How to extract quantitative information on electronic transitions from

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Quantum Electronic Structure

How to extract quantitative information on electronic transitions from the DFT “black box” Agostino Migliore J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.9b00518 • Publication Date (Web): 17 Jul 2019 Downloaded from pubs.acs.org on July 23, 2019

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Journal of Chemical Theory and Computation

How to extract quantitative information on electronic transitions from the DFT “black box” Agostino Migliore* Department of Chemistry, Duke University, Durham, NC 27708, USA.

KEYWORDS: DFT, exchange-correlation functional, charge transfer, electronic coupling, vertical excitation energy.

ABSTRACT: Electronic couplings and vertical excitation energies are crucial determinants of charge and excitation energy transfer rates in a broad variety of processes ranging from biological charge transfer to charge transport through inorganic materials, from molecular sensing to intracellular signaling. DFT is generally used to calculate these critical parameters, but the quality of the results is unpredictable because of the semi-empirical nature of the available DFT approaches. This study identifies a small set of fundamental rules that enables accurate DFT computation of electronic couplings and vertical excitation energies in molecular complexes and materials. These rules are applied to predict efficient DFT approaches to coupling calculation. The result is an easy-to-use guide for reliable DFT description of electronic transitions.

DFT provides the theoretical-computational framework of choice to model electronic materials1 and molecules.2-3 The development of Nanoscience and Nanotechnology has focused growing attention on DFT ability to describe electronic properties involved in charge or energy transfer.2, 4-6 The electronic coupling between the initial and final electronic states of a dynamical process is, generally, a crucial determinant of the process rate constant, either charge/energy transfer7-8 or electronic relaxation9 is at play. In fact, the electronic coupling is the nanoscale determinant of the electronic transition probability in all such cases (Figure 1). Furthermore, the strategies for its calculation involve common themes in charge and excitation energy transfer reactions.7 DFT is the best compromise between accuracy and feasibility for the calculation of the couplings in molecular complexes and materials, although the lacking knowledge of the exact functional gives DFT the partial nature of a ‘black box’ (also regarding the calculation of electronic couplings) due to the empirical or semiempirical approximations inherent in any available density functional. From this ‘black box’ many studies have attempted to extract the necessary information for reliable coupling computation, in terms of unphysical electron self-interaction10 removal, proper description of electron correlation effects,11 inclusion of dispersion corrections,12 definition of the localized (diabatic) electronic states,13 description of multistate effects,14-15 appropriate use of DFT electronic wave functions despite the DFT conception.16-19

A major limitation of DFT approaches to coupling computation in charge-transfer (CT) systems is the excessive spreading of the excess (transferring) charge that results from unphysical electron self-interaction energy in the exchange-correlation functional.20 Another weakness of the most popular DFT methods is their inability to adequately describe long-range dynamic correlation and thus dispersion interactions.12 Self-interaction errors profoundly impact on the study of CT reactions, causing an overestimation of the electronic coupling between charge donor (D) and acceptor (A) by several orders of magnitude, resulting in a dramatic overestimation of the CT rate (Figure 1). Global and range-separated hybrid functionals are the main strategies aiming to address this problem. The former include a global component of Hartree-Fock (HF) exchange; the latter contain different percentages of shortrange (SR) and long-range (LR) HF exchange.12, 21 The inclusion of 100% LR-HF exchange ensures the correct asymptotic behavior of the exchange-correlation potential but is not a sufficient condition for reliable coupling computation. Looking at the different charge localizations before and after a CT reaction (diabatic representation in Figure 1), it is clearly seen that the electronic coupling depends on the overlap between the tail, on A, of the electronic wave function localized on D and the wave function localized on A (or vice versa). Thus, the coupling value depends on both LR and SR behaviors of a functional, and entails the unknown exact description of electronic correlation. Functionals with less than 100% LR-HF exchange may produce reliable coupling values at relatively short D-

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A distances, while their performances often degrade at larger distances. On the contrary, functionals with the correct asymptotic behavior may properly quantify electronic

D

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couplings for far apart D and A, while showing poorer performances when the molecular sizes of D and A are comparable with their distance.

