Photodetachment spectroscopy of carbon doped anionic boron cluster, CB9- : A theoretical study

Accepted Manuscript Photodetachment spectroscopy of carbon dopped anionic boron cluster, CB9− : A theoretical study Rudraditya Sarkar, Daradi Baishya, S. Mahapatra PII: DOI: Reference:

S0301-0104(18)30526-3 https://doi.org/10.1016/j.chemphys.2018.07.017 CHEMPH 10081

To appear in:

Chemical Physics

Received Date: Accepted Date:

15 May 2018 17 July 2018

Please cite this article as: R. Sarkar, D. Baishya, S. Mahapatra, Photodetachment spectroscopy of carbon dopped

anionic boron cluster, CB9− : A theoretical study, Chemical Physics (2018), doi: https://doi.org/10.1016/j.chemphys.

2018.07.017

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Photodetachment spectroscopy of carbon dopped anionic boron † cluster, CB− 9 : A theoretical study

Rudraditya Sarkar, Daradi Baishya and S. Mahapatra∗ School of Chemistry, University of Hyderabad, Hyderabad 500 046, India (Dated: July 18, 2018)

Abstract The photodetachment spectroscopy of CB− 9 is theoretically studied in this paper. Extensive ab initio quantum chemistry calculations are carried out to construct the potential energy surfaces of the electronic ground and excited states of CB9 . With the aid of these calculated adiabatic electronic energies, a vibronic coupling model is developed in a diabatic electronic basis and in terms of normal coordinates of vibrational modes following the standard vibronic coupling theory originally developed by K¨oppel, Domcke, and Cederbaum [Adv. Chem. Phys. 57 (1984) 59]. Employing the developed diabatic electronic model, first principles nuclear dynamics study is carried out to calculate the vibronic structure of photodetachment bands of CB− 9 . A systematic study is performed to elucidate the impact of electronic nonadiabatic effects on the photodetachment bands. The vibronic structure of the latter is assigned in terms of excitation of vibrational modes and the results are compared with the experimental findings. The internal conversion dynamics through conical intersections of electronic states is examined by analyzing the time-dependent electronic populations. The theoretical results are found to be in good agreement with the experimental data. The detailed theoretical study carried out here unambiguously supports the prediction of the global minimum structure of CB− 9.

∗

Corresponding author, E-mail: [email protected] †

Dedicated to Professor Dr. Wolfgang Domcke on the ocassion of his 70th birthday.

1

I.

INTRODUCTION

Boron is uniquely known for its unusual bonding properties [1, 2]. As permitted by its electronic configuration the three-centre two-electron bonding of boron makes it unique and different from the other elements in the Periodic table. The structural and geometrical properties of numerous pure boron and dopped boron clusters are explained by chemical bonding analyses [3–21]. The concept σ and π aromaticity/anti-aromaticity [27] is used to explain the stability of various conformers of boron clusters. The joint experimental and computational studies over the last decades showed bare anionic boron clusters up to B− 30 [3– 26] and carbon dopped ten-atom Cx B− 10−x (x= 3-10) carbon-boron mixed clusters [28–47] possess planar or quasi-planar wheel to monocyclic ring structure. It is well established that bonding pattern of peripheral boron atoms of these boron clusters is two-center twoelectron (2c-2e) localized σ bond type, while the central boron atom is connected with the peripheral boron atoms via delocalized multi-center σ and π bonds [48–51]. It is also found from the theoretical study [47, 51] that the carbon prefers to stay at the peripheral position rather than the center of a mixed carbon-boron cluster due to its higher electronegativity as compared to boron. As a result, carbon atom avoids the central position in the global minimum structure of the mixed carbon-boron clusters. The mentioned clusters are prepared in the plasma reaction between laser vaporized boron and carbon [47]. Photoelectron spectroscopy measurement at different laser wavelengths is extensively used to characterize these clusters [47]. The structural characterization of these clusters is performed by searching various local minimum structures and global minimum structure. The vertical and adiabatic detachment energies (VDEs/ADEs) of several electronic states at the local minimum structures and global minimum of the cluster calculated by ab initio methods are compared with the experimental recording. A close agreement between the two leads to the assignment of photodetachment bands, structure of the global minimum and identification of energetically close isomers of the cluster [8, 9, 11, 12, 15, 47]. These computational exercises however do not lead to a detailed understanding of the structure of the photodetachment bands and the topography of the underlying potential energy surfaces. In the recent past, we have made some efforts towards this endeavor for bare boron − − − clusters, viz., B− 3 [52], B4 , B5 [53] and B7 [54] and partially hydrogenated boron cluster

H2 B− 7 and its deuterated isomer [55]. It turned out that different types of vibronic coupling 2

mechanism of electronic states play crucial role in the structure of the detachment bands. In addition multiple energetically close isomers often makes contribution in the overall band structures recorded in the experiment [54]. A similar attempt is made in the present study aiming to develop a theoretical model to examine detailed nuclear dynamics and to provide an understanding of the observed photodetachment bands of CB− 9 [47]. The photodetachment spectrum of CB− 9 recorded by Galeev et al. [47] at 266 nm using e band with average Nd:YAG laser showed a vibronically resolved ground electronic state (X)

vibrational progression of 340±50 cm−1 and another relatively broad band due to the first e with an energy gap of ∼0.8 eV between the two [cf. Fig. 2(a) excited electronic state (A)

of Ref. [47]]. The experimentally measured vertical detachment energies (VDEs) of these two bands are 3.61±0.03 eV and 4.49 eV, respectively. Another recording at 193 nm using ArF excimer laser by the same authors showed two more prominent bands due to second e and third (C) e excited states of CB9 with VDEs of 4.88 eV and 5.74 eV, respectively (B)

(cf. Fig. 2(b) of Ref. [47]). Electronic structure calculations performed by the same group

revealed a global minimum structure of CB− 9 with C2v point group symmetry. The recorded photodetachment band is predicted to be solely originating from this isomer as the second lowest energy isomer of CB− 9 of Cs symmetry point group possesses significantly higher energy of ∼9.9 kcal/mol (cf. I.1 and I.2 of Fig. 4 in Ref. [47]). On the basis of theoretically calculated and experimentally observed VDEs, the peaks in the observed spectrum are e A, e B e and C e electronic states of CB− identified due to X, 9 . The adaptive natural density partitioning (AdNDP) chemical bonding analysis [56] by the same group indicates that the CB− 9 cluster possesses seven 2c-2e B-B σ bonds and two 2c-2e C-B σ bonds in the periphery of the cluster, whereas four 3c-2e delocalized σ bonds present between the central atom and peripheral atoms. As a result, delocalization in the σ bond produces the σ antiaromaticity in the cluster. On the other hand, one 3c-2e and two 4c-2e π bonds between the central atom and the peripheral atoms produces π aromaticity in the cluster. While the joint experimental and computational work discussed above helped to identify the electronic state and the underlying structure of the isomer, detailed vibronic structure of each band and the progressions therein still remain to be understood. The present effort is made towards this direction. To this effort we set out to study the electronic structure and nuclear dynamics in the first four low-lying electronic states taking the global minimum structure of CB− 9 as reference. The optimized equilibrium structures, harmonic frequency 3

and coordinate of vibrational modes are calculated at the second-order Møller-Plesset perturbation (MP2) level of theory [57]. The adiabatic electronic energies are calculated by complete active space self consistent field (CASSCF)-muti-reference configurational interaction (MRCI) methods [58]. A diabatic vibronic Hamiltonian of four low-lying electronic states of CB9 is constructed and nuclear dynamics calculations are performed by timeindependent and time-dependent quantum mechanical methods. The vibronic structure of the photodetachment bands is examined at length and the impact of electronic nonadiabatic coupling on them is elucidated. The theoretical results are shown to be in good accord with the experimental findings [47].

