Chinese Journal of Chemical Physics  2020, Vol. 33 Issue (1): 13-22

#### The article information

Kaito Takahashi
Takahashi Kaito
Theoretical Study on Direction of Vibrational Transition Dipole Moment of XH Stretching Vibration in HXD
HXD中XH拉伸振动的振动跃迁偶极矩方向的理论研究
Chinese Journal of Chemical Physics, 2020, 33(1): 13-22

http://dx.doi.org/10.1063/1674-0068/cjcp1910191

### Article history

Accepted on: December 10, 2019
Theoretical Study on Direction of Vibrational Transition Dipole Moment of XH Stretching Vibration in HXD
Kaito Takahashi
Dated: Received on October 11, 2019; Accepted on December 10, 2019
Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617
Abstract: Experimental vibrational spectra of heavy light XH stretching vibrations of simple molecules have been analyzed using the local mode model. In addition, the bond dipole approach, which assumes that the transition dipole moment (TDM) of the XH stretching mode is aligned along the XH bond, has helped analyze experimental spectra. We performed theoretical calculations of the XH stretching vibrations of HOD, HND$^-$, HCD, HSD, HPD$^-$, and HSiD using local mode model and multi-dimensional normal modes. We found that consistent with previous notions, a localized 1D picture to treat the XH stretching vibration is valid even for analyzing the TDM tilt angle. In addition, while the TDM of the OH stretching fundamental transition tilted away from the OH bond in the direction away from the OD bond, that for the XH stretching fundamental of HSD, HND$^-$, HPD$^-$, HCD, and HSiD tilted away from the OH bond but toward the OD bond. This shows that bond dipole approximation may not be a good approximation for the present systems and that the heavy atom X can affect the transition dipole moment direction. The variation of the dipole moment was analyzed using the atoms-in-molecule method.
Key words: Heavy light stretching vibration    Transition dipole moment    Vibrational mode coupling    Quantum chemistry
Ⅰ. INTRODUCTION

For the vibrational spectra of heavy light XH (X=O, N, C, $\cdots$) bonds in molecules, the local mode (LM) model has shown great success in calculating the peak positions and absorption intensities [1-6]. Along with this LM model, experimental absorption intensities are sometimes analyzed using an effective dipole moment function in one direction. In the approaches pioneered by Mecke and coworkers, the transition dipole moment (TDM) of the XH stretching vibration is assumed to be along the XH bond [7, 8]. Such a bond dipole concept has been used to analyze the sum frequency spectra of the free OH stretching bonds at air-water interface [9, 10]. However, as mentioned by Kjaergaard et al. and Amrein et al., these bond dipole functions are effective mathematical tools to analyze the experimental results, and should not be considered as physical dipole moments along a certain direction of the molecule [5, 6, 11]. In addition, for the OH bond, experimental studies for gas phase HOD, HCOOH, C$_6$H$_5$OH, and HNO$_3$ have shown that the OH stretching TDM is tilted from the OH bond. Fair et al. [12] have analyzed the ratio between the a and b band transition intensities of the OH stretching transition of HOD and have shown that the TDM is tilted by 19$^{\circ}$, 28$^{\circ}$, and 38$^{\circ}$ from the OH bond for the fundamental ($\Delta \nu$=1), second overtone ($\Delta \nu$=3), and third overtone ($\Delta \nu$=4). Bauer and Badger found that the TDM of the OH stretching second overtone ($\Delta \nu$=3) has a complex contribution of a and b bands for organic acids [13]. For formic acid, Hurtmans et al. analyzed the tilting of the TDM by using the dipole moment obtained by optimizing the molecular geometry as a function of OH bond length [14]. They saw a large correlation between the COH bond angle and the OH bond length to conclude the importance of the coupling to the bending motion for the tilting of the TDM. Ishiuchi et al. showed that the direction of the transition moment varied as a function of the excitation quantum number for the fundamental and overtone OH and OD stretching vibration in phenol [15]. For HONO$_2$, Konen et al. obtained a TDM with the titling of 45$^{\circ}$ from the OH bond from the high-resolution spectra for the first OH stretching overtone ($\Delta \nu$=2) band [16]. Takahashi et al. have used the LM model to calculate the OH stretching TDM direction for H$_2$O as well as simple acids and alcohols, and confirmed the tilting of the TDM from the OH bond [17, 18]. Furthermore, in their calculation, the absolute value of the tilting angle increased as the transition vibrational quantum number ($\Delta \nu$) increased. On the other hand, for ice, Whalley and Klug have shown that the fundamental OH stretching TDM is aligned along the OH bond [19]. In a previous study, we hypothesized that the electronic lone pair of the oxygen atom is important for the tilting of the TDM in the HOD molecule and analyzed the effect of binding molecules and ions [20]. We studied the OH stretching mode of HOD$\cdots$X, X=H$_2$O, CH$_3$OH, F$^-$, Cl$^-$, and Br$^-$, and showed that stronger the binding the smaller the tilting of the TDM from the OH bond.

