Nonadiabatic Photochemistry Induced by Inaccessible Conical

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A: Kinetics, Dynamics, Photochemistry, and Excited States

Nonadiabatic Photochemistry Induced by Inaccessible Conical Intersections Camille A Farfan, and Daniel B. Turner J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b07739 • Publication Date (Web): 22 Aug 2019 Downloaded from pubs.acs.org on August 29, 2019

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The Journal of Physical Chemistry

Nonadiabatic Photochemistry Induced by Inaccessible Conical Intersections Camille A. Farfan and Daniel B. Turner Department of Chemistry, New York University, 100 Washington Square East, New York NY 10003, USA E-mail: [email protected]

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Abstract Conical intersections are ubiquitous in photochemical processes, where nonadiabatic transfer induces ultrafast nonradiative decay from an excited state. Although they eluded experimental detection until the 1990s, today three diagnostic attributes are generally associated with photochemical reactions through conical intersections: ultrafast electronic dynamics, negligible fluorescence, and coherent wavepacket transfer. Here, we use generalized quantum dynamics simulations to show that coherent nonadiabatic transfer of excited vibrational wavepackets can occur even without reaching the conical intersection region. Instead, the wavepacket remains distant from the conical intersection throughout. In some topographies an energetically inaccessible conical intersection can be completely avoided, yet still induce substantial nonadiabatic transfer with ultrafast transfer efficiencies that are nearly identical to those of direct transfer through a conical intersection. These results reveal that the diagnostic properties of conical intersections are not actually specific to decay pathways traveling directly through the intersection funnel, as is the common interpretation, but can also arise from alternative pathways that do not reach the intersection. This suggests that the diagnostic features and experimental signals associated with conical intersections should be reassessed and the concept of pathways through a conical intersection as the “paradigm of photochemistry” may need to be adjusted.

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Introduction Although first theorized as early as the 1930s, 1,2 the significance of conical intersections is still being explored and established today, following technological advancements in the 1990s that made spectroscopic measurements 3–5 and computational modeling 6–8 of these quantum processes possible. Known for their role in the “paradigm of photochemistry,” 5 conical intersections are ubiquitous features in photochemical decay pathways, in which they enable internal conversion through nonadiabatic transfer between electronic states. 6,9 This decay process is remarkable because it is both ultrafast (sub-picosecond–scale) and radiationless. 10–12 The origin of this extreme efficiency is passage through the conical intersection, which often acts as a minimum on the potential energy surface to form a topographical “funnel” that is very favorable for decay. 9,13,14 These pathways are most often examined by assessing the propagation of an excited vibrational wavepacket over the surfaces. 15,16 Because the topographies of conical intersections and potential energy surfaces determine relaxation pathways, 17–20 examination of both the topography and its dynamic effects is critical to elucidating photochemical reaction mechanisms, corresponding changes in molecular geometry, and reaction outcomes. 7,14,21 Recent progress in modeling nonadiabatic dynamics, 22–24 calculating potential energy surfaces, 25–27 detecting conical intersections in a wider variety of molecules, 28–30 and controlling conical intersection-mediated processes 31–34 continues to improve our understanding of the impact and scope of this phenomenon. In this article, we illustrate through simulations with a convenient, generalizable model for nonadiabatic quantum dynamics that inaccessible conical intersections can induce ultrafast nonadiabatic transfer, and can be just as efficient for transfer as directly accessible conical intersections. These simulations show that in the presence of an energetically inaccessible conical intersection, modest levels of nonadiabatic coupling promote transfer without the trajectory ever reaching regions around the conical intersection at any point in time. In contrast to previous studies that have shown nonadiabatic transfer that occurs while the wavepacket is approaching or leaving the conical intersection region, 35–37 our simulations demonstrate that transfer is also possible when the wavepacket remains distant from the intersection region throughout the dynamics. Notably, the nonadiabatic transfer in these simulations is substantial and occurs within ultrafast timescales comparable to those of

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direct transfer through energetically favorable conical intersections. Furthermore, as with direct transfer through a conical intersection, the coherence of vibrational wavepackets is preserved. 5,38,39 Overall, our simulations of inaccessible conical intersections demonstrate nonadiabatic transfer that mimics transfer directly through accessible conical intersections and the characteristics typically used to identify it: ultrafast excited-state decay, absence of fluorescence, and transfer of coherent wavepackets. 12 The key difference in the decay trajectories in these two cases of nonadiabatic transfer implies consequences for the interpretation of both experimental signals of nonadiabatic processes, and their photochemical reaction pathways.

