Conical Intersections in Organic Molecules: Benchmarking Mixed

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A: Kinetics, Dynamics, Photochemistry, and Excited States

Conical Intersections in Organic Molecules: Benchmarking Mixed-Reference Spin-Flip Time-Dependent DFT (MRSF-TD-DFT) vs. Spin-Flip TD-DFT Seunghoon Lee, Svetlana A. Shostak, Michael Filatov, and Cheol Ho Choi J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b06142 • Publication Date (Web): 08 Jul 2019 Downloaded from pubs.acs.org on July 17, 2019

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The Journal of Physical Chemistry

Conical Intersections in Organic Molecules: Benchmarking Mixed-Reference Spin-Flip Time-Dependent DFT (MRSF-TD-DFT) vs. Spin-Flip TD-DFT Seunghoon Lee,†,¶ Svetlana Shostak,‡,¶ Michael Filatov,∗,‡ and Cheol Ho Choi∗,‡ †Department of Chemistry, Seoul National University, Seoul 151-747, South Korea ‡Department of Chemistry, Kyungpook National University, Daegu 702-701, South Korea ¶S.L. and S.S. contributed equally E-mail: [email protected]; [email protected]

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Abstract The mixed-reference spin-flip time-dependent density functional theory (MRSFTD-DFT) method eliminates the erroneous spin-contamination of the SF-TD-DFT methodology while retaining conceptual and practical simplicity of the latter. The availability of the analytic gradient of the energy of the MRSF-TD-DFT response states enables automatic geometry optimization of the targeted states. Here, we apply the new method to optimize the geometry of several S1 /S0 conical intersections occurring in typical organic molecules. We demonstrate that MRSF-TD-DFT is capable of producing the correct double cone topology of the intersections and to describe the geometry of the lowest energy conical intersections and their relative energy with an accuracy matching the best multi-reference wavefunction ab initio methods. In this regard, MRSF-TD-DFT differs from many popular single-reference methods, such as, e.g., the linear response TD-DFT method, which fail to produce the correct topology of the intersections. As the new methodology completely eliminates the ambiguity with the identification of the response states as proper singlets or triplets, which is plaguing the SF-TD-DFT calculations, it can be used for automatic geometry optimization and molecular dynamics simulations not requiring constant human intervention.

Introduction Conical intersections (CIs) are molecular geometries at which two (or more) adiabatic electronic states of the molecule become degenerate. 1–4 As the intersecting electronic states at a CI are coupled by the non-vanishing non-adiabatic coupling, 4–6 CIs provide efficient funnels for the state to state population transfer mediated by the nuclear motion. 7–15 The degeneracy of the intersecting states at a CI is lifted along two directions in the space of internal molecular coordinates Q, 4–6 which are defined by the gradient

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difference and derivative coupling vectors (GDV and DCV, respectively) given by

x1 =

1 ∇Q ES1 − ∇Q ES0 2

x2 = hΨS1 |∇Q |ΨS0 i

(1a) (1b)

for the case of a crossing between the ground (S0 ) and the lowest excited S1 singlet states. The degeneracy is lifted linearly along the GDV and DCV directions, which lends the potential energy surfaces (PESs) of the intersecting states the topology of a double cone, see Fig. 1; hence the name. 1,3,4 The remaining 3N − 8 internal coordinates leave the degeneracy intact, thus defining the crossing seam (or the intersection space) of the CI. Proper description of CIs and their crossing seams is of an utmost importance for accurate modeling of the non-adiabatic processes occurring in the excited states of molecules. 12,13 Not all computational tools at the disposal of computational chemists are capable of correctly describing the topology of CIs. 16–18 The most significant requirement is that the computational method in question should produce non-vanishing non-adiabatic coupling between the intersecting states; the requirement that, for the S1 /S0 CIs, is violated by most of the single-reference methods of quantum chemistry. 17–19 Multi-reference computational methods 20–24 are capable of producing the correct topology of CIs (and the correct dimensionality of the CI seams), 8,17,25,26 however at the expense of very high cost of computations recovering the dynamic electron correlation. Methods of density functional theory (DFT) 27,28 include the dynamic electron correlation from the outset. However, the (spin-conserving) linear-response (LR) time-dependent DFT (TD-DFT) methodology, 29–31 which is, perhaps, the most popular approach to obtaining excited electronic states of molecules, fails to yield the correct dimensionality of the S1 /S0 CI seam and predicts a linear crossing (i.e., 3N − 7 dimensional seam) instead, 16,17,32 see Fig. 1. The problem is rooted in the absence of coupling between the S0 state (the reference state in the LR formalism) and the excited states (the response

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Figure 1: Examples of conical intersection (left panel) and linear intersection (right panel) of two potential energy surfaces. states). 16,17,32 The failure of the LR TD-DFT can be corrected by the use of the spin-flip (SF) TD-DFT formalism, 33–36 which operates with the SF transitions from a reference state of different multiplicity, e.g. triplet. However, the SF-TD-DFT response excited states are often strongly spin-contaminated (i.e., include contributions of states of different multiplicity), 37–41 which obstruct identification of the respective states as singlets. This considerably hinders application of the SF-TD-DFT methodology for automatic geometry optimization and non-adiabatic molecular dynamics (NAMD) simulations, as the targeted state cannot be correctly identified. Recently, a simple remedy for the spin-contamination problem of SF-TD-DFT was proposed by some of us. 41 The mixed-reference (MR) SF-TD-DFT method operates with a reference state comprising two high-spin components, MS = +1 and MS = −1, of the triplet state; thus eliminating the bulk of the spin-contamination of the response states. 41 By contrast to other approaches to eliminating the spin-contamination of SF response methodologies, the MRSF-TD-DFT method is conceptually very simple and enables straightforward computation of the analytic gradient of the response state energies; 42 thus enabling automatic geometry optimization and NAMD simulations. However, before applying the MRSF-TD-DFT methodology on a wide scale it seems appropriate to evaluate the ability of the method to accurately describe the geometries and relative energies of CIs. In this