In this study, we examined the effect of the vibrational mode coupling of the XH stretching mode to other vibrational modes, and evaluated how it may affect the direction of the XH stretching TDM. For this, we have performed vibrational calculations for the HOD molecule using different degrees of freedom. This follows the previous analysis performed by Fair et al. on HOD [12]. Furthermore, to understand the sensitivity of the TDM tilt angle toward the central heavy atom, we also studied HSD, HND$^-$, HPD$^-$, HCD, and HSiD. From a systematic evaluation, we show that the TDM tilt angle is very sensitive toward slight variations in the dipole moment function. Quantum chemistry methods are used to understand the physical origin of the change in the dipole moment.

Ⅱ. METHOD

For the LM1D model vibrational calculation for HXD, we first obtained the one dimensional (1D) potential energy surface (PES) and dipole moment function (DMF) as a function of XH bond length using quantum chemistry methods. The DMF mentioned here is a vector, thus can have contributions from three directions, however for the HXD triatomic molecules studied here, we only have two directions which we will label as parallel $\parallel$ and perpendicular $\perp$ to XH. We performed single point calculations for every 0.1 Å between ($r_{\textrm{XH, eq}}$-0.5) Å and ($r_{\textrm{XH, eq}}$+0.8) Å, and two additional points at ($r_{\textrm{XH, eq}}$$\pm0.05) Å. The remaining degrees of freedom were kept at the optimized equilibrium geometry. We solved the 1D vibrational Schrodinger equation in atomic units  \begin{eqnarray} \left[-\frac{1}{2\mu_{\textrm{XH}} } \frac{ d^2}{{d}r_{\textrm{XH}}{^2}} +V(r_{\textrm{XH}} )\right] \psi_i (r_{\textrm{XH}} )=E_i \psi_i (r_{\textrm{XH}}) \end{eqnarray} (1) where, r_{\textrm{XH}} and \mu_{\textrm{XH}} are the bond distance and the reduced mass of the XH bond, respectively. Here, similar to previous calculations, we have used the mass of X and H to calculate the reduced mass and ignored the effect of D [17, 21]. The TDM is given as the expectation value of the transition to the ith vibrational state from the zero-point vibration state  \begin{eqnarray} \vec{\mu}_{0i}&=&\vec{e}_{\parallel} \int_0^\infty \psi_0^* (r_{\textrm{XH}}) \mu_\parallel (r_{\textrm{XH}} ) \psi_i (r_{\textrm{XH}} ) \textrm{d}r_{\textrm{XH}}+\nonumber\\ &&\vec{e} _\bot \int_0^\infty \psi_0^* (r_{\textrm{XH}}) \mu_\bot (r_{\textrm{XH}}) \psi_i (r_{\textrm{XH}} )\textrm{d}r_{\textrm{XH}} \end{eqnarray} (2) where \vec{e}_{\parallel, \bot} and \mu_{\parallel, \bot}(r_{\textrm{XH}}) are the unit vector and dipole moment function in the direction \parallel and \bot to the XH bond, respectively. To solve the vibrational Schrodinger equation, we used the grid variational method based on fifth-order finite difference approximations, which was used in our previous study for the LM XH stretching vibration of dichloroethylene as well as simple acids and alcohols [17, 18, 21]. The grid spacing was 0.005 Å, between (r_{\textrm{OH, eq}}-0.5) Å and (r_{\textrm{OH, eq}}+0.8) Å, thereby giving 261 grid points. We have confirmed that this amount of grid points will give converged results for the peak position and integrated absorption coefficient [17, 18, 21]. To compare the effect of vibrational degrees of freedom, we solved the vibrational Hamiltonian for the the normal modes (NMs) {q} given below:  \begin{eqnarray} H= \sum\limits_{i=1}^n\left[-\frac{1}{2m_i } \frac{\partial^2}{\partial q_i{^2}} +V(q_1, q_2, \cdots, q_n)\right] \end{eqnarray} (3) where m_i