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Constructive and Destructive Interferences in Nonadiabatic Tunneling via Conical Intersections Changjian Xie, Brian K. Kendrick, David R. Yarkony, and Hua Guo J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00124 • Publication Date (Web): 31 Mar 2017 Downloaded from http://pubs.acs.org on April 2, 2017

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Journal of Chemical Theory and Computation

Submitted to J. Chem. Theo. Comput. 2/6/2017

Constructive and Destructive Interferences in Nonadiabatic Tunneling via Conical Intersections

Changjian Xie,1 Brian K. Kendrick,2 David R. Yarkony,3 and Hua Guo1,* 1

Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131 2

Theoretical Division (T-1, MS B221), Los Alamos National Laboratory, Los Alamos, New Mexico 87545 3

Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218

*: corresponding author, [email protected]

1 ACS Paragon Plus Environment


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Abstract As a manifestation of the molecular Aharonov-Bohm effect, tunneling-facilitated dissociation under a conical intersection (CI) requires the inclusion of the geometric phase (GP) in order to ensure a single-valued adiabatic wavefunction encircling the CI. In this work, we demonstrate using a simple two-dimensional model that the GP induces a destructive interference for vibrational states with even quanta in the coupling mode, but a constructive interference for those with odd quanta. The interference patterns are manifested in tunneling wavefunctions and clearly affect the tunneling lifetime. It is further shown that the inclusion of the diagonal Born-Oppenheimer correction is necessary for agreement with exact results.


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I.

Introduction The Born-Oppenheimer approximation is a pillar of the conventional quantum

mechanical description of spectroscopy and dynamics in molecular systems. Using perturbation theory, Born and Oppenheimer (BO) derived separate equations of motion for the electronic and nuclear degrees of freedom in the limit of an infinite mass ratio between the nuclei and electrons.1 This separation is justified by the relatively large mass disparity in real systems and often large energy separation between electronic states. Indeed, this intuitive and appealing approximation was instrumental in establishing the concept of the adiabatic potential energy surface (PES) for describing molecular spectroscopy and reaction dynamics. However, it is now well known that the BO approximation is inadequate if the coupling among different electronic states, induced by the nonadiabatic couplings (NACs) ignored in this approximation,2, 3 is large, as is the case near an electronic degeneracy such as a conical intersection (CI).4-7 In these cases an accurate characterization of the nonadiabatic dynamics requires going beyond the BO approximation.8-16 Even when the dynamics is ostensibly restricted to the lower adiabatic PES, an adiabatic treatment near a CI has to deal with an additional difficulty. It was shown by Herzberg and Longuet-Higgins that a real adiabatic electronic wavefunction changes sign when transported in a closed path around a CI.17 This sign change ( −1 = eiπ ) is a special case of the geometric phase (GP) discussed by Mead and Truhlar18 and later in a broader context by Berry.19-21 The GP has been shown to have a decisive influence on the adiabatic dynamics even at energies much lower than that of the CI, as amply demonstrated in both bound22-37 and unbound systems.18, 38-50 In simple cases the single valuedness of the total (electronic and nuclear) wavefunction, can be achieved by subjecting the adiabatic nuclear wavefunction to a double valued boundary 3 ACS Paragon Plus Environment


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condition. As pointed out by Mead and Truhlar, this condition can also be enforced by the introduction of a vector potential.18 In this work, we focus on the tunneling-facilitated dissociation near a CI of electronically excited molecules prepared by photoexcitation. The presence of such an electronic degeneracy is quite prevalent for electronically excited states.9, 11-13 A recent example is the photodissociation of phenol, in which its low-lying vibrational states in the S1←S0 excitation are well below the CI and their dissociation51,

