Density Functional Physicality in Electronic Coupling Estimation

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Letter

Density Functinoal Physicality in Electronic Coupling Estimation: Benchmarks and Error Analysis Hyungjun Kim, Theodore Goodson III, and Paul M. Zimmerman J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b01434 • Publication Date (Web): 29 Jun 2017 Downloaded from http://pubs.acs.org on July 3, 2017

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The Journal of Physical Chemistry Letters

Density Functional Physicality in Electronic Coupling Estimation: Benchmarks and Error Analysis Hyungjun Kim, Theodore Goodson, III, and Paul M. Zimmerman∗ Department of Chemistry, Ann Arbor, Michigan 48109, United States E-mail: [email protected]

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Abstract Electronic coupling estimates from constrained density functional theory configuration interaction (CDFT-CI) depend critically on choice of density functional. In this letter, the orbital multi-electron self-interaction error (OMSIE), vertical electron affinity (VEA), and vertical ionization potential (VIP) are shown to be the key indicators inherited from the density functional that determine the accuracy of electronic coupling estimates.

An error metric η is derived to connect the three

properties, based on the linear proportionality between electronic coupling and overlap integral, and the hypothesis that slope of this line is a function of VEA/VIP, test Xset 1 η= |−VERef ×OMSIE+∆VE−∆VE×OMSIE|i . Based on η, BH&HLYP Ntest set i and LRC-ωPBEh are suggested as the best functionals for electron and hole transfer, respectively. Error metric η is therefore a useful predictor of errors in CDFT-CI electronic coupling, showing that the physical correctness of the density functional has a direct effect on the accuracy of the electronic coupling.

Graphical TOC Entry E(N-1)

Localization Error Energy

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Hab

Delocalization Error E(N) N-1

e-

VEA

Number of electrons

N

VIP

e-


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Electron transfer (ET) is a fundamental chemical reaction that is vital in batteries, 1 solar cells, 2 organic semiconductors, 3 and biological processes. 4 The rate of ET can be expressed 1 (λ + ∆G◦ )2 2π |Hab |2 √ exp(− ) , where the via the celebrated Marcus theory, 5,6 kET = h ¯ 4λkb T 4πλkb T electronic coupling, Hab , is the key quantity of interest alongside the reorganization energy, λ, and the change in free energy, ∆G◦ . Hab is the off-diagonal element of the two-state, ˆ b >, and represents the strength of interaction across diabatic Hamiltonian, Hab =< ψa |H|ψ the donor-acceptor pair. Hab , as well as λ and ∆G◦ , have clear physical interpretations, giving an excellent correspondence principle connecting theoretical models for ET to physical reality. To maintain this validity, estimating Hab should also depend on clear physical principles. In recent years, there has been special interest in using quantum chemical simulation to predict Marcus theory parameters. 7–12 In addition to attaining highly accurate results for λ and ∆G◦ , estimating electronic coupling to a useful degree of accuracy remains a challenging issue. Whereas λ and ∆G◦ are straightforward to compute, a wide variety of techniques has been examined for their ability to provide electronic couplings. 8,10,13–28 Of particular note are two recent studies by Kubas et al. that provided high-level ab initio benchmarks of electronic couplings in a variety of conjugated organic molecules. 12,29 These studies looked at hole transfer (HT) and ET, and produced two databases, respectively called HAB11 and HAB7− (here, we extend HAB11 to include the oligoacene family). Once formed, these benchmark sets (see SI Section 1) were used to evaluate density functional theory (DFT) as a useful, low-cost, and extensible alternative to the expensive wave function simulations. Given the huge variety of density functionals in existence, researchers must make difficult choices which have large effects on the overall accuracy. Kubas et al. and subsequent work have shown that – to no surprise – the accuracy of Hab varies with density functional. 12,29 Hab accuracy particularly correlates with amount of Hartree-Fock exchange in the functional, 24,29 but recommendations range from using pure density functionals to those with 50 % exact exchange. These recommendations depend in part on the choice of method, be it constrained DFT configuration interaction (CDFT-CI), 10,30–32 fragment molecular orbital DFT, 9,10,28,33,34



