Effects of DisplacementâDistortion of Potential Energy Surfaces on

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Effects of Displacement-Distortion of Potential Energy Surfaces on Nonadiabatic Electron Transfers via Conical Intersections: Application to SO2 and Trans-1,3,5-Hexatriene Leila Zeidabadinejad, and Maryam Dehestani J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b01849 • Publication Date (Web): 27 May 2016 Downloaded from http://pubs.acs.org on May 28, 2016

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The Journal of Physical Chemistry

Effects of Displacement-Distortion of Potential Energy Surfaces on Nonadiabatic Electron Transfers via Conical Intersections: Application to SO 2 and trans-1,3,5-Hexatriene Leila Zeidabadinejad,†and Maryam Dehestani*,† †

Department of Chemistry, Shahid Bahonar University of Kerman, Kerman, Iran

*Corresponding author: Department of Chemistry, Shahid Bahonar University of Kerman, 22 Bahman Boulevard, P.O. Box 76169-133, Kerman, Iran. Tel: +98 34-31322106; Fax number: +98 34-33257433 E-mail: [email protected]

1 ACS Paragon Plus Environment


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ABSTRACT We show that, the time-correlation function formalism can be applied to calculate nonadiabatic electronic population dynamics on the two vibronically coupled diabatic displaced-distorted harmonic potential energy surfaces through conical intersection. We present general formulas for the time-evolved electronic populations at finite temperature with initial sampling from both initial thermal equilibrium and nonequilibrium nuclear distributions. The validity of our formalism is verified through comparison with previous work in a certain limit of our results for case of displaced harmonic oscillator. Finally for illustration, the derived expressions have been applied to determine the electronic population dynamics at conical intersections for SO 2 and trans-1, 3, 5-hexatriene molecules.

Keywords: Conical intersection; Nonadiabatic; Nonequilibrium; Electron transfers; Trans-1, 3, 5-hexatriene; SO 2 .


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1.

INTRODUCTION

Nonadiabatic dynamics of photoexcited molecules via conical intersections (CIs) is one of the most fundamental problems in processes such as electron transfer1,2 photochemistry, linear and nonlinear spectroscopies3,4 and relaxation rate.5-9 At a multidimensional CI of electronic potential energy surfaces where the electronic states approach close to one another and the mixing between the states is largest, the Born Oppenheimer (BO) approximation (separation of electronic and nuclear motions is valid) breaks down. Therefore, the nonadiabatic (or radiationless) electronic transitions between different potential energy surfaces play a predominant role at the crossings. To understand the nonadiabatic dynamics at the CI, the electronic population probability dynamics is important and exhibits the population transfer between different electronic states. The concepts of “adiabatic” and “nonadiabatic” were introduced for photochemical reactions by Förster.10 A perturbation molecular orbital method with a qualitative view of the breakdown of the BO approximation has been used to develop a picture of photochemical transition states.11 Understanding of the nonadiabatic dynamics at CIs is essential to obtain a complete picture of photoinduced reactions. The methods of wave packets in time-dependent quantum mechanical and multi-configuration time-dependent Hartree (MCTDH) have been used to treat the quantum dynamics via CIs.12-14 The other theoretical approaches to treat nonadiabatic transitions are classical, mixed quantumclassical or semiclassical methods. The internal conversion decay rate constant has been calculated using the displaced- distorted, and rotated harmonic oscillator including temperature by several groups.15,16 The electronic population dynamics in thermal



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equilibrium at the CIs has been computed based on the golden rule.17 Coalson et al. have extended the standard golden rule method for the nonequilibrium initial electronic state as the nonequilibrium golden rule (NGR) formula.18 The linear vibronic coupling model has been used to investigate nonadiabatic dynamics and photoinduced reactions in polyatomic molecules.19-21 The electronic population dynamics were derived using the second-order cumulant approximation in the Liouville space including the displacements, distortions, and rotations of potential energy surfaces.22 Recently Izmaylov et al. have derived the expression for the time evolution of electronic populations at finite temperature using the linear vibronic coupling Hamiltonian.23 The photoinduced electronic transitions dynamics through CIs has recently been formulated using a fully quadratic vibronic model Hamiltonian by Endicott et al..24 Moreover, to verify the effect of distortion and demonstrate properties of non-equilibrium Fermi golden rule method, they have used 2D models. It is our goal of the present article to derive general expressions for the electronic populations at any time on systems involving a CI. The derivations include the effects of the diabatic potential energy surfaces minimum displacements (displaced), diabatic vibrational frequency changes (distorted), temperature, and nonequilibrium nuclear dynamics. At the vicinity of a CI, the adiabatic potential energy surfaces are usually anharmonic and may exhibit double or multiple minima. In contrast to the adiabatic surfaces, the diabatic ones often show a simple structure and may be well approximated by simple harmonic potentials. This means that in this study, the diabatic representation extremely is convenient. The time correlation functions which are involved in both the equilibrium and nonequilibrium electronic population dynamics are calculated based on the 4 ACS Paragon Plus Environment

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nonequilibrium golden rule. In general, the normal modes with low-frequency and molecules with low symmetry show strong mode mixing effects. However, while mixing of normal modes (Duschinsky effect) is important, there are many situations where very little mixing occurs in the modes with the same symmetry. This is the case in the present study that the Duschinsky rotation effect has been ignored. Islampour et al. have calculated the time correlation functions involved in resonance Raman scattering for a displaced–distorted oscillator system at all temperatures.25 The general expressions for multidimensional time-domain integrals that arise in the calculation of absorption and resonance Raman cross sections have been derived with the assumption that the lower and the resonant harmonic potential surfaces are displaced, distorted, and rotated.26 These expressions derived have been applied for calculation of resonance Raman spectra of trans-stilbene27 and ClO 2 .28 The time-correlation function formalism has been applied to include S1–S2 vibronic coupling in trans-1,3,5-hexatriene (THT)29 and SO 2 .30 The latter formalism was based on a perturbation theory that treats the intramolecular couplings in a perturbative manner. Following the seminal work by Izmaylov et al.23 where the electronic population calculation has been carried for the photoinduced electronic transitions through conical intersection, which includes minima separation of the donor and acceptor states, Endicott et al.24 investigated population dynamics using quadratic model Hamiltonian which includes effects such as Duschinsky rotation of the normal coordinates and changes in vibrational frequencies, we represent our formalism which is intermediate between these two previous works. There are several significant advantages to the present formalism compared to works of Izmaylov et al.23 and Endicott et al..24 It has as simple an algebraic form in contrast to formalism of ref 24 with Duschinsky rotation effect being 5 ACS Paragon Plus Environment