52

has to be facilitated by tunneling.53-55 The emergence of full-

dimensional coupled PESs for this system obtained by fitting ab initio data56-58 has stimulated much recent interest in the dissociation dynamics.59-62 Recently, we have shown using a reduced-dimensional model that the tunneling lifetime of the lowest vibronic state calculated with an adiabatic model without the GP can be as much as two orders of magnitude faster than that obtained with the exact multi-state diabatic model.60 Such nonadiabatic tunneling differs fundamentally from its adiabatic counterpart on the lower adiabatic state, as the upper adiabatic state is tacitly involved even when the energy of the vibronic state is significantly lower. This effect has been likened to the Aharonov-Bohm effect, in which a charged particle acquires a phase when circling a small solenoid even though the magnetic field is zero along the path.18,

22

It has been shown by Mead and Truhlar that the

adiabatic approximation fails without considering the GP in the presence of CIs.18 Very recently, it was demonstrated by us that the failure of the adiabatic model in characterizing nonadiabatic tunneling can be largely attributed to the absence of two terms ignored in the BO approximation, namely the GP and diagonal BO correction (DBOC).61 The inclusion of the GP qualitatively changes the dissociative wavefunction by introducing a node,27,

28, 53

reflecting destructive

interference between dissociative “trajectories” on the two sides of the CI.48, 4 ACS Paragon Plus Environment

63, 64

This

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destructive interference retards the tunneling flux. In this work, we present the details of our calculations and extend them to include vibrationally excited states. Interestingly, the GPinduced interference becomes constructive when the coupling mode possesses odd quantum numbers, resulting in an enhancement of the tunneling. Furthermore, we also illustrate how the often ignored DBOC term influences the tunneling lifetimes. This work is organized as follows. The basic theory and its implementation is outlined in Sec. II. The results are presented and discussed in Sec. III. This is followed by a conclusion in Sec. IV. II.

Theory Our focus of this work is on a two-dimensional (2D) model, in which two electronic

states interact through a zero-dimensional CI, namely a point. In this model, the adiabatic NAC can be completely removed by the adiabatic-to-diabatic (AtD) transformation.65,

66

The

Hamiltonian in the diabatic representation65, 67 has the following form ( h = 1 hereafter):  Tˆ Hˆ ( d ) =  0 

0   V11 + Tˆ   V12

V12  , V22 

(1)

where Tˆ = −∇2 / 2 = −(∂ 2 / ∂x 2 + ∂ 2 / ∂y 2 ) / 2 is the kinetic energy operator (KEO) while the potential energy operator (PEO) is a 2×2 matrix. The corresponding adiabatic Hamiltonian  Tˆ + τˆ11 Hˆ ( a ) =   −iτˆ  21

iτˆ12   W− 0 +  Tˆ + τˆ22   0 W+ 

(2)

is obtained via the following unitary AtD transformation



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 cos θ − sin θ  U = ,  sin θ cos θ 

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(3)

where the mixing angle between the two diabatic states is:

1 2

θ = arctan

2V12 . V11 − V22

(4)

The transformation diagonalizes the PEO, giving

W± =

1 1 (V11 + V22 ) ± 2 2

(V11 − V22 )

2

+ 4V122 ,

(5)

and yields NAC terms including the scalar diagonal BO correction (DBOC) 1 2

τˆ11 = τˆ22 = ∇θ ⋅ ∇θ ,

(6)

and vectorial derivative coupling (DC)

τˆ12 = τˆ21 =

where

†

1 2

( ( − i∇ ) ⋅ ∇ θ + ∇ θ ⋅ ( − i∇ ) ) , †

(7)

denotes Hermitian conjugation. Note that the scalar DBOC term can be considered as a

potential and added to the adiabatic potentials. Both the DC and DBOC terms play a vital role in nonadiabatic processes,12 and neglect of the DBOC would lead to the incorrect outcome in most cases.23, 35, 68, 69 As discussed by Mead and Truhlar,18 a position-dependent phase factor e iθ ( x , y ) can be introduced to the adiabatic wavefunction to make the total wavefunction single valued. The result is a GP-corrected adiabatic Hamiltonian


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 Tˆ + τˆ11GP iτˆ12GP  W− 0  (a) −iθ ˆ ( a ) iθ ˆ HGP = e H e =  + ,  −iτˆGP Tˆ + τˆGP   0 W+  22   21

(8)

where

(

)

ˆ iθ − Tˆ , τˆ11GP = τˆ22GP = τˆ11 + e − iθ Te

(9)

τˆ12GP = τˆ21GP = e − iθ τˆ12 e iθ .