Computational Methods We implemented an adaptable computational model for adiabatic potential energy surfaces and propagation of an excited wavepacket under the influence of nonadiabatic coupling. 28,40 The model was designed to make investigations of dynamics resulting from a wide range of conical intersection topographies convenient and tractable. Its purpose is not to mimic the precise potential energy surface topography corresponding to the geometry of any specific molecule, but to represent a variety of topographies that can be generalized to interpret nonadiabatic effects in more complex systems. In general, we use the model to represent “accidental” conical intersections, as their occurrence is very probable and ubiquitous in polyatomic molecules. 41 In addition, we neglect the higher dimensions of the seam space and construct the potential energy surfaces in two dimensions, corresponding to the two nuclear coordinates, g and h, that define the conical intersection branching space, or g–h space. The tuning coordinate, g, and the coupling coordinate, h, are the two coordinates necessary to describe the single degenerate point of a conical intersection. 42,43 In this article, we are primarily focused on the adiabatic representation of the two potential energy surfaces, each corresponding to an electronic state. However, we first construct the model as a two-level system of potential energy surfaces, Vpgdiab and Vpediab , in the diabatic representation,

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with each diabatic potential energy surface modeled as a two-dimensional harmonic oscillator:

pq Vpgdiab pR

pq Vpediab pR

1 2 p2 ωR p R q 2 1 2 p ωR pR R0 q2 2

(1) ∆E .

∆E refers to the energetic displacement between the ground and excited electronic state minima. We use the notation R to represent positions in terms of the g and h coordinates, and ωR to represent the normal mode frequencies that correspond to g and h. Therefore, in Equation 1, R0 designates the displacement of the excited potential energy surface along the branching coordinates. We refer to the two potential energy surfaces in the model as the ground state and excited state surfaces, labeled g and e, but the two electronic states are not actually restricted to be the lowestlying states and we use these terms as shorthand to actually describe the lower electronic state and higher electronic state, respectively, in the adiabatic representation. However, in the diabatic representation, the excited state potential energy surface is not always necessarily higher in energy than the ground state potential energy surface because the diabatic surfaces will cross if a conical intersection is present. To obtain the complete diabatic two-level system, we add a linear diabatic coupling term, dpdiab , on the off-diagonal positions as follows:

p diab p dpdiab Vg pRq pq Vp diab pR pq dpdiab Vpediab pR with dpdiab

µ Rp h

β ,

,

(2)

(3)

where µ and β are parameters for the linear diabatic coupling. Diagonalization of this matrix yields

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the expressions for the adiabatic potential energy surfaces:

p adia p Vg pRq pq Vp diab pR 0 1 p diab Ve Vpgadia Vpeadia

2 1 p diab V 2 e

Vpgdiab Vpgdiab

0

pq Vpeadia pR

c

1 4 dpdiab 2 2 c 1 4 dpdiab 2 2

(4a)

Vpediab Vpgdiab Vpediab Vpgdiab

2 (4b)

2 .

(4c)

Equations 4b and 4c thus enable calculation of the adiabatic potential energy surfaces. Therefore, manipulation of the input parameters R0 , ωR , ∆E, µ, and β allows construction of a variety of potential energy surfaces and conical intersection topographies. In the results to follow, we present four examples of potential energy surface models that demonstrate a range of behavior from that of an avoided crossing to that of a conventional conical intersection. The input parameters used to construct the four models are listed in the Supporting Information, Table S1. In computation of the adiabatic potential energy surfaces, we apply the discrete variable representation (DVR) to make this calculation more efficient and enable visual representation of the surfaces. This method entails calculation of grid-based approximations in the DVR basis. 44–46 We apply this method to the position matrices in the harmonic oscillator basis (HO) to yield the same

p DV R , and the corresponding transformation matrices, T p R: matrices in the DVR basis, R p DV R R

p HO T p :R R pR . T

(5)

