(5A') and 3Î 1

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A: Spectroscopy, Molecular Structure, and Quantum Chemistry

Importance of the Parallel Component of the Transition Moments to the # (5A’) and # (3A’) Excited States of ICN in the Ã-band Photodissociation

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Tatsuhiko Kashimura, and Satoshi Yabushita J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b01127 • Publication Date (Web): 16 Apr 2019 Downloaded from http://pubs.acs.org on April 16, 2019

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The Journal of Physical Chemistry

Importance of the Parallel Component of the Transition Moments to the 11 (5A’) and 31 (3A’) Excited States of ICN in the A-band Photodissociation Tatsuhiko Kashimura and Satoshi Yabushita*

Department of Chemistry, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

*E-mail of corresponding author:

[email protected]　

1 ACS Paragon Plus Environment

The Journal of Physical Chemistry 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

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ABSTRACT ICN is one of the few simple triatomic molecules whose photodissociation mechanisms have been thoroughly investigated. Since it has a linear structure in the electronic ground state, the dissociation follows a photoexcitation at a linear or slightly bent structure. It is generally believed that the A-band consists of the dominant excitation to 3Π0+ (4A’) with the transition dipole moment (TDM) parallel to the molecular axis (z), a slightly weaker transition to 1Π1 (5A’, 4A”), and a much weaker transition to 3Π1 (3A’, 2A”), both of the latter two having perpendicular TDMs. In the present work, we have theoretically studied the geometry dependence of these TDMs and found a pronounced θ (bending angle) dependence in the parallel (z) component of the TDMs to 1Π1 (5A’) and 3Π1 (3A’), both of which should be zero at a linear geometry by symmetry and thus have been previously ignored. We estimated that the z component TDM to 1Π1 (5A’) has a contribution of 15-20% to the total absorption cross-section at 249 nm at room temperature. Interestingly, the TDM to 3Π0+ (4A’) does not exhibit such θ dependency and thus has only z component. We compare the TDMs of ICN and CH3I molecules having similar excited states. The fact that all the TDMs to 3A’, 4A’, and 5A’ have nonnegligible z components implies the importance of the coherent excitation contributions to various observables of CN fragment, such as the anisotropy parameter, the orientation parameter, rotational level distribution as well as the rotational fine structure level distribution.


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1. INTRODUCTION The A-band (λ=210-320 nm) photodissociation of ICN in the gas phase, especially its fragment branching ratios and the stereo dynamics, has suggested the presence of complicated nonadiabatic and interference effects, and prompted many experimental 1-12 and theoretical11-23 studies over the past 30 years, as described in many reviews24-28. This A-band photodissociation of ICN has also been investigated at liquidliquid interface29 and in solvents.30 The following two dissociation channels are established in the gas phase both experimentally and theoretically. I(2𝑃3/2) + CN(X2Σ , 𝑣, 𝑁, 𝐹1/𝐹2) +

ICN + ℎ𝜈→ . + I ∗ (2𝑃1/2) + CN(X2Σ , 𝑣, 𝑁, 𝐹1/𝐹2) Here, 𝑣 and 𝑁 denote the vibrational and rotational quantum numbers, respectively, and 𝐹1 and 𝐹2 are the rotational fine-structure levels (F1: J = N + 1/2, F2: J =N-1/2). Hereafter, we denote I (I ∗ ) as an iodine atom in the spin-orbit ground (excited) state as in other papers. The detailed N-dependent quantum yield (statedependent fraction) of the I* channel ΦI ∗ (𝑁), the rotational distribution P(N), and the anisotropy parameter β of each of the I and I* channels were reported by Wittig et al. at the excitation wavelength λ=266 nm,1 and by Zare et al. at λ=249 nm.2 According to the series of works by the latter group, the β value depends critically on N in both channels.2 The ratios of the perpendicular to parallel transition dipole moments for each N were reported by Black.7 Moreover, since the orientation parameter C(N),5 which is the spatial anisotropy of the rotational angular momentum of the product CN, showed a similar N-dependence as ΦI ∗ (𝑁) and the net orientation parameter of CN(v=2) was 0, they pointed out the importance of the coupling between the CN rotation and the electron spin angular momenta.5,6 Subsequent to the earlier work, Hall et al. published detailed bipolar moments of the CN photofragment with frequency modulation spectroscopy.8-10 In particular, they reported the orientation parameters of the CN fragment for each of the F1 and F2 levels for the coincident I and I* states; they found no notable difference in the orientation parameter between the F1 and F2 levels in both channels, but the opposite sign of the orientation parameter in the I and I* channels.9 The relevant ab initio potential energy surfaces (PESs) were presented by Morokuma and coworkers more than a decade ago and are sometimes called AYM surfaces.16 From their work, the electronic states involved, X1Σ+0 (1A’), 3Π2 (2A’,1A’’), 3Π1 (3A’,2A’’), 3Π0- (3A’’), 3Π0+ (4A’), 1Π1 (5A’,4A’’), 3Σ+0- (5A’’), and 3Σ+

1

(6A’, 6A’’), and the state correlations from the Franck-Condon (FC) region to the dissociation limits were

established. Here, the state order follows that at the FC region, and each state label is based on the irreducible representation of the linear (C∞v) and bent (Cs) structures. (See Figures 1 and 2 which will appear later.) Morokuma et al. also calculated the transition dipole moments (TDMs), but only at the Franck-Condon (FC) region; the oscillator strengths were reported as

3

Π1:3Π0 + :1Π1 = 0.06:0.66:0.28.16 Using their PESs and

TDMs (at FC), the absorption cross section and the channel dependent rotation populations were broadly reproduced by several dynamics calculations. However, the anisotropic parameters and branching ratios calculated by Wang et al.17 and Qian et al.18 did not reproduce the experimental results. Moreover, Hall et al. pointed out that the parallel transition was dominant even at λ=308 nm, the red wing of the A-band, from the 3 ACS Paragon Plus Environment


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experimental anisotropy parameters, and questioned the accuracy of the TDMs of AYM. In this way, many unresolved problems remain concerning the photodissociation reaction of ICN. Photodissociation reactions of a linear triatomic molecule such as ICN are considered to follow a photoexcitation at a linear structure or a slightly bent structure. Therefore, the photoexcitation with the parallel component TDM, which produces a positive anisotropy parameter β value, was considered as an excitation to 3

Π0 + (4A’), and that with the perpendicular component TDM, which produces a negative β value, was

considered as an excitation to 1Π1(5A’, 4A”). Later, Dzegilenko et al. pointed out the importance of the socalled non-Condon effects, that is, the nuclear coordinate dependence of the TDMs.31 A similar non-Condon effect caused by bending motion was emphasized by Suzuki and Nanbu et al. for the OCS molecule, another heavily studied 16 electron linear triatomic molecule.32 Furthermore, the observed orientation of the CN rotational angular momentum vector implied the importance of a one-photon coherent (simultaneous) excitation to multiple excited states in the A-band. Therefore, the above discussion based on the β parameter might be too naive. Actually, in preliminary calculations, we found that some TDMs of ICN showed a remarkable θdependency. Since the wavenumber of the ICN bending vibration is only 304 cm-1,33 the bending vibrational excited states in the ground electronic state can have a nonnegligible population. That is, there is a possibility that the non-Condon effect by bending motion appears in observables such as the absorption cross section. Note the abovementioned dynamics calculations, except for the work by Dzegilenko et al. 31 were carried out by using the TDMs at the equilibrium nuclear configuration, under the assumption of coordinate independence, that is, the Condon approximation. Furthermore, depending on the systems and the electronic structures, TDMs calculated with the previously used nodeless effective potentials (ECPs) sometimes contain a large error, as was exemplified in our study on lanthanide compounds. 34 Such shortcomings of ECPs are easily reduced by using the model core potentials (MCPs). It is recommended to apply several calculation methods to TDMs including the response type method35 for testing the numerical reliability. Recently, we clarified the following two points by applying semiclassical theory to the A-band photodissociation of ICN.23 (i) The nonstatistical population difference observed in the rotational fine structure levels F1 and F2, of the photofragment CN is a consequence of the quantum interference between the wavepackets generated simultaneously on the 3Π1 (3A') and 3Π0+ (4A') PESs through the nonadiabatic transition due to the recoupling of the angular momenta involved in the asymptotic region. In particular, the characteristic oscillation behavior in the rotational fine structure level difference function f(N) defined by Zare et al. was theoretically reproduced. (ii) A quantum interference effect between the 4A’ and 5A’ PESs appears in the orientation parameter of CN through the nonadiabatic transition due to the conical intersection in the molecular region. We have focused on f(N) in that study,23 and the experimentally observed behavior was qualitatively reproduced by the approximate semiclassical calculation. Furthermore, quantitative calculations by using the calculated PESs and TDMs are in progress. In this paper, (a) we calculate the TDMs by using the MCP and corresponding basis functions. As the theoretical expressions for the TDM calculations, we employ the response (resp) type formula35 as well as the usual Hellmann-Feynman (HF) type formula. As discussed above, it has been suggested that various quantum interference, nonadiabatic, and threshold effects can be involved in the ICN photodissociation reaction. Miller 4 ACS Paragon Plus Environment

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et al. applied the semiclassical initial value representation (SC-IVR) method36-38 to the ICN photodissociation with the AYM surfaces and showed the applicability of their method to this system by almost reproducing the results of the full quantum dynamics calculations.36,39 In this paper, with expecting future studies to analyze the various quantum effects involved from semiclassical viewpoints, we adopt the SC-IVR method to calculate the absorption cross section including the non-Condon effect and verify the validity of our PESs and TDMs. (b) We discuss the importance of the non-Condon effect in the context of the f(N) and the orientation parameter 𝐶 (𝑁) as additional information to our previous work.23 (c) In the photodissociation of CH3I with similar electronic excited states, the importance of the interference effect caused by a coherent excitation has not been discussed except for the paper by Freed et al.40 Therefore, by comparing the photodissociation of ICN and CH3I, we clarify their important chemical differences. The purpose of this paper is therefore to point out the importance of the non-Condon effect of TDMs and the quantum interference effect due to the coherent excitation to the multiple PESs in the ICN photodissociation.

