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Dual-Functional Tamm-Dancoff Approximation with SelfInteraction-Free Orbitals: Vertical Excitation Energies and Potential Energy Surfaces Near an Intersection Seam Yinan Shu, Kelsey Anne Parker, and Donald G. Truhlar J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b11400 • Publication Date (Web): 04 Dec 2017 Downloaded from http://pubs.acs.org on December 11, 2017

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The Journal of Physical Chemistry prepared JPC A, Nov. 10, 2017

Dual-Functional Tamm-Dancoff Approximation with Self-Interaction-Free Orbitals: Vertical Excitation Energies and Potential Energy Surfaces Near an Intersection Seam Yinan Shu, Kelsey A. Parker, and Donald G. Truhlar* Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, USA

ABSTRACT. Recently we have developed the dual-functional Tamm-Dancoff approximation (DF-TDA) method. DF-TDA is an alternative to linear-response time-dependent density functional theory (LR-TDDFT) with the advantage of providing a correct double cone topology of S1/S0 conical intersections. In the DF-TDA method, we employ different functionals, which are denoted G and F, for orbital optimization and Hamiltonian construction. We use the notation DF-TDA/G:F. In the current work, we propose that G be the same as F except for having 100% Hartree-Fock exchange. We use the notation F100 to denote functional F with this modification. A motivation for this is that functionals with 100% Hartree-Fock exchange are one-electron self-interaction-free. Here we validate the use of F100/M06 to compute vertical excitation energies and the global potential energy surface of ammonia near a conical intersection to further validate the F100 method for photochemical problems.

Conical Intersections (CIs) of potential energy surfaces are (F – 2)-dimensional seams where two potential energy surfaces (PESs) become degenerate (F is the number of internal degrees of freedom). Regions near these seams are the most important geometries for population transfer between

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PESs.1,2,3,4,5,6,7,8,9,10. The two PESs in the vicinity of a CI have a double-cone shape in the two dimensions that break the degeneracy. Photophysical and photochemical processes are strongly affected by the topography of the PESs near the CI seams.4,11,12 Hence it is critical that an electronic structure theory employed in studying the photochemical processes should yield realistic PESs near CIs. Due to its favorable tradeoff of reasonable quantitative accuracy with low computational cost, the linear response (LR) time-dependent density functional theory (TDDFT) based on the Kohn-Sham ground-state density and adiabatic approximation for response is widely used for computing the excitation energies.13,14,15,16,17 More stable results can be achieved by employing the Tamm-Dancoff approximation (TDA) to this formalsim.18,19,20,21,22,23,24,25,26,27 Throughout the paper, we will denote LR-TDDFT with the TDA as KS-TDA. A key advantage of KS-TDA calculations, in addition to low cost, is that they do not require the selection of an active space as in complete-active-space wave function methods that are useful to describe CIs between ground and excited states, for example, multi-state complete-active-space perturbation theory (MS-CASPT2)28 and multi-configuration quasidegenerate perturbation theory (MC-QDPT)29. However, despite the advantages of LR-TDDFT and KS-TDA, both methods fail at descrbing the topography of PESs near CIs between ground and excited states.30 This is because Brillouin's theorem makes the matrix element connecting the KS ground state to all singly excited states vanish at all geometries.31 As a consequence, LR-TDDFT and KS-TDA predict (F-1)-dimensional S1/S0 CI seams, i.e., the degeneracy is broken in only one coordinate. There have, however, been some DFT-based linear response methods that do describe conical intersections.32,33,34,35,36 For example, we have recently