(10)

Indeed, the introduction of the GP has no effect on W± , but inserts a vector (or gauge) potential ( −iθ ˆ iθ → Pˆ + Aˆ .18 This vector potential is Aˆ ≡ ∇ θ ) into the momentum operator, namely e Pe

singular at the CI. As discussed below, the GP also introduces a scalar potential, which in this model is identical to the DBOC term and thus is also singular at the CI. It is instructive to examine the GP-induced correction term for the KEO in Eq. (9), which can be separated into two parts. The first part ( τˆ11 ) is identical to the DBOC term (Eq. (6)) in the non-GP-corrected Hamiltonian (Eq. (2)), while the second stems from the introduction of the vector potential into the adiabatic Hamiltonian:

(

)

ˆ iθ − Tˆ = 1 ∇θ ⋅∇θ + 1 ( −i∇ )† ⋅∇θ + ∇θ ⋅ (−i∇) . e−iθ Te 2 2

(11)

Interestingly, the GP term in Eq. (11) contains the same term as τˆ11 , in addition to the vector potential modulated momentum operator. As demonstrated below, both terms play important roles in nonadiabatic tunneling. Following our recent work,61 the diabatic PEO in our models is defined as follows



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ω12 

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2

a  ω22 2 V11 = y x+  + 2  2 2 V22 = Ae −α ( x + b ) +

ω22

y2 − ∆ 2 V12 = cy ⋅ exp[ − ( x − xCI ) 2 / 2σ x2 ]

(12)

Here, the x coordinate is the tuning mode and y the coupling mode, corresponding to the g and h vectors spanning the branching space of the CI.6 Note that the diabatic coupling term (V12) is antisymmetric with respect to the coupling coordinate (y) near the CI. To unravel the roles of the GP and DBOC in nonadiabatic tunneling, we examine the following models. In Model I, the dynamics is characterized in the two-state diabatic representation with the Hamiltonian in Eq. (1). The diabatic Hamiltonian is smooth and free of singularities, and as a result is the preferred representation for quantum dynamical studies of nonadiabatic systems.8, 16, 18, 70 Here, this model is exact. Model II is a one-state adiabatic model on the lower adiabatic PES that ignores the NAC term with the upper adiabatic state. Both the GP and DBOC terms are included in the Hamiltonian for this model:

(

)

1 † Hˆ II( a ) = Tˆ + τˆ11GP + W− = Tˆ + ∇θ ⋅∇θ + ( −i∇ ) ⋅∇θ + ∇θ ⋅ ( −i∇) + W− , 2

(13)

It should be noted that the neglect of the upper adiabat and NAC will inevitably introduce errors, as shown in our previous work.61 However, this model is attractive as it deals with only one electronic state and highlights the effect of the GP. Indeed, similar models have been widely used to study the GP effects in scattering processes.18, 38-50


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This Hamiltonian can be further simplified by removing the non-GP-corrected DBOC term ( τˆ11 ), resulting in the Hamiltonian for Model III:

(

)

1 1 † Hˆ III( a ) = Tˆ + ∇θ ⋅∇θ + ( −i∇ ) ⋅∇θ + ∇θ ⋅ (−i∇) + W− 2 2

(14)

Finally, the GP term defined in Eq. (11) is removed in Model IV, leading ultimately to the original BO Hamiltonian:

Hˆ IV( a ) = Tˆ + W− .

(15)

Numerically, the Schrödinger equation is solved using the polar coordinates (ρ, φ), in which x = ρ cos ϕ and y = ρ sin ϕ , and the CI is placed at the origin. The advantage of the polar coordinates is the dense grid near the origin, which is helpful for converging the integrals near the singularity. In this coordinate system, we have: 1  ∂2 1 1 ∂2  Tˆ = −  2 + 2 + 2 , 2  ∂ρ 4ρ ρ ∂ϕ 2 

(16)

and the wavefunction is expanded in the following basis functions:

ψ (ρ ,ϕ ) =

1 2πρ

∑C

mn

Rn ( ρ )eimϕ .

