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Vibronic Coupling in the X # - Ã# Band System of Diacetylene Radical Cation Arpita Ghosh, Samala Nagaprasad Reddy, S. Rajagopala Reddy, and Susanta Mahapatra J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b08892 • Publication Date (Web): 22 Sep 2016 Downloaded from http://pubs.acs.org on September 23, 2016

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The Journal of Physical Chemistry

e 2Πg - A e2Πu Band Vibronic Coupling in the X System of Diacetylene Radical Cation Arpita Ghosh, Samala Nagaprasad Reddy, S. Rajagopala Reddy and S. Mahapatra

∗

School of Chemistry, University of Hyderabad, Hyderabad 500 046, India E-mail: E-mail:[email protected],Ph:+91-40-23134826;Fax:+91-40-23012460

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Abstract e 2 Πg Vibronic interactions in the two energetically lowest electronic states (X e2 Πu ) of diacetylene radical cation (C4 H· + ) are theoretically examined here. The A 2 spectroscopy of these two electronic states of C4 H·2 + has been a subject of considerable interest and measured in laboratory by various groups. Inspired by numerous experimental data, we attempt here a detailed investigation of vibronic interactions within and between the doubly degenerate Π electronic states and their impact on the vibronic structure of each state. A vibronic coupling model is constructed in a diabatic electronic basis and with the aid of ab initio

quantum chemistry calculations.

The vibronic structures of the electronic states are calculated by time-independent and time-dependent quantum mechanical methods. The progression of vibrational modes in the vibronic band is identified, assigned and compared with the literature data. The nonradiative internal conversion dynamics is also examined and discussed.


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I.

Introduction

e 2 Πg - A e2 Πu band system of diacetylene radical cation (C4 H·2 + ) is a subject of intense The X research since 1950. The common visible bands of organic vapors observed by Sch¨ uler and Reinebeck 1 and labelled as “T“ spectrum was assigned to this band system by Callomon. 2 Since then several high resolution experiments carried out attempting to confirm the presence of C4 H·2 + in the interstellar medium as a carrier of 506.9 nm diffuse interstellar band (DIB). Recent gas phase experimental studies of both Motylewski et al. 3 and Krelowski et al. 4 e2 Πu ← X e 2 Πg transition in C4 H·2 + . apparently confirmed the 506.9 nm DIB is due to A However, the most recent gas phase absorption measurements of Maier et al. 5 disagree with this assignment both in terms of absorption wavelength and the band shape and association of 506.9 nm DIB with C4 H·2 + was proposed to be unlikely. Apart from these, C4 H·2 + also proposed to play crucial role in the formation of larger polycyclic aromatic hydrocarbons in the interstellar medium 6 and have relevance in plasma chemistry. The structure and spectroscopy of C4 H·2 + was studied by several research groups over the past decades . Photoelectron spectrum of C4 H2 was measured by Baker et al. 7 Theoretical and experimental ionization energies have been reported and their possible implication in the redox chemistry of planetary atmosphere was discussed. 8 Very recent study of Gronowski et al. 9 reveals that the linear isomer of diacetylene is the most stable structure . In addition to matrix isolation Spectroscopy studies, 10–15 gas phase laboratory 3,14 and stellar spectroscopy e and A e studies 4,5 were also carried out on C4 H·2 + . The progressions of vibrational in the X e band excitation of fundamental of C ≡ C symmetric bands have been reported. In the X stretch (ν2 ) and first overtone of C – H antisymmetric stretch (ν4 ) vibrational modes with energy separation of ∼2121 and ∼5405 cm−1 , respectively, was reported by Baker et al. 7 In e band, progression of the fundamental of central C – C stretch (ν3 ) vibrational mode the A at ∼887 cm−1 was reported by the same authors. Callomon et al. 2 reported excitation of e band. Peak spacings of ∼861, ∼971, four vibrational modes and their combinations in the A ∼2177 and ∼3137 cm−1 were assigned to ν3 , 2ν7 , ν2 and ν1 vibrational modes, respectively. 3 ACS Paragon Plus Environment


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e band and Bondybey et al. 13 reported ∼3143, ∼2177 and ∼865 cm−1 progressions in the X assigned them to the fundamental of ν1 , ν2 and ν3 mode, respectively. In addition, the progression of ∼973 cm−1 was assigned to the overtone of ν7 vibration. 13 The corresponding e band were reported at ∼2821, ∼2002, ∼807 and ∼864 cm−1 , respecprogressions in the A tively, in that order by these authors. The excitation of degenerate vibrational modes in the e2 Πu - X e 2 Πg band system was also reported by various other groups. 3,14,16,17 A The linear C4 H·2 + radical cation belongs to the D∞h symmetry point group at its equilibrium configuration. This linear system is prone to the bending instability and the degeneracy of the Π electronic states would split upon distortion along the bending vibrational modes and would exhibit Renner-Teller (RT) activity. Apart from a few computational studies on the e 2 Πg and A e2 Πu states of C4 H·2 + , a rigorous dynamics study electronic structure of the X including RT coupling and coupling between the Π states to elucidate the detailed vibronic structures has not been carried out so far. The present effort is aimed towards this endeavor. e 2 Πg In order to accomplish the proposed goal, the potential energy surfaces of the coupled X e2 Πu electronic states are constructed with the aid of vibronic coupling theory and ab and A initio quantum chemistry calculations. Employing these potential energy surfaces nuclear dynamics calculations are carried out by time-independent and time-dependent quantum e 2 Πg and mechanical methods. Theoretically calculated vibronic band structures of the X e2 Πu states are compared with the experiment. The progression in the vibronic bands are A e A e coupling on the assigned and compared with the literature data. The effect of RT and Xvibronic dynamics is examined and discussed.

