The main structural effects occurring during the reaction can be observed
displayed in Table . As the rearrangement starts out one
hydrogen atom (H As was already mentioned we will apply now higher-correlated methods for the reactant, product, and transition state system at the CASSCF optimized geometries to account for more accurate relative energies. In any case a small basis set has been used and therefore the goal is not to be extremely accurate. For more complete results see Ref. [224]. We are going to perform calculations with the MP2, MRCI, ACPF, CASPT2, CCSD, and CCSD(T) methods. Starting with dimethylcarbene, we will use the following input file:
Observe in the previous input that we have generated a multiconfigurational
wave function for CASPT2, MRCI, and ACPF wave functions but a single configuration
reference wave function (using OUTOrbitals and CANOnical)
for the CCSD and CCSD(T) wave functions. Notice also
that to compute a multiconfigurational ACPF wave function we have to use
the program, not the MRCI module which does not accept
more than one single reference. In all the highly correlated
methods we have frozen the three carbon core orbitals because of the reasons
already explained in section . For MRCI, ACPF, CCSD, and CCSD(T)
the freezing is performed in the CPF step.
MOTRA
One question that can be addressed is which is the proper reference space
for the multiconfigurational calculations. As was explained when we selected
the active space for the geometry optimizations, we performed several tests
at different stages in the reaction path and observed that the smallest
meaningful active space, two electrons in two orbitals, was sufficient
in all the cases. We can come back to this problem here to select the
reference for CASPT2, MRCI, and ACPF methods. The simple analysis of the
SCF orbital energies shows that in dimethylcarbene, for instance, the
orbital energies of the C-H bonds are close to those of the C-C
bonds and additionally those orbitals are strongly mixed along
the reaction path. A balanced active space including all orbitals necessary
to describe the shifting H-atom properly would require a full valence
space of 18 electrons in 18 orbitals. This is not a feasible space, therefore
we proceed with the minimal active space and analyze later the quality
of the results. The CASSCF wave function will then include for dimethylcarbene
and the transition state structure the ()
The ELECtrons refers to the total number of
electrons that are going to be correlated, that is, all except those
frozen in the previous step.
Keywords MOTRAINACtive and
ACTIve are optional and describe the number of inactive
(occupation two in all the reference configurations) and active
(varying occupation number in the reference configurations) orbitals
of the space. Here ACTIve indicates one orbital of each
of the symmetries. The following keyword CIALl indicates
that the reference space will be the full CI within the subspace
of active orbitals. It must be always followed by symmetry index
(number of the irrep) for the resulting wave function, one here.
For the transition state structure we do not impose any symmetry
restriction, therefore the calculations are performed in the C
Finally we compute the wave functions for the product, propene, in the
C
Table compiles the total and relative energies obtained for the studied reaction at the different levels of theory employed.
We can discuss now the quality of the results obtained and their
reliability (for a more careful discussion of the accuracy of
quantum chemical calculations see Ref. [212]).
In first place we have to consider that a valence
double-zeta plus polarization basis set is somewhat small to obtain
accurate results. At least a triple-zeta quality would be required.
The present results have, however, the goal to serve as an example.
We already pointed out that the CASSCF geometries were very similar
to the MP2 reported geometries [224]. This fact validates
both methods. MP2 provides remarkably accurate geometries using
basis sets of triple-zeta quality, as in Ref. [224], in
situations were the systems can be described as singly configurational,
as the CASSCF calculations show. The Hartree-Fock configuration has
a contribution of more than 95% in all three structures, while the
largest weight for another configuration appears in propene for
() The MRCI calculations provide also one test of the validity of the reference wave function. For instance, the MRCI output for propene is:
The Hartree-Fock configuration contributes to the MRCI configuration with a weight of 85.4%, while the next configuration contributes by 2.8%. Similar conclusions can be obtained analyzing the ACPF results and for the other structures. We will keep the MRCI results including the Davidson correction (MRCI+Q) which corrects for the size-inconsistency of the truncated CI expansion [212].
For CASPT2 the evaluation criteria were already commented in
section . The portion of the
The weight of the CASSCF reference to the first-order wave function is here 87.9%, very close to the weights obtained for the dimethylcarbene and the transition state structure, and there is only a small contribution to the wave function and energy which is larger than the selected thresholds. This should not be considered as a intruder state, but as a contribution from the fourth inactive orbital which could be, eventually, included in the active space. The contribution to the second-order energy in this case is smaller than 1 Kcal/mol. It can be observed that the same contribution shows up for the transition state structure but not for the dimethylcarbene. In principle this could be an indication that a larger active space, that is, four electrons in four orbitals, would give a slightly more accurate CASPT2 energy. The present results will probably overestimate the second-order energies for the transition state structure and the propene, leading to a slightly smaller activation barrier and a slightly larger exothermicity, as can be observed in Table . The orbitals pointed out as responsible for the large contributions in propene are the fourth inactive and 22nd secondary orbitals of the first symmetry. They are too deep and too high, respectively, to expect that an increase in the active space could in fact represent a great improvement in the CASPT2 result. In any case we tested for four orbitals-four electrons CASSCF/CASPT2 calculations and the results were very similar to those presented here.
Finally we can analyze the so-called -diagnostic [225]
for the coupled-cluster wave functions. is defined for closed-shell
coupled-cluster methods as the Euclidean norm of the vector of T
In this case T1aa and T1bb are identical because we are computing a
closed-shell singlet state. The five largest T
There are several considerations concerning the diagnostic
[225]. First, it is only valid within the frozen core
approximation and it was defined for coupled-cluster procedures
using SCF molecular orbitals in the reference function. Second, it is
a measure of the importance of non-dynamical electron correlation effects
and not of the degree of the multireference effects. Sometimes the two
effects are related, but not always (see discussion in Ref. [226]).
Finally, the performance of the CCSD(T) method is reasonably good even
in situations where has a value as large as 0.08.
In conclusion, the use of together with other wave function
analysis, such as explicitly examining the largest T As the present systems are reasonably well described by a single determinant reference function there is no doubt that the CCSD(T) method provides the most accurate results. Here CASPT2, MRCI+Q, ACPF, and CCSD(T) predict the barrier height from the reactant to the transition state with an accuracy better than 1 Kcal/mol. The correspondence is somewhat worse, about 3 Kcal/mol, for the exothermicity. As the difference is largest for the CCSD(T) method we may conclude than triple and higher order excitations are of importance to achieve a balanced correlation treatment, in particular with respect to the partially occupied orbital at the carbenoid center. It is also noticeable that the relative MP2 energies appear to be shifted about 3-4 Kcal/mol towards lower values. This effect may be due to the overestimation of the hyper-conjugation effect which appears to be strongest in dimethylcarbene [227,224].
Additional factors affecting the accuracy of the results obtained
are the zero point vibrational energy correction and, of course,
the saturation of the one particle basis sets. The zero point
vibrational correction could be computed by performing a numerical
harmonic vibrational analysis at the CASSCF level using
In dimethylcarbene both the medium and terminal carbons appear equally charged.
During the migration of hydrogen H Next: 10.5 Excited states.
Up: 10. Examples
Previous: 10.3 Computing a reaction
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