12.7.5 ALMO-EDA for Excited States

So far we have only covered EDA methods for intermolecular interactions between molecules in their ground states. Since electronic excited states are associated with less strongly bound electrons, modified electrostatic multipole moments (due to electron transition), and often larger polarizabilities, effects imposed by other molecules can be even larger as well as less chemically intuitive than those on ground states. Furthermore, there exist systems that are weakly bound in the ground state but much more strongly bound in the electronic excited state (e.g. He${}_2$ versus He${}_2^{\ast }$). Therefore, it is very desirable to develop an interpretation tool that can be utilized to study these important phenomena that are related to intermolecular interactions involving excited-state molecules.

Ge et al. recently extended the ALMO-EDA to treat exciplexes (where the excitation can be assigned to a single molecule within a complex) ⁴⁰⁴ Ge Q., Mao Y., Head-Gordon M.
J. Chem. Phys.
(2018), 148, pp. 064105. Link and excimers (where multiple fragments contribute to the excitation) ⁴⁰³ Ge Q., Head-Gordon M.
J. Chem. Theory Comput.
(2018), 14, pp. 5156. Link computed at the CIS or TDDFT/TDA level of theory. Here we briefly overview the decomposition schemes. In the EDA for exciplexes, one first defines the interaction energy in the excited state ($\mathrm{\Delta }{E}_{\mathrm{INT}}^{\ast }$) as

where $E^{\ast }=E+\omega$ is the energy of the excited supersystem, and $E_{\mathrm{frag}}^{\ast }$ can be expressed as the sum of ground-state fragment energies and the excitation energy of one of the fragments (without losing generality, this excited fragment is denoted as fragment “1”):

Therefore, we can rewrite the excited-state interaction as

which contains contributions from the ground-state interaction energy ($\mathrm{\Delta }E=E-{\sum }_F{E}_F$) and the excitation energy ($\mathrm{\Delta }{\omega }_{\mathrm{INT}}=\omega -{\omega }_1$). Then, as in the first-generation ALMO-EDA for ground states ⁶⁴⁵ Khaliullin R. Z. et al.
J. Phys. Chem. A
(2007), 111, pp. 8753. Link , the excited-state interaction energy can be separated into contributions from frozen interaction (FRZ), polarization (POL), and charge transfer (CT):

Each term on the RHS of Eq. (12.34) can be written in a similar form as Eq. (12.33):

The terms $\mathrm{\Delta }{E}_{\mathrm{FRZ}}$, $\mathrm{\Delta }{E}_{\mathrm{POL}}$, and $\mathrm{\Delta }{E}_{\mathrm{CT}}$ can be obtained by performing a ground-state ALMO-EDA for the supersystem. To compute $\mathrm{\Delta }{\omega }_{\mathrm{FRZ}}$, $\mathrm{\Delta }{\omega }_{\mathrm{POL}}$, and $\mathrm{\Delta }{\omega }_{\mathrm{CT}}$, one needs to define ${\omega }_{\mathrm{FRZ}}$ and ${\omega }_{\mathrm{POL}}$, i.e., excitation energies associated with the frozen and polarized supersystem, respectively. The frozen intermediate state can be viewed as one excited fragment embedded in the environment formed by other ground-state fragments, whose effects on the excited fragment are only through the supersystem Fock matrix. The definition of the polarized intermediate state utilizes the ALMO-CIS model (see Section 12.15), where both MOs and excitation amplitudes are fragment-localized. We also note that the frozen contribution to the excited-state interaction energy, $\mathrm{\Delta }{E}_{\mathrm{FRZ}}^{\ast }$, can be further partitioned into a classical electrostatics term (Coulomb interactions between isolated fragment charge distributions) and a non-electrostatic term (mostly Pauli repulsion if a non-dispersion-corrected model such as CIS is used):

Modifications are needed in order to extend this method to excimers, where different fragments are of degenerate or near-degenerate excited states. In such cases, we choose $M$ reference fragment states as the initial basis. Denote the $s$th excited state on fragment $I$ as the $\kappa$th reference state ($\kappa =1,2,\mathrm{\dots },M$). Similar to Eq. (12.32), we have

The corresponding frozen excited-state wavefunction is then constructed by embedding this excited fragment into the environment formed by other fragments in their ground states:

and the excited-state frozen interaction energy

With $M$ degenerate or near-degenerate frozen excited states, a new intermediate state is then introduced to capture the pure excitonic-splitting (EXSP) effect in the formation of excimers, which can be expressed as a linear combination of the frozen states:

The associated excitation energy ${\omega }_{\mathrm{EXSP}}^{\kappa }$ and the corresponding linear combination coefficients can be obtained by solving a secular equation in the basis of frozen states. As excitonic splitting is purely an excited-state phenomenon, we have

Subsequently, polarization and charge transfer are handled in a similar way as in the excimer case:

One more complication compared to the EDA scheme for exciplexes is that since multiple ($M$) states are considered, extra caution needs be paid to the state-ordering at different stages (EXSP, POL and CT). In order to locate the states of interest (which can be most unambiguously identified at the EXSP stage) correctly during the entire EDA procedure, a state-tracking algorithm based on a maximum-overlap criterion is employed. The reader is referred to Ref.  403 Ge Q., Head-Gordon M.
J. Chem. Theory Comput.
(2018), 14, pp. 5156. Link for more details.

The ALMO-EDA for intermolecular interactions involving excited-state molecules implemented in Q-Chem 5.1 supports CIS and TDDFT within the Tamm-Dancoff approximation (TDA) for closed-shell systems, i.e., excited states calculated by TDDFT and unrestricted systems are currently not supported. The EDA procedure is triggered by setting EX_EDA = TRUE. The code first performs a customized ground-state calculation (using AO-based ALMOs) through the “EDA2” driver. During the isolated fragment calculations in this EDA, the fragment excited states are also computed after its ground-state SCF is converged, which is controlled by a new input section $frgm_cis_n_roots. The format of this input section is as follows:

$frgm_cis_n_roots
¯frgm_idx1¯¯nstates_to_calc¯¯nstates_as_exciton_basis
¯frgm_idx2¯¯nstates_to_calc¯¯nstates_as_exciton_basis
¯.  .  .
$end

Here “nstates_to_calc” specifies the number of states to calculate for each fragment (the value of CIS_N_ROOTS for each fragment calculation), and “nstates_as_exciton_basis” refers to the number of calculated fragment states that are used to construct the EXSP state (whose sum gives $M$ in Eq. (12.40)). When the supersystem is considered as an exciplex where the excitation is assigned to a specific fragment, only one row is needed in this section, and there is no need to specify the number of states used as the basis for the EXSP state.

The other relevant rem variables includes CIS_N_ROOTS, which specifies the number of roots to calculate in the ALMO-CIS/TDA and full CIS/TDA calculations, and EIGSLV_METH (see Section 12.15) that is set to 1 (using the Davidson iterative solver) by default. Note that the number of states that the EDA is concerned with is controlled by the number of isolated fragment states (the exciplex case) or the total number of states that are excitonically coupled (the excimer case). In the latter case, CIS_N_ROOTS is usually set to a value that is larger than $M$ to ensure that all states of interest are captured in the ALMO-CIS/TDA and full CIS/TDA calculations, as changes in state-ordering might occur.