$$2\hat{T}_{\text{main}}^{(j)}|_{ad} = \left[ \hat{\mathbf{J}}_{ad}^{(j)} \right] \begin{bmatrix} \frac{1}{\mu_2^{(j)} R_2^{(j)2}}|_{ad} & 0 & \frac{\cot \theta_1^{BF(j)}}{\mu_2^{(j)} R_2^{(j)2}}|_{ad} \\ 0 & \frac{1}{\mu_2^{(j)} R_2^{(j)2}}|_{ad} & 0 \\ \frac{\cot \theta_1^{BF(j)}}{\mu_2^{(j)} R_2^{(j)2}}|_{ad} & 0 & \frac{1}{\sin^2 \theta_1^{BF(j)} \mu_1^{(j)} R_1^{(j)2}}|_{ad} + \frac{\cot^2 \theta_1^{BF(j)}}{\mu_2^{(j)} R_2^{(j)2}}|_{ad} \end{bmatrix} \left[ \hat{\mathbf{J}}_{ad}^{(j)T} \right]. \quad (237)$$

The 'correction' reads (see Eq. (221) and Eqs. (227,228))

$$\begin{aligned} 2\hat{T}_{\text{corr}}^{(j)}|_{ad} &= - \left[ \hat{\mathbf{J}}_{o}^{(j)} \right] \sigma''^{(j)}|_{ad} \Sigma''^{(j)}|_{ad}^{-1} \sigma''^{(j)}|_{ad}^{T} \left[ \hat{\mathbf{J}}_{ad}^{(j)T} \right] \\ &= - \left[ \hat{\mathbf{J}}_{ad}^{(j)} \right] \begin{bmatrix} 0 & 0 & 0 \\ 0 & \frac{\mu_1^{(j)} R_1^{(j)2}}{\mu_2^{(j)} R_2^{(j)2}(\mu_1^{(j)} R_1^{(j)2} + \mu_2^{(j)} R_2^{(j)2})}|_{ad} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \left[ \hat{\mathbf{J}}_{ad}^{(j)T} \right]. \end{aligned} \quad (238)$$

The final adiabatically constrained operator reads

$$2\hat{T}|_{ad} = \frac{\hat{P}_{R_1}^{\dagger} \hat{P}_{R_1}}{\mu_1} + 2 \sum_{j=1}^{2} \hat{T}^{(j)}|_{ad} \\ + \frac{(\hat{\vec{J}}^{\dagger} - (\sum_{j=1}^{2} \hat{\vec{J}}^{(j)})^{\dagger}) \cdot (\hat{\vec{J}} - (\sum_{j=1}^{2} \hat{\vec{J}}^{(j)}))}{\mu_1 R_1^2}, \quad (239)$$

with the 'flexible' rotor

$$2\hat{T}^{(j)}|_{ad} = \left[ \hat{\mathbf{J}}_{ad}^{(j)} \right] \begin{bmatrix} \frac{1}{\mu_2^{(j)} R_2^{(j)2}}|_{ad} & 0 & \frac{\cot \theta_1^{BF(j)}}{\mu_2^{(j)} R_2^{(j)2}}|_{ad} \\ 0 & \frac{1}{\mu_1^{(j)} R_1^{(j)2} + \mu_2^{(j)} R_2^{(j)2}}|_{ad} & 0 \\ \frac{\cot \theta_1^{BF(j)}}{\mu_2^{(j)} R_2^{(j)2}}|_{ad} & 0 & \frac{1}{\sin^2 \theta_1^{BF(j)} \mu_1^{(j)} R_1^{(j)2}}|_{ad} + \frac{\cot^2 \theta_1^{BF(j)}}{\mu_2^{(j)} R_2^{(j)2}}|_{ad} \end{bmatrix} \left[ \hat{\mathbf{J}}_{ad}^{(j)T} \right]. \quad (240)$$

Let us point out that the functions $q''_i|_{ad}(\mathbf{q}')$, and hence the matrix elements of Eqs (237,238,240) are not known analytically but are numerically defined on a grid. Interestingly enough, in Ref. [192], the extra-potential term of Eq. (217) was neglected: this term is, in general, numerically very small (see Ref. [72, 194, 195], for instance). The numerical
results of Ref. [192] compared to experimental values have proved that the adiabatic model dramatically improves the rigid one for the water dimer.

