Article pubs.acs.org/JPCA
Dynamic Symmetry Breaking Hidden in Fano Resonance of a Molecule: S1 State of Diazirine Using Quantum Wave Packet Propagation Young Choon Park,† Heesun An,‡ Yoon Sup Lee,*,† and Kyung Koo Baeck*,‡ †
Department of Chemistry, KAIST, Daejeon 34141, Republic of Korea Department of Chemistry, Gangneung-Wonju National University, Gangneung 25457, Republic of Korea
‡
S Supporting Information *
ABSTRACT: Fano resonance in the predissociation of the S1 state of diazirine was studied by applying a time-dependent wave packet propagation method, and dynamic symmetry breaking (DSB) around the stationary structure of S1 was disclosed in a detailed analysis of this theoretical result. The DSB was found to originate in coupling between the asymmetric C−N2 stretching and CH2 wagging modes, suggesting that there is a slight time gap between ring opening and the concurrent dragging of two H atoms of the CH2 moiety. Although the depth of the double well due to DSB is just 0.011 eV, its presence noticeably affects the early time dynamics and observed spectrum.
I. INTRODUCTION A detailed understanding of Fano-type resonance in predissociations of molecular systems is an important challenge in the study of molecular dynamics because it involves interference among different dynamic channels on ultrafast time scale.1,2 Dynamic symmetry breaking (DSB) occurring in predissociations of an excited electronic state of a polyatomic molecule is another very delicate and difficult topic in molecular science because it appears only through ultrafast dynamic phenomena, and it is very difficult and not straightforward to study it using theories describing stationary structure.3,4 The involvement of symmetry breaking in Fano resonance of a molecular system, the main point of the present work, has not much been investigated before.5,6 Diazirine (c-CH2N2) has long been studied as a prototype of ultrafast reaction dynamics.7−11 Though the details of overall mechanism leading to the formation of carbene (CH2) from diazirine still remains with some controversies between two extensive theoretical studies,9,10 an important experimental observation directly related to its early time dynamics was reported recently.12 The recent experiment very successfully observed Fano resonances in the predissociation of diazirine in the first excited electronic state (S1) and analyzed the details of its early time dynamics very neatly.12 The skew profiles of the experimental bands were well fitted by a Fano profile function (see eq 1 of ref 12), and their mode-dependent parameters (q and Γ) disclosed Fano-type resonance between bound modes (symmetric ν6 and asymmetric ν9 C−N stretching modes; see Figure S-2 in the Supporting Information of ref 12) and a main dissociation channel, although the latter was not described in © 2016 American Chemical Society
sufficient detail. The involvement of the other internal modes also appeared in the experimental spectrum, but the main features of not only the Fano profiles but also the initial stages of the ring-opening process were well explained using just the two internal modes, ν6 and ν9. The very short lifetimes of the origin and the first overtone of the ν9 bands, 200 and 50 fs, respectively, indicate that the quantum states in the S1 state are only quasi-bound with a very shallow well.12 To verify and reinforce the experimental observation and to provide additional insights on the hidden meaning behind the new finding by an independent theoretical work, the time-dependent propagation of nuclear wave packet method13 is applied in the present work because this theoretical method is enough powerful and reliable on a very detailed level in terms of not only the spatial but also the temporal domain.13,14
II. COMPUTATIONAL DETAILS A. Kinetic Energy Operator. In this study, diazirine was described using just two Jacobi coordinates, R and θ, because the major progressions of the experimental spectrum were well explained qualitatively by the combination of the ν6 and ν9 modes in the previous work.12 R is the distance between the centers of mass of the CH2 and N2 moieties, and θ is the angle between the R axis and the N−N bond, as shown in the upper part of Figure 1. The kinetic energy operator in this Jacobi Received: November 11, 2015 Revised: December 30, 2015 Published: January 28, 2016 932
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B. Potential Energy Surface. The potential energy surfaces (PESs) of both the S0 and the S1 states were constructed using the CASSCF16 and the CASPT217 methods as implemented in the MOLPRO program18 with the 6-311+ +G(d,p) basis sets.19 Not only the nonbonding and bonding but also the four antibonding molecular orbitals (MOs) of cyclic CN2 ring were all included in the active space of the present work in order to cover the wide range of PES including bond dissociations. CASPT2//CASSCF(16,10)/6-311++G(d,p) calculations were carried out at 3266 grid points: 71 points for R from 1.00 to 2.40 Å with 0.02 Å spacing and 46 points for θ from 90° to 180° with 2° spacing. The other geometrical parameters were frozen as their values at the S0 state structure: RCH = 1.07 Å, RNN = 1.24 Å, < HCH = 2β = 120.2°. The main qualitative features of the two numerical PESs for the S0 (1A1) and the S1(1B1) states and their mutual relationships are depicted in the lower part of Figure 1. A combination of several analytic functions was developed to fit the original numerical PES. Simply describing, the main change of PES along R coordinate is fitted by the combination of Morse-type function for the bound part around R = 1.51 Å (2.85 a0), exponential-type function for the dissociation part beyond about R = 1.86 Å (3.52 a0), and a mixing function between the Morse- and the exponential-type functions, as given in the first block of Table S1 in Supporting Information.20 The changes along θ coordinate is handled by treating the parameters in the Morse-, exponential-, and mixing-functions as functions depending on θ, as given in the second block of Table S1.20 Hyperbolic tangent functions are used as the main components of the θ−dependent functions, and the complete list of their parameters are collected in Table S2.20 C. Quantum Wave Packet. The initial wave packet of the ground electronic state S0 was generated by solving the Schrödinger equation with the S0 PES by using the Chebyshev imaginary propagation method21 implemented in the Wavepacket program,22 with the kinetic energy operator, eq 1, and the PESs represented by Fast Fourier Transform grid points for R from 1.10 to 2.38 Å with ΔR = 0.01 Å and Legendre grid points for θ with lmax = 180. As mentioned in the previous experimental work,12 the S1(1B1)−S0(1A1) electronic transition is a C-type transition in the C2 V point group. Our calculations using the EOM-CCSD method23 with the reduced-MOspace24,25 as implemented in the ACES-II program26 confirmed that the direction of the transition dipole moment is parallel to the C-axis shown in Figure 1. It is in accordance with the designation of the S1(1B1)−S0(1A1) transition as n to π* transition because the π* orbital is perpendicular to the ringplane. The effect of the electronic transition on the nuclear motion restricted on the plane of CN2 ring is not expected to be significant. The nuclear wave packet on the S0 state was then vertically excited up to the S1 PES [see the lower part of Figure 1], and the time-dependent Schrödinger equation was solved by using the split-operator method27 implemented in the Wavepacket program22 with time-step of 0.1 fs. The negative imaginary absorbing potential28 of the second order was placed at R = 2.20 Å, θ = 20° and 160° to avoid the artificial reflection of dissipating wave packet at the boundaries. After propagation up to 1000 fs, the photoabsorption spectrum is generated by the Fourier transformation of the autocorrelation function,29 as implemented in the Wavepacket program.22 Other details are the same as used in other recent works.30−33
Figure 1. Jacobi coordinates for diazirine (c-CH2N2) in upper and three-dimensional picture of the PESs of the ground (S0) and first excited (S1) electronic states in lower. The CN2 ring is restricted within the plane defined by the rotational axes A and B. S1-PES here corresponds to CASPT2-SW-PES of the main text.
coordinate, derived by a systematic procedure,15 becomes as follows with the definition of L̂ 2z = −ℏ2 (∂2/∂θ2). T̂ = −
ℏ2 ∂ 2 ℏ2 ∂ 2 ℏ2 ∂ 2 1 ̂2 + − =− Lz 2 2 2μR ∂R 2μθ ∂θ 2μR ∂R2 2μθ (1)
The reduced masses for the R and θ coordinates are defined as follows, respectively.