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taken into account that requires matrix algebra. Our formalism yields an improved accuracy of results compared to ref 23. In this work, we have developed the time correlation function formalism for the electronic population dynamics of excited states with equilibrium and nonequilibrium nuclear distributions, based on the nonequilibrium golden rule and a vibronic model Hamiltonian consisting of two electronic states coupled. Within nonequilibrium golden rule, the electronic population dynamics of excited states with equilibrium and nonequilibrium nuclear distributions can be computed using time correlation formalism. For simplicity, we will ignore Duschinsky rotation between two diabatic states, even though it is important too. In limiting case for displaced harmonic oscillator, we show our results reduce to the simplified analytical expression for time kernel of Izmaylov et al.23 Finally, as an application to the results obtained we consider the population dynamics for THT and SO 2 molecules and illustrate the importance of the distortion effect in nonadiabatic dynamics on these molecules.

2.

THEORY 2.1

ELECTRONIC POPULATION DYNAMICS

Our approach to electronic population dynamics is based on the pioneering work of Coalson et al..18 This dynamic behavior is treated by the standard nonequilibrium golden rule. Briefly, the transition processes can be described using the time dependent Schrödinger equation i

d t = Hˆ t dt

(1)


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� is Hamiltonian of system. The here |𝑡𝑡⟩ is the state of the system at a time t, and 𝐻𝐻

� = 𝐻𝐻 �0 + 𝑉𝑉� where 𝐻𝐻 �0 is the Hamiltonian of the Hamiltonian can be expressed as 𝐻𝐻 unperturbed system in the Born-Oppenheimer approximation, and 𝑉𝑉� is a perturbation applied to the system arising the nonadiabatic couplings by an intramolecular interaction. Carrying out the integration of the eq 1 with respect to time gives t =Uˆ (t , 0 ) t 0

(2)

where t 0 is the state of the system at a time t=0, Uˆ (t , 0 ) is time evolution operator and we obtain a positive time-ordered exponential as following: ˆ ( t , 0 ) exp  t − i Hˆ ( t ′ ) dt ' = U +  ∫0   

(3)

The time evolution operator can also be written as:  i  Uˆ ( t= , 0 ) exp  − Hˆ 0 t  uÎ ( t , 0 )   

(4)

here 𝑢𝑢�𝐼𝐼 (𝑡𝑡, 0) is

 t i  uˆ= exp+  ∫ − VÎ ( t ′ ) dt ' I (t, 0) 0   

(5)

where 𝑉𝑉�𝐼𝐼 (𝑡𝑡) is potential energy evaluated in the interaction representation at time t

i   i  = VÎ ( t ) exp  Hˆ 0 t  Vˆ exp  − Hˆ 0 t      

(6)

and 𝑒𝑒𝑒𝑒𝑒𝑒+ [… ] in eqs 3 and 5 is the positive time ordering exponential, meaning that the integration time variables in the power series expansion of 𝑒𝑒𝑒𝑒𝑒𝑒+ [… ] should be ordered so that time increases from right to left (𝑡𝑡1 > 𝑡𝑡2 ), e.g.



1 2 3 4  t i  5 uˆ I ( t , 0 ) = exp+  − VÎ ( t ′ ) dt ′ = 0 6    7 2 t1 i t ˆ 8  i t 1− VI ( t1 ) dt1 +  −  dt1 dt2 VÎ ( t1 ) VÎ ( t2 ) + … 9 0  0   0 10 (7) 11 Consider a quantum system characterized by vibronic coupling of two excited electronic 12 13 14 states (|𝑏𝑏⟩ and |𝑐𝑐⟩ ) which are close in energy with different dipole couplings to the ground 15 16 electronic state (|𝑔𝑔⟩), while the intramolecular coupling of each excited electronic state 17 18 19 with the well separated ground state can be neglected. The use of diabatic representation, 20 21 rather than adiabatic representation, greatly simplifies the treatment of the electrodynamics 22 23 of the system. In this case we can then represent intramolecular interaction in the simple 24 25 26 fashion and describe the coupling between the states by the off-diagonal elements of the 27 28 potential energy operator which are smooth functions of the nuclear coordinates even at the 29 30 seam of degeneracies of the potential energy surfaces. While the adiabatic potential energy 31 32 33 surfaces of the excited states for the strongly vibronically coupled systems form a CI and 34 35 the coupling between the states is described by the off-diagonal elements of the nuclear 36 37 38 kinetic energy operator, which exhibits a singular behavior at the seam of intersections of 39 40 the potential energy surfaces.31,32 Moreover, previous studies have shown that a nontrivial 41 42 geometric phase (GP) appears in the nuclear and electronic wave functions in the adiabatic 43 44 45 representation for systems with conical intersection.33-35 In such case one needs to account 46 47 this complication otherwise dynamics will be very inadequate. 48 49 In the case of two coupled state, the diabatic Hamiltonian can be given by 50 51 52  b Hˆ b b Hˆ c   TˆN + Eb ( Q ) Λ bc ( Q )  53 (8) = H =    54  c Hˆ b ˆ c   Λ (Q ) ˆ + E (Q )  c H T cb N c     55 56 57 58 59 8 60

∫

∫

∫

∫

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here 𝑇𝑇�𝑁𝑁 stands for the nuclear kinetic energy, and E i (Q) are the diabatic potential energy

surfaces for any electronic state b and c. The donor state |𝑏𝑏⟩ is coupled to the acceptor state |𝑐𝑐⟩ and the coupling strength is determined by the diabatic coupling elements Λ 𝑏𝑏𝑏𝑏 = 𝛬𝛬𝑐𝑐𝑐𝑐 . Both the diagonal and off-diagonal matrix elements of the potential matrix shall depend on

the nuclear coordinates. Taking the Taylor expansion of the matrix elements of H to be about the origin of the normal coordinate of the ground state (𝑸𝑸,, = 0) and limiting the

series to second and first order for diagonal and off-diagonal elements, respectively, the matrix elements of eq 8 can be written:

