10.15.3 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.⁴⁷³ In energy decomposition analysis methods,^{472, 400} 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:

	$|{\psi }_a⟩$	$={\displaystyle \frac{1}{\sqrt{(N-1)!}}}\det \{{\varphi }_{D1}^{(a)},{\varphi }_{D2}^{(a)},\mathrm{\dots },{\varphi }_{D{n}_D-1}^{(a)}{\varphi }_{A1}^{(a)},{\varphi }_{A2}^{(a)},\mathrm{\dots },{\varphi }_{A{n}_A}^{(a)}\}$		(10.109a)
	$|{\psi }_b⟩$	$={\displaystyle \frac{1}{\sqrt{(N-1)!}}}\det \{{\varphi }_{D1}^{(b)},{\varphi }_{D2}^{(b)},\mathrm{\dots },{\varphi }_{D{n}_D}^{(b)}{\varphi }_{A1}^{(b)},{\varphi }_{A2}^{(b)},\mathrm{\dots },{\varphi }_{A{n}_A-1}^{(b)}\}$		(10.109b)

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 SCF-MI technique (see Section 12.4).^{922, 290, 473} These ALMO-based diabatic states are variationally optimized such that the associated nuclear forces can be easily computed.⁶²² 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

(10.110)

and then transform that into the Löwdin-orthogonalized basis

$$𝐇={𝐒}^{-1/2}{𝐇}^{\prime }{𝐒}^{-1/2}$$

(10.111)

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

$$H_{ab}=\frac{1}{1-S_{ab}^2}\left|H_{ab}^{\prime }-\frac{H_{aa}^{\prime }+H_{bb}^{\prime }}{2}{S}_{ab}\right|$$

(10.112)

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

$$S_{ab}=⟨{\psi }_a|{\psi }_b⟩=\det [{({𝐂}_{\mathrm{o}}^{(a)})}^{†}{\mathrm{𝐒𝐂}}_{\mathrm{o}}^{(b)}].$$

(10.113)

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.^{143, 811, 624} For the diagonal elements, it is straightforward to employ the KS energies of the two diabats:

$$H_{aa}^{\prime }=E_a^{\mathrm{KS}}[{𝐏}^{(a)}],H_{bb}^{\prime }=E_b^{\mathrm{KS}}[{𝐏}^{(b)}]$$

(10.114)

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,^{143, 811}

$$H_{ab}^{\prime }=S_{ab}\left[V_{\mathrm{nn}}+{𝐏}_{ab}\cdot 𝐡+\frac{1}{2}{𝐏}_{ab}\cdot \mathrm{𝐈𝐈}\cdot {𝐏}_{ab}+\frac{1}{2}(\mathrm{\Delta }{E}_a^{\mathrm{c}}+\mathrm{\Delta }{E}_b^{\mathrm{c}})\right]$$

(10.115)

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

$${𝐏}_{ab}={𝐂}_{\mathrm{o}}^{(a)}{\left[{({𝐂}_{\mathrm{o}}^{(b)})}^{†}{\mathrm{𝐒𝐂}}_{\mathrm{o}}^{(a)}\right]}^{-1}{({𝐂}_{\mathrm{o}}^{(b)})}^{†}$$

(10.116)

The first three terms on the right-hand side of Eq. (10.115) 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.⁹⁵⁸ 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:

	$\mathrm{\Delta }{E}_a^{\mathrm{c}}$	$=E_a^{\mathrm{KS}}[{𝐏}^{(a)}]-E_a^{\mathrm{HF}}[{𝐏}^{(a)}]$		(10.117a)
	$\mathrm{\Delta }{E}_b^{\mathrm{c}}$	$=E_b^{\mathrm{KS}}[{𝐏}^{(b)}]-E_b^{\mathrm{HF}}[{𝐏}^{(b)}].$		(10.117b)

This approach was denoted as ALMO(MSDFT) in Ref. 624 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. 624, which evaluates the XC contribution using the XC energy of the symmetrized transition density matrix

$$H_{ab}^{\prime }=S_{ab}\left[V_{\mathrm{nn}}+{𝐏}_{ab}\cdot 𝐡+\frac{1}{2}{𝐏}_{ab}\cdot \mathrm{𝐈𝐈}\cdot {𝐏}_{ab}+E_{\mathrm{xc}}[{\widetilde{𝐏}}_{ab}]\right]$$

(10.118)

where

$${\widetilde{𝐏}}_{ab}=\frac{1}{2}({𝐏}_{ab}+{𝐏}_{ba}).$$

(10.119)

Note that in Eq. (10.118), $\mathrm{𝐈𝐈}$ includes only Coulomb integrals and a fraction of exact exchange if hybrid functionals are employed.

