D.54 Derivation of the Hartree-Fock equations

This note derives the canonical Hartree-Fock equations. The
derivation below will be performed under the normally stated rules of
engagement that the orbitals are of the form

First, you can make things a lot less messy by a priori specifying the ordering of the orbitals. The ordering makes no difference physically, and it simplifies the mathematics. In particular, in restricted Hartree-Fock some spatial orbitals appear in pairs, but you can only count each spatial orbital as one unknown function. The easiest way to handle that is to push the spin-down versions of the duplicated spatial orbitals to the end of the Slater determinant. Then the start of the determinant comprises a list of unique spatial orbitals.

So, it will be assumed that the orbitals are ordered as follows:

- 1.
- the paired spatial states in their spin-up version; assume
there are
0 of them; - 2.
- unpaired spin-up states; assume there are
of them; - 3.
- unpaired spin-down states; assume there are
of them; - 4.
- and finally, the paired spatial states in their spin-down version.

The total number of unknown spatial orbitals is

The variational method discussed in chapter 9.1 says that the expectation energy must be unchanged under small changes in the orbitals, provided that penalty terms are added for changes that violate the orthonormality requirements on the orbitals.

The expectation value of energy was in chapter 9.3.3 found to be:

(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

The penalty terms require penalty factors called Lagrangian variables.
The penalty factor for violations of the normalization requirement

will be called

where the first constraint says that the real part of

In those terms, the penalized change in expectation energy becomes, in
the restricted case that all unique spatial orbitals are mutually
orthogonal,

where

But for unrestricted Hartree-Fock, spatial orbitals are not required
to be orthogonal if they have opposite spin, because the spins will
take care of orthogonality. You can remove the erroneously added
constraints by simply specifying that the corresponding Lagrangian
variables are zero:

or equivalently, if

Now work out the penalized change in expectation energy due to a
change in the values of a selected spatial orbital

OK, OK it is a mess. Sums like

Next, note that the second term in each row is just the complex
conjugate of the first. Considering

Now note that if you write out the inner product over the first position coordinate, you will get an integral of the general form

If this integral is to be zero for whatever you take

You can divide by [2]:

(D.35) |

How about those

OK, let’s do the restricted closed-shell Hartree-Fock case first,
then, since it is the easiest one. Every state is paired, so the
lower choice in the curly brackets always applies, and the number of
unique unknown spatial states is

Now, suppose that you define a new set of orbitals, each a
linear combination of the current ones:

where the

where

Now note that the

Consider

matrix of coefficientsmust consist of orthonormal vectors. Mathematicians call such matrices “unitary,” rather than orthonormal, since it is easily confused with

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

Call the complete matrix

Now premultiply the definition of the new orbitals above by

but the sum over

That gives you an expression for the original orbitals in terms of the
new ones. For aesthetic reasons, you might just as well renotate

Now plug that into the noncanonical restricted closed-shell
Hartree-Fock equations, with equivalent expressions for

and use the reduction formula

premultiplying all by

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 value of

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

Since you no longer care about the old orbitals, you can drop the
overlines on the new ones, and revert to sensible roman indices

(D.36) |

In the unrestricted case, the noncanonical equations are

In this case the spin-up and spin-down spatial states are not mutually orthonormal, and you want to redefine the group of spin up states and the group of spin down states separately.

The term in linear algebra is that you want to partition your

and equations for

In these two 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) |

In the restricted open-shell Hartree-Fock method, the partitioning
also needs to include the set P of orbitals

Woof.