Essentials of Computational Chemistry

6.2 BASIS SETS 179
It was Hellmann (1935) who first proposed a rather radical solution to this problem –
replace the electrons with analytical functions that would reasonably accurately, and much
more efficiently, represent the combined nuclear–electronic core to the remaining electrons.
Such functions are referred to as effective core potentials (ECPs). In a sense, we have
already seen ECPs in a very crude form in semiempirical MO theory, where, since only
valence electrons are treated, the ECP is a nuclear point charge reduced in magnitude by the
number of core electrons.
In ab initio theory, ECPs are considerably more complex. They properly represent not only
Coulomb repulsion effects, but also adherence to the Pauli principle (i.e., outlying atomic
orbitals must be orthogonal to core orbitals having the same angular momentum). This being
said, we will not dwell on the technical aspects of their construction. Interested readers are
referred to the bibliography at the end of the chapter.
Note that were ECPs to do nothing more than reduce the scope of the electronic structure
problem for heavy elements, they would still have great value. However, they have another
virtue as well. The core electrons in very heavy elements reach velocities sufficiently near the
speed of light that they manifest relativistic effects. A non-relativistic Hamiltonian operator
is incapable of accounting for such effects, which can be significant for many chemical
properties (see, for example, Kaltsoyannis 2003). A full discussion of modeling relativistic
effects, while a fascinating topic, is well beyond the scope of this book, although some details
are discussed in Section 7.4.4. We note here simply that, to the extent an ECP represents the
behavior of an atomic core, relativistic effects can be folded in, and thereby removed from
the problem of finding suitable wave functions for the remaining electrons.
A key issue in the construction of ECPs is just how many electrons to include in the
core. So-called ‘large-core’ ECPs include everything but the outermost (valence) shell, while
‘small-core’ ECPs scale back to the next lower shell. Because polarization of the sub-valence
shell can be chemically important in heavier metals, it is usually worth the extra cost to
explicitly include that shell in the calculations. Thus, the most robust ECPs for the elements
Sc–Zn, Y–Cd, and La–Hg, employ [Ne], [Ar], and [Kr] cores, respectively. There is less
consensus on the small-core vs. large-core question for the non-metals.
Popular pseudopotentials in modern use include those of Hay and Wadt (sometimes also
called the Los Alamos National Laboratory (or LANL) ECPs; Hay and Wadt 1985), those
of Stevens et al. (1992), and the Stuttgart–Dresden pseudopotentials developed by Dolg and
co-workers (2002). The Hay–Wadt ECPs are non-relativistic for the first row of transition
metals while most others are not; as relativistic effects are usually quite small for this region
of the periodic table, the distinction is not particularly important. Lovallo and Klobukowski
(2003) have recently provided additional sets of both relativistic and non-relativistic ECPs
for these metals. For the p block elements, Check et al. (2001) have optimized polarization
and diffuse functions to be used in conjunction with the LANL double-ζ basis set.
Another recent set of pseudopotentials for the 4p, 5p, and 6p elements has been developed
by Dyall (1998, 2002). These ECPs are designed to be the ECP-equivalent to the
correlation-consistent basis sets of Dunning insofar as (i) prescriptions for double-ζ and