2. COMPUTATIONAL METHODS 2.1 Theoretical Preliminaries of Photoexcitation. The precise definition of the coordinate system is required for the detailed discussion of molecular photodissociation including the coherent effects and the stereo-dynamics of the fragments’ angular momentum vectors. The space-fixed (SF) coordinate system is defined by referring to the electric field polarization vector 𝜺 of the excitation photon. On the other hand, since the TDM vector 𝝁 is usually described in the body-fixed (BF) coordinate system, the scalar product 𝝁 ∙ 𝜺 involved in the interaction Hamiltonian depends on the angle between 𝝁 and 𝜺, for which the Euler angles 𝜴 = (𝜙SF, 𝜃SF,𝛾SF) are used to define the molecular direction and to transform from the SF to the BF coordinate system. If the final state f of a photoproduct is derived from the initial state i through multiple intermediate states n, the amplitude of f is expressed as a superposition of the complex amplitudes through the states n. Therefore, the differential cross section 𝜎𝑓𝑖(𝜴) is proportional to the square of the absolute value of the resultant complex amplitude.41-43 𝜎𝑓𝑖(𝜴) ∝

|∑ 𝑛

|

2

𝑆𝑓𝑛𝑇𝑛𝑖(𝜴) =

∑(𝑆

(𝜴)) ∗ (𝑆𝑓𝑛𝑇𝑛𝑖(𝜴)).

𝑓𝑛′𝑇𝑛′𝑖

(1)

𝑛, 𝑛′

Here, 𝑇𝑛𝑖(𝜴) is the photoexcitation amplitude of the one-photon process, and 𝑆𝑓𝑛 is the matrix elements of the half-collision scattering matrix. The initial state is fixed to the electronic ground state, and the subscript i is omitted hereafter. 𝑇𝑛(𝜴) is given by the overlap of the dissociation continuum wavefunction 𝜒𝑛(𝑅, 𝜃;𝐸), the transition amplitude 𝑡𝑛(𝑅, 𝜃;𝜴), and the stretching and bending vibration wavefunctions of the ground electronic state 𝜓str(𝑅) and 𝜓bend(𝜃) as follows: 44 𝑇𝑛(𝜴) = ⟨𝜒𝑛(𝑅, 𝜃;𝐸)│𝑡𝑛(𝑅, 𝜃;𝜴)│𝜓str(𝑅)𝜓bend(𝜃)⟩.

(2)

Here, R is the distance between I atom and the center of mass of CN, and θ is the polar angle of the CN axis in the Jacobi coordinate system. The bracket means the integration over the nucleus coordinates (𝑅, 𝜃), and E is the total energy. In this paper, CN is regarded as a rigid rotor. The transition amplitude 𝑡𝑛(𝑅, 𝜃;𝜴) is given by the scalar product of the transition dipole moment vector 𝝁𝑛 and the electric field polarization vector 𝜺 of the 5 ACS Paragon Plus Environment


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excitation photon: 𝑡𝑛(𝑅, 𝜃;𝜴) = 𝝁𝑛(𝑅, 𝜃) ∙ 𝜺.

(3)

Additionally, the integrated transition dipole moment vectors 𝑴𝑛 for the respective electronic states n are defined as 𝑴𝑛 = ⟨𝜒𝑛(𝑅, 𝜃;𝐸)│𝝁𝑛(𝑅, 𝜃)│𝜓str(𝑅)𝜓bend(𝜃)⟩.

(4)

When linearly polarized light is used, the direction of 𝜺 is defined as the Z axis of the space-fixed (SF) coordinate system. Therefore, the 𝜺 vector in the SF coordinate system is

[]

0 𝜺SF = 0 . 1

(5)

On the other hand, when circularly polarized light is used, the propagation direction of the circularly polarized light is defined as the Z axis of the SF coordinate system. Therefore, the 𝜺 vector in the SF coordinate system is ± 𝜺SF =∓

[ ]

1 ±𝑖 . 2 0

1

(6)

Here, the + and – signs correspond to clockwise and counterclockwise circularly polarized light, respectively. The differential cross section 𝜎𝑓𝑖(𝜴) is proportional to the square of the absolute value of the transition amplitude given in the following equation. 𝜎𝑓𝑖(𝜴) ∝

∑(𝑆

𝑓𝑛′𝑇𝑛′𝑖)

𝑛, 𝑛′

=

∗

(𝑆𝑓𝑛𝑇𝑛𝑖)

∑𝑆

𝑓𝑛′

∗

∗ 𝑆𝑓𝑛⟨𝜒𝑛(𝑅, 𝜃;𝐸)│𝝁𝑛′(𝑅, 𝜃) ∙ 𝜺│𝜓str(𝑅)𝜓bend(𝜃)⟩ ⟨𝜒𝑛(𝑅, 𝜃;𝐸)│𝝁𝑛(𝑅, 𝜃) ∙ 𝜺│𝜓str(𝑅)𝜓bend(𝜃)⟩.

(7)

𝑛,𝑛′

According to refs 45 and 46, the rightmost expression of eq 7 can be transformed to the sum of the 0th-, first-, and second-rank tensors consisting of the 𝜺 ∗ and 𝜺 vectors. The cross section is defined as the scalar (0thrank tensor) product, and the first- and second-rank tensors are termed the orientation and the alignment, respectively.45 The q-component of the orientation depends on the (𝜺 ∗ × 𝜺)𝑞 and follows the transformation rule as a vector for the coordinate rotation.45 The q-component of the alignment depends on the second-rank 2 tensor product [𝜺 ∗ ⨂𝜺]𝑞.45 With the unitarity of the half-collision S-matrix, the total absorption cross section

𝜎 is obtained from the scalar part after the integration over 𝜴 as 𝜎∝

∑𝑴

𝑛

∗

∙ 𝑴𝑛.

(8)

𝑛

2.2 Electronic Structure Calculation. In this study, we calculated the PESs by using the same MCP, basis functions and the electronic structure calculation methods used in our recent work.23 Briefly, we have chosen the σ’,π, σ, n, σ * , and π * molecular orbitals (MOs) in the active space. These are the bonding σ orbital of CN, bonding π orbital of CN, bonding σ orbital of I-C, nonbonding orbital on I, antibonding σ * orbital of I-C, and antibonding π * orbital of CN, respectively. The set of one-electron MOs was determined by the state-averaged multiconfiguration self6 ACS Paragon Plus Environment

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consistent field (SA-MCSCF) method.23 The TDMs were calculated using the wavefunctions determined by the contracted spin-orbit configuration interaction (cont-SOCI) method and the uncontracted SOCI (SOCI) method. In the cont-SOCI method, first, spin-free CI states were obtained from the CI problem of the spin-free Hamiltonian excluding the spin-orbit interaction (SOI). Then, using the spin-free CI states as the basis functions, the adiabatic potential energies were obtained from the eigenvalue problem of the total Hamiltonian including SOI. As the spin-free states for these calculations, we included not only 1Σ+, 3Π(n→σ*), 1Π(n→σ*), and 3Σ+ (σ→σ*), which correlate to the CN(X 2Σ+) state, but also the higher spin-free excited states that correlate to the CN(A 2) state. In the SOCI method, the adiabatic potential energies and the SOCI wavefunctions were obtained from the eigenvalue problem of the total Hamiltonian including SOI in one step using the spin and spatial symmetry adapted configuration state functions as the basis functions. As the theoretical formulae for the TDMs calculations, we employed the usual sum of the electron coordinates as the operator, which we call Hellmann-Feynman (H-F) type operator denoted 𝜇H𝐽 ― F, 𝝁H𝐽 ― F(𝑅, 𝜃) = ⟨Ψ𝐽│𝒓│Ψ𝐺⟩, as well as the following response (resp) type TDM 𝜇resp ,35 𝐽 ∂𝐶𝐾𝐺 𝝁resp (𝑅, 𝜃) = (𝐸𝐽 ― 𝐸𝐺) 𝐶𝐾𝐽 + 𝐽 ∂𝒆

{∑ 𝐾

(9) ∂𝜑𝑗

∑⟨𝜑 │ ∂𝒆 ⟩𝜌 𝑖

𝑖, 𝑗

}

𝐽𝐺 𝑖𝑗

.