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work, the MRSF-TD-DFT method will be applied to a series of organic molecules, which were studied previously 18 by the SF-TD-DFT method and, perhaps, the most accurate multi-reference wavefunction ab initio technique – the multi-reference configurational interaction with single and double excitations (MRCISD) method. 20–22 The geometries of the S1 /S0 CIs occurring in these molecules will be optimized with the MRSF-TD-DFT method using the same basis set and exchange-correlation (XC) functional as in the previous studies employing SF-TD-DFT and MRCISD.

Computational methods Derivation of the MRSF-TD-DFT methodology 41 is based on the density-matrix formulation of TD Kohn-Sham (TD-KS) theory. 43 In MRSF-TD-DFT, the zeroth-order MR reduced density matrix (RDM) is defined as an equiensemble of the MS = +1 and MS = −1 components of the triplet state, e.g., 0 ρMR 0 ( x, x ) =

o 1 n MS =+1 ρ0 ( x, x 0 ) + ρ0MS =−1 ( x, x 0 ) . 2

(2)

Due to the particular form of MR-RDM, its idempotency condition is not satisfied. However, it was shown 41 that this condition can be recovered by the spinor-like spin functions s1 and s2 s1 =

(1 + i ) α + (1 − i ) β (1 − i ) α + (1 + i ) β , s2 = , 2 2

(3)

where i denotes the imaginary unit. Within the Tamm-Dancoff approximation, the use of MR-RDM in the linear response formalism yields the MRSF-TD-DFT linear response equations for the completely decoupled singlet and triplet excited states 41

∑

(k)(0) A pq,rs

+

0(k) A pq,rs

(k)

(k)

Xrs = Ω(k) X pq ,

rs

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k = S, T

(4)


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(k)(0)

where k = S, T labels singlet and triplet states, A pq,rs is the linear response part of the (k)

orbital Hessian matrix, and A pq,rs is a matrix of coupling terms between the response states originating from different components, MS = +1 and MS = −1, of the mixed refer(k)

ence. 41,42 X pq and Ω(k) are the amplitude vectors and the excitation energies with respect to the MR state, respectively. V O O C

ρ

M S =+1 0

C: Closed-shell MO O: Open-shell MO V: Virtual MO

ρ

MR 0

ρ

V O O C

Idempotency C→O

O→O V O O C

V O O C

O→V V O O C

M S = −1 0

C→V V O O C

Figure 2: Schematic diagram describing electronic configurations originating from linear response of the mixed reference state. MR-RDM is defined with RDMs of the MS = +1 and MS = −1 components of mixed reference state shown in the upper panel. Recovering idempotency of MR-RDM enables one to analyze the linear response in terms of the electronic configurations obtained by one-electron spin-flip transitions from each component of the mixed reference state. In the lower panel, the electronic configurations originating in MRSF-TD-DFT are represented by blue, black, and red arrows, and categorized by different initial and final MOs of a spin-flip excitation. The blue arrows show configurations originating from both components of the reference. The black and red arrows show configurations generated from MS = +1 and MS = −1 components, respectively. In SF-TD-DFT, only the configurations shown by blue and black arrows are generated. Gray arrows show configurations that cannot be recovered by the use of MR-RDM in MRSF-TD-DFT. The effect of using the mixed reference state is illustrated in Fig. 2. Recovering idempotency of MR-RDM of Eq. (2) with the rotated spin functions of Eq. (3) allows to describe linear response of the density in terms of electronic configurations shown in the lower 6


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panel of Fig. 2. Configurations shown with the red arrows are missing in the conventional SF-TD-DFT; their absence leads to spin contamination of the SF-TD-DFT response states. These configurations are recovered in MRSF-TD-DFT and the spin contamination of the response states in MRSF-TD-DFT is nearly perfectly eliminated. 41 Configurations shown in Fig. 2 with the blue arrows originate from both MS = +1 and MS = −1 components of the mixed reference state; the black and red arrows show configurations originating from MS = +1 and MS = −1 components, respectively. Although not all electronic configurations can be recovered by using the MR-RDM, the still missing C → V configurations (gray arrows in Fig. 2) represent high-lying excited states and their effect on the lower part of the excitation spectrum is insignificant. 41 The analytic gradient of the MRSF-TD-DFT excited state energies Ω(k) (k = S, T) is obtained using the Lagrangian formalism. 44,45 Introducing the Lagrangian of an excited state with the energy Ω(k) L(k) [X(k) , Ω(k) , C, Z(k) , W(k) ] = G [X(k) , Ω(k) ] +

∑ ∑ Zpqσ F¯pqσ − 2 ∑ ∑ Wpqσ (S pqσ − δpq ), (k)

p

Conical Intersections in Organic Molecules: Benchmarking Mixed

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