The DVR of the position matrices is used to evaluate the expressions for the adiabatic potential energy surfaces from Equation 4, making this calculation less demanding because the matrices are diagonal in this basis. Then the potential energy surfaces Vpgadia and Vpeadia are transformed back

p R. into the quantum harmonic oscillator basis using T

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Next the adiabatic Hamiltonian is constructed for both states, g and e, as:

p gadia H

p HO q Vpgadia pR

p eadia H

p HO q Vpeadia pR

1 p p HO TN pR q 2 1 p p HO TN pR q , 2

(6a) (6b)

p N is the nuclear kinetic energy operator. The Hamiltonians are solved to obtain their where T adia and E adia , and vibrational eigenfunctions, φadia and φadia : eigenvalues, Eg,m e,n g,m e,n

adia adia Eg,m φg,m adia adia p eadia φadia H e,n Ee,n φe,n .

p gadia φadia H g,m

(7a) (7b)

The linear combination of the vibrational eigenfunctions and the analytic eigenfunction solutions for the harmonic oscillator is used to obtain the complete wavefunctions, Ψadia g,m pRq

¸

¸

φadia g,m piq ψi pRq

(8a)

φadia e,n piq ψi pRq ,

(8b)

i 1

Ψadia e,n pRq

i 1

where the harmonic oscillator analytic eigenfunctions, ψi pRq, are calculated using the two-dimensional form of the Hermite polynomial function. 47 In addition, the vibrational energy eigenvalues from Equation 7 are used to construct the total Hamiltonian matrix as a two-level system of the ground state and excited state harmonic oscillators:

p 0adia H

p adia Hg 0

0

p eadia H

.

(9)

N ADC , also known as the derivative coupling, is evaluated using the p m,n The nonadiabatic coupling, D

conventional expression in which the negligible second-order coupling term is omitted: N ADC p m,n D

¸A

p adia Ψadia e,n pRi q 5 Ψg,m pRi q

i 1

7


E

∆Ri

.

(10)


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Finally we complete the total adiabatic Hamiltonian for a nonadiabatically-coupled two-level sys-

p Tadia tem, H OT , by adding the coupling terms:

p Tadia H OT

N ADC p adia D p m,n Hg N ADC : p m,n p eadia D H

.

(11)

The final component is propagation of a time-dependent wavepacket after excitation into the Franck–Condon region on the upper potential energy surface 15 to examine the resulting nonadiabatic wavepacket dynamics. We construct a standard Gaussian wavepacket,

G pRq

Ae

2σ12 pR R

FC

q2

R

,

(12)

where A is the amplitude, σR is the width of the Gaussian, and RFC is the initial position of the wavepacket. RFC is set to the position of the minimum of the ground state potential energy surface to simulate a wavepacket initially excited within the Franck–Condon region. We calculate the initial time-dependent coefficients of the wavepacket at t0 : cadia e,n pt0 q

xΨadia e,n pRq | G pRqy .

(13)

Here cadia e,n pt0 q indicates the excited-state time-dependent wavepacket coefficients at the initial time point, and we set the ground-state coefficients cadia g,m pt0 q

0 to simulate a well-formed wavepacket

that has been completely excited to the higher-energy state. We now refer to the complete coeffiadia adia cients, composed of both cadia g,m pt0 q and ce,n pt0 q, as cn pt0 q where n indicates the combined indices

for tn, mu. adia p Tadia Diagonalization of the Hamiltonian H OT yields the eigenvalues, ET OT , and the transformation

matrix, Tpadia . We then apply the basis transformation to cadia n pt0 q, which we represent in the alternate basis as cadia n . In this basis we obtain the time-dependent coefficients by applying the

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phase factors, eiET OT,n ptq , and then reversing the basis transformation: adia

padia: cadia cadia n pt0 q T n pt0 q

(14a)

iET OT,nptq adia cadia n ptq cn pt0 q e

(14b)

padia cadia cadia n ptq T n ptq .

(14c)

adia

Finally, we apply the appropriate coefficients and take the linear combination of the wavefunctions to obtain the time-dependent wavepacket,

Ψadia pR, tqE

¸

adia cadia n ptq Ψn pRq

E ,

(15)

n 1

or, the wavepacket population on the ground state and the wavepacket population on the excited state,

Ψadia pR, tqE g Ψadia pR, tqE e

¸

¸

m 1

E

E

adia cadia g,m ptq Ψg,m pRq adia cadia e,n ptq Ψe,n pRq

(16a) .

