Vibronic Coupling in the First Five Electronic States of

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A: Spectroscopy, Molecular Structure, and Quantum Chemistry

Vibronic Coupling in the First Five Electronic States of Dicyanodiacetylene Radical Cation Karunamoy Rajak, Arpita Ghosh, and Susanta Mahapatra J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b08171 • Publication Date (Web): 08 Oct 2018 Downloaded from http://pubs.acs.org on October 11, 2018

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The Journal of Physical Chemistry

Vibronic Coupling in the First Five Electronic States of Dicyanodiacetylene Radical Cation Karunamoy Rajak, Arpita Ghosh and S. Mahapatra∗ School of Chemistry, University of Hyderabad, Hyderabad 500 046, India E-mail: 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

Abstract e 2 Πg - A e2 Πu We examine vibronic coupling in the first five electronic states (X · + e 2 Σ+ e2 + e 2 e2 B g - C Σu - D Πg - E Πu ) of dicyanodiacetylene radical cation (C6 N2 ) in this

article. Prompted by the prediction of its existence in the astrophysical environment, the vibronic band structure of these electronic states of C6 N·2 + has been probed in spectroscopic measurements in laboratory by various groups. Inspired by numerous experimental data, we undertook the task of investigating topographical details of electronic potential energy surfaces, their coupling mechanism and nuclear dynamics on them. The degenarate Π electronic states of this radical are prone to Renner-Teller instability, and in addition symmetry allowed Σ - Π and Π - Π vibronic coupling is expected to play crucial role in the detailed vibronic structure of each of the above electronic states. A vibronic coupling model is developed here and first principles nuclear dynamics study is carried out employing quantum mechanical methods. The vibronic band structure thus calculated is compared with experimental results and the progressions are identified and assigned. The nonradiative internal conversion dynamics among electronic states is also examined and discussed in relation to the various coupling of electronic states.

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1

Introduction

The spectroscopy of cyanopolyacetylene radical cations is a subject of intense research because of their abundance in the interstellar medium (ISM) as identified by radioastronomy. 1 Cyanopolyacetylenes, HC2n+1 N upto n=5, have been detected in the interstellar medium (ISM), 2 and some of them in planetary atmosphere 3 with the aid of terrestrial microwave studies. 4,5 In recent years numerous cyanopolyyne type of molecules, H(CC)n CN have been studied by Fourier transform microwave spectroscopy. 6,7 The abundance of centrosymmetric dicyano derivatives, NC-(C≡C)m -CN also predicted in the ISM. 1,2 They are synthesised in the laboratory (m=1,2), 8,9 however, because of the lack of dipole moment their detection by microwave spectroscopy was difficult. The dicyanopolyacetylenes are also of considerable interest due to the linear arrangement of conjugated triple bonds which results in extended π orbitals and leads to delocalisation of the electrons along the whole molecule. 10,11 For these molecules spectroscopic informations are obtained from high resolution measurements of their vibrational or electronic transitions in the gas phase. 12,13 The dicyanodiacetylene radical cation, C6 N·2 + , is an object of experimental research since 1967, 14 and is a subject of the present rigorous theoretical research.

In 1967, Miller and Lemmon 14 were the first to report the vibrational spectrum of dicyanodiacetylene, N ≡ C – C ≡ C – C ≡ C – C ≡ N. Gas phase emission spectrum of C6 N·2 + was reported by Klöster-Jensen et al. 15 and the transition was identified with the aid of the He I photoelectron spectroscopy measurements. 15 Frequencies of four fundamentals of symmetric vibration of the cationic ground state was reported. 15 Kolos 16 reported the result of photolysis of dicyanodiacetylene in argon matrix together with some prominent infrared absorption peaks of the parent molecule. They have calculated frequencies of fundamental vibrations at the DFT BP86/DN and MP2/6-31G* level of theory and compared with the experimental recording of Miller and Lemon. 14 Gupta et al. 17 carried out quantum chemical calculations based on HF(MP2) and DFT methods using larger basis sets and 3



reported optimized geometry, infrared and Raman spectra and assigned the fundamental vibrations of dicyanodiacetylene. In a latter study Forney and co-workers 18 reported the e2 Πu ← X e 2 Πg absorption spectra of NC6 N.+ in 5 K neon matrices. The band origin for the A transition was located at 659 nm. These authors also recorded the absorption spectra of NC6 N.+ in the gas phase and reported the band origin at 656 nm. 19 Over the past years several research groups have studied the spectroscopy of C6 N·2 + . 17–22

Apart from its astrophysical significance, experimental measurements (vide supra) revealed that the symmetric linear radical cation of dicyanodiacetylene possesses many interesting features that are worthwhile to investigate with a rigorous theoretical model. This system has four degenarate Π and two non-degenarate Σ states within ∼ 5 eV energy range. The degenarate Π states are prone to Renner-Teller (RT) 23 splitting upon distortion along bending vibrational modes. The RT split component states may undergo symmetry allowed pseudo-RT (PRT) type of coupling with their neighboring states. Therefore, it appears to be interesting to investigate as to what extent the effect of various coupling of states manifests in the recorded electronic band structure of C6 N·2 + . Furthermore assignment of experimental data requires a thorough and careful theoretical analysis. With these in mind we here set out to study the detailed electronic structure of the first six electronic states of C6 N·2 + and nuclear dynamics on them. The electronic structure calculations are carried out employing state-of-the art quantum chemistry methods. Multimode vibronic coupling theory and quantum mechanical methods are used to carry out the nuclear dynamics calculations. Theoretically calculated vibronic band structures of electronic states are compared with the experimental data and assigned. The effect of various coupling of electronic states on the vibronic band structure is examined and discussed at length.

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2 2.1

Theory and Methodology The Vibronic Hamiltonian

e2 e2 e2 + e 2 Πg - A e2 Πu - B e 2 Σ+ The vibronic interactions in the coupled X g - C Σ u - D Πg - E Πu electronic states of C6 N·2 + are described by constructing a model Hamiltonian in a diabatic electronic basis. Normal coordinates (dimensionless) of the vibrational modes of electronic 24,25 ground state (1 Σ+ Standard vibronic coupling g ) of C6 N2 is used in the construction.

theory 24 is used and interactions within and among electronic states are described using elementary symmetry rules. The C6 N2 molecule is linear (see later in the text) and possesses D∞h point group symmetry at the equilibrium minimum of its electronic ground state. The 19 vibrational modes of this molecule transform according to,

Γvib = 4σg+ ⊕ 3σu+ ⊕ 3πg ⊕ 3πu ,

(1)

irreducible representations (IREPs) of D∞h point group. The symmetry rule, Γj ⊗ Γk ⊃ Γvib (j and k are electronic state indices) is used to establish the coupling mechanism. The symmetrized direct product of the given electronic terms in the D∞h point group reads Πg ⊗ Πg = δg + σg+ = Πu ⊗ Πu ,

(2a)

Πg ⊗ Σ + g = πg ,

(2b)

Πg ⊗ Σ + u = πu ,

(2c)

Πu ⊗ Σ + g = πu ,

(2d)

Πu ⊗ Σ + u = πg ,

(2e)

+ + Σ+ g ⊗ Σu = σu ,

(2f)

Πg ⊗ Πu = δu + σu+ .

(2g)

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The IREPs of electronic states and vibrational modes are given by the upper and lower case symbols, respectively. The condon active totally symmetric σg+ vibrational modes can not split the degeneracy of Π electronic states. However, first-order splitting of RT degeneracy of the Π state can be caused by the vibrational modes of δg symmetry. Since C6 N2 does not contain the latter modes, first-order RT coupling between the components of Π state vanishes. However, according to symmetry rule, (πg )2 = (πu )2 ⊃ δg , which means the πg or πu modes can be RT active in the Π state in second-order. The RT split components of the Πg and Πu states can undergo PRT coupling according to Eq. (2g) given above. Considering the symmetry rules stated above the following vibronic Hamiltonian is derived,



H

=

              H0 110 +              

WXx f

WXx− f f Xy

WXx− f e Ax

0

WXx− f e B

WXx− f e C

0

0

WXx− f e Ex

WXy f

0

WXy− f e Ay

WXy− f e B

WXy− f e C

0

0

0

WAx− e e Ay

WAx− e e B

WAx− e e C

WAx− e f Dx

0

WAy e

WAy− e e B

WAy− e e C

0

WAy− e f Dy

0

WB e

WB− e C e

WB− e Dx f

WB− e Dy f

WB− e Ex e

WC e

WC− e Dx f

WC− e Dy f

WC− e Ex e

WDx f

WDx− f f Dy

WDx− f e Ex

WAx e

h.c

WDy f

0

0 WEx e

0 WXy−W f

   

e  Ey 

     0    WB−  e Ey e (3).    WC− e Ey e    0    WDy−  f e Ey    WEx− e e Ey  WEy e 0

In the above, H0 = TN + V0 , is the Hamiltonian of the electronic ground (S0 ) state of C6 N2 . The latter is the reference state and H0 is assumed to be harmonic with 1 TN = − 2