1 = H11 Tˆ N + Eb ( 0′′ ) + κ ( b )T Q′′ + Q′′T Γ( b )Q′′ 2 H12 = Q ''T Λ (

(9 a)

bc )

H 21 = Q ''T Λ (

(9 b)

bc )

(9 c)

ˆ + E ( 0′′ ) + κ ( c )T Q′′ + 1 Q′′T Γ( c )Q′′ = H 22 T N c 2

(9 d)

where 𝑬𝑬𝒃𝒃 (𝟎𝟎′′ ) and 𝑬𝑬𝒄𝒄 (𝟎𝟎′′ ) are the vertical excitation energies of the b and c electronic

states, respectively, and 𝜿𝜿(𝒏𝒏) is the N-dimensional (column) vector of the first order (𝑛𝑛)

intrastate coupling constants of an electronic state n for a totally symmetric mode j (𝜅𝜅𝑗𝑗

)

and 𝜞𝜞(𝒏𝒏) is a N × N matrix of second order intrastate coupling constants of an electronic (𝑛𝑛)

state n ( 𝛾𝛾𝑖𝑖𝑖𝑖

) and 𝜦𝜦(𝒃𝒃𝒃𝒃) is the N-dimensional vector of first-order coupling constants (𝑏𝑏𝑏𝑏)

between the diabatic states and the coupling modes (𝜆𝜆𝑗𝑗

(𝑏𝑏𝑏𝑏)

). 𝜆𝜆𝑗𝑗

is nonzero only if the

corresponding vibrational modes belong to irreducible representations Γ b ×Γ c (Γ b and Γ c are symmetry species of two electronic states that could be coupled with each other).



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Using the transforming given in eqs 10 and 11, it is possible to express the normal coordinates of the ground state as a linear combination of the normal coordinates of the excited states |𝑏𝑏⟩ and |𝑐𝑐⟩, respectively (b) Q′=

Q′′ + D( b )

(10)

(c) (11) Q′= Q′′ + D( c ) (𝑛𝑛) (n) where D is the N-dimensional vector of the displacements (𝑑𝑑𝑗𝑗 ) which are the changes in

the nuclear equilibrium positions of the excited electronic state nth with respect to the ground electronic state. By inserting eqs 10 and 11 into eq 9, the diagonal and off-diagonal matrix elements of the Hamiltonian can be expanded into the following equations:

1 bT b b 1 b bT b bT b b b H11 = TN + E b ( 0′′ ) − κ ( ) D( ) + D( ) Γ( ) D( ) + κ ( ) Q′( ) − Q′( ) T Γ( ) D( ) 2 2 1 1 − D( b )T Γ( b )Q′( b ) + Q′( b )T Γ( b )Q′( b ) 2 2

(12)

1 1 H 22 = TN + E c ( 0′′ ) − κ ( c )T D( c ) + D( c )T Γ( c ) D( c ) + κ ( c )TQ′( c ) − Q′( c ) T Γ( c ) D( c ) 2 2 1 1 − D( c )T Γ( c )Q′( c ) + Q′( c )T Γ( c )Q′( c ) 2 2

(13)

′′T Λ ( bc ) Q′( b )T Λ ( bc ) − D( b )T Λ ( bc ) = H12 Q=

(14)

The diabatic potential energy surfaces often show a surprisingly simple structure and may – for the region of interest – be well approximated by simple harmonic potentials, hence the linear and quadratic vibronic couplings, are chosen such that the linear dependence of diabatic potential energies with normal coordinates (linear terms) in eqs (12) and (13) vanish.

κ (b ) − Γ (b ) D (b ) = 0

(15 a)


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1 2 3 4 c c c (15 b) 0, κ ( ) − Γ( ) D ( ) = 5 6 7 Hence due to eqs (15 a) and (15 b), we are in practice mapping our system onto a collection 8 9 of uncoupled harmonic oscillators. 𝜿𝜿(𝑛𝑛) accounts for changes in the nuclear equilibrium 10 11 12 positions from the ground to the excited electronic states. The diabatic vibrational 13 14 ′(𝑏𝑏) ′(𝑐𝑐) frequencies of donor (𝜔𝜔 �𝑖𝑖 ) and acceptor (𝜔𝜔 �𝑖𝑖 ) states are obtained as 15 16 17 2 2 2 18 (b)  1(b ) , ω'  2 (b ) ,…, ω'  N (b ) (16 a) = Γ diag ω' 19 20 21 2 2 2 c 22  1( c ) , ω'  2 ( c ) ,…, ω'  N (c) (16 b) = Γ( ) diag ω' 23 24 (𝑛𝑛) 25 The quadratic intrastate coupling constants 𝛾𝛾𝑖𝑖𝑖𝑖 account for changes in vibrational 26 27 (𝑛𝑛) 28 frequencies in the excited states. The off diagonal coupling constants 𝛾𝛾𝑖𝑖𝑖𝑖 (i≠j) are zero if 29 30 mode mixing (normal coordinate rotation) is ignored. 31 32 33 During nonadiabatic population transfer dynamics, we ignore the Duschinsky rotation, 34 35 therefore, the corresponding Hamiltonian, eq 9 can be written as follows: 36 37 38 1  i (b )2 Q 'i (b )2 (17 a) ω' H11 =Tˆ N + Δ b + 39 2 i 40 41 42 (17 b) λi(bc )Q 'i (b ) − λi(bc ) di(b ) H= H= 12 21 i Γ ⊂Γ ×Γ ( ) i b c 43 44 45 1  i ( c )2 Q 'i ( c ) (17 c) ω' H 22 = TˆN + Δ c + 46 2 i 47 48 49 where 50 51 1 ( b )T ( b ) ( b ) b T (b) 52 = ∆ b E b ( 0′′ ) − κ ( ) ∆∆ + Γ ∆ (18 a) 53 2 54 55 56 57 58 59 11 60

((

) (

((

) (

)

)

(

(

))

))

∑

(

∑

)

∑



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( c ) 1 ( c )T ( c ) ( c ) ∆ c =E c ( 0′′ ) − κ ( c )T ∆∆ + Γ ∆ 2

(18 b)

We can write the H 0 and V matrices as follows:

