Chinese Journal of Chemical Physics  2017, Vol. 30 Issue (6): 776-788

#### The article information

Dan Hou, Xiao-Long Zhang, Yu Zhai, Hui Li

The Role of High Excitations in Constructing Sub-spectroscopic Accuracy Intermolecular Potential of He-HCN: Critically Examined by the High-Resolution Spectra with Resonance States

Chinese Journal of Chemical Physics, 2017, 30(6): 776-788

http://dx.doi.org/10.1063/1674-0068/30/cjcp1712231

### Article history

Accepted on: December 27, 2017
The Role of High Excitations in Constructing Sub-spectroscopic Accuracy Intermolecular Potential of He-HCN: Critically Examined by the High-Resolution Spectra with Resonance States
Dan Houb, Xiao-Long Zhanga, Yu Zhaia, Hui Lia
Dated: Received on December 3, 2017; Accepted on December 27, 2017
1. Laboratory of Theoretical and Computational Chemistry, Institute of Theoretical Chemistry, Jilin University, Changchun 130023, China;
2. Institute of Functional Materials and Agricultural Applied Chemistry, College of Science, Jiangxi Agricultural University, Nanchang 330045, China
*Author to whom correspondence should be addressed. Hui Li, E-mail: Prof-huili@jlu.edu.cn
Abstract: Interpreting high-resolution rovibrational spectra of weakly bound complexes commonly requires spectroscopic accuracy (< 1 cm-1) potential energy surfaces (PES). Constructing high-accuracy ab initio PES relies on the high-level electronic structure approaches and the accurate physical models to represent the potentials. The coupled cluster approaches including single and double excitations with a perturbational estimate of triple excitations (CCSD(T)) have been termed the "gold standard" of electronic structure theory, and widely used in generating intermolecular interaction energies for most van der Waals complexes. However, for HCN-He complex, the observed millimeter-wave spectroscopy with high-excited resonance states has not been assigned and interpreted even on the ab initio PES computed at CCSD(T) level of theory with the complete basis set (CBS) limit. In this work, an effective three-dimensional ab initio PES for HCN-He, which explicitly incorporates dependence on the Q1 (C-H) normal-mode coordinate of the HCN monomer has been calculated at the CCSD(T)/CBS level. The post-CCSD(T) interaction energy has been examined and included in our PES. Analytic two-dimensional PESs are obtained by least-squares fitting vibrationally averaged interaction energies for v1(C-H)=0, and 1 to the Morse/Long-Range potential function form with root-mean-square deviations (RMSD) smaller than 0.011 cm-1. The role and significance of the post-CCSD(T) interaction energy contribution are clearly illustrated by comparison with the predicted rovibrational energy levels. With or without post-CCSD(T) corrections, the value of dissociation limit (D0) is 8.919 or 9.403 cm-1, respectively. The predicted millimeter-wave transitions and intensities from the PES with post-CCSD(T) excitation corrections are in good agreement with the available experimental data with RMS discrepancy of 0.072 cm-1. Moreover, the infrared spectrum for HCN-He complex is predicted for the first time. These results will serve as a good starting point and provide reliable guidance for future infrared studies of HCN doped in (He)n clusters.
Key words: Potential energy surface    Rovibrational spectra    van der Waals complex
Ⅰ. INTRODUCTION

Hydrogen cyanide (HCN) is one of the most basic interstellar molecules [1], and HCN is observed in space, dark cold clouds of circumstellar envelopes, comets, active galaxies and cool carbon stars [2-7]. More importantly, HCN is an essential tracer of the dense molecular gas which is used in the interstellar medium [8]. Calculation of collisional excitation rate coefficients for the HCN molecule was among the first interstellar applications [9]. Hydrogen is generally the most abundant colliding partners in interstellar space, however, helium can also play an important role in energetic regions [7]. Rate coefficients for the rotational excitation of HCN by He atom have been researched by Green and Thaddeus [10] in 1974. Since then, the collision of HCN with He has been extensively studied by more and more astrophysicists and chemists [7, 8, 11, 12].

High-resolution infrared or microwave spectra studies of cold helium clusters or droplets doped with a single chromophore molecule have been used to probe the microscopic superfluid [13-26]. Among the studied chromophor-He$_n$ species, carbon monoxide as a gentle probe molecule [27-31] with quantum solvation and microscopic superfluidity has received numerous attentions because of the well-known less anisotropic interaction and a relatively large rotational constant. HCN, as one of CO isoelectronic, was used as a probe to investigate the microscopic superfluid in the He nanodroplets and test the adiabatic following [32, 33]. Compared with CO, HCN with its relatively large dipole moment of 2.985(2) D [34] is also a better candidate for analogous measurements of HCN-He$_n$ clusters, and its clusters with various isotopologues and spin isomers of molecular hydrogen [35]. The intermolecular potential energy surfaces (PESs) between HCN and a helium atom is a fundamental interest and an essential starting point for the exploration of large clusters.

The He-HCN complex is a typical example of van der Waals molecular complex, and the HCN part rotates almost freely in the complex [36]. Drucker et al. [37] measured the ground-state $J$=1$\leftarrow$0 and $J$=2$\leftarrow$1 transitions at 15893.6108 and 31325.2443 MHz by using millimeter wave/microwave double resonance, and together with ab initio calculations of the He-HCN potential surface at the fourth-order Møller-Plesset (MP4) level. Then, Atkins and Hutson developed two empirical PESs, $1E8$ and $2E8$, based on two different functional forms using eight experimental data of Drucker et al. [38]. Toczylowski and his cooperator [39] calculated the global minimum on the avtz+(33221) PES of He-HCN, and a global minimum is in the linear He-HCN configuration at $R$=7.97 a$_0$ with a well depth of 29.90 cm$^{-1}$. In 2002, the millimeter-wave (MMW) spectroscopy of He-HCN was reported by Harada et al. [36]. They employed a multi-reflection path for MMW spectroscopy to facilitate the observation of weak transitions and improved the empirical PES by extending measurement of internal-rotation transitions. At the same time, Ansari and Varandas [40] determined a single-valued 6-D PES by using the energy method, which utilized a global double many-body expansion (DMBE) and a Legendre polynomial expansion. In 2013, Denis-Alpizer et al. [41] constructed a 4-D He-HCN PES which subjected to bending vibrational motion by calculating more than 43015 single-point. Recently, new experimental results [42] have exhibited the highly excited resonance states which are very sensitive with the accuracy of PES, and also provide an important experimental basis for further rigorous testing of the PES.