X i ∈

σg+ ,

2 ∂2 1 X ∂ ∂2 ωi 2 − ωi + , ∂Qi 2i ∈ π , π ∂Q2ix ∂Q2iy +

σu

g

(4)

u

and

V0 =

1 2

X

ωi Q2i +

+ i ∈ σg+ , σu

1 X ωi Q2ix + Q2iy . 2i ∈ π , π g

(5)

u

The quantity 110 is a 10×10 diagonal unit matrix. The diagonal elements W of the matrix Hamiltonian (cf., Eq. 3) are the diabatic potentials of the given electronic states of C6 N·2 + and off-diagonal elements represent the coupling potentials between states men6


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tioned in the suffix. A Taylor series expansion around the reference equilibrium geometry at, Q = 0, of these elements is carried out in the following way 24 X

Wjx/jy = E0j +

κji Qi +

i ∈ σg+

X

[ξij (Q2ix + Q2iy )2 ]

X

±

X

ηij (Q2ix − Q2iy ) ±

X

Wj = E0j +

X

γij Q2i +

+ i ∈ σg+ , σu

i ∈ σg+

X

X

e A, e D, e E e δij (Q4ix − Q4iy ); j ∈ X,

i ∈ πg , πu

X

κji Qi +

[γij (Q2ix + Q2iy )] +

i ∈ πg , πu

+ i ∈ σg+ , σu

i ∈ πg , πu

i ∈ πg , πu

X

γij Q2i +

[γij (Q2ix + Q2iy )] +

i ∈ πg , πu

e C e [ξij (Q2ix + Q2iy )2 ]; j ∈ B,

(7)

i ∈ πg , πu

X

Wjx−jy =

e A, e D, e E. e 2ηij (Qix Qiy ) + 2δij (Q3ix Qiy + Qix Q3iy ) ; j ∈ X,

(8)

i ∈ πg , πu

X

Wjx−kx/jy−ky =

i ∈

Wj−k =

e − A, eX e − E, e A e − D. e λj−k Qi ; j − k ∈ X i

X i ∈

Wjx−k/jy−k =

X

j

λi x/y

−k

(9)

+ σu

e − C. e λj−k Qi ; j − k ∈ B i

(10)

+ σu

e A, e D, e E e and k ∈ B, e C. e Qix/iy ; jx/y ∈ X,

(11)

i ∈ πg/u

“In the above equations the components of degenerate electronic state and vibrational mode are identified with x/y. The quantity E0 j defines the vertical ionization energy (VIE) of the 7


(6)


electronic state j.” The linear, second-order and fourth-order intrastate coupling parameters of the ith vibrational mode in the j th electronic state are denoted by κji , γij and ξij , respectively. The quadratic and quartic RT coupling parameters of the π modes within the Π electronic states are denoted by ηij and δij , respectively. The quantity λj−k represents the first-order i coupling parameter (PRT coupling) between the states j and k caused by the ith vibrational mode. The summations are carried out over normal modes of vibration of given symmetry. In the series expansion the terms applicable to x and y components of degenarate electronic states are given by + and - sign, respectively. These relative signs of various elements emerge in accordance with the symmetry invariance (with respect to symmetry operations in D∞h point group) of the diabatic electronic matrix Hamiltonian of Eq. 3. 26

2.2

Electronic structure calculations

The geometry optimization of C6 N2 in its electronic ground state is carried out by the second-order Møller-Plesset perturbation (MP2) method employing augmented polarized valence triple-zeta basis set (aug-cc-pVTZ). 27 Gaussian 09 suite of programs 28 is used for the purpose. The optimized geometry belongs to D∞h symmetry point group. Harmonic frequency (ωi ) of vibrational modes is calculated by diagonalizing the kinematic (G) and ab initio force constant (F) matrix of the reference equilibrium geometry. The results are given in Table 1 for all 19 vibrational modes. The IREP and description of the latter are also given in the table including the results available in the literature. 14 It can be seen that the present results are in good accord with the iterature data. The vertical ionization energies (VIEs) of each excited state along each vibrational mode for a range of nuclear geometries are calculated using equation-of-motion coupled-cluster singles and doubles (EOMIP-CCSD) method and aug-cc-pVTZ basis set. CFOUR suite of programs 29 is used for the calculations. The calculated ab initio energies are fit to the adiabatic form of the diabatic electronic Hamiltonian of Eq. 3 by a least squares procedure to obtain various parameters introduced in the Hamiltonian in Sec. 2.1. The set of coupling parameters derived from these fits are 8


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given in Tables 2, 3, 4 and 5. The VIEs obtained at the reference equilibrium geometry are compared with the available experimental data in Table 6. From Table 6, it can be seen e2 + e 2 Σ+ that the B g and C Σu electronic states are quasi-degenerate at the vertical configuration. Vibronic coupling between these two states is expected to be significant.

2.3

Nuclear dynamics

e A, e B, e C, e D e and E e electronic states of C6 N·2 + is calculated The vibronic spectrum of the X, by the Fermi’s golden rule. The spectral intensity is given by 2 X α ˆ i I(E) = hΨv |T |Ψ0 i δ(E − Evα + E0i ).

(12)

v

Here, |Ψi0 i represents the initial wavefunction corresponding to the vibronic ground (v=0) state of C6 N2 and |Ψαv i is the final vibronic wavefunction corresponding to the electronic states of C6 N·2 + (α). The energies of these initial and final states are given by E0i and Evα , respectively. The transition dipole operator is given by Tˆ. The vibronic ground (reference) state is expressed as

|Ψi0 i = |Φ00 i|0i.

(13)

In the above equation the electronic and vibrational components of |Ψi0 i are defined by |Φ00 i and |0i, respectively. As the reference state is assumed to be harmonic the vibrational component |0i is expressed as a direct product of harmonic oscillator functions of vibrational modes. The spectral intensity I(E), is calculated by diagonalizing the Hamiltonian matrix represented in the direct product basis of diabatic electronic state and harmonic oscillator functions. The Lanczos algorithm is used for this purpose. 24,30,31 The eigenvalues of the diagonal matrix represent the vibronic energy levels of the final state and the intensities are calculated from the Lanczos eigenvectors. 24,32,33

9



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In a time-dependent framework the golden rule equation (Eq. 12) transforms to the Fourier transform of time autocorrelation function of wave packet (WP) that evolves on the final electronic state. In this case the intensity is given by 24

I(E) ≈

X

≈

α

eiEt/¯h hΨα (t = 0)|τ † e−iHt/¯h τ |Ψα (t = 0)idt,

2Re

(14)

0

α

X

∞

Z

Z 2Re

∞

e A, e B, e C, e D, e E. e eiEt/¯h C α (t) dt, α ∈ X,

(15)

0

In the above equations, C α (t) = hΨα (t = 0)|Ψα (t)i , is the time autocorrelation function. The WP, |Ψα (t = 0)i, prepared on the αth electronic state at t=0 through vertical excitation of the initial WP, |Ψi (t = 0)i. The elements, τ α = hΦα |Tˆ|Φ0 i, of the transition dipole matrix, τ [τ † = (τ X , τ A , τ B , τ C , τ D and τ E )] are treated as constant within the generalized Condon approximation. 33 The time evolved WP is calculated by numerically solving the time-dependent Schrödinger equation, |Ψα (t)i = e−iHt/¯h |Ψα (t = 0)i. Partial spectra for initial transition to each of the electronic states of C6 N·2 + are calculated. These partial spectra are finally combined to calculate the composite vibronic spectrum of the five electronic states considered here. The multi-configuration time-dependent Hartree (MCTDH) method 34–36 is used to propagate the WP. The MCTDH wavefunction is given by 34–36

Ψ(q1 , ..., qp , t) =

n1 X j1 =1

...

np X

Aj1 ,...,jp (t)

jp =1

p Y

(k)

ϕjk (qk , t).

(16)

k=1

In the above equation, qk = (Qi , Qj , ..) represents the set of vibrational degrees of freedom (DOF) combined into a single particle (p) utilizing the multiset ansatz of MCTDH. 34–36 The MCTDH expansion coefficients are given by, Aj1 ,...,jp . The time-dependent single particle (k)

functions (SPFs) are denoted by ϕjk . The number of SPFs for the k th DOF is given by nk . The SPFs are represented in a primitive harmonic oscillator basis to solve the MCTDH 10


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equations of motion. In the present application Heidelberg MCTDH suite of programs 38 is used to carry out the time-dependent WP calculations.

3

Results and discussion

3.1

Adiabatic potential energy surface

e 2 Πg , A e2 Πu , In this section the topography of the adiabatic PESs of four degenarate (X · + e2 + e 2 Πg and E e 2 Πu ) and two non-degenerate (B e 2 Σ+ is D g and C Σu ) electronic states of C6 N2

examined. According to the symmetry selection rules given in Sec. 2.1, the degeneracy of the Π electronic states can not be lifted by the totally symmetric vibrational modes of σg+ symmetry (ν1 -ν4 ) and the same is true for the modes of σu+ symmetry, (ν5 -ν7 ). The degeneracy e A, e D e and E e electronic states, on the other hand, can be lifted by the πg (ν8 -ν10 ) of the X, and πu (ν11 -ν13 ) bending vibrational modes which are Renner Teller active in second-order.