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proposed a variant of KS-TDA, named dual-functional TDA (DF-TDA).36,37,38 In the DF-TDA/G:F method, we employ one functional, called G, in ground-state KS theory to optimize the orbitals, and another functional, called F, in the construction of the whole TDA Hamiltonian. Therefore, Brollouin’s theorem does not force the coupling to excited states to be zero, and the problem is solved. In KS-TDA, the Hamiltonian is block diagonal with two blocks; the first (of size 1 × 1) corresponds to the ground state and the second (of size nm × nm, where n and m are numbers of occupied and virtual orbitals) correspoinds to all the singly excited states. Here, because we bring in the coupling between the blocks, the TDA Hamiltonian matrix has the dimension of (nm +1) × (nm +1); the extra row and column of the DF-TDA Hamiltonian contain the coupling between ground and excited states. In the current work, we further explore the capability of the DF-TDA method for photochemical problems, and in particular we provide a reasonable way to select the G functional to pair with a given F functional in DF-TDA. Self-Interaction-Free Density. The first question that arises when we employ DF-TDA to compute the global potential energy surfaces (PESs) is the choice of the two functionals, G and F. Our previous work investigating a small database in the Franck-Condon region, has indicated that the accuracy of the DF-TDA is mostly decided by F with some exceptions when G and F are combinations of local and hybrid functionals36 (where local functionals are those with no Hartree-Fock (HF) exchange and hybrid functional are those that do include this ingredient). For photochemistry, it is important to consider regions wider than the Franck-Condon region, for example, regions in the vicinity of CIs. Since the accuracy is mainly determined by F, one should



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choose F by the same criterion (for example, previous validations or experience) that one uses in single-functional KS-TDA or LR-TDDFT. In this section, we propose a systematic way to choose G for a given choice of F. Following the language of Kim et al.39 the errors in density functional theory can be classified as functional error and density-driven error. Substantial density-driven error often shows up in the regions with small orbital energy gaps, i.e., where highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are close in energy.39,40,41,42 Regions close to S0-S1 CIs are also usually regions with small orbital energy gaps.2 Thus, employing well-behaved densities for regions close to CIs, for example can be useful for the purpose of computing global accurate PESs. Here we propose to compute the ground-state orbitals and hence the ground-state density by using a functional with 100% HF exchange while keep the correlation part the same as in the functional F used for the ground state and response state. We call such a functional F100; for example, replacing the density functional exchange by Hartree-Fock exchange in the M0622 functional yields the M06100 functional. This is motivated by the observation, in the context of multiconfiguration pair-density functional theory (MC-PDFT), that using densities computed with full wave function exchange, combined with translated KS density functionals can yield much more accurate Rydberg and charge transfer excitations,43,44 and it can be understood by the physical interpretation of Hartree-Fock exchange that it removes one-electron self-interaction error.45,46 (The distinction between one-electron self-interaction and what is sometimes called many-electron self-interaction is explained elsewhere.47) Thus the self-consistent density computed by KS with an F100 functional can be called one-electron self-interaction free or – for short – self-interaction free. This density,


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combining orbital optimization, full wave function exchange, and correlation energy, can be thought of as an inexpensive analog of methods such as orbital-optimized coupled cluster theory.48,49 Most significantly, though, the use of an F100 functional provides a systematic way to choose the G functional in DF-TDA, as required to make a model chemistry that allows systematic testing and validation.50 In this work, we test DF-TDA/M06100:M06 method to compute the global PESs for ammonia molecule. When we use the F100 approach, we simplify the notation for DF-TDA by denoting DF-TDA/F100:F as simply DF100/F. Thus the tested method is abbreviated as DF100/M06. All calculations in this study employ the 6-311+G(2d,p) basis set51. We select the M06 functional as the functional F to calculate the response because M06 has performed well in previous studies24,32,52,53 of excitation energies. Franck-Condon region. As discussed in the above section, the DF100 method is designed to give the correct surface topology in the vicinity of S0-S1 CIs. However, to be useful it should also give reasonable results in the Franck-Condon region. To test this, we use the same small database as reported in a previous paper.36 In addition, we have also tested two relatively larger molecules, octatetraene and norbonadiene. Table 1 shows the KS-TDA/M06, HF/M06 and DF100/M06 vertical excitation energies of several low-lying excited singlet states. Only the character of the hole orbital is shown in Table 1, as the particle orbital is always π*. The mean unsigned errors (MUEs) for KS-TDA/M06, HF/M06 and DF100/M06 are 0.37 0.36 and 0.40 eV, which are very close. This will always be the case (a few additional tests with other functionals54,55,56,57 are in the Supporting Information), and it is well known that the errors vary from molecule to molecule and from functional to functional, so it is sufficient here to show that the results with DF100 are reasonable,



