(ECP) or a pseudopotential. With a set of valence orbital basis functions optimized
for use with it, it simulates the effect on the valence electrons of the atomic nuclei
plus the core electrons. A distinction is sometimes made between an ECP and a
pseudopotential, the latter term being used to mean any approach limited to the
valence electrons, while ECP is sometimes used to designate a simplified pseudo-
potential corresponding to a function with fewer orbital nodes than the “correct”
functions. However, the terms are usually used interchangeably to designate a
nuclei-plus-core electrons potential used with a set of valence functions, and that
is what is meant here. The use of an ECP stands in contrast to usingall-electron
basis setslike the Pople or Dunning sets discussed above.
So far we have discussednonrelativisticab initio methods: they ignore those
consequences of Einstein’s special theory of relativity that are relevant to chemistry
(Section 4.2.3; [ 57 ]). These consequences arise from the dependence of mass on
velocity [ 58 ]. This dependence causes the masses of the inner electrons of heavy
atoms to be significantly greater than the electron rest mass; since the Hamiltonian
operator in the Schr€odinger equation contains the electron mass (e.g. Eq. 5.4), this
change of mass should be taken into account. Relativistic effects in heavy-atom
molecules affect geometries, energies, and other properties [ 59 ]. Relativity is
accounted for in the relativistic form of the Schr€odinger equation, the Dirac
equation (interestingly, Dirac thought his equation would not be relevant to chem-
istry [ 60 ]). This equation is not commonly used explicitly in molecular calculations,
but is instead used to develop [ 61 ]relativistic effective core potentials(relativistic
pseudopotentials). Relativistic effects can begin to become significant for about
third-row elements, i.e. the first transition metals. For molecules with these atoms
ECPs begin to be useful for speeding up calculations, so it makes sense to take these
effects into account in developing these potential operators and their basis func-
tions, and indeed ECPs are generally relativistic. Such ECPs can give accurate
results for molecules with third-row and beyond atoms by simulating the electronic
relativistic mass increase. Comparing such a calculation on silver fluoride using the
popular LANL2DZ basis set (a split valence basis) with a 3–21G(*)calculation,
using Gaussian 03 for Windows [ 62 ] (on a older machine running under XP):
LANL2DZ basis, 31 basis functions, 2.0 min; Ag–F¼2.064 A ̊.
3–21G(*)basis, 48 basis functions, 2.0 min; Ag-F¼2.019 A ̊.
The experimental bond length is 1.983 A ̊[ 63 ].
In this simple case there is no advantage to the pseudopotential calculation (the
3–21G(*)geometry is actually better!), but more challenging calculations on “very-
heavy-atom” molecules, particularly transition metal molecules, rely heavily on
ab initio or DFT (Chapter 7) calculations with pseudopotentials. Nevertheless,
ordinary nonrelativistic all-electron basis sets sometimes give good results with
quite heavy atoms [ 64 ]. A concise description of pseudopotential theory and
specific relativistic effects on molecules, with several references, is given by Levine
[ 65 ]. Reviews oriented toward transition metal molecules [ 66 a,b,c] and the lantha-
nides [ 66 d] have appeared, as well as detailed reviews of the more “technical”
aspects of the theory [ 67 ]. See too Section 8.3.
252 5 Ab initio Calculations