In Figs. 1 (a-d) one dimensional potential energy curves of the electronic states of C6 N·2 + are plotted along the normal displacement coordinates of symmetric (σg+ ) vibrational mode. Each of these curves represent an one dimensional cut through the multi dimensional PESs of C6 N·2 + . Similar plots along the x component of degenerate πg and πu vibrational mode are shown in Figs. 2(a-c) and (d-e), respectively. In these plots the adiabatic electronic energies obtained from the present vibronic model and those calculated ab initio are shown by solid lines and points, respectively. It can be seen from the above plots that the ab initio energy data are in good agreement with those obtained from by the model Hamiltonian constructed in Sec. 2.1. It can be seen from Fig. 1 that the electronic ground state of the radical cation is energetically well separated from the remaining five excited states which are energetically close. The latter states undergo curve crossings (cf. panel b and c) in the near vicinity of the reference equilibrium geometry (Q=0). The e state is largest along the C≡N symmetric stretchdisplacement of the minimum of the X 11



e state undergoes mild distortion along C≡C symmetric stretch, ing vibration, ν1 , whereas A e and C e states undergo mild distortion along terminal C-C symmetric stretch, ν2 . Both B e state is largest along this mode. The D e state ν4 , and the distortion of the minimum of E undergoes mild distortion along both middle C-C, (ν3 ) and terminal C-C stretching (ν4 ) vibrational modes. The distortions mentioned above are in accordance with the first-order coupling strength of the given vibrational mode in the given electronic state (cf., Tables 2 -4).

The potential energy cuts plotted along the RT active degenarate π vibrational modes in Fig. 2 reveal very weak splitting of electronic degeneracy. It can be seen from Tables 2 -4 that the RT coupling parameters of the relevant vibrational modes of πg and πu symmetry for all four degenarate electronic states are quite small (of the order of 10−3 to 10−4 eV). Therefore, the RT coupling within the degenerate electronic states is not expected to have significant role in the dynamics.

The stationary points in the coupled manifold of electronic states are examined next. These are equilibrium minimum (Vmin ) and the energetic minimum of the seam of various conical intersections (Vcmin ) calculated within second-order coupling model and a constrained optimization using Langrange multiplier. 37 The resuls are reported in Table 7, in which the diagonal entries are Vmin and the off-diagonal ones are (Vcmin ). As discussed before, it can e state is energetically well separated also be seen from the data given in Table 7 that the X from the rest. Minimum of the intersection seam of this state with others also occurs at very high energies. Therefore, the nuclear dynamics on this state is expected to be completely e- B e and A e- C e intersections occurs ∼ 2.6 eV adiabatic. The minimum of the seam of A e state minimum. These seam minima are relatively close (by about an eV) above the A e and C e states. The latter states are quasi-degenerate at the to the minimum of both B e-D e and neighborhood of their equilibrium minimum. The minimum of the seam of both B e-D e intersections occurs ∼ 0.6 eV above both B e and C e state minimum. These intersection C

12


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e state minimum. The intersection of the minima are very close to (∼ 0.15 eV above) the D e state with others occurs at much higher energies (cf. Table 7). The D e -E e intersection E e state minimum. It can be seen from Table minimum is closest (within ∼ 0.2 eV) to the E e state with the B e and C e states are large 5 that the first-order coupling parameters of the X e state are moderate. Despite large coupling strength, these couplings are and with the E not expected to have any major effect on the dynamics because the intersection minima are located at very high energies as compared to the respective equilibrium minimum. These predictions from the electronic structure data calculated at the present level of theory ought to be validated with the dynamical outcome vis-â-vis experimental findings. The coupling e- B e - C e - D e - E e electronic manifold is expected to have some effect on the within the A dynamics as the energetics discussed above and the parameters given in Table 5 suggest. We will revert to this point later in the text.

3.2

e A, e B, e C, e D e and E e electronic Vibronic band structure of the X, states of C6 N·2 +

3.2.1

Spectrum of uncoupled state and Renner-Teller effect

e A, e B, e C, e D e and E e electronic The vibrational energy level spectrum of the uncoupled X, states of C6 N·2 + is examined first in order to demonstrate the effect of RT and PRT type of interactions on them. The RT and PRT coupling between the states is included subsequently and the final results are compared with the experiment and discussed in detail. The spectra e and A, e B e and C e and D e and E e states are shown in Figs. 3(a-c), 4(a-b) of the uncoupled X and 5(a-c), respectively. The stick line spectra presented in the above figures are calculated by the time-independent matrix diagonalization method and the corresponding envelopes are generated by convoluting each stick spectrum with a Lorentzian line shape function of 10 meV full width at the half maximum (FWHM). The numerical details of the calculations viz., number of vibrational basis functions, dimension of the Hamiltonian matrix and number

13



of Lanczos iterations for each figure mentioned above are given in Table S1 of the supporting information (SI).

e 2 Πg , A e2 Πu , In absence of any inter-mode coupling the Hamiltonian for the uncoupled X e 2 Πg and E e 2 Πu states is separable in terms of the totally symmetric (σ + ) and degenarate (πg D g and πu ) vibrational modes. Therefore, in absence of coupling with the neighboring states calculations are carried out considering the totally symmetric and degenerate vibrational modes separately, and the results from these two calculations are convoluted to generate a e and A e states composite vibronic spectrum of these degenarate states. The spectra of the X calculated with ν1 -ν4 modes of σg+ symmetry are shown in the panel a of Fig. 3. The peak spacings of ∼ 2203, ∼ 2061, ∼ 1289 and ∼ 458 cm−1 from the origin peak found in this spectrum are due to the excitation of the fundamental of ν1 , ν2 , ν3 and ν4 vibrational modes, e state. It can be seen from Table. 2 that symmetric C≡N stretching respectively in the X e state, as a result its first overtone can mode ν1 possesses large excitation strength in the X also be seen at ∼ 4406 cm−1 in Fig. 3a. The terminal C-C stretching mode ν4 is the next e band. It forms several combination peaks dominant one in the vibronic structure of the X with ν1 as can be seen from the figure. The excitation of C≡C symmetric stretch ν2 and middle C-C symmetric stretch ν3 is very weak. This can be contrasted to the similar findings e 2 Πg state spectrum of the latter, vibrational in case of diacetylene radical cation. 26 In the X mode ν2 (C≡C stretch) was found to be strongly excited. The vibrational mode ν2 on the e state spectrum of C6 N·2 + and its fundamental and other hand, is strongly excited in the A first overtone can be seen to appear at ∼ 2092 and ∼ 4184 cm−1 , respectively in Fig. 3a. e state as compared to that in the The frequency of this mode is reduced in the cationic A e2 Πu state originates mostly from neutral ground state (cf. Table 1) which confirms that the A an ionization of a bonding electron located in the C≡C moiety of C6 N2 . It can be seen that e state and it forms the mode ν4 is also excited in this state to a similar extent as in the X e state, the excitation of the ν3 vibrational mode is relatively combination peaks. Unlike X

14


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e state. The latter modes also form several stronger and that of ν1 is relatively weaker in the A combination peaks as can be seen from Fig. 3a. In contrast to the above, the vibrational e2 Πu state spectrum of diacetylene radical mode ν3 was found to be strongly excited in the A cation. 26

e and A e states calculated by including the degenerate πg and πu viThe spectra of X brational modes are plotted in panel b of Fig. 3. It can be seen from the figure that the excitation of these RT active modes is very weak in both the states. This is in accordance with the small value of RT coupling parameters (cf. Table 2) and a weak splitting of electronic degeneracy found in Fig. 2. Despite weak excitation it is intriguing to see that the e state and ν9 and ν12 in the A e state degenerate vibrational modes, ν10 , ν12 and ν13 in the X are excited in second-order. Because of weak RT coupling effect the composite spectra (cf. Fig. 3c) essentially remain similar to the symmetric mode spectra shown in panel a. When e and compared with similar spectra it is found that RT effect is relatively stronger in the X e states of diacetylene radical cation. 26 A

e and C e states are shown in panels The vibronic spectra of uncoupled non-degenerate B a and b of Fig. 4, respectively. In both these states, only the symmetric vibrational modes ν2 , ν3 and ν4 are weakly excited. The fundamental of these modes appears at ∼ 2089, ∼ e state and at ∼ 2088, ∼ 1295 and ∼ 461 cm−1 in the C e 1295 and ∼ 460 cm−1 in the B states, respectively. The excitation of the mode ν4 is somewhat stronger on both the states. e and C e states, The first overtone of this mode appears at ∼ 919 and ∼ 921 cm−1 in the B respectively.

e and E e states shown in Fig. 5 reveal progression of ν4 , The spectra of the uncoupled D e state (cf. panel a). The fundamentals of ν3 , ν1 and ν2 symmetric vibrational modes in the D these modes appear at ∼ 467, ∼ 1312, ∼ 2149 and ∼ 2202 cm−1 in that order, respectively.