(17)

m,n

with the following normalization condition: 2π

ρ max

0

0

∫ ∫

* ψ mn ( ρ , ϕ )ψ m′n′ ( ρ , ϕ ) ρ d ρ dϕ = δ nn′δ mm′ .

(18)

The computational details of the matrix elements of the KEO, PEO, DBOC, and GP terms in polar coordinates and the Chebyshev wave packet propagation method71 have been given in our



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recent paper.61 In the current calculations, 45 and 75 basis functions were used for ρ and φ coordinates, respectively. 1000 and 2000 Gauss-Legendre quadrature points were used in the ranges [0.0, 15.0] a.u. and [0.0, 2π] for the ρ and φ coordinates, respectively. The damping function D=exp[-0.0027(ρ-9.0)2] for ρ>9.0 a.u. was used in the Chebyshev propagation to enforce the outgoing boundary condition. The lifetimes were extracted from the complex energies (E-iΓ/2) of the vibronic state determined by a low-storage filter diagonalization method71 To this end, the Chebyshev correlation functions are used to build an energy-localized Hamiltonian matrix, from which the complex energies of the resonances are obtained by diagonalization. 20,000 Chebyshev steps were found to give converged the lifetime. When needed, the energy-dependent wavefunction is obtained from the Chebyshev wave packet via a cosine Fourier transform. III.

Results and discussion To understand how the GP affects the tunneling-facilitated dissociation of the vibrational

ground and excited states, calculations were performed on three different sets of model PESs (parameters in a.u.) with varying coupling strengths (c in Eq. (12)): (a) ω1=1, ω2=1, a=4, A=5.0, b=-11, ∆=12, α=0.1, and σ x=1.274, (b) ω1=0.7, ω2=1, a=6, A=5.0, b=-11, ∆=12, α=0.1, and σ x=1.274,

and (c) ω1=1, ω2=0.3, a=4, A=5.0, b=-11, ∆=12, α=0.1, and σx=1.274. The first set of

PESs have identical frequencies of both the x and y modes, while the other two are different for the two modes. The diagonal and off-diagonal elements of the potential matrix are shown in Figure 1 for a typical set of parameters. The corresponding lower and upper adiabatic PESs are also shown in the figure. For this symmetry-allowed CI system, the lower adiabatic PES features the lower cone of the CI at x=y=0, flanked by two energetically lower and equivalent saddle points on both sides (y0). Table I lists the energies of the CIs and saddle points of the 10 ACS Paragon Plus Environment

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three model PESs in eV. It can be readily seen that for a stronger coupling strength c, a lower saddle point energy is produced. The vibrational states are denoted by (nx, ny), where nx and ny are the two quantum numbers for the two modes. The exact energies (given by Model I) of the (0, 0), (1, 0), and (0, 1) states on the three model PESs are listed in Table II in eV, it is clear from the table that these states are all lower in energy than the CIs and saddle points listed in Table I. Thus, their dissociation can only occur through tunneling. Figure 2 shows the lifetime of the ground (0, 0) and excited (0, 1) states for the three model PESs, respectively, as a function of the coupling strength (c). It can be seen from the figure that for the ground (0, 0) state on all three PESs, the lifetimes obtained from the onestate adiabatic model (Model IV) are much shorter than those for the exact two-state diabatic (Model I). The inclusion of the GP (Model III) significantly improves the results, and the lifetimes of the one-state adiabatic model with both GP and DBOC (Model II) agree best with the exact results. The remaining error is attributed to the neglect of the upper adiabatic state in these adiabatic models, as discussed in our earlier work.61 These results further confirm our recent conclusions,61, 62 namely, that the large difference between nonadiabatic and adiabatic (Model IV) lifetimes is caused by the neglect of the GP. Furthermore, the exact lifetimes decrease with increasing coupling strength (c), underscoring the fact that nonadiabatic tunneling is largely controlled by the V12 term, rather than the height and width of the adiabatic barrier as in the case of adiabatic tunneling.61, 62 In the upper panel of Figure 3, a typical adiabatic wavefunction for the (0, 0) state is shown for Models I, II, III, and IV. All wavefunctions possess nodes in the x coordinate beyond the CI (x>0), which are attributable to the translational momentum acquired by the system when it dissociates. It is also clear that the addition of the GP in the adiabatic Hamiltonian (Models II 11 ACS Paragon Plus Environment