(10)

and compared their performance. Here Ψ𝐺 and 𝐸𝐺 are the CI (either cont-SOCI or SOCI) wavefunction and the eigenvalue of the electronic ground state G, respectively, and Ψ𝐽 and 𝐸𝐽 are the corresponding ones for excited states J. 𝐶𝐾𝐼 is the CI coefficient of the K-th configuration of the CI wavefunction for the state I. 𝜑𝑖,𝑗 are the MOs, and 𝜌𝐼𝐽 𝑖𝑗 is the transition density matrix between the CI states I and J. Here, r and e are the sum of the electron coordinates and the electrostatic field for the numerical differentiation, respectively. To analyze the θ-dependence of TDMs, we evaluated the degree of natural orbital (NOs) mixings caused by molecular bending. To understand the state specific polarization and the electron correlation effects, we diagonalized the density matrices of the 4A’ and 5A’ states obtained from the SOCI wavefunctions using the common set of the SA-MCSCF MOs, and calculated the state specific NOs for each state. These calculations have been carried out with the COLUMBUS program package47 with the MCP integrals passed from the GAMESS program.48

2.3 Semiclassical Initial Value Representation (SC-IVR) Method. We have calculated the dynamics with SC-IVR method of Miller et al.36-38 in consideration of the nonCondon effect of TDMs. In our previous paper, the coordinates in the space-fixed (SF) and the body-fixed (BF) coordinate systems were expressed as (X’, Y’, Z’) and (X, Y, Z), respectively. 23 In this paper, we have changed their coordinates to (X, Y, Z) and (x, y, z), respectively, following the standard representation. 𝜇𝑘, 𝑠(𝑅, 𝜃) denotes the s=(x, y, z) component of the TDM vector of the electronic excited state k at the 𝑠 nuclear coordinates (𝑅, 𝜃), and then the semiclassical initial wavefunction (photo prepared state) Ψ𝑘, 0 (𝐪) on

the excited state k is given as follows: 7 ACS Paragon Plus Environment


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𝑠 str bend Ψ𝑘, (𝜃)𝜙𝑘(𝐱). 0 (𝐪) = 𝜇𝑘, 𝑠(𝑅, 𝜃)𝜓 (𝑅)𝜓

(11)

Here, 𝐪 is the Ndf dimensional coordinates (details will be described below). 𝜙𝑘(𝐱) is the F-dimensional electronic vibration wavefunction of the electronic state k introduced by Stock and Thoss,49 and expressed as the product of the first excited harmonic wavefunction for level k and the ground harmonic wavefunction for the others, that is, 𝐹/4

()

𝜙𝑘(𝐱) = 2

1 𝜋

𝑥𝑘𝑒

1 ― (𝐱 ∙ 𝐱) 2

(12)

.

Here, 𝐱 is the Cartesian electronic coordinates. We consider the five adiabatic electronic states, 3Π0+ (4A’), 1Π1 (5A’), 1Π1 (4A’’), 3Π1 (3A’), and 3Π1 (2A’’), which are labeled as 1, 2, 3, 4, and 5. In the SC-IVR method, the wavefunction for the electronic state k at time t is given as Ψ𝑘,𝑠 𝑡 (𝑅 ′ , 𝜃 ′ ) =

∑∫𝑑𝐱 𝑑𝐪𝜙 ′

∗ 𝑘

𝑘′, 𝑠 (𝐱′)𝐾SC 𝑡 (𝐪′, 𝐪)Ψ0 (𝐪).

(13)

𝑘′

Here, 𝐾SC 𝑡 (𝐪′, 𝐪) is the semiclassical Feynman kernel operator which evolves the system on the coordinate q at t=0 to q’ at t. We use a time evolution kernel introduced by Herman and Kluk50 as Miller et al.36 To consider the nonadiabatic interaction between 4A’ and 5A’, we adopted the Meyer-Miller (MM) classical analog (CA) Hamiltonian.51 We set the total angular momentum JICN=0, then the MM CA Hamiltonian is given as follows.38 (𝑃 + Δ𝑃)2 1 1 1 + 𝐻(𝑅, 𝑃, 𝜃, 𝑁, 𝑥1, 𝑝1, 𝑥2, 𝑝2) = 𝐻(𝐪, 𝐩) = + 𝑁2 + 2 2 2𝑀 2 2𝑀𝑅 2𝑚𝑟

)

(

2

∑(𝑝

2 𝑘

+ 𝑥𝑘2 ― 1)𝑊𝑘(𝑅, 𝜃).

(14)

𝑘

Here, p is the momentum conjugate to q, and P is the kinetic momentum along the dissociation axis. Δ𝑃 is the same nonadiabatic coupling term (NACT) as in eq 2.13 of ref 38, which depends on the nonadiabatic matrix element between adiabatic states 1 and 2. M and m are the reduced mass of ICN and CN, respectively. 𝑊𝑘 is the adiabatic potential energy surfaces of the electronic state k. Since there is only one nonadiabatic coupling term between the adiabatic states 1 and 2, it is only necessary to include these states in the sum for the electronic degrees of freedom in eq 14. Moreover, since we treat the CN molecule as the rigid rotor, the canonical coordinates are given as 𝐪 = (𝑅, 𝜃, 𝑥1, 𝑥2), and thus the dimension of the coordinates is set as Ndf=4. The electronic Cartesian coordinate 𝑥𝑘 and momentum 𝑝𝑘 are transformed from the classical action-angle variables (n, q) as 𝑥𝑘 = 2𝑛𝑘 + 1cos 𝑞𝑘, 𝑝𝑘 = ― 2𝑛𝑘 + 1sin 𝑞𝑘.

(15) (16)

The autocorrelation function 𝜉𝑘, 𝑠(𝑡) for each electronic state k and the component s=(x, y, z) of TDM is calculated as follows, 𝑠 SC 𝑘, 𝑠 𝜉𝑘, 𝑠(𝑡) = ⟨Ψ𝑘, 0 (𝐪′)│𝐾𝑡 (𝐪′, 𝐪)│Ψ0 (𝐪)⟩.

The total absorption cross section 𝜎(𝜔) is calculated by the Fourier transform as follows, ∞ 4𝜋2𝜔 1 𝜎(𝜔) = 𝜉𝑘, 𝑠(𝑡)𝑒𝑖𝜔𝑡𝑑𝑡. 3𝑐 2𝜋 ―∞

∑ ∫

(17)

(18)

𝑘, 𝑠 = 𝑥, 𝑦, 𝑧

From eq 8, since nonadiabatic effects do not appear in the total absorption cross section 𝜎(𝜔), we ignore the nonadiabatic coupling term Δ𝑃 in eq 14 in the actual calculation of 𝜎(𝜔).


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3. Results and Discussion 3.1 The Transition Dipole Moments and the Absorption Cross Section. Figure 1 shows the potential energy curves at colinear structure, and Figure 2 shows the potential energy curves for bending motion at R=5 bohr.

Figure 1.

Potential energy curves (PECs) of X1Σ0+ (black), 3Π2 (orange), 3Π1 (green), 3Π0+ (blue), 1Π1 (red),

and 3Σ+1 (gray) for collinear structures. The conical intersection exists at R≅6.2 bohr.

Figure 2.

PECs of the 1A’, 2A’, 3A’, 4A’, 5A’, and 6A’ states versus bending angle θ at R=5.0 bohr.

We label each of the electronic states based on the dominating electronic configurations at the FC 9 ACS Paragon Plus Environment


Page 10 of 31

region and set the energy zero at the dissociation limit of the I channel. The conical intersection point exists at R=6.2 bohr in Figure 1. The calculated equilibrium distance 𝑅e and the vibrational frequencies (𝜔R for C-I stretching and 𝜔θ for bending) are 𝑅e = 5.01 bohr, 𝜔R = 478 cm ―1, and 𝜔θ = 296 cm ―1. These are in 52 𝜔exp = 498 cm ―1, and reasonable agreement with the respective experimental values, 𝑅exp e = 4.99 bohr, R

―1 33 𝜔exp . θ = 304 cm

Table 1 shows the calculated values of the H-F type and resp type TDMs at the equilibrium configuration (R=5, θ=0). First, examining the TDM values with the usual calculation method of the H-F type formula, eq 9, we see that both the TDM values of 1Π1 and 3Π1 are insensitive to the two CI methods, cont-SOCI and SOCI. However, there are notable differences in the H-F type TDM values of 3Π0 + , as is also shown graphically in Figure 3.

Table 1. H-F type and resp type TDMs (e∙a0) calculated with the cont-SOCI and SOCI methods at (R=5, θ=0). 𝜇H𝑥 ― F(3Π1)

𝜇H𝑧 ― F(3Π0 + )

𝜇H𝑥 ― F(1Π1)

𝜇r𝑥esp(3Π1)

3 𝜇resp 𝑧 ( Π0 + )

1 𝜇resp 𝑥 ( Π1)

cont-SOCI

0.033

0.003

0.094

0.035

0.103

0.103

SOCI

0.033

0.101

0.097

0.035

0.105

0.104

Figure 3. R-dependence of the 3Π0+ H-F type TDM obtained by the SOCI (blue solid line) and cont-SOCI (blue broken line) methods. Δ𝑑 was calculated from eq 29 using the difference of the dipole moment 1

functions for the Σ0+ and 3Π states, as defined.

3

Π

Analyzing their transition density matrix elements, there was a marked difference in the elements 𝜌𝜎 ∗0𝜎+ 10 ACS Paragon Plus Environment

1

←X Σ0+

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≡ ⟨3Π0 + │𝑋𝜎†∗ 𝑋𝜎│X Σ0+ ⟩ obtained by the cont-SOCI and SOCI methods. Here, 𝑋 † and 𝑋 are the creation 1

and the annihilation operators of the second quantization, respectively. 𝜎 and 𝜎 ∗ are the bonding σ, and the antibonding σ * orbitals of the I-C bond, respectively. We have denoted the nonbonding π orbital on I as nx and ny. The leading electron configurations contributing to the ground state X0+ in the Franck-Condon (FC) 0

1

1

1

region were 𝜎2𝑛2𝑥𝑛2𝑦𝜎 ∗ , 𝜎2𝑛2𝑥𝑛1𝑦𝜎 ∗ , 𝜎2𝑛1𝑥𝑛2𝑦𝜎 ∗ , and 𝜎1𝑛2𝑥𝑛2𝑦𝜎 ∗ . The configuration functions generated by 1

2

2

operating 𝑋𝜎†∗ 𝑋𝜎 onto the above four configurations are 𝜎1𝑛2𝑥𝑛2𝑦𝜎 ∗ , 𝜎1𝑛2𝑥𝑛1𝑦𝜎 ∗ , 𝜎1𝑛1𝑥𝑛2𝑦𝜎 ∗ , and 𝜎0𝑛2𝑥𝑛2𝑦 2