15



While both the vibrational modes ν3 and ν4 have comparable excitation strength, the excitation of ν1 and ν2 is relatively weak (cf. Table 4). The vibrational mode ν4 forms an extended e state (cf. panel a). This is on par with its large coupling strength (cf. progression on the E e states. The fundamental, first and second overtone of this mode appears Table 4) on the E at ∼ 475, ∼ 950 and ∼ 1425 cm−1 , respectively, in this state. Weak excitation of ν1 , ν2 and ν3 vibrational modes at ∼ 2178, ∼ 2179 and ∼ 1374 cm−1 , respectively, can also be seen in e state. The RT effect is even weaker in the D e and E e state [cf. Fig. the spectrum of the E e and A e state [cf. Fig. 3(b)]. Very weak excitation of 5(b)] as compared to the same in the X e and E e states. two quantum of ν12 vibrational mode can be seen in the spectrum of both D Because of the weak RT effect the composite vibronic spectrum remains essentially similar to the symmetric mode spectrum of both the states (cf. panel c of Fig. 5).

The low-lying vibrational energy levels energies (relative to the origin peak) of all six electronic states and their assignments are given in Table 8 along with the available literature (experimental) data. Klöster-Jensen et al. 15 reported the fundamental of ν1 vibrational e and A e bands, respectively. These results are mode at 2100 and 1940 cm−1 found in the X in fair agreement with the present theoretical findings which locates them at ∼ 2203 and ∼ 2191 cm−1 , respectively. Agreiter et al. 41 reported the fundamental of the symmetric vibrae state at ∼ 2185.3 (ν1 ), ∼ 2093.8 (ν2 ), ∼ 1353.6 (ν3 ) and tional modes of C6 N·2 + in the X ∼ 475.8 (ν4 ) cm−1 in their neon matrix studies. As can be seen from Table 8, these results for ν1 , ν2 and ν4 are closest to our findings. Miller and Lemon 14 reported the fundamental e state at ∼ 1287 and ∼ 2570 cm−1 , and the first overtone of the vibrational mode ν3 in the X respectively. These results compare well with their location found in the present study at ∼ 1288 and ∼ 2577 cm−1 , respectively. The fundamental of ν2 and ν4 vibrational modes was e state reported to appear at ∼ 2028 and ∼ 449 cm−1 , respectively, in the spectrum of the A recorded by Agreiter et al. 41 The present study locates these fundamentals at ∼ 2092 and ∼ 463 cm−1 , respectively.

16


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e state, ν2 in the A e state, We note that apart from strong excitation of ν1 and ν4 in the X e C e and E e states and ν3 and ν4 in the D e state, excitation of the remaining symmetric ν4 in B, vibrational modes in each state is weak. The assignments given in Table 8 are carried out by examining the nodal pattern of the corresponding vibrational wavefunction in a reduced dimensional space. These wavefunctions are calculated by the block improved relaxation 39,40 method as implemented in the MCTDH 38 program modules. In order to illustrate, some of e and A e states mostly for which experimental data are available the wavefunctions of the X are plotted in Figs. 6 and 7. The vibrational wavefunctions of the fundamental of ν1 , ν2 , ν3 e state spectrum are shown in panels a-d of Fig. and ν4 vibrational modes excited in the X 6, respectively. The probability density of the wavefunctions is plotted along the normal coordinate of vibrational modes. It can be seen from these wavefunctions that each of them contains one node along the corresponding normal coordinate and represents the fundamental of these modes. The wavefunctions in panel e and f reveal two quantum excitation along Q1 and Q4 coordinates, respectively, and hence represent the first overtone of mode ν1 and ν4 . The wavefunction in panel g reveal one quantum excitation along both Q1 and Q4 coordinates and represents the combination level ν1 10 ν4 10 . The vibrational wavefunctions of e state spectrum are the fundamental of ν1 , ν2 and ν4 vibrational modes excited in the A shown in panels a-c of Fig. 7, respectively. The first overtones of ν2 and ν4 are shown in panels d and e, respectively. The wavefunctions of the combination levels ν2 10 ν4 10 and ν2 10 ν3 10 e state are shown in panels f and g of Fig. 7, respectively. excited in the A 3.2.2

Coupled state spectrum and time-dependent dynamics

The symmetry rules given in Sec. IIA reveal that all six states are coupled with each other through the available vibrational modes of σu+ , πg and πu symmetry. The interstate coupling e B, e parameters given in Table 5 reveals that the coupling strength in some cases, e.g. Xe C, e AeB e and AeC e are quite large. However, a large energy gap between the states often X-

17



quenches the strong coupling effect in the dynamical outcome. In order to assess the effect e Ae Be Ce De E e couof the coupling between states in the nuclear dynamics, we carried out Xpled states dynamics calculations employing the complete Hamiltonian given in Sec. IIA. In the coupled states situation the Hamiltonian for the Π states can not be separated in terms of σg+ and πg and πu vibrational modes. Full dimensional calculations are therefore carried out with six coupled electronic states and 19 vibrational degrees of freedom. The MCTDH program modules 38 are used to propagate WPs in the dynamics calculations. Separate calculations are carried out for the transition of the initial WP to the Σ and each components of the Π states. The vibrational basis functions and the parameters used in the calculations to obtain converged results are given in Table S2 of SI. The WP is propagated for 400 fs in each coupled states calculations. At each time the autocorrelation function and electronic populations (both diabatic and adiabatic) are recorded. The time autocorrelation functions from ten separate calculations are combined and Fourier transformed to generate the spectrum. Prior to Fourier transformation the autocorrelation function is damped with a function, e(−t/τr ) (with τr = 66 fs), in order to obtain close agreement with observed spectral broadening. The spectral envelopes as calculated above are shown in panel b of Fig. 8. The experimental results reproduced from Ref. 15 and are plotted in panel a. It can be seen that the agreement between the theoretical and experimental results is very good. The intensity pattern also follow the same trend as observed in the experiment. However, the observed spectral broadening differs to some extent with the theoretical result. When compared with the latter results presented in Figs. 3(c), Fig. 4(a-b) and Fig. 5(c) where interstate couplings are not included, it can be seen that the various possible conical intersections (as discussed in sec. III A) has insignificant effects on the overall structure of vibronic bands of the electronic states of C6 N·2 + . It is already discussed that RT effect is weak and there is not much effect due to interstate coupling despite some of them are particularly strong. This is on par with the topographical details of the potential energy surfaces discussed in Sec. III A. It can be

18


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e and E) e is largest. seen from Fig. 8 that the intensity of the origin line in each band (except D This indicates a very little geometry change of the cation in those states as compared to the e and C e states form an overlapping band and some neutral ground state. As expected, the B e and E e states. effect of interstate coupling prevails in the D

e2 Πu band occurs at ∼ 582 nm relative to the origin peak of the The origin line of the A e band. This result, although not strictly favorable, may be compared with the reported X DIB at 656 nm (in gas phase) 19 and 659 nm (in Ne matrix). 18 A more meaningful comparison with the laboratory spectroscopy data requires a study of the absorption spectrum of C6 N·2 + and is out of scope of the present contribution.

The comparison of the above coupled states spectra with their uncoupled counterparts (cf. Figs. 8) does not give much insight of the effect of various nonadiabatic coupling terms of the Hamiltonian into the dynamics. In order to get a better insight the time-dependence of the electronic populations in the coupled states situation is examined and a few representative population curves are shown below. It is observed that electronic population does e or not flow to the other electronic states when the dynamics is initiated either on the X e state. In these situations the population flows back and forth between the components A of these degenarate states only, and therefore, the dynamics is solely governed by the RT coupling within these electronic states. To illustrate, the time dependence of the diabatic and adiabatic populations shown in Fig. 9. In this case the WP is initially excited to one of e state. It can be seen from the panel a that the population of the x component of the the A e state starts from 1.0 and monotonically decreases untill ∼ 140 fs and saturates at diabatic A longer times. A nearly mirror image behavior of the two population curves indicates no major population transfer to the remaining electronic states included in this study. Very little e and C e states can be seen from the diabatic population curves. population transfer to the B The variation of the adiabatic electronic population shown in panel b of Fig. 9 reveals, a

19



e state at t=0. At longer times 50:50 population of both the components of the adiabatic A e state only the population flows back and forth between the RT split components of the A and no population transfer occurs to the other states. e state Time-dependence of electronic populations for an initial excitation to the diabatic B is shown in panel a of Fig. 10. It can be seen that the population primarily flows back and e and C e states in this situation and very little population transfer occurs forth between the B e state. The B e and C e states are very weakly coupled to the degenarate components of the A through the vibrational modes of σu+ symmetry (cf. Table 5). Despite a weak coupling rapid e and C e oscillations of populations in Fig. 10 a is a consequence of quasi-degeneracy of the B states at their equilibrium minimum (cf. Sec.III A). The recurrence of the oscillations in the population curves translates to a vibrational frequency of 900 cm−1 .