II.

COMPUTATIONAL DETAILS OF ELECTRONIC STRUCTURE CALCULA-

TIONS

The optimized equilibrium geometry of the electronic ground state of CB− 9 (the reference state) is calculated by the MP2 method employing the correlation-consistent polarized valence triple zeta (cc-pVTZ) basis set of Dunning [59]. GAUSSIAN-09 [60] suite of programs is used for this purpose. The optimized equilibrium structure of the CB− 9 in the electronic ground state converges to C2v point group symmetry and leads to 1 A1 electronic term for this closed shell system. The equilibrium harmonic vibrational frequencies of this equilibrium configuration (treated as the reference state), ωi , are calculated by diagonalizing the kinematic (G) and ab initio force constant (F) matrix at the same level of theory. The eigenvectors of the GF matrix yield the mass-weighted normal coordinate of the vibrational √ modes. The latter is transformed to the dimensionless form Q by multiplying with ωi (in a. u.) [61]. In an analogous way the geometry of neutral CB9 in its ground electronic state is optimized. Since this neutral molecule has open shell configuration, its optimized structure is calculated by the UMP2 method and cc-pVTZ basis set. The optimized neutral ground state structure of CB9 also converges to the C2v symmetry point group. The adiabatic energies of the electronic states are calculated by the CASSCF-MRCI methods with cc-pVTZ basis set. The calculations are carried out using MOLPRO [62] suite of programs. In order to find a suitable active space, VDE calculations are carried out at the optimized configuration of CB− 9 with (10,10), (10,11), (10,12), (12,10), (12,11), (12,12), (14,10), (14,11) and (14,12) active spaces to find out VDE closest to the experiment 4

[47]. The results of these calculations are given in Table S1 in the supporting information (SI). Among these chosen active spaces, (10,12), (10,11) and (10,10) active spaces provide nearly the same VDEs in closest agreement with the experiment [47]. Therefore we employ the (10,10) active space in the rest of the electronic energy calculations in order to have a balance between the accuracy and computational cost. This active space includes five valence orbitals and five virtual orbitals with ten electrons for CB− 9 . The neutral states have open shell configuration and a (10,9) active space is used to calculate the single point electronic energies at various distorted geometries along the normal coordinate of vibrational modes.

III.

A.

THEORETICAL FRAMEWORK

Vibronic Hamiltonian

e 2 A2 ) and first three excited states (A e2 A1 , B e 2 B2 and C e2 B1 ) of Electronic ground (X

CB9 are considered here to carry out nuclear dynamics study. Therefore, a model 4⊗4 vibronic Hamiltonian is constructed in a diabatic electronic basis using dimensionless normal

displacement coordinates of the vibrational modes of the reference CB− 9 equilibrium configuration calculated in Section II. The standard vibronic coupling theory and the symmetry rules [63–67] are used to determine the non-vanishing elements of the Hamiltonian. The anion CB− 9 has twenty four vibrational modes. At the reference C2v symmetry configuration they decompose into the following irreducible representations (IREPs):

Γ = 9a1 ⊕ 3a2 ⊕ 4b1 ⊕ 8b2 .

(1)

In a diabatic electronic basis the vibronic Hamiltonian can be written as

H = H0 14 + ∆H,

(2)

where, H0 and ∆H represents the unperturbed Hamiltonian of the reference electronic

ground state of CB− 9 and the change in electronic energy upon electron detachment, respec5

tively. 14 represents a (4×4) unit matrix. The unperturbed Hamiltonian of Eq. 2 within a harmonic representation can be written as

24

1X ωi H0 = − 2 i=1



∂2 ∂Q2i



24

+

1X ωi Q2i 2 i=1

(3)

The quantity ∆H in a diabatic electronic representation can be symbolically given by 

WXX WXA WXB WXC

  WAA WAB WAC  ∆H =   h.c. WBB WBC  WCC

      

(4)

Where, X, A, B and C represents the first four low-lying electronic states of CB9 . Using elementary symmetry rules, the elements of this Hamiltonian matrix are expanded in a Taylor series around the reference equilibrium geometry at Q=0, as follows:

Wjj = Ej0 +

X

i=a1

κji Qi +

X 1 2! i=a ,a ,b 1

2

γij Q2i + 1 ,b2

X 1 X j 3 1 C i Qi + Dij Q4i 3! i=a 4! i=a ,a ,b ,b 1 1 2 1 2 X 1 X j 5 1 E i Qi + F j Q6 + ... (5) + 5! i=a 6! i=a ,a ,b ,b i i 1

and, ∗ = Wjk = Wkj

X

λj−k Qi , i

1

2

1

2

(6)

i

where, j and k, are the electronic state indices and i represents the vibrational modes. The Hamiltonian parameters introduced in Eqs. 5-6 have the following definitions. The vertical e A, e B e and C e states are defined by Ej0 . The quanelectron detachment energy of the X, tity κji represents the linear intrastate coupling parameter and γij is the diagonal quadratic

intrastate coupling parameter of vibrational mode i in the j th electronic state. The quanis linear interstate coupling parameter between j th and k th state coupled through tity, λj−k i ith vibrational mode of appropriate symmetry. The quantity C, D, E and F are third-order, fourth-order, fifth-order and sixth-order intrastate coupling parameter, respectively. All the parameters introduced above are estimated from the calculated ab initio electronic energies 6

and is discussed in Sec. IV.B below.

B.

Nuclear Dynamics

The vibronic energy level spectrum of CB9 is calculated by a time-independent matrix diagonalization approach [68] using Fermi’s golden rule equation for the spectral intensity P (E) =

X n

2 |<Ψfn |Tˆ|Ψi0 >| δ(E − Enf + E0i ),

(7)

where, P (E) represents spectral intensity. |Ψi0 > and Ψfn > are the initial and final vibronic states with energy Ei0 and Efn , respectively. The operator Tˆ is the transition dipole operator. The reference electronic ground state |Ψi0 > is assumed to be vibronically decoupled from the excited electronic states and is given by |Ψi0 i = |Φi0 i|χi0 i,

(8)

where |Φi0 i and |χi0 i represent the electronic and vibrational components of this state, respectively. As stated above this state is assumed to be harmonic and the vibrational component of the above wavefunction is given by the eigenfunctions of reference harmonic Hamiltonian, H0 (cf. Eq. 3). In the normal coordinate representation of vibrational modes, the vibrational wavefunction is a direct product of one-dimensional oscillator function along each mode. The final vibronic state of CB9 can be expressed as |Ψn i = |Φm i|χm n i,

(9)

e 2 A2 , A e2 A1 , B e 2 B2 and C e2 B1 electronic states of where the superscript m represents the X

CB9 , respectively. With the above definitions the spectral intensity of Eq. 7 can be written as P (E) =

X n

2 f i |τ m hχm n |χ0 i| δ(E − En + E0 ),

(10)

τ m = hΦm |Tˆ|Φ0 i,

(11)

where,

7

represents the transition dipole matrix elements. These are treated as constant assuming the general applicability of Condon approximation in a diabatic electronic basis [64]. The time-independent Schr¨odinger equation of the vibronically coupled states is solved by representing the Hamiltonian (cf. Eq. 2) in the direct product harmonic oscillator (HO) basis of the reference state. The final vibronic states, |Ψfn i, can be expressed as X

|Ψfn i =

|Ki i,m

anki ,...,kf ,m |Ki i...|Kf i|Φm i.