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and frozen density embedding. 35,36 While these numerical benchmarks shed some light on the accuracy of DFT, the physical reasons for variance in DFT accuracy for Hab couplings have not been examined in detail. The physical correctness aspect of DFT itself has received significant attention due to the wide efforts for developing new functionals. Some evidence suggests that density functionals are straying from physically correct properties, relying instead on statistical accuracy across a training set. 37,38 DFT training sets do not include electronic couplings, so there is no reason to believe a purely statistical approach would lead to uniformly accurate Hab values. This aspect of DFT is troubling, so we sought to show specific physical properties determined by the functional – vertical electron affinity (VEA), vertical ionization energy (VIP), and self-interaction error – can provide an estimate of the Hab prediction accuracy. To achieve this goal, the three physical properties and Hab were evaluated across a range of density functionals for the extended HAB11 set, called HAB15, and the HAB7− benchmark set. These test sets include a number of organic molecules with conjugated π systems, resulting in a wide range of values for the three chosen properties. While the electronic coupling value is also heavily affected by relative orientation, 39 these test sets benefit from symmetry-preserving displacements. To verify the results, experimental VIPs are used as benchmarks whenever available, and high-level coupled cluster (CC) computations were used to fill in missing values of VIPs and VEAs. (See the Computational Methods, and SI Section 2 and 3 for full descriptions of all DFT functionals and reference ab initio CC calculations) Delocalization errors in DFT can be evaluated with the orbital many-electron self-interaction error (OMSIE). This error quantifies the deviation from linearity of the DFT energy with reZ 1 40–42 spect to fractional charges: OMSIE = ∆E(N +δ)dδ, where ∆E(N +δ) = E(N +δ)− 0

[E(N )+{E(N +1)−E(N )}δ]. Negative (Positive) deviation comes from the over-stabilization of delocalized (localized) states, which results in favouring fractional (integer) charges/spins, respectively. 41 As a first indicator of physical correctness for Hab coupling computations, the


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OMSIE metric (for neutral/anionic dimers in the ET set, and neutral/cationic for HT) was evaluated for a suite of density functionals. These global hybrid (GH) functionals span 10 to 56 % exact exchange, plus long-range-corrected (LRC) density functions, with ∼ 20 % exact exchange at short range and 100 % at long range. Further details are provided in the Computational Methods section. While our test suite includes essentially integer charge transfers confirmed by well-behaved exponential decaying, note that it has been shown CDFT-CI is not trustworthy with fractional charge transfers, 43 likely due to significant OMSIE. 0.05 0.00 -0.05

OMSIE [eV]

-0.10 -0.15

ET

HT

ωB97X-D (22.2 -100 %)

-0.25

LRC-ωPBEh (20 - 100 %)

-0.20

M05-2X (56 %)

BH&HLYP (50 %)

PBE0 (25 %)

B97-1 (21 %)

-0.30 B3LYP (20 %)

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Figure 1: OMSIE value of seven DFT functionals with 6-311++G** for ET (blue square) and HT (orange triangle) subset, respectively. Contribution of exact exchange is given next to the functional name. Black line is given as baseline. Figure 1 shows the OMSIEs for the seven evaluated density functionals. At low values of exact exchange (B3LYP, B97-1, PBE0), the errors across the ET and HT benchmark sets are significant and negative, corresponding to the well-known over-delocalization that is present in these functionals. 42 When ∼ 50 % exact exchange is included, as in the BH&HLYP and M05-2X functionals, the absolute OMSIE decreases by ∼ 0.1 eV, indicating improved energy evaluation for the delocalized density. The LRC functionals produce low OMSIE, and give the most physically valid electron densities according to this metric. These results are in line with previous studies of DFT SIE. 42,44 The second and third metrics, VEA and VIP, were evaluated across the same test set 5 ACS Paragon Plus Environment


0.55 0.45

(A)

Deviation in Vertical Energy [eV]