In our previous work [20], we reported a 5-D ab initio PES of CO-H$_2$ that explicitly incorporated dependence on the stretch coordinate of the CO monomer, the resulting vibrationally averaged PESs provide good representations of the experimental data. HCN molecule as an isoelectronic molecule of CO, has a triple bond with nonnegligible high excitations effects. In order to predict the IR spectra, the C$-$H stretching mode ($Q_1$) of HCN is also considered in calculating the intermolecular potential. In this work, we construct an accurate three-dimensional ab initio PES for He-HCN complex, which is computed at CCSD(T) level of theory with the complete basis set (CBS) limit. The triple and quadruple high-order of excitation energies are included by executing CCSDT(Q) calculations, which cannot be ignored when the systems containing triple bond or isolated electron pair. The resulting PESs after vibrationally averaging over $Q_1$ intramolecular coordinate, are fitted to two dimensional Morse/Long-Range (MLR) function. The MLR form explicitly incorporates the theoretically known inverse-power long-range behavior and its anisotropy dependence on the intramolecular stretch coordinates. Using these PESs, we predict the rovibrational bound states and the millimeter-wave spectrum which are in good agreement with previous experimental data [36, 37]. Moreover, we provide the first prediction of the infrared spectra and band origin shift for He-HCN dimer.

Ⅱ. POTENTIAL ENERGY SURFACES A. Ab initio calculations and reduced-dimension treatment

The geometry of a He-HCN complex in which HCN is kept linear can be described naturally using Jacobi coordinates $(R, \, \theta, \, Q_1)$ shown in FIG. 1, where $R$ is the distance from the center of mass of HCN to the He atom, $\theta$ is the angle between the vector pointing from the center of mass of HCN to He and vector pointing from N to H, and $Q_1$ is the normal mode coordinate for the $\nu_1$ (C$-$H stretch) vibration of HCN. Our recent work showed that when one of the intramolecular vibrational stretch modes of the chromophore is excited, the effect of the associated change in the average values of the other coordinates cannot be ignored [44, 45]. Hence, in our ab initio calculations for the He-HCN complex, the equilibrium bond lengths for C$-$H and C$-$N were fixed at the average bond lengths implied by the experimental moments of inertia for ground state $(v_1, v_2, v_3)$=(0, 0, 0) and for the first excited (1, 0, 0) level of HCN [46].

 FIG. 1 Jacobi coordinates for He-HCN complex.

In a full four-dimensional (4D) treatment, which takes into account the coordinate $Q_1$, the total potential energy for He-HCN would be written as

 $\begin{eqnarray} \label{eq:4DVtot} V(R, \, \theta, \, Q_1, \, Q_3)&=& V_{\rm{HCN}}(Q_{1}, \, Q_3) +\nonumber\\ &&{}\Delta V(R, \, \theta, \, Q_1, \, Q_3) \end{eqnarray}$ (1)

here $V_{\rm{HCN}}(Q_{1}, \, Q_3)$ is the effective two-dimensional (2D) potential energy for the C$-$H and C$-$N stretching of an isolated linear HCN molecule, and $\Delta V(R, \, \theta, \, Q_1, \, Q_3)$ is the intermolecular interaction potential. However, our previous results for the He-CO$_2$ system showed that a reduced-dimension treatment with the symmetric stretch coordinate $Q_1$ fixed at its average values for the appropriate $v_3$ vibrational level of CO$_2$ was a good approximation which led to very accurate predicted vibrational frequency shifts for CO$_2$ in (He)$_n$ clusters [44, 45]. Following that approach, our effective 3D potentials for He-HCN can be defined as

 $\begin{eqnarray} \label{eq:3DVtot} &&\langle \psi_{v_3}^{\{v_1\}}(Q_{3}) |(R, \theta, Q_1, Q_3) |\psi_{v_3}^{\{v_1\}}(Q_{3}) \rangle\nonumber \\ &&\approx V(R, \theta, Q_1; \overline Q_3^{\, \{v_1\}})\nonumber \\ &&= V_{\rm{HCN}}(Q_{1};\overline{Q}_3^{\{v_1\}})+ \Delta V(R, \theta, Q_1;\overline Q_3^{\, \{v_1\}})\quad\quad \end{eqnarray}$ (2)

in which the notation reminds us that the average value of $Q_3$ depends on the C$-$N stretch vibrational quantum number $v_1\,$.

Hence, both the effective 1D potentials $V_{\rm{HCN}}$ $(Q_1;\overline Q_3^{\, \{v_1\}})$ which govern the $Q_1$ vibration of a free HCN monomer and the intermolecular potential $\Delta V(R, \theta, Q_1;\overline Q_3^{\, \{v_1\}})$, depend not only on $Q_{1}$, but also on the associated averaged value of the $Q_3$ stretch coordinate $\overline Q_3^{\, \{v_1\}}$. The vibrationally average values of the C$-$H and C$-$N bond lengths in the ground $\, (v_1, v_2, v_3)$=(0, 0, 0) ($r_{\rm{CH}}$=1.069064 Å and $r_{\rm{CN}}$=1.155525 Å) and the (1, 0, 0) excited state ($r_{\rm{CH}}$=1.099040 and $r_{\rm{CN}}$=1.154432 Å) of HCN are obtained by fitting the experimental rotational constants of the main isotopic species of the HCN monomer [47].