(12)

In the above equation the K th level of the ith vibrational mode is denoted by |Ki i and |Φm i

denotes the mth electronic state of the interacting electronic manifold of CB9 . The size of

the oscillator basis is chosen based on the numerical convergence of the vibronic eigenvalue spectrum. The Hamiltonian matrix represented in the above basis is diagonalized by Lanczos algorithm [69–71] to calculate the eigenvalue spectrum. In a time-dependent picture, the spectral intensity is calculated by Fourier transforming the time autocorrelation function of the WP propagating on the final electronic state [72]

P (E) ≈

4 X

2Re

m=1

≈

4 X

Z

∞ 0

2Re

m=1

eiEt/~ hχ0 |τ † e−iHT /~ τ |χ0 idt, Z

(13)

∞

eiEt/~ C m (t)dt,

(14)

0

where, Cm = hΨ(0)|Ψ(t)i, represents the time autocorrelation function of the WP, initially prepared on the electronic state m. The time-dependent WP propagation is carried out within the multi-configuration time dependent Hartree approach using Heidelberg MCTDH program modules [73–76].

IV. A.

RESULTS AND DISCUSSION Electronic structure of the ground state of CB− 9 and CB9

It was found by Galeev et al. [47] that the global minimum structure of the closed shell CB− 9 belongs to C2v symmetry point group by about 9.9 kcal/mol more stable than the next isomeric configuration of Cs symmetry. As stated above, we also found the same global 8

minimum structure of CB− 9 of C2v symmetry in our calculations. This optimized structure is shown in panel a of Fig. 1 with atom numbering. The geometrical parameters of this structure are presented in Table I and the vibrational frequencies of the global minimum structure of CB− 9 are given in Table II. The data given in Tables I and II are compared with the available literature data. Total fifty two electrons of CB− 9 are distributed among twenty six occupied molecular orbitals (MOs). The symmetry of the six penultimate occupied MOs is as follows: ...8(b2 )2 13(a1 )2 2(b1 )2 9(b2 )2 14(a1 )2 1(a2 )2 . These MOs are shown in Fig. S1 of SI. A huge energy gap of ∼6.92 eV between the highest occupied MO (HOMO) and lowest

unoccupied MO (LUMO) makes CB− 9 highly stable closed-shell electronic system. The MO

diagrams presented in Fig. S1 in the SI reveals six delocalized π electrons among the three π MOs [HOMO, HOMO-3 and HOMO-5, cf. panels b, e and g of Fig. S1 in the SI], which introduces π aromaticity. This finding is in well accord with the AdNDP analysis performed by Galeev et al. [47]. The present MO analysis also gives an indication of the delocalization of σ electrons in CB− 9 . Overall, both LUMO-HOMO energy gap and π aromaticity introduces stability in the CB− 9 electronic system, whereas, σ antiaromaticity reduces its stability. The four low-lying electronic states of CB9 originate from the HOMO (a2 ), HOMO-1 (a1 ), HOMO-2 (b2 ) and HOMO-3 (b1 ) of CB− 9 upon an electron detachment. The VDEs of these four electronic states are given in Table III along with the available literature data. The e 2 A2 , A e2 A1 , B e 2 B2 and C e2 B1 electronic term. Hereafter, first four states of CB9 belong to X e A, e B e and C. e we will refer these electronic states as X,

The open shell structure of the electronic ground state of CB9 is optimized by UMP2

level of theory using cc-pVTZ basis set. This optimized structure is an energy minimum and is confirmed by subsequent frequency calculation at the same level of theory. The energy minimum structure of CB9 also possesses same atomic arrangement and point group symmetry as in Fig. 1. The geometry parameters of this structure are also presented in Table I. The frequencies of the energy minimum structure of CB9 are presented in Table II. As stated above, the ground electronic state of CB9 results from removal of one electron from the HOMO of CB− 9 , the latter reveals delocalization of π electrons over 8B-6B-4B-2B and 9B-7B-5B-3B atoms (cf. panel b of Fig. S1 in the SI). Therefore, the geometry parameters connecting the above atoms are expected to change in the equilibrium electronic ground state structure of CB9 . It can be seen from Table I that all geometry parameters containing 8B, 6B, 4B, 2B, 9B, 7B, 5B and 3B atom centres are altered for CB9 as compared to those 9

for CB− 9 . The major change for the B2-B4-B10/B3-B5-B10 bond angles, while the increment of B2-B4/B3-B5 and B4-B6/B5-B7 bond lengths in CB9 are minimal (∼0.02 ˚ A ).

B.

Hamiltonian parameters

In order to calculate the parameters of the Hamiltonian of Eq. 4, the adiabatic form of the diabatic electronic Hamiltonian [Eq. 4] is fit to the adiabatic electronic energies calculated by the CASSCF(10,10)/MRCI methods using cc-pVTZ basis set. The Hamiltonian parameters of the totally symmetric (a1 ) vibrational modes are given in Table IV. The strength of 2

κ the first-order intrastate coupling measured in terms of Huang-Rhys parameter ( 2ω 2 ) is

given in parentheses in Table IV. The extent of progression of a vibrational mode in an electronic state is proportional to this parameter. The values given in Table IV reveal that the intrastate coupling strength of the ν9 vibrational mode is highest in all four electronic e and B e electronic states of CB9 , whereas, that of ν3 vibrational mode is moderate in the X e and states. Likewise the vibrational mode ν5 is expected to be moderately active in both A

e electronic states. A strong intrastate coupling of ν8 vibrational mode can be found in C e state of CB9 . Overall, it is expected from the data given in Table IV that ν3 and the A e and B e state dynamics, whereas, ν9 vibrational modes would play important role in the X

ν5 , ν6 , ν8 and ν9 , and ν4 , ν5 , ν7 and ν9 vibrational modes would be important in the e and C e state, respectively. The change in frequency (increment or reduction) dynamics of A e A, e B e and C e states of CB9 depends on the as compared to anionic reference state of the X,

value of the second-order intrastate coupling parameter (γ). The data presented in Table IV

reveals both increase and decrease of the frequency of the totally symmetric modes in the e A, e B e and C e states of CB9 as compared to its reference anionic state. The other higherX, order Hamiltonian parameters related to the anharmonicity of the potential energy surfaces

(PESs) are also given in Table IV. It can be seen that the higher order coupling parameters are generally very small. The bilinear coupling parameters also have small value (of the order of 10−3 eV) and are not included in the table. The intrastate coupling parameters for the nontotally symmetric vibrational modes are given in Table V. It can be seen that the parameters beyond second order are small in this case also. The interstate coupling parameters (λ) for the coupling vibrational modes and their 2

λ excitation strength ( 2ω 2 ) are given in Table VI. The symmetry of the vibrational modes are

10

e A, e Xe B e and Xe C e couplings are also given in the table. It can be seen from the table that X-

generally stronger and caused by the vibrational modes ν12 , ν16 and ν24 , respectively. The eB e coupling is weak and BeC e coupling is moderate. AC.