is the reduced mass of each mode q_i. Here, we have ignored the mass dependent Watson term as well as the vibrational-rotational coupling terms [22]. We obtained the Cartesian representation of the NM from the Gaussian 09 [23] output using the freq=hpmodes keyword. For the NM vibrational calculation, we used the discrete variable representation (DVR) [24] of the harmonic oscillator basis set reported in our previous studies [25, 26]. For the present systems, we can use up to three NM degrees of freedom, and depending on the number of freedom used in the calculation, n, we will label the calculation as NMnD. We note that the DVR method can obtain the same results as the grid variational method used for the LM1D using smaller grid points (basis sets); therefore, it is suitable for performing multidimensional calculations. The integrated absorption coefficient base e in km/mol, from state 0 to state i, was calculated by [27],  \begin{eqnarray} A= \frac{N_\textrm{A} \pi}{3\varepsilon_0 \hbar c} |\langle\psi_0|{ \hat{\vec{\mu}}} |\psi_i \rangle|^2{\tilde{\nu}}_{0i}=2.506| \vec{\mu}_{0i}|^2\tilde{\nu}_{0i} \end{eqnarray} (4) where the constant 2.506 is in km\cdotmol^{-1}$$\cdot$cm$\cdot$D$^{-2}$, $\hat{{\vec{\mu}}}$ is the DMF vector operator, $\tilde{\mathcal{\nu}}_{0 i}$ is the transition energy in cm$^{-1}$ and $|{{\vec{\mu}}}_{0i}|^2$ is the square of the absolute value of the TDM in Debye (D) squared units.

For the calculation of the PES and DMF, we used the density functional theory method based on Becke's three-parameter hybrid functional B3LYP [28, 29] with Dunning's augmented correlation consistent polarization valance triple zeta, aug-cc-pVTZ (aVTZ) [30-32] basis set. In our previous work, we have shown that for HOD, this method gives TDM directions that are consistent with experimental results [20]. Furthermore, it was shown that the basis set dependence is small, and a triple zeta basis is large enough to describe the TDM. Furthermore, the B3LYP results and the coupled cluster singles and doubles with perturbative triples (CCSD(T)) [33-35] method with the aVTZ basis set gave very similar results. Therefore, we performed most of our preliminary calculations using B3LYP/aVTZ. As mentioned later, we noticed that slight changes in the DMF could give large variations in the TDM angles; thus, we also performed NM1D calculation using CCSD(T). In FIG. 1, we show a schematic diagram and geometries for the molecules we calculated in the present study. Consistent with previous studies, HCD had a triplet ground electronic state, while all remaining molecules had a singlet ground state. For the HND$^-$ and HPD$^-$, we have set the center of mass as the origin of the dipole moment. All the B3LYP and CCSD(T) calculations were performed using the Gaussian 09 program. We note that in the Gaussian 09 calculation, the DMF is actually obtained at the CCSD level of calculation, while the PES is at the CCSD(T) level, we will use the label "CCSD(T)" for these calculations. We also list the available experimental values obtained from the Computational Chemistry Comparison and Benchmark DataBase (CCCBDB) [36]. Before ending, we note that in the default output of Gaussian 09, the dipole moment values are given up to 10$^{-4}$ in Debye units, while we can also find dipole moment values given up to 10$^{-6}$ in atomic units. We have found that the use of the dipole moment with less digits results in slight errors: at most 0.5$^{\circ}$ in tilt angle for the system studied here. See supplementary materials for further details.