A

l

tunneling

G 0 q

q QI

I

F

VDA



Q' Qt QF

S IF

Q PI→F(t)

PIF  f (VDA , Ev ) k DA 

  G 0  λ 2  2 π VDA  exp    λkBT 4 λkBT   

t

Figure 1. Significance of the electronic coupling in coherent electronic transitions and molecular charge transfer. The central parabolas describe the free energies associated with the initial (I) and final (F) electronic states of a CT reaction as 0

functions of the nuclear reaction coordinate Q. λ and G are the reorganization energy and reaction free energy, respectively. At any given coordinate Q ', the ground-state wave function is written   a I  b F (dark yellow curve in the bottom-left inset) with a > b (a < b) if state I is lower (higher) in energy than F. Self-interaction errors cause an excessive spreading of  over D and A (orange curve, corresponding to more similar a and b), resulting in coupling overestimation (eq S1). The parabolas cross at the transition state coordinate Qt , where the electronic energy levels in the D and A potential wells, along the electron coordinate q, have equal energy, and electron tunneling can occur conserving the energy. For fixed nuclei, the system evolution starting from state I would be described by a coherent Rabi’s oscillation with a transition probability PI  F that is never zero for non-orthogonal states22 (blue line in the bottom-right inset). Coherent electron dynamics may be observed on a sufficiently short timescale, but the interplay with nuclear motion causes a different evolution on the reaction timescale. The thermal fluctuations leading to Qt and the coupled electronic-nuclear dynamics near 2

Qt produce the rate constant k DA in the high-temperature nonadiabatic limit.23 k DA is proportional to VDA . If D and A are sufficiently distant, VDA is proportional to the overlap S IF between  I and  F ,22 which is unduly overestimated by functionals without 100% LR-HF exchange.

Some fundamental queries emerge from the above scenario: (i) what is the cutoff distance, if any, for the proper performance of a given functional? (ii) How much of such performance depends on the functional form, the accuracy of the computational setup (e.g., the basis set used to expand the electronic density), and the specific electronic properties of the given system? (iii) Is it possible to set criteria to discern the relative merits of different functionals

to study CT systems within given D-A distance intervals? (iv) Even more importantly, can we identify general guidelines to construct functionals enabling reliable computation of the electronic couplings and related vertical excitation energies at any distance? The lack of knowledge of the exact functional has hindered the answers to such questions and has motivated many studies interrogating the DFT evolution toward the exact functional24-26 and the


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functional ‘physicality’ in estimating couplings and related energy gaps.27 Through a vast analysis of DFT methods, this study contributes to answer the fundamental questions above, in order to enable the reliable use of DFT for computing electronic couplings and related energy gaps. MATERIALS AND METHODS Electronic coupling in paradigmatic CT systems. The DFT investigation is carried out on redox couples in which the transferring charge is either an electron hole or an excess electron, because the two types of CT pose different challenges to DFT methods. Short-distance hole transfer is studied in the guanine-guanine (GG) and guanine-thymine (GT) nucleobase stacks from B-DNA. GG is a basic unit in DNA charge transport because the hole hopping preferentially occurs through guanines.28 GT is the base pair with the largest intrastrand coupling.29-30 Longdistance hole transfer is studied through a G_G couple obtained removing the adenine in a GAG triad.30 Ref 30 provides the benchmark coupling values here used to assess the quality of the DFT methods. Previous investigations suggested that functionals with full HF exchange tend to underestimate the coupling for long-distance CT,18 while functionals with relatively little HF exchange can unduly overestimate the coupling.31 Excess-electron transfer poses a critical challenge to DFT methods because self-interaction errors are exacerbated by the presence of the excess electron. Here, the excess-electron transfer is studied for an ortho-semiquinone redox couple (which results from proton transfer between a semiquinone radical anion and a quinol), denoted QQ. The electronic couplings are calculated using the theoretical formula32

VDA

 ab  2 2  a b   Ev  2



a b



2 ab

EIF  1 

2

2



S IF 

1

 1  S IF 2

( EIF  0)

(1) ( EIF  0)