The effective 1D potentials $V_{\rm{HCN}}(Q_{1};\overline Q_3^{\, \{v_1\}})$ governing the $Q_1$ vibration of the isolated HCN monomer were calculated using the explicitly correlated CCSD(T)-F12a [48] approximations with augmented correlation-consistent polarized core-valence triple-zeta (cc-pCVTZ) basis sets [49]. The HCN $Q_1$ normal mode coordinate was obtained by performing geometry optimization numerically at CCSD(T)-F12a/aug-cc-pCVTZ level. The optimized equilibrium C$-$H and C$-$N bond distances were found to be 1.06561 and 1.15341 Å, respectively, in good agreement with the experimental values of $\, r_0({\rm{CH}})$=1.083 Å and $\, r_0({\rm{CN}})$=1.1527 Å [50]. For chosen fixed values for the C$-$H and C$-$N bond lengths at the vibrationally averaged ground $(v_1, v_2, v_3)$=(0, 0, 0) and first excited state (1, 0, 0), the potential energy was computed at 51 values of $Q_1$ ranging from $-$1.0 Å to 1.0 Å, and spline interpolation was used to provide values of the effective 1D potentials at configurations between those grid points.

The intermolecular potential energies of He-HCN were calculated at using single-and double-excitation coupled-cluster theory with a non-iterative perturbation treatment of triple excitations (CCSD(T)) [43]. The basis set used was the augmented correlation-consistent polarized n-zeta basis set of Woon and Dunning (denoted as aug-cc-pVNZ or AVNZ for $N$=4 and 5) [51], supplemented with an additional set of bond functions (3s3p2d1f1g) (where the exponent coefficient $\alpha$=0.9, 0.3, 0.1 for 3s and 3p; $\alpha$=0.6, 0.2 for 2d; $\alpha$=0.3 for f and g) placed at the mid-point of the intermolecular axis $R$ [52, 53]. The supermolecule approach was used to produce the intermolecular potential energies $\Delta V(R, \theta, Q_1;\overline Q_3^{\, \{v_1\}})$, which is defined as the difference between the energy of the He-HCN complex and the sum of the energies of the HCN and He monomers. The full counterpoise procedure was employed to correct for basis set superposition error (BSSE) [54]. All calculations were carried out using the MOLPRO package [55].

The total intermolecular interaction potential $\Delta V_{\rm int}$ can be expressed as

 $\begin{eqnarray} \label{eq:DVint} \Delta V_{\rm int}\, ~=~\, \Delta V_{\rm int}^{\rm HF}\, +\, \Delta V_{\rm int}^{\rm corr} \end{eqnarray}$ (3)

in which,

 $\begin{eqnarray} \label{eq:DVcorr} \Delta V_{\rm int}^{\rm corr}\, ~=~\, \Delta V_{\rm int}^{\rm CCSD(T)} \, +\, \Delta V_{\rm int}^{\rm T(Q)} \end{eqnarray}$ (4)

The Hartree-Fock part $\Delta V_{\rm int}^{\rm HF}$ is extrapolated by using two-point formula of $(n+1)\cdot\, \exp(-9\sqrt n )$ [56], where $n$ is the so called cardinal number of aug-cc-pVNZ basis set with $N$=4 and 5. The correlation energy $\Delta V_{\rm int}^{\rm CCSD(T)}$ is obtained directly from the two-point $1/n^3$ extrapolation. The electronic structure of HCN with triple bond, which makes the He-HCN complex is particularly sensitive to electronic correlation in high orders of perturbation theory [57]. Noga et al. has examined the importance of higher order contributions to the interaction potential of He-HCN at most important minima of the interaction energy surface. The results indicate that both the missing contributions from the triple excitations and from the quadruple excitations affect the final potential appreciably and anisotropically. Their total value in the vicinity of the global minimum amounts to about 3 cm$^{-1}$, whereas only about 1 cm$^{-1}$ at the secondary minimum [57]. In this work, electron correlation energies from the triple and quadruple excitations $\Delta V_{\rm int}^{\rm T(Q)}$ are calculated at CCSDT(Q) level using the aug-cc-pVDZ basis set without bond function. Convergence study has been performed in Ref.[58, 59]. All calculations were carried out using the MOLPRO package [55], and MRCC program of Kallay [60].

FIG. 2 shows the comparisons of the one-dimen-sional cuts of the interaction potentials for selected angular orientations of the He-HCN complex. These interaction potentials were calculated at CCSD(T)/AVXZ(X=Q, 5), CCSD(T)/CBS(complete basis set) and CCSD(T)/CBS+$\Delta V^{\rm T(Q)}$ levels, respectively. As seen in FIG. 2, for the T-shaped configurations ($\theta$=90$^\circ$), within the standard coupled cluster computations, the depth of the potential wells increases as the size of the basis set increases, and those computed without the $\Delta V_{\rm int}^{\rm T(Q)}$ have higher energies than those with the $\Delta V_{\rm int}^{\rm T(Q)}$.

 FIG. 2 One-dimension cuts of the interaction potentials for one orientation of He-HCN complex computed at different levels of theory.

FIG. 3 presents the intermolecular potentials at four stationary points including the global minimum, the local minimum, saddle point and T-shape configurations. The energies calculated at CCSD(T)/AV5Z, CCSD(T)/CBS and CCSD(T)/CBS+$\Delta V^{\rm T(Q)}$ levels are shown in green, gray and blue, respectively. Relative to the intermolecular potentials calculated at the reference level of CCSD(T)/CBS, the energy differences $\Delta V_{\rm diff}$, for each stationary point are shown in the upper panel of FIG. 3. It is clear that the energy differences between CCSD(T)/AV5Z and CCSD(T)/CBS for all four stationary points are smaller than 0.1 cm$^{-1}$, which indicates that the potentials at CCSD(T)/CBS level are convergent. However, the absolute energy differences between CCSD(T)/CBS+$\Delta V^{\rm T(Q)}$ and CCSD(T)/CBS for all four stationary points are almost seven times larger than those between CCSD(T)/AV5Z and CCSD(T)/CBS, and the differences are also not systematically changed. Thus, $\Delta V^{\rm T(Q)}$ term plays a nonnegligible role in our calculations. CCSD(T)/CBS+$\Delta V^{\rm T(Q)}$ approach will give us more confidence in the accuracy of the PES.

 FIG. 3 The intermolecular potentials at four stationary points including the global minimum, the local minimum, saddle point and T-shape configuration. The energies calculated at four stationary points with CCSD(T)/AV5z, CCSD(T)/CBS, and CCSD(T)/CBS+$\Delta V^{\rm T(Q)}$ levels are shown in green, gray and blue, respectively.