e and A e states, the exchange of population between the RT split components of Unlike X e or E e states does not take place. This is because of very weak RT coupling within the D e B e and C e states when the dynamics is started these states. Population mostly flows to the A, e state. In contrast, population mostly flows to the D e state when the dynamics is on the D e state. The population recorded in the started on one of the component of the diabatic E latter situation is shown in the panel b of Fig. 10. It can be seen from the figure that more e state in this case. It therefore emerged from than 60% of the population flows to the D e and A e state the above discussion that the RT coupling is relatively stronger within the X e and E e states. The PRT coupling of the later states as compared to the same within the D e and A e states. The with other states, on the other hand, is quite stronger than that of the X e and E e states is well above the minimum minimum of the conical intersections between the D e state and therefore population does not flow to the E e state when the dynamics is of the D e state. started on the D

20


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4

Summarizing remarks

A comprehensive account of the electronic structure and nuclear dynamics of astrophysically important dicyanodiacetylene radical cation is presented in this paper. The electronic structure calculations are carried out with the available state-of-the-art quantum chemistry softwares and the nuclear dynamics is treated fully quantum mechanically from first principles. Energetically low-lying four doubly degenerate and two non-degenarate electronic states of C6 N·2 + are considered in this study. The adiabatic energies of these electronic states along all vibrational modes are calculated and based on them a coupled vibronic Hamiltonian is constructed in a diabatic electronic basis employing the standard vibronic coupling theory and symmetry rules. Both the RT and PRT type of coupling of the states are studied in detail and their effect on the vibronic structure and electronic population dynamics is discussed at length. The theoretical results presented here are in good accord with various experimental results available in the literature. It is found that the RT effect is e and A e states whereas, it is much weaker within the D e and E e states. stronger within the X e and A e states is weak and of the D e and E e In contrast, the PRT coupling effect of the X states is relatively strong with the remaining states in each case.

Supporting Information Table S1: Numerical details of time-independent calculations. Table S2: Numerical details of time-dependent calculations.

Acknowledgement This study is financially supported in part by a grant from the Department of Science and Technology (DST), New Delhi, INDIA, through Grant No.SB/S1/PC-052/2013. K. R acknowledges UGC, New Delhi for Dr. D. S. Kothari Postdoctoral Fellowship.

21



References (1) Bell, M. B.; Feldman, P. A.; Matthews, H. E. Cyanopolyyne Absorption in the Direction of Cassiopeia A. Astron. Astrophys. 1981, 101, L13-L16. (2) Kroto, H. W. The Spectra of Interstellar Molecules. Int. Rev. Phys. Chem. 1981, 1, 309-376. (3) Coustenis, A.; Bezard, B.; Gautier, D.; Marten, A.; Samuelson, R. Titan’s atmosphere from Voyager infrared observations. III - Vertical contributions of hydrocarbons and nitriles near Titan’s north pole. Icarus 1991, 89, 152-167. (4) Kroto, H. W. Smaller carbon species in the laboratory and space. Int. j. mass spectrom. ion processes. 1994, 138, 1-15. (5) Winnewiser, M. Interstellare Molekle und Mikrowellenspektroskopie I. Chem. Zeit. 1984, 18, 1-16. (6) McCarthy, M. C.; Grabow, J. U.; Travers, M. J.; Chen, W.; Gottlieb, C. A.; Thaddeus, P. Laboratory detection of the carbon chains HC15 N and HC17 N. Astrophys. J. 1998, 494, L231-L234. (7) McCarthy, M. C.; Thaddeus, P. Microwave and laser spectroscopy of carbon chains and rings. Chem. Soc. Rev. 2001, 30, 177-185. (8) Moureau, C.; Bongrand, J. C. New researches on carbon subnitride : Action of halogenes, halogen acids and alcohols. Bull. Soc. Chim. Fr. 1909, 5, 846. Ann. Chem. (Leipzig) 1920, 14, 5. (9) Moureau, C.; Bongrand, J. C. Le Cyanoacetylene C3 NH. Ann. Chem. (Paris) 1920, 14, 47. (10) Connors, R. E.; Roebber, J. L.; Weiss, K. Vacuum ultraviolet spectroscopy of cyanogen and cyanoacetylenes. J. Chem. Phys. 1974, 60, 5011-5024. 22


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(11) Brown, K. W.; Nibler, J. W.; Hedberg, K.; Hedberg, L. Structure of Dicyanoacetylene by Electron Diffraction and Coherent Rotational Raman Spectroscopy J. Phys. Chem. 1989, 93, 5679-5684. (12) Khanna, R. K.; Perera-Jarmer, M. A.; Ospina, M. J. Vibrational infrared Raman spectra of dicyanoacetylene. Spectrochim. Acta A. 1987, 43, 421-425. (13) Wayne, R. P. Chemistry of atmospheres (Clarendon Press, Oxford, 1991). (14) Miller, F. A.; Lemon, D. H. The infrared and Raman spectra of dicyanoacetylene. Spectrochimica Acta, 1967, 23A, 1415-1423. e2 Πu → X e 2 Πg emis(15) Klöster-Jensen, E.; Maier, J. P.; Marthaler, O.; Mohraz, M. The A sion and the photoelectron spectrum of dicyanodiacetylene radical cation. J. Chem. Phys. 1979, 71, 3125-3128. (16) Kolos, R. Photolysis of dicyanoacetylene in argon matrices. Chem. Phys. Lett. 1999, 299, 247-251. (17) Gupta, V. P.; Thul, P; Tandon, P. Quantum chemical study of infrared and Raman spectra of dicyanodiacetylene. Indian J. Pure Appl. Phys. 2011, 49, 162-167. (18) Forney, D.; Freivogel, P.; Fulara, J.; Maier, J. P. Electronic absorption spectra of cyano-substituted polyacetylene cations in neon matrices. J. Chem. Phys., 1995, 102, 1510-1514. (19) Nagarajan, R.; Maier, J. P. Electronic spectra of carbon chains and derivatives. Int. Rev. Phys. Chem. 2010, 29, 521-554. (20) Linnartz, H.; Pfluger, D.; Vaizert, O.; Cias, P.; Birza, P.; Khoroshev, D.; Maier, J. P. e2 Πu ← X e 2 Πg electronic transition of NC6 N+ . J. Chem. Phys. Rotationally resolved A 2002, 116, 924-927.

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(21) Lee, S. Density Functional Theory Study of Cyano- and Dicyanopolyacetylene Cations. J. Phys. Chem. 1996, 100, 13959-13962. (22) Zhao, Y.; Guo, J.; Zhang, J. Electronic spectra of the linear cationic chains NC2n N+ (n=17) : an ab initio study. Theor. Chem. Acc. 2011 129, 793-801. (23) Renner, R. Zur Theorie der Wechselwirkung zwischen Elektronen- und Kern-bewegung bei dreiatomigen, stabförmigen Molek¨ ulen. Z. Physik 1934, 92, 172-193. (24) Köppel, H.; Domcke, W.; Cederbaum, L. S. Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation. Adv. Chem. Phys. 1984, 57 , 59-246. (25) Wilson Jr., E. B.; Decius, J. C.; Cross, P. C. Molecular vibrations; McGraw-Hill, New York, 1955. e 2 Πg (26) Ghosh, A.; Reddy, S. N.; Reddy,S. R.; Mahapatra, S. Vibronic Coupling in the X e2 Πu Band System of Diacetylene radical cation. J. Phys. Chem. A, 2016, 120, 7881A 7889. (27) Dunning, T. H. Jr. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron Through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007-1023. (28) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B. ; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J. ; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann,

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R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, Revision C.01, Wallingford CT2010. (29) CFOUR, Coupled-Cluster techniques for Computational Chemistry, a quantumchemical program package written by Stanton, J.F.; Gauss, J.; Cheng, L.; Harding, M. E.; Matthews, D. A.; Szalay, P. G.; with contributions from Auer, A. A.; Bartlett, R. J.; Benedikt, U.; Berger, C.; Bernholdt, D. E.; Bomble, Y. J.; Christiansen, O.; Engel, F.; Faber, R.; Heckert, M.; Heun, O.; Hilgenberg, M.; Huber, C.; Jagau,T-C.; Jonsson, D.; Jusélius, J.; Kirsch, T.; Klein, K.; Lauderdale, W. J.; Lipparini, F.; Met´ Neill, D. P.; Price, D. R.; Prochnow, E.; Puzzarini, C.; zroth, T.; M¨ uck, L. A.; O Ruud, K.; Schiffmann, F.; Schwalbach, W.; Simmons, C.; Stopkowicz, S.; Tajti, A.; Vázquez, J.; Wang, F.; Watts, J. D. and the integral packages MOLECULE (Almlöf, J.; Taylor, P. R.), PROPS (Taylor, P. R), ABACUS (Helgaker, T.; Aa. Jensen, H. J.; Jør-gensen, P.; Olsen, J.), and ECP routines by Mitin, A. V.; van W¨ ullen, C. For the current version, see http://www.cfour.de (30) Lanczos, C. An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators. J. Res. Nat. Bur. Stand. 1950, 45, 255-282. (31) Cullum, J.; Willoughby. R. Lanczos Algorithms for Large Symmetric Eigenvalue Problems (Birkhäuser, Boston, 1985), Vols. I and II. (32) Domcke, W.; Köppel, H. Encyclopedia of Computational Chemistry-Theoretical and Physical chemistry; edited by H.F.Schacfer. III. ( Wiley. New York, 1998). (33) Domcke, W.; Köppel, H.; Cederbaum, L. S. Spectroscopic Effects of Conical Intersections of Molecular Potential Energy Surfaces. Mol. Phys. 1981, 43, 851-875.

