10.15.1 Eigenstate-Based Methods

Under the two-state approximation, the diabatic reactant and product states are assumed to be a linear combination of the eigenstates. For ET, the choice of such linear combination is determined by a zero transition dipoles (GMH) or maximum charge differences (FCD). In the latter, a $2\times 2$ donor–acceptor charge difference matrix, $\mathrm{\Delta }𝐪$, is defined, with elements

where ${\rho }_{mn}(𝐫)$ is the matrix element of the density operator between states $|m⟩$ and $|n⟩$.

For EET, a maximum excitation difference is assumed in the FED, in which an excitation difference matrix is similarly defined with elements

where ${\rho }_{\mathrm{ex}}^{(mn)}(𝐫)$ is the sum of attachment and detachment densities for transition $|m⟩\to |n⟩$, as they correspond to the electron and hole densities in an excitation. In the FSD, a maximum spin difference is used and the corresponding spin difference matrix is defined with its elements as,

where ${\sigma }_{mn}(𝐫)$ is the spin density, difference between $\alpha$-spin and $\beta$-spin densities, for transition from $|m⟩\to |n⟩$.

Since Q-Chem uses a Mulliken population analysis for the integrations in Eqs. (10.89), (10.90), and (10.91), the matrices $\mathrm{\Delta }𝐪$, $\mathrm{\Delta }𝐱$ and $\mathrm{\Delta }𝐬$ are not symmetric. To obtain a pair of orthogonal states as the diabatic reactant and product states, $\mathrm{\Delta }𝐪$, $\mathrm{\Delta }𝐱$ and $\mathrm{\Delta }𝐬$ are symmetrized in Q-Chem. Specifically,

The final coupling values are obtained as listed below:

• •

  For GMH,

  $$V_{\text{ET}}=\frac{(E_n-E_m)\left|{\overrightarrow{\mu }}_{mn}\right|}{\sqrt{{({\overrightarrow{\mu }}_m-{\overrightarrow{\mu }}_n)}^2+4{\left|{\overrightarrow{\mu }}_{mn}\right|}^2}}$$

  (10.93)

• •

  For FCD,

  $$V_{\text{ET}}=\frac{(E_n-E_m){\overline{\mathrm{\Delta }q}}_{mn}}{\sqrt{{(\mathrm{\Delta }{q}_m-\mathrm{\Delta }{q}_n)}^2+4{\overline{\mathrm{\Delta }q}}_{mn}^2}}$$

  (10.94)

• •

  For FED,

  $$V_{\text{EET}}=\frac{(E_n-E_m){\overline{\mathrm{\Delta }x}}_{mn}}{\sqrt{{(\mathrm{\Delta }{x}_m-\mathrm{\Delta }{x}_n)}^2+4{\overline{\mathrm{\Delta }x}}_{mn}^2}}$$

  (10.95)

• •

  For FSD,

  $$V_{\text{EET}}=\frac{(E_n-E_m){\overline{\mathrm{\Delta }s}}_{mn}}{\sqrt{{(\mathrm{\Delta }{s}_m-\mathrm{\Delta }{s}_n)}^2+4{\overline{\mathrm{\Delta }s}}_{mn}^2}}$$

  (10.96)

Q-Chem provides the option to control FED, FSD, FCD and GMH calculations after a single-excitation calculation, such as CIS and TDDFT. To obtain ET coupling values using GMH (FCD) scheme, one should set $rem variables STS_GMH (STS_FCD) to be TRUE. Similarly, a FED (FSD) calculation is turned on by setting the $rem variable STS_FED (STS_FSD) to be TRUE. In FCD, FED and FSD calculations, the donor and acceptor fragments are defined via the $rem variables STS_DONOR and STS_ACCEPTOR. It is necessary to arrange the atomic order in the $molecule section such that the atoms in the donor (acceptor) fragment is in one consecutive block. The ordering numbers of beginning and ending atoms for the donor and acceptor blocks are included in $rem variables STS_DONOR and STS_ACCEPTOR.

The couplings will be calculated between all choices of excited states with the same spin. In FSD, FCD and GMH calculations, the coupling value between the excited and reference (ground) states will be included, but in FED, the ground state is not included in the analysis. It is important to select excited states properly, according to the distribution of charge or excitation, among other characteristics, such that the coupling obtained can properly describe the electronic coupling of the corresponding process in the two-state approximation.