A total of 1710 ab initio points were calculated for both the ground ($v_1$=0) and first excited ($v_1$=1) states, with C$-$H and C$-$N fixed at the averaged values defined by the experimental inertial rotational constants for the (0, 0, 0) and (1, 0, 0) levels of HCN, respectively. The calculations were performed on regular grids for all three degrees of freedom. Five potential optimized discrete variable representation (PODVR) grid points corresponding to $Q_1$=$-$0.533037, $-$0.305233, $-$0.111046, 0.072396, and 0.263342 Å for the ground (0, 0, 0) and $Q_1$=$-$0.482687, $-$0.253978, $-$0.059161, 0.124794, and 0.316206 Å for the excited (1, 0, 0) state were chosen, respectively. A relatively dense grid of 21 points ranging from 2.2 Å to 10.0 Å was used for the $R$ intermolecular coordinate. The angular coordinate $\theta$ ranged from 0$^\circ$ to 180$^\circ$ with step sizes of 10$^\circ$. The CCSDT(Q) calculations scale with the number of basis functions $N$ as $N^9$, which is two order of magnitude greater than the scale of CCSD(T) calculations with $N^7$, and hence full two-dimensional calculations at CCSDT(Q) level become soon computationally not executable even for small systems and basis sets. In this work, correlation energies $\Delta V_{\rm int}^{\rm T(Q)}$ are only calculated on 2D intermolecular coordinates with C$-$H stretch fixed at its averaged distance of vibrational ground state. A relatively sparse grid point of 18 points range from 2.4 Å to 10.0 Å was used for the $R$. The angular coordinates $\theta$ range from 0$^\circ$ to 180$^\circ$ with step sizes of 10$^\circ$. This gives a total of 342 ab initio points to yield a 2D PESs for $\Delta V_{\rm int}^{\rm T(Q)}$.

Since the $v_1$ vibrational mode of HCN has a much higher frequency than the intermolecular modes, Born-Oppenheimer separation type arguments suggest that it should be a good approximation to introduce such a separation, as long as the off-diagonal vibrational coupling is sufficiently small [61]. In this approximation, the total intra-and intermolecular vibrational wave function would be written as the product

 $\begin{eqnarray} \label{eq:psiBO} \Psi_{v_1}(R, \theta, Q_1;\overline Q_3^{\, \{v_1\}})~=~ \phi_{v_1}(R, \, \theta, )~\psi_{v}(Q_{1};\overline Q_3^{\, \{v_1\}}) \end{eqnarray}$ (5)

in which $v_1$ is the quantum number for a specific C$-$H stretch vibrational state of the free HCN molecule, and the associated 1D vibrational wavefunction $\psi_{v} (Q_{1};\overline Q_3^{\, \{v_1\}})$ is obtained by solving the 1D Schrödinger equation:

 $\begin{eqnarray} &&\left[\frac{-1}{2M}\frac{{\rm{d}}^{2}}{{\rm{d}}{Q_{1}}^{2}}+ V_{\rm{HCN}}(Q_{1}; \overline Q_3)\, \right] \, \psi_{v}(Q_{1};\overline Q_3)\nonumber\\ &&= E_{v}\, \psi_{v} (Q_{1};\overline Q_3) \label{eq:HQ1} \end{eqnarray}$ (6)

The present work focuses on complexes formed from HCN in the ground ($v_1$=0) and first excited ($v_1$=1) C$-$H stretch states of HCN. Using Eq. (5), the vibrationally averaged He-HCN interaction potential for HCN in vibrational level $v_1$ is

 $\begin{eqnarray} \label{eq:Vbar} \overline V_{[v_1]}(R, \theta)&=& \int_{-\infty}^{\infty} \psi_{v_1}^{*} (Q_{1};\overline Q_3^{\{v_1\}}) \Delta V(R, \theta, Q_{1};\overline Q_3^{\{v_1\}})\cdot\nonumber\\ &&{}\psi_{v_1}(Q_{1};\overline Q_3^{\{v_1\}}){\rm{d}}Q_{1} \end{eqnarray}$ (7)

Note that the vibrationally averaged intermolecular potentials $\overline V_{[v_1]}(R, \, \theta)$ for different values of $v_1$ differ both because the wavefunctions $\psi_{v_1}(Q_1)$ are associated with different values of $v_1$, and because they were obtained from effective 1D potentials associated with different values of $\overline Q_3^{\{v_1\}}$.

B. Analytic potential energy function

The vibrationally averaged ab initio intermolecular potential energies $\overline V_{[v_1]}(R, \theta)$ for He-HCN obtained from Eq. (7) were fit to a generalization of the two-dimensional Morse/Long-Range (MLR) potential function form [44, 62],

 $\begin{eqnarray} \overline V_{\rm MLR}(R, \theta) ={\mathfrak D}_e(\theta)\times \bigg[1-\frac{u_{\rm LR}(R, \theta)} {u_{\rm LR}(R_e, \theta)}\cdot\nonumber\\ {\rm{e}}^{-\beta(R, \theta)\cdot y_p^{\rm{eq}}(R, \theta)} \bigg]^2 \label{eq:V2DMLR} \end{eqnarray}$ (8)

In Eq.(8), ${\mathfrak D}_e(\theta)$ is the well depth and $R_e(\theta)$ is the position of the minimum along a radial cut through the potential at angle $\theta$, while $\, u_{\rm LR}(R, \theta)\,$ is a function that defines the (attractive) limiting long-range behaviour of the effective 1D potential along that cut

 $\begin{eqnarray} \overline V (R, \theta)\simeq{\mathfrak D}_e(\theta) u_{\rm LR}(R, \theta) \label{eq:VMLRlim} \end{eqnarray}$ (9)

Since HCN is polar while He is non-polar, the appropriate functional form for $u_{\rm LR}(R, \theta)$ is

