The problems with CT excitations are alleviated but not solved by partially
substituting the DFT exchange by nonlocal HF exchange. Using such hybrid
functionals, the dependency of excitation energies on the amount of HF exchange
represents a good test for the validity of the results. An a priori criterion has been
proposed by Peach et al. [25]. To completely eliminate the problem, rangeseparated hybrid methods have been proposed [26], which evaluate exchange
integrals using the short-range part of a local density functional and combine it
with the long-range part of HF exchange. Known as Coulomb-attenuation method
(CAM) [27] or long-range corrected (LC) functionals [28], this approach has been
shown to produce accurate results for CT, Rydberg, and local excitations [29–31].
4.2.2 Single-Reference Methods
Single-reference methods use the HF wave function as a starting point to describe
electron correlation. Although variational methods based on truncated configuration interaction (CI) expansions are in principal more robust than many-body
perturbation theory (MBPT), they are not size-extensive, i.e., total energies are
not size consistent. For example, the energy of two separated molecules is not the
sum of the monomer energies. Coupled-cluster methods pursue the goal to be both
size-extensive and achieve a faster convergence towards the full-CI limit. Excitation energies are obtained from linear-response or equation-of-motion (EOM)
theory. In analogy to the CI series, including singles, doubles, triples, etc.
excitations (CIS, CISD, CISDT, . . .), they build a hierarchy towards full-CI:
CCS, CC2, CCSD, CC3, CCSDT, . . . [32]. Their scaling with system size N is to
the power of 4, 5, 6, 7, 8, respectively. For most applications to larger molecules,
CC2 and CCSD are the methods of choice nowadays, while CC3 is feasible for
smaller dyes. The accuracy of excitation energies depends on the character of the
excited state. For excited states that are dominated by single excitations, CC2 gives
energies that are correct to second order and competes well with CCSD [33]. For
states with strong contributions of double excitations, CCSD is the lowest level of
theory that is applicable. Efficient parallelised implementations of CC2 make this
method attractive as an alternative to TDDFT [34]. The performance is usually
superior for local valence excitations and describes CT and Rydberg excitations
reasonably well [33, 35, 36]. In close vicinity to state crossings, EOM-CC methods
produce artifacts in the PES due to their non-Hermiticity, which may cause
problems in the optimisation of conical intersection seams [37].
4.2.3 Multi-Reference Methods
The idea of multi-reference methods is to substitute the HF reference wave function
in single-reference methods by a (small) multiconfigurational one to achieve a
qualitative description of ground and excited states. The remaining part of the
correlation, referred to as dynamic correlation, is described by extending the
50
M. Wanko and A. Rubio
substituting the DFT exchange by nonlocal HF exchange. Using such hybrid
functionals, the dependency of excitation energies on the amount of HF exchange
represents a good test for the validity of the results. An a priori criterion has been
proposed by Peach et al. [25]. To completely eliminate the problem, rangeseparated hybrid methods have been proposed [26], which evaluate exchange
integrals using the short-range part of a local density functional and combine it
with the long-range part of HF exchange. Known as Coulomb-attenuation method
(CAM) [27] or long-range corrected (LC) functionals [28], this approach has been
shown to produce accurate results for CT, Rydberg, and local excitations [29–31].
4.2.2 Single-Reference Methods
Single-reference methods use the HF wave function as a starting point to describe
electron correlation. Although variational methods based on truncated configuration interaction (CI) expansions are in principal more robust than many-body
perturbation theory (MBPT), they are not size-extensive, i.e., total energies are
not size consistent. For example, the energy of two separated molecules is not the
sum of the monomer energies. Coupled-cluster methods pursue the goal to be both
size-extensive and achieve a faster convergence towards the full-CI limit. Excitation energies are obtained from linear-response or equation-of-motion (EOM)
theory. In analogy to the CI series, including singles, doubles, triples, etc.
excitations (CIS, CISD, CISDT, . . .), they build a hierarchy towards full-CI:
CCS, CC2, CCSD, CC3, CCSDT, . . . [32]. Their scaling with system size N is to
the power of 4, 5, 6, 7, 8, respectively. For most applications to larger molecules,
CC2 and CCSD are the methods of choice nowadays, while CC3 is feasible for
smaller dyes. The accuracy of excitation energies depends on the character of the
excited state. For excited states that are dominated by single excitations, CC2 gives
energies that are correct to second order and competes well with CCSD [33]. For
states with strong contributions of double excitations, CCSD is the lowest level of
theory that is applicable. Efficient parallelised implementations of CC2 make this
method attractive as an alternative to TDDFT [34]. The performance is usually
superior for local valence excitations and describes CT and Rydberg excitations
reasonably well [33, 35, 36]. In close vicinity to state crossings, EOM-CC methods
produce artifacts in the PES due to their non-Hermiticity, which may cause
problems in the optimisation of conical intersection seams [37].
4.2.3 Multi-Reference Methods
The idea of multi-reference methods is to substitute the HF reference wave function
in single-reference methods by a (small) multiconfigurational one to achieve a
qualitative description of ground and excited states. The remaining part of the
correlation, referred to as dynamic correlation, is described by extending the
50
M. Wanko and A. Rubio