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and III) produces a node in the y coordinate beyond the CI (x>0), which is also present in the exact wavefunction in Model I. This nodal structure, which is a hallmark of the GP,27, 28, 35, 36, 48 suggests a destructive interference for “trajectories” passing around the CI on the two opposite sides (y0). As shown in previous studies35, 48, 61 and discussed below, the destructive interference causes the corresponding wavefunctions to be out of phase at y=0, x>0, thus the node in the wavefunction. It is this destructive interference that retards the tunneling, resulting in a reduction of the tunneling lifetime when GP is included in the adiabatic treatment.60 The situation is quite different for the excited state (0, 1). As shown in Figure 2, the inclusion of the GP in Model III increases the tunneling rate compared to the one-state adiabatic model (Model IV). Again, the inclusion of the DBOC (Model II) further improves the results. Interestingly, this behavior is opposite to that of the ground state. The clue to the origin of such behavior can be found in Figure 3 (middle panel), where the corresponding adiabatic wavefunctions are plotted. Beyond the CI (x>0), the wavefunctions still have nodes in the x direction, but different nodal structures in the y direction are apparent. Consistent with the exact Model I, the (0, 1) wavefunctions from Models II and III possess at y=0 one node before the CI (x0), which differ qualitatively from that from Model IV in which the GP is absent. This suggests that the GP is again responsible for the change in the nodal structure of the wavefunction. It appears that the inclusion of the GP introduces a constructive interference at y=0 for the (0, 1) state, rather than the destructive interference seen for the (0, 0) state. Because of the constructive interference, the inclusion of the GP enhances the tunneling, leading to a shorter lifetime, as shown in Figure 2 for all PESs. In the lowest row of Figure 3, the wavefunctions of the (1, 0) state are shown for all the models. The nodal pattern of this state beyond the CI is essentially the same as that of the (0, 0) 12 ACS Paragon Plus Environment

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state. In particular, the emergence of a node at y=0 outside the CI can be considered as the result of a destructive interference. Clearly, the nodal structure of the wavefunction beyond the CI is determined by the nodal structure of the wavefunction in the y, rather than x, direction. This is easily understood as the V12 term is antisymmetric in y. To illustrate the different interference patterns in these two cases, we adapt the approach of Althorpe and coworkers,48,

63, 64

in which the wavefunctions are unwound based on their

reaction paths. In this unwinding scheme, the GP-corrected adiabatic wavefunction is decomposed into wavefunctions encircling the CI with even ( Φ e ) and odd ( Φ o ) loops:63, 64

Φ GP =

1 ( Φe − Φo ) , 2

(19)

which correspond to Feynman paths passing around the CI clockwise and counter-clockwise. The corresponding wavefunctions are thus capable of interference in the symmetry plane (y=0) outside the CI, as initially predicted by Mead and Truhlar for the H + H2 reaction.18, 38 Their phase difference ( ∆φ = φe − φo ) can be obtained from the adiabatic wavefunctions computed using the two-state adiabatic model with and without the GP ( ΦGP and Φ NGP ).48 In Figure 4, the phase differences of the two time-dependent wave packets are plotted for both the (0, 0) and (0, 1) states, and it is clear that Φ e and Φ o are in phase ( cos( ∆φ ) = 1 ) and out of phase (

cos( ∆φ ) = −1 ) at y=0 for x>0, respectively. As a result, the GP-corrected wavefunction in Eq. (19) has either a node or no node at y=0 for x>0. The details of the construction of the timedependent wave packet can be found in our earlier work.61 Such a constructive/destructive interference can also be found for higher vibrationally excited states. With the same PES as that in Figure 3, the energies of the (0, 0), (0, 1), (0, 2), and 13 ACS Paragon Plus Environment