Adiabatic potential energy curves and conical intersections

One dimensional cuts of the multi-dimensional adiabatic potential energy surfaces of CB9 are plotted in Fig. 2 along the normal displacement coordinate of totally symmetric vibrational modes. The solid lines in the figure represent the potential energy obtained from the vibronic model developed above and the asterisks represent the potential energy calculated ab initio. It can be seen that the calculated ab initio energy data are reproduced well by the model Hamiltonian with the parameter set of Tables III and IV. The ground e of CB9 is well separated from the other three electronic states (A, e electronic state (X) e and C) e mainly at the Franck-Condon (FC) zone, whereas, energetic proximity between B e state and A e state can be seen beyond the FC zone along ν3 , ν6 , ν8 and ν9 vibrational the X

modes (cf. panels c, f, h and i of Fig. 2). Several curve crossings and quasi-degeneracy e B e and C e electronic states can be seen from Fig. 2. In multi-dimensional space, among A,

these curve crossings and quasi-degeneracy of the electronic states acquire the topography of conical intersections (CIs) [66]. The energetic location of these CIs is estimated and their possible impact on the nuclear dynamics is discussed below. The energetic locations of the equilibrium minimum of the four electronic states and

the minimum of the seam of CIs are calculated within the quadratic vibronic coupling model using the parameters of the Tables III and IV. The results are given in Table VII. The diagonal entries of this table represent the energetic location of the equilibrium minimum of the electronic states, whereas, the off-diagonal entries correspond to the energy minimum of the seam of the CIs. It is noted that the data given in Table VII are calculated by solving constrained minimization problem between two interacting electronic states employing Lagrangian mulitiplier as implemented in Mathematica [77]. It can be seen later in the text that these stationary points have direct impact on the vibronic structure of the recorded e A e CIs is electronic spectra. The data given in Table VII reveals that the minimum of X-

e state minimum and ∼0.08 eV above the minimum located more than an eV above the X e state. The CIs of X e state with B e and C e states located far above the minimum of of A 11

eB e and AeC e CIs on the other hand is located ∼0.01 all electronic states. The minimum Ae and C e states, respectively. Therefore, these eV and ∼0.32 eV above the minimum of B

intersections are expected to have profound impact on the dynamics of all three states. The eC e CIs is located ∼0.01 eV above the minimum of the C e state will also have minimum of B-

e considerable impact on the dynamics. Therefore, it appears that the dynamics on the X e Be C e states would state would be adiabatic only, whereas the group of energetically close Areveal nonadiabatic features.

D.

Nuclear dynamics

e A, e B e and C e electronic states of CB9 In this section the vibronic structure of the X,

is calculated both in absence and presence of coupling with their neighbours. The decay rates of electronic states and internal conversion dynamics are studied in terms of change e of electronic population in time. To start with the vibronic structure of the uncoupled X, e B e and C e states is examined. The effect of electronic state coupling on the vibronic A, structure and relaxation dynamics is examined subsequently. Both time-independent and time-dependent quantum mechanical methods are employed for the purpose.

1.

e A, e B e and C e electronic states of CB9 Vibronic structure of the uncoupled X,

e A, e B e and C e electronic states of CB9 The vibronic band structures of the uncoupled X,

are presented in Fig. 3. The nine totally symmetric vibrational modes are considered and the results of calculations carried out by time-independent matrix diagonalization and timedependent WP propagation methods are plotted in panel a and b, respectively. The 266 nm experimental recording is reproduced from Ref. [47] and shown in panel c. The numerical details of calculations are presented in Table S2 of SI. The stick vibronic spectrum obtained in the time-independent matrix diagonalization calculations (cf. panel a) are convoluted with a Lorentzian function of 50 meV full width at the half maximum to generate the spectral envelopes plotted in panel a. The time autocorrelation function calculated in the −t

WP propagation method is damped with an exponential function e τ (with τ =26 fs) before Fourier transformation to generate the spectral envelopes shown in panel b. 12

e and A e electronic states At 266 nm laser wavelength the experiment could probe only X

of CB9 . It can be seen from Fig. 3 that the theoretical results obtained by the two different e methods agree with each other and also corresponds well to the vibronic structure of the X

e B e and C e state recorded in the experiment (cf. panel c). The vibronic structure of the A,

states are compared with better resolved experiment recorded using 193 nm laser later in the e A, e B e and C e states are given in Table text. The low energy vibronic levels of uncoupled X, VIII along with their assignment. The latter is carried out by examining the nodal pattern of the corresponding vibronic wavefunction calculated by using block-improved-relaxation method as implemented in the MCTDH program module [73–76].

It can be seen from the data given in Table VIII that fundamental of all symmetric vibrational modes, ν1 -ν9 , is excited in each of the given electronic state of CB9 . Strong e A, e B e and C e state of CB9 , respectively, excitation of ν5 -ν9 , ν6 , ν6 -ν8 and ν6 -ν9 on the X, is revealed by the data. The energy level corresponding to first overtone of the above vibrational modes in the mentioned electronic states are also given in the table. Several combination energy levels of these strongly excited modes are also found.

As mentioned above the assignments of all energy levels given in Table VIII are done based on the observed nodal pattern of the corresponding vibronic wavefunction. In order to illustrate, the reduced dimensional probability density plots of a few vibronic wavefunctions are given in Figs. 4(a-h). The wavefunctions in panels a and b of the figure represent e state at ∼324 the fundamental and the first overtone of the mode ν9 excited in the X

and 648 cm−1 , respectively. It can be seen that these wavefunctions possess one and two

nodes, respectively, along the coordinate of ν9 vibrational mode. The wavefunctions of the e state at ∼571 and fundamental of ν6 and the combination level, ν6 +ν7 , excited in the A ∼1181 cm−1 , respectively, are shown in panels c and d, respectively. The nodal pattern in

the wavefunction (as illustrated above) justifies the assignment. Likewise, the wavefunction e state at ∼620 and 1240 of the fundamental and the first overtone of ν7 excited in the B

cm−1 , respectively, are shown in panels e and f. The wavefunctions of the fundamental e state of ν8 and a combination level of it with ν7 found at ∼485 and 1103 cm−1 on the C

are, respectively, shown in panels g and h of the figure. The strongest excitation of the ν9 vibrational mode is in accord with its large excitation strength given in Table IV. 13

2.