 FIG. 1 Schematic plot of the molecules calculated in the present study, as well as their equilibrium geometries obtained from B3LYP/aug-cc-pVTZ: (a) HOD, (b) HND$^-$, (c) HCD, (d) HSD, (e) HPD$^-$, and (f) HSiD. Available experimental results obtained from CCCBDB [36] are given in parenthesis. Computational results by Huang and Lee [37] are given in square brackets

In our previous work, we have shown that for the fundamental transition, the ratio between the symmetric and antisymmetric transitions of H$_2$X can be used to estimate the tilt angle for the fundamental TDM for the OH stretching vibration [20]. Therefore, we also used the intensity ratios obtained from the double harmonic approximation using B3LYP/aVTZ to obtain the fundamental tilt angle.

Lastly, as given in FIG. 1, the angle is defined positive if it tilts toward the XD bond while negative if it tilts away.

Ⅲ. RESULTS AND DISCUSSION

Before addressing our results concerning the vibrational calculation, it is important to quantify the accuracy of the calculated equilibrium geometries. As given in FIG. 1, the equilibrium bond lengths and bond angles are in good agreement with the experimental results for H$_2$O, H$_2$S, H$_2$C, and H$_2$Si obtained from the CCCBDB [36]. As for H$_2$N$^-$, the present values obtained by B3LYP are consistent with those obtained from CCSD(T)/aug-cc-pV5Z by Huang and Lee [37]. We were not able to find results for PH$_2$$^-$, but the results for the other HXD molecules give us confidence that the present method can give decent geometries.

In our previous work, we have evaluated the convergence for the LM1D calculations [17, 21], but have not evaluated it for the NM calculations. Therefore, first, we evaluate the grid points (basis sets) required for the multidimensional vibrational calculation based on NM. In Table Ⅰ, we present the tilt angle $\theta$, for HOD calculated using several different DVR grid points using the Hamiltonian for one NM, the mode that is mainly the XH stretching mode. As one can see, the values for the transitions up to $\Delta \nu$=4 are converged by using 15 grid points. Therefore, we performed the NM2D calculation by considering the NMs corresponding to the XH and XD stretching vibration along with the full NM3D calculation using 15 grid points for each NM. Thereby for these calculations, we performed 15$^n$ single point quantum chemistry calculations for each NM$n$D calculation to obtain the PES and DMF.

Table Ⅰ Vibrational quantum number dependence of the tilt angle obtained for the NM1D OH stretching vibration of HOD calculated using different grid points. Experimental results of Fair et al. [12] are also given, and the values in parenthesis are their uncertainties. We used B3LYP/aVTZ to calculate the PES and DMF

