subspaces of weakly and strongly perturbing configurations, corresponding to their
contribution to the MRMP2 energy. The reference and the strongly perturbing
configurations build the interacting space for the final MRCI calculation, which
gives the variational part of the correlation energy. The contribution of the weakly
perturbing configurations is then determined from a second, relaxed, MRMP2
calculation, which uses the same FOIS as the first one. Due to the variational
treatment of strongly perturbing configurations, excitation energies from SORCI
are much less sensitive to the size of the reference than those of CASPT2. For a
2-root calculation, e.g., good results can be obtained with less than 100
configurations. The latter are selected by their weights in a preliminary CAS or
RAS calculation. In the MRCI calculations, certain types of excitations are
excluded that have little influence on excitation energies, following the concept
of difference-dedicated CI (DDCI). In the final DDCI3 calculation, e.g., double
excitations involving only inactive orbitals that are not part of the reference space
are excluded. The Davidson correction is applied to the MRCI part and G3 zerothorder Hamiltonian in the MRMP2 part.
4.2.4 Semiempirical Methods
Due to the advancement of computational power and implementation of efficient
(linear scaling, resolution-of-identity approximation), parallelised single-reference
methods like TDDFT and CC2, semiempirical methods have lost their dominance
in excited-state calculations of large molecules. In view of the hype for “ab initio”
methods and the limitations addressed above, semiempirical methods should be
redeemed as a valuable tool for excited-state applications. Their low computational
cost allows for excited-state dynamics on the ps to ns time scale, configurational
sampling of the excitation spectrum to obtain inhomogeneous line broadening and
applications to extended systems, like light-harvesting complexes or molecules
absorbed on a solid-state surface. They can also give satisfying results where
TDDFT is not applicable, e.g., for CT excitations or systems involving static
correlation. In general, the accuracy of excitation energies from semi-empirical
Hamiltonians parameterised for the ground state, like AM1 or PM3, is not great.
They are commonly used in combination with small CI expansions (usually CIS) to
calculate excitation energies, because in the full-CI limit they underestimate excitation energies considerably [45]. This error can be fixed by a special parametrization for CIS excitation energies (ZINDO/S [46]), or by adding orthogonalisation
corrections to the matrix elements (OM1 [47], OM2 [45], OM3 [48]). The latter
approach is very appealing as it fixes another error of NDDO-type Hamiltonians,
the underestimation of torsional barriers, in particular around double bonds
[48–52]. As a by-product, the underestimation of excitation energies in the fullCI limit is removed, without the need for a special parametrisation for excited
states. The use of the OMx Hamiltonians for MRCI calculations makes sense
because of the fast convergence of excitation energies with respect to the size of
the CI expansion. Compared with ab initio MRCI, very small reference wave
functions and active orbital spaces are sufficient to obtain results close to the
52
M. Wanko and A. Rubio
contribution to the MRMP2 energy. The reference and the strongly perturbing
configurations build the interacting space for the final MRCI calculation, which
gives the variational part of the correlation energy. The contribution of the weakly
perturbing configurations is then determined from a second, relaxed, MRMP2
calculation, which uses the same FOIS as the first one. Due to the variational
treatment of strongly perturbing configurations, excitation energies from SORCI
are much less sensitive to the size of the reference than those of CASPT2. For a
2-root calculation, e.g., good results can be obtained with less than 100
configurations. The latter are selected by their weights in a preliminary CAS or
RAS calculation. In the MRCI calculations, certain types of excitations are
excluded that have little influence on excitation energies, following the concept
of difference-dedicated CI (DDCI). In the final DDCI3 calculation, e.g., double
excitations involving only inactive orbitals that are not part of the reference space
are excluded. The Davidson correction is applied to the MRCI part and G3 zerothorder Hamiltonian in the MRMP2 part.
4.2.4 Semiempirical Methods
Due to the advancement of computational power and implementation of efficient
(linear scaling, resolution-of-identity approximation), parallelised single-reference
methods like TDDFT and CC2, semiempirical methods have lost their dominance
in excited-state calculations of large molecules. In view of the hype for “ab initio”
methods and the limitations addressed above, semiempirical methods should be
redeemed as a valuable tool for excited-state applications. Their low computational
cost allows for excited-state dynamics on the ps to ns time scale, configurational
sampling of the excitation spectrum to obtain inhomogeneous line broadening and
applications to extended systems, like light-harvesting complexes or molecules
absorbed on a solid-state surface. They can also give satisfying results where
TDDFT is not applicable, e.g., for CT excitations or systems involving static
correlation. In general, the accuracy of excitation energies from semi-empirical
Hamiltonians parameterised for the ground state, like AM1 or PM3, is not great.
They are commonly used in combination with small CI expansions (usually CIS) to
calculate excitation energies, because in the full-CI limit they underestimate excitation energies considerably [45]. This error can be fixed by a special parametrization for CIS excitation energies (ZINDO/S [46]), or by adding orthogonalisation
corrections to the matrix elements (OM1 [47], OM2 [45], OM3 [48]). The latter
approach is very appealing as it fixes another error of NDDO-type Hamiltonians,
the underestimation of torsional barriers, in particular around double bonds
[48–52]. As a by-product, the underestimation of excitation energies in the fullCI limit is removed, without the need for a special parametrisation for excited
states. The use of the OMx Hamiltonians for MRCI calculations makes sense
because of the fast convergence of excitation energies with respect to the size of
the CI expansion. Compared with ab initio MRCI, very small reference wave
functions and active orbital spaces are sufficient to obtain results close to the
52
M. Wanko and A. Rubio