H 0  H 0 = = 0  0 Hc  0 b

1 ˆ  i (b )2 Q 'i (b )2  TN + Δ b + 2 ∑ω' i   0  

   1  ( )2 ( )2 c c  i Q 'i  Tˆ N + Δ c + ∑ω' 2 i 

0

(19)  0  V =  ∑ λi(bc )Q 'i (b ) − λi(bc ) di(b )  i( Γ ⊂Γ ×Γ )  i b c

(

∑ (λ(

)

i ( Γ i ⊂ Γ b ×Γ c )

i

bc )

)

Q 'i (b ) − λi(bc ) di(b )     0  

(20)

where ˆ  1 0  i (b )2 Q 'i (b )2 =  TN + ∆ b + ∑ω'  b > Ebb b > 2 i  

(21 a)

ˆ  1  i ( c )2 Q 'i ( c )2 = γ > Eb0γ | γ >  TN + ∆ c + ∑ω' 2 i  

(21 b)

and 1   i (b ) Eb0b =∆ b + ∑  βi +   ω' 2 i 

(22 a)

1   i (c) Ec0γ =∆ c + ∑  γ i +   ω' 2 i 

(22 b)

�𝑏𝑏0 and 𝐻𝐻 �𝑐𝑐0 , respectively. Here |𝛽𝛽⟩ and |𝛾𝛾⟩ are eigenstates of the 𝐻𝐻 Thereby, the V I (t) is given by

V = V= 0 I ( t )11 I ( t ) 22

(23 a)


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1 2 3 4   i  i  5 = VI ( t )12 VI ( t )21 exp  H b0t  i ( Γ ⊂Γb×Γc )  λi(bc )Qi′ (b ) − λi(bc ) di(b ) exp  − H c0   (23 b) 6 i       7 8 We shall be interested in calculating P b (t) the time dependent population of the excited 9 10 11 donor electronic state |𝑏𝑏⟩ in photoinduced nonadiabatic transitions which population 12 13 transfer from donor to acceptor electronic state |𝑐𝑐⟩can be carried out via CIs: 14 15 16 = Pb ( t ) t= b bt t0 uˆ +I ( t, 0 ) b b uˆ I ( t, 0 ) t 0 (24) 17 18 Inserting eqs 7 and 23 into eq 24, the electronic population probability is given by 19 20 21 t t1 0 Pb ( t ) = 1 − 2 Re dt2 dt2 b eiH b t1 /  i( Γ ⊂ Γ ×Γ ) λ i( bc ) Q′i ( b ) − λ i( bc ) d (i b ) 22 0 0 i b c b 23 (25) − iH 0c ( t1− t 2 ) /  − iH 0b t 2 /  24 ( bc ) ( b) ( bc ) ( b ) e λ j Q′j − λ j d j e b + ... j( Γ j ⊂ Γ b × Γc ) 25 26 27 The population function given in eq 25 can be simplified by using the cumulant expansion 28 29 method and truncating higher order terms than the second. 30 31 32 t (26) Pb( 2) ( t ) = exp dt1 k ( t1 ) 33 0 34 35 where 36 37 38 0  0t / t  ( bc ) ˆ ' ( b ) ( bc ) ( b )  −iH c ∑ b ρ b (T ) eiH bt / ∑i( Γ ⊂Γ ×Γ )  λi −2 Re ∫01 dt2ei( ∆b −∆ c )t / >>> −λi k ( t1 ) = Qi di  e 39 i b c   b 40 t  ( bc ) ˆ ' ( b ) ( bc ) ( b )  41 ∑ j ( Γ j ⊂Γb ×Γc )  λ j Q j −λ j d j |b > = −2 Re ∫01 dt2ei( ∆b −∆ c ) t / > f (t )   42 43 (27) 44 45 �𝑏𝑏0 and 𝐻𝐻 �𝑐𝑐0 are the vibrational Hamiltonians (excluding the ∆𝑏𝑏 and ∆𝑐𝑐 energies) of the here 𝐻𝐻 46 47 diabatic electronic states b and c, respectively, ρ β (T) is the equilibrium initial state nuclear 48 49 50 density operator and t 1 -t 2 =τ. Eq 26 shows the time-dependent total (summed over all 51 52 vibrational levels of electronic state b) population probability of the electronic state |𝑏𝑏⟩ 53 54 (2) (2) 55 normalized to 𝑃𝑃𝑏𝑏 (0)= 1. 𝑃𝑃𝑏𝑏 (𝑡𝑡) is of particular interest as it reflects explicitly the non56 57 58 59 13 60

)

∑

∫

∫

∑

(

(∫

)

∑

∑

(

)


)


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(2)

BO dynamics in the diabatic representation: 𝑃𝑃𝑏𝑏 (𝑡𝑡)=1 in the absence of vibronic coupling (𝑏𝑏𝑏𝑏)

(V bc =0 or 𝜆𝜆𝑖𝑖

= 0).

2.2 THE NONEQUILIBRIUM NUCLEAR DISTRIBUTION

We consider the nonequilibrium effects of the nuclear distribution on the donor potential surface. We obtain a nonequilibrium Fermi-Golden rule formula based on the time correlator formalism for the displaced-distorted harmonic oscillator model. The nonequilibrium population on the donor surface is created by an initial ultrafast laser pulse that excites a Boltzmann density on the ground state potential energy surface to the donor state. Because we are considering a nonequilibrium population on the donor state, it is necessary to introduce the nonequilibrium density instead of Boltzmann equilibrium density (𝜌𝜌𝛽𝛽 (𝑇𝑇)) as

ρ b ( t1 ) = ei H t /  ρ g (T ) e − iH t /  ˆ0

ˆ0

b 1

where

ρ g (T ) = e

(28)

b 1

− Hˆ g / k T

/ Z g= with Z g

∏e

− ω "i /2 k T

(1 − e

− ω "i / k T

i

)

−1

, 𝜔𝜔𝑖𝑖′′ is the ground

�𝑔𝑔 is the ground electronic state electronic state vibrational frequency of mode i and 𝐻𝐻 Hamiltonian.