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(34) Meyer, H. -D.; Manthe, U.; Cederbaum, L. S. The Multi-Configurational Timedependent Hartree Approach. Chem. Phys. Lett. 1990, 165, 73-78. (35) Manthe, U.;, Meyer, H. -D.; Cederbaum, L. S. Wave-packet Dynamics Within the Multiconfiguration Hartree Framework: General Aspects and Application to NOCl. J. Chem. Phys. 1992, 97, 3199-3213. (36) Beck, M. H.; Jäckle, A.; Worth, G. A.; Meyer, H. -D. The Multiconfiguration Timedependent Hartree (MCTDH) Method: a Highly Efficient Algorithm for Propagating Wavepackets. Phys. Rep. 2000, 324, 1-105. (37) Gindensperger, E.; Bâldea, I.; Franz, J.; Köppel, H. Multi-state vibronic interactions in the fluorobenzene radical cation: The importance of quadratic coupling terms. Chem. Phys. 2007, 338, 207-219. (38) Worth, G. A.; Beck, M. H.; Jäckle, A.; Meyer, H. -D. The MCTDH Package, Version 8.4, University of Heidelberg, Heidelberg, Germany (2007) http://www.pci.uniheidelberg.de (39) Meyer, H. -D.; Quéré, F. L.; Léonard, C.; Gatti, F. Calculation and selective population of vibrational levels with the Multiconfiguration Time-Dependent Hartree (MCTDH) algorithm. Chem. Phys. 2006, 329, 179-192. (40) Roriol, L. J.; Gatti, F.; Lung, C.; Meyer, H. -D. Computation of vibrational energy levels and eigenstates of fluoroform using the multiconfiguration time-dependent Hartree method. J. Chem. Phys.. 2008, 129, 224109. (41) Agreiter, J.; Smith, A. M.; Bondybey, V. E. Laser-induced fluorescence of matrix+ isolated C6 N+ 2 , and of C8 N2 . Chem. Phys. Lett. 1995, 241, 317-327.

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Table 1: Symmetry, harmonic frequency (in cm−1 ) and the nature of the vibrational modes of the electronic ground electronic state of dicyanodiacetylene, C6 N2 . The experimental results represent fundamental frequencies. Symmetry Mode Expt. 14 2235 2183 1287.5 571

Frequency DFT BP86/DN* 16 2209 2194 919 473

MP2/6-31G* 16 2208 2124 909 468

Schematic description C≡N symmetric stretch C≡C symmetric stretch middle C-C symmetric stretch terminal C-C symmetric stretch

σg+

ν1 ν2 ν3 ν4

MP2/aug-cc-pVTZ 2173 2105 1302 463

σu+

ν5 ν6 ν7

2202 2018 900

2266 2097 717

2261 2090 1336

2229 2039 1314

C≡N antisymmetric stretch C≡C antisymmetric stretch C-C antisymmetric stretch

πg

ν8 ν9 ν10

511 460 160

501 455 156

509 459 172

507 448 168

N≡C-C antisymmetric bend N≡C-C antisymmetric bend C≡C-C antisymmetric bend

πu

ν11 ν12 ν13

506 280 61

490.5 276 61.5

503 289 79

496 265 65

N≡ C-C symmetric bend middle symmetric bending terminal symmetric bending

27



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Table 2: Linear (κi ), quadratic (γi or ηi ) and quartic (ξi and δi ) coupling parame 2 Πg and A e2 Πu electronic states eters of the Hamiltonian [cf., Eqs.(3-8)]for the X · + of C6 N2 . Dimensionless excitation strengths are given in the square brackets. All quantities are given in the eV unit. Mode Frequency Symmetry σg+

κi or ηi (δi )

γi (ξi )

e 2 Πg X [0.47] [0.00] [0.01] [0.10]

κi or ηi (δi )

0.0038 -0.0053 -0.0017 -0.0006

-0.0925 0.1652 -0.0476 -0.0233

γi (ξi )

e2 Πu A [0.06] [0.20] [0.04] [0.08]

ν1 ν2 ν3 ν4

0.2694 0.2610 0.1615 0.0574

-0.2618 0.0250 0.0252 0.0257

σu+

ν5 ν6 ν7

0.2730 0.2502 0.1116

-

-0.0163(0.0002) -0.0010 -0.0013

πg

ν8 ν9 ν10

0.0634 0.0570 0.0198

0.0041(0.0000) -0.0016(0.0000) 0.0002(0.0000)

0.0002(0.0000) -0.0022(0.0000) -0.0032(0.0000)

πu

ν11 ν12 ν13

0.0628 0.0347 0.0076

0.0015 (0.0000) -0.0023 (0.0000) -0.0029(0.0000) 0.0017 (0.0000) -0.0126 (0.0000) -0.0006(0.0000) -0.0002 (0.0000) -0.0028 (0.0000) -0.0000(0.0000)

-

0.0022 -0.0016 -0.0041 0.0000

0.0159(-0.0001) -0.0135(0.0002) -0.0007

0.0000 (0.0000) -0.0053 (0.0000) 0.0005 (0.0000) -0.0071 (0.0000) 0.0005 (0.0000) -0.0122 (0.0000) -0.0062(0.0000) -0.0112(0.0000) -0.0082(0.0000)

Table 3: Linear (κi ), quadratic (γi ) and quartic (ξi ) coupling parameters of the e2 Σ+ electronic states of C6 N·2 + . e 2 Σ+ and C Hamiltonian [cf., Eqs.(3-8)]for the B u g Dimensionless excitation strengths are given in the square brackets. All quantities are given in the eV unit. Mode Frequency

κi

γi (ξi )

κi or ηi (δi )

γi (ξi )

ν1 ν2 ν3 ν4

0.2694 0.2610 0.1615 0.0574

e 2 Σ+ B g -0.0497 [0.02] -0.0233 [0.00] -0.0305 [0.02] 0.0284 [0.12]

σu+

ν5 ν6 ν7

0.2730 0.2502 0.1116

-

πg

ν8 ν9 ν10

0.0634 0.0570 0.0198

-0.0009 0.0039 0.0028

0.0041 0.0075 0.0032

πu

ν11 ν12 ν13

0.0628 0.0347 0.0076

0.0073 0.0016 0.0051

0.0000(-0.0001) 0.0012 -0.0052

Symmetry σg+

28

-0.0056 -0.0020 -0.0009 -0.0004 -0.0038 0.0013 -0.0007

e 2 Σ+ C u -0.0497 [0.02] -0.0236 [0.00] -0.0308 [0.02] 0.0294 [0.13]


-

-0.0057 -0.0021 -0.0009 -0.0003 -0.0038 -0.0017 -0.0007

Page 29 of 66 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


Table 4: Linear (κi ), quadratic (γi or ηi ) and quartic (ξi and δi ) coupling parame 2 Πg and E e 2 Πu electronic states eters of the Hamiltonian [cf., Eqs.(3-8)]for the D · + of C6 N2 . Dimensionless excitation strengths are given in the square brackets. All quantities are given in the eV unit. Mode Frequency Symmetry σg+

κi or ηi (δi )

γi (ξi ) e 2 Πg D -0.1002 [0.07] -0.0029 0.1151 [0.10] 0.0123 0.1115 [0.24] 0.0012 -0.0381 [0.22] 0.0005

κi or ηi (δi )

γi (ξi ) e 2 Πu E -0.0945 [0.06] 0.0007(0.0000) 0.0313 [0.01] 0.0087 (0.0007) 0.0212 [0.01] 0.0085(-0.0003) -0.0759 [0.87] 0.0015(0.0000)

ν1 ν2 ν3 ν4

0.2694 0.2610 0.1615 0.0574

σu+

ν5 ν6 ν7

0.2730 0.2502 0.1116

πg

ν8 ν9 ν10

0.0634 0.0570 0.0198

0.0013(0.0000) -0.0034(0.0000) 0.0000(0.0000) -0.0090(0.0000) 0.0009(0.0000) -0.0042(0.0000) -0.0003(0.0000) -0.0077(0.0000) 0.0001(0.0000) -0.0132(0.0000) 0.0000(0.0000) -0.0095(0.0000)

πu

ν11 ν12 ν13

0.0628 0.0347 0.0076

0.0029(0.0000) -0.0060(0.0000) 0.0002(0.0000) -0.0111(0.0000) 0.0000(0.0000) -0.0172(0.0000)

