12.11.1 Theory

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${}_2$ vs. 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) ²⁸⁶ and excimers (where multiple fragments contribute to the excitation) ²⁸⁵ 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

$$\mathrm{\Delta }{E}_{\mathrm{INT}}^{\ast }=E^{\ast }-E_{\mathrm{frag}}^{\ast }$$

(12.27)

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"):

$$E_{\mathrm{frag}}^{\ast }=\sum _F{E}_F+{\omega }_1$$

(12.28)

Therefore, we can rewrite the excited-state interaction as

$$\mathrm{\Delta }{E}_{\mathrm{INT}}^{\ast }=\mathrm{\Delta }{E}_{\mathrm{INT}}+\mathrm{\Delta }{\omega }_{\mathrm{INT}}$$

(12.29)

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 ⁴⁷², the excited-state interaction energy can be separated into contributions from frozen interaction (FRZ), polarization (POL), and charge transfer (CT):

$$\mathrm{\Delta }{E}_{\mathrm{INT}}^{\ast }=\mathrm{\Delta }{E}_{\mathrm{FRZ}}^{\ast }+\mathrm{\Delta }{E}_{\mathrm{POL}}^{\ast }+\mathrm{\Delta }{E}_{\mathrm{CT}}^{\ast }$$

(12.30)

Each term on the RHS of eq. 12.30 can be written in a similar form as eq. 12.29:

(12.31)

$\mathrm{\Delta }{E}_{\mathrm{FRZ}}$, $\mathrm{\Delta }{E}_{\mathrm{POL}}$, 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 Sec. 12.19), 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 (e.g. CIS) is used]:

$$\mathrm{\Delta }{E}_{\mathrm{FRZ}}^{\ast }=\mathrm{\Delta }{E}_{\text{CLS-ELEC}}^{\ast }+\mathrm{\Delta }{E}_{\text{NON-ELEC}}^{\ast }$$

(12.32)

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 $s^{\mathrm{th}}$ excited state on fragment $I$ as the ${\kappa }^{\mathrm{th}}$ reference state ($\kappa =1,2,\mathrm{\dots },M$). Similar to eqn. 12.28, we have

$$E_{\mathrm{frag}}^{\kappa }=\sum _F{E}_F+{\omega }_I^s$$

(12.33)

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:

$$|{\mathrm{\Phi }}_{\mathrm{FRZ}}^{\kappa }⟩=|{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2\mathrm{\cdots }{\mathrm{\Psi }}_I^s\mathrm{\cdots }{\mathrm{\Psi }}_N⟩$$

(12.34)

and the excited-state frozen interaction energy

$$\mathrm{\Delta }{E}_{\mathrm{FRZ}}^{\kappa }=\mathrm{\Delta }{E}_{\mathrm{FRZ}}+\mathrm{\Delta }{\omega }_{\mathrm{FRZ}}^{\kappa }=\mathrm{\Delta }{E}_{\mathrm{FRZ}}+({\omega }_{\mathrm{FRZ}}^{\kappa }-{\omega }_I^s)$$

(12.35)

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:

$$|{\mathrm{\Phi }}_{\mathrm{EXSP}}^{\kappa }⟩=\sum _{{\kappa }^{\prime }}^M{c}^{\kappa {\kappa }^{\prime }}|{\mathrm{\Psi }}_1{\mathrm{\Psi }}_2\mathrm{\cdots }{\mathrm{\Psi }}_I^s\mathrm{\cdots }{\mathrm{\Psi }}_N⟩$$

(12.36)

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

$$\mathrm{\Delta }{E}_{\mathrm{EXSP}}^{\kappa }=\mathrm{\Delta }{\omega }_{\mathrm{EXSP}}^{\kappa }={\omega }_{\mathrm{EXSP}}^{\kappa }-{\omega }_{\mathrm{FRZ}}^{\kappa }$$

(12.37)

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

(12.38)

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. 285 for more details.