We introduce normal coordinate in the Heisenberg picture with respect to excited state ˆ 0 /  '( b ) − i Hˆ 0 /  dynamics b Qˆ i'(b ) (τ ) ≡ ei Hbττ Qˆ i e b which is given by

(

)

(

(b) (b) (b) (b) 1 1 ' ' Qˆ i ( b )= + e − iω' i − Pˆ ( b ) eiω' i − e − iω' i (τ ) Qˆ i'(b ) eiω' i ττττ (b ) i  'i 2 2i ω


)

(29)

Page 15 of 52

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


′(𝑏𝑏)

′(𝑏𝑏)

is the conjugate momentum to normal coordinate 𝑄𝑄�𝑖𝑖

here 𝑃𝑃�𝑖𝑖

.

Substitution of the eq 28, into eq 27 and using eq 29 and the following elementary relations which hold for harmonic oscillator systems:

 2 ω 'i (b )

b Qˆ i'( ) bi =

(

bi + 1 bi + 1 + bi bi − 1

 'i (b )  ω Pî '(b ) bi = 2

(

)

(30)

)

bi + 1 bi + 1 − bi bi − 1

(31)

gives

k (t 1 ) = −2 Re ∫ dt 2e t1

i ( ∆b −∆c ) t / 

0

f (t )

(32)

where − ω ( )τ /2 e ∑i i f (τ ) = 2 Zg 'b

×∑∑∑∑λi( ) λ j( bc

b′

b

i

(e

bc )

j

' 1

i

1

i

1

' i

' 1

' i

' j

bi' ( b j + 1) g b' b b b  (τ ) g b b −1b

'b i ω i( )τ

 '(b )τ

' 1

i

1

i

1

1

' i

' 1

' i

i

1

' j b1b i b j

j

+1

(τ )

bi' b j g b' b b b  (ττ ) g b b −1b b b b −1 ( )

+e i ωi

1

(b

 '(b )τ

b

'b i ω i( )τ

b)

' i

' i

' 1

' i

' j

1

i

j

+ 1) g b' b b ' b '  (ττ ) g b 'b ' +1b ' b b b  ( ) 1

1

i

1

i

i

j

1

i

bi' g b' b b b  (ττ ) g b b −1b b b b  ( ) 1

− 2 d i(

' 1

i

− 2 d j( )e − i ωi −d j(b )e

+ 1) ( b j + 1) g b' b b ' b '  (ττ ) g b 'b ' +1b ' b b b +1 ( )

' i

b j ( bi' + 1) g b' b b b  (ττ ) g b b +1b b b b −1 ( )

 '(b )τ

+e − i ωi e

(b

'b − i ω i( )τ

i

' 1

' i

' 1

' i

' j

1

i

j

b j + 1g b' b b b  (ττ ) g b b b b b b +1 ( ) 1

i

' 1

' i

' 1

' i

' j

1

i

j

− 2 d i(b ) b j g b' b b ' b '  (ττ ) g b 'b ' b ' b b b −1 ( ) 1

i

1

1

i

i

j

1

i

j

− d i( )d j( ) g b' b b ' b '  (ττ ) g b ' b ' b ' b b b  ( ) b

b

1

i

1

i

1

i

j

1

i

j

)


j

i

j

1

i

j


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

− ω ( )τ /2 e ∑i i = 2 Zg

(

'b

+e

'b i ω i( )τ

∑∑λ ( i

i

bc )

'(b ) i τ

λ j(bc ) e −i ω

j

Page 16 of 52

( ) G bb ′b ′+1b +1 + e − i ωi τ G bb ′b ′+1b −1 'b

( ) ( ) G bb ′b ′−1b +1 + e i ωi τ G bb ′b ′−1b −1 − 2d j(b )e − i ωi τ G bb ′b ′+1b 'b

'b

( ) − 2 d j(b )e i ωi τ G bb ′b ′−1b − 2 d i(b )G bb ′b ′b +1 − 2 d i(b )G bb ′b ′b −1 + d i(b )d j(b )G bb ′b ′b 'b

) (33)

here the multi-dimensional time domain integrals are /  − iH c /  g b1′bN′ b1bN (τ ) = b1′ b N′ e iH b ττ e b1  b N 0

0

(34)

, and N is the number of vibrational degrees of freedom g b' b b ' b '  (τ ) = b1  bi  e iH b τ / e 0

1

i

− Hˆ g / kT

i

1

b1'  bi' 

(35)

and G bb ′b ′b = ∑∑e b

i

∑ωi( ) ( bi t 2 − bi' t1 ) 'b

i

g b' b b ' b '  (t ) g b ' b ' b ' b b b 1

b′

i

1

i

1

i

j

1

i

j

(t )

(36)

N

It

follows

that

for

a

displaced-distorted

' oscillator g ββββββ = ∏g ' ' and '   ' 1

N

1

N

i =1

i

i

N

g ββββββ = ∏g ' '  '  1

N 1

N

i =1

i i

We have developed the time correlator formalism introduced by Islampour et al.25 for the electronic population dynamics of excited states. Using the harmonic oscillator wavefunctions, introducing integral representation of Hermite polynomials, and integrating over coordinates, the multi-dimensional time domain integrals become:


Page 17 of 52

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 S i (1 + m i )  +∞ ( −2 i ) i b ω ( )τ − e g exp   ∫ d x1x1b e −x ( ) (ττ ) 00 1   −∞ − m 1 ( i ) b +b ' 2   bi + bi'

g b'b i

2

i

i

(

p 2

' i

i

bi ! bi ! f i m i

'b i

i

2 1

)

2  1− f i 1 2 bi' −x 22 × − − + + − + − d e exp S m m i m 2 1 1 1 x x x x (x1 − x 2 )  ( ) ( )( ) ( )  1 2 i i i i ∫−∞ 2 2 2 fi  2 m i (1 − m i )  (37)

(

+∞

)

where  iω i' (b )τ  exp  − Si (1 + mi )   2 

g 00 (τ ) h =

1/2 i

(38)

and

Si =

(

ω i'(b ) di( c ) − di(b )

)

2

(39 a)

2

ω i'(b ) − ω i'( c ) bi = '(b ) ω i + ω i'( c )

(39 b)

−b − e ) ( m = (1 + b e )

(39 c)

−b + e ) ( f = (1 − b e )

(39 d)

' c − iω i ( )τ

i

' c − iω i ( )τ

i

i

' c − iω i ( )τ

i

' c − iω i ( )τ

i

i

(1 − b ) e h = i

(

2 i

' c − iω i ( )τ ' (c)