Eq 1 is an exact formulation of the effective22, 33-34 electronic coupling within the two-state approximation (namely, whenever only two electronic states are at play, as it is for all systems considered here), as is obtained from solution of the secular equation for diabatic electronic states that generally have nonzero overlap. In particular, eq 1 is exact to all orders in the overlap between the two diabatic electronic states. Clearly, the value of VDA does not depend on the choice of such states within the infinite set of diabatic state pairs related by the Löwdin transformation that provide a good basis for the expansion of the system ground state.22, 35 Therefore, errors in the parameter values can only result from the DFT approaches used to calculate the quantities in eq 1. The method of eq 1 also avoids the use of the adiabatic excited state, and hence the approximations inherent in any available time dependent density functional

theory (TDDFT) approach (yet, it is worth mentioning that good performance in the electronic coupling calculation for small symmetric systems at resonance was achieved using TDDFT in combination with optimally-tuned rangeseparated hybrid density functionals36 that are not considered in this study). Further details on eq 1 are provided in refs 22, 32 and in the SI. The energy parameters studied here determine the transition probability between electronic states (for example, between two localized states near the transition state of a CT reaction, as in Figure 1), with relevance to the growing research area on uses of quantum coherence to enhance the molecular function.37 In fact, near the transition state of a molecular complex, the CT dynamics is governed by the interplay between the nuclear dynamics and the Rabi’s oscillation describing the coherent evolution of the electronic transition probability at fixed nuclear coordinates. In the frequent case of nonorthogonal (yet, distinguishable) electronic states involved in chemical reactions,34 this transition probability is given by22



PI  F (t )  S I F   2

2

4VDA

 ( Ev )



 S I F  sin 2

2



2

 Ev  t   2 

(2)

In eq 2, I (F) denotes the initial (final) diabatic electronic state, with the charge localized in D (A); VDA is the electronic coupling between the two states; E v is the vertical excitation energy, which depends on VDA and the (average) energy difference between the diabatic states, EIF : Ev



EIF 2

1

2 S IF

 4VDA 2

(3)

Apart from the theoretical significance of the energy parameters studied, the identification of DFT methods able to properly describe both short- and long-distance CT through DNA is important not only for nanotechnological applications of DNA and its derivatives,38 but also because of the possible implications of DNA charge transport for intracellular redox signaling.39-40 Computational methods. All details on the DFT computational methods used are provided in the Supporting Information. RESULTS AND DISCUSSION Electronic couplings and vertical excitation energies. The VDA values for the GG, GT and G_G redox couples calculated using 50 functionals (Figure 2) are com0

pared (Figure 3) with the benchmark values VDA produced by second-order complete active space perturbation theory (CAS-PT2).30 The deviations of the DFT VDA values from


1.4

1.2

1.2

1.0

VGT (eV)

1.0 0.8 0.6 0.4

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0.8 0.6 0.4 0.2

0.2

0.0

0.0

0

20 140 40 20 40 60 80 100 120

0

A

B

% HF exchange

104

103

103

102 101 100

102 101

10-1

100

10-2

10-1 0

0

20 40 60 80 100 120 20 140 40

C

D

% HF exchange

20 140 40 20 40 60 80 100 120

% HF exchange

104

VQQ (meV)

VG_G (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

VGG (eV)


20 140 40 20 40 60 80 100 120

% HF exchange

Figure 2. Dependence of the electronic coupling value on the density functional. The abscissa in black spans the amount of HF exchange in global pure (dark yellow) and hybrid (pink) functionals. The cyan data points refer to range-separated functionls. The amount of SR-HF exchange in the range-separated functionals with 100% LR-HF exchange is spanned by the abscissa in blue color. A vertical line separates the two regions of the diagram. The results using the best performing functionals (of any kind) for the DNA systems are represented in yellow. The couplings were obtained using the cc-pVDZ basis set for (A) GG and (B) GT, and a mixed cc-pVDZ/aug-cc-pVDZ basis set (SI section S2.2) for (C) G_G and (D) QQ. 0

VDA are quantified by the magnitudes of the relative errors δVDA  (VDA  VDA ) VDA . The overall quality of a func0

0

tional is assessed through the sum δV of the relative errors in the VDA values for the selected DNA molecular systems, using different basis sets (sections S2-S3):

literature. However, the estimates of the QQ effective electronic couplings provided here, together with the previous ones in the literature,22, 41 help define a probable range for the value of the coupling in this system. In Table 1, the functionals are classified, depending on their HF exchange components, as pure exchange-correlation, global hybrid and range-separated hybrid functionals. Functional’s ranking is based on the δV value.