μR = μθ =
MCH2M N2 MCH2 + M N2
(2)
Ir(IR + Is) Ir + IR + Is
(3)
The momenta of inertia in above equations are as follows:
IR = μR R2 Ir = μr rNN 2 =
(4)
MNMN 1 rNN 2 = MNrNN 2 MN + MN 2
Is = μs (rCHcosβ)2 =
MC2MH (rCHcosβ)2 MC + 2MH
(5)
(6)
rNN and rCH are the bond lengths of N2 and C−H, respectively, and β is the angle between R axis and C−H bond as shown in the upper part of Figure 1. MCH2, MN2, MC, MN, and MH are the masses of CH2, N2, C, N, and H, respectively. Note that μR is a constant, whereas μθ depends on R. 933
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III. RESULTS AND DISCUSSION According to previous studies,11,12 the S1 state has a stationary structure with C2V symmetry, but Figure 2a shows that
Figure 2. (a−c) Contours (with a gap of ΔE = 0.001 au) of the PES around the stationary structure of the S1 state, shown by the blue rectangle in the “0 fs” panel of Figure 5. The x and y axes of (a−c) represent the R and θ coordinates, respectively. (d) Effect of the CH2 wagging mode (γ) on the energy. The points A in (b) and A in (c) both correspond to the point A1 in (d). The point B in (b) corresponds to the point B1 in (d) whereas the point B in (c) corresponds to the point B2 in (d).
Figure 3. Overlap between experimental (red line with black dots) and simulated (blue line) spectra using (a) numerical CASSCF-PES, (b) numerical CASPT2-PES having a single well (SW), (c) analytically fitted CASPT2-SW-PES, (d) analytically fitted CASPT2-SW-PES adjusted to fit ν6, (e) analytically fitted CASPT2-SW-PES adjusted further to also match the intensities of nν6, and (f) CASPT2-DW-PES having a double well (DW). Panels on right show the norm of the autocorrelation function, ⟨χ(0)|χ(t)⟩.
CASSCF-PES by the present CASSCF(16,10)/6-311++G(d,p) calculations does not have a stationary structure, and the dissociation proceeds without any barrier. The previous study with the CASPT2/cc-pVTZ//CASSCF(12,10)/cc-pVTZ methodology also found no local stationary minimum of the S1 state.10 As will be discussed at the end of this work, the existence of any local stationary structure near the Franck− Condon (FC) region is not clear yet. In spite of this unclearness, our simulated spectrum using CASSCF-PES reproduces the main progressions of the symmetric stretching mode ν6 rather successfully [Figure 3a]. As stated in the previous experimental work,12 the total emission from 1CH2 (B), the red line with black dots in Figure 3, reflects the absorption cross section multiplied by the yield of the 1CH2 (B) fragments. Meanwhile, the earlier electronic absorption spectrum6 reported not only the first series of the nν6 progressions but also the second series of the nν6+2ν9 progressions, but their resolution was not enough to detect anything about the Fano-type resonance. Therefore, the absorption cross section generated by the present theoretical work was directly compared with the experimental profile of the total emission of the previous experimental work.12 The relative intensities of the main progressions nν6 were reproduced reasonably well, and the interval between the bands was also qualitatively proper. Even the asymmetry of the first two bands of the Fano-type profile was well reproduced qualitatively. This result implies that the coupling between the dissociation channel and the symmetric C−N stretching plays the main role in the Fano resonance here and that the minimum energy path disclosed in the morphology shown in Figure 2a reflects the main feature of the dissociation path, which corresponds to the asymmetric stepwise dissociation of two C−N bonds as suggested in the previous experimental work.12 However, no signals due to the asymmetric stretching mode ν9 were generated in the simulated spectrum in Figure 3a.
The numerical PES produced by our CASPT2//CASSCF(16,10)/6-311++G(d,p) calculations, shown in Figure 2b, shows one local minimum (a single well, SW) of the S1 state with a well depth of 0.069 eV and will be referred to as numerical CASPT2-SW-PES. The well corresponds to the stationary structure reported in previous experimental and theoretical works.11,12 The spectrum simulated using CASPT2SW-PES includes small bands due to combination with the asymmetric mode, nν6+2ν9 [Figure 3b]. Only the even quanta of ν9 combined with ν6 because of its asymmetry. However, the relative intensities of the main progressions of the ν6 mode in the simulated spectrum noticeably differ from the corresponding experimental ones. Another deviation is the gap (Δ2ν9) between the main band (nν6) and its combination with the asymmetric mode (nν6+2ν9); the simulated Δ2ν9 is about twice the experimental Δ2ν9 [Figure 3b]. The dissociation rate was also slower [Figure S120], which caused the main bands to be narrower; compare the blue lines in Figure 3a,b. More importantly, the asymmetry of the main bands almost disappeared. The resonance interaction between the bound symmetric C−N stretching mode (ν6) and the dissociation channel seems to be broken by the involvement of the asymmetric stretching mode (2ν9). The involvement of another degree of freedom in the Fano resonance tends to make the theoretical reproduction of the asymmetric profile more difficult, as shown by the long list of theoretical works to reproduce the Fano profiles of the S1 state of FNO.34 To investigate the effect of the detailed characteristics of the PES on the simulated spectrum, combinations of many analytic functions were developed to fit the numerical CASPT2-SWPES before systematic adjustments. The functions and their parameters are given in Tables S1 and Table S2, respectively, of 934
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The Journal of Physical Chemistry A Supporting Information.20 The simulated spectrum generated using this analytically fitted CASPT2-SW-PES, reconstructed by the analytic functions and their parameters, is given in Figure 3c, which is almost identical to Figure 3b. The population dynamics little changed by replacing the numerical CASPT2SW-PES with the analytically fitted CASPT2-SW-PES, as shown in Figure S1.20 CASPT2-SW-PES will be used for the analytically fitted CASPT2-SW-PES hereafter. The interval between the main progressions of ν6 can be adjusted by changing the curvature along R around the local minimum. After slight adjustment of just one parameter of the analytic functions,20 the positions of the main progressions of the ν6 mode match the corresponding experimental positions well [Figure 3d]; P1014 in Table S2 is multiplied by 0.97 to adjust the curvature of the PES along R. However, the relative intensities of the main bands in the simulated spectrum still differ from the experimental ones. This result indicates that the minimum of CASPT2-SW-PES is too close to the FC region of S0, and the wave packet sweeps only the lower-energy areas of the S1 state PES. To adjust the relative intensities, the entire PES of S1 is moved by 0.12 a0 toward larger values of R. The band heights in the simulated spectrum in Figure 3e now reproduce the experimental relative intensities quite well, but the simulated Δ2ν9 is still about twice the experimental Δ2ν9. The Δ2ν9 can be reduced gradually toward its experimental value by adjusting just a few parameters of the analytic functions. However, according to our study, the intensity of the nν6+2ν9 bands decreases gradually as Δ2ν9 approaches the experimental value and finally disappears entirely right before the simulated Δ2ν9 reaches the experimental magnitude. Even though all possible systematic adjustments were made to CASPT2-SW-PES, the simulated spectrum cannot reproduce both the position and the intensity of the combination bands simultaneously as long as the PES has just an SW. An interesting result emerges here. Both the position and intensity of the combination bands were finally reproduced simultaneously and successfully [Figure 3f] only when the PES has two local minima (a double well, DW), as shown in Figure 2c; this will be referred to as CASPT2-DW-PES hereafter. The DW was introduced assuming a symmetry breaking, as discussed in the case of cyclic C3C2H radical.4 To introduce the DW, the harmonic function ν122(θ) in the original set of functions in Table S1 is replaced by the double-well function ν122d(θ), as given in Table S3,20 and their parameters were optimized again. ν122(θ ) = p1221 (θ − p1222 )2 + p1223
respectively. All the other parts of the PES changed little, as represented by contours in Figure S2.20 Figure 3f shows that the positions of not only the main bands but also the combination bands become appropriate, and the relative intensities among the bands are also reproduced very well. Five other experimental bands at 1392, 1493, 1711, 2186, 2265, and 2548 cm−1 cannot be reproduced by the present simulation because they correspond to combinations involving other vibration modes (ν3, ν4, and ν8). Their effects on the Fano-resonance were expected to be very small in the previous work,12 but their effects could be a possible source of the DW of the CASPT2-DW PES. Among the nine vibrations of diazirine (see Figure S2 of ref 12), three modes (ν1, ν5, and ν8) are asymmetric to the CN2 plane and cannot affect our twodimensional (2D) PES. The other three modes (ν2, ν3, and ν4) also cannot affect our 2D PES because they are all perpendicular to the ring-opening plane. Except for the two modes (ν6 and ν9) consisting of the 2D space of present theoretical study, only the CH2 wagging mode (ν7) remains as a possible source of the DW. Another rationale comes from the fact that only two modes (ν7 and ν9) belong to b1 when diazirine maintains C2 V symmetry. To investigate the effect of the ν7 mode, two points, designated as A and B in Figure 2b, were chosen. The change from A to B corresponds to the combination of ν6 and ν9 to follow the minimum-energy path (see the dashed line in Figure S2 in SI20) of ring opening. The main motion of ν7 is described by changing just the wagging angle γ of the CH2 plane (the plane is represented by a vertical line in Figure 4) while the
Figure 4. Change in the wagging angle γ of the CH2 moiety. The plane of the CH2 moiety is shown using just a single C−H line here. See the main text and the caption of Figure 2 for the positions of A and B on the contour representations of PES in Figure 2.