Impact of electronic nonadiabatic coupling on the vibronic structure

We performed several coupled two states calculations in order to assess the impact of nonadiabatic coupling on the vibronic structure of the individual state. To save space and for bravity we do not show the results, however, discuss the findings here. The overall e state does not change upon inclusion of its coupling with structure of the spectrum of the X

e B e and C e states despite having its fairly strong coupling with these states (cf. Table the A, e state is vertically well separated from the rest (cf. Table III) VI). This is because the X and the minimum of the seam of its intersections with them lies well above its equilibrium e state does not minimum (cf. Table VII). Therefore, the WP initially prepared on the X access the neighbourhood of various CIs which is revealed by very little electronic population e flow to the other states in this situation. Understandably, the vibronic structures of the A,

e and C e states are also not significantly affected by their coupling with the X e state in this B

situation.

eB e and BeC e states have quite some In contrast to the above, the coupling between AeB e coupled states the energetic impact on their individual vibronic structure. In case of Aminimum of the intersection seam is quasi-degenerate with the estimated equilibrium minie state (cf. Table VII). The equilibrium minimum of these states are slightly mum of the B

more than half an eV apart. The VDEs of these states are within ∼0.4 eV (cf. Table III).

Therefore, although the coupling between these states caused by the b2 vibrational modes is not very strong, a large population exchange takes place between these states. In order to illustrate, time-dependence of the diabatic electronic population for an initial transition e state in the AeB e coupled states situation is shown in Fig. 5. It can be seen that to the B e state monotonically decays to ∼0.2 and that of the A e state grows the population of the B

to ∼0.8 in about 200 fs. Such a huge population exchange causes a large increase in the spectral line density and broadening of the vibronic spectrum of both the states. Similar eC e intersections and equilibrium minimum of quasi-degeneracy between the minimum of B-

e state and energetic proximity of the two equilibrium minimum (cf. Table VII) causes the C e and C e states. a broadening of the vibronic spectrum of both the B e AeB e coupled states situation is examined next. The vibronic band structure in the X-

The symmetric vibrational modes ν3 , ν6 -ν9 and non-totally symmetric vibrational modes ν10 , ν12 , ν14 , ν16 , ν18 and ν24 are included and calculations are carried out by both matrix 14

diagonalization and WP propagation methods. The composite band structures obtained in this situation are plotted in panels a and b of Fig. 6, respectively, along with 193 nm experimental recording reproduced from Ref. [47] in panel c. It can be seen from the figure that the results obtained by the two theoretical methods agrees with each other and also to the experiment. Both the location of individual band and the broadening are well reproduced by the theoretical results. The numerical details of the theoretical calculations are given in Table S2 of SI. The low energy part of the vibronic energy levels of Fig. 6a and their assignments are given in Table IX. As before, the assignment of the vibronic levels is carried out based on the nodal pattern of the corresponding vibronic wavefunction. In comparison with the vibronic energy levels obtained in the uncoupled state calculations (cf. Table VIII), it can be seen that the excitation of the non-totally symmetric modes takes place in the coupled states situation (cf. Table IX). These modes also form combination levels among themselves and also with symmetric vibrational modes. Such a mixing of symmetry causes a large increase in spectral line density and contributes to the broadening of spectral band observed in the experiment. In order to demonstrate, the probability density plots of a few vibronic wavefunctions are shown in Figs. 7(a-d). The wavefunction density shown in panel a of Fig. 7 exhibits a node along the ν12 mode of a2 symmetry and corresponds to the fundamental of this mode obtained at ∼165 cm−1 . The harmonic frequency of this mode in the electronic

ground state of CB9 is ∼157 cm−1 (cf. Table II). Likewise the fundamentals of ν16 and ν24

found at ∼104 and ∼227 cm−1 , respectively, are shown in panel b and c. A combination level of ν12 and ν24 revealing one node along each Q12 and Q24 coordinate and occuring at

∼392 cm−1 is shown in panel d of Fig. 7. Furthermore, the fundamentals of the symmetric vibrational modes remain almost at the same energetic location in the coupled states results of Table IX as compared to their location in the uncoupled state data presented in Table VIII. Experimental results of Galeev et al. revealed a vibrational progression with an average e band [47]. They assigned it to the excitation of totally spacing of 340±50 cm−1 in the X

symmetric vibrational mode ν9 (defined as ν1 in Ref [47]) based on the harmonic frequency of ∼346 cm−1 of this mode calculated at the C2v global minimum structure of CB9 . It can

be seen from Table IX that the present theoretical results predict the location of this line at ∼325 cm−1 (which is more closer to the fundamental at this level of theory) and assigned to the fundamental of ν9 based on the nodal pattern of the vibrational wavefunction. 15

Finally, the coupling among all four states are included in the dynamics calculations carried out by the WP propagation method using the MCTDH program module [73–76]. A matrix diagonalization calculation could not be performed in this situation owing to a e Ae BeC e coupled huge increase in the dimensionality. The complete band structure of the X-

states obtained in the present calculations is plotted in panel b of Fig. 8 along with the 193 nm experimental recording reproduced from Ref [47] in panel a. Sixteen vibrational modes,

ν2 -ν9 , ν10 , ν12 -ν14 , ν17 , ν18 , ν20 and ν24 are included in the theoretical calculations. The numerical details of the calculations are given in Table S3 of SI. The time-autocorrelation −t

function recorded during WP propagation is damped with an exponential function, e τr

(with τr =20 fs), and Fourier transformed to generate the spectral envelope. It can be seen from the figure that the theoretical results agree very well to the experimental recording. While plotting the theoretical results of Fig. 8 the location of the 0-0 peak is placed at the experimentally estimated adiabatic detachment energy of ∼3.61±0.03 eV [47]. Because of unique bonding properties of boron atom, the cluster formed by it exists in many energetically close-lying isomeric form. Confirmation of global minimum structure often becomes cumbersome and cannot be relied on theoretical data alone. Theoretical results in comparison with experimental observations serve as the tool for an unbiased structure determination. It is discussed by Galeev et al. [47] that when one boron atom of B− 10 is replaced by a carbon atom in CB− 9 , the quasi planar eight membered ring structure enclosing two inner atom of B− 10 undergoes significant structural change. The carbon atom occupies the peripheral position and one of the inner boron atom is pushed to a peripheral position resulting in a global minimum distorted wheel type of structure of CB− 9 as found in the present work as well (cf. Fig.1(a)). This structure is confirmed by comparing the estimated experimental vertical detachment energies with those obtained from theoretical calculations [47]. Furthermore, chemical bonding analyses revealed that this structure contains seven 2c-2e B-B σ bonds, two 2c-2e C-B σ bonds, four 3c-2e σ bonds, one 3c-2e π bond and two 4c-2e π bonds [47]. A thorough and more complete analysis of the recorded detachment bands in the present work unequivocally confirms the above prediction of the global minimum structure. The calculated vibronic structures of the detachment bands are in good accord with the recorded ones, which also establishes the coupling mechanism of the electronic states of CB9 developed in this study. 16

V.

SUMMARIZING REMARKS

The photodetachment spectrum of anionic carbon doped boron cluster, CB− 9 , is theoretically studied in order to establish its global minimum structure. Adiabatic energies of the ground and first three excited electronic states of neutral CB9 are calculated by the ab initio quantum chemistry methods. Based on the calculated adiabatic electronic energies, a vibronic coupling model is developed using standard vibronic coupling theory. Employing this model first principles nuclear dynamics calculations are carried out and vibronic structure of the photodetachment bands is explained. The progressions in the photodetachment bands are identified by analyzing the vibronic wavefunctions. The impact of the electronic nonadiabatic coupling arising due to multiple conical intersections of electronic states on the vibronic band structure and time-dependent internal conversion dynamics is studied at length. The theoretical results are shown to be in excellent accord with the recorded structure of the photodetachment bands in the experiment [47]. In addition to the progression of symmetric vibrational modes, the nonadiabatic coupling among electronic states leads to the excitation of non-totally symmetric vibrational modes and their combination levels with symmetric modes. It causes a huge increase of the spectral line density and broadening of the photodetachment bands. A very good agreement between the theoretical and experimental results unequivocally confirms the C2v global minimum structure of both CB− 9 and CB9 .