(16b)

n 1

For analysis, the quantity of primary importance is the fraction of the wavepacket population that is transferred to the ground state. We also examine the wavepacket trajectory over the potential energy surfaces to assess its proximity to the conical intersection and the magnitude of nonadiabatic coupling encountered, both of which are correlated to the extent of nonadiabatic wavepacket transfer that occurs. Because the topography of the potential energy surfaces dictates the wavepacket trajectory, we examine the energy values at key points on the surfaces to quantify this relationship. The energetic properties for the four models presented in the results are summarized in Table 1. The ground state potential energy surface minimum in all four example models is roughly 0 eV. To maintain the generalizability of the simulations, the model does not incorporate wavepacket decoherence due to dissipation through interactions with solvent (intramolecular vibrational relaxation), which contributes to overall dephasing of spectroscopic signals. 3,48 This prevents the model conditions from being highly specific to one molecule in a particular solvent. Although over time

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Table 1: Properties of potential energy surfaces EFC – energy of the Franck–Condon region on the adiabatic excited state potential energy surface, Eemin – minimum of the adiabatic excited state potential energy surface, ECI – energy of the conical intersection location on the adiabatic excited state potential energy surface, ∆ECI FC – energy difference between the Franck–Condon region and the conical intersection on the excited state, ∆ECI emin – energy difference between the excited state minimum and the conical intersection on the excited state, Ewp t0 – initial energy of the wavepacket at t0 after excitation in the Franck–Condon region

p q

Ñ

Ñ

dissipation impacts wavepacket preservation and is a critical factor for spectroscopic measurements in solution, it is an external perturbation of the wavepacket and not a determinant of nonadiabatic effects. Furthermore, previous work has shown that dissipation primarily affects wavepacket dynamics after a nonadiabatic transition, when it damps wavepacket oscillations and reduces reverse transfer back to the excited state, and that it does not affect the rate or efficacy of the transfer itself. 38,49,50 These properties are instead determined by the potential energy surface and conical intersection topographies, and it is this relationship that we investigate using this model.

Results and Discussion The ubiquitous presence of conical intersections and the rarity of avoided crossings has been thoroughly established in previous work. 6,51 This concept is evidenced even in our simple model: mathematically, two curved surfaces will always intersect at some point in space, except in the case where the surface origins are placed at equivalent locations along the coordinate axes. Therefore, a conical intersection will always occur if the branching space coordinates are unlimited and the potential energy surfaces are displaced from each other. Likewise, we can only model a true avoided crossing if the potential energy surfaces are undisplaced. An example of an avoided crossing is shown in Figure 1, for comparison to the cases of inaccessible conical intersections. As seen in Figure 1a, the adiabatic potential energy surfaces do not meet regardless of the size of the branching coordinate axes, and there is no region of energetic degeneracy between the two surfaces as shown in Figure 1b, resulting in an inconsequential magnitude of average nonadiabatic coupling

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of 0.016 eV in the area around the wavepacket. As expected for a conventional avoided crossing, the wavepacket distribution on the ground state and excited state potential energy surfaces in Figure 1c shows no nonadiabatic transfer to the ground state, implying that the excited-state population would eventually decay radiatively.

Figure 1: (a) 3D potential energy surfaces of the ground and excited adiabatic electronic states for an avoided crossing that does not promote nonadiabatic transfer. The Franck–Condon region, labeled FC, corresponds to the excited state minimum at 1.80 eV indicated by an orange diamond ( ), which is reached by excitation from the ground state minimum indicated by a teal green diamond ( ). (b) Potential energy difference between the two surfaces in (a). The teal diamond indicates the minima for both the ground and excited states because their location along the g, h coordinates is the same. Therefore excitation in the Franck–Condon region promotes the wavepacket directly to the excited state minimum, where it then oscillates without moving across the potential energy surface. The region where the potential energy difference is minimized coincides with the Franck–Condon region, so the minimum energy difference is also 1.80 eV. The average nonadiabatic coupling (NADCavg ) in the area around the wavepacket location (black box) is 0.016 eV. (c) Percentage of the total wavepacket (%) occupying the excited state (orange) and ground state (teal) potential energy surfaces from (a) from 0-2 ps after excitation in the Franck–Condon region. The alternative case in which a conical intersection is present is shown in Figure 2, but in this topography the intersection is completely energetically inaccessible and therefore mimics the effects of an avoided crossing, as expected for an inaccessible conical intersection that is remote. In assessing a wide range of conditions and conical intersection topographies, we found that the occurrence of an energetically inaccessible conical intersection was fairly common. In these cases, often the conical intersection region does not coincide with the excited state minimum as in the archetypal case of the photochemical funnel with the intersection at its apex. In contrast, a steep energetic slope and a fairly large distance between the location of Franck–Condon excitation and the conical intersection prevents wavepacket access to the intersection region, effectively prohibiting nonadiabatic transfer from the excited state to the ground state. As seen in the example in Figure