1 − bi2 e −2 iωi

τ

)

(39 e)

A result similar to eq 37 can be obtained for 𝑔𝑔𝛽𝛽′ 𝛽𝛽′ (𝜏𝜏) as following: 𝑖𝑖 𝑖𝑖



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

( −2 i ) i b ω ( )τ ' e g 00 ( ) (ττ ) 1 bi + bi'

'

g b b' i

i

' i

(

p 2b + b bi ! bi' ! f i ' m i' ' i

i

)

' b i

2

Page 18 of 52

 S ' (1 + m ' )2  +∞ +∞ ' '2 i i ' 'bi −x1'2   exp − d x1x1 e d x 2' x 2'bi e −x2 ∫ ∫ '  (1 − m i )  −∞ −∞ 

2   1− f i' ' 1 ' ' ' ' ' ' 2  2 1 1 1 exp  S m m i m x x x x − − + + − + − − ( ) ( ) ( )( ) ( ) 1 2 1 2 i i i i ' ' ' 2 fi  2 m i (1 − m i )  (40)

)

(

here  iω i'(b )τ  g (τ ) h exp  = − Si' (1 + mi' )   2  ' 00

'1/2 i

(41)

and

Si' =

( )

ω i'(b ) di(b )

2

(42 a)

2 

ω i'(b ) − ωi'' b = '(b ) ω i + ωi'' ' i

(42 b)

m i'

( −b − e = (1 + b e

f i'

( −b + e = (1 − b e

' i

' i

h = ' i

' i

' i

− ωi'' / kT

− ωi'' / kT

− ωi'' / kT −ωi'' / kT

(1 − b )e

(

'2 i

)

) )

(42 c)

)

(42 d)

− ωi'' / kT

1 − bi'2e −2  ωi / kT ''

)

(42 e)

Inserting eqs 37 and 40 into eq 36 and carrying out the summations over β i and 𝛽𝛽𝑖𝑖′ and then integrating over 𝜉𝜉 1 , 𝜉𝜉 2 , 𝜉𝜉1′ and 𝜉𝜉2′ , an equation for G ββββββββ is obtained '   ' ' '  R

R

1

N 1

N 1


N 1

N

Page 19 of 52

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


G ββββ ′ ′ =

g

00

' (ττ ) g 00 ( ) a i

)(

(

S

× exp ∑ i

i 2a i

)

(

)

(

)

2 2 2 2 2 2  '  e e e e e +e2 − 2 f f 'm ' 1+ m − 2 h ' 1+ m  k i 1 + mi + i i i i i i i i i i i i 1, 2, 1, 2, 1, 2,   ' S  2 2 2 ∑ i 2 ai  k i 1 + m i' e1,2 i + e 2,2 i − 2 f i f i' m i 1 + m i' e1,2 i e 2,2 i − hi 1 + m i' e1, i e 2, i i 

)(

(

∑

S S' i i 1+ m i a i

(

i

)

(

)

(

)

  − 

) (1 + mi' ) (1 − f i f i'e1, i e 2, i )(e1, i + e 2, i ) (43)

′(𝑏𝑏)

where 𝜀𝜀1,𝑖𝑖 = 𝑒𝑒 −𝑖𝑖𝜔𝜔�𝑖𝑖

𝑡𝑡2

′(𝑏𝑏)

, 𝜀𝜀2,𝑖𝑖 = 𝑒𝑒 𝑖𝑖𝜔𝜔�𝑖𝑖

𝑡𝑡1

and

2 2 ' 2 2 ' ai = 1 + m i' f i ' m i f i εεεεεε 1,i 2,i − k i k i ( 1,i + 2,i ) − 2 hi hi 1,i 2,i

ki =−bi ×

(

(1 − e

)

' c −2 iω i ( )τ

' (c)

1 − bi2 e −2 iωi

(1 − e =−b × (1 − b e

τ

)

−2  ωi'' / kT

k i'

' i

'2 −2  i

(44 a)

(44 b)

)

ωi'' / kT

)

(44 c)

The other terms in eq 33 can be evaluated in the similar way, and are given in Appendix. Under displaced harmonic approximation, assuming same frequencies for ground and excited

states ′

′(𝑏𝑏)

(𝜔𝜔𝑖𝑖′′ = 𝜔𝜔 �𝑖𝑖

′(𝑐𝑐)

= 𝜔𝜔 �𝑖𝑖

= 𝜔𝜔 �𝑖𝑖′ ),

′

−𝑚𝑚𝑖𝑖 = 𝑒𝑒 −𝑖𝑖𝜔𝜔�𝑖𝑖 𝜏𝜏 , 𝑓𝑓𝑖𝑖′ = ℎ𝑖𝑖′ = −𝑚𝑚𝑖𝑖′ = 𝑒𝑒 −ℏ 𝜔𝜔�𝑖𝑖 ⁄𝑘𝑘𝑘𝑘 ,

𝑏𝑏𝑖𝑖 = 𝑏𝑏𝑖𝑖′ = 0,

′

𝑘𝑘𝑖𝑖 = 𝑘𝑘𝑖𝑖′ = 0,

𝑎𝑎𝑖𝑖 = (1 − 𝑒𝑒 −ℏ 𝜔𝜔�𝑖𝑖 ⁄𝑘𝑘𝑘𝑘 )2

time

𝑓𝑓𝑖𝑖 = ℎ𝑖𝑖 =

correlation

functions can be simplified and are given in Appendix. Then, after simplifying by aid of eqs A11-A19, eq 33 reduces to:



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

= f (t )

∑∑λ(

bc )

i

i

(( n + 1) e

λ (j bc ) di(b ) + di( c ) d (jb ) + d (jc ) + di(b ) − di( c ) d (jb ) − d (jc )

j

− iω 'jt

j

)(

) (

)(

((

Page 20 of 52

− nje

iω 'jt

) + ( d

(b)

i

)(

+ di( c ) d (jb ) − d (jc )