 $\begin{eqnarray} u_{\rm LR}(R, \theta)= \frac{\overline C_6(\theta)}{R^6} +\frac{\overline C_7(\theta)}{R^7} +\frac{\overline C_8(\theta)}{R^8} \label{eq:VLR678} \end{eqnarray}$ (10)

in which the long range coefficients $\overline C_n$ have also been averaged over the HCN stretch coordinate $Q_1$. In Eq.(8), the denominator factor $u_{\rm LR}(R_e, \theta)$ is the function $u_{\rm LR}(R, \theta)$, Eq. (10), evaluated at $R$=$R_e(\theta)$. The radial distance variable in the exponent in Eq.(8) is the dimensionless quantity

 $\begin{eqnarray} y_p^{\rm{eq}}(R, \theta)=\frac{R^p - {R_{e}(\theta)}^p} {R^p + {R_{e}(\theta)}^p} \label{eq:ypeq} \end{eqnarray}$ (11)

where $p$ is a small positive integer which must be greater than the difference between the largest and smallest (inverse) powers appearing in Eq.(10) [62]. The exponent coefficient function $\beta(R, \theta)$ is a (fairly) slowly varying function of $R$, which is written as the constrained polynomial

 $\begin{eqnarray} \beta(R, \theta)&=& y_p^{{\rm ref}}(R, \theta) ~\beta_\infty(\theta) + \left[1-y_p^{{\rm ref}}(R, \theta)\right]\cdot\nonumber\\&&{} \sum\limits_{i=0}^N \beta_i(\theta) y_q^{{\rm ref}}(R, \theta)^i \label{eq:betapq} \end{eqnarray}$ (12)

In Eq. (12), two new radial variables have been introduced

 $\begin{eqnarray} \begin{array}{l} y_p^{{\rm{ref}}}(R, \theta ) =\displaystyle \frac{{{R^p} - R_{{\rm{ref}}}^p}}{{{R^p} + R_{{\rm{ref}}}^p}}\\ [2ex] y_q^{{\rm{ref}}}(R, \theta ) = \displaystyle\frac{{{R^q} - R_{{\rm{ref}}}^q}}{{{R^q} + R_{{\rm{ref}}}^q}} \end{array} \label{eq:ypref} \end{eqnarray}$ (13)

in which, $R_{{\rm ref}}$$\equiv$$f_{{\rm ref}}$$\times$$R_e(\theta)$. In the potential function model used in the present work: $p$=$q$=3, and $f_{{\rm ref}}$=0.9.

Note that the definition of $y_p(R, \theta)$ and the algebraic structure of Eq.(8) and Eq.(12) mean that $\lim_{R\rightarrow \infty}\beta(R, \theta)$$\equiv$$\beta_\infty(\theta)$=$\ln[2\, {\mathfrak D}_e(\theta)/ u_{\rm LR}(R_e, \theta)]$. The parameters ${\mathfrak D}_e(\theta)$, $R_e(\theta)$, and the various $\beta_i(\theta)$ are all expressed as Legendre expansions in $\theta$, written in the form

 $\begin{eqnarray} A(\theta) = \sum\limits_{\lambda=0} A^{\lambda} P_{\lambda}(\cos\theta) \label{eq:Apoly} \end{eqnarray}$ (14)

where $A$=${\mathfrak D}_{e\, }$, $R_e$, or $\beta_{i\, }$.

Following our previous work [44], the leading vibrationally averaged van der Waals coefficients $\overline C_6(\theta)$ in specific vibrational level $v_1$ are calculated from

 $\begin{eqnarray} \label{eq:C6bar} \overline C_{6, [v_1]}(\theta)&=& \int_{-\infty}^{\infty} \psi_{v_1}^{*} (Q_{1};\overline Q_3^{\, \{v_1\}})~ C_{6}(\theta, \, Q_{1};\overline Q_3^{\, \{v_1\}})\cdot\nonumber\\ &&{}\psi_{v_1}(Q_{1};\overline Q_3^{\, \{v_1\}}){\rm{d}}Q_{1} \end{eqnarray}$ (15)

in which $C_6(\theta, Q_1;\overline Q_3^{\{v_1\}})$ is expanded as

 $C_6(\theta, Q_1;\overline Q_3^{\{v_1\}})=\sum\limits_{\lambda=0(2)}^2 [C_{\rm 6, ind}^{\lambda}(Q_1;\overline Q_3^{\{v_1\}})+\nonumber\\ \;\;\;\;\;\;\;\;\;C_{\rm 6, disp}^{\lambda}(Q_1;\overline Q_3^{\{v_1\}})] P_{\lambda}(\cos\theta)\quad\quad$ (16)

The induction term is approximated as

 $\begin{eqnarray} C_{\rm 6, ind}^{0}(Q_1;\overline Q_3^{\, \{v_1\}})~=~ \left[\mu_{\rm{HCN}}(Q_1;\overline Q_3^{\, \{v_1\}})\right]^2 \alpha_{\rm He} \label{eq:C6D} \end{eqnarray}$ (17)

where $\mu_{\rm{HCN}}(Q_1;\overline Q_3^{\, \{v_1\}})$ is the stretching-dependent HCN dipole moment and $\alpha_{\rm He}$ is the polarizability of atomic He. The equilibrium value of the isotropic dispersion coefficient $C_{\rm 6, disp}^0$($Q_1$=0) can be calculated as follows:

 $\begin{eqnarray} C_{\rm 6, dis}^{0}&=&\frac{3}{\pi}\sum\limits_{i}{\rm{d}}\omega_i\alpha_{\|}(Q_1;\overline Q_3^{\{v_1\}})(i\omega_i)\cdot\nonumber\\ &&{}\alpha_{\bot}(Q_1;\overline Q_3^{\{v_1\}})(i\omega_i) \end{eqnarray}$ (18)