According to the benchmark results in Ref. 624, 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. (10.118) may become inaccurate when the overlap between two states becomes near-singular, as

$${𝝈}_{ba}={({𝐂}_{\mathrm{o}}^{(b)})}^{†}{\mathrm{𝐒𝐂}}_{\mathrm{o}}^{(a)}$$

(10.120)

is inverted when constructing the transition density [Eq. (10.116)]. To circumvent this numerical issue, one can replace the inverse in Eq. (10.116) with the Penrose pseudo-inverse, which was suggested for a similar objective in Ref. 729.

Besides ALMO-based diabatization method, we also implemented two other fragment-based approaches: the projection operator diabatization (POD) and the fragment orbital DFT (FODFT) approach. The POD method ⁴⁹⁴ 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:

(10.121)

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

$${\mathit{ϵ}}_d={𝐃}_d^{†}{\widetilde{𝐅}}_{dd}{𝐃}_d,{\mathit{ϵ}}_a={𝐃}_a^{†}{\widetilde{𝐅}}_{aa}{𝐃}_a,$$

(10.122)

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

(10.123)

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

$${\overline{𝐅}}_{da}={𝐃}_d^{†}{\widetilde{𝐅}}_{da}{𝐃}_a$$

(10.124)

yielding

(10.125)

The couplings between these single-particle diabats 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. 494 and 1080, where a closed-shell reference system is used to generate the Fock matrix to be processed, i.e., $𝐅$ in Eq. (10.121).

Fragment orbital DFT (FODFT) ^{699, 865, 853} 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.⁸⁵³ 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

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

  (10.126)

  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)

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

  (10.127)

• •

  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)

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

  (10.128)

The approach described above is denoted as FODFT($2\mathrm{n}-1$)@$D^{+}A$ (HT) / FODFT($2\mathrm{n}+1$)@$D^{-}A$ (ET) ⁸⁵³ 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($2n$)@$DA$:⁸⁶⁵ 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): ⁶⁹⁹ 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,^{853, 624} 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. ⁶²⁴ 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 implementation of ALMO(MSDFT) is limited to ground state electron/hole transfer involving two states. We will further generalize the implementation of these methods in future releases.

$molecule
1 2
--
1 2
  C      0.000000    0.000000    0.000000
  C      1.332000    0.000000    0.000000
  H     -0.574301    0.000000   -0.928785
  H     -0.574301    0.000000    0.928785
  H      1.906301    0.000000    0.928785
  H      1.906301    0.000000   -0.928785
--
0 1
  C     -0.000000    4.000000    0.000000
  C      1.332000    4.000000   -0.000000
  H     -0.574301    4.000000    0.928785
  H     -0.574301    4.000000   -0.928785
  H      1.906301    4.000000   -0.928785
  H      1.906301    4.000000    0.928785
$end

$rem
  JOBTYPE  SP
  METHOD   PBE0
  BASIS    6-31+G(D)
  UNRESTRICTED  TRUE
  THRESH  14
  SCF_CONVERGENCE  8
  SYMMETRY   FALSE
  SYM_IGNORE  TRUE
  SCFMI_MODE   1
  FRGM_METHOD  STOLL
  FRAG_DIABAT_METHOD ALMO_MSDFT
$end

$almo_coupling
  1  2
  0  1
  --
  0  1
  1  2
$end

$molecule
0 1
--
0 1
  C      0.000000    0.000000    0.000000
  C      1.332000    0.000000    0.000000
  H     -0.574301    0.000000   -0.928785
  H     -0.574301    0.000000    0.928785
  H      1.906301    0.000000    0.928785
  H      1.906301    0.000000   -0.928785
--
0 1
  C     -0.000000    4.000000    0.000000
  C      1.332000    4.000000   -0.000000
  H     -0.574301    4.000000    0.928785
  H     -0.574301    4.000000   -0.928785
  H      1.906301    4.000000   -0.928785
  H      1.906301    4.000000    0.928785
$end

$rem
  jobtype sp
  frag_diabat_method pod
  method  lrc-wpbeh
  basis   6-31+g(d)
  scf_convergence 8
  thresh 14
  symmetry false
  sym_ignore true
$end

$molecule
1 2
--
1 2
  C      0.000000    0.000000    0.000000
  C      1.332000    0.000000    0.000000
  H     -0.574301    0.000000   -0.928785
  H     -0.574301    0.000000    0.928785
  H      1.906301    0.000000    0.928785
  H      1.906301    0.000000   -0.928785
--
0 1
  C     -0.000000    4.000000    0.000000
  C      1.332000    4.000000   -0.000000
  H     -0.574301    4.000000    0.928785
  H     -0.574301    4.000000   -0.928785
  H      1.906301    4.000000   -0.928785
  H      1.906301    4.000000    0.928785
$end

$rem
  jobtype sp
  method  wb97x-d
  basis   6-31+g(d)
  unrestricted true
  scf_convergence 8
  thresh 14
  symmetry false
  sym_ignore true
  frag_diabat_method fodft
  fodft_method 1
$end