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Figure 2: (a) 3D potential energy surfaces of the ground and excited adiabatic electronic states for an

energetically inaccessible conical intersection (, labeled CI) at 7.67 eV that mimics the behavior of a conventional avoided crossing. Franck–Condon excitation, labeled FC, promotes the wavepacket from the ground state minimum indicated by the teal green diamond ( ) to the excited state Franck–Condon region at 1.68 eV. The orange diamond ( ) indicates the excited state minimum, which is slightly displaced from the Franck–Condon region. (b) Potential energy difference between the two surfaces in (a), with the same indicator symbols. The wavepacket trajectory from 0-500 fs is plotted as a gray line, where lighter shades indicate earlier time points and darker shades indicate later time points. The average nonadiabatic coupling (NADCavg ) in the area containing the wavepacket trajectory (black box) was calculated as 0.07 eV. (c) Percentage of the total wavepacket (%) occupying the excited state (orange) and ground state (teal) potential energy surfaces from (a) from 0-2 ps after Franck–Condon excitation.

2a, the Franck–Condon region is fairly close to the excited state minimum, both of which are far from the conical intersection point. Because the nonadiabatic coupling value always approaches infinity at the exact conical intersection point, instead we show the energy difference between the two potential energy surfaces as a more informative graphical representation in Figure 2b. As shown in this plot, the wavepacket momentum is not great enough to allow access to the conical intersection region, and the wavepacket becomes trapped in the energetic minimum on the excited state. Therefore despite the presence of a conical intersection, the average nonadiabatic coupling encountered by the wavepacket remains insignificant, at 0.07 eV. The result is that the wavepacket distributions on the ground state and excited state potential energy surfaces shown in Figure 2c exhibit only a negligible amount of internal conversion through nonadiabatic transfer, as is also observed in cases of avoided crossings. It is well-established that transfer can also occur in regions surrounding the exact conical intersection point where nonadiabatic coupling is still present, however, this example from Figure 2 reiterates that this is definitely not always the case, as regions with large enough coupling are not always accessible.

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energetically inaccessible conical intersection (, labeled CI) at 2.52 eV that results in low levels of nonadiabatic transfer. Franck–Condon excitation, labeled FC, promotes the wavepacket from the ground state minimum indicated by the teal green diamond ( ) to the excited state Franck–Condon region at 1.27 eV. The orange diamond ( ) indicates the excited state minimum. (b) Potential energy difference between the two surfaces in (a), with the same indicator symbols. The wavepacket trajectory from 0-500 fs is plotted as a gray line, where lighter shades indicate earlier time points and darker shades indicate later time points. The average nonadiabatic coupling (NADCavg ) in the area containing the wavepacket trajectory (black box) was calculated as 0.43 eV. (c) Percentage of the total wavepacket (%) occupying the excited state (orange) and ground state (teal) potential energy surfaces from (a) from 0-2 ps after Franck–Condon excitation. (d) Color-shaded contours show the wavepacket population at selected time points on the excited state (top) and the ground state (bottom) potential energy surfaces from (a), which are depicted as contour lines and have the same indicator symbols as in (a).

