The article information
 Dan Hou, XiaoLong Zhang, Yu Zhai, Hui Li
 侯丹, 张晓龙, 翟羽, 李辉
 The Role of High Excitations in Constructing Subspectroscopic Accuracy Intermolecular Potential of HeHCN: Critically Examined by the HighResolution Spectra with Resonance States^{†}
 高次激发相关能在构建HCNHe亚光谱精度势能面中的角色:通过实验高分辨率光谱进行严格检测
 Chinese Journal of Chemical Physics, 2017, 30(6): 776788
 化学物理学报, 2017, 30(6): 776788
 http://dx.doi.org/10.1063/16740068/30/cjcp1712231

Article history
 Received on: December 3, 2017
 Accepted on: December 27, 2017
2. Institute of Functional Materials and Agricultural Applied Chemistry, College of Science, Jiangxi Agricultural University, Nanchang 330045, China
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 [27]. 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].
Highresolution 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 [1326]. Among the studied chromophorHe
The HeHCN 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 groundstate
In our previous work [20], we reported a 5D ab initio PES of COH
The geometry of a HeHCN complex in which HCN is kept linear can be described naturally using Jacobi coordinates
In a full fourdimensional (4D) treatment, which takes into account the coordinate
$ \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
$ \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
Hence, both the effective 1D potentials
The effective 1D potentials
The intermolecular potential energies of HeHCN were calculated at using singleand doubleexcitation coupledcluster theory with a noniterative perturbation treatment of triple excitations (CCSD(T)) [43]. The basis set used was the augmented correlationconsistent polarized nzeta basis set of Woon and Dunning (denoted as augccpVNZ or AVNZ for
The total intermolecular interaction potential
$ \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 HartreeFock part
FIG. 2 shows the comparisons of the onedimensional cuts of the interaction potentials for selected angular orientations of the HeHCN complex. These interaction potentials were calculated at CCSD(T)/AVXZ(X=Q, 5), CCSD(T)/CBS(complete basis set) and CCSD(T)/CBS+
FIG. 3 presents the intermolecular potentials at four stationary points including the global minimum, the local minimum, saddle point and Tshape configurations. The energies calculated at CCSD(T)/AV5Z, CCSD(T)/CBS and CCSD(T)/CBS+
A total of 1710 ab initio points were calculated for both the ground (
Since the
$ \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
$ \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 (
$ \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
The vibrationally averaged ab initio intermolecular potential energies
$ \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),
$ \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 nonpolar, the appropriate functional form for
$ \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
$ \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
$ \begin{eqnarray} \beta(R, \theta)&=& y_p^{{\rm ref}}(R, \theta) ~\beta_\infty(\theta) + \left[1y_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,
Note that the definition of
$ \begin{eqnarray} A(\theta) = \sum\limits_{\lambda=0} A^{\lambda} P_{\lambda}(\cos\theta) \label{eq:Apoly} \end{eqnarray} $  (14) 
where
Following our previous work [44], the leading vibrationally averaged van der Waals coefficients
$ \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\}})=\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
$ \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
Finally, the vibrationally averaged potentials are fit to 2D Morse/LongRange (2DMLR) form Eq.(8). In the fits at this time, all the longrange coefficients are fixed, and the input vibrationally averaged ab initio, energies were weighted by assigning them uncertainties of
FIG. 4 displays contour plot of the vibrationally averaged 2D MLRQ potential energy surface (CCSD(T)/ CBS+
Table Ⅱ summarizes the geometries and energies of these global minima along with barriers of VMLRQ PES for HeHCN 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
For the vibrationallyaveraged excitedstate (
Within the BornOppenheimer approximation, the twodimensional intermolecular Hamiltonian of the HeHCN complex in the Jacobi coordinate system
$ \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_xi\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
A directproduct discrete variable representation (DVR) grid was used in the rovibrational level energy calculation for the HeHCN complex [71]. A 100point sincDVR grid from 2.2 Å to 15.0 Å was used for the radial
PES is listed in Table Ⅲ for HeHCN(
For PES of CBS+
Microwave transitions calculated from our vibrationally averaged ground state VMLRQ(
For
$ \begin{eqnarray} v_{1} ~=~ v_{1}({\rm HCN}) ~+~ E_{v_{1}=1}^{\rm {upper}} ~~ E_{v_{1}=0}^{\rm {lower}} \end{eqnarray} $  (20) 
where,
As shown in Table Ⅴ, the infrared
We present accurate analytic vibrationally averaged 2D potential energy surfaces for HeHCN complexes for
Rovibrational energy levels for HeHCN were obtained by the DVR method. The RMSD for bound and metastable states are 0.072 and 0.070 cm
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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