0.35 0.25 0.15 0.05

-0.05 -0.15 -0.25

VEA of ET Subset

-0.35

VIP of HT Subset ωB97X-D (22.2 - 100 %)

LRC-ωPBEh (20 - 100 %)

M05-2X (56 %)

BH&HLYP (50 %)

PBE0 (25 %)

B97-1 (21 %)

B3LYP (20 %)

-0.45

0.50 0.45

Absolute Deviation in Vertical Energy [eV]

(B)

0.40 0.35 0.30 0.25 0.20 0.15 0.10

VEA of ET Subset

0.05

VIP of HT Subset ωB97X-D (22.2 - 100 %)

LRC-ωPBEh (20 - 100 %)

M05-2X (56 %)

BH&HLYP (50 %)

PBE0 (25 %)

B97-1 (21 %)

0.00

B3LYP (20 %)

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Figure 2: (A) Mean signed deviation of VEA for the ET subset (orange square) and VIP for the HT subset (blue circle). (B) Mean absolute deviation of VEA for the ET subset (orange square) and VIP for the HT subset (blue circle). Contribution of exact exchange is given next to the functional name. Black line is given as baseline.


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as the SIE. Compared to the systematic trend in SIE vs density functional, the relationship between functional and VEA/VIP error shown in Figure 2 is less obvious. While functionals with little HF exchange perform poorly for VEA and VIP, BH&HLYP has a low average error for VEA, while the Minnesota double exact exchange (M05-2X and M06-2X) functionals perform well for VIP. The LRC functionals have significant errors (> 0.25 eV) for VEAs, but are reasonably accurate for VIPs (∼ 0.15 eV). These results were partially anticipated by Vydrov et al., who observed that inclusion of exact exchange improved the prediction quality for IP, but not for EA. 45 Similarly, the good performance of the Minnesota 2X functional family and LRC functionals for VIP has been noted. 46,47 Atalla et al. have pointed out that large amount of exact exchange (50–80 %) is needed to reach accurate HOMO energies, which are closely related to VIP. 48 Predicting EA challenges LRC functionals (which are nearly SIE-free 42 ), which may be because short-range exchange-correlation energies are as important as SIE. BH&HLYP (LRC functionals and Minnesota 2X functionals) is (are) considered to strike the balance between correction of SIE and delocalization of electron densities for VEA (VIP), respectively. The three metrics therefore show considerable variation amongst density functionals for physical property evaluation. In each category, one class of functional stands out: 1. for OMSIE, LRC functionals have the lowest errors, 2. for VEA, BH&HLYP is best, and 3. for VIP, LRC and 2X functionals are superior. The use of any one density functional therefore is not expected to produce reliable results for both ET and HT models, but individual functionals may still be useful for either ET or HT. Having screened DFT functionals for their physical reliability, we now connect these metrics to the electronic coupling. Intuitively, having the correct density locality is important for producing accurate Hab values because the coupling is proportional to the degree of overlap. 26,49 For example with overly delocalized densities, the overlap will be too large, and so will Hab , and vice versa. Figure 3 shows this situation quantitatively using the pentacene dimer cation as an example. Here, the overlap integral is plotted against the electronic



450 400

0.11

(A)

(B) 0.10

350 300

0.09 Overlap Integral

Electronic Coupling [meV]

250 200 150

0.08

0.07

100 0.06

0.05

ωB97X-D (22.2 - 100 %)

0.40

LRC-ωPBEh (20 - 100 %)

0.30

BH&HLYP (50 %)

0.20

Overlap Integral

PBE0 (25 %)

0.10

B97-1 (21 %)

0 0.00

M05-2X (56 %)

50

B3LYP (20 %)

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Figure 3: (A) Proportional linear correlation between electronic coupling and overlap integral of pentacene dimer cation at four intermolecular distance, 5.0, 4.5, 4.0 and 3.5 ˚ A (from left to right). C(BH&HLYP)-CI/6-311++G** calculation is employed. (B) Inversely proportional linear correlation between overlap integral and list of selected DFT functionals. The GH functionals is sorted as ascending order of HF exchange, and two LRC functionals are inserted at the end of list. Pentacene dimer cation at intermolecular distance of 5.0 ˚ A is used. CDFTCI is used as a diabatization procedure.