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In contrast, Figure 3a shows a model of potential energy surface topography in which the conical intersection location is significantly closer to the Franck–Condon region, but still energetically inaccessible to the wavepacket. We recently investigated this topography as a model for Pfr to Pr photoisomerization in the phytochrome Cph1∆, which was hypothesized to proceed nonadiabatically through a conical intersection. 28 However, the only topographies that successfully reproduce the characteristic vibrational coherence, photoproduct quantum yield, and timescale of the isomerization were those in which the conical intersection is actually inaccessible to the wavepacket and the nonadiabatic transfer does not proceed directly through the intersection. As shown by the trajectory in Figure 3b and the dynamics plots in Figure 3d, the wavepacket does not ever traverse more than halfway across the distance between the Franck–Condon region and the conical intersection on the excited state, and becomes trapped around the favorable excited-state minimum after a few oscillations. This is consistent with the parameters of the simulation, in which the wavepacket has an initial energy of 0.49 eV after excitation to the Franck–Condon region at 1.27 eV, while the conical intersection is located at 2.52 eV. The excited state minimum is much lower, at 0.71 eV. However, as seen in the corresponding ground state dynamics from Figure 3d and the plot of the state populations over time in Figure 3c, despite the significant distance between the wavepacket location and the conical intersection, the wavepacket does encounter enough of the modest levels of nonadiabatic coupling in regions farther from the conical intersection to undergo ultrafast transfer between states. The wavepacket experiences an average nonadiabatic coupling of 0.43 eV, resulting in overall nonadiabatic transfer of 25% of the wavepacket to the ground state within 1 ps, with only some of the ground-state wavepacket population ultimately forming the isomerized photoproduct. 28 Of note in this example is that the wavepacket transfer proceeds very gradually and induces significant wavepacket bifurcation. 52 These dynamics are similar to those observed in cases of direct transfer through sloped conical intersections, 8,17,53,54 despite the observation in this example that the wavepacket trajectory is quite distinct and neither travels toward the conical intersection nor passes directly through it. As also observed for sloped conical intersections, there is a greater extent of reverse transfer, in which the wavepacket can oscillate between the two potential energy surfaces. 16 This behavior can be identified by the oscillations in the wavepacket population

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occupying each state, as seen in Figure 3c. Altogether, these dynamic effects contribute to the longer time duration over which transfer occurs, although the transition still occurs on an ultrafast timescale as characteristic of nonadiabatic processes, in under 500 fs.


energetically inaccessible conical intersection (, labeled CI) at 4.43 eV that results in substantial nonadiabatic transfer. Franck–Condon excitation, labeled FC, from the ground state minimum indicated with the teal green diamond ( ) promotes the wavepacket to the excited state Franck–Condon region at 3.83 eV. The orange diamond ( ) indicates the excited state minimum. (b) Potential energy difference between the two surfaces in (a), with the same indicator symbols. The wavepacket trajectory from 0-500 fs is plotted as a gray line, where lighter shades indicate earlier time points and darker shades indicate later time points. The average nonadiabatic coupling (NADCavg ) in the area containing the wavepacket trajectory (black box) was calculated as 0.11 eV. (c) Percentage of the total wavepacket (%) occupying the excited state (orange) and ground state (teal) potential energy surfaces from (a) from 0-2 ps after excitation in the Franck–Condon region. (d) Color-shaded contours show the wavepacket population at selected time points on the excited state (top) and the ground state (bottom) potential energy surfaces from (a), which are depicted as contour lines and have the same indicator symbols as in (a). The plot at 23 fs shows the wavepacket at its minimum distance from the conical intersection achieved throughout the simulation.

An example of more substantial nonadiabatic transfer around an inaccessible conical intersection is shown in Figure 4. Like the previous case in Figure 3, the conical intersection remains

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energetically inaccessible to the wavepacket as demonstrated by the excited-state wavepacket dynamics in Figure 4d. However in this example, the conical intersection location and regions where the potential energy surfaces are close in energy are less energetically unfavorable for wavepacket dynamics, which experience an average nonadiabatic coupling of 0.11 eV. This is consistent with the topography of the potential energy surfaces in Figure 4a and their energy difference in Figure 4b, as well as the energy values characterizing the surfaces: on the excited state, the wavepacket initial energy is 0.59 eV at its starting point in the Franck–Condon region located at 3.83 eV, whereas the conical intersection is located at 4.43 eV and the excited state minimum is 3.00 eV. Despite these conditions, a substantial fraction of the wavepacket undergoes ultrafast nonadiabatic transfer and internal conversion to the ground state, with 50% of the wavepacket already transferred by 30 fs and ultimately 72% occupying the ground state within 200 fs, as shown in Figure 4c. Consistent with previous studies, 35–37 portions of the wavepacket are able to transfer to the ground state at early time points while the wavepacket is approaching the conical intersection, but still distant, as shown in Figure 4d, particularly at 5 fs. The key difference in this simulation is that the wavepacket does not propagate all the way to the intersection region throughout the dynamics. As evidenced here, the nonadiabatic transfer resulting from cases in which the wavepacket does not ever contact the conical intersection region can be just as substantial and rapid as transfer achieved when the wavepacket does reach the conical intersection and passes directly through it. For example, the rate of transfer in this simulation is almost identical to that of one of the most efficient known conical intersections, rhodopsin, 55 which is strikingly different in that it has a peaked topography with the familiar funnel shape that is well-known for being highly effective for nonadiabatic transfer. A key result from the last two cases is that even in the absence of direct access to the conical intersection point or even the nearby surrounding regions, the nonadiabatic transfer maintains the coherence of the wavepacket. This has important implications for ultrafast spectroscopy measurements, which are the primary experimental method for detecting conical intersections, because the coherent wavepacket will therefore have a corresponding coherent oscillation that is observable in the time-resolved spectrum after transfer to the lower state. 5,15,39,56 As our results indicate, transfer dynamics far from a conical intersection can be indistinguishable from direct passage through an intersection based on transfer rate, fluorescence quantum yield, and the spectroscopic signatures,