II. A.

Theoretical Framework The Vibronic Hamiltonian

In this section we construct a Hamiltonian describing vibronic interactions in the Πg and Πu electronic states of C4 H·2 + . The Hamiltonian is constructed in terms of the dimensionless 4 ACS Paragon Plus Environment

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normal displacement coordinates of the vibrational modes of the electronic ground state of neutral C4 H2 . 18,19 Standard vibronic coupling theory in a diabatic electronic representation and elementary symmetry rules are utilized for the purpose. The 13 vibrational modes of neutral C4 H2 molecule transform according to, Γvib = 3σg+ ⊕ 2σu+ ⊕ 2πg ⊕ 2πu ,

(1)

irreducible representations of D∞h symmetry point group. The activity of vibrational modes in the electronic states j and k is governed by the symmetry rule, Γj ⊗ Γk ⊃ Γvib . The symmetrized direct product of Πg and Πu states in the D∞h point group reads Πg ⊗ Πg = δg + σg+ = Πu ⊗ Πu

(2)

The totally symmetric σg+ vibrational modes can not split the degeneracy of either Πg or Πu electronic states. These modes are Condon active within these electronic states. The δg vibrational modes can split the orbital degeneracy of the Π state in first-order. Because of lack of vibrational modes of δg symmetry in a linear molecule, the first order coupling between the components of doubly degenerate Π state vanishes. However, (πg )2 = (πu )2 ⊃ δg , and therefore, the πg or πu modes can couple them in second-order. This gives rise to RT coupling and a splitting of the Π degeneracy. The components of the split Πg or Πu states can also undergo coupling according to Πg ⊗ Πu = δu + σu+ + σu− .


(3)


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Considering the symmetry rules given above the following vibronic Hamiltonian is derived,   W W W 0 e e e e e Xx− Xy Xx− Ax  Xx      WXy 0 WXy− e e Ay e   H = (TN + V0 )14 +  .   W W  e e Ay e  Ax Ax−   h.c WAy e

(4)

In the above, H0 = TN + V0 , represents the Hamiltonian (assumed to be harmonic) of the reference electronic ground (S0 ) state of C4 H2 with

1 TN = − 2

( 2 ) ∂2 1 ∑ ∂ ∂2 ωi 2 − ωi + , 2 2 ∂Q 2 ∂Q ∂Q i ix iy + i ∈ π , π

∑

i ∈ σg+ , σu

g

(5)

u

and

V0 =

1 2

∑ i ∈

σg+ ,

ωi Q2i + + σu

( ) 1 ∑ ωi Q2ix + Q2iy . 2i ∈ π , π g

(6)

u

The quantity 14 is a 4×4 diagonal unit matrix. The matrix Hamiltonian (with elements W ) in Eq. 4 represents the diabatic energies of the given electronic states of the radical cation (diagonal elements) and their coupling energies (off-diagonal elements). The elements of this matrix are expanded in a standard Taylor series around the reference equilibrium geometry at, Q = 0, in the following way 18

Wjx/jy = E0j + ∑ i ∈ πg , πu

∑

κji Qi +

i ∈ σg+

[ξij (Q2ix + Q2iy )2 ]

±

∑

∑

γij Q2i +

+ i ∈ σg+ , σu

ηij (Q2ix − Q2iy ) ±

∑

i ∈ πg , πu

i ∈ πg , πu


∑

[γij (Q2ix + Q2iy )] +

i ∈ πg , πu

e A e δij (Q4ix − Q4iy ); j ∈ X,

(7)

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Wjx−jy =

∑

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

(8)

i ∈ πg , πu

Wjx−kx/jy−ky =

∑ i ∈

e and k ∈ A e λj−k Qi ; j ∈ X i

(9)

+ σu

In the above equations the quantity E0 j represents the vertical ionization energy (VIE) of the j th electronic state. The two components of the degenerate states and modes are labeled with x/y throughout this study. The quantities κji , ηij and δij represent linear intrastate, quadratic and quartic RT coupling parameters for the symmetric (σg+ ) and degenerate (πg , πu ) vibrational modes, respectively, for the j th electronic state. The first-order coupling parameter of the ith vibrational mode between the electronic states j and k is given by λij−k (this is like pseudo-Jahn-Teller coupling of Jahn-Teller split component electronic states) and γij and ξij are the second-order and fourth-order intrastate coupling parameters respectively, of the ith vibrational mode for the j th electronic state. The summations run over the normal modes of vibration of symmetry specified in the index. The + and - sign applicable to the x and y components of the degenerate state, respectively. We note that the relative sign of various elements of the diabatic electronic Hamiltonian matrix is determined by checking its invariance with respect to various symmetry operations of D∞h symmetry point group and following similar works on benzene and cyclopropane radical cations. 20–22

B.