𝜎 ∗ , respectively. Therefore, it is expected that the contributions of

(𝜎, 𝑛𝑦→𝜎 ∗ 2), and

1 +

Σ

(𝜎2→𝜎 ∗ 2)

SOCI calculations. Since

2 Σ (𝜎→𝜎 ∗ ), 3Π(𝜎, 𝑛𝑥→𝜎 ∗ ),3Π

1 +

type configuration may be underestimated in the more approximate cont-

Π(𝜎, 𝑛𝑥→𝜎 ∗ ),3Π(𝜎, 𝑛𝑦→𝜎 ∗ ), 2

3

2

and

1 +

Σ

(𝜎2→𝜎 ∗ 2)

are doubly excited

configurations and have exceedingly higher energies, it is especially inefficient to include these SF states as the basis in the cont-SOCI calculation. We also found notable differences in the transition density matrix elements 3

Π

𝜌𝜋 ∗0𝑛+

1

←X Σ0+

3

Π

and 𝜌𝜋 ∗0𝜋+

1

←X Σ0+

obtained by the cont-SOCI and SOCI methods. Because this transition is of 3

3

parallel type, the contributions of the Σ ― (𝑛→𝜋 ∗ ) and Σ ― (𝜋→𝜋 ∗ ) configurations can be underestimated in the more approximate cont-SOCI calculations. Conversely, it can be said that H-F type TDMs can be calculated more efficiently using the uncontracted SOCI method. Next, response type TDMs with the formula of eq 10 are discussed. Table 1 shows that those with the cont-SOCI and SOCI calculations agree well including the difficult case of 3Π0 + . Moreover, these TDMs are in good agreement with the corresponding H-F type TDMs, except for 3Π0 + obtained with the cont-SOCI method. Thus, the resp type formula for TDMs and for dipole moment expectation values for the cont-SOCI calculations (see later eq. (29)) has generally a small dependency on the calculation methods and turns out to be numerically stable. From the above calculated results, ICN has 𝜇3Π0 + :𝜇1Π1x:𝜇1Π1𝑦≅1:1:1 at the equilibrium nuclear configuration. If the absorption intensity is determined only by the transition moments at this equilibrium nuclear configuration (Codon approximation), and the effect of coherent excitation is negligible, the product angular distribution would be almost isotropic (β=0). This is of course inconsistent with most experimental results that the A-band photoexcitation is dominated by parallel transition(s). Therefore, in this study, the magnitude of the actual transition amplitude is more accurately evaluated by including the non-Condon effect, that is, by including the initial state vibration wavefunctions and the TDM functions as described before, and the theoretically constructed absorption spectrum is compared with the experiment results to evaluate the accuracy of the TDM functions. Figure 4 shows the H-F and resp type TDMs calculated by the SOCI method and the ground stretching vibrational wavefunction 𝜓str 0 (𝑅) as a function of R at θ=0. We will use just A’ and A’’ as the labels of the electronic states.



Page 12 of 31

Figure 4. R-dependence of 3A’x (green), 4A’z (blue), and 5A’x (red) TDMs in H-F (solid lines) and resp type (broken lines).

The ground stretching vibrational wavefunction 𝜓str 0 (𝑅) is also shown.

Figure 5 shows the H-F and resp type TDMs calculated by the SOCI method and the ground bending (𝜃) as a function of θ at R=5. Since the x component of 4A’ TDM, 𝜇4A′𝑥, was vibrational wavefunction 𝜓bend 0 less than 0.001 for all θ, it is not shown in Figure 5. In addition, since the y components of 2A’’ and 4A’’ TDMs, 𝜇2A′′𝑦, 𝜇4A′′𝑦 were nearly equal to 𝜇3A′𝑥, 𝜇5A′𝑥, these y components were also omitted. Here, the θ-dependence of the z component of 5A’ TDM, 𝜇5A′z, is noteworthy. In the axial recoil approximation for this direct dissociation, if the ground state ICN is at θ=0 or at a very small bending angle, 𝜇5A′z = 0 due to symmetry reason, then the photoexcitation to 5A' yields only the perpendicular transition with the angular distribution of photofragment with β = -1, as almost every study on ICN has assumed so far. However, at θ>10°, since 𝜇5A′𝑧 is greater than 𝜇5A′x, an electron excitation to 5A' also gives the products with the parallel component with β > 0. In Figure 5, a similar θ-dependent behavior is seen between 𝜇3A′𝑥 and 𝜇5A′𝑥, and another similar θ-dependence is seen between 𝜇3A′𝑧 and 𝜇5A′𝑧. Their similarities are reasonable because the intensity of the 3A’ state comes from the 5A’ state by the intensity borrowing mechanism caused by the SOI. Another interesting behavior is that the x component contribution of 4A', 𝜇4A′𝑥, is negligible even if ICN is highly bent. These results, summarized in Figure 5, are among the most important findings in this paper.


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Figure 5. θ-dependence of 3A’x (green), 3A’z (gray), 4A’z (blue), 5A’x (red), and 5A’z (orange) TDMs in HF (solid lines) and resp type (broken lines) calculations. The 4A’x TDM is smaller than 0.001 for all θ and not (𝜃) is also shown. shown. The ground bending vibrational wavefunction 𝜓bend 0 Figure 6a shows the total absorption cross section obtained by assuming the initial vibration distribution at room temperature, 300 K, and the partial cross sections of each TDM as explained above. Figure 6b shows the partial cross sections of 4A’, 3A’, and 2A’’ in the low excitation energy region. In Figure 6c, the theoretical total absorption cross section shifted by +600 cm-1 is compared with the experimental results by Leone et al.53 and Pitt et al.54 Compare to these experimental cross sections, our theoretical total absorption cross sections seem to have a slightly larger width.

Table 2 compares our theoretical absorption cross section with experimental ones by Leone et al.,53 Pitts et al.,54 and McGlynn et al.55, with regard to the band max, the maximum value of the absorption cross section 𝜎max, the full width at half maximum value, and 𝜎max × FWHM as an alternative to the integrated intensity.



Page 14 of 31

Figure 6. (a) Calculated total (black solid) and partial cross sections with 4A’z (blue), 5A’x (red solid), 4A’’y (red dashed), 5A’z (orange solid), and 3A’+2A’’ (purple) components. (b) The partial cross section of 4A’z(blue), 3A’z(gray), 3A’x(green solid), and 2A’’y(green broken) in the lower energy region of A-band photodissociation. (c) Calculated total (black solid) cross section shifted by +600 cm-1 and the experimental cross sections of Leone et al.53 and of Pitts et al.54 Table 2. Comparison of the theoretical and the experimental absorption cross sections. This study

Leone et al.53

Pitts et al.54

McGlynn et al.55

Band max/ cm-1

39400

40000

40000

39800

𝜎max/ l mol-1 cm-1

105

84

105

125

9700

8400

7000

8200

102

70.6

73.5

103

FWHM/

cm-1

𝜎max × FWHM/ × 104 l mol-1 cm-2


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Our theoretical 𝜎max, 105 l mol-1 cm-1, overestimates the experimental one by Leone et al.,53 but corresponds to that by Pitts et al.54 Our theoretical FWHM overestimates the previous experimental values, but the theoretical value for 𝜎max × FWHM is in good agreement with the experimental one by McGlynn et al.55 For the accurate and balanced TDM calculations, the present study employed the multireference first-order CI method with the complete active space reference functions. For this reason, the electron correlation effects in the ground state might be slightly underestimated relative to the excited states at the FC region. Then the vertical excitation energies were slightly underestimated. By the same reasoning, the dissociation energy was underestimated by about 7 kcal/mol, and the dissociative PESs at the FC region have larger slopes, thus the absorption cross section has larger widths as seen in Table 2.

As eqs 4 and 8 show, the contribution to the total absorption cross section from each excited state n is proportional to the 𝑴𝑛 ∗ ∙ 𝑴𝑛, where 𝑴𝑛 is the integration of the product consisting of the initial state vibration wavefunctions, the transition dipole moment 𝝁𝑛, and dissociative continuum wavefunction on the excited state n. According to the reflection principle discussed by Schinke,56 the amplitude of the dissociative continuum wavefunction is proportional to the reciprocal of the absolute value of the gradient for the excited state potential at the classical turning point. According to Figure 1, since there is no difference between the gradients of the 0+(II) (4A’) and 1(II) (5A’) potential curves, we will discuss the magnitude of the contribution to the cross section only by the product of the vibrational wavefunctions of the ground state and the TDM functions. We have emphasized before a remarkable θ-dependence of 𝜇5A′𝑧. According to Figure 6a, this 𝜇5A′𝑧 contribution to the parallel component always has an intensity more than half of 𝜇5𝐴′𝑥 in a wide region of the A -band. Therefore, the observed positive β values are likely to be due to the following two types of non-Condon effect via the bending vibration and stretching vibration. (i) Since the bending vibrational wavenumber of ICN is only 304 cm-1, their hot band contribution cannot be ignored at room temperature. Actually, considering the degeneracy of the first excited bending vibrational states, the population ratio of the two states with the vibrational quantum number 0 to 1 is approximately 7:3. Since the hot bending vibration increases the contribution of 5A’ TDM 𝜇5A′𝑧, as seen from Figure 5, a part of the electronic transition to 5A' that was previously assigned only to the perpendicular transition, actually contributes to the parallel component. (ii) Even if the vibrational states of the two modes are both in the zero-point state in the electronic ground state, bend (𝜃), 𝜇4A′z and 𝜇5A′x in Figures 4 and 5, and the expression considering the functional shapes of 𝜓str 0 (𝑅), 𝜓0

in eq 4, it is clear that 𝑀4A′z is greater than that for 𝑀5A′x. After all, due to the above two kinds of non-Condon effect, the parallel transition has been observed as the dominant component of the ICN A-band photoexcitation. The leading electron configurations of the 4A' and 5A' states at the colinear structure are