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most notably vibrational coherence, 55 used to identify nonadiabatic transfer. Therefore experimental measurements must collect additional information or detect more specific signals of direct pathways through conical intersections to accurately elucidate the physical processes and reaction mechanisms occurring. Possible solutions could be the assessment of spectroscopic signatures of wavepacket bifurcation, 50,57 or ideally through spectroscopic measurement of the geometric phase, 58 which is a defining characteristic unique to passage through a conical intersection. 59,60

Conclusions The nuances of conical intersections and their related phenomena continue to be refined based on the many advancements made within the past twenty years. 42,43 With the objective of improving clarity and understanding in studies of nonadiabatic transfer mediated by conical intersections, these results suggest that inaccessible conical intersections should be considered, as well as the possibility of indirect transfer during excited-state dynamics that do not lead directly through a conical intersection. These findings reiterate the influence of the energetic landscape and conical intersection topography on nonadiabatic processes and their outcomes. 17,19,20,52 Furthermore, they reemphasize the importance of assessing dynamics around a conical intersection, as the topography alone is not a definite indicator of the photochemical processes taking place. 7,19–21 Notably, the wavepacket trajectories we observed in some simulations with nonadiabatic transfer were dramatically distinct from possible pathways leading through the conical intersection, as in Figure 3, and show that nonadiabatic transfer can occur in regions that are significantly distant from the intersection. Such differences in decay pathways are of primary importance in deciphering photochemical mechanisms. 9,13,14 Because movement across potential energy surfaces corresponds to motion along specific vibrational modes, and different locations on the surfaces correspond to specific molecular geometries, a different wavepacket trajectory can indicate that the configurational changes a molecule undergoes during excited-state decay are quite different. 9 Also significant is that in some cases, the simulations show that inaccessible conical intersections can promote nonadiabatic transfer as rapidly and effectively as accessible conical intersections through which the wavepacket is directly funneled. The simulations here demonstrate that inaccessible conical intersections can in-

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duce a wide range of nonadiabatic dynamics, mimicking the behavior of different topologies from avoided crossings to sloped and peaked conical intersections, which are synonymous with highly efficient transfer and the “paradigm of photochemistry.” Furthermore, nonadiabatic transfer via inaccessible conical intersections reproduced the experimental signatures of vibrational coherence that are conventionally associated with conical intersections. These findings demonstrate that the standard diagnostic features of conical intersections—ultrafast dynamics, anomalous fluorescence, and coherent wavepacket transfer—are not specific to nonadiabatic transfer directly through the conical intersection, and that inaccessible conical intersections are not always inconsequential for nonadiabatic transfer.

Associated Content Supporting Information Available Input parameters for potential energy surface computational models (S1), and additional contour plot figures of the potential energy surfaces and nonadiabatic coupling magnitude from the models (S2) Animated simulations of wavepacket dynamics for each model (MP4)

Author Information The authors declare no competing financial interest.

Acknowledgement D.B.T. acknowledges support from the Alfred P. Sloan Foundation and from the National Science Foundation under CAREER Grant No. CHE–1552235. This work was supported in part through the NYU IT High Performance Computing resources, services, and staff expertise.

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Nonadiabatic Photochemistry Induced by Inaccessible Conical

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