Electronic structure calculations

The molecular geometry of C4 H2 in the electronic ground state is optimized within the second-order Møller-Plesset perturbation (MP2) level of theory employing Dunning’s augmented polarized valence triple-zeta basis set (aug-cc-pVTZ) 23 using Gaussian 03 suite of programs. 24 The equilibrium geometry converged to D∞h symmetry point group. The equilibrium geometry parameters are given in Table 1. Harmonic vibrational frequency (ωi ) and mass-weighted normal displacement coordinate of vibrational modes are calculated at the 7 ACS Paragon Plus Environment


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optimized molecular geometry. The normal coordinates are multiplied by

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√

ω i (in atomic

units) to transform them to their dimensionless form (Qi ). 19 In this definition equilibrium geometry of C4 H2 occurs at Q=0. The calculated harmonic frequency of 13 vibrational modes (including degeneracy), their symmetry and description along with the available literature data are given in Table 2. The vertical ionization energies (VIEs) of C4 H2 along each vibrational mode for various nuclear geometries, Qi = ±0.10 and ± 0.25 to ± 4.0 with a spacing of 0.25, are computed with the equation-of-motion coupled-cluster singles and doubles (EOMIP-CCSD) 25 method using aug-cc-pVTZ basis set. The EOMIP-CCSD 25 calculations are performed using CFOUR suite of programs. 26 This gives rise to the adiabatic energies of the cationic state when the harmonic contribution [Eq. 6] of the reference neutral state is added. The ab initio energies are then fitted to the adiabatic counterpart of the diabatic electronic Hamiltonian of Eq. 4 by a least squares procedure to estimate the parameters of the Hamiltonian defined in Sec. II.A. The coupling parameters (defined above) and VIEs estimated from the present electronic structure data are presented in Tables 2 and 3, respectively.

C.

Nuclear dynamics

e and A e electronic states of C4 H·2 + is examined in this study. The vibronic spectrum of the X The spectral intensity as a function of final state energy is calculated using Fermi’s golden rule,

2 ∑ i α ˆ I(E) = ⟨Ψv |T |Ψ0 ⟩ δ(E − Evα + E0i ).

(10)

v

Here, |Ψi0 ⟩ represents the wavefunction of the initial vibronic ground state of reference C4 H2 (with energy E0i ), and |Ψαv ⟩ is the wavefunction of the final (α) vibronic state with energy Evα . The quantity Tˆ defines the transition dipole operator. Action of this operator 8 ACS Paragon Plus Environment

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promotes this reference wavefunction to a given electronic state of C4 H·2 + . The vibronic ground (reference) state is expressed as

|Ψi0 ⟩ = |Φ00 ⟩|0⟩

(11)

where |Φ00 ⟩ and |0⟩ represent the electronic and vibrational components of the initial wavefunction, respectively. Within the harmonic approximation the vibrational component (|0⟩) is taken as the direct product of the harmonic oscillator functions along the vibrational modes. In the time-independent framework, I(E) is calculated by a matrix diagonalization method. The Hamiltonian matrix, represented in a direct product basis of diabatic electronic state and one-dimensional harmonic oscillator eigenfunctions of the reference Hamiltonian (TN + V0 ), is diagonalized by using the Lanczos algorithm. 27,28 The exact location of the vibronic energy levels and their relative intensities are the eigenvalues and the squared first components of the Lanczos eigenvectors, respectively. 18,29,30 The spectral envelope can be obtained within the time-dependent framework by Fourier transform of the time autocorrelation function of a wave packet (WP) evolving on the final electronic state as ∫

∞

P (E) ≈ 2Re

eiEt/¯h ⟨Ψα (0)|τ † e−iHt/¯h τ |Ψα (0)⟩dt,

(12)

0

∫ ≈ 2Re

∞

eiEt/¯h C α (t) dt.

(13)

0

Here, C α (t) = ⟨Ψα (t = 0)|Ψα (t)⟩ represents the time autocorrelation function of the WP initially prepared on the αth electronic state of C4 H·2 + . The initial WP at t = 0, i.e., |Ψi0 >, is vertically promoted to the final state α, |Ψα (0) >, and its time-evolution is described by, Ψα (t) = e−iHt/¯h Ψα (0). The quantity τ represents the transition dipole matrix: τ † =



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(τ X , τ A ), where τ α = ⟨Φα |Tˆ|Φ0 ⟩. Within the generalized Condon approximation in a diabatic electronic basis, the matrix elements of Tˆ are set to be independent of the nuclear coordinates and are treated as constant. 30 Finally, the composite spectrum is calculated by combining the partial spectra obtained by propagating WP on each of the electronic states. The WP propagation calculations are carried out within the framework of multi-configuration time-dependent Hartree (MCTDH) method. 31–33 The multiset formalism of this method allows to combine several vibrational degrees of freedom (DOF) into a ”particle” (p). With such a combination the MCTDH wavefunction can be expressed as

Ψ(q1 , ..., qp , t) =

n1 ∑ j1 =1

...

np ∑ jp =1

Aj1 ,...,jp (t)

p ∏

(k)

φjk (qk , t).

(14)

k=1

Here, qk = (Qi , Qj , ..) is the set of DOF combined together in a single particle and Aj1 ,...,jp (k)

denote the MCTDH expansion coefficients. The time-dependent basis functions, φjk , are the single-particle functions (SPFs) and nk is the number of SPFs used to describe the motion of the k th DOF. The MCTDH equations of motion are solved by representing SPFs in a primitive time-independent basis. A harmonic oscillator discrete variable representation (DVR) is used for the primitive basis. All time-dependent WP calculations are carried out using the Heidelberg MCTDH suite of program modules. 31–33

III. A.

Results and discussion e -A e conAdiabatic potential energy surface : RT effect and X ical intersections

In this section we examine the topography of the adiabatic PESs of the degenerate ground and first excited doublet electronic states of C4 H·2 + obtained by diagonalizing the diabatic electronic Hamiltonian (Eq. 4). According to the symmetry selection rules given in Sec. II.A, the totally symmetric vibrational modes ν1 -ν3 cannot split the degeneracy of the 10 ACS Paragon Plus Environment