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(0, 3) are 17.65, 25.72, 33.96, and 42.10 eV, respectively. All of them are lower than the CI (70.84 eV) and saddle point (64.80 eV). Figure 5 plots the lifetimes as a function of the quantum number ny of the vibrational mode in the y direction in the two-state diabatic and one-state adiabatic models. It can be seen that the lifetime from the two-diabatic (Model I), one-state with the GP and DBOC (Model II), and one-state adiabatic without the GP and DBOC (Model IV) models decreases slowly but monotonically with the quantum number ny, due apparently to the fact that the corresponding energy becomes higher as ny increases. Interestingly, however, the lifetime from the one-state adiabatic model with the GP (Model III) is oscillatory with ny, due apparently to the alternating GP-induced destructive/constructive interferences in the y direction outside the CI (x>0). The modulus of the corresponding wavefunctions are plotted in Figure 6. As shown in Figures 2 and 5, the addition of the DBOC ( τ 11 ) always increases the tunneling lifetime relative to the GP Model III. This is because this term is very small everywhere except near the CI. In fact, it is singular (∞) at the CI. In Figure 7, this term is plotted for a typical PES with different coupling strengths. Its inclusion thus adds to the adiabatic PES an additional potential barrier to block the tunneling flux. As shown in the figure, the DBOC becomes more repulsive as the coupling strength decreases, which is consistent with the results (Models I and II) in Figure 2 that the lifetimes decay monotonically as c increases. The DBOC term was long known to affect vibrational energy levels in molecules, but its impact is often considered quite small in molecular spectroscopy.72 In the current case, however, the DBOC plays a much more dominant role because it significantly modifies the tunneling path near the CI by providing a potential barrier. As a result, it cannot be ignored in an accurate adiabatic treatment of the nonadiabatic tunneling dynamics. We note in passing that the DBOC term requires special numerical attention, because it is singular at the CI. The quadrature points near 14 ACS Paragon Plus Environment

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the CI thus need to be quite dense in order to converge the integral approaching the CI, but exclude the CI point. As alluded to above, the GP term in Eq. (11) contains the same term as the DBOC ( τ 11 ). Thus, the inclusion of this term of the GP correction is expected to reduce tunneling as it provides an additional potential barrier. However, the second term of GP (Eq. (13)),

(( −i∇ ) ⋅∇θ + ∇θ ⋅ (−i∇) ) / 2 , introduces the phase shift in the wavefunction via the vector †

potential ( ∇θ ). This term is thus responsible for the constructive and destructive interferences seen in the wavefunctions of different ny states. As a result, the GP effect is the largest for the ground state, because both terms work in tandem to retard tunneling. For higher excited state, the effect becomes smaller. IV.

Conclusions In this work, we examined nonadiabatic tunneling facilitated dissociation dynamics using

a 2D two-state model. It is shown that the inclusion of the GP in an adiabatic treatment of the dynamics is essential to produce the correct dissociative wavefunction. It also impacts the tunneling lifetime significantly. Specifically, it is shown that the GP induces a constructive interference for vibronic states that have even quanta in the coupling mode, but a destructive interference for those with odd quantum in the same mode. These interferences manifest themselves by the presence or absence of a node in the coupling coordinate outside the CI, and by the oscillation of the corresponding tunneling lifetime. It is also shown that the often ignored DBOC term is vital in describing the tunneling dynamics correctly as it forms a potential barrier near the CI that attenuates the tunneling flux. These results clearly suggest that the BO treatment of nonadiabatic tunneling near a CI is inadequate. 15 ACS Paragon Plus Environment


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Acknowledgments: H.G. and D.R.Y. thank DOE (DE-SC0015997) and NSF (CHE-1361121) for financial support. B.K.K. acknowledges that part of this work was done under the auspices of the US Department of Energy under Project No. 20170221ER of the Laboratory Directed Research and Development Program at Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Los Alamos National Security, LLC, for the National Security Administration of the US Department of Energy under contract DE-AC52-06NA25396.


