10.14.3 Fragment-Based Methods for Electronic Coupling

One can use absolutely localized molecular orbitals (ALMOs, see Chapter 12) to construct charge-localized diabatic states directly from DFT calculations. The ALMOs on each fragment are expanded by the AO basis functions belonging to the same fragment alone, whose corresponding MO coefficient matrix is fragment block-diagonal.⁵⁶⁴ In energy decomposition analysis methods,^{563, 486} ALMOs are utilized to separate the effects of polarization and charge transfer in intermolecular binding, because they have the useful property that they do not allow for charge transfer between fragments under the Mulliken definition of charge population. Making use of this property, one can construct charge-localized diabats for hole and electron transfer. For example, considering the initial and final states of a hole transfer process, $|D^{+}A⟩$ and $|D{A}^{+}⟩$, the two diabats can be represented in the following form:

For systems where the donor and acceptor moieties are well-separated, one can construct the ALMO-based diabats by simply concatenating orbitals obtained from isolated fragment calculations: $D^{+}$ and $A$ for one diabat, and $D$ and $A^{+}$ for the other. The energy of each ALMO diabat can then be variationally optimized with respect to orbital rotations on fragment, using the SCFMI technique (see Section 12.4).^{1085, 359, 564} These ALMO-based diabatic states are variationally optimized such that the associated nuclear forces can be easily computed.⁷³⁶ The mutual polarization of donor and acceptor moieties in the presence of each other is also taken into account.

To calculate the electronic coupling between two ALMO diabats, one should first construct the diabatic Hamiltonian in the ALMO state basis

and then transform that into the Löwdin-orthogonalized basis

whose off-diagonal element, $H_{ab}$, corresponds to the diabatic coupling to be evaluated. In the 2-state case, we have

which requires the overlap between two ALMO diabats and the diagonal and off-diagonal elements of ${𝐇}^{\prime }$. The interstate overlap is given by

where ${𝐂}_{\mathrm{o}}^{(a)}$ and ${𝐂}_{\mathrm{o}}^{(b)}$ are MO coefficients for the occupied orbitals in diabats $|{\psi }_a⟩$ and $|{\psi }_b⟩$, respectively, and $𝐒$ is the AO overlap matrix.

The elements of the diabatic Hamiltonian matrix can be evaluated using the multi-state DFT (MSDFT) approach.^{179, 955, 740} For the diagonal elements, it is straightforward to employ the KS energies of the two diabats:

where ${𝐏}^{(a)}$ and ${𝐏}^{(b)}$ are the one-electron density matrices associated with two ALMO states $|{\psi }_a⟩$ and $|{\psi }_b⟩$, respectively. The approximation for the off-diagonal element is theoretically more challenging. In the original MSDFT scheme,^{179, 955}

where ${𝐏}_{ab}$ is the one-particle transition density matrix between two ALMO states

The first three terms on the right-hand side of Eq. (10.134) correspond to the contributions from nuclear repulsion, one-electron Hamiltonian (kinetic energy and nuclei-electron attraction), and full two-electron integrals (Coulomb and full HF exchange), which can be derived as in non-orthogonal CI.¹¹²⁰ The last term accounts for the contribution from exchange-correlation (XC) functional as a correction to the HF coupling, which is given by the average of the difference between the KS and HF energies calculated from the same one-electron density matrix for each diabat:

This approach was denoted as ALMO(MSDFT) in Ref. 740 and it was found to overestimate the electronic couplings for the tested hole and electron transfer systems. A modified approach, denoted as ALMO(MSDFT2), was proposed in Ref. 740, which evaluates the XC contribution using the XC energy of the symmetrized transition density matrix

where

Note that in Eq. (10.137), $\mathrm{𝐈𝐈}$ includes only Coulomb integrals and a fraction of exact exchange if hybrid functionals are employed.

According to the benchmark results in Ref. 740, ALMO(MSDFT2) shows better accuracy than the original MSDFT method for hole and electron transfer, and thus it is implemented as the default approach to compute electronic couplings between ALMO diabats in Q-Chem. We note that the results given by Eq. (10.137) may become inaccurate when the overlap between two states becomes near-singular, as

is inverted when constructing the transition density [Eq. (10.135)]. To circumvent this numerical issue, one can replace the inverse in Eq. (10.135) with the Penrose pseudo-inverse, which was suggested for a similar objective in Ref. 863.

Besides ALMO-based diabatization method, other fragment-based diabatization methods are available in Q-Chem. The projection operator diabatization (POD) method ⁵⁸² starts from a standard KS-DFT calculation of the system and post-processes the converged Fock matrix. It first transforms the Fock matrix into the Löwdin-orthogonalized AO basis and then partitions that into the donor and acceptor blocks, assuming that these orthogonalized AO basis functions still retain their original fragment tags:

One then diagonalizes ${\widetilde{𝐅}}_{dd}$ and ${\widetilde{𝐅}}_{aa}$ separately

where the eigenvectors ${𝐃}_d$ and ${𝐃}_a$ define the single-particle “diabatic states”:

and transforms the off-diagonal block of the Fock matrix into this diabatic basis

yielding

The couplings between these single-particle diabatic orbitals can then be directly read off from the elements of ${\overline{𝐅}}_{da}$.