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e 2 Πg and A e2 Πu electronic states. One dimensional view graphs of the multidimensional X PESs of C4 H·2 + along a given totally symmetric (σg+ ) vibrational mode, keeping others at their equilibrium values are shown in Fig. 1. In the latter the adiabatic electronic energies calculated ab initio

and those obtained from the present vibronic model are shown as

points and lines, respectively. It can be seen from this figure that the minimum of the e state displaced considerably away from the equilibrium minimum of the reference state X e state along ν3 . occurring at Q=0 along ν2 . A similar observations can be made for the A e and A e The impact of these shift of equilibrium minimum on the vibronic dynamics of the X e and states of C4 H·2 + is discussed later in the text. The RT effect splits the degeneracy of X e electronic states when the molecule is distorted along πg (ν6 -ν7 ) and πu (ν8 -ν9 ) vibrational A modes. The potential energy curves along the x component of πg and πu modes and are plotted in Figs. 2(a-b) and Figs. 2(c-d), respectively. Like in Fig.1 the points and lines in these figure represent the adiabatic electronic energies calculated ab initio and those obtained from the present vibronic model, respectively. It can be seen from Figs. 1 and 2 that the computed ab initio energies compare very well to those obtained from the vibronic coupling model. The RT splitting of electronic degeneracy is small, quite obvious from Fig. 2. The quartic RT coupling terms of the Hamiltonian are very small (Table. 2), they make only minor contribution to the electronic energies in Fig. 2(d). The relevant stationary points (c) e A e conical interactions of the PESs, viz., the energetic minimum (Vmin ) of the seam of X-

and the energetic minimum (Vmin ) of the electronic states are calculated. The energetic e and A e states of C4 H·2 + occurs at ∼ 10.23 and ∼ 12.85 eV, respectively. The minimum of X e A e seam minimum occurs at a very high energy. Therefore the coupling of the X e and A e Xstates is expected to have a negligible role in the nuclear dynamics.



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B.

e and A e states of C4 H· + Vibronic band structure of the X 2

1.1

Uncoupled state spectrum and Renner-Teller effect

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e A) e on the In order to demonstrate the effect of nonadiabatic coupling (both RT and Xvibronic structure of the photoionization bands of C4 H2 , we first examine the vibrational ene and A e states of C4 H·2 + . The Xe A e coupling is treated ergy level structure of the uncoupled X next. The uncoupled state spectra shown in Fig. 3 are calculated by the time-independent matrix diagonalization approach as discussed in Sec. II. C. The vibrational basis used in each calculation is given in Table 4. The theoretical stick line spectrum of each electronic state (shown in panel a) is calculated with the vibronic Hamiltonian of Eq. 4 and including all three totally symmetric vibrational modes ν1 -ν3 . A Lorentzian line shape function of 20 meV full width at the half maximum (FWHM) is used to convolute the stick line spectrum e A e in each case and to generate the respective spectral envelop. In a situation without Xcoupling and in absence of any intermode coupling terms, the Hamiltonian for the degene 2 Πg and A e2 Πu states is separable in terms of the σ + , πg and πu modes. Therefore, erate X g the partial spectra calculated with the totally symmetric and degenerate vibrational modes separately, can be convoluted to get the composite vibronic structure of the electronic states. e and A e states obtained by including the σ + (ν1 -ν3 ) modes only is The spectrum of the X g e state, excitation of ν1 , ν2 and ν3 mode is weak, strong and shown in Fig. 3 (a). In the X moderate, respectively. The fundamentals of ν1 , ν2 , and ν3 vibration are ∼3484, ∼2227 and e band. The X e 2 Πg state of ∼885 cm−1 spaced, respectively, from the origin peak of the X C4 H·2 + originates from ionization of an electron from an antibonding (along ν2 ) molecue state of lar orbital of C4 H2 . The vibrational frequency of ν2 therefore increases in the X C4 H·2 + as compared to the same in the electronic ground state of C4 H2 (Table 1). Since the coupling strength of the ν2 mode is particularly strong [Table 2], this mode forms an exe state spectrum, the same trend can be observed for the ν3 mode tended progression in the X e state. Peak spacing of ∼3481, ∼2225 and ∼932 cm−1 due to the fundamental of in the A


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e state spectrum. In this state ν1 , ν2 and ν3 vibrational modes, respectively, is found in the A the coupling strength [cf., Table. 2] of ν1 , ν2 and ν3 vibrational modes is weak, moderate and strong, respectively, in that order. The partial spectra calculated with the degenerate RT modes (πg and πu ) are shown in panel b of Fig. 3. The composite spectrum obtained by convoluting the symmetric and e and A e states presented in panel a and panel b of Fig. 3, degenerate mode spectrum of X is shown in panel c of this figure. The spectra presented in panel b of Fig. 3 reveal a weak e and A e states. As a result, the contribution of RT coupling in the vibronic structure of X overall spectral envelop (panel c of Fig. 3) appears to be very similar and close to that shown in panel a of Fig. 3. The weak RT coupling within each of these states however, has some effects on the spectral broadening and possibly excitation of additional vibrational modes as the line structure within each spectral peak in panel c suggests. To this end some of the spectral progressions reported in experiments are compared with our e band is reported by Baker theoretical findings. Excitation of the ν2 vibrational mode in the X et al. . 7 From Table 2 it can be seen that this mode has the strongest excitation strength e state. Our theoretical analysis reveals that compared to all other symmetric modes in the X the fundamental of ν2 appears at ∼2227 cm−1 as compared to its value ∼2121 cm−1 reported in the experiment of Baker et al. . 7 Bondybey et al.

