Chinese Journal of Chemical Physics  2020, Vol. 33 Issue (1): 8-12

#### The article information

Masaaki Baba, Ayumi Kanaoka, Akiko Nishiyama, Masatoshi Misono, Takayoshi Ishimoto, Taro Udagawa
Baba Masaaki, Kanaoka Ayumi, Nishiyama Akiko, Misono Masatoshi, Ishimoto Takayoshi, Udagawa Taro
Large Amplitude Motion in 9-Methylanthracene: High-Resolution Spectroscopy and Ab Initio Theoretical Calculation

Chinese Journal of Chemical Physics, 2020, 33(1): 8-12

http://dx.doi.org/10.1063/1674-0068/cjcp1910188

### Article history

Accepted on: December 13, 2019
Large Amplitude Motion in 9-Methylanthracene: High-Resolution Spectroscopy and Ab Initio Theoretical Calculation
Masaaki Babaa , Ayumi Kanaokaa , Akiko Nishiyamab , Masatoshi Misonoc , Takayoshi Ishimotod , Taro Udagawae
Dated: Received on October 27, 2019; Accepted on December 13, 2019
a. Division of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan;
b. Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Toruń, Toruń 87-100, Poland;
c. Department of Applied Physics, Faculty of Science, Fukuoka University, Jonan-ku, Fukuoka 814-0180, Japan;
d. Association of International Arts and Science Institute of Natural Science, Yokohama City University, Kanazawa-ku, Yokohama 236-0027, Japan;
e. Department of Chemistry and Biomolecular Science, Faculty of Engineering, Gifu University, Yanagido, Gifu 501-1193, Japan
Abstract: CH$_3$ internal rotation is one of the typical large amplitude motions in polyatomic molecules, the spectral analysis and theoretical calculations of which, were developed by Li-Hong Xu and Jon Hougen. We observed a Doppler-free high-resolution and high-precision spectrum of 9-methylanthracene (9MA) by using the collimated supersonic jet and optical frequency comb techniques. The potential energy curve of CH$_3$ internal rotation is expressed by a six-fold symmetric sinusoidal function. It was previously shown that the barrier height ($V_6$) of 9MA-$d_{12}$ was considerably smaller than that of 9MA-$h_{12}$ [M. Baba, et al., J. Phys. Chem. A 113, 2366 (2009)]. We performed ab initio theoretical calculations of the multi-component molecular orbital method. The barrier reduction by deuterium substitution was partly attributed to the difference between the wave functions of H and D atomic nuclei.
Key words: CH3 internal rotation    9-Methylanthracene    High-resolution spectroscopy    Optical frequency comb    Ab initio caclulation
Ⅰ. INTRODUCTION

Large amplitude motion plays an important role in the excited-state dynamics of a polyatomic molecule. CH$_3$ internal rotation (torsion) is of great interests and is a very attractive research subject for molecular spectroscopists and theoreticians. Li-Hong Xu and Jon Hougen finely analyzed the high-resolution spectra of small molecules, such as methanol (CH$_3$OH) and acetaldehyde (CH$_3$CHO), and developed an elegant theory to reproduce the experimentally determined level energies and spectral intensities [1-10]. As another example, we observed high-resolution spectrum of 9-methylanthracene (9MA) and performed ab initio calculations of multi-component molecular orbital (MC$_-$MO) method. In this article, we present the results of experiments and theoretical calculations and discuss the vibrational and rotational level structure for CH$_3$ internal rotation.

For 9MA, the potential energy curve shows a six-fold barrier because of CH$_3$ three-fold symmetry and a two-fold symmetry axis in the aromatic planar moiety. This situation is similar to toluene, whose high-resolution spectrum was previously analyzed by Borst and Pratt [11]. It should be noted that the barrier height ($V_6$) to rotation is extremely low ($\sim$5 cm$^{-1}$) in the S$_0$ state of toluene. In contrast, the $V_6$ value in the S$_0$ state of 9MA is considerably large. The experimental values of $V_6$ for 9MA-$h_{12}$ and for 9MA-$d_{12}$ were estimated to be 118 and 85 cm$^{-1}$, respectively [12]. The six-fold barrier arises mainly from steric hindrance and hyperconjugation [13, 14]. This hyperconjugation is expected to have significant energy variation with the CH$_3$ rotational angle in 9MA. This is supported by the fact that the $V_6$ values in the S$_1$ state of 9MA-$h_{12}$ and 9MA-$d_{12}$ are 33 and 54 cm$^{-1}$, respectively, markedly smaller than those in the S$_0$ state.

