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Part 3
Benchmarking for Radicals
R FCI (CC) = 1.2367Å and R FCI (CH) = 1.0802Å.
The error in the resulting geometry is a sum of the error from the finite difference approximations
and the error from the Newton step. The gradient and Hessian carry an error of O(δ 4 ) where δ =
0.01Å, this is an error in the order of 10 -8 Å. The Newton step has an error of O((H -1 G) 2 ), in this
case H -1 G is of the size 10 -3 Å and so the error is in the order of 10 -6 Å. The error in total is thus in
the order of 10 -6 Å.
The gradient for the FCI equilibrium geometry has been found as above, making single-point
calculations at the FCI geometry and at geometries distorted in steps of 0.01Å from the FCI
geometry. The same finite-difference expressions as before are used. The gradient is found to be
⎡
FCI 1.8593 10
E
⎢
;3.0661 10
⎣
Å
⎤
Å⎥⎦
G −5 h
−5
= − ⋅ ⋅ h , (3.2)
thus verifying the correctness of the FCI geometry.
Since the geometry was determined at the CCSDT level to be R CCSDT (CC) = 1.23448Å and
R CCSDT (CH) = 1.07924Å, the error due to truncation of the many-electron basis in CCSDT is in the
order of 10 -3 Å. This is similar to the results obtained for CN. This also suggests that the quadruples
correction to the equilibrium geometry is in the order of 0.001-0.002Å.
3.4 Conclusion
Full configuration interaction (FCI) and coupled cluster (CC) calculations have been carried out on
CN using the cc-pVDZ and cc-pVTZ basis sets. The equilibrium bond distance, harmonic
frequency, atomization energy, and vertical electron affinity have been evaluated on the various
levels of theory.
As expected, the cc-pVDZ basis set does not provide accurate geometries and frequencies and
CCSD is insufficient for prediction of equilibrium properties. Apparently, the CCSDT method is a
better approximation than CCSDTQ for obtaining the equilibrium geometry and the harmonic
frequency. This is due to a favorable cancellation of errors for CCSDT calculations in small basis
sets. Also the vertical electron affinities are affected by cancellation of errors, and already at the
CCSD level, the error is less than 1mE h compared to the FCI value.
The convergence patterns for the CI and CC hierarchies are studied for CN and it is found similar to
the convergence patterns previously reported for N 2 . 74 Thus, it does not seem that the open-shell
nature of CN leads to slow convergence of the CI and CC hierarchies compared to closed shell
cases.
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84