13

also identified the fundamental of

ν1 , ν2 , ν3 modes and the overtone of ν7 vibrational mode at ∼3143, ∼2177, ∼865 and ∼973 cm−1 , respectively. These results are in good accord with the progressions found at ∼3484, ∼2227, ∼885 and ∼842 cm−1 , respectively, in that order in the present study. In order to confirm the assignments, vibronic wavefunctions are calculated by the block-improvedrelaxation method using the MCTDH program modules. 38–40 The wavefunction probability density plots corresponding to ∼3484, ∼2227, ∼885 and ∼842, ∼1295, ∼1442 and ∼3112 cm−1 lines in a given coordinate space are shown in Figs. 4(a-g), respectively. It can be seen from these plots that the wavefunction in panels a, b and c reveals one node along ν1 , ν2 and ν3 , respectively, and therefore corresponds to the fundamental along these modes. The plot



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in panel d, e and f confirms two quantum excitation along ν7 , ν6 and ν8 , respectively. The wavefunction of panel g, on the otherhand, reveals one quantum excitation each along ν2 and ν3 and hence represents a combination peak. e state major progressions are formed by ν1 , ν2 , ν3 and the overtone of ν7 , ν6 and ν8 In the A e state and assigned them to vibrational modes. Many groups identified progressions in the A the corresponding vibrational modes. A comparative account of all these results along with the present findings is given in Table 5.

1.2

e -A e Coupled state spectrum and time-dependent dynamics X

e and A e states in first-order The symmetry rule (stated in Sec.II.A) allows the coupling of X through the vibrational modes of σu+ symmetry. It can be seen from Table. 2 that the vibrational modes ν4 and ν5 have nonzero first-order coupling. It is mentioned in sec. III.A e and A e states occurs that the energetic minimum of the seam of conical interactions of X e A e coupling is, therefore, not expected to affect the nuclear at much higher energy and Xe A e coupled states nuclear dynamics on these states. In order to confirm, we carried out Xdynamics calculations. In the coupled states situation the separability of the Hamiltonian (as discussed in the previous section) in terms of symmetric and degenerate vibrational modes is no longer valid. We, therefore, carried out full dimensional calculations with two degenerate coupled electronic states including all thirteen vibrational degrees of freedom. The dynamics calculations are carried out by propagating WPs employing the MCTDH program modules. 34 Four different calculations are carried out by initially preparing the WP on each component of the Π states separately. The vibrational basis functions used in the calculations are given in Table 6. The WP is propagated for 200 fs in the coupled manifold of electronic states and the time autocorrelation function, diabatic and adiabatic electronic populations are recorded during the propagation. The time autocorrelation functions obtained from four different calculations are combined, damped with an exponential function, e(−t/τr ) (with τr = 66 fs) and Fourier transformed to generate the spectral envelopes of the electronic states.


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The numerical details of converged MCTDH calculations are given in Table 6. The resulting theoretical spectral envelopes are shown in panel a of Fig. 5 along with the experimental results of 11 in panel b. It can be seen that the theoretical results compare very well with the experiment. When compared with the theoretical results presented in e A e intersection has negligible effects on the vibronic Fig. 3(c), it can be seen that the Xband structure of both the states. On the other hand, the RT effect is weak but it causes spectral broadening by increasing the vibronic line density as can be seen from Fig. 3(c). e state relative to From the results presented in Fig. 5, the position of the origin line of the A e state is estimated ∼ 471 nm. A comparison of the latter with 506.9 the origin line of the X nm 3,4 is not very favorable. However, a more meaningful comparison requires a study of the absorption spectrum of C4 H·2 + . e A e coupled states dynamics is The time-dependence of electronic population in the Xshown in Fig. 6. The diabatic (panel a) and adiabatic (panel b) electron populations are shown in this figure for an initial excitation of the WP to one component of the diabatic e state. While the diabatic population of the initially prepared state starts from 1.0, its A adiabatic counterpart has a population of ∼0.5 at t = 0. It can be seen from both the e state during the entire course of the panels of Fig. 6 that the WP does not move to the X dynamics. The RT coupling drives the WP motion back and forth between the RT split component states only. The population dynamics retains all the features mentioned above e state. In this case the WP when the WP is initially prepared on one component of the X e state. It therefore emerges that the X e-A e coupling has insignificant does not move to the A role on the dynamics of both the states. The RT coupling is weak, however, has noticeable e and A e state of diacetylene radical cation. effect on the vibronic dynamics of both X



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IV.

Summary

e 2 Πg and A e2 Πu band system of diacetylene radical cation is theoretically investigated The X with the aid of ab initio quantum chemistry calculations, vibronic coupling theory and timeindependent and time-dependent quantum dynamics methods. The broad band vibronic spectra calculated theoretically compare well with the experiments. The progression of vibrational modes in each electronic states is identified and discussed in relation to various results available in the literature. It is found that ν1 , ν2 and ν3 vibrational modes form e ans A e states. While the mode ν2 forms an extended progressions progression in both X e band, the mode ν3 forms such a progression in the A e band. It is established in the X e and A e state is weak. However, this coupling triggers that the RT coupling within each X the excitation of degenerate vibrational modes within each electronic states. Two quantum excitation of degenerate vibrational mode ν6 , ν7 and ν8 is found in the vibronic structure of e and A e band. The pseudo-Jahn-Teller type of X e -A e coupling, through the vibrational both X modes of σu+ symmetry, has in particular, no effect on the dynamics.

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. Council of Scientific and Industrial Research (CSIR), New Delhi is also acknowledged by A. G. for a doctoral research fellowship. We thank Rudraditya Sarkar for his help in calculating the vibronic wavefunctions.