The Q-Chem implementation of the POD method follows the description in Refs. 582 and 1250, where a closed-shell reference system is used to generate the Fock matrix to be processed, i.e., $𝐅$ in Eq. (10.140). By default, only the D(HOMO) - A(HOMO) coupling is calculated for the hole transfer cases, and the D(LUMO) - A(LUMO) coupling for the electron transfer cases. To calculate the couplings between multiple pairs of donor and acceptor orbitals, the user can set $rem variable POD_MULTI_PAIRS to TRUE and control the number of orbitals pairs through POD_WINDOW. See the instruction in Sec. 10.14.3.4.

Because of the use of globally Löwdin-orthogonalized orbitals in Eq. (10.140), the diabatic orbitals created by POD cannot be strictly localized on fragments. This renders the POD results unstable with the change of employed AO basis sets: when larger basis sets are used, the mixing between AO basis functions on different fragments becomes stronger, and the resulting $H_{ab}$ decreases. To alleviate this problem, a revised POD method, which was named as “POD2”, was proposed by Ghan et al.. ³⁵⁶ It avoids the global Löwdin-orthogonalization of the AO basis; instead, it separately diagonalizes the the donor and acceptor blocks of the Fock matrix (in the original AO basis):

The obtained diabatic MO coefficient matrix is fragment-block-diagonal in the AO basis:

Transforming the AO Fock matrix into this diabatic MO basis, the D-D and A-A blocks of the resulting matrix are diagonal matrices:

Using the matrix elements in the off-diagonal block (${\overline{𝐅}}_{da}$) directly would yield overestimated couplings since the diabatic MOs ${𝐂}_d$ and ${𝐂}_a$ are not orthogonal to each other. Therefore, a final orthogonalization step is required to obtain the diabatic coupling between a pair of orbitals that are located on the donor and acceptor, respectively. Denoting this pair of orbital as ${\varphi }_d$ and ${\varphi }_a$, one can construct the $2\times 2$ Hamiltonian and overlap matrices:

Two orthogonalization schemes have been investigated by Ghan et al.. ³⁵⁶ The first approach performs a Löwdin orthgonalization on ${\varphi }_d$ and ${\varphi }_a$, which is denoted as POD2L. The resulting coupling between the orthogonalized diabatic orbitals are

The second approach employs the Gram-Schmidt orthogonalization, which keeps one of the two orbitals (${\varphi }_d$ or ${\varphi }_a$) intact while ensures that the other is strictly orthogonal to it. This approach is denoted as POD2GS, and it might be better choice for asymmetric cases (e.g. surface and adsorbates) where one can choose to retain the orbital on the less sizable fragment. These two POD2 variants afford significantly improved accuracy over the original POD method, especially in terms of the robustness with regard to the use of extensive basis sets.

Fragment orbital DFT (FODFT) ^{828, 1018, 1005} is an approach to compute the diabatic couplings for hole and electron transfer between fragments. There have been several different flavors of FODFT approaches developed in literature, and here we introduce the most recent variant by Schober et al.¹⁰⁰⁵ Considering a hole transfer process $D^{+}+A\to D+A^{+}$ or an electron transfer process $D^{-}+A\to D+A^{-}$, where the donor ($D$) and acceptor ($A$) fragments have $n_D$ and $n_A$ electrons, respectively, the procedure is as follows:

• •

  Perform KS-DFT calculations for isolated donor and acceptor fragments; collect the converged fragment orbitals:
  $\{{\varphi }_{D1},{\varphi }_{D2},\mathrm{\dots },{\varphi }_{D{n}_D\pm 1}\}$ and $\{{\varphi }_{A1},{\varphi }_{A2},\mathrm{\dots },{\varphi }_{A{n}_A}\}$

• •

  Löwdin-orthogonalize the occupied orbitals on two fragments. The reactant diabat ($D^{+}A$ or $D^{-}A$) can be represented as

  $$|{\overline{\psi }}_a⟩=\frac{1}{\sqrt{(N-1)!}}\det \{{\overline{\varphi }}_{D1},{\overline{\varphi }}_{D2},\mathrm{\dots },{\overline{\varphi }}_{D{n}_D\pm 1}{\overline{\varphi }}_{A1},{\overline{\varphi }}_{A2},\mathrm{\dots },{\overline{\varphi }}_{A{n}_A}\}$$

  (10.150)

  where “$\overline{\varphi }$” denotes Löwdin-orthogonalized orbitals, and $N=n_D+n_A$. Note that the lowest unoccupied orbital where the electron is transferring to, ${\varphi }_{D{n}_D}$ in the case of HT or ${\varphi }_{A{n}_A+1}$ in the case of ET, also needs to be made orthogonal to the space spanned by all occupied orbitals.