## 2. The photo-induced non-adiabatic dynamics of ethene

In Ref. [196], Krawczyk et al. have computed the three lowest potential energy surfaces of ethene. Ethene has six atoms and its geometric configuration is, therefore, described by 12 internal coordinates. However, in order to make the investigation feasible, the authors considered only the six most important or 'active' coordinates. The determination of the active coordinates is achieved by screening many geometric parameters for the most important geometries (for instance, the stationary points of the potential energy surfaces). The geometric parameters and the corresponding coordinates that change the most must be retained as active coordinates, whereas the coordinates that remain approximately constant can be considered as inactive and frozen during the dynamics. In Ref. [196], the six active modes were torsion, C-C stretching, left and right scissoring and left and right wagging (pyramidalization). The left and right rockings and the four C-H stretches were ignored and their coordinates were fixed to their equilibrium positions, i.e. we have the case of a rigid constraint studied in Section IVA 1. The six active modes are parametrized by six coordinates, which are: $\phi$, the torsional angle; $r$, the C-C distance; $\alpha_r$, the right scissor angle; $\alpha_l$, the left scissor angle; $\vartheta_r$, the right pyramidalization angle; $\vartheta_l$, the left pyramidalization angle. The PES were transformed to a diabatic representation and dynamical calculations were performed in Ref. [197] using the multi-configuration time-dependent Hartree (MCTDH) method. Unfortunately the KEO, obtained by inversion of the 6 × 6 matrix: $\sum_{\alpha=1}^{18} \frac{\partial x_\alpha}{\partial q_i} \frac{\partial x_\alpha}{\partial q_j}$, where the $x_\alpha$ are mass-weighted Cartesian coordinates of the six atoms of the molecule, and $q_i \in \{\phi, r, \alpha_r, \vartheta_r, \vartheta_l\}$ (see Eqs. (2,5,6) of Section IIA), is unsuitable for the MCTDH propagation. Indeed, it cannot be factorised as a sum of products of one-dimensional operators. Therefore, several simplifications have been added to the KEO.

In Ref. [126], we have revisited the problem and proposed a set of five vectors, which are depicted in Fig. 22, and the corresponding 12 standard polyspherical coordinates: ($R_1, R_2$,
<figure><img src="image_23.png" /><figcaption>FIG. 22: Mixed Jacobi/valence description of the C₂H₄ system (see the text).</figcaption></figure>

R₃, R₄, R₅, θ₁BF, θ₂BF, θ₃BF, θ₄BF, ϕ₁BF, ϕ₂BF, ϕ₃BF). The corresponding matrix M reads

$$ M = \begin{bmatrix} \frac{1}{m_C} + \frac{1}{2m_H} & 0 & 0 & 0 & -\frac{1}{m_C} \\ 0 & \frac{1}{m_C} + \frac{1}{2m_H} & 0 & 0 & -\frac{1}{m_C} \\ 0 & 0 & \frac{2}{m_H} & 0 & 0 \\ 0 & 0 & 0 & \frac{2}{m_H} & 0 \\ -\frac{1}{m_C} & -\frac{1}{m_C} & 0 & 0 & \frac{2}{m_C} \end{bmatrix}. \qquad (241) $$

This particular choice of the vectors was motivated by the attempt to obtain coordinates that are as close as possible to the ones used in Ref. [197] (φ, r, αᵣ, αᵢ, ϑᵣ, ϑᵢ). Indeed, since the potential energy surface is given only in terms of six degrees of freedom, we subject the system to the following constraints:
$$
\begin{align}
\theta_3^{BF} &= \frac{\pi}{2}, \nonumber \\
\theta_4^{BF} &= \frac{\pi}{2}, \nonumber \\
\varphi_2^{BF} &= \frac{\pi}{2}, \nonumber \\
\varphi_1^{BF} &= \frac{3\pi}{2} + \varphi_3^{BF}, \nonumber \\
R_3 &= 2\sqrt{R_{CH}^2 - R_1^2}, \nonumber \\
R_4 &= 2\sqrt{R_{CH}^2 - R_2^2}, \tag{242}
\end{align}
$$