ACKNOWLEDGEMENT

This study is supported in part through a research grant (Grant no. SB/S1/PC-052/2013) from the DST, New Delhi. D. B. acknowledges the University of Hyderabad for a Doctoral Fellowship.

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TABLE I. Bond length (R in ˚ A) and bond angle (∠ in degree) of the optimized equilibrium structure of CB− and CB [cf. Fig. 1]. The geometry parameters available in the literature are also given 9 9 in the table for comparison.

Parameters R(C1-B2,C1-B3) R(B2-B3) R(B2-B4,B3-B5) R(B4-B6,B5-B7) R(B6-B8,B7-B9) R(B6-B10,B7-B10) R(B8-B10,B9-B10) R(B8-B9) ∠(C1-B2-B4,C1-B3-B5) ∠(B3-B2-B4,B2-B3-B5) ∠(B2-B4-B6,B3-B5-B7) ∠(B4-B6-B8,B5-B7-B9) ∠(B6-B8-B9,B7-B9-B8) ∠(B6-B10-B9,B7-B10-B8) ∠(B2-B4-B5,B3-B5-B4) ∠(B2-B4-B10,B3-B5-B10) ∠(B4-B5-B7,B5-B4-B6) ∠(B4-B10-B7,B5-B10-B6) ∠(B4-B10-B8,B5-B10-B9) ∠(B4-B10-B9,B5-B10-B8) ∠(B6-B10-B7) ∠(B4-B10-B5)

CB− 9 1.41 1.84 1.60 1.53 1.55 1.78 1.75 1.57 160.34 111.01 168.47 132.03 128.50 105.07 68.99 106.58 99.48 153.93 101.01 154.17 156.97 104.83

22

This work CB9 1.40 1.82 1.62 1.55 1.55 1.77 1.76 1.57 157.82 108.32 173.44 130.27 127.96 105.07 71.67 111.12 101.76 151.98 102.95 155.95 157.14 101.10

Ref. [47] CB− 9 1.40 1.84 1.60 1.51 1.54 1.78 1.75 1.56

TABLE II. Normal vibrational modes (symmetry) and harmonic frequency [in cm−1 (eV)] of the equilibrium structure of the ground state of CB− 9 and CB9 . The frequency data available in the literature (Ref. [47]) are also given for comparison.

Modes (symmetry) ν1 (a1 ) ν2 (a1 ) ν3 (a1 ) ν4 (a1 ) ν5 (a1 ) ν6 (a1 ) ν7 (a1 ) ν8 (a1 ) ν9 (a1 ) ν10 (a2 ) ν11 (a2 ) ν12 (a2 ) ν13 (b1 ) ν14 (b1 ) ν15 (b1 ) ν16 (b1 ) ν17 (b2 ) ν18 (b2 ) ν19 (b2 ) ν20 (b2 ) ν21 (b2 ) ν22 (b2 ) ν23 (b2 ) ν24 (b2 )

This CB− 9 1536 (0.1905) 1427 (0.1769) 1223 (0.1516) 944 (0.1171) 744 (0.0923) 655 (0.0812) 621 (0.0770) 500 (0.0620) 332 (0.0411) 462 (0.0573) 404 (0.0501) 183 (0.0227) 473 (0.0587) 418 (0.0519) 307 (0.0380) 132 (0.0164) 1583 (0.1962) 1413 (0.1752) 1102 (0.1366) 787 (0.0976) 614 (0.0761) 585 (0.0725) 416 (0.0516) 295 (0.0366)

Ref. [47] CB9 1556 1448 1267 958 778 671 631 514 346 492 378 161 488 334 274 132 1623 1345 1068 839 623 595 423 318

work CB9 1553 1461 1240 944 774 658 630 504 344 487 380 157 481 340 271 129 1598 1341 1061 832 606 586 416 311

TABLE III. Vertical detachment energy (in eV) of the first four electronic states of CB9 . The theoretical and experimental results available in the literature [47] are also given.

State ˜ 2 A2 X A˜2 A1 ˜ 2 B2 B C˜ 2 B1

This work MRCI 3.46 4.84 5.21 5.85

Ref.[47] TD-PBE0 3.53 4.64 4.88 5.67

23

ROVGF 3.54 4.49 4.88 5.73

CCSD(T) 3.64 4.57 4.93 5.74

Experiment 3.61 4.49 4.88 5.74

TABLE IV. The first-order (κ), second-order (γ), third-order (C) and fourth-order (D) intrastate e A, e B e and C e electronic coupling parameters (in eV) of the symmetric vibrational modes for the X, states of CB9 . The parameters are calculated from the calculated CAS(10,10)SCF/MRCI electronic energy data. The numbers given in the parantheses represent the excitation strength of the respective vibrational mode.

Mode

κji



(κji ) 2ωi2

2



ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9

0.0251 (0.0087) 0.0389 (0.0242) 0.0877 (0.1672) 0.0138 (0.0069) 0.0250 (0.0367) 0.0029 (0.0006) -0.0253 (0.0540) 0.0132 (0.0228) 0.0490 (0.7010)

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9

-0.0255 (0.0089) 0.0286 (0.0130) 0.1143 (0.2841) 0.0055 (0.0011) -0.0098 (0.0056) -0.0157 (0.0186) -0.0275 (0.0639) 0.0169 (0.0372) 0.0760 (1.7077)

γij ˜ 2 A2 X 0.0058 -0.0003 0.0063 -0.0003 0.0023 -0.0011 -0.0004 -0.0056 -0.0018 ˜ 2 B2 B 0.0061 -0.0018 0.0076 -0.0034 -0.0076 -0.0009 -0.0003 0.0142 -0.0119

C

D

κji



(κji ) 2ωi2

2



0.0003 (0.0000) 0.0377 (0.0227) -0.0337 (0.0247) -0.0001 -0.0000 0.0279 (0.0284) -0.0585 (0.2006) -0.0001 -0.0000 -0.0585 (0.2594) -0.0114 (0.0101) -0.0001 0.1263 (2.0762) -0.1205 (4.3010) -0.0003 0.0004

-0.0292 (0.0118) -0.0721 (0.0831) -0.0109 (0.0026) 0.1028 (0.3851) -0.0004 0.0001 0.0498 (0.1454) 0.0001 -0.0001 -0.0225 (0.0385) -0.0458 (0.1773) 0.0001 -0.0001 -0.0159 (0.0329) -0.0503 (0.7487)

24

γij A˜2 A1 0.0097 -0.0069 -0.0017 0.0019 -0.0060 -0.0195 -0.0028 -0.0036 -0.0023 C˜ 2 B1 -0.0635 -0.0661 -0.0763 -0.0161 0.0015 -0.0177 -0.0007 -0.0040 -0.0110

C

D

0.0002 0.0005 0.0002 -0.0001 -0.0001

-0.0007

0.0007 0.0006 0.0008 0.0006 0.0023 0.0004 -0.0013 0.0002 0.0004 -0.0001 0.0018 -0.0003

TABLE V. Same as in Table IV for the non-totally symmetric vibrational modes.