other parts of the geometry are kept unchanged, and its effects on the PES are shown in Figure 2d. The energy profile at position A (blue curve) is symmetric with respect to the wagging angle γ = 0°. However, the energy profile at position B [red curve in Figure 2d] is no longer symmetric with respect to γ = 0° because the distance from H2 to one N atom becomes shorter than the distance to the other N atom at γ = 0°: see B1 in Figure 4. Note that the lowest energy in the red curve in Figure 2d is located about 0.01 eV below that of the blue curve, and the molecular structures at B1 and B2 in Figure 4 show that the potential energy is lowered as two H atoms are dragged slightly behind the ring opening. This result implies that the local minimum point along the ring-opening path could be moved from point A to point B, and as a result, the SW shown in Figure 2b changes into the DW shown in Figure 2c. The points A in (b) and (c) of Figure 2 both correspond to the A1 in Figure 2d, but the point B in Figure 2b corresponds to the point B1 in Figure 2d whereas the point B in Figure 2c corresponds to the point B2 in Figure 2d. The energy scale of the y axis in Figure 2d is very small, and the change from an SW
(7)
ν122d(θ ) = p1221 [(θ − p1222 )/p1223 ]4 − p1224 [(θ − p1222 )/p1223 ]2 + p1225
(8)
Two other parameters, P1014 and P1113, were also slightly adjusted, as given in Table S4.20 The important feature is that the change from CASPT2-SW-PES used in the production of the spectrum in Figure 1e to CASPT2-DW-PES is very small, and the depth of the DW is extremely small. The largest change along the main dissociation path is just 0.04 eV, as shown in Figure S2.20 The barrier between the two wells of DW is 0.011 eV, whereas the barrier from the well toward the ring-opening path is 0.033 eV, which is within the range of 0.027−0.054 eV suggested by the experiment12 on the basis of the lifetimes of the origin and the first overtone of the ν9 bands, 200 and 50 fs, 935
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The population within the 2D window space (defined by R = 1.20−2.20 Å, θ = 20−160°) starts to decrease at ∼35 fs, see Figure S1,20 but the majority of the wave packet remains around the DW and reaches approximately the initial FC region at ∼40 fs (corresponding to ν6 = 802 cm−1), as indicated by the first peak in the autocorrelation function ⟨χ(0)|χ(t)⟩ [righthand panel of Figure 3f]. In other words, the first round of this delicate dynamics is completed in ∼40 fs, and the second round begins after that. The regular appearance of the main peaks in ⟨χ(0)|χ(t)⟩ in Figure 3f implies that the main feature of the early time dynamics does not change greatly. Note that the first recurrence of the wave packet near the FC region due to the DW appears at around 120 fs and causes the shoulder of the third peak of ⟨χ(0)|χ(t)⟩ as shown in the right-hand part of Figure 3f. The effect of the DW, however, already begins to appear starting at ∼35 fs and at ∼80 fs (corresponding to one quantum, ν9 = 108 cm−1) in the change of population; that is, the overall dissociation rate is noticeably affected by the presence of the delicate small DW, as shown by the black dots in Figure S2.20 The slowed rate of the population decrease after 80 fs could be a reason for the increased q value of the Fano fitting parameter from 4.5 at 802 cm−1 to 6.0 at 1072 cm−1 (see Table 1 in ref 12).
to a DW involves only a tiny adjustment in the energy scale, see Figure S2.20 Despite the smallness of this change, the simulated spectrum is affected dramatically, as disclosed in this study. It is well-known that even a tiny difference in a PES can cause huge changes in the dynamics, the fine structure of the corresponding spectrum, and even in final branching ratios.13,34,35 This type of small change is very hard to detect even with the highest level of electronic structure theories in conjunction with extremely large size of basis sets, as discussed elsewhere.4,5,36 Even after such elaborated calculations under the Born-Oppenheimer approximation for stationary structures, the symmetry breaking may not be detected without applying dynamical study including quantum effects on nuclear behavior as carried out in the present work. Now, the microscopic and quantum mechanical details of the dynamics producing the final simulated spectrum can be discussed further using the 12 snapshots in Figure 5. The initial
IV. CONCLUSIONS The morphology of the 2D PES in this work confirms that combination of the symmetric and asymmetric stretching modes of the C−N2 ring excites the dissociation channel of diazirine excited to its S1 state. More importantly, this work shows that both the positions and relative intensities of the experimentally observed bands can be generated only when the morphology of the S1 state PES has a DW around the FC region. The depth of the DW is just 0.011 eV, but its presence noticeably affects the main features of the absorption spectrum and dissociation rate. The delicate DW, which was not disclosed directly by an experimental observation, reflects the dynamic symmetry breaking (DSB) due to coupling between the asymmetric C−N2 stretching mode ν9 and the CH2 wagging mode ν7. However, the skew feature of the Fano-type profile was not successfully reproduced in this work, suggesting that the full details of coupling between the dissociation path and the bound internal modes were not incorporated. Further efforts to elucidate the delicate coupling are desirable. It is understandable that this work, which considered a 2D space, cannot completely reproduce the Fano profiles of the S1 state of diazirine, which has nine degrees of freedom, considering the long list of theoretical works and the extremely complicated analytic functions for the PES used to reproduce the fine details of the Fano profiles of the S1 state of FNO,34 which has just three degrees of freedom. Although the existence of a local stationary structure of diazirine in the first excited electronic state S1 seems still unclear in spite of all previous theoretical works,9−11 the recent experimental12 and the present theoretical studies show that the stationary structure, if it exist, has an extremely small barrier for a dissociation pathway. We now expect that the magnitude of the energy barrier will exhibit extreme sensitivity on not only the level of dynamic electron correlations but also the size of basis sets. The DW in the present study was introduced assuming a symmetry breaking due to the influence from a nearby upper electronic state, as discussed in the case of cyclic C3C2H radical.4 The present work shows that the combination
Figure 5. Twelve snapshots showing propagation of the wave packet on CASPT2-DW-PES represented by red contours with a gap of 0.002 au. The x and y axes represent the R and θ coordinates, respectively.
position of a vertically excited wave packet is shown by the blue contours in the “0 fs” panel. The main dynamics during the first 10 fs is just the symmetric stretching of the C−N bond corresponding to the ν6 mode, and the wave packet arrives at the saddle point between the DWs. The wave packet propagates further, and the spreading toward larger angle θ is negligible until 20 fs, at which it arrives at R = 3.3 a0 and starts to bifurcate. The bifurcation here implies that the bond lengths of two C−N bonds start to differ asymmetrically. The direction along the R coordinate is reversed at this point, and the split wave packets start to spread along θ, but the degree of angular spreading is still marginal. The centers of the two split wave packets propagate slightly out of the two respective local minima, as shown in the “25 fs” and “30 fs” panels of Figure 5. As it proceeds further to 35 fs, some portion of the wave packet begin to leak out toward the ring-opening channel, as shown in the three panels at 35, 40, and 45 fs. 936
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The Journal of Physical Chemistry A of the asymmetric C−N2 stretching mode ν9 and the wagging mode ν7 is the main component responsible for the dynamic symmetry breaking influenced by a nearby electronic state. The second excited state 1B2 (3s) of diazirine10,11 could be the upper state mainly responsible, but it is not clear yet.
