connection with Figs.5.20and5.21). There is some limited evidence thatwhen a
correlation method is already being used, one tends to get improved geometries by
using a bigger basis set rather than by going to a yet higher correlation level [ 103 ].
Figure5.21shows the results of HF and MP2 methods applied to chemical reac-
tions. The limitations and advantages of numerous such methods are shown in a
practical way in the Gaussian 94 workbook by Foresman and Frisch [ 1 e]. Energies
and times for some correlated calculations are given in Table5.6.
5.4.3.1 Size-Consistency
Two factors that should be mentioned in connection with post-HF calculations are
the questions of whether a method issize-consistentand whether it isvariational.A
method is size-consistent if it gives the energy of a collection ofnwidely-separated
atoms or molecules as beingntimes the energy of one of them. For example, the HF
method gives the energy of two water molecules 20 A ̊apart (considered as a single
system or “supermolecule”) as being twice the energy of one water molecule. The
example below gives the result of HF/3–21G(*)geometry optimizations on a water
molecule, and on two water molecules at increasing distances (with the two-H 2 O
supermolecule the O/H internuclear distancerwas held constant at 10, 15. ... A ̊
while all the other geometric parameters were optimized):
HOHrH HOEnergy of H 2 O¼#75.58596
2 'Energy or H 2 O¼#151.17192
Energy of (H 2 O) 2 ¼#151.17206, at r¼10 A ̊
Energy of (H 2 O) 2 ¼#151.17196, at r¼15 A ̊
Energy of (H 2 O) 2 ¼#151.17194, at r¼20 A ̊
Energy of (H 2 O) 2 ¼#151.17193, at r¼25 A ̊
Energy of (H 2 O) 2 ¼#151.17193, at r¼30 A ̊
As the two water molecules are separated a hydrogen bond (equilibrium bond
lengthr¼ca. 2.0 A ̊) is broken and the energy rises, levelling off at 20–25 A ̊ to
twice the energy of one water molecule. With the HF method we find that for any
numbernof molecules M, at large separation the energy of a supermolecule (M)n
equalsntimes the energy of one M. The HF method is thus size-consistent. We
might say that a size-consistent method is one that scales in a way that makes sense.
Now, it is hard to see why,physically, the energy ofnidentical molecules so
widely-separated that they cannot affect one another shouldnotbentimes the
energy of one molecule. Anymathematicalmethod that does not mimic this
276 5 Ab initio Calculations