Mode ν10 ν11 ν12 ν13 ν14 ν15 ν16 ν17 ν18 ν19 ν20 ν21 ν22 ν23 ν24 ν10 ν11 ν12 ν13 ν14 ν15 ν16 ν17 ν18 ν19 ν20 ν21 ν22 ν23 ν24

γij

D F γij D F 2 2 ˜ A2 X A˜ A1 0.0007 0.0000 -0.0048 0.0000 -0.0079 0.0000 -0.0082 0.0001 -0.0073 -0.0083 0.0000 -0.0113 0.0000 0.0035 -0.0005 -0.0273 0.0002 -0.0022 -0.0007 0.0000 -0.0075 0.0037 -0.0001 -0.0077 0.0001 -0.0000 0.0004 -0.0000 0.0000 0.0087 -0.0110 0.0000 0.0084 0.0000 -0.0173 0.0001 -0.0071 -0.0012 0.0000 0.0123 -0.0229 -0.0043 0.0052 -0.0000 0.0064 -0.0011 0.0000 -0.0080 0.0077 -0.0002 -0.0013 -0.0000 -0.0222 0.0001 2 ˜ B B2 C˜ 2 B1 -0.0074 0.0000 -0.0066 -0.0001 -0.0047 0.0001 -0.0000 -0.0035 -0.0001 -0.0108 0.0000 -0.0143 -0.0001 0.0049 -0.0611 0.0005 0.0184 -0.0399 0.0000 -0.0063 -0.0267 0.0001 -0.0031 -0.0087 -0.0006 0.0203 -0.0001 0.0055 -0.0000 0.0222 -0.0001 0.0092 -0.0000 -0.0021 0.0127 -0.0187 0.0070 -0.0000 -0.0012 0.0000 -0.0009 0.0000 -0.0048 -0.0034 -0.0000 -0.0180 0.0002 -0.0020 0.0001 0.0082 -0.0001 -0.0031 -0.0000

25

j−k TABLE VI.  interstate coupling parameter (λ ) (in eV) and corresponding excitation  Linear 2 j−k (λ ) between pairs of electronic states j-k. strength 2ω 2 i

Mode

λ

ν10 ν11 ν12

0.0824 0.0553 0.0553

ν13 ν14 ν15 ν16

0.0853 0.1425 0.0443 0.0465

ν17 ν18 ν19 ν20 ν21 ν22 ν23 ν24



j−k

(λj−k ) 2ωi2

2



e A e X1.0340 0.6092 2.9673 e e X-B 1.0558 3.7693 0.6795 4.0196 e C e X0.2772 0.2158 0.6142 1.4604 0.8703 2.2530 2.3305 6.2886

0.1461 0.1151 0.1514 0.1668 0.1004 0.1539 0.1114 0.1298

λ



j−k

(λj−k ) 2ωi2

0.0243 0.0383 0.0324

eC e B0.0899 0.2922 1.0186

0.0514 0.0592

eB e A0.0343 0.0571

0.0149

0.0116

0.0498

0.0926

2



TABLE VII. Estimated energy (in eV) of the equilibrium minimum (diagonal entries) and minimum of the seam of various CIs (off-diagonal entries) of the electronic states of CB9 calculated with the Hamiltonian truncated at the second order level (cf. Sec. III.A).

˜ 2 A2 X A˜2 A1 ˜ 2 B2 B C˜ 2 B1

e 2 A2 X 3.387

e2 A1 A

4.534

e 2 B2 B

7.529

6.604

-

4.454 -

5.071 5.057 -

6.012 5.705 5.691

26

e 2 B1 C

TABLE VIII. Energy (in cm−1 ) and assignment of the low-lying vibronic levels of the uncoupled e A, e B e and C e electronic states of CB9 . X,

No. Energy Assignment e X 1 0.0 0 2 324 ν9 3 477 ν8 4 619 ν7 5 648 2ν9 6 649 ν6 7 754 ν5 8 801 ν8 +ν9 9 943 ν4 10 944 ν7 +ν9 11 955 2ν8 12 973 ν6 +ν9 13 1097 ν7 +ν8 14 1126 ν6 +ν8 15 1239 2ν7 16 1248 ν3 17 1268 ν6 +ν7 18 1297 2ν6 19 1426 ν2 20 1507 2ν5 21 1560 ν1 22 1572 ν3 +ν9 23 1725 ν3 +ν8 24 1867 ν3 +ν7 25 1897 ν3 +ν6

Energy Assignment e A 0.0 0 424 ν9 571 ν6 576 ν8 610 ν7 720 ν5 951 ν4 995 ν6 +ν9 1000 ν8 +ν9 1034 ν7 +ν9 1143 2ν6 1144 ν5 +ν9 1147 ν6 +ν8 1181 ν6 +ν7 1186 ν7 +ν8 1217 ν3 1291 ν5 +ν6 1329 ν5 +ν6 1375 ν4 +ν9 1396 ν2 1506 ν4 +ν6 1527 ν4 +ν8 1561 ν4 +ν7 1575 ν1 1671 ν4 +ν5

27

Energy Assignment e B 0.0 0 320 ν9 552 ν8 620 ν7 650 ν6 713 ν5 872 ν8 +ν9 931 ν4 940 ν7 +ν9 970 ν6 +ν9 1033 ν5 +ν9 1101 2ν8 1171 ν7 +ν8 1201 ν6 +ν8 1240 2ν7 1251 ν4 +ν9 1253 ν3 1265 ν5 +ν8 1269 ν6 +ν7 1298 2ν6 1333 ν5 +ν7 1418 ν2 1427 2ν5 1483 ν4 +ν8 1560 ν1

Energy Assignment e C 0.0 0 316 ν9 485 ν8 581 ν6 618 ν7 627 2ν9 750 ν5 801 ν8 +ν9 885 ν3 898 ν6 +ν9 908 ν4 935 ν7 +ν9 968 2ν8 1066 ν6 +ν8 1067 ν5 +ν9 1103 ν7 +ν8 1162 2ν6 1177 ν2 1199 ν6 +ν7 1201 ν3 +ν9 1235 ν5 +ν8 1236 2ν7 1281 ν1 1332 ν5 +ν6 1369 ν5 +ν7

TABLE IX. Vibronic energy (in cm−1 ) levels and assignment of the lower part of (starting from e AeB e coupled states spectrum of CB9 . the origin) the X-

No. Energy Assignment No. 1 0.0 0 21 2 104 ν16 22 3 165 ν12 23 4 205 2ν16 24 5 227 ν24 25 6 264 ν14 26 7 325 ν9 27 8 330 2ν12 28 9 361 ν14 +ν16 29 10 392 ν12 +ν24 30 11 427 ν12 +ν14 31 12 453 2ν24 32 13 461 ν10 33 14 481 ν8 34 15 522 2ν14 35 16 561 ν10 +ν16 36 17 585 ν8 +ν16 37 18 624 ν7 38 19 627 ν10 +ν12 39 20 643 ν8 +ν12 40