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coupling for a series of intermolecular distances (3.5 to 5 ˚ A), and a nearly linear dependence is observed. For the same system at a fixed intermolecular distance (5 ˚ A), the overlap integral is computed across the test suite of density functionals. The overlap decreases with increasing exact exchange, as anticipated by the resulting localization of the density. Since the electronic coupling linearly depends on the overlap, and the overlap depends on errors in the functional, we sought for a useful expression to quantify this overall dependence. Prior studies have correlated Hab with the overlap, Sab , by a scalar, using Hab = CSab . 26,49 Rather than assume that C is constant across a range of DFT functionals, we hypothesized that C depends on the VIP and VEA. Knowing that the overlap integral depends on OMSIE and assuming linear behavior for both dependencies, we arrive at the following estimate, Hab = C × Sab Calc = c(VECalc ) × Sab Ref

= c(VE

+ ∆VE) ×

(1) Ref Sab (1

− OMSIE)

Ref = cSab × [VERef + (−VERef × OMSIE + ∆VE − ∆VE × OMSIE)]

where ∆VE is the error in VEA and VIP (∆VE=VECalc −VERef ), Ref and Calc indicate the reference and calculated value, respectively. The right-hand side, in brackets, of Equation 1 measures the relative error from a given density functional for a single species, where an error of zero returns the exact electronic coupling. The error estimate over the benchmark test Xset 1 | − VERef × OMSIE + ∆VE − ∆VE × OMSIE|i set is therefore expressed as η = Ntest set i where Ntest set is the total number of molecules in the test set. Since the error metric relies on ∆VE, accurate estimates of VERef must be available.


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All numerical values are given in eV.

VEA (HAB7−) VIP (HAB15) OMSIE (HAB7−) OMSIE (HAB15) ηET (HAB7−) ηHT (HAB15) ηET&HT (HAB7− and HAB15)

Property MAD of VEA (HAB7−) MAD of VIP (HAB15) OMSIE (HAB7−) OMSIE (HAB15) ηET (HAB7−) ηHT (HAB15) ηET&HT (HAB7− and HAB15)

BH&HLYP 0.147 0.347 −0.090 −0.134 0.177 0.225 0.211

M05-2X 0.300 0.154 −0.081 −0.123 0.485 0.423 0.441

LRC-ωPBEh 0.308 0.170 −0.003 −0.016 0.287 0.202 0.222

ωB97X-D 0.367 0.185 0.022 0.010 0.256 0.208 0.226

Functional listed in the performance order: Best to worst BH&HLYP > B97-1 > ωB97X-D > B3LYP > LRC-ωPBEh > PBE0 > M05-2X M05-2X > LRC-ωPBEh > ωB97X-D > PBE0 > B3LYP > B97-1 > BH&HLYP LRC-ωPBEh > ωB97X-D > M05-2X > BH&HLYP > PBE0 > B97-1 > B3LYP ωB97X-D > LRC-ωPBEh > M05-2X > BH&HLYP > PBE0 > B97-1 > B3LYP BH&HLYP > ωB97X-D > LRC-ωPBEh > B97-1 > B3LYP > M05-2X > PBE0 LRC-ωPBEh > ωB9X-D> BH&HLYP > M05-2X > PBE0 > B97-1 > B3LYP BH&HLYP > LRC-ωPBEh > ωB97X-D > M05-2X > B97-1 > PBE0 > B3LYP

B3LYP B97-1 PBE0 0.273 0.213 0.308 0.217 0.251 0.211 −0.183 −0.181 −0.168 −0.275 −0.272 −0.250 0.406 0.401 0.526 0.797 0.733 0.698 0.685 0.638 0.648

Table 1: Performance of seven functionals for VEA, VIP, OMSIE and corresponding value of η.