In the above expression, the $Q_1$-dependent functions representing $\mu_{\rm HCN}(Q_1; \overline Q_3^{\, \{v_1\}})$, $\alpha_{\|}(Q_1;\overline Q_3^{\, \{v_1\}})$, and $\alpha_{\bot}(Q_1;\overline Q_3^{\, \{v_1\}})$ were all calculated at CCSD(T)/aug-cc-pVQZ level using the finite-field method [63], which has been successfully employed in our previous work [64]. When we got the leading vibrational averaged coefficient $\overline C_{\rm 6}^0$ and $\overline C_{\rm 6}^2$, the other vibrational averaged long-rang coefficients for $\overline C_{\rm 7}^1$, $\overline C_{\rm 7}^3$, $\overline C_{\rm 8}^0$, $\overline C_{\rm 8}^2$ and $\overline C_{\rm 8}^4$ are obtained by fitting the vibrationally averaged intermolecular potentials with $R$$\geq6 to Eq.(10), while keeping \overline C_{\rm 6}^0 and \overline C_{\rm 6}^2 fixed at the calculated values. All of the vibrationally averaged long-range coefficients or ratios relative to \overline C_{\rm 6}^0 for He-HCN complex with HCN in its vibrational ground (0, 0, 0) and excited (1, 0, 0) states are summarized in Table Ⅰ. Table Ⅰ Expansion coefficients \overline{\mathfrak D}_e^{\, \lambda} in cm^{-1}, \overline{R}_e^{\, \lambda} in Å, \overline{\beta}_i^{\, \lambda} and Lang-Rang coefficients \overline{C_n} in cm^{-1}Å^6 defining our 2D vibrationally averaged potential energy surfaces for He-HCN (v_1=0 and 1). Finally, the vibrationally averaged potentials are fit to 2D Morse/Long-Range (2D-MLR) form Eq.(8). In the fits at this time, all the long-range coefficients are fixed, and the input vibrationally averaged ab initio, energies were weighted by assigning them uncertainties of u_i=0.1 cm^{-1} for the attractive region where \Delta V(R, \theta)$$\leq$0.0 cm$^{-1}$, and $u_i$=($\Delta V(R, \theta)$+1.0)/10.0 cm$^{-1}$ for the repulsive region where ($\Delta V(R, \theta, \phi$)$>$0.0 cm$^{-1}$). Using these weights, a fit with a root-mean-square residual discrepancy of 0.011 cm$^{-1}$ is obtained on fitting the 342 ab initio points at energies $\Delta V(R, \theta)$$< 1000 cm^{-1} to a 2D-MLR potential defined by only 49 fitting parameters. The values of the resulting set of potential parameters are presented in Table Ⅰ. C. Features of the vibrationally averaged two-dimensional potential energy surfaces FIG. 4 displays contour plot of the vibrationally ave-raged 2D MLRQ potential energy surface (CCSD(T)/ CBS+\Delta V^{\rm T(Q)}) for He-HCN(v_1=0) in cylindrical coordinates with x=R\cos\theta, y=R\sin\theta. As shown in FIG. 4, the global minimum of -30.740 cm^{-1} is located at the hydrogen end with \theta=0.0^\circ and R=4.234 Å. One local minimum with energy appears at the collinear geometries with \theta=108.7^\circ and R=3.572 Å. Along the lowest-energy isomerization path between the global minimum and local minima, there is a saddle point with a barrier of height 22.643 cm^{-1} located at \theta=90.8^\circ with R=3.533 Å in a planar T-shape geometry. FIG. 4 also describes the minimum energy path joining these local minima to the global minima, which is more clearly shown in the FIG. 5, and FIG. 5 illustrates the energy (upper) and radial position (lower) along the minimum energy path which joins the one saddle point to the global minimum. Compared with the V_{\rm harada} surface [65], as shown in the upper panel of FIG. 5, the present VMLRQ(v_1=0) potential has lower energies than those of V_{\rm harada} throughout the whole path. The energies at the stationary points are somewhat closer to the values of the two optimized potentials of V_{\rm harada}. The radial positions on the VMLRQ(v_1=0) PES are slightly shorter than those on V_{\rm harada} as shown in the lower panel of FIG. 5. The differences between the VMLRQ(v_1=0) and V_{\rm harada} potentials are mostly attributed to a combination of the effects of different ab initio method and basis sets, and slightly different HCN geometries employed in the calculations.  FIG. 4 Contour plot of the 2D MLR potential energy surface for He-HCN in cylindrical coordinates.  FIG. 5 Energies (upper) and radial positions (lower) along the minimum energy path in the Jacobi angular coordinate θ on the vibrationally averaged 2D MLR PES for He-HCN(v1=0), and compared with those obtained from Vharada PES 65. Table Ⅱ summarizes the geometries and energies of these global minima along with barriers of VMLRQ PES for He-HCN complex, all of which can be compared with previous literature results for this system [36, 37, 39, 41, 66]. Our results are consistent with previous empirical and ab initio surfaces, showing essentially the same anisotropy and similar interaction strength. However, on our 2D VMLRQ PES, the well depth at the global minimum is deeper than those from the previous reports [36, 37, 39, 41, 66]. The global and local minimum are -30.740 and -22.894 cm^{-1}, which are slighly deeper than corresponding values of -30.35 and -22.08 cm^{-1} obtained from V_{13} [41]. Table Ⅱ Properties of stationary points of the He-HCN potential energy surface, and comparisons with the results from previously reported surfaces. All entries are given as ( R in Å, \theta in degree , \Delta V in cm^{-1} ). For the vibrationally-averaged excited-state (v_1=1) surface, the contour plots look almost the same as those for the ground state (v_1=0), and as shown in Table Ⅱ, the positions and energies of the stationary points are only slightly shifted. Ⅲ. MICROWAVE AND INFRARED SPECTRA FOR He-HCN COMPLEX A. Hamiltonian and bound states calculations Within the Born-Oppenheimer approximation, the two-dimensional intermolecular Hamiltonian of the He-HCN complex in the Jacobi coordinate system (R, \theta) with the total angular momentum represented in the body-fixed reference frame can be written as [67-69]  \begin{eqnarray}\label{eq:Ham} \hat{H}&=&-\frac{\hbar^{2}}{2\mu}~\frac{{\partial}^{2}}{\partial{R}^{2}}~+~ \left( \frac{\hbar^{2}}{2\mu{R}^2}~+~ B_{v_1}\right)\cdot\nonumber\\ &&{}\left( \frac{-1}{\sin\theta}~ \frac{\partial~} {\partial\theta}\, \sin\theta~ \frac{\partial~}{\partial\theta}~+~ \frac{\hat{J_z^2}}{\sin^2\theta} \right)+\frac{\hat{J^2} -\, 2\, \hat{J_z^2}}{2\mu\, {R}^2} + \nonumber \\ &&{}\frac{\cot\theta}{2\mu\, {R}^2} \left[\, (\hat{J_x} +i\hat{J_y}) ~+~ (\hat J_x-i\hat J_y)\, \right]\hat{J_z} + \nonumber \\ &&{}\frac{\hbar}{2\mu\, {R}^2}~\frac{\partial~} {\partial\theta} ~\left[\, (\hat{J_x}+ i\hat{J_y})~-~(\hat{J_x}-i\hat{J_y}) \, \right]~+\nonumber \\ &&{} \overline V_{[v_1]}(R, \theta) \end{eqnarray} (19) in which \, \mu^{-1}=m_{\rm{He}}^{-1}+(m_{\rm{N}}+m_{\rm{C}}+m_{\rm{H}})^{-1} where m_{\rm{He}}, m_{\rm{N}}, m_{\rm{C}} and m_{\rm{H}} are the masses of the He, N, C and H atoms [70], respectively, B_{v_1} is the inertial rotational constant of HCN, and \overline V_{[v_1]}(R, \theta) is the vibrationally averaged intermolecular potentials of the He-HCN system. The operators \hat{J_x}, \hat{J_y} and \hat {J_z} are the components of the total angular momentum operator \hat J in the body-fixed frame, and the z axis of the body-fixed frame lies along the Jacobi radial vector R. A direct-product discrete variable representation (DVR) grid was used in the ro-vibrational level energy calculation for the He-HCN complex [71]. A 100-point sinc-DVR grid from 2.2 Å to 15.0 Å was used for the radial R stretching coordinate, and a 100-point Gauss-Legendre grid used for the angular variable. The Lanczos algorithm was used to calculate the ro-vibrational energy levels by recursively diagonalizing the discretized Hamiltonian matrix [72]. Eigenvalue calculations reported below utilized the experimental HCN rotational constants [73] B_{v_1}=1.478219 and 1.467799 cm^{-1} for v_1=0 and 1, respectively. Only 2000 Lanczos steps are required to fully converge the bound state energies by less than 0.00001 cm^{-1}. PES is listed in Table Ⅲ for He-HCN(v_1=0), respectively. The rovibrational energy levels are labeled with four quantum numbers (j, l, J, P) in free internal rotor model, where J is the total angular momentum for the whole complex, j denotes the angular momentum of the HCN, l stands for the end-over-end rotation of the complex, P stands for the parity e or f symmetry with P=0 or P=1. Table Ⅲ lists intermolecular rovibrational energy levels (J=0-6) and the dissociation limit (D_0) in ground state for He-HCN from four PESs with different basis sets, and compares them with results obtained from the observed values [42]. The four PESs of He-HCH are obtained using CCSD(T)/AVQZ, CCSD(T)/AV5Z, CCSD(T)/CBS and CCSD(T)/CBS+\Delta$$V^{\rm T(Q)}$, respectively. From Table Ⅲ, the RMSDs with CCSD(T)/AVQZ, CCSD(T)/AV5Z and CCSD(T)/CBS for the bound state and metastable state are from 0.351 cm$^{-1}$ to 0.495 cm$^{-1}$ which are on the same order of magnitude. However, the RMSDs for states without metastable states or with metastable states are 0.072 or 0.070 cm$^{-1}$ based on CCSD(T)/CBS+$\Delta V^{\rm{T(Q)}}$, respectively, which are more than five times better than those of 0.376 or 0.351 cm$^{-1}$ obtained from PES with CCSD(T)/CBS level. The dissociation limit ($D_0$) for CCSD(T)/CBS+$\Delta$V$^{\rm T(Q)}$ is 9.403 cm$^{-1}$, which in good agreement with the observed value of 9.420 cm$^{-1}$ from Harada [36]. These results clearly indicate that high excitations of $\Delta V^{\rm T(Q)}$ play an important role in constructing subspectroscopic accuracy intermolecular potential of He-HCN.