It is of note that the barrier height is considerably different between 9MA-$h_{12}$ and 9MA-$d_{12}$. We performed MC_MO calculations in which the quantum effect on the H and D nuclei such as shrinking of nuclear wavefunction by deuterium substitution was taken into account. The spacial distribution of the wave function of D nucleus is slightly smaller than that of H nucleus.

Ⅱ. EXPERIMENTS

The detail of our experimental setup is described here briefly. The light source for high-resolution spectroscopy was a single-mode Ti:Sapphire laser (Microlase, MBR-110) pumped by a CW Nd:YVO$_4$ laser (Spectra Physics, Millennia Xs). The wavelength, output power, and full width at half maximum are approximately 742 nm, 650 mW, and 400 kHz, respectively. The output of the laser was input to a second harmonics generator (Spectra-Physics, Wavetrain SC) with an LBO crystal. The UV power at 371 nm was approximately 60 mW.

In order to accurately stabilize and to control the laser light wavelength, we employed a home-made system of an optical frequency comb locked to the GPS signal. The uncertainties in determining the transition frequencies of observed spectral lines were approximately 10 kHz in our measurement system.

The laser light beam was crossed with a pulsed supersonic jet collimated by a conical skimmer (orifice diameter 2 mm) at right angles to get rid of Doppler broadening. Fluorescence from the excited molecules was collected to the cathode's surface of a photomultiplier (Hamamatsu R595) and the output was processed by a gated photon counter (Stanford Research SR400). The change in fluorescence intensity with the laser light wavelength was recorded as a Dopper-free fluorescence excitation spectrum using the Labview system.

Ⅲ. LEVEL STRUCTURE OF CH$_\textbf{3}$ INTERNAL ROTATION

In CH$_3$OH and CH$_3$CHO, CH$_3$ internal rotation is generally three-fold symmetric with respect to the CH$_3$ rotation angle. The one-dimensional Schrödinger equation is expressed as

 $\begin{eqnarray} \left[ -F \frac{\partial^2}{\partial \phi^2} + \frac{V_3}{2} (1-\cos 3\phi) \right]\Psi = E_m \Psi \label{eq:01 Hamiltonian-CH3} \end{eqnarray}$ (1)

where $V_3$ and $\phi$ are the barrier height to CH$_3$ internal rotation and the rotation angle, respectively. $F$ is the CH$_3$ rotational constants (5.4 cm$^{-1}$). If $V_3$=0, this system is considered to be a free rigid-rotor, and the eigenvalues are given by [15, 16]

 $\begin{eqnarray} E_{m} = F{m}^2 \;, \; \:\:\:\: m = 0, \pm1, \pm2, \cdots \label{eq:02 V=0-Eigenvalue} \end{eqnarray}$ (2)