(𝑛𝑥, 𝑦→𝜎 ) ∗

and Π(𝑛𝑥→𝜎 1

∗

3

Π

), respectively. Although their spin multiplicities are different, they have the same 1

electron configuration 𝑛3𝜎 ∗ . For this reason, as shown in Figures 1 and 2, these PESs show a similar behavior. On the other hand, as mentioned before, 𝜇5A′𝑧 rapidly increases and 𝜇5A′𝑥 slowly decreases with 𝜃, whereas 𝜇4A′𝑥 remains zero. To analyze the remarkable difference in their θ-dependences, natural orbital (NO) analysis was performed. First, we add a prime ' to the labels of the state-specific NOs obtained from the density matrix evaluated using the SOCI wavefunction of each state and distinguish them from the SA-MCSCF orbitals used 15 ACS Paragon Plus Environment


Page 16 of 31

as input for the SOCI calculation. Figures 7a and 7b are contour plots of the state-specific natural orbitals 𝑛′𝑥

(4A′) and 𝑛′𝑥(5A′) of the 4A' and 5A'states in the xz molecular plane. Note first that the direction of each orbital is defined in the BF coordinate system, θ is the polar angle of CN in the Jacobi coordinate system, and the x axis is perpendicular to the z axis. At a glance, we can see that the general behavior of 𝑛′𝑥(4A′) and 𝑛′𝑥

(5A′) differs greatly at this θ=20°. 𝑛′𝑥(4A′) shows a small change in the direction of the nonbonding atomic 5px orbital on I atom by 𝛼4≅10°, but 𝑛′𝑥(5A′) shows a larger rotation of the atomic 5px orbital by 𝛼5≅20°.

Figure 7 (a) The 𝑛′𝑥 natural orbital of 4A’(3Π0+) at θ=20°. (occupation number = 1.5724). In this natural orbital, the 5p nonbonding atomic orbital on I is rotated clockwise by 𝛼4≅10°. (b) The 𝑛′𝑥 natural orbital of 5A’(1Π1) at θ=20°. (occupation number =1.1872). The constituent 5p nonbonding atomic orbital is rotated clockwise by 𝛼5≅20°.

Our natural orbital analysis shows that each state-specific NO can be expanded in terms of the common SA-MCSCF MOs, 𝑛𝑥 and 𝜎 ∗ , as follows: 𝑛′𝑥(4A′)(𝜃 = 20°)≅𝑛𝑥,

(19)

𝑛′𝑥(5A′)(𝜃 = 20°)≅0.95 × 𝑛𝑥 + 0.27 × 𝜎 ,

(20)

𝜎 ∗ ′(5A′)(𝜃 = 20°)≅0.95 × 𝜎 ∗ ― 0.27 × 𝑛𝑥.

(21)

∗

The above eqs 19 and 20 show a remarkable difference between 𝑛′𝑥(4A′) and 𝑛′𝑥(5A′); 𝑛′𝑥(5A′) consists of both 𝑛𝑥 and 𝜎 ∗ . In eq 21, 𝜎 ∗ ′(5A′) is the state-specific NO of 𝜎 ∗ for the 5A’ state. From the orthogonality constraint, 𝜎 ∗ ′(5A′) must be mixed with 𝑛𝑥 with the opposite sign as is evident by comparing eqs 20 and 21. In this way, we see that the mixing between the 𝑛𝑥 and the 𝜎 ∗ orbitals due to the bending motion is easily occurs in the 5A' state where the leading spin multiplicity is singlet, but such a mixing does not occur in the 4A' state, the dominant spin multiplicity of which is triplet. It is easy to understand this phenomenon concretely by expressing these many-electron wavefunctions using the Slater determinants. As explained above and seen in Figure 7(b), at the bent structure with 𝜃 = 20°, the 5px atomic orbital rotates by the same angle 𝛼5 = 20°. This relation 𝛼5≅𝜃 held not only at 𝜃 = 20°, but even at other angles 𝜃. Thus, for the 5A’ state, we can relate the 16 ACS Paragon Plus Environment

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rotated 𝑛𝑥′ and 𝜎𝑧∗′ orbitals to the unrotated 𝑛𝑥 and 𝜎𝑧∗ orbitals with the angle θ using the following unitary transformation and keeping the orthogonality of the natural orbitals:

[𝜎𝑛 ] = [ ―cossin𝜃𝜃

][ ]

sin 𝜃 𝑛𝑥 cos 𝜃 𝜎𝑧∗ .

𝑥′ ∗ 𝑧′

(22)

Then, the 1Π𝑥′ and 3Π𝑥′ states of the bent structure by θ are expressed as follows: 1

(

)

(

)

‖𝑛𝑥′𝜎𝑧∗′ ‖ + ‖𝜎𝑧∗′ 𝑛𝑥′‖ = ― sin 2𝜃 ‖𝑛𝑥𝑛𝑥‖ ― ‖𝜎𝑧∗ 𝜎𝑧∗ ‖ 2 + cos 2𝜃 ‖𝑛𝑥𝜎𝑧∗ ‖ + ‖𝜎𝑧∗ 𝑛𝑥‖ 2,

Π𝑥′(𝑛𝑥′→𝜎𝑧∗′ ) =

1

(

1

Π𝑥′(𝑛𝑥′→𝜎𝑧∗′ ) =

3

2

)

(‖𝑛𝑥 𝜎𝑧∗ ‖ ― ‖𝜎𝑧∗ 𝑛𝑥 ‖) = (‖𝑛𝑥𝜎𝑧∗ ‖ ― ‖𝜎𝑧∗ 𝑛𝑥‖) ′

2

′

′

′

2.

(23)

(24)

In eqs 23 and 24, we have omitted other doubly occupied orbitals. Here, || || expresses the normalized Slater determinant. For example, 𝑛𝑥 and 𝑛𝑥 in || || express the spin orbitals 𝑛𝑥α and 𝑛𝑥β, respectively. As eq 23 shows, the singlet 1Π𝑥′ includes the ground state configuration

‖𝑛𝑥𝑛𝑥‖

due to the mixing of 𝑛𝑥 and 𝜎𝑧∗ as

in eq 22. However, eq 24 shows that the triplet 3Π𝑥′ cannot include the doubly occupied configuration and equals 3Π𝑥 at θ=0. This difference in the transformation of the constituent one-electron orbitals is a well-known property of the Slater determinant and is just a mathematical realization of the Pauli exclusive principle.57 Since

‖𝑛𝑥𝑛𝑥‖

is the ground state configuration, it is expressed as Ψ𝐺; then, the TDMs 𝜇𝑥(5A′) and 𝜇𝑧(5A′) are

given as follows. 𝜇𝑥(5A′) = ⟨1Π𝑥(𝑛𝑥′→𝜎𝑧∗′ )│𝑥│Ψ𝐺⟩ = 2cos 2𝜃⟨𝜎 ∗ │𝑥│𝑛𝑥⟩,

(

𝜇𝑧 5A′

)=⟨

1

(

Π𝑥 𝑛𝑥′→𝜎𝑧∗′

)│𝑧│Ψ𝐺⟩ = ―

2sin 2𝜃⟨Ψ │𝑧│Ψ ⟩. 𝐺

𝐺

(25) (26)

Therefore, 𝜇5A′𝑧 increases rapidly in proportion to sin 2𝜃, and 𝜇5A′𝑥 decreases in proportion to cos 2𝜃 as seen in Figure 5. On the other hand, 𝜇4A′𝑥 does not change from the value at θ=0. Thus, the different 𝜃dependent behaviors of 𝜇5A′ and 𝜇4A′ are reflection of their difference in the dominant spin multiplicities. The 𝜃-dependent behaviors of 𝜇3A′𝑥 and 𝜇3A′𝑧, especially their respective similarities to 𝜇5A′𝑥 and 𝜇5A′𝑧, are understandable since their intensities come from 𝜇5A′𝑥 and 𝜇5A′𝑧 through the intensity borrowing mechanism. As shown in Table 1, the ratio of the TDMs of 4A’ and 5A’ at the equilibrium nuclear configuration is 𝜇( Π0 + ):𝜇(1Π1) = 0.105:0.104≅1:1 for ICN. On the other hand, for the CH3I molecule, the 3𝑄0 + TDM 3

is 𝜇(3𝑄0 + ) = 0.1825 a.u. and is much larger than the 1𝑄1 TDM 𝜇(1𝑄1) = 0.0861 a.u.58 Therefore, we next consider the large difference between 𝜇(3Π0 + ) and 𝜇(3𝑄0 + ) and relate this difference to the electrical properties of ICN and CH3I. Since both ICN and CH3I contain strong SOI, significant mixing occurs between the electronic states with the same Ω value. In the case of ICN, to the zeroth order approximation, the ground state

|𝑋0 + ⟩

and the excited state

|0 + (II)⟩

are expressed as in the following form of the unitary

transformation from the two spin-free (SF) configurations, the singlet X1Σ0+ and the triplet 3Π0 + ,

|𝑋0 + ⟩ = 𝑎|1Σ0+ ⟩ ― 𝑏|3Π0 + ⟩,

(27)

|0 + (II)⟩ = 𝑏|1Σ0+ ⟩ + 𝑎|3Π0 + ⟩.

(28)

Since the dipole moment operator is spin-independent, the TDM between eqs 27 and 28 can be approximated 17 ACS Paragon Plus Environment


Page 18 of 31

as the difference between the two dipole moment expectation values, 𝑑𝑧( Σ0+ ) and 𝑑𝑧(3Π), for the two SF 1

states appearing in the right-hand side of eqs 27 and 28, as follows,

⟨0 + (II)│𝑧│𝑋0 + ⟩ = Δ𝑑 = 𝑎𝑏(𝑑𝑧(1Σ0+ ) ― 𝑑𝑧(3Π)).