Table Ⅲ Energies (in cm−1) for the vibrational levels of the vibrationally averaged 2D-MLR potential energy surface for He-HCN (expressed relative to the relevant asymptote), compared with observed results [36, 42].

For PES of CBS+$\Delta V^{\rm T(Q)}$, the lowest rovibrational energy for the ground state is equal to $-$9.403 cm$^{-1}$ indicating that the zero-point energy is 21.237 cm$^{-1}$, about 2/3 of the global well depth $D_e$ (30.740 cm$^{-1}$). The zero-point energy of He-HCN which is significantly higher than the saddle point height of 8.097 cm$^{-1}$. The wave functions for bound state and metastable state levels of He-HCN are shown in FIG. 6. The wave function is along $\theta$ coordinate, which indicates that the levels are all vibrational bending states. The calculated band origin shift predicted by our 2D VMLRQ surfaces is $\Delta v_0$=0.195 cm$^{-1}$ for He-HCN, which is a blue shift.

 FIG. 6 Wave functions for the $J$=0 energy levels and metastable-state levels of He-HCN ($v_1$=0) as functions of $\theta$ and $R$.
B. Microwave and infrared transitions

Microwave transitions calculated from our vibrationally averaged ground state VMLRQ($v_1$=0) PES for He-HCN are listed and compared with experiment and with previous theoretical predictions in Table Ⅳ. The calculated transition frequencies relative to the corresponding $j$ stacks involving P-, Q-and R-branches are listed in columns 4 for $j$=0$\leftarrow$0, $j$=1$\leftarrow$1, $j$=1$\leftarrow$0 and $j$=2$\leftarrow$1 which agree very well with the corresponding experimental values as listed in columns 2 [36, 37] The differences listed in columns 3 are very small, and the RMSD is only 0.072 cm$^{-1}$. This enhances our confidence to extend the calculated microwave transitions which have not been recorded experimentally such as $j$=2$\leftarrow$1. The exact selection rules associated with infrared spectra are $\Delta{J}$=0$\pm$1, and $\Delta{P}$=${\rm e}$$\leftrightarrow$${\rm f}$. In addition, to the extent that $j$ and $l$ are conserved quantities, the approximate selection rules of $\Delta{j}$=$\pm$1 and $\Delta{l}$=0 are also observed. The relative intensities at temperature 3 K are calculated, and expressed relative to $R(0)$ for $j$=1$\leftarrow$0, whose intensity is set to 1. A total of 27 strongest transition for $j$=1$\leftarrow$0 and $j$=2$\leftarrow$1 are kept and listed in Table Ⅳ, and 13 transitions for $j$=1$\leftarrow$0 are compared with the available experimental data [36, 37]. FIG. 7 shows all lines strengths for $j$=1$\leftarrow$0 and $j$=2$\leftarrow$1 subbands, and all the transition energies are shifted relative to $R$(0). Some line strengths are weaker, especially for $j$=2$\leftarrow$1, which are multiplied by 10 and labelled with * in FIG. 7.