The eigenvalues for $V_3$$\neq0 are obtained by Mathieu's equation, but are actually calculated by diagonalizing the energy matrix, in which nonvanishing matrix elements are  \begin{eqnarray} \langle m | H | m \rangle \hspace{-0.15cm}&=&\hspace{-0.15cm} F m^2 + \frac{V_3}{2}\end{eqnarray} (3)  \begin{eqnarray} \langle m | H | m \pm 3 \rangle \hspace{-0.15cm}&=&\hspace{-0.15cm} - \frac{V_3}{4}\end{eqnarray} (4) The G_{6} permutation-inversion group is isomorphous with the C_{3\textrm{v}} point group, and the symmetry of torsional levels is represented by a_1, a_2, and e [17]. The actual energy levels in the S_0 state of acetaldehyde (V_3=400 cm^{-1}) are schematically depicted in FIG. 1(a).  FIG. 1 Energy levels for CH_3 internal rotation in the S_0 state of (a) acetaldehyde and (b) 9MA. The barrier height is defined by the energy difference between the potential minima and maxima In the case of 9MA, the potential energy curve is six-fold symmetric with respect to the CH_3 rotation angle \phi, and the Schrödinger equation is expressed as  \begin{eqnarray} \left[ -F \frac{\partial^2}{\partial \phi^2} + \frac{V_6}{2} (1-\cos 6\phi) \right] \Psi = E_m \Psi \end{eqnarray} (5) The eigenvalues are obtained by diagonalizing the energy matrix, in which nonvanishing matrix elements are  \begin{eqnarray} \langle m | H | m \rangle &=& F m^2 + \frac{V_6}{2} \end{eqnarray} (6)  \begin{eqnarray} \langle m | H | m \pm 6 \rangle &=& - \frac{V_6}{4} \end{eqnarray} (7) The G_{12} permutation-inversion group is isomorphous with the D_{3\textrm{h}} point group, and the symmetries of torsional levels are represented by a_1', a_1'', a_2', a_2'', e', and e''. The actual energy levels in the S_0 state of 9MA (V_6 = 115 cm^{-1}) are schematically shown in FIG. 1(b). The energy levels sensitively vary with the V_6 value, and it is thus possible to estimate the V_6 value from the experimentally obtained level energies. Two conformers, staggered (FIG. 2(a)) and eclipsed (FIG. 2(b)), are considered to be the stable structures at the potential minima. The rotational constants are, however, identical because the CH_3 moiety is a symmetric top, and it is impossible to determine the stable conformer from the spectral analysis. In addition, the rotational energy levels are affected by the vibrational-rotational interaction, which also varies the rotational constants. It is essential to observe a high-resolution and high-precision spectrum and to analyze the level structure of the isolated molecule, and ab initio calculations are necessary to accurately estimate the level energies and molecular structure.  FIG. 2 Two conformers of 9MA: (a) Staggered and (b) Eclipsed Ⅳ. RESULTS AND DISCUSSION FIG. 3 shows the Doppler-free fluorescence excitation spectrum of 9MA-h_{12} in a collimated supersonic jet. Frequency calibration markers generated by optical frequency comb and conventional Fabry-Perot cavity are shown in FIG. 3 (a) and (b), respectively. The observed peaks are assigned to the rotational lines of the S_1 (a_1')$$\leftarrow$S$_0 (a_1')$ or S$_1 (e')$$\leftarrow S_0 (e') transition. These two bands are expected to exhibit a b-type spectrum, in which the transition moment is parallel to the molecular in-plane short axis. The linewidth is approximately 10 MHz, and a number of lines are completely resolved. The rotational temperature was estimated to be approximately 15 K. Most of strong peaks are somewhat broad, however, because of the overlap of several rotational lines. Each transition energy, which is given by the light energy at the line center, can be determined with the uncertainty of approximately 300 kHz, because the laser frequency is completely controlled by our system of an optical frequency comb locked to the GPS signal. We observed approximately 1000 lines and are now undertaking fine analysis using the PGOPHER program [18]. The prominent peaks with regular splitting are assigned to ^R R_{0} (J) and so on, which obey the selection rules for the b-type transition, \Delta J =0, \pm1, \Delta K_a = \pm 1, and \Delta K_c=\pm 1. In this case, strong Q transitions near the band origin are missing in the spectrum. The electronic structure of 9MA is similar to that of anthracene [19]. The S_1$$\leftarrow$S$_0$ transition of 9MA corresponds to HOMO$\rightarrow$LUMO one-electron excitation, and its transition moment is large and is parallel to the $b$ axis (short in-plane axis) [12]. Our experimental results are consistent with these theoretical considerations.

 FIG. 3 (a) Trace of the beat frequency between stabilized optical frequency comb and scanning Ti:Sapphire laser, which was used for the frequency calibration of Doppler-free spectrum, (b)transmission of temperature stabilized Fabry-Perot cavity, and (c) a part of the Doppler-free fluorescence excitation spectrum of the S$_1 (a_1')$$\leftarrowS_0 (a_1') transition of 9MA in a collimated supersonic jet First, we performed theoretical calculations at the standard level with geometry optimization, MP2/6-31G(d, p) for the S_0 state, and TD-DFT/6-31G(d, p) for the S_1 state. The results are summarized in Table Ⅰ. The calculated excitation energy for the S_1$$\leftarrow$S$_0$ transition is sufficiently close to the experimental value of the band origin of the S$_1 (a_1')$$\leftarrowS_0(a_1') transition, 26932 cm^{-1} (371.20 nm). We calculated the high-resolution spectrum with these rotational constants, and the result was sufficiently close to the observed spectrum. We are now trying to obtain the accurate rotational constants by a least-squares fit of the transition energies of observed spectral lines for both the S_0 and S_1 states. Table Ⅰ Calculated rotational constants and excitation energy for the S_1$$ \leftarrow$S$_0$ transition of 9MA

We found that the observed high-resolution spectrum was not simply reproduced only with the rigid-rotor rotational constants, $A$, $B$, and $C$, suggesting that the vibrational-rotational interaction is significant in the CH$_3$ internal-rotation levels. The vibronic level energies could not be reproduced well by the simple model described in Section Ⅲ. The barrier to CH$_3$ internal rotation is significantly varied by deuterium substitution. In order to better understand these facts, we performed theoretical calculations of the MC_MO method, which takes nuclear quantum effect into account [20-25]. The Hamiltonian is expressed by