(29)

As shown in Figure 3, H-F type TDMs calculated by the cont-SOCI methods and the approximate expression Δ 𝑑 given in eq 29 exhibit a similar behavior. In this way, the TDM is proportional to the expectation value of the difference 𝑑𝑧( Σ0+ ) ― 𝑑𝑧(3Π). The 𝜇(3𝑄0 + ) of CH3I is similarly approximated by the difference between 1

the dipole moment of the SF 3Q0+ state and that of the SF 𝑋𝐴1 state. The products 𝑎𝑏 in eq 29 of ICN and CH3I have approximately the same magnitudes 0.0877 and 0.0784, respectively, and their R-dependence is also very similar. Therefore, the difference in the magnitudes of their TDMs between ICN and CH3I is due to the difference in the dipole moment difference between the SF ground singlet and the excited triplet states. Now, the CH3 group has an electron donating property while the I atom has an electron accepting property. However, both the CN group and I atom have an electron accepting property. Therefore, the dipole moment of the SF ground state is greater for CH3I than for ICN. On the other hand, since both the 3𝑄0 + and 3Π0 + states have covalent properties due to triplet, the dipole moments of the excited triplet states of CH3I and ICN are close to zero. From the above discussion, it is found that 𝜇(3Π0 + ) of ICN is smaller than 𝜇(3𝑄0 + ) of CH3I. The TDM of 1Π1(1𝑄1) is proportional to ⟨𝜎 ∗ │𝑥│𝑛𝑥⟩ as in eq 25. Since the 𝜎 ∗ and 𝑛𝑥 orbitals are the antibonding σ * orbital of I-C and the nonbonding orbital on I, they have no notable difference between ICN and CH3I. In this way, the difference in electric properties of CH3 and CN groups makes the difference in the magnitude of parallel TDMs 𝜇(3𝑄0 + ) and 𝜇(3Π0 + ).

3.2 Discussion. First, we examine the consistency of considering the TDM component 𝜇5A′𝑧 from the viewpoint of the product distributions observed in the experimental studies by Black et al. and Hall et al. Black measured the B, C, and E fractions out of the A to E fractions defined by Dixon.59 Defining the transition moment vector 𝝁, the dissociation velocity vector 𝒗, and the rotational angular momentum vector J, the A, B, C, D, and E fractions are respectively proportional to the product yields satisfying the following vector correlation relations; A: 𝝁 ∥ 𝒗 ∥ 𝑱, B: 𝝁 ∥ 𝒗 ⊥ 𝑱, C:𝝁 ∥ 𝑱 ⊥ 𝒗, D:𝝁 ⊥ 𝒗 ∥ 𝑱, and E: 𝝁 ⊥ 𝒗 ⊥ 𝑱. If the dissociation axis is defined as the z axis and the molecular plane is defined as the xz plane, vectors v and J are parallel to the z and y axes, respectively. Therefore, if photodissociation proceeds maintaining the dissociation axis immediately after the excitation (axial recoil approximation), the molecule excited by the 𝜇𝑧, 𝜇𝑦, and 𝜇𝑥 components should contribute to the B, C, and E fractions, respectively. The A and D fractions are terms representing deviation from the axial recoil approximation, for example, by Coriolis interaction.8 The ratio of our theoretical partial cross section at λ=249 nm, which was used by Black et al.2,7 is given as follows, 𝜎3𝐴′𝑧:𝜎3𝐴′𝑥:𝜎2𝐴′′𝑦:𝜎4𝐴′𝑧:𝜎5𝐴′𝑧:𝜎5𝐴′𝑥:𝜎4𝐴′′𝑦 = 0.062:0.076:0.061:0.32:0.18:0.26:0.24. 18 ACS Paragon Plus Environment

(30)

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Since the band maximum of the theoretical absorption cross section in Figure 6a underestimates the experimental results of Leone et al. by 600 cm-1 (Table 2), the values in eq 30 were taken at λ=252.8 nm in Figure 6a. The contributions of 3A’ and 2A’’ are relatively very weak as eq 30 shows. This is so even at λ=300 nm and longer wavelengths, as shown in Figure 6b and discussed below. Therefore, we ignore the contributions of 3A’ and 2A’’ in this section and argue that the sum of the experimental partial cross sections of the remaining excited states is normalized to unity as follows: 𝜎4A′𝑧 + 𝜎5A′𝑧 + 𝜎5A′𝑥 + 𝜎4A′′𝑦 = 1.

(31)

According to the overall rotational distribution given in Figure 10 of ref 7, the ratio of the perpendicular component to the parallel component was 38:62. By using the axial recoil approximation and the theoretically derived relation, 𝜎5A′𝑥 ≈ 𝜎4A′′𝑦, as in eq 30, the following values are obtained for the experimental perpendicular components. 𝜎5A′𝑥 = 𝜎4A′′𝑦 = 0.19.

(32)

Then, we obtain the following relation for the parallel component cross sections including 𝜎5A′𝑧 found in this study. 𝜎4A′𝑧 + 𝜎5A′𝑧 = 0.62.

(33)

To further divide the absorption cross section into these two parallel components, it is necessary to consider the nonadiabatic transition. According to both Black et al.2 and Leone et al.,53 the overall quantum yield of the I* channel 〈ΦI ∗ 〉 was 0.43 at λ=249 nm. Moreover, the perpendicular component of the I* channel was 10%, and the parallel component was 90% from Figure 8 of ref 7. From the above discussion, the following simultaneous equations are obtained for each of the perpendicular and parallel components to determine the nonadiabatic transition probability 𝑝LZ:

(1 ― 𝑝LZ) × 𝜎5A′𝑥 = 0.43 × 0.1, (1 ― 𝑝LZ) × 𝜎5A′𝑧 + 𝑝LZ𝜎4A′𝑧 = 0.43 × 0.9.

(34) (35)

Substituting 𝜎5A′𝑥 = 0.19 from eq 32 into eq 34 yields 𝑝LZ = 0.77. Then, by using eq 35, we obtain the following values from the experimental results by Black et al., 𝜎4A′𝑧≅0.45, 𝜎5A′𝑧≅0.17.

(36)

Compared with these experimental results in eq 36, our theoretical value in eq 30 for the parallel component 𝜎4A′𝑧 was slightly underestimated but our theoretical calculation is consistent with the previous experimental results and the contribution of the 5A’z component is 15~20%. Next, we estimate the contribution of 5A’z by using the experimental values of the A to E fractions given by Hall et al. in their Table VI of ref 8 for 248 nm. In the following discussion, we used 〈ΦI ∗ 〉 = 0.43 observed by Black et al.2 and Leone et al.,53 because the difference between λ=249 nm and λ=248 nm is negligibly small as shown in Figure 6a. Furthermore, the relatively small A and D fractions in Table VI of ref 8 due to the Coriolis interaction were ignored. According to the value of the B fraction of the I channel there, and the rotational distribution of the I channel in Figure 9 of ref 7, the parallel component of the I channel can be estimated to be 49% and the perpendicular component can be estimated to be 51%. Furthermore, according to the value of the B fraction of the I* channel in Table VI of ref 8 and the rotational distribution of the I* channel in Figure 8 of ref 7, the parallel component of the I* channel can be estimated to be 85% and the 19 ACS Paragon Plus Environment


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perpendicular component can be estimated to be 15%. Therefore, the sum of the perpendicular components is given as (37) 𝜎5A′𝑥 + 𝜎4A′′𝑦 = 0.57 × 0.51 + 0.43 × 0.15 = 0.36. Furthermore, assuming the axial recoil approximation, we divide the perpendicular component and obtain the following values: 𝜎5A′𝑥 = 𝜎4A′′𝑦 = 0.18.

(38)

Assuming that the nonadiabatic transition probability is 𝑝LZ, the following two equations hold, as in the above analysis for Black et al.

(1 ― 𝑝LZ) × 𝜎5A′𝑥 = 0.43 × 0.15, (1 ― 𝑝LZ) × 𝜎5A′𝑧 + 𝑝LZ𝜎4A′𝑧 = 0.43 × 0.85.

(39) (40)

From eqs 38 and 39, 𝑝LZ = 0.64 is obtained. Substituting this into eqs 40 and the following 41, 𝑝LZ × 𝜎5A′𝑧 + (1 ― 𝑝LZ)𝜎4A′𝑧 = 0.57 × 0.49,

(41)

we obtain the following values. 𝜎4A′𝑧≅0.47, 𝜎5A′𝑧≅0.17.

(42)

In this way, the partial cross sections of 4A’ and 5A’ estimated from the experimental results of Black et al.2,7 are in reasonable agreement with those of Hall et al.8 North et al. further studied the temperature dependence of the B, C, and E fractions and ΦI ∗ (𝑁) at λ=262 nm.60 According to their work, the difference between 296 K and 550 K in the B, C and E fractions of the I and I* channels was quite small, but ΦI ∗ (𝑁) increased from 296 K to 550 K for the intermediate N values. The ratios of our theoretical partial cross sections at λ=262 nm for 296 K and 500 K were given as follows. 𝜎4A′𝑧:𝜎5A′𝑧:𝜎5A′𝑥:𝜎4A′′𝑦 = 0.52:0.14:0.15:0.15 at 296 K,

(43)

𝜎4A′𝑧:𝜎5A′𝑧:𝜎5A′𝑥:𝜎4A′′𝑦 = 0.48:0.18:0.15:0.15 at 550 K.

