To derive the Hartree-Fock equations, we begin with the time independent Schrodinger wave equation in the Schrodinger picture for the ab initio quantum chemistry calculations. This equation can be written as follows:
where H is the Hamiltonian energy function describing the system and
is the wave function. For a single molecule, the total Hamiltonian
can be written as follows:
where
First re-write the Hamiltonian in Eq. (1.2) using atomic units (For further information about atomic units see the Appendix: Units.) and then express the Hamiltonian as a sum of nuclear HN and electronic He parts,
and let the wave function from Eq. (1.1) be a product of a nuclear and
electronic wave functions,
.
Note that
,
is a function of
only, the
are treated as constants due to the Born-Oppenheimer approximation.
That is, rewiting the wave function separates the problem into
electronic and nuclear regimes, and the electronic problem is
solved given a specific nuclear configuration. Then the energy
eigenvalues determined from
called
,
depend on
.
Let
be the eigenvalues of the electronic Hamiltonian.
Using Eq. (5), we can rewrite the left side of Eq. (1) as,
If the external force on the nuclei is zero, then
is just the kinetic energy operator and,
where,
the ratio of the electron mass to the mass to the proton, is on the order
of 1/1836 and can be neglected. Thus,
where,
is the kinetic energy of the nuclei.
In the Born-Oppenheimer approximation, the motion of the nuclei is decoupled from the motion of the electrons, and for most cases the motion of the nuclei can be ignored. This approximation is valid because the ratio of the mass of the proton to the mass of the electron is 1/1836. Thus, the motion of the nuclei is negligible compared to the motion of the electrons.
Now, to solve for the total electronic wave function
of N particles, write
as a product of N single particle orbitals,
in 3N dimensional space. This is called MO-LCAO, Molecular
Orbital as Linear Combination of Atomic
Orbitals. The total electronic wave function must be antisymmetric
with respect to an interchange of electron coordinates due to the Pauli
exclusion principle. The antisymmetrized wave function can be written as,
where
denotes the single particle quantum state,a state vector in 3N dimensional
space and
is the particle label (position) and
is the permutation operator. The sum is over all permutations of the states
,
and
indicates a -1 for odd permutations and a +1 for even permutations.
This can also be written as a determinant, known as the Slater Determinant
or single determinant wave function. Note: each electron is in its own 3
dimensional space. The product space is 3N dimensional and derivatives with
respect to 3N dimensional space.
The total electronic energy of the system E, is written as
Note, the symbols
and
denotes a state vector in Hilbert space, and
and
is the complex conjugate of
.
Now apply the variational principle to
subject to the orthonormality of the total wave function
,
to determine
(For further information about the
Born-Oppenheimer approximation, see the appendix:
Derivation of the
Hartree-Fock Equations This constraint appears as Lagrange multipliers,
written as
,
in the minimization and are identified as the energy required to remove the
particle from the system. This is known as Koopman's theorem. The result
of minimizing E gives a set of integro-differential equations known as the
Hartree-Fock Equations (HF).
One can then expand the single particle orbitals in terms of a
set of basis functions as follows.
above.
where is a dummy index labelling the single particle orbitals.
Now minimize the total energy with respect to the variational parameters
subject to orthonormality of the total wave function
.
This constraint appears as Lagrange multipliers, written as
,
in the minimization and are identified as the energy required to remove the
particle from the system. This minimization plus the expansion in terms
on Hilbert Space basis functions, gives the Hartree-Fock
Roothaan (HFR) algebraic equations to solve for the variational
parameters
,
written as,
where the Fock matrix
is given by,
,
is the density matrix and
is the overlap matrix which arises from the non-orthogonality of the
basis functions.
In matrix form, the HFR equations are of the following form,
where C is the expansion coefficient matrix
and E is the energy. To solve Eq.(1.14), transform to a standard
eigenvalue problem, solve, and then transform back.
The types of integrals needed are, one electron integrals giving
the overlap between different states,
and
one electron kinetic energy integrals,
coulomb attraction between a single electron and the nuclei,
and two electron integrals, one for the coulomb repulsion and one for the quantum mixing due to indistinguishability of particles.
The system of HFR equations are solved iteratively and might be outlined as follows:
(a) make an initial guess for
.
(b) calculate
and
.
(c) solve HFR equations for
and
.
(d) repeat steps (a)-(d) until
and/or
converge.
One major concern of the HFR equations are, of course, the basis functions, which can be summarized as:
The derivation above makes no distinction between closed shells or open shells. In fact, the distinction is in how one treats the basis sets. The easiest is the Restricted Hartree Fock (RHF) where only closed shells are considered. This removes the spin symmetry problem. Open shell calculations are called Unrestricted Hartree Fock (UHF) or any of its variants (e.g. Spin Unrestricted Hartree Fock, and many others.) For all of these calculations, the equations are nearly the same.
Up to this point, no reference to the particular functions to be used as basis functions has been made. First some terminology. A minimum basis set is a set of functions that assigns one function to each orbital. An extended basis set assigns a linear combination of functions to each orbital.
Basis sets need only be any complete set that spans Hilbert Space. However, for computational ease, Cartesian Gaussians, Gaussian Type Functions or Gaussian Type Orbitals (GTF or GTO) are used. These are written as,
A physically more accurate basis set are the Slater Type Orbitals (STO), written as,
To regain the computational ease of the GTF with the accuracy of the STO's, a basis set may be constructed from a linear combination of n GTF's fitted (in the least square sense or energy minimized) to STO's. This type of basis set is denoted as STO-nG. A common variant of these is the STO-3G basis set.
Various extensions and combinations of these basis sets have been used. The double zeta basis set is two STO's per orbital to increase accuracy near the nucleus. To increase accuracy at high energy where the valance electrons contribute more, different basis sets were developed to include the valence shells. These are denoted 6-31G, 4-31G, 4-31G*, etc. In practice, most of the common basis sets are incorporated in ab initio codes, or can be put in explicitly.
The variants of the Hartree-Fock SCF procedure are all single determinant solutions. Other approaches include the MCSCF Multi Configuration SCF. This procedure consists of using many (more than one) Slater determinants. One implementation is the CASSCF , the Complete Active Space SCF
Either a single or multi-determinant approach is an all electron procedure. To treat larger systems (more electrons) one can use an effective core potential to represent the core electrons with a single function and the valance electrons with individual orbital functions.
All of the derivation thus far, is non-relativistic. For example, spin-orbit coupling is not included. There are relavisitic forulations known as Dirac-Fock to address this issue. (see the Dirac Fock discussion here )