 $\begin{eqnarray} \hat{H} & = & - \frac{1}{2m_e} \sum\limits_i \nabla_i{^2} - \sum\limits_i \sum\limits_A \frac{Z_A}{r_{iA}} + \sum\limits_{i>j} \frac{1}{r_{ij}} \nonumber \\ & & -\frac{1}{2 M_p} \sum\limits_p \nabla_p{^2} + \sum\limits_{p} \sum\limits_{A} \frac{Z_p Z_A}{r_{pA}} +\sum\limits_{p>q} \frac{Z_p Z_q}{r_{pq}} \nonumber \\ &&- \sum\limits_i \sum\limits_p \frac{Z_p}{r_{ip}} + \sum\limits_{A>B}^M \frac{Z_A Z_B}{R_{AB}} \; \label{eq:07 Hamiltonian-MCMO} \end{eqnarray}$ (8)

The indices of $i$ and $j$ refer to the electrons, $A$ and $B$ refer to the classical nuclei, and $p$ and $q$ refer to the quantum nuclei. $m_e$ and $M_p$ are the masses of an electron and the $p$-th quantum nucleus, respectively. $Z_A$ is the charge of the $A$-th nucleus, and $Z_p$ represents the charge of the $p$-th quantum nucleus. The fourth to sixth terms are the quantum nuclear terms, and the seventh term is Coulomb term between an electron and a quantum nucleus, and the last term is the classical nucleus-nucleus repulsion.

We calculated the barrier height to CH$_3$ internal rotation, which is given by the energy difference between the eclipsed and staggered conformers ($\Delta E$=$E(eclipsed) $$-$$ E(staggered)$), at the MC_MO-MP2/6-31G(d, p) level. The results are summarized in Table Ⅱ. The barrier-height reduction by the deuterium substitution is attributable to the difference between nuclear quantum nature of the proton and deuteron.

Table Ⅱ Calculated and experimental values of barrier height to CH$_3$ internal rotation of 9MA in the S$_0$ state

Although the MC_MO-MP2 calculations reproduced the barrier-height reduction in 9MA-$d_{12}$ qualitatively, the magnitude of the reduction was underestimated compared to the experimental values. It is likely that the vibration-vibration interaction affects the energies of rotational levels, for example, CH$_3$ internal rotation and out-of-plane wagging [26]. We are carrying out detailed calculations at a higher level.

It should be noted that the main cause of barrier to CH$_3$ internal rotation is hyperconjugation [13, 14]. The sp$^3$ orbitals in the CH$_3$ group spread in the out-of-plane direction, partly act as the $\pi$ orbital, and conjugate with the aromatic $\pi$ orbital. This is called hyperconjugation. It was experimentally shown that the $V_6$ value varied upon electronic excitation, suggesting that hyperconjugation is important for the barrier height to CH$_3$ internal rotation in 9MA, which depends on the change in the aromatic $\pi$ orbital. In toluene, the $V_6$ value is extremely small, and this is also considered to mainly depend on hyperconjugation. It is very interesting to solve this problem by means of high-resolution spectroscopy and theoretical calculations.

In summary, the vibrational and rotational level structure of 9-methylanthracene has been investigated by observing and analyzing the high-resolution and high-precision spectrum, and it is well understood with the basic theoretical considerations originally established by Li-Hong Xu and Jon Hougen for small molecules such as methanol and acetaldehyde. The final goal is to perform a global fit of all vibrational and rotational levels relatied to the CH$_3$ internal rotation. The fine analysis of spectroscopic results and ab initio theoretical calculations are further desired to accurately reproduce the experimentally obtained level energies and to determine the accurate and reliable molecular constants and equilibrium and average molecular structure.

Ⅴ. ACKNOWLEDGMENTS

Masaaki Baba is deeply grateful to the late Dr. Li-Hong Xu and Dr. Jon T. Hougen for their kind help and encouragement for this work.

We all thank Li-Hong and Jon for their great contributions to establishing the Asian Workshop on Molecular Spectroscopy.

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Masaaki Babaa , Ayumi Kanaokaa , Akiko Nishiyamab , Masatoshi Misonoc , Takayoshi Ishimotod , Taro Udagawae
a. 日本京都大学理学研究院化学系，京都 606-8502;
b. 波兰哥白尼大学物理、天文和信息学院物理研究所，托伦 87-100;
c. 日本福冈大学理学院应用物理系，福冈市城南区 814-0180;
d. 日本横滨市立大学国际艺术与自然科学研究所协会, 横滨金泽区 236-0027;
e. 日本岐阜大学工程学院化学与生物分子科学系, 岐阜柳堂 501-1193