Essentials of Computational Chemistry

230 7 INCLUDING ELECTRON CORRELATION IN MO THEORY
Table 7.2 Valence correlation energies (−Ecorr, mEh) from standard and R12 CCSD calculations and
from extrapolation using Eq. (7.57) for seven closed-shell singlet molecules
CH2 H2O HF Ne CO N2 F2 MUE a
Standard calculations b
D 138.0 211.2 206.8 189.0 294.5 309.4 402.7 107.9
T 164.2 267.4 273.9 266.3 358.3 372.0 526.0 39.8
Q 171.4 286.0 297.6 294.7 380.6 393.2 569.7 16.2
5 173.6 292.4 306.3 305.5 388.5 400.7 586.1 7.7
6 174.5 294.9 309.7 309.9 391.7 403.7 592.8 4.2
Extrapolation protocol c
DT 175.2 291.1 302.2 298.9 385.1 398.3 577.9 11.2
TQ 176.7 299.5 314.9 315.4 396.9 408.7 601.6 1.0
Q5 176.0 299.2 315.3 316.8 396.8 408.6 603.4 1.3
56 175.7 298.3 314.4 316.0 396.1 407.9 601.9 0.5
R12 benchmarks
175.5 297.9 313.9 315.5 395.7 407.4 601.0
a Mean unsigned error over all molecules compared to R12 energies.
b First column gives n for cc-pVnZ basis set.
c First column gives x and y equivalents for Eq. (7.57).
calculation has been completed. Indeed, even the simplest DT extrapolation is on average
30 percent more accurate than the single, much more expensive CCSD/cc-pVQZ level.
Having discussed extrapolation in the context of correlation energy, it is appropriate to
recognize that if one is going to estimate the infinite basis correlation energy, one wants to
add this to the infinite basis HF energy. Parthiban and Martin (2001) have found the analog
of Eq. (7.57) involving the fifth power of the angular momentum quantum numbers to be
highly accurate, i.e.,
EHF,∞ = x5EHF,x − y5EHF,y x5 − y5 (7.58)
7.6.2 Sensitivity to Reference Wave Function
For single-reference correlated methods, there are several issues associated with the HF
reference that can significantly affect the interpretation of the correlated calculation. First,
there is the degree to which the wave function can indeed be reasonably well described by a
single configuration, i.e., the extent to which the HF determinant dominates in the expansion
of Eq. (7.1). When a non-trivial degree of multireference character exists, perturbation theory
is particularly sensitive to this feature, and can give untrustworthy results. To appreciate this,
recall the TMM example with which this chapter began (Figure 7.1). Let us take the singleconfiguration
HF wave function represented by Eq. (7.2), and consider the MP2 energy
contribution from the double excitation taking both electrons from occupied orbital π2 to
virtual orbital π3. From Eq. 7.48, we see that the denominator associated with this term is
the difference in orbital energies. However, since these orbitals are formally degenerate, the
denominator is zero and the perturbation theory expression for the energy associated with