δV  (δVGG ) cc-pVDZ  (δVGG ) 6 311g**  (δVGT ) cc-pVDZ  (δVGT ) 6 311g**

(4)

 (δVG_G ) cc-pVDZ  (δVG_G ) cc-pVDZ / aug-cc-pVDZ Similar definitions are used for the error δ(Ev ) basis set and total error δ E associated with the vertical excitation energies. No errors are associated with the estimates of the coupling in QQ because a benchmark value of accuracy comparable to those for the DNA systems is not available in the

As shown in Figures 2 and 3, all non-hybrid functionals unduly overestimate the couplings and hence the vertical excitation energies (section S3) because they are unable to describe properly the charge distribution in the electronic grounds state (Figure 1 and section S1). This problem is exacerbated for the description of long-distance charge selfexchange in G_G and QQ, as is clearly shown by a comparison between Figures 2A-B and 2C-D, despite the absence of a benchmark value for the electronic coupling in QQ. In fact, while the electron tunneling distances in G_G and QQ are approximately double those in GG and GT, the G_G and QQ electronic coupling values obtained using non-hybrid


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functionals are several orders of magnitude larger than the GG and GT values, which is at odds with any tunneling barrier model. A significant improvement for the systems with shorter D-A distance, GG and GT, is achieved using Becke’s hybrid functional theory, which prescribes the use of 50% HF exchange.42 Yet, this recipe fails for the G_G and QQ couplings (similarly to the use of pure density functionals), for which global or range-separated hybrid functionals with more LR-HF exchange (e.g., M08-SO43 and CAMB3LYP) work depending on the basis set accuracy (section S3). The comparison between the results in panels A, B and C, D of Figure 2, and the errors in Figure 3, highlight, overall, a huge gap in performance between the functionals with full LR-HF exchange and the other functionals for the CT systems with sufficiently large D-A distances. However, two important facts show that using 100% LR-HF exchange is not a panacea to achieve sufficiently accurate electronic coupling calculations at all (or even only at large) D-A distances: (a) Global hybrid functionals with HF exchange percentages around 50% or larger (e.g., BHLYP, some Minnesota functionals, and CAM-B3LYP) can be more accurate than some range-separated functionals with full LR-HF exchange to estimate couplings at short and even relatively large D-A distances. BHLYP, e.g., outperforms LRC-PBEh, B97X-D and M05-D for the GG (Tables S3-S4) and GT (Table S5-S6) couplings; M05-2X outperforms these three functionals, as well as B97, B97X and B97X-V, for the G_G system, if diffuse basis functions are used to describe the electronic density around the most electronegative atoms (Table S8). Full LR-HF exchange is, instead, indispensable to correct the large self-interaction errors at play in (relatively) long-distance excess-electron transfer systems (Figures 2C-D and Table S9). (b) Several functionals with 100% LR-HF exchange show relatively poor performances in systems such as GG and G_G (Figures 2A,C), leading to overestimation of the corresponding CT rates (Figure 1) by an order of magnitude. This fact can be ascribed to the description of the electronic exchange and correlation within D and A, which also plays a role since the size of guanine is not negligible compared to the D-A distance. The coupling values obtained using long-range corrected functionals and popular functionals such as B3LYP differ by orders of magnitude. Since all functionals produce similar EIF values, these large differences in VDA have remarkable effects on the E v values (Tables S3-S9), with enormous implications for the estimates of the coherent transition probability (eqs 2-3) or CT rate (Figure 1). On the origin of functional quality for coupling calculations. M11, M06-HF, B97M-VB97X, B97X-V and

B97 are the best performing functionals, with errors

δVDA  1 for all molecular systems studied. Figure 4 shows the corresponding total error δV . The prevalence of rangeseparated functionals in the best set agrees with previous results.44 M06-HF is the only global hybrid functional in this set, built using a recipe of the type45

EXC  EXC  EX local

HF

(5)

Because of the presence of full HF exchange, this functional includes both local exchange from a global DFT functional and SR-HF exchange. As was argued in previous studies,45-46 although this "double counting" does not allow the full cancellation of electron self-interaction errors in the SR, the main LR character of the exact exchange and an appropriate choice of the local functional can minimize these errors and even provide a better approximation to the correct (self-interaction-free) exchange energy in the medium range. In fact, the exchange hole produced by the local spin density or generalized gradient approximation is not sufficiently extended to properly describe the exchange interactions in the short-to-medium range, and improvement can be obtained by mixing with SR-HF exchange.47 This useful excess of HF exchange is also used in longrange corrected (LC) functionals of the form12, 47