Table Ⅳ Microwave transition frequencies (in cm−1) of He-HCN calculated from our vibrationally averaged 2D MLRQ PES and comparisons with experiment [36, 37].
 FIG. 7 Calculated microwave spectra of He-HCN complex involving the P-, Q-and R-with $j$=1$\leftarrow$0 and $j$=2$\leftarrow$1 subbands for $T$=3 K, and compared with the experimental values [36]. The * means the line strength values are multiplied by 10.

For $v_1$ transitions, the predicted infrared (IR) transition frequencies (cm$^{-1}$) from our 3D-VMRLQ potential are listed in Table Ⅴ. For infrared transitions, we calculate the IR rovibrational transitions from:

 $\begin{eqnarray} v_{1} ~=~ v_{1}({\rm HCN}) ~+~ E_{v_{1}=1}^{\rm {upper}} ~-~ E_{v_{1}=0}^{\rm {lower}} \end{eqnarray}$ (20)
Table Ⅴ Predicted rotational transition frequencies (in cm−1), line strengths and relative intensities for He-HCN(v1=0←1) from our vibrationally averaged 2D-VMLRQ PESs.

where, $v_1$(HCN) [73]$=$3311.396 cm$^{-1}$ is the experimental fundamental C$-$H stretch vibrational transition frequencies of a free HCN molecule.

As shown in Table Ⅴ, the infrared $v_1$=1$\leftarrow$0 transitions of HCN are $\Delta j$=$-$1 and $\Delta j$=1, respectively. To select most possible IR transitions of He-HCN, the relative intensities at temperature 6.0 K are calculated, and expressed relative to the strongest transitions of $\Delta j$=1 for 101$\leftarrow$000 (labeled with ($j$, $l$, $J$)), whose intensities are set to 1. The strengths and intensities for the possible transitions including P-, Q-, and R-branches with $J$ values up to 5 for $\Delta j$=$-$1 and $\Delta j$=1 are shown in FIG. 8 and FIG. 9, respectively. As seen in Table Ⅶ, the transitions for $\Delta j$=$-$1 and $\Delta j$=1 have strong line strengths and intensities at 6 K, which may be observed transitions in future experimental studies.

 FIG. 8 Calculated infrared spectra (labeled with free rotor quantum number $j'l'J'$-$j''l''J''$) of He-HCN involving the P-, Q-and R-branches with $\Delta j$=1 subband, shown with line strength (upward pointing lines) and intensity at 6 K (downward pointing lines), respectively.
 FIG. 9 Calculated infrared spectra (labeled with free rotor quantum number $j'l'J'$-$j''l''J''$) of He-HCN involving the P-, Q-and R-branches with $\Delta j$=$-$1 subband, shown with line strength (upward pointing lines) and intensity at 6 K (downward pointing lines), respectively.
Ⅳ. DISCUSSION AND CONCLUSIONS

We present accurate analytic vibrationally averaged 2D potential energy surfaces for He-HCN complexes for $v_1$=0 and 1 which were obtained from three-dimensional ab initio potential energies that explicitly incorporates the dependence of C$-$H stretching coordinate. The ab initio interaction energies were obtained at the CCSD(T) level using a complete basis set extrapolated from aug-cc-pVQZ and aug-cc-pV5Z basis sets, and with bond functions placed at the mid-point on the intermolecular axis. The vibrationally averaged potential energies were fitted to a 2D generalization of the Morse/Long-Range (MLR) potential form [62, 74]. With the high excitation correction term of $\Delta V_{\rm int}^{\rm T(Q)}$, the fitted 2D potential energy surfaces are denoted as VMLRQ. The global 2D fit to the 342 interaction energies had a root-mean-square residual of only 0.011 cm$^{-1}$ for $v$=0 of VMLRQ potentials, and only required 49 fitting parameters.

Rovibrational energy levels for He-HCN were obtained by the DVR method. The RMSD for bound and metastable states are 0.072 and 0.070 cm$^{-1}$ with CCSD(T)/CBS+$\Delta V^{\rm T(Q)}$, respectively, which are more than five times better than those obtained from AVQZ, AV5Z and CBS basis sets. The dissociation limit ($D_0$) for CCSD(T)/CBS+$\Delta V^{\rm T(Q)}$ is 9.403 cm$^{-1}$ which is in the best agreement with the observed value 9.316 cm$^{-1}$ [42]. The calculated microwave transition frequencies are based on CCSD(T)/CBS+$\Delta V^{\rm T(Q)}$ level, and the transitions relative to the corresponding $j$ stacks involving P-, Q-and R-branches are in agreement very well with the corresponding experimental values and the RMSD is only 0.072 cm$^{-1}$. Finally, we first presented the infrared spectra of He-HCN on the vibrationally quantum state-specific PESs.

Ⅴ. ACKNOWLEDGMENTS

We are grateful to Professor K. Harada and Professor K. Tanaka at Kyushu University for their helpful discussions. This work was supported by the National Key Research and Development Program (No.2016YFB0700801 and No.2017YFB0203401) and the National Natural Science Foundation of China (No.21533003, No.21773081 and No.91541124).

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1. 吉林大学理论化学研究所, 长春 130023;
2. 江西农业大学理学院功能材料和农业应用化学研究所, 南昌 330045