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(11) Fedorov, I.; Koziol, L.; Mollner, A. K.; Krylov, A. I.; Reisler, H. Multiphoton Ionization and Dissociation of Diazirine: A Theoretical and Experimental Study. J. Phys. Chem. A 2009, 113, 7412−7421. (12) Ahn, D.-S.; Kim, S.-Y.; Lim, G.-I.; Lee, S.; Choi, Y. S.; Kim, S. K. Mode-Dependent Fano Resonances Observed in the Predissociation of Diazirine in the S1 State. Angew. Chem., Int. Ed. 2010, 49, 1244− 1247. (13) Schinke, R. Photodissociation Dynamics; Cambridge University Press: Cambridge, 1993. (14) Tannor, D. J. Introduction to Quantum Mechanics: A TimeDependent Perspective; University Science Books: South Orange, NJ, 2007. (15) Hadder, J. E.; Frederick, J. H. Molecular Hamiltonians for Highly Constrained Model Systems. J. Chem. Phys. 1992, 97, 3500− 3520. (16) Knowles, P. J.; Werner, H.-J. An Efficient Second-Order MC SCF Method for Long Configuration Expansions. Chem. Phys. Lett. 1985, 115, 259−267. (17) Werner, H.-J.; Knowles, P. J. A Second Order Multiconfiguration SCF Procedure with Optimum Convergence. J. Chem. Phys. 1985, 82, 5053−5063. (18) Werner, H.-J.; et al., MOLPRO, version 2012.1, a package of ab initio programs; http://www.molpro.net. (19) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. SelfConsistent Molecular Orbital Methods. A Basis Set for Correlated Wave Functions. J. Chem. Phys. 1980, 72, 650−654. (20) More details are given in the Supporting Information of this work. (21) Kosloff, R.; Tal-Ezer, H. A Direct Relaxation Method for Calculating Eigenfunctions and Eigenvalues of the Schrö dinger Equation on a Grid. Chem. Phys. Lett. 1986, 127, 223−230. (22) Schmidt, B.; Lorenz, U. WavePacket4.7: a program package for quantum-mechanical wavepacket propagation and time-dependent spectroscopy; http://wavepacket.sourceforge.net. (23) Stanton, J. F.; Bartlett, R. J. The Equation of Motion CoupledCluster Method. A Systematic Biorthogonal Approach to Molecular Excitation Energies, Transition Probabilities, and Excited State Properties. J. Chem. Phys. 1993, 98, 7029−7030. (24) Baeck, K. K. The Analytic Gradient for the Equation-of-Motion Coupled-Cluster Energy with a Reduced Molecular Orbital Space: An Application for the First Excited State of Formaldehyde. J. Chem. Phys. 2000, 112, 1−4. (25) Baeck, K. K.; Jeon, S. I. The Analytic Gradient with a Reduced Molecular Orbital Space for the Equation-of-Motion Coupled-Cluster Theory: Systematic Study of the Magnitudes and Trends in Simple Molecules. Bull. Korean Chem. Soc. 2000, 21, 720−726. (26) Stanton, J. F.; et al., ACES-II, Release 2.5.0, 2006, a program product of the Quantum Theory Project; University of Florida: Gainesville, FL, 2006. (27) Feit, M. D.; Fleck, J. A.; Steiger, A. Solution of the Schrödinger Equation by a Spectral Method. J. Comput. Phys. 1982, 47, 412−433. (28) Kosloff, R.; Kosloff, D. Absorbing Boundaries for Wave Propagation Problems. J. Comput. Phys. 1986, 63, 363−376. (29) Heller, E. J. The Semiclassical Way to Molecular Spectroscopy. Acc. Chem. Res. 1981, 14, 368−375. (30) Lan, Z.; Domcke, W.; Vallet, V.; Sobolewski, A. L.; Mahapatra, S. Time-Dependent Quantum Wave-Packet Description of the σ* Photochemistry of Phenol. J. Chem. Phys. 2005, 122, 224315. (31) An, H.; Baeck, K. K. Quantum Wave Packet Propagation Study of the Photochemistry of Phenol: Isotope Effects (Ph-OD) and the Direct Excitation to the 1πσ* State. J. Phys. Chem. A 2011, 115, 13309−13315. (32) Park, Y. C.; An, H.; Choi, H.; Lee, Y. S.; Baeck, K. K. WavePacket Propagation Study of the Early-Time Non-Adiabatic Dissociation Dynamics of NH3Cl: Diabatic Picture, Effects of Isotope Substitution and Varying the Initial Vibration Levels. Theor. Chem. Acc. 2012, 131, 1212.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b11052. Part 1 contains the analytic functions and their optimized parameters: Table S1. Analytic functions used for CASPT2-SW-PES; Table S2. Parameters of functions for the analytically f itted CASPT2-SW-PES; Table S3. Analytic functions used for CASPT2-DW-PES; Table S4. Parameters of functions for the CASPT2-DW-PES. Part 2 provides supplementary figures: Figure S1. The population dynamics with different PESs; Figure S2. Difference between PESs used for Figure 3e,f; Figure S3 and S4 for the same Figure 3 and 5, respectively, on a larger scale for clarity. (PDF)
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AUTHOR INFORMATION
Corresponding Authors
*E-mail: [email protected]. *E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the National Research Foundation of Korea (NRF-2015R1D1A3A03019217 to K.K.B. and NRF2012R1A1A2000915 to Y.S.L). The authors thank Professor S. K. Kim of KAIST for the initial suggestion of this work and helpful discussions.