In Table Ⅱ, we summarize the peak position, absorption intensity, and the tilt angle calculated by NM1D using B3LYP/aVTZ. The peak position, TDM, integrated absorption coefficients, and tilt angle values for the LM, as well as the other NM$n$D calculations, are all given in supplementary materials. As can be seen with the comparison with the available experimental results for HOD by Fair et al. [12], the peak positions are off by 70 cm$^{-1}$ for higher overtones. As for the tilt angle, which we are mainly interested in, the values are not perfect but show the general trend of increasing in absolute value as the excitation quanta increases. Concerning the XH stretching integrated absorption coefficients of HXD molecules, we are not aware of absolute values, thus will compare with experimental values of H$_2$X by adding the symmetric and antisymmetric contributions then dividing by 2. For H$_2$O, the sum of the fundamental symmetric stretching (2.99) and antisymmetric stretching (43.3) is 46.29 km/mol; thus, we would expect a value of $\sim$23 km/mol. Our calculated value for the OH stretching fundamental of HOD is 34.3 km/mol, which overestimates this value. For the first overtone, the reported experimental values are 0.37 and 3.39 km/mol for the symmetric and antisymmetric modes, respectively. By dividing the sum by 2, we obtain $\sim$1.9 km/mol, which is fairly consistent with the 1.72 km/mol given in Table Ⅱ. Thereby, we believe the general trend in the intensities is obtained by using B3LYP/aVTZ.

Table Ⅱ Vibrational quantum number dependence of the peak position (in cm$^{-1}$), integrated absorption coefficient (in km/mol), and tilt angle (in $^{\circ}$), obtained for the NM1D XH stretching vibration of HXD calculated using B3LYP/aVTZ. We also list the experimental peak positions and tilt angles obtained by Fair et al. [12] for HOD in parenthesis

Now comparing the values obtained for different heavy atoms, one can notice a decrease in vibrational frequency when going down the periodic table: HOD to HSD, HND$^-$ to HPD$^-$, and HCD to HSiD. As for the integrated absorption coefficients, other than the case of HCD, the values decrease by orders of magnitude following the increase in the excitation quanta. This is consistent with the trends that have been seen previously for XH stretching vibrations [1, 5, 8, 17, 18]. When one looks at the tilt angles, one can see that unlike the case of OH stretching vibration in HOD, which shows an increase in the absolute value of the tilt angle as a function of excitation quantum number, tilt angles jump from positive to negative and do not follow a simple trend. Even looking at the fundamental transition, we see that for HOD, the tilt angle is negative, away from the XD bond, while for the others, it is positive toward the XD bond. As will be shown later, this is because the sign of the DMF gradient at equilibrium geometry is opposite for $\mu_{\parallel}$ versus $\mu_\perp$ for HOD, but is the same for all other HXD molecules along $Q_1$, the NM corresponding to the XH stretching mode.

In FIG. 2, we compare the tilt angles obtained by all vibrational methods for all the systems studied in the present work (see supplementary materials for a full list of data). As one can see, from the similarity with the linear black line, NM1D gives tilt angles consistent with NM2D as well as NM3D. We note that as given in supplementary materials, all peak positions obtained by the LM and NM$n$D calculations are all within 1% of each other. Furthermore, the integrated absorption coefficients of the NM1D can give values within 10% error of the NM3D results. This signifies that the localized 1D picture for describing the XH stretching vibration in HXD is valid. A similar conclusion was obtained by Fair et al. for HOD previously [12]. We note that for the fundamental CH stretching transition for HCD the NM3D gave very different results from LM1D, NM1D as well as NM2D. This is because the bending mode of HCD has a fundamental frequency of $\sim$800 cm$^{-1}$, while the CD stretching mode has a fundamental frequency of $\sim$2300 cm$^{-1}$. Therefore, there is a strong mixing of the CD stretching and HCD bending combination mode with the CH stretching mode which has a fundamental frequency of 3100 cm$^{-1}$. This strong coupling causes the TDM of the NM3D results to be very different from the NM1D and NM2D which assumes that the state is mainly a CH stretching mode. All in all, the general trend of the tilt angle of the NM3D results can be obtained from the NM1D results; thus, we will compare the NM1D results with the LM1D calculations, and also analyze the results using NM1D. It is interesting to note that the estimation using the double harmonic results for the fundamental vibration of H$_2$X (pink triangles) is also able to give values very close to the NM3D results. This shows that the double harmonic approximation can give good estimation in determining the direction of the TDM.