) (( n + 1) e

− iω 'jt

j

− nje

iω 'jt

( d ( ) − d ( ) ) ( d ( ) + d ( ) ) (( n + 1) e − n e ) + d ( ) d ( ) ( e + e ) (e + e ) − d( ) (( d( ) + d( ) ) + ( d( ) − d( ) ) (( n + 1) e − n e )) (e + e ) − d( ) (e + e ) (( d( ) + d( ) ) + ( d( ) − d( ) ) (( n + 1) e b

c

i

b

i

iω 'j t2

− iω i't

c

j

j

− iω 'j t2

b

− iω 'j t2

c

i

b

iω i' t1

b

(

− iω 'jt

+ nje

iω 'jt

ij

b

j

(b ) j

c − d (j )

j

)

)+

i

j

)d ) × exp  −∑ ( d

'

iω i't

i

c

j

− ni eiωit

− iω i' t1

iω i' t1

− iω i't

i

− iω i't

i

j

c

i

− iω i' t1

b

i

b

i

i

+ ( n j + 1) e

b

i

b

j

iω 'j t2

iω i't

i

) (( n + 1) e

c

j

− iω 'jt

) ((1 − e ) + n (1 − e 2

− iω 'jt

)) ) + n (1 − e )) 

− nje

j

iω 'jt

j

iω 'jt

− iω 'jt

j

(45) where

(𝑘𝑘) 𝑑𝑑̃𝑖𝑖

=

(𝑘𝑘) −�𝜔𝜔�𝑖𝑖′⁄2 ℏ 𝑑𝑑𝑖𝑖

,

(𝑏𝑏𝑏𝑏) 𝜆𝜆̃𝑖𝑖

��2 = 𝜆𝜆(𝑏𝑏𝑏𝑏) 𝑖𝑖

� 𝑖𝑖′ ℏ 𝜔𝜔

and 𝑛𝑛𝑖𝑖 = 1⁄(𝑒𝑒

′(𝑏𝑏) � 𝑖𝑖 �𝑘𝑘 𝑇𝑇 ℏ 𝜔𝜔

− 1).

This equation is identical to the one reported in the study of Izmaylov et al. (eq 11 in ref (23). In the special case where 𝐻𝐻𝑔𝑔 = 𝐻𝐻𝑏𝑏0 , we obtain the population dynamics for the equilibrium

nuclear distribution. In this case, the final form of the relationship f(τ) becomes eq 46: 1 + εεε + 2 i m j − (1 + m j ) (1 − m i ) (b ) c b c f (τll di − di( ) d j( ) − d j( ) ) ∑∑ i(bc ) j(bc )  i j (1 + m i εε  i j i ) (1 + m j j ) (1 − m i ) + 2 ε i d (b ) − d (c ) d (b ) + d (c ) + d (b ) + d (c ) d (b ) + d (c ) − i i j j i i j j (1 + m i ε i )

(

(

+

)(

2 2 hi (1 − εε i ) + (1 − m i − 2 k i + 2 m i ) i

(1 + m i εε i )(1 + f i i )

) (

)(



d ij  ∏ 

)(

l

)

)

hl

(1 + n l ) (1 − εε l f l )(1 + l m l ) 2

2   S j (1 − m j ) ε j 'b × εxp  ∑( − S j (1 − m j ) + i ω j( )τ / 2)   j 2 + ε j (1 − m j )   

(46) where e i =

ni  ′( b ) e i ωi τ . 1 + ni


Page 21 of 52

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3.

RESULTS AND DISCUSSION

This section is devoted to the application of the time-correlation function formalism to the study of the time-dependent electronic population probability of real molecules. The general expressions derived in this paper allow us to derive nonadiabatic transfer of population between two diabatic electronic states for two real molecular systems: SO 2 and THT. Before turning to our main task, which is applying eqs 32, 33, 45 and 46 for calculating time-dependent population of excited electronic states inclusion of equilibrium and nonequilibrium characters of the nuclear distribution in these molecules, we briefly outline some concepts from the electronic structure that are needed to understand their population dynamics. The near UV absorption spectrum of SO 2 molecule shows three major bands.36 Among these, the first band occurring in the wavelength range between 390 and 340 nm was identified to be due to the spin-forbidden transition 𝑎𝑎� 3B 1 ← 𝑋𝑋� 1A 1 . This band is very weak P

P

and results from spin-orbit mixing with higher singlet states.37,38

The

second

band

appears as one band of moderate intensity in the region 340–260 nm.39-46 It is interpreted to be originating from a transition to the excited electronic states 1A 2 and 1B 1 . The transition nπ*1B 1 ← 1A 1 has low oscillator strength, but the transition ππ*1A 2 ← 1A 1 has zero oscillator strength and becomes allowed by strong vibronic coupling between the 1A 2 and 1B 1 states.41,42 Finally, the third band spanning the 240–160 nm wavelength range, is attributed to 1B 2 ← 𝑋𝑋� 1A 1 transition. This transition is very strong and appears very P

intensely in the absorption spectrum.43,47-52

Elementary symmetry selection rules suggest that in the C 2v configuration, two electronic



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states (A 2 and B 1 ) of SO 2 can couple to each other through only b 2 mode (the asymmetric stretching vibrational mode) and form CIs. Internal conversion process caused by nonadiabatic coupling was found occurring on a fast time scale 15 fs.53 The broad electronic absorption band (340 – 260 nm) and the fast electronic relaxation are ascribed to the vibronic coupling between 1A 2 and 1B 1 excited electronic states via the only mode of symmetry b 2 . Based on ab initio calculations, Lévêque et al. described photodynamics of the SO 2 molecule on the manifold of two lowest singlet states (1A 2 , 1B 1 ) and indicated a rapid population transfer (55%) from the 1B 1 diabatic state to the 1A 2 diabatic state during the early stage of the dynamics (15 fs).54 They found that a large amount of population (30%) was transferred to the 1A 2 diabatic state during 40 to 60 fs. In a theoretical study, the role of the three lowest triplet states (3B 1 , 3A 2 , and 3B 2 ) and the two lowest singlet excited states (1B 1 and 1A 2 ), in the photophysics of SO 2 was investigated based on surface-hopping technique and a constant spin-orbit coupling.55 The fast electronic population decay to the triplet states occurred on a time scale of only 30 fs, and efficient 40%. A rather detailed picture of the photo-excitation dynamics of SO 2 has been developed in terms of combining CIs, and spin-orbit coupling between a manifold of states.56 It has been shown that the state 3B 2 plays an important role in the photophysics of SO 2 . Recent time-resolved photoelectron spectroscopy and time-resolved photoelectron-photoion studies gave more insight into the excited state dynamics of SO 2 .57 The main characteristics for four lowestlying excited state potential energy surfaces (minima, CI) required for the first two UV absorption bands have been determined through extensive ab initio calculations based on multireference configuration interaction (MRCI) level of theory.58 SO 2 is a convenient model system for the study of nonadiabatic effects occurring at CIs. 22 ACS Paragon Plus Environment