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REFERENCES
(1) Fano, U. Effect of Configuration Interaction on Intensities and Phase Shifts. Phys. Rev. 1961, 124, 1866−1878. (2) Kim, B.; Yoshihara, K.; Lee, S. Complex Resonances in the Predissociation of Cs2. Phys. Rev. Lett. 1994, 73, 424−427. (3) Bersuker, I. B. The Jahn-Teller Effect; Cambridge University Press: Cambridge, 2006. (4) Mintz, B.; Crawford, T. D. Symmetry Breaking in the Cyclic C3C2H radical. Phys. Chem. Chem. Phys. 2010, 12, 15459−15467. (5) Simons, J. Finding Transition States When Second-Order JahnTeller Instability Occurs. Int. J. Quantum Chem. 1993, 48, 211−218. (6) Johnson, A.; Myers, A. B. Solvent Effects in the Raman Spectra of the Triiodide Ion: Observation of Dynamic Symmetry Breaking and Solvent Degree of Freedom. J. Phys. Chem. 1996, 100, 7778−7788. (7) Graham, W. H. A Direct Method of Preparation of Diazirine. J. Am. Chem. Soc. 1962, 84, 1063−1064. (8) Robertson, L. C.; Merritt, J. A. The Electronic Absorption Spectra of n-Diazirine, N115-Diazirine, d1-Diazirine, and d2-Diazirine. J. Mol. Spectrosc. 1966, 19, 372−388. (9) Yamamoto, N.; Bernardi, F.; Bottoni, A.; Olivucci, M.; Robb, A. A.; Wilsey, S. Mechanism of Carbene Formation from the Excited States of Diazirine and Diazomethane: An MC-SCF Study. J. Am. Chem. Soc. 1994, 116, 2064−2074. (10) Arenas, J. F.; Lopez-Tocon, I.; Otero, J. C.; Soto, J. Carbene Formation in Its Lower Singlet State from Photoexcited 3H-Diazirine or Diazomethane. A Combined CASPT2 and ab initio Direct Trajectory Study. J. Am. Chem. Soc. 2002, 124, 1728−1735. 937
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The Journal of Physical Chemistry A (33) An, H.; Baeck, K. K. Branching Ratio Between Proton Transfer and Electron Transfer Channels of a Bidirectional Proton-Coupled Electron Transfer. J. Phys. Chem. Lett. 2014, 5, 1307−1311. (34) Dobbyn, A. J.; von Dirke, M.; Schinke, R.; Fink, R. The Photodissociation of FNO in the S1 state: Three-Dimensional Calculation on a New Potential Energy Surface. J. Chem. Phys. 1995, 102, 7070−7079 and references therein.. (35) An, H.; Choi, H.; Lee, Y. S.; Baeck, K. K. Factors Affecting the Branching Ratio of Photodissociation: Thiophenol Studied through Quantum Wavepacket Dynamics. ChemPhysChem 2015, 16, 1529− 1534. (36) Malinovskaya, S. A.; Cederbaum, L. S. The Role of Coherence and Time in the Mechanism of Dynamical Symmetry Breaking and Localization. Int. J. Quantum Chem. 2000, 80, 950−957.
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Dynamic Symmetry Breaking Hidden in Fano Resonance of a Molecule: S1 State of Diazirine Using Quantum Wave Packet Propagation Young Choon Park,† Heesun An,‡ Yoon Sup Lee,*,† and Kyung Koo Baeck*,‡ †
Department of Chemistry, KAIST, Daejeon 34141, Republic of Korea Department of Chemistry, Gangneung-Wonju National University, Gangneung 25457, Republic of Korea
‡
S Supporting Information *
ABSTRACT: Fano resonance in the predissociation of the S1 state of diazirine was studied by applying a time-dependent wave packet propagation method, and dynamic symmetry breaking (DSB) around the stationary structure of S1 was disclosed in a detailed analysis of this theoretical result. The DSB was found to originate in coupling between the asymmetric C−N2 stretching and CH2 wagging modes, suggesting that there is a slight time gap between ring opening and the concurrent dragging of two H atoms of the CH2 moiety. Although the depth of the double well due to DSB is just 0.011 eV, its presence noticeably affects the early time dynamics and observed spectrum.
I. INTRODUCTION A detailed understanding of Fano-type resonance in predissociations of molecular systems is an important challenge in the study of molecular dynamics because it involves interference among different dynamic channels on ultrafast time scale.1,2 Dynamic symmetry breaking (DSB) occurring in predissociations of an excited electronic state of a polyatomic molecule is another very delicate and difficult topic in molecular science because it appears only through ultrafast dynamic phenomena, and it is very difficult and not straightforward to study it using theories describing stationary structure.3,4 The involvement of symmetry breaking in Fano resonance of a molecular system, the main point of the present work, has not much been investigated before.5,6 Diazirine (c-CH2N2) has long been studied as a prototype of ultrafast reaction dynamics.7−11 Though the details of overall mechanism leading to the formation of carbene (CH2) from diazirine still remains with some controversies between two extensive theoretical studies,9,10 an important experimental observation directly related to its early time dynamics was reported recently.12 The recent experiment very successfully observed Fano resonances in the predissociation of diazirine in the first excited electronic state (S1) and analyzed the details of its early time dynamics very neatly.12 The skew profiles of the experimental bands were well fitted by a Fano profile function (see eq 1 of ref 12), and their mode-dependent parameters (q and Γ) disclosed Fano-type resonance between bound modes (symmetric ν6 and asymmetric ν9 C−N stretching modes; see Figure S-2 in the Supporting Information of ref 12) and a main dissociation channel, although the latter was not described in © 2016 American Chemical Society
sufficient detail. The involvement of the other internal modes also appeared in the experimental spectrum, but the main features of not only the Fano profiles but also the initial stages of the ring-opening process were well explained using just the two internal modes, ν6 and ν9. The very short lifetimes of the origin and the first overtone of the ν9 bands, 200 and 50 fs, respectively, indicate that the quantum states in the S1 state are only quasi-bound with a very shallow well.12 To verify and reinforce the experimental observation and to provide additional insights on the hidden meaning behind the new finding by an independent theoretical work, the time-dependent propagation of nuclear wave packet method13 is applied in the present work because this theoretical method is enough powerful and reliable on a very detailed level in terms of not only the spatial but also the temporal domain.13,14
II. COMPUTATIONAL DETAILS A. Kinetic Energy Operator. In this study, diazirine was described using just two Jacobi coordinates, R and θ, because the major progressions of the experimental spectrum were well explained qualitatively by the combination of the ν6 and ν9 modes in the previous work.12 R is the distance between the centers of mass of the CH2 and N2 moieties, and θ is the angle between the R axis and the N−N bond, as shown in the upper part of Figure 1. The kinetic energy operator in this Jacobi Received: November 11, 2015 Revised: December 30, 2015 Published: January 28, 2016 932
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B. Potential Energy Surface. The potential energy surfaces (PESs) of both the S0 and the S1 states were constructed using the CASSCF16 and the CASPT217 methods as implemented in the MOLPRO program18 with the 6-311+ +G(d,p) basis sets.19 Not only the nonbonding and bonding but also the four antibonding molecular orbitals (MOs) of cyclic CN2 ring were all included in the active space of the present work in order to cover the wide range of PES including bond dissociations. CASPT2//CASSCF(16,10)/6-311++G(d,p) calculations were carried out at 3266 grid points: 71 points for R from 1.00 to 2.40 Å with 0.02 Å spacing and 46 points for θ from 90° to 180° with 2° spacing. The other geometrical parameters were frozen as their values at the S0 state structure: RCH = 1.07 Å, RNN = 1.24 Å, < HCH = 2β = 120.2°. The main qualitative features of the two numerical PESs for the S0 (1A1) and the S1(1B1) states and their mutual relationships are depicted in the lower part of Figure 1. A combination of several analytic functions was developed to fit the original numerical PES. Simply describing, the main change of PES along R coordinate is fitted by the combination of Morse-type function for the bound part around R = 1.51 Å (2.85 a0), exponential-type function for the dissociation part beyond about R = 1.86 Å (3.52 a0), and a mixing function between the Morse- and the exponential-type functions, as given in the first block of Table S1 in Supporting Information.20 The changes along θ coordinate is handled by treating the parameters in the Morse-, exponential-, and mixing-functions as functions depending on θ, as given in the second block of Table S1.20 Hyperbolic tangent functions are used as the main components of the θ−dependent functions, and the complete list of their parameters are collected in Table S2.20 C. Quantum Wave Packet. The initial wave packet of the ground electronic state S0 was generated by solving the Schrödinger equation with the S0 PES by using the Chebyshev imaginary propagation method21 implemented in the Wavepacket program,22 with the kinetic energy operator, eq 1, and the PESs represented by Fast Fourier Transform grid points for R from 1.10 to 2.38 Å with ΔR = 0.01 Å and Legendre grid points for θ with lmax = 180. As mentioned in the previous experimental work,12 the S1(1B1)−S0(1A1) electronic transition is a C-type transition in the C2 V point group. Our calculations using the EOM-CCSD method23 with the reduced-MOspace24,25 as implemented in the ACES-II program26 confirmed that the direction of the transition dipole moment is parallel to the C-axis shown in Figure 1. It is in accordance with the designation of the S1(1B1)−S0(1A1) transition as n to π* transition because the π* orbital is perpendicular to the ringplane. The effect of the electronic transition on the nuclear motion restricted on the plane of CN2 ring is not expected to be significant. The nuclear wave packet on the S0 state was then vertically excited up to the S1 PES [see the lower part of Figure 1], and the time-dependent Schrödinger equation was solved by using the split-operator method27 implemented in the Wavepacket program22 with time-step of 0.1 fs. The negative imaginary absorbing potential28 of the second order was placed at R = 2.20 Å, θ = 20° and 160° to avoid the artificial reflection of dissipating wave packet at the boundaries. After propagation up to 1000 fs, the photoabsorption spectrum is generated by the Fourier transformation of the autocorrelation function,29 as implemented in the Wavepacket program.22 Other details are the same as used in other recent works.30−33
Figure 1. Jacobi coordinates for diazirine (c-CH2N2) in upper and three-dimensional picture of the PESs of the ground (S0) and first excited (S1) electronic states in lower. The CN2 ring is restricted within the plane defined by the rotational axes A and B. S1-PES here corresponds to CASPT2-SW-PES of the main text.