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Table I. Energies (in eV) of the CI and saddle point (SP) for the three model PESs for different c values (in a.u.). PES (a) c 0.2 0.4 0.6 0.8 1.0

CI

70.84

PES (b) SP 70.30 68.67 65.94 62.13 57.24

c 0.6 0.8 1.0 1.1 1.2

CI

71.30

PES (c) SP 66.40 62.59 57.69 54.84 51.71

c 0.1 0.2 0.3 0.4

CI

70.84

SP 69.33 64.80 57.24 46.65

Table II. Energies (in eV) of the (0, 0)，(1，0) and (0, 1) vibrational states on the three model PESs for different c values (in a.u.).

c 0.2 0.4 0.6 0.8 1.0

PES (a) (0, 0) (0, 1) 27.20 54.38 27.17 54.26 27.11 54.05 27.03 53.74 26.93 53.30

(1, 0) 54.39 54.29 54.11 53.84 53.46

c 0.6 0.8 1.0 1.1 1.2

PES (b) (0, 0) (0, 1) 23.11 50.28 23.10 50.22 23.08 50.15 23.07 50.10 23.06 50.05

(1, 0) 42.09 42.02 41.92 41.86 41.78

c 0.1 0.2 0.3 0.4


PES (c) (0, 0) 17.68 17.65 17.59 17.51

(0, 1) 25.82 25.72 25.54 25.27

(1, 0) 44.86 44.75 44.55 44.24


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Figure Captions: Figure 1. Contour plots of the diabatic PESs, coupling surface, and adiabatic PESs: ω1=0.7, ω2=1, c=0.8, a=6, A=5.0, b=-11, ∆=12, α=0.1, and σ x=1.274. Figure 2. Lifetimes (ps) of the (0, 0) and (0, 1) states in the two-state diabatic and one-state adiabatic models for different c, other parameters (a.u.) in model PESs: (a) ω1=1, ω2=1, a=4, A=5.0, b=-11, ∆=12, α=0.1, and σ x=1.274, (b) ω1=0.7, ω2=1, a=6, A=5.0, b=-11, ∆=12, α=0.1, and σ x=1.274, and (c) ω1=1, ω2=0.3, a=4, A=5.0, b=-11, ∆=12, α=0.1, and σ x=1.274. Figure 3. Modulus of the adiabatic wavefunction for the (0, 0) (upper panel), (0, 1) (middle panel), and (1, 0) (lower panel) states for Models I, II, III, and IV on the PES: ω1=1, ω2=0.3, a=4, c=0.2, A=5.0, b=-11, ∆=12, α=0.1, and σ x=1.274. Red dots denote the CIs. Figure 4. Relative phase between the clockwise and counter-clockwise wave packets (t=5000 a.u. as an example) at x=3.0 and 2.5 a.u. for the (0, 0) and (0, 1) states, respectively. Figure 5. Lifetimes (ps) of the (0, ny=0, 1, 2, and 3) states (ny denotes the quanta of the stretching mode in y direction) in the two-state diabatic and one-state adiabatic models other parameters (a.u.) in model PESs: ω1=1, ω2=0.3, a=4, c=0.2, A=5.0, b=-11, ∆=12, α=0.1, and σ x=1.274. Figure 6. Modulus of the adiabatic wavefunctions in the one-state adiabatic models with GP of the (0, ny=0, 1, 2, and 3) states on the PES: ω1=1, ω2=0.3, a=4, c=0.2, A=5.0, b=-11, ∆=12, α=0.1, and σ x=1.274. Figure 7. DBOC terms for different c of the model PES: ω1=1, ω2=1, a=4, A=5.0, b=-11, ∆=12, α=0.1, and σ x=1.274.


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Fig. 1



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Fig. 2


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Fig. 3



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Fig. 4


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Fig. 5



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Fig. 6


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Fig. 7



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TOC graphic


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