-

-0.0064(0.0002) 0.0194(-0.0002) -0.0144(0.0003)

29


-

0.0001(0.0000) 0.0001(0.0000) 0.0000(0.0000)

0.0174(-0.0002) 0.0117 (0.0000) 0.0191(-0.0003)

-0.0083(0.0000) -0.0126(0.0000) -0.0113(0.0000)


Page 30 of 66

Table 5: Interstate coupling parameters (in eV) of the vibronic Hamiltonian e 2 Πg , A e2 Πu , B e 2 Σ+ , C e2 Σ+ , D e 2 Πg and E e 2 Πu electronic states of Eqs. 9-11 for the X g u · + of C6 N2 estimated from the ab initio electronic structure results (see text for details). Symmetry Mode Frequency(eV)

σu+

ν5 ν6 ν7

0.2730 0.2502 0.1116

πg ν8 ν9 ν10

0.0634 0.0570 0.0198

πu ν11 ν12 ν13

0.0628 0.0347 0.0076

λX−A i

λX−E i

λA−D i

λB−C i

0.1844(0.23) 0.0261(0.03)

0.2593(0.45) 0.1622(0.21) 0.2010(1.62)

λX−B i

λA−C i

λB−D i

λC−E i

0.0626(0.49) 0.1085(1.81) 0.0928(10.98)

0.0634(0.5) 0.0797(0.98) 0.0813(8.43)

-

-

λX−C i

λA−B i

λB−E i

λC−D i

-

-

λD−E i

0.0345(0.01) 0.0827(0.05) 0.1501(0.18) 0.0314(0.01) 0.0419(0.07) 0.0998(0.40)

0.0547(0.38) 0.0857(0.93) 0.1333(7.38) 0.0771(2.47) 0.1113(107.02) 0.0770(51.31)

e A, e B, e C, e D e and E e states Table 6: Vertical ionization energies (in eV) of the X, · + of C6 N2 calculated at the reference equilibrium geometry. State Term e 2 Πg X e2 Πu A e 2 Σ+ B g e 2 Σ+ C u e 2 Πg D e 2 Πu E a

Vertical Ionization Energy (VIE) Present Study Expt a 11.1025 10.88 13.2286 12.77 14.0912 13.63 14.0978 13.63 14.5910 14.05 15.1639 14.65

Ref. 15

30


Page 31 of 66 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


Table 7: Estimated, Vmin (diagonal entries) and Vcmin (off-diagonal entries) of the electronic states of C6 N·2 + within a quadratic coupling model (see text for details). All quantities are given in eV.

e 2 Πg X e2 Πu A e 2 Σ+ B g e2 Σ+ C u e 2 Πg D e 2 Πu E

e 2 Πg X

e2 Πu A

e 2 Σ+ B g

e2 Σ+ C u

e 2 Πg D

e 2 Πu E

10.97 -

21.41 13.15 -

26.37 15.79 14.07 -

26.35 15.80 17.38 14.08 -

25.85 14.67 14.66 14.50 -

49.38 18.43 18.16 17.06 15.10

31



Page 32 of 66

e Table 8: Energetically low-lying vibrational energy levels (in cm−1 ) of the X, · + e B, e C, e D e and E e electronic states of C6 N2 A, alongwith their assignment. The experimental data available in the literature are given in the parentheses. e 2 Πg X Vibronic energy level 0 458 (475.8a ) 916 1288 (1353.6a ,1287b ) 1374 1746 1832 2062 (2093.8a ) 2203 (2185.3a , 2100c ) 2520 2577 (2570b ) 2661 3119 3351 3492 4123 4265 4406 4865 931 845 257 939 215 53 e2 Σ+ C u Vibronic energy level 0 461 921 1295 1382 1756 1842 2088 2126 2216 2549 2587 2591 3047 3383 3422 4176 4214 4253

a

= Ref. 41

b

= Ref. 14

c

= Ref. 15

Assignment 000 ν4 10 ν4 20 ν3 10 ν4 30 ν4 10 ν3 10 ν4 40 ν2 10 ν1 10 ν4 10 ν2 10 ν3 20 ν4 10 ν1 10 ν4 20 ν1 10 ν2 10 ν3 10 ν1 10 ν3 10 ν2 20 ν1 10 ν2 10 ν1 20 ν4 10 ν1 20 ν8 20 ν9 20 ν10 20 ν11 20 ν12 20 ν13 20 Assignment 000 ν4 10 ν4 20 ν3 10 ν4 30 ν4 10 ν3 10 ν4 40 ν2 10 ν1 10 ν4 20 ν3 10 ν4 10 ν2 10 ν4 10 ν1 10 ν3 20 ν4 20 ν1 10 ν2 10 ν3 10 ν1 10 ν3 10 ν2 20 ν1 10 ν2 10 ν1 20

e2 Πu A Vibronic energy level 0 463 (449a ) 926 1269 1389 1732 1852 2092 (2028a ) 2191 (1940c ) 2195 2538 2555 2653 3361 3460 4184 4283 4381 4647 933 807 833 356

e 2 Πg D Vibronic energy level 0 467 934 1312 1779 2149 2202 2246 2616 2624 2669 3091 3136 3462 3514 3937 4299 4351 4404 997 826 836 343

32

Assignment 000 ν4 10 ν4 20 ν3 10 ν4 30 ν4 10 ν3 10 ν4 40 ν2 10 ν1 10 ν4 20 ν3 10 ν3 20 ν4 10 ν2 10 ν4 10 ν1 10 ν3 10 ν2 10 ν3 10 ν1 10 ν2 20 ν2 10 ν1 10 ν1 20 ν2 20 ν4 10 ν8 20 ν9 20 ν11 20 ν12 20

Assignment 000 ν4 10 ν4 20 ν3 10 ν4 10 ν3 10 ν1 10 ν2 10 ν4 20 ν3 10 ν1 10 ν4 10 ν3 20 ν4 10 ν2 10 ν4 10 ν3 20 ν4 20 ν2 10 ν3 10 ν1 10 ν3 10 ν2 10 ν3 30 ν1 20 ν1 10 ν2 10 ν2 20 ν8 20 ν9 20 ν11 20 ν12 20

e 2 Σ+ B g Vibronic energy level 0 460 919 1295 1379 1755 1839 2089 2127 2215 2549 2587 2591 3047 3384 3422 4178 4216 4254

e 2 Πu E Vibronic energy level 0 475 950 1374 1425 1849 1900 2178 2179 2324 2374 2653 2654 2799 3128 3129 3603 3604 4357 865 793 871 297


Assignment 000 ν4 10 ν4 20 ν3 10 ν4 30 ν4 10 ν3 10 ν4 40 ν2 10 ν1 10 ν4 20 ν3 10 ν4 10 ν2 10 ν4 10 ν1 10 ν3 20 ν4 20 ν1 10 ν3 10 ν2 10 ν3 10 ν1 10 ν2 20 ν2 10 ν1 10 ν1 20

Assignment 000 ν4 10 ν4 20 ν3 10 ν4 30 ν4 10 ν3 10 ν4 40 ν1 10 ν2 10 ν4 20 ν3 10 ν4 50 ν4 10 ν2 10 ν4 10 ν1 10 ν4 30 ν3 10 ν4 20 ν2 10 ν4 20 ν1 10 ν4 30 ν2 10 ν4 30 ν1 10 ν2 20 ν8 20 ν9 20 ν11 20 ν12 20

Page 33 of 66

20

a)

20

b) -1

-1

υ1 ( 2173 cm )

υ2 ( 2105 cm )

18

18

16

16

~ E

~ E

14

~ D

~ D

~ ~ B&C

~ ~ B&C

~ A

14

~ A

12

12 ~ X

~ X

10 Energy (in eV)

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


-6

-4

-2

0

20

2

4

6 -6

-4

-2

0

c) -1

16

6

-1

18

16 ~ E ~ D

~ D

14

~ ~ B&C

~ ~ B&C ~ A

~ A

12

12 ~ X

~ X

10

10

20

υ4 ( 463 cm )

~ E

14

4

d)

υ3 ( 1302 cm )

18

2

-6

-4

-2

0

2

4

6 -6

-4

-2

Q

0

2

4

6

10

Q

e A, e B, e C, e D e and E e electronic states of Figure 1: Adiabatic potential energies of the X, C6 N·2 + along the totally symmetric (σg+ ) vibrational modes (ν1 -ν4 ). The energies obtained from the present vibronic model and computed ab initio energies are shown by lines and points, respectively.

33



e A, e B, e C, e D e and E e electronic states of Figure 2: Adiabatic potential energies of the X, · + C6 N2 along the degenarate (πg , panel a-c and πu , panel d-f) vibrational modes (ν7 -ν13 ). The energies obtained from the present vibronic model and computed ab initio are shown by lines and points, respectively.

34


Page 34 of 66

Page 35 of 66 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


e and A e electronic states Figure 3: Vibrational energy level spectrum of the uncoupled X · + + of C6 N2 computed with (a) totally symmetric vibrational modes (σg ) and (b) degenerate vibrational modes (πg and πu ). The convoluted spectra of symmetric and degenerate vibrational modes are shown in panel c.