Energy Assignment No. 651 2ν9 41 661 ν6 42 710 ν8 +ν24 43 728 ν7 +ν16 44 783 ν9 +ν10 45 791 ν7 +ν12 46 804 ν8 +ν9 47 822 ν6 +ν12 48 847 ν7 +ν24 49 885 ν6 +ν24 50 922 2ν10 51 942 ν8 +ν10 52 949 ν7 +ν9 53 962 2ν8 54 986 ν6 +ν9 55 1086 ν7 +ν10 56 1105 ν7 +ν8 57 1122 ν6 +ν10 58 1142 ν6 +ν8 59 1245 2ν7 60

28

Energy Assignment 1250 ν3 1285 ν6 +ν7 1322 2ν6 1351 ν3 +ν16 1411 ν3 +ν12 1446 ν18 1478 ν3 +ν24 1547 ν16 +ν18 1569 ν3 +ν9 1614 ν12 +ν18 1672 ν18 +ν24 1730 ν3 +ν8 1769 ν9 +ν18 1874 ν3 +ν7 1910 ν3 +ν6 1929 ν8 +ν18 2067 ν7 +ν18 2107 ν6 +ν18 2500 2ν3 2894 2ν18

FIG. 1. Arrangement of atoms at the C2v equilibrium geometry of carbon dopped boron cluster. The equilibrium geometry parameters of CB− 9 and CB9 are given in Table I.

29

Potential Energy (eV)

8 (a)

8

ν1

(b)

8

ν2

7

7

7

6

6

6

5

5

5

4

4

4

3

3 -4

7

-2

(d)

0 ν4

2

4

~X ~A ~B ~ C

ν3

3

-4 (e) 7

-2

0

2

4

ν5

-4 (f)

-2

-4 (i)

-2

-4

-2

0

2

4

0 ν9

2

4

0

2

4

ν6

6

6

6

5

5

4

4

4

3

3

3

7

(c)

-4 -2 (g)

0 ν7

2

4

5

-4 (h)

-2

0

2

4

ν8

6

6

6

5

5

5 4

4

4

3

3 -4

-2

0

2

4

3 -4

-2

0

2

4

Q e A, e B e and C e electronic states (plotted with FIG. 2. Adiabatic potential energy curves of the X, different colors and given in the legend) of CB9 along the dimensionless normal coordinate (Q) of totally symmetric vibrational modes. Potential energies obtained from the present vibronic model using the CASSCF-MRCI parameter values of Tables III and IV are shown by the solid lines and the calculated energies ab initio shown by the points in the diagram.

30

~ X

~ B

Relative Intensity

~ A

3

4

5 ~ B

~ X ~ A

3

4

(a) Theory (ν1−ν9) Time-independent un-coupled

~ C

6 ~ C

5

(b) Theory (ν1−ν9) Time-dependent un-coupled

6

~ X

(c) Experiment 266 nm

~ A

3

4

5

6

Energy (eV) e A, e B e and C e electronic states of CB9 calculated FIG. 3. Vibronic structure of the uncoupled X, including all nine symmetric vibrational modes by the matrix diagonalization (panel a) and WP propagation (panel b) methods. The 266 nm experimental recording of Ref. [47] is reproduced in panel c. The intensity in arbitrary units is plotted as a function of the vibronic energies of the respective state. The zero of energy corresponds to the energy of the equilibrium minimum configuration of CB− 9.

31

Density

Density

2

1

2 0

-1

Q1

-2

-3

-2

2

1

0

-1

3

1

0

Q1

Q9

-1

-2

-4

2

1

0

4

3

Q9

e X

(b)

Density

Density

(a)

-1

-2

-3

-4

-3

-2

3 -1

0

1

Q3

2

3

4 -4

-3

-2

-1

2

1

0

3

4

2

1

Q7

0

-1

-2

-3 -4

2

1

0

-1

-2

-3

3

4

Q6

Q6

(d)

Density

Density

(c)

e A

-3

-2

-1

0

Q4

1

2

3 -3

-2

-1

1

0

2

-3 -2 -1

3

Q4

Q7

0

1

2

3

Q7

e B

(f)

Density

Density

(e)

2

1

0

-1

-2

3 -3

-3 -2 -1

Q4

0

1

2

3 -3

-2

(g)

-1

1

0

Q8

2

3

3 2 1 0 -1 -2 Q8 -3 -3

-2

(h)

-1

0

1

2

3

Q7

e C

FIG. 4. Reduced density plots of the wavefunctions of selected vibrational levels of the uncoupled e [fundamental (a) and first overtone of ν9 (b)], A e [fundamental of ν6 (c) and its combination X e [fundamental (e) and first overtone of ν7 (f)] and C e [fundamental of ν8 (g) and its with ν7 (d)], B 32 combination with ν7 (h)] states of CB9 (see text for details). The assignment of the vibrational levels given in Table VIII is based on the nodal pattern of the corresponding wavefunction (for example fundamental of ν9 contains one node along its normal coordinate Q9 ) as shown in the figure.

1

~ B

Electronic population

0.8

~ A 0.6

~ ~ A-B coupled 0.4

0.2

0 0

100

50

150

200

Time (fs)

eB e state dynamics. FIG. 5. Time-dependence of diabatic electronic population in the coupled Ae The WP is initially located on the B state.

33

~ B

(a) Theory Matrix diagonalization

Relative Intensity

~ X

3

~ A

4

5 ~ B

(b) Theory WP propagation ~ X

3

~ A

4

5 ~ B

(c) Experiment 193 nm ~ X ~ A

3

4

5

Energy (eV) e A e and B e coupled electronic states of CB9 calculated by FIG. 6. The vibronic structures of the X, the matrix diagonalization (panel a) and WP propagation (panel b) methods. The experimental results of Galeev et al. [47] recorded with 193 nm laser is reproduced in panel c.

34

Density

Density

-2 -1

0

Q3

1

2

2

1

0

-3

-2

-1

-2

-1

0

Q9

Q12

1

-1

-2

3

0

1

2

Q16

(b)

Density

Density

(a)

2

2

Q6

-3 1

0

-1

-2

-2

-1

0

1

2

-2

Q12

Q24

(c)

-1

0

1

2

3 -3

-2

-1

0

1

2

3

Q24

(d)

FIG. 7. Reduced density plots of the wavefunctions of the fundamental of ν12 , ν16 and ν24 vie state brational modes (panels a-c) and a combination level of ν12 +ν24 (panel d) excited in the X e AeB e states dynamics. spectrum of CB9 in the coupled X-

35

~ B

(a) Experiment 193 nm ~ X

~ C

Relative Intensity

~ A

~ D

3

4

5

(b) Theory WP propagation ~ X

3

6 ~ B ~ C

~ A

4

5

6

Energy (eV) e e e FIG. 8. The photodetachment bands of CB− 9 revealing the vibronic structure of the X, A, B and e C electronic states of CB9 . The theoretical results plotted in panel (b) are obtained by propagating e Ae BeC e electronic states including 16 relevant vibrational modes WPs in the coupled manifold of X(see text for details). Intensity in arbitrary units is plotted as a function of the energy of the final vibronic state. The zero of energy corresponds to the energy of the equilibrium minimum configuration of CB− 9 . The 193 nm experimental recording is reproduced from Ref. [47] and plotted in panel (a).

36