:48.210 --> 01:16:49.770
So you do these
two calculations.
01:16:49.770 --> 01:16:51.090
You take the difference.
01:16:51.090 --> 01:16:53.340
And that will be our
ionization energy.
01:16:53.340 --> 01:16:54.990
And the affinity will
be the difference
01:16:54.990 --> 01:16:58.920
between the calculation
with 27 or 26 electrons.
01:16:58.920 --> 01:17:02.130
But you can actually
do very well
01:17:02.130 --> 01:17:05.670
without having to
do two calculations,
01:17:05.670 --> 01:17:11.220
but having just one calculation,
if you make the hypothesis
01:17:11.220 --> 01:17:14.670
that really your
single-particle electrons do not
01:17:14.670 --> 01:17:16.000
change in the process.
01:17:16.000 --> 01:17:17.580
So if you make the
hypothesis that
01:17:17.580 --> 01:17:22.050
in going from 26 to 25 the
shape of electron 1, electron 2,
01:17:22.050 --> 01:17:24.090
electron 3 do not change--
01:17:24.090 --> 01:17:25.290
and that's an approximation.
01:17:25.290 --> 01:17:27.070
They will change a little bit.
01:17:27.070 --> 01:17:29.100
But if you make
this approximation,
01:17:29.100 --> 01:17:32.700
you can actually prove
that the difference
01:17:32.700 --> 01:17:36.030
between the system
with 26 electrons
01:17:36.030 --> 01:17:38.160
and the system
with 25 electrons--
01:17:38.160 --> 01:17:40.890
the difference in
energy is just given
01:17:40.890 --> 01:17:44.460
by the eigenvalue of
the 26th electron.
01:17:44.460 --> 01:17:46.950
So basically, a
single calculation
01:17:46.950 --> 01:17:50.670
gives you already an estimate
of ionization energies
01:17:50.670 --> 01:17:52.063
and electron affinities.
01:17:52.063 --> 01:17:53.730
Although, in principle,
you could always
01:17:53.730 --> 01:17:55.200
do two calculations.
01:17:55.200 --> 01:17:58.830
But these, if you find them, are
called the Koopmans' theorems.
01:18:01.360 --> 01:18:03.520
What is missing in Hartree-Fock?
01:18:03.520 --> 01:18:06.430
What is this correlation that
we are trying to recover?
01:18:06.430 --> 01:18:10.030
Well, often we think
at it in two ways.
01:18:10.030 --> 01:18:14.710
We can think that part of
it is dynamical correlation.
01:18:14.710 --> 01:18:18.040
It's what I described to you
in the case of the helium atom.
01:18:18.040 --> 01:18:21.370
That is, when we have two
electrons interacting,
01:18:21.370 --> 01:18:25.420
they like to keep each other
as far away as possible
01:18:25.420 --> 01:18:27.670
from each other instantaneously.
01:18:27.670 --> 01:18:30.130
And because we have a
mean field solution,
01:18:30.130 --> 01:18:33.550
we are actually overestimating
the Hartree energy.
01:18:33.550 --> 01:18:36.670
We tend to put electrons
too close to each other
01:18:36.670 --> 01:18:38.830
because we have one
electron interacting
01:18:38.830 --> 01:18:41.210
with the average
field of the other.
01:18:41.210 --> 01:18:43.797
And so in that
energy term, you have
01:18:43.797 --> 01:18:45.380
that there are a lot
of configurations
01:18:45.380 --> 01:18:48.040
in which the electrons are
too close to each other.
01:18:48.040 --> 01:18:50.360
That raises the
energy of your system.
01:18:50.360 --> 01:18:52.690
So we call that
dynamical correlation.
01:18:52.690 --> 01:18:55.390
And these are heuristic terms.
01:18:55.390 --> 01:18:58.840
And then there is
another class of errors
01:18:58.840 --> 01:19:02.980
that we are making that often
are called static correlations.
01:19:02.980 --> 01:19:05.560
And those have to do
more with the fact
01:19:05.560 --> 01:19:08.620
that a single determinant
solution doesn't have
01:19:08.620 --> 01:19:10.220
the flexibility that you need.
01:19:10.220 --> 01:19:13.630
And this was the case of the
breaking apart of the hydrogen
01:19:13.630 --> 01:19:14.560
molecule.
