Hartree-Fock Theory: Tutorial and Assignments

Ab Initio Quantum Mechanics: the Basis Set

Slater Functions for Atomic Calculations

The radial wavefunction for the hydrogen atom in the groud state is exponential function. It can be shown that the radial wave function for any system where a single electron moves in the central field has an exponential form. The complete wave function in which the exponential function describes the radial part is called the Slater function. For exampe, true wave functions of He⁺, Li²⁺, and C⁵⁺ ions are Slater functions. Treatment of two-electron systems such as He atom, or Li⁺ requires furter approximations, such as the Hartree-Fock approximation, in which each of the electrons feels a central field due to the positively charged nucleus, and the average cental field due to all other electrons. This is still a cental field problem, and the Hartree-Fock wave function for each of the electrons is a Slater function. In the helium atom ground state, both of these electron share an 1s orbital and a single Slater function is sufficient for the Hartree-Fock calculation of a helium atom in its ground singlet state. Two Slater functions are needed to describe the ground state of second row atoms. In summary, Slater functions describe accurately the electron distributions in atoms within the Hartree-Fock approximation, and quantum mechanical calculations of atoms with Slater functions are practicable. One example of atomic Hartree-Fock calculations that employes Slater functions is available here.

Gaussian Functions for Molecular Calculations

Slater-type functions are not well suited for evaluation of two-electron multi-center integrals that arise in molecular calculations. Samuel Boys proposed in 1950 to use Gaussian-type functions instead in molecular calculations. The key idea here was that each Slater-type function can be approximated by several Gaussian-type functions because evaluation of two-electron multi-center integrals with Gaussian functions is not too difficult. Modern electronic structure calculations almost always use such Gaussian-type functions to represent atomic orbitals from which molecular orbitals are constructed. Gaussia functions are also widely used to carry out calculations with atoms. The details about the the use of Gaussian functions in quantum chemistry is well explained in the Simplified Introduction to Ab Initio Basis Sets by Dr. Jan Labanowsy, the maintainer of the Computational Chemistry List.

Basis Functions and Basis Sets

A set of Gaussian functions that descibes an atom is called a basis set for this atom. An example of a Gaussian basis set is the Slater Type Orbital via Three Gaussian (STO-3G) set for hydrogen. Here, a single Slater function for the 1s atomic orbital is approximated as a linear combination of three Gaussian functions. Fitting the combination of these Gaussian functions to hydrogen atom's Slater function yields exponent values of 3.4252, 0.6239, and 0.1688 that yield a good fit. Such a compact basis set allows very rapid calculations, but the truth is that three Gaussians do not represent a Slater function very well. Furthermore, the STO-3G provides centrosymmetric description that is appropriate for the description of isolated in hydrogen atom, but not for the description of directional hydrogen bonding in molecules. One can construct basis sets that allows for more accurate description by including p, d, f ... type functions. Most commonly used Gaussian-type basis sets are available via EMSL Basis Set Exchange.

Basis functions in an atomic basis set can be characterized by the principal quantum number (n) and the angular quantum number (l). The principal quantum number characterizes the size (radial extent) of the orbital; the angular quantum number describes its shape. For example, the spherical basis function that describes the 1s atomic orbital in the hydrogen atom has the principal quantum number n = 1 and the orbital quantum number l = 0. A balanced description of electron distribution in atoms and molecules is achieved by a basis sets that include some functions with high principal quantum number and some functions with high angular quantum number. Such basis functions are especially important to describe the dynamic electron correlation, which is ignored in the Hartree-Fock theory. For example, some advanced correlated calculations of the Ne atom have employed bases providing a 25s orbital as well as some k-orbitals (l=7).

Correlation Consistent Basis Sets

Ideally, a basis that has many functions with large principal quantum number as well as with large angular quantum number should be employed. In practice, such complete bases are computationally too taxing and may suffer from linear dependencies. One practical solution was suggested by Thom Dunning in 1989: let's employ a family of basis sets, in which each member systematically improves on the previous member. The correlation consistent basis sets of Dunning are constructed such that each basis in the series supplements the previous basis with a complete shell of functions that go with the current principal quantum number. For example, the carbon basis set with the maximum principal quantum number l_max = 4 adds a single s, p, d, f, and g function to a basis that already has four s, three p, two d, and one f function. Traditionally, bases in the cc-pVXZ series are characterized by the cardinal number X, which is related to the maximum angular momentum function present in the basis set For example, the largest angular momentum function in the cc-pV5Z basis (X=5) for carbon is h, thus l_max=5 and X = l_max . However, the largest angular momentum function in the cc-pV5Z basis for hydrogen and helium is g, thus l_max=4 and now X = l_max + 1.

Systematic Convergence of Energies in Correlation Consistent Basis Sets

Consider a set of calculations performed on the same molecule but using a series of correlation consistent basis sets. It turns out that the Hartree-Fock energy converges exponentially in such a series and the limiting value can be obtained by extrapolation. Because the exponential decay (E_X = E_inf + A * exp(-B*X) is described by three parameters (E_inf, A, and B), three energy values (say, with cc-pVDZ, cc-pVTZ, and cc-pVQZ bases) are needed to extrapolate E_inf. In practice, good results are obtained by using cc-pVTZ, cc-pVQZ, and cc-pV5Z data.

Which Basis Set Should Be Used?

The large number of available basis sets might be confusing at first. 3-21G? 6-31+G(d,p)? MIDI? Sadlej pVTZ? Correlation consistent? Polarization consistent? Including h-finctions on heavy atoms? With diffuse functions on hydrogens? With tight functions of sulfur? Fully Uncontracted? The choice may seem difficult. It is important to make a wise choice because otherwize the results are significantly incorrect or the calculation may take too long to complete. Here are some general suggestions:

Use the largest basis set that is feasible for a given molecule considering the time and resource constraints. In practice, if you are planning for an overnight Hartree-Fock single point calculation on a modern personal computer, limit the total number of basis functions to less than 600
Take advantage of basis set extrapolation. If you can afford a calculation with cc-pV5Z basis, perform also relatively cheap cc-pVQZ and cc-pVTZ calculations and extrapolate the Hartree-Fock energy. The result is likely as accurate than a considerably more expensive cc-pV6Z calcalulation,
For larger systems, 6-31G(d,p) may be a good first choice for Hartree-Fock calculations. Extrapolation with larger basis is not useful here but it is worthwhile checking how much the results differ when the basis is changed to, say 6-311+G(2d,p).
Some systems and properties require diffuse functions. Examples include anions, and calculation of polarizabilities.
Tight functions improve results for some systems, e.g. in sulfur-containing molecules. Special tight functions are needed to describe electron correlation effects involving core orbitals.
Consider that calculations in which the Hartree-Fock step can be performed fully in the memory is much faster than the conventional algorithm that reads integrals from the disk at every iteration, or the direct algorithm that recomputes integrals every time they are needed. If possible, use the largest basis set for which in core calculation is possible. For example, you can carry out all calculations with neon atom in core when using the remote workstation, Gaussian's InCore algorithm with Ne atom requires 2 MW of memory with cc-pVDZ basis, 3 MW with cc-pVTZ, 8 MW with cc-pVQZ, 48 MW with cc-pV5Z, and 300 MW of memory with cc-pV6Z basis set.