(44)

Note that the above data were actually taken at λ=266.2 nm, for the same reason as before. As is easily supposed from Figures 4 and 5 by replacing the respective vibrational wavefunction from the ground state to the first excited state, as temperature increases from 296 K to 550 K, the relative partial cross section of 𝜎5A′𝑧 increases by 4% while that of 𝜎4A′𝑧 decreases by the same amount, and those of 𝜎5A′𝑥 and 𝜎4A′′𝑦 remain the same. Although the need for a more detailed comparison between experiment and theory will necessitate our future dynamics study, the above results, especially the temperature independent behavior of 𝜎4A′𝑧 + 𝜎5A′𝑧, are consistent with the experimental result by North et al.60 From the above discussion, it is possible to justify the importance of the parallel component in the transition moment of 5A’ in a consistent manner with the previous representative experimental results. To the best of our knowledge, no one has ever considered the importance of the parallel component cross sections 𝜎5A′𝑧 in the history of ICN investigations over 30 years. El-Sayed et al. reported positive anisotropy parameters at λ=304 nm (32900 cm-1).61 Hall et al. also reported positive anisotropy parameters at λ=308 nm (32500 cm-1).8,62 Furthermore, since the transition to 3Π1, which contributes to the perpendicular transition, was predominant at λ=304 and 308 nm in the past theoretical absorption cross section18, they pointed out problems in the old TDMs by AYM from the discrepancies between theory and experiment. Figure 6b shows the calculated result of the partial absorption cross sections of 4A’z, 20 ACS Paragon Plus Environment

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3A’z, 3A’x, and 2A’’y from 32000 to 35000 cm-1 with the current TDMs. Thus, the parallel contribution from 4A’z remains dominant even in these low energy regions at λ=304 and 308 nm. A more interesting finding now is that the parallel transition of 3A’z shows a greater contribution than the perpendicular transition of 3A’x and 2A’’y. The importance of hot-band, and non-Condon effects of the 3A’ transition dipole moments has been pointed out by several workers, including Hall et al.62 In this work, we have clarified that the 3A’ intensities are borrowed from 1Π1 and not from the vibronic interaction with 3Π0+. The above results support the experimental results showing that the anisotropy parameter is positive even at λ=304 and 308 nm. In our recent study,23 we derived semiclassical expressions for the rotational fine-structure level population difference function f(N) of the photofragment CN. Especially, f(N) with the I* channel being the closed channel was given by the following formula. 𝑓(𝑁) =

𝑃(𝐹1) ― 𝑃(𝐹2) 𝑃(𝐹1) + 𝑃(𝐹2)

= 2𝑴3A′ ∙ 𝑴4A′cos [𝜙3A′(𝑅max, 𝑅0) ― 𝜙4A′(𝑅max, 𝑅0) + 𝜙𝑆] 𝜎I.

(45)

Here, 𝑃(𝐹1) and 𝑃(𝐹2) are the F1 and F2 level populations, respectively, and 𝑴3A′ and 𝑴4A′ are the integrated transition dipole moment vectors for the respective electronic states n=3A’ and 4A’ defined by eq 4. Note that 𝑴𝑛 defined above is identical to 𝑡𝑛 in eq 69 of ref 23. In eq 45, 𝜙3A′(𝑅max, 𝑅0) and 𝜙4A′(𝑅max, 𝑅0) are the phases on each of the 3A’ PES and 4A’ PES from the photoexcitation point 𝑅0 to the nonadiabatic transition point, 𝑅max = 13 bohr, due to the angular momentum recoupling.23 𝜙𝑆 is the Stokes phase, and 𝜎I is the cross section of the I channel. f(N) is proportional to the quantity 𝑀3𝐴′𝑥𝑀4𝐴′𝑥 + 𝑀3𝐴′𝑧𝑀4𝐴′𝑧, since f(N) was evaluated from the total cross section obtained by integrating the angular distribution of the products over the solid angle. However, since the transition moment of 4A’ has a nonzero value only for the z component (see Figure 6a), the coefficient of eq 45 includes only the product of the z component 𝑀3A′𝑧𝑀4A′𝑧. Since the transition moment of the z component of 3A’ is 0 when ICN has a linear structure, the combination of the present and previous23 studies suggests that the non-Condon effect is of essential importance for f(N). In other words, the non-Condon effects of 3A’z caused f(N) to have nonzero values. In addition, we also derived semiclassical expressions for the orientation parameter of the rotational angular momentum of CN. The orientation parameter 𝑂(𝐽) is defined as the following formula.20 𝑂

(𝐽)

=

𝐽𝑍

〈〉 |𝐽|

=

∑ 𝑀SF 𝑁

𝑀SF 𝐽

SF. 𝜌(𝐽) 𝑀SF 𝐽 𝑀𝐽

𝐽(𝐽 + 1)

(46)

Here, 𝐽𝑍 ≡ 𝑀SF 𝐽 is the SF Z component of the total angular momentum 𝑱 = 𝑵 + 𝑺 of CN, where S is the spin SF SF is the relative population observed in the level 𝑀 angular momentum of CN, and 𝜌(𝐽) 𝑀SF 𝐽 and is normalized 𝐽 𝑀𝐽 (𝐽)SF SF ∑ as SF𝜌𝑀𝐽 𝑀𝐽 = 1. When the angle formed by the J and Z axes is denoted by 𝜃𝐽, the orientation stands for

𝑀𝐽

cos 𝜃𝐽 and takes a value from 1 to -1. As is apparent from the expression for the first-rank tensor derived from 𝜺, linearly polarized light has a real vector 𝜺 in eq 5 and gives only (𝜺 × 𝜺)𝑞 = 0, yielding no orientation phenomena, but circularly polarized light with a complex electric field vector in eq 6 can have nonzero components for (𝜺 ∗ × 𝜺)𝑞 and thus causes the orientation phenomena. Note that Zare et al. defined the orientation parameter 𝐶(𝑁) as the intensity difference between the two cases of using clockwise circularly polarized light and counterclockwise polarized light for the excitation light. They pointed out that the orientation 21 ACS Paragon Plus Environment


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parameter 𝐶(𝑁) thus defined correlates well with the N-dependent quantum yield of the I* channel ΦI ∗ (𝑁).5 According to Figure 4 of ref 5, the orientation parameter 𝐶(𝑁) becomes 0 when ΦI ∗ (𝑁) = 0.5, and the sign of 𝐶(𝑁) is reversed in the cases of ΦI ∗ (𝑁) > 0.5 and ΦI ∗ (𝑁) < 0.5.5 Here, 𝐶(𝑁) defined by Zare et al. is identical to the 𝑂(𝑁) formulated by Beswick et al. in eq 46. Although the behavior of 𝐶(𝑁) was roughly reproduced by the theoretical calculation by Beswick et al.,20 its interpretation was rather difficult. Below, we explain its behavior with our semiclassical expression. To formulate the transition amplitude 𝑡𝑛 to an electronic state n with circularly polarized excitation ± light, we first transform the electric field polarization vector in the SF coordinate system, 𝜺SF in eq 3, to that ± in the BF coordinate system, 𝜺BF , using the Euler angles (𝜙SF, 𝜃SF,𝛾SF) defining the molecular direction relative ± to the SF coordinate system. Then, the x, y and z components of 𝜺BF are derived as follows,

± 𝜺BF

[

]

cos 𝜃SFcos 𝛾SF ± 𝑖sin 𝛾SF ― cos 𝜃SFsin 𝛾SF ± 𝑖cos 𝛾SF . = 2 sin 𝜃SF 1

(47)

Here, the common phase factor ∓ 𝑒 ± 𝑖𝜙SF has been removed. Then, with eq 3, it is easy to have the transition ± amplitude of 4A’ denoted by 𝑡4A ′ by taking the scaler product of the transition moment vector 𝝁4A′ and the ± above 𝜺BF . In the same way, we have the following transition amplitude expressions for 4A’ and 5A’, ± 𝑡4A ′ =

1

(48)

𝜇4A′𝑧sin 𝜃SF, 2

± 𝑡5A ′ = Re[𝑡5A′] ± 𝑖 ∙ Im[𝑡5A′].

(49)

Here, the real part of the 5A’ transition amplitude Re[𝑡5A′] and imaginary part of the 5A’ transition amplitude Im[𝑡5A′] are given as follows: Re[𝑡5A′] =

1

1 𝜇5A′𝑥cos 𝜃SFcos 𝛾SF + 𝜇5A′𝑧sin 𝜃SF, 2 2

Im[𝑡5A′] =

(50)

1

(51)

𝜇5A′𝑥sin 𝛾SF. 2

According to eqs C5 through C8 in ref 23, which were obtained by semiclassical connection formula, the orientation parameter is given as follows: 𝐶(𝑁) = 2 𝑝LZ(𝜃CI)(1 ― 𝑝LZ(𝜃CI))𝑀5A′𝑥𝑀4A′𝑧sin [𝜙4A′(𝑅CI, 𝑅0) ― 𝜙5A′(𝑅CI, 𝑅0) + 𝜙𝑆 ― 𝜎𝐿𝑆 0 ]∙

𝜎I(𝑁) ― 𝜎I ∗ (𝑁) 𝜎I(𝑁) + 𝜎I ∗ (𝑁)

(52)

.