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and Table 1 does show that. The small MUE for the DF100/M06 method validates this method near the Franck-Condon region. Global PESs. We use the same S0 minimum geometry for ammonia as in the previous study.36 The S0-S1 minimal energy CI (MECI) geometries are optimized with MS-CASPT2, and the DF100/M06 methods using the CIOpt algorithm58. For MS-CASPT2 we employ an active space of eight electrons and seven orbitals, and we do state averaging of two states in the SCF step and diagonalization in a two-state subspace in the quasidegenerate perturbation step.59,60 The MS-CASPT2 calculations are performed with the Molpro61,62 software package. The DF-TDA calculations are performed with a locally modified version of GAMESS37,63,64,65. The Gamess+DF routines implemented for the general DF-TDA method and the DF100 method are available online.38 For accurate computation of the global PESs, in the present DF100 calculations we employ spin-unrestricted ground states66 and the spin-unrestricted version of KS-TDA18 (although the method could also be used with the spin-restricted scheme). Note that for the vertical excitation calculations reported here, one would obtain the same results with the restricted and unrestricted formalisms since there is no spin polarization. For comparison, we have also performed DF-TDA/HF:M06 calculations, which will be denoted as HF/M06 in the following text. At the ground-state equilibrium geometry, MS-CASPT2, KS-TDA/M06, HF/M06 and DF100/M06 predict the vertical excitation energy to be 6.78 eV, 6.57eV, 6.70 eV and 6.45 eV, respectively, as compared to an experimental value of 6.39 eV67. First we consider the optimized S0-S1 MECI geometries. The MECI geometries optimized by


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MS-CASPT2, HF/M06 and DF100/M06 are very similar. To clarify the geometrical parameters, three hydrogen atoms are labeled as H(d), H(1) and H(2) as shown in Figure 1(a), where H(d) simply denotes the hydrogen that is dissociated from the N atom at the MECI geometry. We will use the same labeling of the atoms throughout the text. Table 2 shows the six internal coordinates of the optimized MECI geometries. Both DF100/M06 and MS-CASPT2 optimized MECI geometries are very close to the planar geometry, as the H(d)-N-H(2)-H(1) dihedral angles are close to 180.0 degrees. The HF/M06 optmized MECI is close to planar geometry as well with 179.5 degree of H(d)-N-H(2)-H(1) dihedral angle. DF100/M06 and HF/M06 predict a relatively shorter N-H(d) distance, in particular 1.834 Å and 1.844 Å as compared to 1.941 Å for the MECI optimized by MS-CASPT2. The energy levels of DF100/M06, HF/M06 and MS-CASPT2 optimized MECIs are 5.22, 5.34 and 5.04 eV above the ground state minimum. This small difference indicates that all three methods are comparable for the ammonia molecule near MECIs. The S1 and S0 PESs are computed by both MS-CASPT2 and DF100/M06 methods and are shown in Figure 2(a) and (b). The geometries considered in these PESs are generated by starting at the DF100/M06 optimized MECI and displacing along two coordinates, in particular the N out-of-plane coordinate and the N-H dissociation coordinate, as shown in Figure 1(b). These two coordinates are chosen because they are close to the g and h vectors of the branching plane.36 In the N-H dissociation coordinate, the H atom corresponds to the H(d) atom in Figure 1(a). We can see that the branching-coordinates PESs include geometries both in the Franck-Condon region and in the S0-S1 MECI region. For example, the bottom middle insert in Figure 2 is a geometry that is close to the ground-state minimum geometry. Figure 2 shows that DF100 provides qualitatively correct PESs



