Lecture: ab initio Methods Hartree-Fock-Roothan Equations

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:

equation21

where H is the Hamiltonian energy function describing the system and tex2html_wrap_inline565 is the wave function. For a single molecule, the total Hamiltonian can be written as follows:

equation23

where

The first term in Eq.(1.2) is the kinetic energy of each nucleus, the third term is the electrostatic repulsion between the nuclei, and the second term is any external force acting on the nuclei (usually zero). The next term is the electrostatic attraction between electron i and nuclei A. The last two terms are the kinetic energy and electostatic repulsion of the electrons, respectively. At this stage the problem is intractable except for a small number of physically uninteresting cases (from a molecular point of view), so we need to reduce the complexity of Eq. (1.2). Note; physically uninteresting cases refers to solving H2. The first step is the Born-Oppenheimer approximation. (For further information about the Born-Oppenheimer approximation, see the appendix: Derivation of the Hartree-Fock Equations

The Born-Oppenheimer Approximation

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,

equation43

equation48

and let the wave function from Eq. (1.1) be a product of a nuclear and electronic wave functions, tex2html_wrap_inline577. Note that tex2html_wrap_inline579, tex2html_wrap_inline581 is a function of tex2html_wrap_inline583 only, the tex2html_wrap_inline585 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 tex2html_wrap_inline581 called tex2html_wrap_inline589, depend on tex2html_wrap_inline585. Let tex2html_wrap_inline593 be the eigenvalues of the electronic Hamiltonian.

equation64

Using Eq. (5), we can rewrite the left side of Eq. (1) as,

equation66

If the external force on the nuclei is zero, then tex2html_wrap_inline595 is just the kinetic energy operator and,

equation68

where, tex2html_wrap_inline597 the ratio of the electron mass to the mass to the proton, is on the order of 1/1836 and can be neglected. Thus,

equation74

where, tex2html_wrap_inline599 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.

The MO-LCAO Approximation

Now, to solve for the total electronic wave function tex2html_wrap_inline601 of N particles, write tex2html_wrap_inline601 as a product of N single particle orbitals, tex2html_wrap_inline605 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,

equation82

where tex2html_wrap_inline607 denotes the single particle quantum state,a state vector in 3N dimensional space and tex2html_wrap_inline583 is the particle label (position) and tex2html_wrap_inline611 is the permutation operator. The sum is over all permutations of the states tex2html_wrap_inline607, and tex2html_wrap_inline615 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

equation88

Note, the symbols tex2html_wrap_inline617 and tex2html_wrap_inline626 denotes a state vector in Hilbert space, and tex2html_wrap_inline619 and tex2html_wrap_inline630 is the complex conjugate of tex2html_wrap_inline617.

The Hartree-Fock Equations

Now apply the variational principle to tex2html_wrap_inline623 subject to the orthonormality of the total wave function tex2html_wrap_inline625, to determine tex2html_wrap_inline605 (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 tex2html_wrap_inline629, in the minimization and are identified as the energy required to remove the tex2html_wrap_inline631 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. tex2html_wrap_inline633 above.

equation97

where tex2html_wrap_inline646 is a dummy index labelling the single particle orbitals.

Now minimize the total energy with respect to the variational parameters tex2html_wrap_inline635 subject to orthonormality of the total wave function tex2html_wrap_inline637. This constraint appears as Lagrange multipliers, written as tex2html_wrap_inline639, in the minimization and are identified as the energy required to remove the tex2html_wrap_inline641 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 tex2html_wrap_inline635, written as,

equation111

where the Fock matrix tex2html_wrap_inline645 is given by,

equation117

tex2html_wrap_inline647, is the density matrix and tex2html_wrap_inline649 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,

equation129

where C is the expansion coefficient matrix tex2html_wrap_inline635 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, tex2html_wrap_inline653 and tex2html_wrap_inline655

equation132

one electron kinetic energy integrals,

equation139

coulomb attraction between a single electron and the nuclei,

equation149

and two electron integrals, one for the coulomb repulsion and one for the quantum mixing due to indistinguishability of particles.

equation158

The system of HFR equations are solved iteratively and might be outlined as follows:

(a) make an initial guess for tex2html_wrap_inline657.
(b) calculate tex2html_wrap_inline645 and tex2html_wrap_inline649.
(c) solve HFR equations for tex2html_wrap_inline663 and tex2html_wrap_inline665.
(d) repeat steps (a)-(d) until tex2html_wrap_inline663 and/or tex2html_wrap_inline657 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,

equation190

A physically more accurate basis set are the Slater Type Orbitals (STO), written as,

equation201

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 )



Author: Ken Flurchick