coordinate, derived by a systematic procedure,15 becomes as follows with the definition of L̂ 2z = −ℏ2 (∂2/∂θ2). T̂ = −
ℏ2 ∂ 2 ℏ2 ∂ 2 ℏ2 ∂ 2 1 ̂2 + − =− Lz 2 2 2μR ∂R 2μθ ∂θ 2μR ∂R2 2μθ (1)
The reduced masses for the R and θ coordinates are defined as follows, respectively.
μR = μθ =
MCH2M N2 MCH2 + M N2
(2)
Ir(IR + Is) Ir + IR + Is
(3)
The momenta of inertia in above equations are as follows:
IR = μR R2 Ir = μr rNN 2 =
(4)
MNMN 1 rNN 2 = MNrNN 2 MN + MN 2
Is = μs (rCHcosβ)2 =
MC2MH (rCHcosβ)2 MC + 2MH
(5)
(6)
rNN and rCH are the bond lengths of N2 and C−H, respectively, and β is the angle between R axis and C−H bond as shown in the upper part of Figure 1. MCH2, MN2, MC, MN, and MH are the masses of CH2, N2, C, N, and H, respectively. Note that μR is a constant, whereas μθ depends on R. 933
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III. RESULTS AND DISCUSSION According to previous studies,11,12 the S1 state has a stationary structure with C2V symmetry, but Figure 2a shows that
Figure 2. (a−c) Contours (with a gap of ΔE = 0.001 au) of the PES around the stationary structure of the S1 state, shown by the blue rectangle in the “0 fs” panel of Figure 5. The x and y axes of (a−c) represent the R and θ coordinates, respectively. (d) Effect of the CH2 wagging mode (γ) on the energy. The points A in (b) and A in (c) both correspond to the point A1 in (d). The point B in (b) corresponds to the point B1 in (d) whereas the point B in (c) corresponds to the point B2 in (d).
Figure 3. Overlap between experimental (red line with black dots) and simulated (blue line) spectra using (a) numerical CASSCF-PES, (b) numerical CASPT2-PES having a single well (SW), (c) analytically fitted CASPT2-SW-PES, (d) analytically fitted CASPT2-SW-PES adjusted to fit ν6, (e) analytically fitted CASPT2-SW-PES adjusted further to also match the intensities of nν6, and (f) CASPT2-DW-PES having a double well (DW). Panels on right show the norm of the autocorrelation function, ⟨χ(0)|χ(t)⟩.
CASSCF-PES by the present CASSCF(16,10)/6-311++G(d,p) calculations does not have a stationary structure, and the dissociation proceeds without any barrier. The previous study with the CASPT2/cc-pVTZ//CASSCF(12,10)/cc-pVTZ methodology also found no local stationary minimum of the S1 state.10 As will be discussed at the end of this work, the existence of any local stationary structure near the Franck− Condon (FC) region is not clear yet. In spite of this unclearness, our simulated spectrum using CASSCF-PES reproduces the main progressions of the symmetric stretching mode ν6 rather successfully [Figure 3a]. As stated in the previous experimental work,12 the total emission from 1CH2 (B), the red line with black dots in Figure 3, reflects the absorption cross section multiplied by the yield of the 1CH2 (B) fragments. Meanwhile, the earlier electronic absorption spectrum6 reported not only the first series of the nν6 progressions but also the second series of the nν6+2ν9 progressions, but their resolution was not enough to detect anything about the Fano-type resonance. Therefore, the absorption cross section generated by the present theoretical work was directly compared with the experimental profile of the total emission of the previous experimental work.12 The relative intensities of the main progressions nν6 were reproduced reasonably well, and the interval between the bands was also qualitatively proper. Even the asymmetry of the first two bands of the Fano-type profile was well reproduced qualitatively. This result implies that the coupling between the dissociation channel and the symmetric C−N stretching plays the main role in the Fano resonance here and that the minimum energy path disclosed in the morphology shown in Figure 2a reflects the main feature of the dissociation path, which corresponds to the asymmetric stepwise dissociation of two C−N bonds as suggested in the previous experimental work.12 However, no signals due to the asymmetric stretching mode ν9 were generated in the simulated spectrum in Figure 3a.
The numerical PES produced by our CASPT2//CASSCF(16,10)/6-311++G(d,p) calculations, shown in Figure 2b, shows one local minimum (a single well, SW) of the S1 state with a well depth of 0.069 eV and will be referred to as numerical CASPT2-SW-PES. The well corresponds to the stationary structure reported in previous experimental and theoretical works.11,12 The spectrum simulated using CASPT2SW-PES includes small bands due to combination with the asymmetric mode, nν6+2ν9 [Figure 3b]. Only the even quanta of ν9 combined with ν6 because of its asymmetry. However, the relative intensities of the main progressions of the ν6 mode in the simulated spectrum noticeably differ from the corresponding experimental ones. Another deviation is the gap (Δ2ν9) between the main band (nν6) and its combination with the asymmetric mode (nν6+2ν9); the simulated Δ2ν9 is about twice the experimental Δ2ν9 [Figure 3b]. The dissociation rate was also slower [Figure S120], which caused the main bands to be narrower; compare the blue lines in Figure 3a,b. More importantly, the asymmetry of the main bands almost disappeared. The resonance interaction between the bound symmetric C−N stretching mode (ν6) and the dissociation channel seems to be broken by the involvement of the asymmetric stretching mode (2ν9). The involvement of another degree of freedom in the Fano resonance tends to make the theoretical reproduction of the asymmetric profile more difficult, as shown by the long list of theoretical works to reproduce the Fano profiles of the S1 state of FNO.34 To investigate the effect of the detailed characteristics of the PES on the simulated spectrum, combinations of many analytic functions were developed to fit the numerical CASPT2-SWPES before systematic adjustments. The functions and their parameters are given in Tables S1 and Table S2, respectively, of 934
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The Journal of Physical Chemistry A Supporting Information.20 The simulated spectrum generated using this analytically fitted CASPT2-SW-PES, reconstructed by the analytic functions and their parameters, is given in Figure 3c, which is almost identical to Figure 3b. The population dynamics little changed by replacing the numerical CASPT2SW-PES with the analytically fitted CASPT2-SW-PES, as shown in Figure S1.20 CASPT2-SW-PES will be used for the analytically fitted CASPT2-SW-PES hereafter. The interval between the main progressions of ν6 can be adjusted by changing the curvature along R around the local minimum. After slight adjustment of just one parameter of the analytic functions,20 the positions of the main progressions of the ν6 mode match the corresponding experimental positions well [Figure 3d]; P1014 in Table S2 is multiplied by 0.97 to adjust the curvature of the PES along R. However, the relative intensities of the main bands in the simulated spectrum still differ from the experimental ones. This result indicates that the minimum of CASPT2-SW-PES is too close to the FC region of S0, and the wave packet sweeps only the lower-energy areas of the S1 state PES. To adjust the relative intensities, the entire PES of S1 is moved by 0.12 a0 toward larger values of R. The band heights in the simulated spectrum in Figure 3e now reproduce the experimental relative intensities quite well, but the simulated Δ2ν9 is still about twice the experimental Δ2ν9. The Δ2ν9 can be reduced gradually toward its experimental value by adjusting just a few parameters of the analytic functions. However, according to our study, the intensity of the nν6+2ν9 bands decreases gradually as Δ2ν9 approaches the experimental value and finally disappears entirely right before the simulated Δ2ν9 reaches the experimental magnitude. Even though all possible systematic adjustments were made to CASPT2-SW-PES, the simulated spectrum cannot reproduce both the position and the intensity of the combination bands simultaneously as long as the PES has just an SW. An interesting result emerges here. Both the position and intensity of the combination bands were finally reproduced simultaneously and successfully [Figure 3f] only when the PES has two local minima (a double well, DW), as shown in Figure 2c; this will be referred to as CASPT2-DW-PES hereafter. The DW was introduced assuming a symmetry breaking, as discussed in the case of cyclic C3C2H radical.4 To introduce the DW, the harmonic function ν122(θ) in the original set of functions in Table S1 is replaced by the double-well function ν122d(θ), as given in Table S3,20 and their parameters were optimized again. ν122(θ ) = p1221 (θ − p1222 )2 + p1223