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in the branching space. Figure 2(c) and (d) show the S1-S0 energy gap for the same geometries as in Figure 2. Near the MECI geometry, MS-CASPT2 and DF100/M06 predict very similar S1-S0 energy gaps as indicated by the dark blue color. This is the key result of the present letter. Very close examination of Figure 2(b) shows some places where the surfaces are not smooth; these are where the Kohn-Sham ground state breaks symmetry (analogous to the Coulson-Fischer68 point of H2). Figure 2(d) shows that this is not a serious problem, as its effect on the topography in the branching space is small. In addition to the branching space PESs, we computed potential energy cuts along two other coordinates, namely the H(d) wagging coordinate and N-H(1) dissociation coordinate. These coordinates are shown in Figure 3, and the S1 and S0 PESs along these coordinates are shown in Figures 4 and 5. Plots (a) and (b) in Figure 4 show the PES cuts computed by MS-CASPT2 and DF100/M06 methods respectively with the N-H(d) distance fixed at 1.834 Å. The geometries in (a) and (b) were generated by changing the H(d)-N-H(2) angle and fixing the other internal coordinates at the DF100/M06 optimized MECI geometry. The two PES cuts are similar, again demonstrating good agreement of MS-CASPT2 and DF100/M06 for the shapes of the PESs. However, we see that the S1-S0 gaps are quantitatively different between (a) and (b), especially in the region where the H(d)-N-H(2) angle is between 110 and 140 degrees. But this is not surprising when we recall that the DF100/M06 optimized MECI geometry has the N-H(d) distance equal to 1.834 Å, whereas the MS-CASPT2 optimized MECI has an N-H(d) distance of 1.941 Å. Thus we are comparing the two surfaces at different distance from their respective CI seams. Therefore there is a coordinate


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perpendicular to the N-H(1) coordinate that reduces the MS-CASPT2 S1-S0 gap. For this reason, we recomputed the MS-CASPT2 PESs along the H(d) wagging coordinate with the N-H(d) distance changed to 1.941 Å, which corresponds to the value of the MS-CASPT2 optimized MECI. This is shown in Figure 4(c), which shows that the S1-S0 gap near the 110 to 140 degrees of H(d)-N-H(2) angle is much reduced, and Figure 4(c) is very similar to Figure 4(b). We conclude that theDF100 surface has a reasonable gap opening along this angular coordinate in the vicinity of its CI seam. If we compare Figure 4(a) and (b), we find that the S1-S0 energy gaps near 110-140 degrees of N(d)-N-H(2) angle are quantitatively different. That is because we use the same set of geometries along the H(d) wagging coordinate, while the PESs are computed by two different methods, MS-CASPT2 and DF100/M06. One can see that the S1-S0 energy gap is larger when computed by MS-CASPT2 than DF100/M06 for the N(d)-N-H(2) angle in the range 110-140 degrees. This means that this set of geometries is farther from the CI seam for MS-CASPT2 than DF100/M06. This is caused by the N-H(d) distance. Both (a) and (b) are computed by fixing the N-H(d) distance at 1.834 Å. This distance corresponds to the N-H(d) distance of the DF100/M06-optimized MECI. That is why one sees that the S1-S0 energy gap for DF100/M06 PES is small for the N(d)-N-H(2) angle in the range 110-140 degrees. If we change this N-H(d) distance to 1.941 Angstrom (corresponding to the MS-CASPT2 optimized MECI), while keeping the rest of the coordinates the same as before, this re-computed MS-CASPT2 PES is shown in Figure 4(c). In Figure 4(c), this gap becomes small. So one can imagine that the N-H(d) distance coordinate is perpendicular to the H(d) wagging cut. If we move along this N-H(d) distance closer to CI seam, the S1-S0 energy gap will reduce. The S1 and S0 PESs along the N-H(1) dissociation coordinate are shown in Figure 5. The