Within the ambit of the single excitation theory such as the CIS or TDDFT, one can easily obtain analytical expressions for the matrix elements of the excitation density and can therefore, use Eq. 10.95 to compute electronic couplings between adiabatic states. However, for multiexcitation wavefunctions such as those obtained from RAS-CI no simple expressions exist for the off-diagonal elements in the excitation difference ($\mathrm{\Delta }{x}_{mn}$ in Eq. 10.95). To circumvent this challenge, a new scheme was developed known as $\theta$-FED  ⁶⁴⁶ Kue K. Y., Claudio G. C., Hsu C.-P.
J. Chem. Theory Comput.
(2018), 14, pp. 1304. Link ^{, 729} Lin H.-H. et al.
J. Chem. Theory Comput.
(2019), 15, pp. 2246. Link ^{, 769} Manjanath A. et al.
J. Chem. Theory Comput.
(2022), 18, pp. 1017. Link . In this approach, the diabatic states are assumed to be functions of a mixing angle $\theta$. Consequently, the excitation difference density ($𝚫𝐱$ in Eqs 10.90 and 10.95) is dependent on $\theta$. In order to obtain ‘ideal’ diabatic states, a scan of $\theta$ is performed from $-\pi /4$ to $\pi /4$ to maximize the difference of the excitation, i.e.,

with ‘i’ and ‘f’ indicating the initial and final diabatic states, respectively. The corresponding $\theta$-dependent coupling can then be written as

Fortunately, one can still use Eq. 10.94 to compute ET couplings between two adiabatic states for FCD with RAS-CI. This is because the charge difference matrix ($𝚫𝐪$ in Eqs 10.89 and 10.94) depends on the one-particle (for $\mathrm{\Delta }{q}_{m/n}$) and transition density matrices (for $\mathrm{\Delta }{q}_{mn/nm}$), which are also easily obtainable with the RAS-CI wavefunctions.

The $rem variables STS_FED, STS_FCD, STS_DONOR, and STS_ACCEPTOR also apply to FCD and FED calculations with RAS-CI.

When dealing with multiple charge or electronic excitation centers, diabatic states can be constructed with Boys ¹¹⁶⁰ Subotnik J. E. et al.
J. Chem. Phys.
(2008), 129, pp. 244101. Link or Edmiston-Ruedenberg ¹¹⁵⁶ Subotnik J. E. et al.
J. Chem. Phys.
(2009), 130, pp. 234102. Link localization. In this case, we construct diabatic states $\left\{|{\mathrm{\Xi }}_I⟩\right\}$ as linear combinations of adiabatic states $\left\{|{\mathrm{\Phi }}_I⟩\right\}$ with a general rotation matrix $𝐔$ that is $N_{\mathrm{state}}\times N_{\mathrm{state}}$ in size:

The adiabatic states can be produced with any method, in principle, but the Boys/ER-localized diabatization methods have been implemented thus far only for CIS, TDDFT or RASCI (section 7.12.6) methods in Q-Chem. In analogy to orbital localization, Boys-localized diabatization corresponds to maximizing the charge separation between diabatic state centers:

Here, $\overrightarrow{\mu }$ represents the dipole operator. ER-localized diabatization prescribes maximizing self-interaction energy:

where the density operator at position $\overrightarrow{ℛ}$ is

Here, ${\overrightarrow{r}}^{(j)}$ represents the position of the $j$th electron.

These models reflect different assumptions about the interaction of our quantum system with some fictitious external electric field/potential: $(i)$ if we assume a fictitious field that is linear in space, we arrive at Boys localization; $(ii)$ if we assume a fictitious potential energy that responds linearly to the charge density of our system, we arrive at ER localization. Note that in the two-state limit, Boys localized diabatization reduces nearly exactly to GMH. ¹¹⁶⁰ Subotnik J. E. et al.
J. Chem. Phys.
(2008), 129, pp. 244101. Link

As written down in Eq. (10.100), Boys localized diabatization applies only to charge transfer, not to energy transfer. Within the context of CIS or TDDFT calculations, one can easily extend Boys localized diabatization ¹¹⁵⁹ Subotnik J. E. et al.
J. Phys. Chem. A
(2010), 114, pp. 8665. Link by separately localizing the occupied and virtual components of $\overrightarrow{\mu }$, ${\overrightarrow{\mu }}^{\mathrm{occ}}$ and ${\overrightarrow{\mu }}^{\mathrm{virt}}$:

where

and the occupied/virtual components are defined by

Note that when we maximize the Boys OV function, we are simply performing Boys-localized diabatization separately on the electron attachment and detachment densities.

Finally, for energy transfer, it can be helpful to understand the origin of the diabatic couplings. To that end, we now provide the ability to decompose the diabatic coupling between diabatic states into Coulomb (J), Exchange (K) and one-electron (O) components: ¹²³⁷ Vura-Weis J. et al.
J. Phys. Chem. C
(2010), 114, pp. 20449. Link