respectively. All the other parts of the PES changed little, as represented by contours in Figure S2.20 Figure 3f shows that the positions of not only the main bands but also the combination bands become appropriate, and the relative intensities among the bands are also reproduced very well. Five other experimental bands at 1392, 1493, 1711, 2186, 2265, and 2548 cm−1 cannot be reproduced by the present simulation because they correspond to combinations involving other vibration modes (ν3, ν4, and ν8). Their effects on the Fano-resonance were expected to be very small in the previous work,12 but their effects could be a possible source of the DW of the CASPT2-DW PES. Among the nine vibrations of diazirine (see Figure S2 of ref 12), three modes (ν1, ν5, and ν8) are asymmetric to the CN2 plane and cannot affect our twodimensional (2D) PES. The other three modes (ν2, ν3, and ν4) also cannot affect our 2D PES because they are all perpendicular to the ring-opening plane. Except for the two modes (ν6 and ν9) consisting of the 2D space of present theoretical study, only the CH2 wagging mode (ν7) remains as a possible source of the DW. Another rationale comes from the fact that only two modes (ν7 and ν9) belong to b1 when diazirine maintains C2 V symmetry. To investigate the effect of the ν7 mode, two points, designated as A and B in Figure 2b, were chosen. The change from A to B corresponds to the combination of ν6 and ν9 to follow the minimum-energy path (see the dashed line in Figure S2 in SI20) of ring opening. The main motion of ν7 is described by changing just the wagging angle γ of the CH2 plane (the plane is represented by a vertical line in Figure 4) while the
Figure 4. Change in the wagging angle γ of the CH2 moiety. The plane of the CH2 moiety is shown using just a single C−H line here. See the main text and the caption of Figure 2 for the positions of A and B on the contour representations of PES in Figure 2.
other parts of the geometry are kept unchanged, and its effects on the PES are shown in Figure 2d. The energy profile at position A (blue curve) is symmetric with respect to the wagging angle γ = 0°. However, the energy profile at position B [red curve in Figure 2d] is no longer symmetric with respect to γ = 0° because the distance from H2 to one N atom becomes shorter than the distance to the other N atom at γ = 0°: see B1 in Figure 4. Note that the lowest energy in the red curve in Figure 2d is located about 0.01 eV below that of the blue curve, and the molecular structures at B1 and B2 in Figure 4 show that the potential energy is lowered as two H atoms are dragged slightly behind the ring opening. This result implies that the local minimum point along the ring-opening path could be moved from point A to point B, and as a result, the SW shown in Figure 2b changes into the DW shown in Figure 2c. The points A in (b) and (c) of Figure 2 both correspond to the A1 in Figure 2d, but the point B in Figure 2b corresponds to the point B1 in Figure 2d whereas the point B in Figure 2c corresponds to the point B2 in Figure 2d. The energy scale of the y axis in Figure 2d is very small, and the change from an SW
(7)
ν122d(θ ) = p1221 [(θ − p1222 )/p1223 ]4 − p1224 [(θ − p1222 )/p1223 ]2 + p1225
(8)
Two other parameters, P1014 and P1113, were also slightly adjusted, as given in Table S4.20 The important feature is that the change from CASPT2-SW-PES used in the production of the spectrum in Figure 1e to CASPT2-DW-PES is very small, and the depth of the DW is extremely small. The largest change along the main dissociation path is just 0.04 eV, as shown in Figure S2.20 The barrier between the two wells of DW is 0.011 eV, whereas the barrier from the well toward the ring-opening path is 0.033 eV, which is within the range of 0.027−0.054 eV suggested by the experiment12 on the basis of the lifetimes of the origin and the first overtone of the ν9 bands, 200 and 50 fs, 935
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The population within the 2D window space (defined by R = 1.20−2.20 Å, θ = 20−160°) starts to decrease at ∼35 fs, see Figure S1,20 but the majority of the wave packet remains around the DW and reaches approximately the initial FC region at ∼40 fs (corresponding to ν6 = 802 cm−1), as indicated by the first peak in the autocorrelation function ⟨χ(0)|χ(t)⟩ [righthand panel of Figure 3f]. In other words, the first round of this delicate dynamics is completed in ∼40 fs, and the second round begins after that. The regular appearance of the main peaks in ⟨χ(0)|χ(t)⟩ in Figure 3f implies that the main feature of the early time dynamics does not change greatly. Note that the first recurrence of the wave packet near the FC region due to the DW appears at around 120 fs and causes the shoulder of the third peak of ⟨χ(0)|χ(t)⟩ as shown in the right-hand part of Figure 3f. The effect of the DW, however, already begins to appear starting at ∼35 fs and at ∼80 fs (corresponding to one quantum, ν9 = 108 cm−1) in the change of population; that is, the overall dissociation rate is noticeably affected by the presence of the delicate small DW, as shown by the black dots in Figure S2.20 The slowed rate of the population decrease after 80 fs could be a reason for the increased q value of the Fano fitting parameter from 4.5 at 802 cm−1 to 6.0 at 1072 cm−1 (see Table 1 in ref 12).
to a DW involves only a tiny adjustment in the energy scale, see Figure S2.20 Despite the smallness of this change, the simulated spectrum is affected dramatically, as disclosed in this study. It is well-known that even a tiny difference in a PES can cause huge changes in the dynamics, the fine structure of the corresponding spectrum, and even in final branching ratios.13,34,35 This type of small change is very hard to detect even with the highest level of electronic structure theories in conjunction with extremely large size of basis sets, as discussed elsewhere.4,5,36 Even after such elaborated calculations under the Born-Oppenheimer approximation for stationary structures, the symmetry breaking may not be detected without applying dynamical study including quantum effects on nuclear behavior as carried out in the present work. Now, the microscopic and quantum mechanical details of the dynamics producing the final simulated spectrum can be discussed further using the 12 snapshots in Figure 5. The initial
IV. CONCLUSIONS The morphology of the 2D PES in this work confirms that combination of the symmetric and asymmetric stretching modes of the C−N2 ring excites the dissociation channel of diazirine excited to its S1 state. More importantly, this work shows that both the positions and relative intensities of the experimentally observed bands can be generated only when the morphology of the S1 state PES has a DW around the FC region. The depth of the DW is just 0.011 eV, but its presence noticeably affects the main features of the absorption spectrum and dissociation rate. The delicate DW, which was not disclosed directly by an experimental observation, reflects the dynamic symmetry breaking (DSB) due to coupling between the asymmetric C−N2 stretching mode ν9 and the CH2 wagging mode ν7. However, the skew feature of the Fano-type profile was not successfully reproduced in this work, suggesting that the full details of coupling between the dissociation path and the bound internal modes were not incorporated. Further efforts to elucidate the delicate coupling are desirable. It is understandable that this work, which considered a 2D space, cannot completely reproduce the Fano profiles of the S1 state of diazirine, which has nine degrees of freedom, considering the long list of theoretical works and the extremely complicated analytic functions for the PES used to reproduce the fine details of the Fano profiles of the S1 state of FNO,34 which has just three degrees of freedom. Although the existence of a local stationary structure of diazirine in the first excited electronic state S1 seems still unclear in spite of all previous theoretical works,9−11 the recent experimental12 and the present theoretical studies show that the stationary structure, if it exist, has an extremely small barrier for a dissociation pathway. We now expect that the magnitude of the energy barrier will exhibit extreme sensitivity on not only the level of dynamic electron correlations but also the size of basis sets. The DW in the present study was introduced assuming a symmetry breaking due to the influence from a nearby upper electronic state, as discussed in the case of cyclic C3C2H radical.4 The present work shows that the combination
Figure 5. Twelve snapshots showing propagation of the wave packet on CASPT2-DW-PES represented by red contours with a gap of 0.002 au. The x and y axes represent the R and θ coordinates, respectively.