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Using the error metric η, the test suite of density functionals were ranked according to their expected efficacy for predicting electronic coupling on the ET and HT test sets. This benchmark is shown in Table 1 for each functional. On the ET and HT test sets, the LRC density functionals and BH&HLYP show the best performance according to η. The LRC are overall better options for HT, while BH&HLYP is best for ET. ET 80.0 B3LYP

70.0 60.0

MAD in Hab [meV]

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B97-1

50.0

B3LYP

B97-1 PBE0

PBE0

40.0 LRC-ωPBEh

30.0

M05-2X

20.0

M05-2X

LRC-ωPBEh ωB97X-D

η eta

10.0

OMSIE

BH&HLYP

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Error metric [eV]

HT

90.0 B97-1

80.0

B97-1

B3LYP

B3LYP

70.0

MAD in Hab [meV]

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60.0

PBE0

PBE0

50.0 ωB97X-D

40.0

M05-2X BH&HLYP

30.0

M05-2X

20.0

eη

ωB97X-D

10.0

OMSIE

LRC-ωPBEh

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Error metric [eV]

Figure 4: Correlation between MAD in Hab calculated by CDFT CI and error metrics, η and OMSIE given in eV for (A) ET and (B) HT test sets. To determine the reliability of the a priori error estimator η, electronic couplings using 11 ACS Paragon Plus Environment


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CDFT-CI were computed. (See SI Section 4 and 5 for numerical values of Hab ) Figure 4 compares η to the absolute error in electronic coupling across the range of density functionals. In the same figure, the error in Hab is also compared to OMSIE. The η metric is seen to be more predictive than just OMSIE in quantifying deviation of Hab , but OMSIE alone of course has some predictive value (R 2 of 0.82 for ET and 0.89 for HT set). The most noticeable improvement of η over OMSIE is for BH&HLYP ET coupling, reflecting the functional’s outstanding accuracy in VEA. OMSIE-based prediction gives reasonable correlation with the CDFT-CI calculation for HT, but like ET, does not sufficiently predict the accuracy of BH&HLYP. Along similar lines, the two LRC functionals have nearly the same OMSIE, so the difference in Hab cannot be explained via the overlap integral alone. Inclusion of vertical energy terms in η, however, nicely differentiates their capabilities While η creates an excellent ranking of density functionals for Hab prediction, it is somewhat less useful for functionals with large errors in electronic coupling. This effect can be seen for large values of η in Figure 4, where errors in Hab do not monotonically increase with η (R 2 on this subset is 0.39 for ET). Nevertheless, η still provides a useful ranking of the best functionals for computing Hab , as the low-error cases are most important. (R 2 is improved up to 0.94 for HT), and correlation for the entire set can be found in SI Section 6) This comparison (Figure 4) shows that physical properties coming from the density functional have a direct relationship with the electronic coupling, with errors in localization and electron attachment/detachment energies being specifically important. These relationships can be easily justified: given that HT occurs via cation exchange, VIP should clearly play a role in determining Hab , and similarly, ET requires VEA to be correct. The overlap and localization dependencies of Hab are similarly straightforward, as delocalization increases overlap and interaction between molecules. Interestingly, all of these proportionalities can be justified by considering the electronic coupling produced by CDFT-CI.


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The off-diagonal coupling for CDFT-CI can be expressed as

Hab =

1 0 0 (Hab − Sab Hbb ) 2 1 − Sab

(2)

0 0 After plugging the explicit expression of Hab and Hbb into Equation 2 (See SI Section 7),

Blumberger and McKenna have shown that 50 this can be simplified to

Hab ' Va Na

Sab Wa 2 1 − Sab Sab

(3)

Va Na can be expressed in terms of ∆Eif , which is the energy required to constrain the charge onto one monomer. 50 This gives

Hab ' 2∆Eif Sab

(4)

when the overlap is relatively small (the largest Sab in our test set is 0.36 for perylene dimer at d=3.5 ˚ A, producing an error of ∼ 13 %). Next, upon recognizing that ∆Eif is proportional to VE plus a (smaller) intermonomer interaction energy (∆e), Hab ' 2(VE + ∆e)Sab = 2VE(1 +