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geometries in parts (a) and (b) are generated by varying the N-H(1) distance while keeping the rest of the internal coordinates the same as at the DF100/M06 optimized MECI. The MS-CASPT2 computed PESs, as shown in Figure 5(a), predict a small S1-S0 gap for the N-H(1) distance around 1.3 to 1.5 Å, whereas DF100/M06 has a small S1-S0 gap around 0.7 to 1.1 Å. Again, this difference is caused by comparing the two surfaces at different distances from their CI seams because the MS-CASPT2 optimized MECI geometry has a longer N-H(d) distance. Hence we recomputed the MS-CASPT2 PESs for a geometry that has the N-H(d) distance corresponding to the MS-CASPT2 optimized MECI. This is shown in Figure 5(c), which is very similar to Figure 5(b). Again we see a similar behavior of DF100/M06 PESs and MS-CASPT2 when they are compared in regions around their own CI seams. Discussion. As an inexpensive and accurate electronic structure theory for excited states, KS-TDA is widely used in predicting the spectroscopic properties for molecules and chromophores. Theses simulations are very often performed at the geometries that are close to the Frank-Condon region. For absorption spectroscopy, one is primarily interested in the Frank-Condon region, and for a closed-shell ground state, the spin-restricted scheme and the spin-unrestricted scheme should provide identical results, and for this reason it is frequently not stated in the literature which scheme is used for the calculation. For photochemical dynamics and sometimes for fluorescence, the situation is more complicated because strong structural deformations (like bond-breaking or incipient bond breaking), may induce open-shell character and spin polarization of the ground state. In such a case it may be essential to use the spin-unrestricted scheme. Although the paper by Levine et al.30, pointing out that LR-TDDFT is not suitable for describing the PESs near S0–S1 CIs, is based on the


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results from spin-restricted LR-TDDFT calculations, Brillouin’s theorem still holds for spin-unrestricted LR-TDDFT or KS-TDA, and so the spin-unrestricted versions also predict the wrong topology for conical intersections of the the ground and excited states. The DF100 method corrects the problem in both the restricted and unrestricted cases. Concluding remarks. We have proposed a way to choose the orbitals for DF-TDA. In particular, we suggest using self-interaction-free exchange combined with the same correlation functional as used for calculating the response. The functional with the same correlation as functional F but 100% Hartree-Fock exchange (to eliminate one-electron self-interaction) is called F100. Using the F100 functional to compute the orbitals and using functional F in the DF-TDA method to compute the response is called the DF100 method. There are two main advantages of the DF100 method. (1) The self-consistent densities computed with KS/F100 are free of one-electron self-interaction error. (2) The proposed scheme reduces the arbitrariness of the selection of the orbital functional in the DF-TDA scheme, so that the method can be tested systematically, as is done here. The tests here have two aspects. First we showed that vertical excitation energies obtained by DF100/M06 are reasonably accurate for a small test set of excitation energies. Second, we investigated the S1 and S0 PESs of ammonia molecule, and compared them with the MS-CASPT2 method. We found that the optimized S1/S0 MECI geometries are very similar for DF100/M06 and MS-CASPT2, except DF100/M06 predicts a longer N-H(d) distance (by 0.107 Å) compared with MS-CASPT2. The energy levels of the optimized S0-S1 MECI for both DF100/M06 and MS-CASPT2 are very close. Then we compared the computed PESs as far as their dependence on



both branching coordinates (with the geometries varied from those close to the Franck-Condon region to the MECI region and beyond) and nonbranching coordinates, and we found very similar behavior between DF100/M06 and MS-CASPT2. We conclude that the DF100 method is very promising for photochemistry.


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Table 1. The vertical excitation energies (eV) of singlet low-lying excited states of five small organic molecules as calculated by KS-TDA/M06, HF/M06 and DF100/M06, basis set = 6-311+G(2d,p), compared to best estimates from the literature. The last row shows the mean unsigned error (MUE) compared to the best estimates Molecules

Acetone

Butadiene Ethylene

Formaldehyde

Furan

Octatetraene Norbornadiene

MUE

States

Best Estimates

KS-TDA/

DF-TDA/

DF-TDA/

M06

HF/M06

DF100/M06

A2(n)

4.4

4.31

4.23

4.26

B1(σ)

9.1

8.51

8.50

8.54

A1(π)

9.4

8.54

8.63

8.50

Bu(π)

6.21

5.81

5.94

5.85

Ag(π)

6.39

6.33

6.52

6.38

B1u(π)

8.02

7.48

7.62

7.47

A2(n)

