Hartree Fock Equations

The electronic energy is a functional of the spin orbitals, and we want to minimize it subject to some set of constraints. This can be done using the calculus of variations applied to functionals. So lets look at a general example of functional variation applied to E, a functional of some trial wavefunction that can be linearly varied under a single constraint.
  
By equation 74, we see that , depending on the form of the wavefunction, and by equation 75 that can be linearly expanded (hence linearly varied) in some set of N functions. This is directly analogous to expanding the asymmetric top rotational wavefunctions in a complete set of symmetric top rotational wavefunctions. The task is to minimize E subject to the single constraint that the wavefunction remain normalized, or
 
Writing the energy as

we want , so
 
However, due to the normalization constraint, there is one linear dependancy in the expansion coefficients. If we simply solve equation 78 for the coefficients, they may no longer be normalized, and if we solve for N-1 of the coefficients and invoke the normalization condition to determine the , the energy may not be stationary about it. So we construct the function

Taking the differential of ,
 
Since and are arbitrary, the bracketed parts of equation 81 must be zero. Thus

It is clear that this can be written as a matrix product, and is in fact an eigenvalue equation in the form

Knowing that H and S are hermetian, this matrix eigenvalue equation can be rewritten as
 
The matrix SHS is symmetric and easily diagonalized, with eigenvectors Sc. These can be transformed (multiply on the left by S) to give the optimal coefficients for stationary state. This is very powerful, since in one fell swoop we've got the entire energy spectrum and the appropriate wave functions, properly orthonormal, for all the states. This should illustrate the general technique we will be employing to develop the Hartree-Fock equations and from them the algebraic Roothaan equations, which you will be programming later this summer.

On to the true problem. Assume we have a wave function in the form of a Slater determinant of spin orbitals, . We state the problem as:

Please minimize the electronic energy of this single determinant subject to the constraint that the spin orbitals all remain orthonormal to one another.

We already understand the energy fine by equation 37, and the constraint can be simply stated as

There are N spin orbitals, so there are N(N+1)/2 independent constraints (note: ), so we need that many undetermined multipliers in our lagrangian function, for which we present
 
The restricted sum will prove inconvenient, but it can be eliminated. By taking the complex conjugate of the constraints and the lagrangian function, equation 86,

we realize we can unrestrict the summation by restraining to be a hermitian matrix, such that . This introduces no new undetermined multipliers into the equation, and creates a form more amenable to further derivation. Thus the Lagrangian function we will be working with is
 
The differential of this function must be set to zero as before, giving us

Since we have an electronic energy in terms of spin orbitals

we can write the variance of the energy as

It takes some thought to realize that there are only two unique two electron integrals in this list, and that it can be written

The other variance we need is , which can be expanded as

So the whole thing boils down to one neat statement,

If we conveniently define a coulomb operator and an exchange operator as
  
we can rewrite the two electron integral

This allows for a more compact notation to be employed in writing the variance in the lagrangian function

Like before, the part in brackets is forced to be zero, since can be anything. Setting it equal to zero and rearranging to make it look like some sort of eigenvalue equation yields

These are the glorious Hartree-Fock equations derived in general in the spin orbital basis. But wait - there's a problem. These are coupled integro-differential equations, and while they are not strictly unsolvable, they're a pain. It would be nice to at least uncouple them, so let's do that.

If we apply a unitary rotation to the full set of spin orbitals, generating a new set

where U is unitary, ie. U = U, what changes? The rotation can be written as a matrix product, if we define A as the matrix resembling the slater determinant for the system, ie. det(A) = . In that case,

However, since this matrix U is unitary, , and the new wavefunction differs from the old by a phase factor, affecting nothing observable. How does it affect f(1) and ?



So f(1)' = f(1)! How about ? Start by realizing that are matrix elements of the fock operator.

This last result can be written as a martix product as well, and it is seen that this is now a unitary transformation to the matrix . We are free to choose U to be whatever we please, and if we choose it to make diagonal, we can rewrite the Hartree-Fock equations as

When this is done, the resulting spin orbitals are termed the Hartree-Fock canonical orbitals. Read section 3.3 in Szabo and Ostlund for various fun things to do with the Hartree-Fock equations.

The problem still remains, though. These are integro-differential equations, which computers (and computer programmers) balk at. That is why Roothaan is a HERO! Through his results, we can transform these into a set of matrix formulated algebraic equations that computers and programmers dig. The general case is too troublesome for now, so let's limit ourselves to closed shell, RHF orbitals. To take advantage of the simplifications this can afford, we need to return to the spatial orbital basis. We've derived the spin orbital based Fock operator

Without further ado, we'll introduce the spatial orbital based Fock operator and be done with it.
   
With the properties

Now we introduce a basis set expansion to bring the HF integro-differential equations to soluable algebraic equations. Letting ,

Multiplying by and integrating over electron 1 gives

Identifying the integrals as matrix elements of the fock operator and the unit operator (overlap) respectively,

Using the fact that is diagonal, this can be written as the matrix product
 
So what is F, this so called Fock Matrix? We've defined before as the matrix element of the one electron fock operator, f(1) in equation 119. Writing out this integral and expanding,
 
This is a quantity which can be easily constructed given a set of molecular orbitals (the coefficients ) and a precalculated set of atomic orbital integrals. At this point, the Hartree-Fock equations have been reduced to a matrix eigenvector problem, , but not in a computationally convenient form. Following the analysis leading to equation 84, we first define the transformed Fock matrix as

We can then take

to be equivalent to equation 127. Since is diagonal by choice, diagonalizing the transformed Fock matrix gives a set of transformed molecular orbital coefficients, . The matrix can be back transformed to give the true MO coefficient matrix, . The density matrix for these coefficients is formed by the product , and can subsequently be used to construct a new fock matrix via equation 129. Since the overlap matrix does not depend on the MO coefficients, the same unitary transformation can be applied to the new fock matrix to give a new transformed fock matrix. This can be diagonalized to produce new MO coefficients, and the process repeated until convergance. As an initial guess for the fock matrix, one generally uses the core hamiltonian, ignoring all the two electron integrals.

From the core hamiltonian, an initial C is obtained and a much improved Fock matrix can be built including the two electron integrals. And that's about it for the Hartree-Fock Self Consistent Field Method. These last few pages will be the most important when you get around to programming the closed shell SCF method for the specific case of water, as you will be given the integrals in a file, and you can begin the process by building the core hamiltonian as described above.