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References ¨ (1) H. Sch¨ uler and L. Reinebeck, Uber Neue Spektren in der Glimmentladung Mit Naph¨ thalindampf. Z. Naturforsch. 1951, 6a, 271-275 ; Uber ein Spektrum im Sichtbaren beobachtet bei Acetylen, thylen und aromatischen Moleklen 1952, 7a, 285-259. (2) Callomon, J. H. An Emission Spectrum of the Diacetylene Ion, A Study of Sch¨ uler ”T” Spectrum Under High Resolution Can. J. Phys. 1956, 34, 1046-1074. (3) Motylewski, T.; Linnartz, H.; Vaizert, O.; Maier, J. P.; Galazutdinov, G. A.; Musaev, F. A.; Krelowski, J.; Walker, G. A. H.; Bohlender, D. A. Gas-Phase Electronic Spectra of Carbon-Chain Radicals Compared With Diffuse Interstellar Band Observations. Astrophys. J. 2000, 531, 312-320. (4) Krelowski, J.; Beletsky, Y.; Galazutdinov, G. A.; Kolos, R.; Gronowski, M.; Locurto, G. Evidence for Diacetylene Cation as the Carrier of a Diffuse Interstellar Band. Astrophys. J. 2010, 714, L64-L67. (5) Maier, J. P.; Chakrabarty, S.; Mazzotti, F. J.; Rice, C. A.; Dietsche, R.; Walker, G. A. H.;Bohlender, D. A.; Assignment of 5069 A0 Diffuse Interstellar Band to HC4H+: Disagreement With Laboratory Absorption Band Astrophys. J. 2011, 729, L20(4pp). (6) Maier, J. P.; Walker, G. A. H.; Bohlender, D. A.; On The Possible Role of Carbon Chains as Carriers of Diffuse Interstellar Bands Astrophys. J. 2004 602, 286-290. (7) Baker, C.; Turner, D. W. Photoelectron Spectra of Acetylene, Diacetylene, and their Deutero-derivatives Chemical Communications 1967, 797-799. (8) Kaiser, R. I.; Sun, B. J.; Lin, H. M.; Chang, A. H. H.; Mebel, A. M.; Kostko, O. An experimental and Theoretical Study on the ionization Energies of Polyyenes (H(CC)nH; n = 19). Astrophys. J. 2010, 719, 1884-1889.



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(9) Gronowski, M.; Kolos, R.; Krelowski, J. A Theoretical Study on Structure And Spectroscopy of C4 H2+ isomers. Chem. Phys. Lett. 2013, 582, 56-59. (10) Fulara, J.; Gr¨ utter, M.; Maier, J. P. Higher Excited Electronic Transitions of Polyacetylene Cations HC2n H + n = 2-7 in Neon Matrixes. J. Phys. Chem. A 2007 111, 11831-11836. (11) Allan, M.; Kloster-Jensen, E.; Maier, J. P. Emission Spectra of the Radical Cation of e2 Πu → X e 2 Πg ), Triacetylene (A e2 Πg → X e 2 Πu ), and Tetraacetylene (A e2 Πu Diacetylene (A e 2 Πg , 00 , and the Lifetimes of Some Vibronic Level of the A e States. Chem. Phys. →X 0 1976, 7, 11-18. (12) Bally, T.; Tang, W.; Jungen, M. The Electronic Structure of the Radical Cations of Butadiene, Vinylacetylene and Diacetylene: Similarities and Differences. Chem. Phys. Lett. 1992, 190, 453-459. (13) Bondybey, V. L.; English, J. H. Electronic Spectrum of the Diacetylene Radical Cation in Solid Rare Gases. J. Chem. Phys. 1979, 71, 777-782. (14) Schmidt, T. W.; Pino, T.; van Wijngaarden, J.; Tikhomirow, K.; Guthe, F.; J. P Maier, J. P. Electronic Photodissociation Spectra of the Arn C4 H2+ (n= 1 4) Weakly Bound Cationic Complexes. J. Mol. Spectrosc. 2003, 222, 86-92. (15) Freivogel, P.; Fulara, J.; Lessen, D.; Forney, D.; Maier, J. P. Absorption Spectra of Conjugated Hydrocarbon Cation Chains in Neon Matrices. Chem. Phys. 1994, 189, 335-341. (16) Smith, W. L. The Absorption Spectrum of Diacetylene in The Vacuum Ultraviolet. Proc. R. Soc. A. 1967, 300, 519-533. (17) Baker, C.; Turner, D. W. High Resolution Molecular Photoelectron Spectroscopy. III. Acetylenes and Aza-acetylenest. Proc. R. Soc. A. 1968, 308, 19-37. 18 ACS Paragon Plus Environment