SF = 1. Note that the normalization of the above orientation parameter has been changed to satisfy ∑𝑀SF𝜌(𝑁) 𝑀SF 𝑁 𝑀𝑁

𝑁

Since we have derived the above eq. 52 by assuming a positive nonadiabatic coupling term (NACT) in the reduced scattering matrix 𝑂LZ in eq B5 of ref 22, the signs of the electronic TDMs, 𝜇4A′𝑧 and 𝜇5A′𝑥, must be chosen in a consistent manner.63 We have made sure this point by fixing the phases of the 4A’ and 5A’ electronic CI wavefunctions in such a way that these TDMs are both positive at the FC region. With keeping the phase continuity of the CI wavefunctions, the NACT between these states turned out a positive value at the avoided crossing points. In eq. 52, 𝑝LZ(𝜃CI) is the nonadiabatic transition probability of the trajectory passing the avoided crossing seam with the bending angle 𝜃CI. 𝜙4A′(𝑅CI, 𝑅0) and 𝜙5A′(𝑅CI, 𝑅0) are the phases on each of 4A’ PES and 5A’ PES from the excitation point 𝑅0 to the nonadiabatic transition point 𝑅CI due to the conical 22 ACS Paragon Plus Environment

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intersection. 𝜎𝐿𝑆 0 is the additional phase due to the nonadiabatic transition, and 𝜎I ∗ is the cross section of the I* channel. Since the x component of the 4A’ transition moment is 0, 𝐶(𝑁) depends only on the product of 𝑀5A′𝑥𝑀4A′𝑧 as in eq 52. At first, we consider eq 52 qualitatively as follows. In the actual calculations, 𝑝LZ(𝜃CI) had values ranging 𝑝LZ(𝜃CI)(1 ― 𝑝LZ(𝜃CI)) had values always close to 1/2 for

from 0.4 to 0.6 for 20 < 𝑁 < 50, yet the term

these N, since it is the geometric mean of 𝑝LZ(𝜃CI) and 1 ― 𝑝LZ(𝜃CI). Our evaluated phase difference 𝜙4A′

(𝑅CI, 𝑅0) ― 𝜙5A′(𝑅CI, 𝑅0) showed little N dependence, which is a point originally supposed experimentally by Hall et al.10 and can be explained by almost the same θ-dependences of the 4A’ and 5A’ PESs in the FC region as seen in Figure 2. Therefore, the overall behavior of the 𝐶(𝑁) parameter is controlled by the last

𝜎I(𝑁) ― 𝜎I ∗ (𝑁) 𝜎I(𝑁) + 𝜎I ∗ (𝑁)

term. With the definition of the N-dependent quantum yield for the I* channel given in eq 53, ΦI ∗ (𝑁) =

𝜎I ∗ (𝑁) 𝜎I(𝑁) + 𝜎I ∗ (𝑁)

,

(53)

the following relation is satisfied: 𝜎I(𝑁) ― 𝜎I ∗ (𝑁) 𝜎I(𝑁) + 𝜎I ∗ (𝑁)

= (1 ― 2ΦI ∗ (𝑁)).

(54)

Substituting eq 54 into eq 52, we have the following relation. 𝐶(𝑁) ∝ (1 ― 2ΦI ∗ (𝑁)).

(55)

In this way, with the semiclassical expressions previously derived, we could qualitatively explain the behavior of the orientation parameter by Zare et al.5 Next, as a preliminary SC-IVR calculation, we evaluate 𝐶(𝑁) of semiclassical eq 52 with the phase integrals obtained by classical trajectory calculation using the Meyer-Miller classical analog Hamiltonian.51 The initial conditions for the trajectory calculation were determined by regarding the stretching and bending vibrational motions as the classical harmonic oscillators as in Morokuma et al.16 We then calculate the 𝜙4A′

(𝑅CI, 𝑅0) and 𝜙5A′(𝑅CI, 𝑅0) along each trajectory by the classical action integral, 𝑅CI

𝜙𝑖(𝑅CI, 𝑅0) =

∫𝐿 (𝑡)𝑑𝑡. 𝑖

(56)

𝑅0

Here, 𝐿𝑖(𝑡) is the classical Lagrangian. Figure 8 shows the C(N) calculated by the approximate semiclassical approach and compared with the experimental result by Zare et al.5 Thus, we could reproduce the N-dependence of the orientation parameter C(N) much better than Beswick et al.20 and Carrington et al.21 This successful simulation of C(N) suggests that the 𝑀4A′𝑧 and 𝑀5A′𝑥 values involved as proportional constants in eq. 52 have been evaluated in a reasonable accuracy from the electronic TDMs 𝜇4A′𝑧 and 𝜇5A′𝑥. For N higher than around 25, our theoretical C(N) values are shifted to the higher N side by 5 to 10 units, relative to the experimental values. This shift is probably caused by the underestimation of the dissociation energy, which then results in the overestimation of the fragment rotational energies.



Page 24 of 31

Figure 8. The N-dependence of the theoretical (black square) and the experimental5 (red circle) orientation parameter C(N). We also simulate the dependence of C(N) on the vibrational quantum number v of CN fragment as reported by Hall et al.9 As clarified in the previous AYM potential,16 the excited state PESs have only a weak coupling between the r- and R-dependent terms. Therefore, the CN stretching vibration, represented by the r coordinate, plays only a spectator role in the photodissociation dynamics. This is simply because, in the A-band photodissociation, the CN triple bond is not disturbed significantly both in the photoexcitation and subsequent dissociation processes. For example, the wavenumber of the CN stretching vibration is 𝜔CN = 2068.7 cm ―1 in the CN fragment,64 and is 𝜔1 = 2158 cm ―1 in the ICN molecule,52 and the vibration can be treated as an independent uncoupled mode. In such a case, the fragment vibrational populations can be treated as the ‘elastic case’ in the terminology of ref 56, that is, the vibrational population of the CN fragment is simply determined by the Franck-Condon factor. Since the CN stretching frequency is quite large, the vibrational level in the electronic ground state ICN can be assumed in v = 0. Following the photoexcitation, the CN vibration can have only small populations for v = 1 and 2. With the energy conservation, the initial conditions for classical trajectories to start on the excited PESs were generated by subtracting the CN vibrational excitation energy 𝐸vib = 𝑣𝜔CN from the total available energy, that makes the 𝑅0 value in eq. 56 longer, and the phase difference 𝜙4A′(𝑅CI, 𝑅0) ― 𝜙5A′(𝑅CI, 𝑅0) smaller as v increases. Figure 9 shows the v-dependence of the calculated orientation parameters 𝐶(𝑁 = 48) and 𝐶(𝑁 = 53) and the experimental results for v = 0 and v = 1 reported by Hall et al. and for v = 2 reported by Zare et al.6


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Figure 9. The v-dependence of the theoretical (black) and the experimental10 (red) orientation parameter C(N = 48) (square) and C(N = 53) (circle). Since the orientation parameter 𝐶(𝑁) obtained from P branch and R branch measurements have opposite signs,5,10 we have employed the sign convention for R branch in Figure 8 and for P branch in Figure 9, respectively to facilitate comparison with the previous experimental data. We reproduced the general experimental v-dependence of the C(N). As explained by Hall et al.10 vibrationally excited CN fragments have the reduced dissociative kinetic energy, then the phase difference 𝜙4A′(𝑅CI, 𝑅0) ― 𝜙5A′(𝑅CI, 𝑅0) decreases with v. Since our calculation yielded the 𝜙4A′(𝑅CI, 𝑅0) ― 𝜙5A′(𝑅CI, 𝑅0) values slightly larger than 3π for v = 1, the sign change of C(N) occurs between v=1 and 2. Our C(N) results for v=2 apparently seem to be too small and outside the experimental error bar in Figure 8 of reference 10. A part of the reason may be attributed to some threshold effects which were not included in the present semiclassical treatment.6

4. CONCLUSIONS In this paper, TDMs and PESs of ICN were calculated by using MCP and corresponding basis functions. As a result, it was clarified that the most popular H-F type formula with the cont-SOCI method yielded too small TDM 𝜇4A′𝑧 and that the calculated values of TDMs with the response type formula with the SOCI wavefunctions were insensitive to small errors that might be involved in the wavefunctions. With the SCIVR method, we calculated the A-band absorption cross section including the non-Condon effects. Then, we showed theoretically that the z component contribution of 1Π1 (5A’) TDM, 𝜇5A′𝑧, to the total absorption cross 25 ACS Paragon Plus Environment


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section had a similar magnitude to the x and y components due to the non-Condon effect. Furthermore, by analyzing the experimental results of Black et al.2,7 and Hall et al.,8 it was shown that at 248 nm, some 15-20 % contribution to the total A-band absorption cross section might be originated from the z component of 5A’ from the experimental results. The strong θ-dependence of the z component of 5A’ TDM 𝜇5A′𝑧 and θ-independence of the x component of 4A’ TDM, thus 𝜇4A′𝑥 = 0, are caused by the spin multiplicity difference. The significant difference in the TDMs of the 3Π0+ of ICN and 3Q0+ of CH3I was shown to be a reflection of the chemical property difference between the electron donating CH3 group and the electron accepting I atom. According to our previous study, the quantum interference effects between the wavepackets generated coherently on the 3A’ and 4A’ PESs appear in the rotational fine-structure level population difference function f(N). Since the 4A’ TDM has a nonzero value only in the z component 𝜇4A′𝑧, it was emphasized that the presence of the z component of 3A’ TDM 𝜇3A′𝑧, in other words, the non-Condon effect is essential for f(N). Since 𝜇4A′𝑥 is zero and has no θ-dependence, the orientation parameter of the rotational angular momentum of CN 𝐶(𝑁) is expressed as the product of 𝑀4A′𝑧 and 𝑀5A′𝑥. We could reproduce essential features of the N- and vdependences of the 𝐶(𝑁) using semiclassical connection formula with the phase integrals obtained by preliminary classical trajectory calculations. Since the 5A’ and 4A’ TDMs have nonzero values in the common z component due to the non-Condon effect for 𝜇5A′𝑧, the quantum interference effect between wavepackets generated coherently on the 4A’ and 5A’ PESs can appear not only in 𝐶(𝑁) but also in some scalar quantities such as the rotational distribution and the anisotropy parameter.63 In this way, ICN photodissociation continues to be a very important system where the quantum interference effects due to coherent excitations onto the multiple PESs appears conspicuously.

ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI Grant Number16K05668.　The computations were partly performed using the computer facilities at the Research Center for Computational Science, Okazaki National Institutes.


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TOC Graphic


(5A') and 3Î 1

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