 FIG. 2 The correlation between the TDM tilt angle calculated by different vibrational calculation methods. All tilt angles were calculated using the PES and DMF calculated by B3LYP/aVTZ. Results for NM2D (red cross), NM1D (blue circle), LM1D (black square), and from the ratio of the intensity of symmetric and antisymmetric vibration from the harmonic approximation for H$_2$X (pink triangle) are given. The linear black line is drawn to guide the eyes

For HOD, the tilt angles obtained from LM1D and NM1D methods only differ by a few degrees at the maximum; however, for the other HXD species, differences of several tens of degrees were seen. To investigate the origin of this difference, we plot the DMF calculated along the LM1D ($r_{\textrm{XH}}$) and NM1D (along mode $Q_1$, which mainly consists of XH stretching motion) in FIG. 3. One can see that the difference is very small. Both pictures give similar trends for the $\parallel$ direction, but for the $\perp$ direction HPD$^-$, HCD, and HSiD show obvious differences. As given in Table Ⅲ, the NM1D is mainly a XH stretching motion but does include some bending and XD contraction contributions. This slight contribution of the other degrees of freedom can give enough changes to the $\perp$ direction DMF which results in a variation in the tilt angle. This sensitivity shows that the tilt angle can help analyze slight changes in the DMF, which is related to the variation of the electron density following the change in the XH bond length. It is interesting to notice that for HOD, the $\parallel$ direction has a positive gradient at equilibrium, while for all other molecules, it has a negative gradient. Furthermore, if one looks at the $\perp$ direction, HOD has a negative gradient, similar to other HXD molecules.

 FIG. 3 Dipole moment functions as a function of $\Delta r_{\textrm{XH}}$ and the XH stretching NM $\Delta Q_1$ for (a) HOD, (b) HND$^-$, (c) HCD, (d) HSD, (e) HPD$^-$, and (f) HSiD. The results for the bond length are given in solid and dashed lines, while the results for NM are given in symbols. The parallel contribution is given in red, while the perpendicular is given in blue. The symbol $\Delta$ signifies that the equilibrium value is set to zero
Table Ⅲ Internal coordinate representation of $Q_1$ normal mode

To gain more understanding of the trends of the DMF, we use Bader's atom-in-molecule (AIM) method [38] using the idea of Bruns and coworkers [39, 40]. We use capital letters to define nuclei position ($\vec{R}$) and small letters to define electron position ($\vec{r}$). In this analysis, we divide the dipole moment to come from two contributions. First the atomic charge (AC) term

 $\begin{eqnarray} \vec{\mu} ^{\textrm{AC}} (\vec{R})&=&\sum\limits_\alpha^{N_{\textrm{atom}}}\left[Q_\alpha-\int_{V_\alpha}\rho(\vec{r} _\alpha; \vec{R} )\textrm{d}\vec{r}_\alpha \right] \vec{R} _\alpha \nonumber\\ &=&\sum\limits_\alpha^{N_{\textrm{atom}}}\vec{\mu}_\alpha^{\textrm{AC}} (\vec{R}) \end{eqnarray}$ (5)

where atom $\alpha$ has a volume and attractor position ($V_\alpha$, and $\vec{R}_\alpha$) determined by the AIM method, and $\rho(\vec{r}_\alpha$; $\vec{R}$) is the electron density at the electron position $\vec{r}_\alpha$ (a vector for electrons around atom $\alpha$, which has its origin at $\vec{R}_\alpha$). $Q_\alpha$ is the nuclear charge of atom $\alpha$. This $\vec{\mu}^{\textrm{AC}}$ gives the dipole moment contribution of the total charge around atom $\alpha$, centered at $\vec{R}_\alpha$. We note that this attractor position usually coincides with the nuclear position. Next is the atom dipole (AD) term

 $\begin{eqnarray} \vec{\mu}^{\textrm{AD}} (\vec{R})&=&-\sum\limits_\alpha^{N_{\textrm{atom}}}\int_{V_\alpha}\rho(\vec{r}_\alpha; \vec{R}) \vec{r}_\alpha\textrm{d}\vec{r}_\alpha \nonumber\\ &=&\sum\limits_\alpha^{N_atom}\vec{\mu}_\alpha^{\textrm{AD}}(\vec{R} ) \end{eqnarray}$ (6)

which gives the sift of the electron position expectation value from the attractor position $\vec{R}_\alpha$. So if an atom has a lone pair sticking out in one direction, this contribution will be seen in this AD term.