EXC  EX  aEX LC

SR

SR-HF

 EX

LR-HF

 EC

(6)

where the SR and LR terms are separated through the splitting 1



r

erf ( r ) r



erfc( r )

(7)

r

If the range-separation parameter  is relatively small 1

(e.g.,   0.3bohr for B97X, B97X-V, and B97M-V), the SR region determined by eq 7 is relatively large, and the exchange in such region is not well described by the local or semilocal approximation, which is cured by adding SRHF exchange. The success of this formulation for electronic coupling computation depends on the balance between the a and  values. In B97, a is zero, but a larger  value 1

( 0.4 bohr ), compared to other B97-based LC functionals, produces a correspondingly narrower SR region in which the generalized gradient approximation works better. Yet, this recipe seems to provide less flexibility for use in different systems, thus leading to a slightly reduced performance of B97 compared to B97X, B97X-V, and B97M-V. The exchange "double counting" is avoided in the range1

separated M11 functional. In M11,  is 0.25bohr , but the relatively large percentage of SR-HF exchange (42.8%) enables an appropriate description of the hole in the SR region.



Page 6 of 10

Table 1. Exchange-correlation functionals, their HF properties, errors, and years. The HF properties shown are: (i) the percentage of HF exchange for the global functionals (pink); (ii) the percentages of SR- and LR-HF exchange followed by the  value in bohr

1

for the range-separated functionals (cyan). The total relative error in the vertical excitation energy

( δ E ) is defined similarly to the total error δV in VDA . The pure functionals (dark yellow) show the worst performances. functional

HF properties

V

E

M11 42.8-100, 0.25 0.71 0.24 M06-HF 100 1.43 0.21 wB97M-V 15-100, 0.3 1.52 0.25 wB97X 15.77-100, 0.3 1.54 0.24 wB97X-V 16.7-100, 0.3 1.63 0.29 wB97 0-100, 0.4 1.83 0.22 wM05-D 36.96-100, 0.2 5.83 0.48 CAM-B3LYP 19-65, 0.33 8.44 0.94 wB97X-D 22.2-100, 0.2 8.96 0.77 LRC-wPBEh 20-100, 0.2 1.02·101 0.67 M05-2X 56 5.01·103 5.86 M08-SO 56.79 5.22·103 6.24 BHLYP 50 7.91·103 7.85 BHH 50 8.01·103 7.96 M06-2X 54 1.09·104 12.19 M08-HX 52.23 1.20·103 13.77 SOGGA11-X 40.15 1.48·104 17.65 M05 28 1.90·104 23.60 M06 27 2.15·104 27.83 B1B95 28 2.21·104 28.80 mPW1PW91 25 2.18·104 28.31 mPW1PBE 25 2.18·104 28.30 PBE0 25 2.22·104 28.77 B97-3 26.93 2.23·104 29.05 B98 21.98 2.43·104 32.31 X3LYP 21.8 2.44·104 32.43 B97-2 21 2.46·104 32.79 B97-1 21 2.50·104 33.46 B3PW91 20 2.53·104 33.64 B3LYP 20 2.54·104 34.04 HSE06 25-0, 0.11 2.55·104 33.32 TPSSh 10 2.92·104 39.95 O3LYP 11.61 3.13·104 42.86 M11-L 0-0, 0.25 3.23·104 45.11 SOGGA11 0 3.24·104 48.08 BLYP 0 3.32·104 48.52 PBE 0 3.34·104 48.59 revPBE 0 3.38·104 48.47 PW91 0 3.42·104 49.50 M06-L 0 3.43·104 47.55 TPSS 0 3.51·104 49.43 Slater 0 3.57·104 51.48 BP91 0 3.75·104 53.03 BP86 0 3.75·104 53.16 SLYP 0 3.77·104 54.80 SPW91 0 3.82·104 54.65 SVWN 0 3.86·104 55.52 SP86 0 3.88·104 55.64 SVWN1RPA 0 3.93·104 56.32 SOP a 0 LRC-wPBE 0-100, 0.3 1.80 0.19 mLC-wPBEh 20-100, 0.25 1.32 0.24 a Convergence of the diabatic state calculations using constrained DFT was unachievable.