35



~ B state

0

a)

Relative Intensity (arb. unit)

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

00

ν40

1

2

ν40

b)

00

Page 36 of 66

1

ν30

1

ν20

~ C state

0

1

ν40

ν40

13.5

13.6

2

1

ν30

13.7 13.8 Energy (eV)

1

ν20

13.9

14

e (panel a)and C e (panel Figure 4: Vibrational energy level spectrum of the uncoupled B · + b)electronic states of C6 N2 computed with totally symmetric vibrational modes (σg+ ).

36


Page 37 of 66

a)

00

0

~ E state

~ D state 00

0

ν40 1

ν30 ν40

ν40

1

ν30 ν40


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


+

σg

1

1

1

ν40 1

ν10 ν20

ν30

2

1

1

ν40

2

ν10

3

ν20

1 1

b) πg - πu

ν120 ν120

2

2

c) +

σg - πg - πu

13.8

14

14.2

14.4 14.6 Energy (eV)

14.8

15

e and E e electronic states Figure 5: Vibrational energy level spectrum of the uncoupled D · + + of C6 N2 computed with (a) totally symmetric vibrational modes (σg ) and (b) degenerate vibrational modes (πg and πu ). The convoluted spectra of symmetric and degenerate vibrational modes are shown in panel c. 37


density

-4-3 -2 Q2-1012 34 -4 -3 -2 -1 0Q11 2 3 4

density

(b)

density

(a)

-4-3 -2 Q1-1012 34 -4 -3 -2 -1 0Q21 2 3 4

-4-3 -2 Q1-1012 34 -4 -3 -2 -1 0Q31 2 3 4

(d)

density

(c)

-4-3 -2 Q1-1012 34 -5 -4 -3 -2 -1 0Q41 2 3 4 5

density

-4-3 -2 Q2-1012 34 -4 -3 -2 -1 0Q11 2 3 4

-4-3 -2 Q1-1012 34 -5 -4 -3 -2 -1 0Q41 2 3 4 5

(e)

(f)

density

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

density


-4-3 -2 Q1-1012 34 -5 -4 -3 -2 -1 0Q41 2 3 4 5

(g)

Figure 6: Probability density of the wavefunction of the fundamentals of ν1 , ν2 , ν3 and ν4 (panel a, b, c and d, respectively) and first overtone of ν1 and ν4 ( panel e and f, respece state spectrum of C6 N·2 + . The wavefunction in panel g represents a tively) excited in the X combination level, ν1 + ν4 . 38


Page 38 of 66

density

-4 -3 -2

-4-3 -2 Q1-1012 34 -6 -4 -2

234 -1 -1 0 1Q3 -2 Q1 0 1 2 3 -3 4 -4

0Q2 2

4

6

0Q2 2

4

6

(b)

density

(a)

density

-4-3 -2 Q1-1012 34 -4 -3 -2 -1 0Q41 2 3 4

(d)

density

(c)

-4-3 -2 Q1-1012 34 -6 -4 -2

density

-4-3 -2 Q1-1012 34 -4 -3 -2 -1 0Q41 2 3 4

-6-4 Q2-2 0 2

(e)

4 6 -4 -3 -2 -1 0Q41 2 3 4

(f)

density

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


density

Page 39 of 66

-6-4 Q2-2 0 2

4 6 -4 -3 -2 -1 0Q31 2 3 4

(g)

Figure 7: Probability density of the wavefunction of the fundamentals of ν1 , ν2 and ν4 (panel a, b and c, respectively) and first overtone of ν2 and ν4 ( panel d and e, respectively) excited e state spectrum of C6 N·2 + . The wavefunction in panel f and g represent combination in the A levels, ν2 + ν4 and ν2 + ν3 , respectively. 39



e Ae Be Ce De E e coupled electronic states of C6 N·2 + . Figure 8: Vibronic band structure of the XRelative intensity (in arbitrary units) is plotted as a function of the energy of the vibronic states of C6 N·2 + . The present theoretical results are shown in panel b and the experimental results of Ref. 15 are reproduced in panel a.

40


Page 40 of 66

Page 41 of 66

1 ~ Diabatic A

a)

0.8 Ax 0.6

0.4 B

Ay

0.2 Population

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


C

01

~ Adiabatic A

b)

0.8

0.6

0.4

0.2

0

0

100

200 Time (fs)

300

400

Figure 9: Time dependence of diabatic (panel a) and adiabatic (panel b) electron populations e Ae Be Ce De E e coupled states nuclear dynamics of C6 N·2 + . The initial WP (at t=0) in the Xe state. is located on one of the components of the diabatic A

41



1 ~ Diabatic B

B

a)

0.8

0.6 Ax

0.4

Ay

C 0.2 Population

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

Page 42 of 66

01

~ Diabatic E

b)

0.8 Ex 0.6 Ax 0.4

B C

Dx 0.2

0 0

100

200

300

400

Time (fs)

e Ae Be Ce De E e coupled Figure 10: Time dependence of diabatic electron populations in the X· + e state (panel states nuclear dynamics of C6 N2 . The initial WP (at t=0) is located on the B e state (panel b). a) and one of the component of the degenarate E

42


Page 43 of 66 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


43


20

Energy (in eV)

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


a)

-1

20

Page 44 of 66

b) -1

υ1 ( 2173 cm )

υ2 ( 2105 cm )

18

18

16

14

16

~ E

~ E ~ D

~ D

~ ~ B&C

~ ~ B&C

~ A

14

~ A

12

12 ~ X

~ X

10

-6

-4

-2

0

20

2

4

6 -6

-4

-2

0

c) -1

16

6

20 -1

υ4 ( 463 cm )

18

16

~ E

~ E ~ D

~ D

14

4

d)

υ3 ( 1302 cm )

18

2

14

~ ~ B&C

~ ~ B&C ~ A

~ A

12

12 ~ X

~ X

10

10

-6

-4

-2

0

2

4

6 -6

-4

-2

0


Q

Q

2

4

6

10

Page 45 of 66 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



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


Page 46 of 66

0 Page 47 a) of 66 The Journal ~ Chemistry 00 of Physical

B state


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

ν40

1

2

ν40

b)

00

1

ν30

1

ν20

~ C state

0

1

ν40

ν40

13.5

2

1

ν30

ACS Paragon Plus Environment 13.6 13.7 13.8 Energy (eV)

1

ν20

13.9

14

a)


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

00

0

~ The Journal of Physical Chemistry ~ D state E state 00

0

ν40 1

ν30 ν40

ν40

1

ν30 ν40

Page 48 of 66

+

σg

1

1

1

ν40 1

ν10 ν20

ν30

2

1

1

ν40

2

ν10

3

ν20

1 1

b) πg - πu

ν120 ν120

2

2

c)

13.8

+

σg - πg - πu

14

14.2ACS Paragon 14.4Plus Environment 14.6 Energy (eV)

14.8

15

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


density

Page 49 of 66

-4-3 -2-1 Q2 012 4 3 2 1 0 34 -4 -3 -2 -1 Q1 ACS Paragon Plus Environment

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

density



Page 50 of 66

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


density

Page 51 of 66


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

density


-4-3 -2-1 Q1 012 5 4 3 2 1 0 34 -5 -4 -3 -2 -1 Q4 ACS Paragon Plus Environment

Page 52 of 66

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


density

Page 53 of 66


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

density



Page 54 of 66

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


density

Page 55 of 66


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

density


-4 -3 -2

4 3 2 1 0 -1 -1 Q3 -2 Q1 0 1 2 3 4 -4 -3 ACS Paragon Plus Environment

Page 56 of 66

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


density

Page 57 of 66

-4-3 -2-1 Q1 012 34 -6

-4

-2


0Q2 2

4

6

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

density



Page 58 of 66

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


density

Page 59 of 66

-4-3 -2-1 Q1 012 34 -6

-4

-2


0Q2 2

4

6

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

density



Page 60 of 66

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


density

Page 61 of 66

-6-4 -2 Q2 0 2

4 6 -4 -3 -2 -1 0Q41 2 3 4 ACS Paragon Plus Environment

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

density


-6-4 -2 Q2 0 2

4 6 -4 -3 -2 -1 0Q31 2 3 4 ACS Paragon Plus Environment

Page 62 of 66

Page 63 of 66 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



1 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


Page 64 of 66

~ Diabatic A

a)

0.8 Ax

0.6

0.4

0.2

Population

B

Ay

01

C

~ Adiabatic A

b)

0.8

0.6

0.4

0.2

0

0

100


200 Time (fs)

300

400

1

Page 65 of 66

Population

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


~ Diabatic B

B

a)

0.8

0.6 Ax

0.4

Ay

C 0.2

01

~ Diabatic E

b)

0.8 Ex 0.6 Ax 0.4

B C

Dx 0.2

0 0

100

200


Time (fs)

300

400


338x190mm (96 x 96 DPI)


Page 66 of 66

Vibronic Coupling in the First Five Electronic States of

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