position of a vertically excited wave packet is shown by the blue contours in the “0 fs” panel. The main dynamics during the first 10 fs is just the symmetric stretching of the C−N bond corresponding to the ν6 mode, and the wave packet arrives at the saddle point between the DWs. The wave packet propagates further, and the spreading toward larger angle θ is negligible until 20 fs, at which it arrives at R = 3.3 a0 and starts to bifurcate. The bifurcation here implies that the bond lengths of two C−N bonds start to differ asymmetrically. The direction along the R coordinate is reversed at this point, and the split wave packets start to spread along θ, but the degree of angular spreading is still marginal. The centers of the two split wave packets propagate slightly out of the two respective local minima, as shown in the “25 fs” and “30 fs” panels of Figure 5. As it proceeds further to 35 fs, some portion of the wave packet begin to leak out toward the ring-opening channel, as shown in the three panels at 35, 40, and 45 fs. 936
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The Journal of Physical Chemistry A of the asymmetric C−N2 stretching mode ν9 and the wagging mode ν7 is the main component responsible for the dynamic symmetry breaking influenced by a nearby electronic state. The second excited state 1B2 (3s) of diazirine10,11 could be the upper state mainly responsible, but it is not clear yet.
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(11) Fedorov, I.; Koziol, L.; Mollner, A. K.; Krylov, A. I.; Reisler, H. Multiphoton Ionization and Dissociation of Diazirine: A Theoretical and Experimental Study. J. Phys. Chem. A 2009, 113, 7412−7421. (12) Ahn, D.-S.; Kim, S.-Y.; Lim, G.-I.; Lee, S.; Choi, Y. S.; Kim, S. K. Mode-Dependent Fano Resonances Observed in the Predissociation of Diazirine in the S1 State. Angew. Chem., Int. Ed. 2010, 49, 1244− 1247. (13) Schinke, R. Photodissociation Dynamics; Cambridge University Press: Cambridge, 1993. (14) Tannor, D. J. Introduction to Quantum Mechanics: A TimeDependent Perspective; University Science Books: South Orange, NJ, 2007. (15) Hadder, J. E.; Frederick, J. H. Molecular Hamiltonians for Highly Constrained Model Systems. J. Chem. Phys. 1992, 97, 3500− 3520. (16) Knowles, P. J.; Werner, H.-J. An Efficient Second-Order MC SCF Method for Long Configuration Expansions. Chem. Phys. Lett. 1985, 115, 259−267. (17) Werner, H.-J.; Knowles, P. J. A Second Order Multiconfiguration SCF Procedure with Optimum Convergence. J. Chem. Phys. 1985, 82, 5053−5063. (18) Werner, H.-J.; et al., MOLPRO, version 2012.1, a package of ab initio programs; http://www.molpro.net. (19) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. SelfConsistent Molecular Orbital Methods. A Basis Set for Correlated Wave Functions. J. Chem. Phys. 1980, 72, 650−654. (20) More details are given in the Supporting Information of this work. (21) Kosloff, R.; Tal-Ezer, H. A Direct Relaxation Method for Calculating Eigenfunctions and Eigenvalues of the Schrö dinger Equation on a Grid. Chem. Phys. Lett. 1986, 127, 223−230. (22) Schmidt, B.; Lorenz, U. WavePacket4.7: a program package for quantum-mechanical wavepacket propagation and time-dependent spectroscopy; http://wavepacket.sourceforge.net. (23) Stanton, J. F.; Bartlett, R. J. The Equation of Motion CoupledCluster Method. A Systematic Biorthogonal Approach to Molecular Excitation Energies, Transition Probabilities, and Excited State Properties. J. Chem. Phys. 1993, 98, 7029−7030. (24) Baeck, K. K. The Analytic Gradient for the Equation-of-Motion Coupled-Cluster Energy with a Reduced Molecular Orbital Space: An Application for the First Excited State of Formaldehyde. J. Chem. Phys. 2000, 112, 1−4. (25) Baeck, K. K.; Jeon, S. I. The Analytic Gradient with a Reduced Molecular Orbital Space for the Equation-of-Motion Coupled-Cluster Theory: Systematic Study of the Magnitudes and Trends in Simple Molecules. Bull. Korean Chem. Soc. 2000, 21, 720−726. (26) Stanton, J. F.; et al., ACES-II, Release 2.5.0, 2006, a program product of the Quantum Theory Project; University of Florida: Gainesville, FL, 2006. (27) Feit, M. D.; Fleck, J. A.; Steiger, A. Solution of the Schrödinger Equation by a Spectral Method. J. Comput. Phys. 1982, 47, 412−433. (28) Kosloff, R.; Kosloff, D. Absorbing Boundaries for Wave Propagation Problems. J. Comput. Phys. 1986, 63, 363−376. (29) Heller, E. J. The Semiclassical Way to Molecular Spectroscopy. Acc. Chem. Res. 1981, 14, 368−375. (30) Lan, Z.; Domcke, W.; Vallet, V.; Sobolewski, A. L.; Mahapatra, S. Time-Dependent Quantum Wave-Packet Description of the σ* Photochemistry of Phenol. J. Chem. Phys. 2005, 122, 224315. (31) An, H.; Baeck, K. K. Quantum Wave Packet Propagation Study of the Photochemistry of Phenol: Isotope Effects (Ph-OD) and the Direct Excitation to the 1πσ* State. J. Phys. Chem. A 2011, 115, 13309−13315. (32) Park, Y. C.; An, H.; Choi, H.; Lee, Y. S.; Baeck, K. K. WavePacket Propagation Study of the Early-Time Non-Adiabatic Dissociation Dynamics of NH3Cl: Diabatic Picture, Effects of Isotope Substitution and Varying the Initial Vibration Levels. Theor. Chem. Acc. 2012, 131, 1212.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b11052. Part 1 contains the analytic functions and their optimized parameters: Table S1. Analytic functions used for CASPT2-SW-PES; Table S2. Parameters of functions for the analytically f itted CASPT2-SW-PES; Table S3. Analytic functions used for CASPT2-DW-PES; Table S4. Parameters of functions for the CASPT2-DW-PES. Part 2 provides supplementary figures: Figure S1. The population dynamics with different PESs; Figure S2. Difference between PESs used for Figure 3e,f; Figure S3 and S4 for the same Figure 3 and 5, respectively, on a larger scale for clarity. (PDF)
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AUTHOR INFORMATION
Corresponding Authors
*E-mail: [email protected]. *E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the National Research Foundation of Korea (NRF-2015R1D1A3A03019217 to K.K.B. and NRF2012R1A1A2000915 to Y.S.L). The authors thank Professor S. K. Kim of KAIST for the initial suggestion of this work and helpful discussions.
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REFERENCES
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