Year 2011 2006 2016 2008 2014 2008 2012 2004 2008 2009 2006 2008 1993 1993 2008 2008 2011 2005 2008 1996 1998 1998 1999 2005 1998 2004 2001 1998 1993 1993 2006 2003 2001 2012 2011 1988 1996 1998 1992 2003 2003 1972 1992 1988 1988 1992 1980 1986 1980 1999 2008 2019

Page 7 of 10

rationalized as resulting from their relatively small 

102

1

VGG

101 100 10-1 10-2 10-3 0

A

20 140 40 20 40 60 80 100 120

% HF exchange

1

101

VGT

1

value of 0.2 bohr , against   0.25bohr in M11, and the correspondingly insufficient amounts of SR-HF exchange (20% and 37%, respectively, versus 42.8% in M11). Point (b) also explains why the performances of these two functionals correlate with the SR-HF percentage. Guidelines (a-c) imply that a functional such as LRCPBE (not included in the initial set) has a performance similar toB97 for coupling computation, although the two functionals differ by the description of the correlation and the SR exchange. This is confirmed by the results in Figure 4 and section S3. Criteria (a-c) also imply that a relatively small increase in  should improve LRC-PBEh performance. In fact, using   0.25bohr (midway between the recommended and alternate  values in ref 48) in this functional form gives a functional (which I will call mLRC-PBEh) with an excellent general performance for

102

VDA computations (Figure 4). Further analysis in SI section

100

S3.3 also shows that criterion (a), as a stand-alone condition, is insufficient to ensure optimal functional performances and would cause coupling overestimation.

10-1

2.0

10-2 0

B

20 40 60 80 100 120 20 140 40

1.5

V

% HF exchange 1.0

104

VG_G

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.5

102 100

0.0 M11 1

M06-HF mLRC-PBEh 2 3 4 5 6 LRC-PBE 7 8 B97M-V B97X B97X-V B97

XC functional

10-2

Figure 4. Total errors associated with the best performing functionals for coupling calculation. The DFT errors δV

10-4 0

C

20 40 60 80 100 120 20 140 40

(eq S3) are calculated with respect to the CAS-PT2 (full circles) and CASSCF (empty circles) benchmark values.30 δV

% HF exchange

Figure 3. Dependence of the relative error in the coupling on the density functional. The abscissa is the same as in Figure 2. (A) δVGG , (B) δVGT and (C) δVG_G measure the relative deviations of the DFT coupling values for the indicated systems from the benchmark values in ref 30. Simple functional recipes for reliable coupling calculations. The analysis above identifies the few crucial ingredients to build exchange-correlation functionals able to produce reliable electronic couplings whatever the CT system: (a) 100% LR-HF exchange, (b) sufficient SR-HF for proper description of the nonlocal exchange in the shortto-medium range or (c) sufficiently large range parameter  where less or zero SR-HF exchange is used. Based on this set of rules, the reduced performances of functionals such as LRC-PBEh and M05-D compared to M11 is easily

is also shown for LRC-PBE and mLRC-PBEh. CONCLUSIONS Although the unknown form of the exact functional partly makes DFT a "black box" with empirical or semiempirical components, this study identifies the combinations of factors that determine the reliability of functionals for the calculation of electronic couplings in molecular systems of diverse size and donor-acceptor distances. This study disentangles a pre-existing inconvenient scenario characterized by discordant conclusions on functionals with large amount of HF exchange18, 44 and the alarming perception that the application of DFT to a given system could just provide a continuum of coupling values depending on the amount of HF exchange used, with the real coupling value remaining thus elusive. This study identifies the small set of fundamental rules that, once taken together, allow to build functionals with


7

Journal of Chemical Theory and Computation robust performance for coupling computation. The significance of using such rules as a set is explained, against the partial, non-general value of considering them separately. The use of such rules was exemplified through a successful modification of the LRC-PBEh functional. M11 turned out to be the most performing functional on the selected set of systems, followed by M06-HF, and the results produced by these two functionals are robust with respect to the basis set size (section S2.2). Several B97-based range-separated functionals showed good performances despite the moderate size of the basis sets used.49 The theoretical method used emphasizes the central role of the DFT description of the ground state for coupling calculations, in relation to the choice of physically meaningful diabatic states.22 The use of the ground state determines a critical role for the HF exchange in the electronic coupling evaluation. Theoretical methods that do not use the adiabatic ground state (such as, e.g., the CDFT approach in ref 13) or use closed-shell wave functions (thus neglecting the orbital relaxation that results from the excess charge50) are less sensitive to the percentage of HF exchange and may be used with the generalized gradient approximation or with smaller (system-dependent) percentages of HF exchange.51 The method of eq 1 removes all theoretical approximations (thus limiting the possible sources of errors to the computational setup employed) and allows to verify the physical reliability of the diabatic states used, since such states need to provide a good basis set for the description of the adiabatic ground state (once the latter is obtained using hybrid DFT approaches that suitably address the electron self-interaction problem20). The results obtained for the GG and GT systems confirm the generally good performance of hybrid functionals that use the Becke approach42 for -conjugated CT systems with relatively small D-A distances.13, 52-53