01:19:14.560 --> 01:19:19.660
You really want, when you you're
breaking apart a molecule,
01:19:19.660 --> 01:19:22.690
to have a two determinant
kind of flexibility
01:19:22.690 --> 01:19:26.650
with both bonding and
anti-bonding combinations.
01:19:26.650 --> 01:19:28.730
And all of this is missing.
01:19:28.730 --> 01:19:33.310
And we can systematically
build it up,
01:19:33.310 --> 01:19:37.690
improving, say, the flexibility
of the wave function.
01:19:37.690 --> 01:19:42.520
And one of the
conceptually simplest way,
01:19:42.520 --> 01:19:45.340
but computationally
more expensive ways,
01:19:45.340 --> 01:19:48.430
is actually to look
at the wave function
01:19:48.430 --> 01:19:52.210
that now, instead of being
given by a single determinant,
01:19:52.210 --> 01:19:55.780
is given by a combination
of determinants
01:19:55.780 --> 01:19:59.830
with different coefficients in
which, say, the determinants
01:19:59.830 --> 01:20:05.230
have been constructed with
a number of orbitals that
01:20:05.230 --> 01:20:07.890
include also excited orbitals.
01:20:07.890 --> 01:20:11.050
Our original Hartree-Fock
determinant, say,
01:20:11.050 --> 01:20:15.010
for iron was given by
the 26th lowest solution.
01:20:15.010 --> 01:20:16.060
And this is it.
01:20:16.060 --> 01:20:17.920
But then you could
add a second term
01:20:17.920 --> 01:20:22.450
that contains 25 of those 26
lower solutions and then one
01:20:22.450 --> 01:20:23.380
excited state.
01:20:23.380 --> 01:20:25.160
Or you could do a
number of things.
01:20:25.160 --> 01:20:27.790
But basically, you could
increase the variation
01:20:27.790 --> 01:20:29.710
of flexibility of your problem.
01:20:29.710 --> 01:20:32.020
And the more
flexible you become,
01:20:32.020 --> 01:20:34.370
the closer you get to
the right solution.
01:20:34.370 --> 01:20:36.850
But you pay an enormous
price for this.
01:20:36.850 --> 01:20:39.460
And this general
approach would be called
01:20:39.460 --> 01:20:41.110
configuration interaction.
01:20:41.110 --> 01:20:45.040
That has actually this horrific
scaling of n to the 7th.
01:20:45.040 --> 01:20:47.260
And so you can really
do it for 10 electrons.
01:20:47.260 --> 01:20:50.410
But you can't do it for
11 electrons on your best
01:20:50.410 --> 01:20:51.000
computer.
01:20:51.000 --> 01:20:55.120
Probably now, we can get
to 15 electrons or so.
01:20:55.120 --> 01:20:57.460
And I think with
this, I'll conclude.
01:20:57.460 --> 01:21:00.980
So this was a panorama
on Hartree-Fock methods.
01:21:00.980 --> 01:21:02.860
One of the best books
is the Jensen book
01:21:02.860 --> 01:21:05.380
of computational chemistry
in the literature.
01:21:05.380 --> 01:21:07.120
And then with the
next class, we'll
01:21:07.120 --> 01:21:09.880
start looking at density
functional theory and again
01:21:09.880 --> 01:21:14.560
the Nobel Prize for
chemistry in 1998.
01:21:14.560 --> 01:21:19.720
Next Tuesday is-- being
Monday Presidents' Day,
01:21:19.720 --> 01:21:23.060
we'll have on Tuesday
Monday's schedule of classes.
01:21:23.060 --> 01:21:25.720
That means, in practice,
that we have no class.
01:21:25.720 --> 01:21:28.720
So you have time until
Thursday of next week
01:21:28.720 --> 01:21:29.950
to brood over this.
01:21:29.950 --> 01:21:34.120
I've posted a few
readings on the website.
01:21:34.120 --> 01:21:37.300
If you are really
wanting to more
01:21:37.300 --> 01:21:40.030
about the functional theory,
one of those readings
01:21:40.030 --> 01:21:42.700
is the Kohanoff
paper on fundamentals
01:21:42.700 --> 01:21:44.230
of density financial theory.
01:21:44.230 --> 01:21:46.690
And if you want to know
more of Hartree-Fock,
01:21:46.690 --> 01:21:50.090
we won't really see any more
in the rest of the class.
01:21:50.090 --> 01:21:52.360
You should go to one
of the references.
01:21:52.360 --> 01:21:55.380
And otherwise, see
you next Thursday.