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Therefore, we determine the electronic population dynamics at CI for this molecule. The values of the parameters entering the definition of the vibronic coupling model Hamiltonians used for the population dynamics calculations obtained from our previous work.30 In the present work, the rotation of normal modes upon excitation has been ′(𝑘𝑘)

′(𝑘𝑘)

�2 neglected. Let us note 𝜔𝜔 �1 , 𝜔𝜔

′(𝑘𝑘)

and 𝜔𝜔 �3

are diabatic vibrational frequencies of SO 2 in

the electronic state of kth (k=b donor, c acceptor) for the bending, the symmetric stretching, and the antisymmetric stretching modes, respectively. Table 1 presents the values of ′(𝑏𝑏)

diabatic harmonic vibrational frequencies in the electronic states 1B 1 as donor (𝜔𝜔 �𝑖𝑖 ′(𝑐𝑐)

A 2 as acceptor (𝜔𝜔 �𝑖𝑖

1

(𝜔𝜔𝑖𝑖′′ ).

) and

) as well as vibrational frequencies of the ground electronic state

Table 1. Diabatic harmonic vibrational frequencies (cm-1) of 1A 1 , 1A 2 and 1B 1 electronic states of SO 2 . Modes

1

1

A1

1

A2

B1

υ1

519.10

87.76

240.87

υ2

1178.5

971.49

920.17

υ3

1376.9

587.69

517.63

The parameters required for the calculation of time dependent population of the 1B 1 electronic state including displacements corresponding to the totally symmetric modes υ 1 and υ 2 as well as Huang–Rhys coupling constants S j and 𝑆𝑆𝑗𝑗′ , calculated from eqs 39a and 42a, respectively, are listed in Table 2. The 1A 2

--

1

B 1 first-order nonadiabatic vibronic



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coupling constant induced by b 1 vibration is 0.13 eV.

(𝑘𝑘)

and Huang–Rhys coupling constants 𝑆𝑆𝑗𝑗 and 𝑆𝑆𝑗𝑗′ for the 1A 1 -

Table 2. Displacements 𝑑𝑑𝑗𝑗 1

B 1 , 1A 1 – 1A 2 and 1B 1 – 1A 2 electronic states of SO 2 (𝒌𝒌)

Modes 1

A 1 – 1A 2

𝒅𝒅𝒋𝒋 (amu1/2 Å) A 1 - 1B 1

𝑺𝑺′𝒋𝒋

Sj

1

1

B 1 – 1A 2

υ1

-0.3003

-2.5398

-2.2395

17.878 0.3202

υ2

0.2933

-0.2641

-0.5574

4.2441 1.1753

The energy gap between the isolated acceptor and donor (𝐸𝐸𝑏𝑏 (0′′ ) − 𝐸𝐸𝑐𝑐 (0′′ )) is 0.5 eV. In

an attempt to understand the contribution of vibrational frequency changes (distortion effect) to population, we have calculated the population for two cases: displaced, and displaced-distorted models, taking into account all three modes. For the displaced model, ′(𝑏𝑏)

the geometric means of the vibrational frequencies 𝜔𝜔 �𝑖𝑖

′(𝑐𝑐)

and 𝜔𝜔 �𝑖𝑖 ′(𝑏𝑏)

and the geometric means of the vibrational frequencies 𝜔𝜔 �𝑖𝑖

′(𝑏𝑏)

, that is, ��𝜔𝜔 �𝑖𝑖 ′(𝑐𝑐)

and 𝜔𝜔 �𝑖𝑖

′(𝑐𝑐)

𝜔𝜔 �𝑖𝑖

�,

, and 𝜔𝜔𝑖𝑖′′ that is,

′(𝑏𝑏) ′(𝑐𝑐) ′′ �𝑖𝑖 𝜔𝜔𝑖𝑖 � are used at equilibrium and nonequilibrium nuclear distributions, ��𝜔𝜔 �𝑖𝑖 𝜔𝜔

respectively. To perform the calculation, we have used eqs 32, 33, 43, 45 and A3-A10 for

nonequilibrium case and eq 46 for the equilibrium case. All calculations were performed using Mathematica programs.59 The temperature used in calculations was 0 K. Figure 1 (a) shows the time-dependence of the diabatic electronic state populations on the 1B 1 state over the period of 100 fs after electronic excitation at a nonequilibrium


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nuclear distribution. The result obtained of ab initio quantum study by of Lévêque et al.54 (grey line) is compared with those obtained by our calculations with displaced (blue line) and displaced-distorted (red line) models. Figure 1 (a) depicts the effect of difference the frequency of modes between the donor, acceptor and ground states in population dynamics. One can note that our result with displaced-distorted model is in very good agreement with Lévêque et al.54 result. Since initial population is located on the 1B 1 diabatic state, the population of this state is 1.0 at t=0 as shown in Figure 1 (a). The population of this state sharply decreases to 0.4 at 10 fs. During 15 fs the majority of the singlet population is transferred to 1A 2 through CI. At initial time, population decays quickly to 1A 2 diabatic state, in agreement with the theoretical work of Lévêque et al.54 and study of Xie et al..58



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Figure 1. Diabatic population probability of 1B 1 state of SO 2 calculated with the displaced (blue line) and displaced-distorted harmonic oscillator (red line) models for (a) nonequilibrim and (b) equilibrium nuclear distributions. The population obtained in the work of Lévêque et al.54 is indicated by the grey line. A stepwise population transfer can be seen after fast population transfer to 1A 2 diabatic state. The stepwise shape of the dynamics of the singlet/singlet transfer demonstrates the fundamental role of the crossing of the potential energy surface, due to the smallness of the nonadiabatic coupling elements, which makes possible internal conversion only at localized geometries, so that transfer at 15

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