∆e )Sab VE

(5)

' cVE × Sab With these approximations, the connection to error metric η is obtained. This derivation relies on two assumptions: 1. the overlap is small, and 2. the energy of constraining the charge is dominated by VE. Numerical results (i.e. Figure 4) suggest the two assumptions are reasonable for single-electron charge transfers between well-separated monomers. Using these principles, errors across the two best functionals, BH&HLYP for ET and LRC-ωPBEh for HT, are examined in more detail. The overall trends and outliers are summarized in Table 2 (see also SI Section 8). BH&HLYP gives a reasonably well-bounded 13 ACS Paragon Plus Environment


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Hab prediction with an error distribution centered between −20 and 20 meV. Even though BH&HLYP has significant OMSIE, its contribution to the error in Hab is suppressed in error metric η due to its accurate VE prediction. Since the range of errors in VE are roughly centered at zero, the errors in Hab are also centered around zero. In contrast, Hab prediction for HT using LRC-ωPBEh, while overall accurate, suffers from some outliers in the positive and negative directions. Significant underestimations result in the oligoacene family (4- and 5-rings) due to over-localization, whereas overestimation occurs in acetylene and cyclopropene due to over-delocalization. Table 2: Error distribution of CDFT-CI calculations and physical properties (OMSIE and VEA/VIP) ET: BH&HLYP −20 meV ≤ Err(Hab ) ≤ 20 meV ∼ −0.09 −0.42 ∼ 0.13

CDFT-CI Result OMSIE [eV] ∆VEA [eV]

HT: LRC-ωPBEh CDFT-CI Result Underestimation Overestimation Err(Hab ) ≤ −30 meV 30 meV ≤ Err(Hab ) Molecule Tetracene Pentacene Acetylene Cyclopropene OMSIE [eV] 0.025 0.031 −0.080 −0.049 ∆VIP [eV] −0.27 −0.35 −0.11 −0.25 a Err(Hab ) [meV] −32.17 −38.23 38.26 59.44 a) Intermonomer distance at 3.5 ˚ A

The Marcus theory description of ET and HT relies on accurate and reliable estimates of electronic coupling. We have shown in this article that highly accurate estimates of Hab are only available to density functionals that show the correct physical behaviors of electron locality and electron attachment/detachment energies. In addition to the physical metrics of this letter, fractional charge/spin transfers are also important metrics for success, 43 due to their connection to SIE. These insights, while useful in themselves for measuring the expected accuracy of Hab estimates, also further motivates the construction of physically correct density functionals. The latter goal, however difficult, can in part be attained by 14 ACS Paragon Plus Environment

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seeking functionals with excellent EA, IP, and low SIE. Simultaneous fulfillment of these conditions within one functional is hard to achieve since the recent DFT development have often relied upon statistical fitting rather than fulfilling rigorous physical properties. 37,38 The physical requirements for accurate electronic coupling estimation derived herein, based on minimizing η, highlight a specific subset of specific conditions that must be fulfilled for new functionals to be most useful.

Computational Methods CDFT-CI calculations and reference calculations of equation-of-motion coupled-cluster with single and double substitutions for ionized system (IP-EOM-CCSD) are performed using QChem 4.0. 51 VEA reference calculations of domain based local-pair natural orbital (DLPNO)CCSD(T) 52 are conducted using ORCA 4.0. 53 All OMSIE calculations have been conducted using PSI4 1.0. 54 More detailed information can be found in SI.

Supporting Information Test set molecules, description of DFT functionals and reference calculations, computational details, numerical value of electronic coupling, correlation between CDFT-CI values and error metrics, explicit expression of Hamiltonian matrix element, and benchmark of best functional.

Acknowledgement The authors thank the University of Michigan Energy Institute and Office of Naval Research (N00014-14-1-055) for funding.



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Density Functional Physicality in Electronic Coupling Estimation

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