This allows us to evaluate which terms dominate the DMF, and as given in FIGs. 4, 5, and 6, for $\mu_{\parallel}$ the AD term dominates for HOD and HSiD, while the AC term dominates for HND$^-$ and for HSD, HPD$^-$ and HCD the two terms cancel out each other. As for $\mu_\perp$, we see a cancelation of the two terms for HSD, HPD$^-$ and HCD, while the AD term dominates for HOD, HND$^-$, and HSiD. Next, from the definition of a dipole moment, decreasing dipole moment in one direction means that electron distribution transfers to atoms in that direction (AC term), or shifting of the electron density around an atom in that direction (AD term), and vice versa. For HOD the DMF has a positive gradient for $\mu_{\parallel}$ and a negative gradient for $\mu_\perp$, and the DMF in both directions is dominated by the AD term of the oxygen atom. This means that the electron distribution around oxygen is decreasing toward the OH bond and increasing toward the OD bond, as the OH bond is elongated. On one hand, HND$^-$ has a negative gradient for $\mu_{\parallel}$ which is dominated by the AC term. This means that the electron charge is being transferred from N to H as the NH bond is elongated. On the other hand, the negative gradient for $\mu_\perp$, which is dominated by the AD term can be interpreted that the electron distribution around the nitrogen is being pulled toward the D atom when the NH bond is elongated. At equilibrium, the AD in the NH $\perp$ direction is positive, which is due to the lone pair electron of nitrogen pointing out in the negative direction. The pulling of the electron distribution toward the D atom direction symbolizes that the lone pair electron becomes closer to N as the NH bond is elongated. For HSD, HPD$^-$ and HCD, we see that for $\mu_{\parallel}$ the positive AC contribution is canceled by the negative AD contribution. This means that as the XH bond is elongated, the electron transfers from H to X, but the electron density distribution around X is pulled toward the H direction. We can interpret this as the shrinking of the electron density around X in the XH direction. Similar to HND$^-$ above, the $\mu_{\parallel}^{\textrm{AD}}$ for atom X in these three molecules is positive, once again signifying that the lone pair electron sticking out in the negative direction gets pulled toward X when XH is elongated. What is interesting is that for $\mu_\perp$, the contributions are similar to that of $\mu_{\parallel}$ for HSD and HPD$^-$, but are reversed for HCD. This can be summarized that for HSD and HPD$^-$, the electron density is shrinking toward S and P in both directions while for HCD a large shrinking is seen for the XH direction, but there is a small amount of electron density shifting toward XD as XH is elongated.

 FIG. 4 Dipole moment functions as a function of NM $\Delta Q_1$ for (a) HOD $\mu_{\parallel}$, (b) HOD $\mu_\perp$, (c) HSD $\mu_{\parallel}$, and (d) HSD $\mu_\perp$. The atomic charge contribution is given in blue solid lines, while the atomic dipole term is given in red dashed lines, and the total value is given in black lines with a cross. The symbol $\Delta$ signifies that the equilibrium value is set to zero
 FIG. 5 Dipole moment functions as a function of NM $\Delta Q_1$ for (a) HND$^-$ $\mu_{\parallel}$, (b) HND$^-$ $\mu_\perp$, (c) HPD$^-$ $\mu_{\parallel}$, and (d) HPD$^-$ $\mu_\perp$. The atomic charge contribution is given in blue solid lines, while the atomic dipole term is given in red dashed lines, and the total value is given in black lines with a cross. The symbol $\Delta$ signifies that the equilibrium value is set to zero
 FIG. 6 Dipole moment functions as a function of NM $\Delta Q_1$ for (a) HCD $\mu_{\parallel}$, (b) HCD $\mu_\perp$, (c) HSiD $\mu_{\parallel}$, and (d) HSiD $\mu_\perp$. The atomic charge contribution is given in blue solid lines, while the atomic dipole term is given in red dashed lines, and the total value is given in black lines with a cross. The symbol $\Delta$ signifies that the equilibrium value is set to zero