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(18) Köppel. H.; Domcke. W.; Cederbaum. L. S. Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation. Adv. Chem. Phys. 1984, 57 , 59-246. (19) E.B. Wilson Jr., E. B.; Decius, J. C.; Cross, P. C. Molecular vibrations (McGraw-Hill, New York, 1955). (20) Döscher, M.; Köppel, H.; Szalay, P. Multistate Vibronic Interactions in the Benzene Radical Cation. I. Electronic Structure Calculations. J. Chem. Phys. 2002 117, 26452656. (21) Venkatesan, T. S.; Meyer, H. -D.; Köppel, H.; Cederbaum, L. S.; Mahapatra, S. Multimode JahnTeller and Pseudo-JahnTeller Interactions in the Cyclopropane Radical Cation: Complex Vibronic Spectra and Nonradiative Decay Dynamics. J. Phys. Chem. A 2007, 111, 1746-1761. (22) Mondal, T.; Reddy, S. R.; Mahapatra, S. Photophysics of Fluorinated benzene. III. Hexafluorobenzene. J. Chem. Phys. 2012, 137, 054311-054317. (23) 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. (24) Frisch, M. J.; Trucks, G. W.; Schlegel H. B. et al. , GAUSSIAN 03, Revision B. 05, Gaussian, Inc., Pittsburgh PA, 2003. (25) Stanon, J. F.; Gauss, J. Analytic Energy Derivatives for Ionized States Described by the Equation-of-motion Coupled cluster Method. J. Chem. Phys. 1994, 101, 8938-8944. (26) CFOUR, a quantum chemical program package written by Stanton, JF.; Gauss, J.; Harding, M. E.; Szalay, P. G.; with contributions from Auer, A. A.; Bartlett, R. J.; Benedikt, U.; Berger, C.; Bernholdt, D. E.; Bomble, Y. J.; Cheng, L.; Christiansen, O.; Heckert, M.; Heun, O.; Huber, C.; Jagau,T-C.; Jonsson, D.; Jusélius, J.; Klein, ´ Neill, D. P.; K.; Lauderdale, W. J.; Matthews, D. A.; Metzroth, T.; Mck, L. A.; O 19 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 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Price, D. R.; Prochnow, E.; Puzzarini, C.; Ruud, K.; Schiffmann, F.; Schwalbach, W.; Simmons, C.; Stopkowicz, S.; Tajti, A.; Vzquez, J.; Wang, F.; Watts, J. D. and the integral packages MOLECULE (Almlf, 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 Wllen, C. For the current version, see http://www.cfour.de (27) 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. (28) Cullum. J.; Willoughby. R. Lanczos Algorithms for Large Symmetric Eigenvalue Problems (Birkhäuser, Boston, 1985), Vols. I and II. (29) Domcke. W.; Köppel. H. Encyclopedia of Computational Chemistry-Theoretical and Physical chemistry, edited by H.F.Schacfer. III. ( Wiley. New York, 1998). (30) Domcke. W.; Köppel. H.; Cederbaum. L. S. Spectroscopic Effects of Conical Intersections of Molecular Potential Energy Surfaces. Mol. Phys. 1981, 43, 851-875. (31) Meyer. H. -D.; Manthe. U.; Cederbaum. L. S. The Multi-Configurational Timedependent Hartree Approach. Chem. Phys. Lett. 1990, 165, 73-78. (32) 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. (33) 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. (34) Worth. G. A.; Beck. M. H.; Jäckle. A.; Meyer. H. -D. The mctdh package, Version 8.4, (2007), University of Heidelberg, Heidelberg, Geramny. See: http://mctdh.uni-hd.de.


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(35) S. Thorwirth, S.; Harding, M. E.; Muders, D.; Gauss, J. The Empirical Equilibrium Structure of Diacetylene. J. Mol. Spectrosc. 2008, 251, 220-223. (36) Williams, G. A.; Macdonald, J. N. An ab initio and Experimental Study of the Harmonic Force Field of Diacetylene. J. Mol. Struct. 1994 320, 217-225. (37) G. Guelachvili, A.M. Craig, and D.A. Ramsay, High-Resolution Fourier Studies of Diacetylene in the Regions of the ν4 and ν5 Fundamentals. J. Mol. Spectrosc. 1984, 105, 156-192. (38) 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. (39) Doriol, L. J.; Gatti, F.; Iung, 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, 224109-1. (40) Sarkar, R.; Mahapatra, S. Vibronic Dynamics of Electronic Ground State of CH2 F2+ and Its Deuterated Isotopomer. J. Phys. Chem. A. 2016, 120, 3504 (2016).



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Table 1: Symmetry and harmonic frequency (in cm−1 ) of vibrational modes of the ground electronic state of C4 H2 . The experimental results represent fundamental frequencies. Symm. σg+

Mode ν1 ν2 ν3

MP2/aug-cc-pVTZ 3475 2185 897

(fc)CCSD(T)/cc-pVQZ 35 3457 2235 892

Ref. 36† 3489 2222 885

Expt. 37 3332 2189 872

Description C-H symmetric stretch C≡C symmetric stretch C-C symmetric stretch

σu+

ν4 ν5

3475 2001

3458 2057

3490 2050

3333 2019

C-H antisymmetric stretch C≡C antisymmetric stretch

πg

ν6 ν7

608 448

632 481

638 490

626 483

H-C≡C antisymmetric bend C≡C-C antisymmetric bend

πu

ν8 ν9

619 219

634 220

641 223

628 220

H-C≡C symmetric bend C≡C-C symmetric bend

Table 2: Linear (κi and λX−A ), quadratic (γi and ηi ) and quartic (ξi and δi ) i e 2 Πg and A e2 Πu eleccoupling constants of the Hamiltonian [cf., Eqs.(4-9)]for the X tronic states of C4 H·2 + . Dimensionless excitation strengths are given in the squarebrackets. All quantities are given in the eV unit. Symmetry Mode Freq σg+ σu+ πg πu

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9

0.4309 0.2710 0.1112 0.4309 0.2481 0.0754 0.0556 0.0768 0.0271

κi or ηi (δi )

γi (ξi )

κi or ηi (δi )

γi (ξi )

λX−A i

e 2 Πg e2 Πu X A -0.0136 [0.0005] 0.0011 -0.0203 [0.0011] 0.0007 -0.2393 [0.3899] 0.0052 -0.0646 [0.0284] 0.0049 0.0396 [0.0634] -0.0015 -0.1373 [0.7623] 0.0045 0.0006 0.0010 0.0234 [0.0014] -0.0041 0.0176 0.1737 [0.2451] -0.0045 (0.0001) -0.0009 (0.0000) 0.0027 (0.0000) -0.0036 (0.0000) -0.0044 (0.0000) 0.0026 (0.0000) -0.0025 (0.0000) -0.0129 (0.0000) -0.0079 (0.0001) 0.0025 (0.0001) 0.0037 (0.0000) -0.0037 (0.0000) -0.0004 (0.0000) -0.0210 (0.0001) -0.0005 (0.0000) -0.0189 (0.0000) -