• •

  Construct the product diabat ($D{A}^{+}$ or $D{A}^{-}$), simply by moving the hole from ${\overline{\varphi }}_{D{n}_D}$ to ${\overline{\varphi }}_{A{n}_A}$ (HT), or the excess electron from ${\overline{\varphi }}_{D{n}_D+1}$ to ${\overline{\varphi }}_{A{n}_A+1}$ (ET)

  $$|{\overline{\psi }}_b⟩=\frac{1}{\sqrt{(N-1)!}}\det \{{\overline{\varphi }}_{D1},{\overline{\varphi }}_{D2},\mathrm{\dots },{\overline{\varphi }}_{D{n}_D}{\overline{\varphi }}_{A1},{\overline{\varphi }}_{A2},\mathrm{\dots },{\overline{\varphi }}_{A{n}_A\pm 1}\}$$

  (10.151)

• •

  Compute the electronic coupling between $|{\overline{\psi }}_a⟩$ and $|{\overline{\psi }}_b⟩$, which is approximated by the coupling of the orthogonalized fragment orbitals through the Kohn-Sham Fock operator (built from the reactant diabat)

  $⟨{\overline{\psi }}_a|\widehat{H}|{\overline{\psi }}_b⟩\approx \{\begin{array}{cc}⟨{\varphi }_{D{n}_D}|{\widehat{f}}_{\mathrm{KS}}|{\varphi }_{A{n}_A}⟩,\mathrm{HT}\hfill & \\ ⟨{\varphi }_{D{n}_D+1}|{\widehat{f}}_{\mathrm{KS}}|{\varphi }_{A{n}_A+1}⟩,\mathrm{ET}\hfill & \end{array}$

  (10.152)

The approach described above is denoted as FODFT($2\mathrm{n}-1$)@$D^{+}A$ (HT) / FODFT($2\mathrm{n}+1$)@$D^{-}A$ (ET) ¹⁰⁰⁵ as the charged fragment is explicitly taken into account when preparing the fragment orbitals and the KS Fock matrix is built from $2n\mp 1$ occupied orbitals. Besides this, there are two other variants of FODFT:

1. 1.

   FODFT($2n$)@$DA$:¹⁰¹⁸ fragment orbitals prepared with $D$ and $A$ both closed-shell; KS Fock operator constructed from $2n$ occupied orbitals

2. 2.

   FODFT($2\mathrm{n}-1$)@$DA$ (HT) / FODFT($2\mathrm{n}+1$)@$D^{-}{A}^{-}$ (ET): ⁸²⁸ fragment orbitals prepared with the system having one excess electron ($DA$ for HT and $D^{-}{A}^{-}$ for ET), while one occupied orbital is removed when building the KS Fock operator

According to the benchmark results,^{1005, 740} FODFT($2\mathrm{n}-1$)@$D^{+}A$ (HT) / FODFT($2\mathrm{n}+1$)@$D^{-}A$ (ET) is the best-performing method, possibly because of its explicit account for charged fragments and consistent electron count in the preparation of fragment orbitals and in the construction of Fock matrix.

One issue associated with the FODFT methods is that for asymmetric systems, the results would depend on how one chooses the initial and final states for an electron or hole transfer process (e.g. $D^{+}A$ vs. $D{A}^{+}$), especially for the two variants that build the Fock matrix with $2n\pm 1$ occupied orbitals.⁷⁴⁰ The Q-Chem implementation of FODFT($2\mathrm{n}-1$)@$DA$ / FODFT($2\mathrm{n}+1$)@$D^{-}{A}^{-}$ automatically computes $H_{ab}$ in both ways and then reports the average, as it only requires an extra Fock matrix build. This, however, is not automatically done for FODFT($2\mathrm{n}-1$)@$D^{+}A$ / FODFT($2\mathrm{n}+1$)@$D^{-}A$.

POD, FODFT, and ALMO(MSDFT) calculations in Q-Chem require specification of fragments in the $molecule section (see Sec. 12.2). For ALMO(MSDFT) calculations, one also needs to specify the charge and multiplicity of each fragment in each diabatic state in the $almo_coupling section, where two hyphens indicate the separation of different diabats:

$almo_coupling
   charge_frag_1     mult_frag_1        !diabat 1
   charge_frag_2     mult_frag_2
   --
   charge_frag_1     mult_frag_1        !diabat 2
   charge_frag_2     mult_frag_2
$end

The current implementation of FODFT is limited to hole transfer between the HOMOs of two fragments or electron transfer between the LUMOs, and the current simplementation of ALMO(MSDFT) is limited to ground state electron or hole transfer involving two states.