3.88

3.84

3.65

3.64

B1(σ)

9.1

8.80

8.64

8.59

A1(π)

9.3

9.46

9.37

9.26

B2(π)

6.32

5.98

6.21

6.09

A1(π)

6.57

6.57

6.68

6.61

A1(π)

8.13

7.75

8.05

7.84

Ag(π)

4.47

4.92

5.90

5.98

Bu(π)

4.66

4.30

5.25

5.33

A2(π)

5.34

4.75

5.60

5.57

B2(π)

6.11

5.40

6.26

6.22

0.37

0.36

0.40



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Table 2. The geometrical parameters for S0-S1 MECI geometries optimized by DF100/M06, HF/M06 and MS-CASPT2. Geometric Parameters

DF100/M06 optimized MECI

HF/M06 Optimized MECI

MS-CASPT2 optimized MECI

N-H(d) distance / Å

1.834

1.844

1.941

H(d)-N-H(2)-H(1) dihedral / Degree

179.96

179.53

179.99

N-H(1) distance / Å

1.019

1.029

1.023

N-H(2) distance / Å

1.026

1.035

1.026

H(d)-N-H(1) Angle / Degree

111.87

113.09

111.91

H(d)-N-H(2) Angle / Degree

141.87

138.38

137.90


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Figure 1. (a) The labeling of the hydrogen atoms of ammonia. The blue and gray colors represent nitrogen and hydrogen atoms respectively. H(d) represents the hydrogen atom that is from NH2 along the reaction path. (b). The coordinates for the branching-coordinates surfaces. The red arrow indicates the N-H dissociation coordinate. The vertical blue arrow indicates the out-of-plane displacement of N. The parallelogram represents the plane formed by the four atoms.



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Figure 2. The S1 and S0 PESs computed by (a) MS-CASPT2 and (b) DF100/M06. The pictured molecules represent typical geometries in the regions of the PESs indicated by the arrows. The blue and gray colors represent nitrogen and hydrogen atoms respectively. (c) and (d) are the S1-S0 energy gap contour graph correspond to (a) and (b) We computed the PES in Figure 2 by displacing the two coordinates (N-H distance and N out of plane); however, the rest of the coordinates are fixed at MECI geometry, which is away from the ground state minimum geometry.


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Figure 3. (a) H(d) wagging coordinate and (b) N-H(1) dissociation coordinate. The yellow dashed curve in (a) and red dashed arrow in (b) indicate the directions of the movement of the H atoms.



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Figure 4. Cuts through the S1 and S0 PESs along the H(d) wagging coordinate: (a) MS-CASPT2; (b) DF100/M06. The geometries along the H(d) wagging coordinate are generated by changing the H(d)-N-H(2) angle while fixing the rest of the internal coordinates at the DF100/M06 optimized MECI. Part (c) is like part (a) except that we fix the N-H(d) distance at 1.941 Å, which corresponds to the distance of the MS-CASPT2 optimized MECI geometry. The zero of energy for figs. 2, 4, and 5 is the same, namely the ground-state equilibrium geometry


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Figure 5. The S1 and S0 PESs along the N-H(1) dissociation coordinate: (a) MS-CASPT2; (b) DF100/M06. The geometries along the N-H(1) dissociation coordinate are generated by changing the N-H(1) distance while fixing the rest of the internal coordinates at the DF100/M06 optimized MECI. Figure (c) is similar to that of Figure (a) except that we fix the N-H(d) distance at 1.941 Å, which corresponds to the distance of the MS-CASPT2 optimized MECI geometry.



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! AUTHOR INFORMATION Corresponding Author: *Email: [email protected] Notes The authors declare no competing financial interest. ORCID Yinan Shu: 0000-0002-8371-0221 Kelsey Parker: 0000-0002-9176-3681 Donald G. Truhlar: 0000-0002-7742-7294

! ACKNOWLEDGMENTS K. P. acknowledges a 2016-2017 Excellence Fellowship in Chemistry at the University of Minnesota. This work was supported in part by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under award number DE-SC0015997.

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