Considering the sensitivity of the tilt angle to the DMF, it is important to evaluate the quantum chemistry method dependence for systems other than HOD. Therefore, we used the NM coordinates calculated by B3LYP but used energy correction by CCSD(T) and CCSD dipole moments to obtain the PES and DMF for the NM1D vibrational calculation. Looking at the similarity in the DMF given in FIG. 4, we would expect small changes. As seen in Table Ⅳ, results for HOD and HSiD have a small variation of only $\sim$5$^{\circ}$. On the other hand, those for HSD and HCD have very large differences, even in the fundamental transition. As shown in FIG. 7, for HSD the general trend of $\mu_\bot$ is very different with two quantum chemistry methods. As for HCD, there is a 2nd order like $Q_1$ coordinate dependence for the DMF in both $\parallel$ and $\perp$ directions. We hope that experimental determination for the tilt angles of the vibrational TDM for these two molecules can help us evaluate which quantum chemistry method is correct.

Table Ⅳ Vibrational quantum number dependence of the tilt angle obtained for the NM XH stretching vibration of HXD calculated using CCSD(T)/aVTZ
 FIG. 7 Dipole moment functions as a function of the XH stretching NM $\Delta Q_1$ for (a) HOD, (b) HND$^-$, (c) HCD, (d) HSD, (e) HPD$^-$, and (f) HSiD calculated using two different quantum chemistry methods. The results for the CCSD(T) are given in lines, while the results for B3LYP are given in symbols. The parallel contribution is given in red, while the perpendicular contribution is given in blue. The symbol $\Delta$ signifies that the equilibrium value is set to zero
Ⅳ. CONCLUSION

We performed theoretical calculations of the XH stretching vibrations of HOD, HND$^-$, HCD, HSD, HPD$^-$, and HSiD using local mode and multi-dimensional normal modes. We paid extra effort to understand how the XH stretching vibrational transition dipole moment is affected by mode coupling. From the comparison between full dimensional NM3D and LM1D as well as NM1D calculations, we found that consistent with previous notions, a localized 1D picture to treat the XH stretching vibration is valid even for analyzing the TDM tilt angle. In addition, while the TDM of the OH stretching fundamental transition tilted away from the OH bond in the direction away from the OD bond, that for the XH stretching fundamental of HSD, HND$^-$, HPD$^-$, HCD, and HSiD tilted away from the OH bond but toward the OD bond. This shows that the bond dipole approximation is not a good approximation for the present systems, and also that trends seen for OH bond are different from other XH bonds. Using the atoms-in-molecule method we gave physical pictures based on electron density movement and its effect toward the linear term of the DMF along the XH stretching normal mode. Before ending, we note that in the present calculation, we ignored the effect of rotation. It was recently shown that even for linear molecules the rotational averaging could cause the dipole moment to vary from its linear geometry value [41, 42]. We hope to study how such rotational effect can affect the direction of the transition dipole moment for the XH stretching vibrational of HXD molecules in the future.

Supplementary materials: The peak position, transition dipole moment, integrated absorption coefficients, and tilt angle values for the local mode model, as well as the multidimensional normal mode calculations are provided.

Ⅴ. ACKNOWLEDGMENTS

Kaito Takahashi thanks support from Academia Sinica (AS-CDA-106-M05), Ministry of Science and Technology (MOST 108-2113-M-001-011-), and National Center for High Performance computing of Taiwan.

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HXD中XH拉伸振动的振动跃迁偶极矩方向的理论研究
Kaito Takahashi