D.54 Derivation of the Hartree-Fock equations

This note derives the canonical Hartree-Fock equations. It will use
some linear algebra; see the Notations section under
matrix

for some basic concepts. The derivation will
be performed under the normally stated rules of engagement that the
orbitals are of the form

The derivations must allow for the fact that in restricted Hartree-Fock, it is required that pairs of spin-up and spin-down orbitals have the same spatial orbital. So there are three possible kinds of spatial orbitals. A spatial orbital may produce a single unpaired spin orbital that is spin-up, or a single unpaired spin orbital that is spin-down, or a pair of spin-up and spin-down orbitals with the same spatial orbital. These three types of spatial orbitals will be referred to as unpaired spin-up, unpaired spin-down, and restricted. Note that these names do not refer to properties of the spatial orbits themselves, of course, but to the properties of the spin orbits that these spatial orbitals produce.

Assume that there are

and that is the total number of unknown spatial orbitals to find. A corresponding number of

However, the total number of spin orbitals,

Things become a bit easier if the ordering of the orbitals is specified a priori. The ordering makes no difference physically. So it will be assumed that the spatial orbitals are ordered with the unpaired spin-up ones first, the unpaired spin-down ones second, and the restricted ones last. The ordering of the spin orbitals will be the same as that of the spatial orbitals, but with the restricted orbitals at the end appearing twice; first in the spin-up versions and then in the spin-down versions.

To find the spatial orbitals, the variational method as discussed in
chapter 9.1.3 says that the expectation energy

To do so, first note that

This condition is real too. However, the condition that any spatial mode

In general this condition has both a real and an imaginary component. But it can be written as two real conditions;

The reason is that if you swap the sides in an inner product, you get the complex conjugate; therefore the first equation above is the real part of the inner product and the second the imaginary part.

Since we now have a completely real problem in real independent
variables, the penalty factors (the Lagrangian multipliers) in the
problem will be real too. For reasons evident in a second, the
penalty factor for the normalization condition above will be called

The reason for these notations is that in terms of them, the penalized
variational condition that the spatial orbitals must satisfy, chapter
9.1.3, takes the simple form

where

Note however that two spatial orbitals do not have to be orthogonal if
one is a unpaired spin-up one and the other an unpaired spin-down one.
In that case the spins take care of orthogonality. This can be
accomodated by stipulating that the penalty factors of the
corresponding constraints are zero,

Next the variational condition is to be evaluated for a small change

(From here on, the argument of the first orbital of a pair in either side of an inner product is taken to be the first inner product integration variable

Taking that into account, the variational condition for the

Here

Note that the difference between

The second term in each row in the expression above is just the
complex conjugate of the first. These second terms can be thrown out
using the same trick as in chapter 9.1.3. (In other
words, average with the same equation with

Now write out the inner product over the first position coordinate

If this integral is to be zero for whatever is

Unavoidably then, the following equations, one for each value of

This can be cleaned up a bit by dividing by [2?]:

(D.35) |

These are the general Hartree-Fock equations, one for each

Note that the general Hartree-Fock equation above includes eigenvalues

The restricted closed-shell Hartree-Fock case will be done first,
since it is the easiest one. Every spatial orbital is restricted, so
the lower choice in the curly brackets always applies. The summation
upper limits

Now the reason why all these

Each orbital in the special set will be some combination of the
orbitals in the typical set above. In particular, any orbital in the
special set, call it

where the numbers

matrix,a table of numbers. This matrix will be indicated by

The multiples

where

Substituting in the expression for the special orbitals above, making
sure not to use the same name

or noting that numbers come out of the left side of an inner product as complex conjugates,

Now since the set of typical orbitals

What does that mean? Well, for given values of

unit,and that keeps mathematicians in business explaining all the confusion.

The Hermitian adjoint matrix

where

This can be used to find the typical orbitals in terms of the special
ones. To do so, premultiply the expression for the special orbitals
as given earlier by

As seen above, the sum over

That then gives any typical orbital

Now plug that into the non canonical restricted closed-shell
Hartree-Fock equations given earlier. Be careful not to use the same
summation index name twice in the same term; this derivation will use

for

Note that the only thing that has changed more than just by symbol
names is the matrix in the right hand side. Now for each separate
value of

So the right hand side becomes

So, in terms of the special orbitals defined by the requirement that

Since the old typical orbitals are no longer of interest, the
overlines on the special orbitals can be dropped to save typing, and
the Greek index names

(D.36) |

Turning now to the case of (fully) unrestricted Hartree-Fock (UHF),
you might make the same simple argument as above and be done. But it
is worthwhile to go through the full mathematics anyway, to better
understand open-shell restricted Hartree-Fock later. In the
unrestricted case, the non canonical equations are

In this case, there are two different types of spatial orbitals; those appearing in spin-up spin orbitals, and those appearing in spin-down spin orbitals. You cannot just make arbitrary combinations of all these orbitals. If you combine spin-up and spin-down orbitals, they correspond to spin orbitals of uncertain spin. That would make the assumptions used to derive the Hartree-Fock equations invalid.

However, combinations of purely spin-up orbitals can still be made
without problems, and so can combinations of purely spin down
orbitals. To do the mathematics, the spatial orbitals can be
separated into two sets. The set of orbital numbers

and equations for

In these two types of equations, the fact that the up and down spin states are orthogonal was used to get rid of one pair of sums, and another pair was eliminated by the fact that there are no Lagrangian variables

Now separately replace the orbitals of the up and down states by a
modified set just like for the restricted closed-shell case above, for
each using the unitary matrix of eigenvectors of the

(D.37) |

That leaves only the restricted open-shell Hartree-Fock method. Here,
the partitioning also needs to include the set R of of restricted
orbitals besides U and D. There is now a problem, because you cannot
make combinations of restricted orbitals with spin-up or spin-down
orbitals. That means that the

Woof.