VDA

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coupling computation (Figure 5)54 and identifies the reasons for such trend, thereby also fostering developments towards the exact exchange-correlation functional. The results presented here enable quantitative DFT computations of electronic couplings (especially regarding DNA systems), which impact on the evaluation of charge transition probabilities and rates in physical and chemical processes. Future studies that apply and validate the set of rules proposed here over broader sets of molecules are desirable.

ASSOCIATED CONTENT Supporting Information. Detailed description of the theoretical formulation of the electronic coupling and of the DFT methods used; tests of computational accuracy using different basis sets; details on computations using the predicted functionals; Figures. S1 to S7; Tables S1 to S10. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author * [email protected]

ORCID: 0000-0001-7780-2296 Notes The author declares no competing ﬁnancial interest.

ACKNOWLEDGMENT

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Computations were performed on the ET cluster at Duke University, the Hydrogen cluster at Tel Aviv University and the Helium cluster at the University of Pennsylvania. This research was supported by the National Science Foundation (grant DMR-1608454) and the German Research Foundation (grant DFG TH 820/11-1, for which I am grateful to Prof. Abraham Nitzan).

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REFERENCES AND NOTES

100 10-2 10-4 1970 1980 1990 2000 2010 2020

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B3LYP for ground states. J. Phys. Chem. A 2006, 110, 1312613130. 46. Zope, R. R.; Dunlap, B. I., Slater's exchange parameters alpha for analytic and variational X alpha calculations. J. Chem. Theory Comput. 2005, 1, 1193-1200. 47. Chai, J.-D.; Head-Gordon, M., Systematic optimization of long-range corrected hybrid density functionals. J. Chem. Phys. 2008, 128, 084106. 48. Rohrdanz, M. A.; Martins, K. M.; Herbert, J. M., A longrange-corrected density functional that performs well for both ground-state properties and time-dependent density functional theory excitation energies, including charge-transfer excited states. J. Chem. Phys. 2009, 130, 054112. 49. B97X-V and B97M-V were trained with large basis sets. This causes B97M-V to be incompatible with the cc-pVDZ and aug-cc-pVDZ basis sets for themochemistry and other energy parameter calculations. In fact, I observed difficult convergence of self-consistent field calculations using these two functionals. However, the main features required for reliable coupling computation are built-in these functionals, thus determining their good performance for electronic coupling computation with ccpVDZ and aug-cc-pVDZ basis sets for the molecular systems considered. 50. Voityuk, A. A.; Rosch, N., Fragment charge difference method for estimating donor-acceptor electronic coupling: Application to DNA pi-stacks. J. Chem. Phys. 2002, 117, 56075616. 51. Blumberger, J.; McKenna, K. P., Constrained density functional theory applied to electron tunnelling between defects in MgO. Phys. Chem. Chem. Phys. 2013, 15, 2184-2196. 52. Kubas, A.; Gajdos, F.; Heck, A.; Oberhofer, H.; Elstner, M.; Blumberger, J., Electronic couplings for molecular charge transfer: benchmarking CDFT, FODFT and FODFTB against high-level ab initio calculations. II. Phys. Chem. Chem. Phys. 2015, 17, 1434214354. 53. Futera, Z.; Blumberger, J., Electronic Couplings for Charge Transfer across Molecule/Metal and Molecule/Semiconductor Interfaces: Performance of the Projector Operator-Based Diabatization Approach. J Phys Chem C 2017, 121, 19677-19689. 54. The HF exchange components of Minnesota functionals more recent than M11 are not advantageous for coupling computation and were not included in the analysis.

Table of Contents artwork

 ground state

DFT e–  DFT

donor

acceptor


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How to extract quantitative information on electronic transitions from

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