interacting space using truncated CI (MRCI), coupled-cluster theory (MRCC), or
perturbation theory (MRPT). The latter requires the reference to already include all
important configurations such that the perturbations are small. The most prominent
version is the CASPT2 method, which employs a complete active space selfconsistent field (CASSCF) reference. The CASSCF wave function uses a full-CI
expansion for a limited set of “active” orbitals, the “inactive” orbitals are doubly
occupied or unoccupied. The energy is minimised with respect to both orbital
rotations and CI coefficients. If several states are calculated, the orbitals are often
optimised for the average of the selected CI root energies (SA-CASSCF), to avoid
root-flipping problems. In principle, the CASSCF wave function can provide a
robust description of nearly degenerate states (static correlation), but the number of
active orbitals is limited due to the factorial scaling of the full-CI problem.
Therefore, the description of electron correlation is incomplete and the missing
part of the correlation energy, usually referred to as dynamic correlation, must be
calculated to obtain quantitative results.
The most widely used way to do this is via second-order perturbation theory, in
terms of the CASPT2 method. CASPT2 can be considered size-consistent if the
active orbital space increases naturally with the system, e.g., comprises all orbitals
of an irreducible representation. Truncated MRCI and MRMP2 methods are generally not size-consistent and empirical correction schemes, e.g., the Davidson
correction [38] are applied to alleviate the problem. If the CASSCF states form a
good reference for the perturbative treatment, CASPT2 is comparable in accuracy
with CC3 [33]. Dynamic correlation can strongly contribute to excitation energies
and alter the order of excited states. In some cases, states that are well separated in
energy in the exact solution can be nearly degenerate and strongly mix in the
CASSCF solution. In these cases, the CASSCF states do not represent a good
reference for perturbation theory and the error of CASPT2 can be considerably
larger than 0.1 eV. This situation typically arises close to (avoided) crossings
between states of different character, e.g., valence and Rydberg states. The multistate CASPT2 method (MS-CASPT2) solves this problem by calculating
perturbatively an effective Hamiltonian in the space of the reference wave functions
from a SA-CASSCF calculation [39]. A recent example where a MS-CASPT2
treatment is required is the spectrum of the HBDI chromophore of the greenfluorescent protein (GFP) [40]. The same article also reports a large influence of
the zeroth-order Hamiltonian, for which several modifications have been suggested
to improve the convergence behavior of the perturbation series. Recently, the IPEA
ansatz [41] has been proposed, which eliminates a systematic underestimation of
excitation energies of the previously common G1 and G3 Hamiltonians [42], but
introduces an empirical parameter.
The spectroscopy-oriented CI (SORCI) method [43, 44] combines the concepts
of MRCI and MRPT. More flexible than CASPT2, arbitrary molecular orbitals and
reference wave functions can be used. Instead of multiconfigurational SCF orbitals,
approximate natural orbitals are used by default, which are obtained in a preliminary MRCI calculation by diagonalizing the state-averaged density matrix. After a
first MRMP2 calculation, the first-order interacting space (FOIS) is divided into
4 Theoretical Methods
51
perturbation theory (MRPT). The latter requires the reference to already include all
important configurations such that the perturbations are small. The most prominent
version is the CASPT2 method, which employs a complete active space selfconsistent field (CASSCF) reference. The CASSCF wave function uses a full-CI
expansion for a limited set of “active” orbitals, the “inactive” orbitals are doubly
occupied or unoccupied. The energy is minimised with respect to both orbital
rotations and CI coefficients. If several states are calculated, the orbitals are often
optimised for the average of the selected CI root energies (SA-CASSCF), to avoid
root-flipping problems. In principle, the CASSCF wave function can provide a
robust description of nearly degenerate states (static correlation), but the number of
active orbitals is limited due to the factorial scaling of the full-CI problem.
Therefore, the description of electron correlation is incomplete and the missing
part of the correlation energy, usually referred to as dynamic correlation, must be
calculated to obtain quantitative results.
The most widely used way to do this is via second-order perturbation theory, in
terms of the CASPT2 method. CASPT2 can be considered size-consistent if the
active orbital space increases naturally with the system, e.g., comprises all orbitals
of an irreducible representation. Truncated MRCI and MRMP2 methods are generally not size-consistent and empirical correction schemes, e.g., the Davidson
correction [38] are applied to alleviate the problem. If the CASSCF states form a
good reference for the perturbative treatment, CASPT2 is comparable in accuracy
with CC3 [33]. Dynamic correlation can strongly contribute to excitation energies
and alter the order of excited states. In some cases, states that are well separated in
energy in the exact solution can be nearly degenerate and strongly mix in the
CASSCF solution. In these cases, the CASSCF states do not represent a good
reference for perturbation theory and the error of CASPT2 can be considerably
larger than 0.1 eV. This situation typically arises close to (avoided) crossings
between states of different character, e.g., valence and Rydberg states. The multistate CASPT2 method (MS-CASPT2) solves this problem by calculating
perturbatively an effective Hamiltonian in the space of the reference wave functions
from a SA-CASSCF calculation [39]. A recent example where a MS-CASPT2
treatment is required is the spectrum of the HBDI chromophore of the greenfluorescent protein (GFP) [40]. The same article also reports a large influence of
the zeroth-order Hamiltonian, for which several modifications have been suggested
to improve the convergence behavior of the perturbation series. Recently, the IPEA
ansatz [41] has been proposed, which eliminates a systematic underestimation of
excitation energies of the previously common G1 and G3 Hamiltonians [42], but
introduces an empirical parameter.
The spectroscopy-oriented CI (SORCI) method [43, 44] combines the concepts
of MRCI and MRPT. More flexible than CASPT2, arbitrary molecular orbitals and
reference wave functions can be used. Instead of multiconfigurational SCF orbitals,
approximate natural orbitals are used by default, which are obtained in a preliminary MRCI calculation by diagonalizing the state-averaged density matrix. After a
first MRMP2 calculation, the first-order interacting space (FOIS) is divided into
4 Theoretical Methods
51