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e and A e states of C4 H·2 + . Table 3: Vertical ionization energies (in eV) of the X States e 2 Πg X e2 Πu A

EOMIP-CCSD(aug-cc-pVTZ) Ref. 7 Ref. 8 10.33 10.17 10.03 12.94 12.62 -

Table 4: Number of Harmonic oscillator (HO) basis functions for vibrational modes, the dimension of the secular matrix and the number of Lanczos iterations e and A e states of used to calculate the converged theoretical stick spectra of X C4 H·2 + shown in various figures. Vibrational modes( HO basis functions)

Dimension of the matrix Lanczos iterations

ν1 , ν2 , ν3 (4,8,10) ν6x , ν6y , ν7x , ν7y (4,4,6,6) ν1 , ν2 , ν3 , ν6x , ν6y ,ν7x , ν7y , ν8x , ν8y , ν9x , ν9y (3,6,8,4,4,4,4,4,4,4,4)

320 576 9437184

10000 10000 10000

Figure(s) panel a of 3 panel b of 3 panel c of 3

e2 Πu elecTable 5: A comparative account on the vibrational progressions in the A tronic state of diacetylene radical cation. The numbers represent the frequency of the vibrational mode in cm−1 . * Experimental data. Mode ν1 ν2 ν3 2ν6 2ν7 2ν8

Present results 3481 2225 932 1154 757 1172

Ref. 2∗ 3137 2177 861 971 -

Ref. 16∗ 3158 2096 987 -

Ref 14∗ 1961 806 862 -

Ref. 13∗ 2821 2002 807 861 -

Table 6: Normal mode combinations, sizes of the primitive and the single particle basis used in the wave packet propagation within the MCTDH framework in the e − A) e electronic manifold using the complete vibronic Hamiltonian four coupled (X of Eq.(4-9). First column denotes the vibrational degrees of freedom (DOF) which are combined to particles. Second column gives the number of primitive basis functions for each DOF. Third column gives the number of single particle functions (SPFs) for each electronic state. Normal modes ν1 , ν4 , ν6y , ν8x , ν9y ν2 , ν5 , ν7x , ν8y ν3 , ν6x , ν7y , ν9x

Primitive Basis (4,4,6,6) (8,8,6,6) (10,6,6,6)

SPF Basis [12,12,8,12] [12,10,8,10] [8,12,10,10]



20

~ ~ A

(a)

X

16

v1

~3475 cm

-1

12

8 ~

(b)

~ X ~ A

16 v2

V [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

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~2185 cm

-1

12

8 ~

(c)

~ X ~ A

16 v3 ~897 cm-1 12

8

-3

0

3

Q e and A e electronic states of Figure 1: Adiabatic potential energies of the X · + C4 H2 along the totally symmetric vibrational modes: (a) ν1 , (b) ν2 and (c) ν3 ). The energies obtained from the present vibronic model and the computed ab initio energies are shown by lines 24 and points, respectively. ACS Paragon Plus Environment

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14 (a) v6 ~608 cm-1

14

(b) v7

~ A

-1

13

~

~

11

~448 cm ~ A

13

~

~ 11

~ X

10

~ X

10 -4

-2

0

2

4

-4

-2

14 (c) v8 ~619 cm-1

V (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


~

4

-1 (d) v9 ~219 cm

12.8

~

~

~ X

10.3 11

2

~ A

12.9

~ A

13 12 ~

0

~ X

10

10.2 -4

-2

Q 0

2

4

-4

-2

Q0

2

4

Figure 2: Same as in Fig. 1, along degenerate πg (ν6 -ν7 ) and πu (ν8 -ν9 ) vibrational modes shown in panel (a-b) and (c-d), respectively.



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e and A e electronic Figure 3: Vibrational energy level spectrum of the uncoupled X · + states of C4 H2 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.


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(a)

(b)

(c)

(d)

Figure 4: Reduced density plots of the vibronic wavefunctions of the fundamental of ν1 , ν2 and ν3 (panel a, b and c, respectively) and first overtone of ν7 , ν6 and ν8 (panel d, e and e state spectrum of C4 H·2 + . The wavefunction in panel g f, respectively) excited in the X represent the combination peak ν2 + ν3 .



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)

(f)

(g)

Figure. 4.(contd.)


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e A e coupled electronic states of C4 H·2 + . Figure 5: Vibronic band structure of the XRelative intensity (in arbitrary units) is plotted as a function of the energy of the vibronic states of C4 H·2 + . The present theoretical results are shown in panel a and the experimental results of Ref. 7 are reproduced in panel b.



~ Diabatic A

(a)

0.8 Ax Ay

0.4

Population

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0 ~ Adiabatic A (b) 0.8

0.4

0 0

50

100 Time (fs)

150

200

Figure 6: Time dependence of diabatic (panel a) and adiabatic (panel b) electron e A e coupled states nuclear dynamics of C4 H·2 + . The initial populations in the Xe state. (at t=0) WP is located on one of the components of the diabatic A


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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

X 2g

.+

h

-e

Vibronic band structure of the X 2g− A 2u coupled electronic states of C4H2.+ .

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A 2u

Vibronic Coupling in the XÌ2Î g - American

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