with the distance R<sub>CH</sub> being fixed to its equilibrium geometry. After a simple change of coordinates (for instance $\varphi_1^{BF'} = \varphi_1^{BF} - \varphi_3^{BF}$) that does not affect the 'main' term of the final KEO, the constraint for $\varphi_1^{BF'}$, R<sub>3</sub>, and R<sub>4</sub> can be expressed as a rigid one. The relation between our polyspherical coordinates and the six degrees of freedom used in Ref. [197] is then simply given by

$$
\begin{align*}
r &= R_5, \\
\vartheta_r &= \theta_2^{BF}, \\
\vartheta_l &= \theta_1^{BF}, \\
\alpha_r &= 2 \arccos(R_2/R_{CH}), \\
\alpha_l &= 2 \arccos(R_1/R_{CH}), \\
\varphi &= \varphi_3^{BF},
\end{align*}
$$

where $R_{CH}$ denotes the C-H equilibrium distance.

The fact that we use $R_1$ and $R_2$ (see Figure 22 for their definition) instead of the two scis-
sor angles, $\alpha_r$ and $\alpha_l$, between the C-H bonds allows us to reformulate the problem in terms
of polyspherical coordinates. The use of polyspherical coordinates allows a direct obtaining
of the main term of the KEO in a form perfectly adapted to MCTDH. For $C_2H_4$, when using
the elementary volume element $d\tau = dR_3 dR_4 dR_5 d\theta_1^{BF} d\theta_1^{BF} d\theta_2^{BF} d\varphi_3^{BF}$ and assuming $J =
0$, the main term, $1/2 [\hat{P}'|_o^\dagger] \Sigma'_o [\hat{P}'|_o^T]$, is given by Eq. (11) of Ref. [126] (we used
Eqs. (111,C4)). The 'correction', $-1/2 [\hat{P}'|_o^\dagger] \Omega|_o^T \Sigma''|_o^{-1} \Omega|_o [\hat{P}'|_o^T]$ for $J = 0$, cannot
be written as a sum of products of mono-mode operators. In Ref. [126], this problem was
removed by introducing an additional reasonable approximation. We fix $\Omega|_o^T \Sigma''|_o^{-1} \Omega|_o$ to
its value at the equilibrium geometry: it is not only the inactive coordinates but also the
active coordinates that are fixed in the correction. This (approximate) correction term was provided by TNUM [68] and reads:

$$ T_{corr'} = 0.0000019889 \frac{\partial^2}{\partial R_5^2} - 0.0000138299 \frac{\partial}{\partial R_2} \frac{\partial}{\partial R_5} - 0.0000138299 \frac{\partial}{\partial R_1} \frac{\partial}{\partial R_5} + 0.0000480826 \frac{\partial^2}{\partial \theta_2^{BF2}} + 0.0000480826 \frac{\partial^2}{\partial \theta_1^{BF2}}, \quad (243) $$

where atomic units are assumed.

An improvement, in the future, could be to make a Taylor expansion of $\Omega|_o^T \Sigma''|_o^{-1} \Omega|_o$ using the numerical values given by TNUM or to calculate this term analytically and to refit it, with Mathematica for instance, as a sum of products of mono-mode operators. In Ref. [126], it was shown that the effect of this correction on the dynamics is small. This study emphasizes that 'the general structure' of the polyspherical coordinates should be kept in mind when choosing the active coordinates. In other words, when the active coordinates are a subset of polyspherical coordinates, it is straightforward to obtain the part of the KEO that carries the essential part of the physics and in a form that can be very easily implemented.

### 3. General strategies for very large systems

(i) The example of the cationic complex $[CpRh(PH_3)H(C_2H_4)]^+$

In Section IVA, we have presented a rigorous way to derive the KEO of systems subjected to constraints, i.e. for which the dynamics is performed within a subset of active coordinates. We have explained, in Section IVB, why it is advantageous to combine the constraints with the polyspherical approach. Finally, we have given some illustrations in Section IVC. If we summarize our findings, we can say that:

(1) the constrained operator can be split into two parts: the 'main' term and the
'correction'.

(2) If the choice of the coordinates, active and inactive, is judicious, the 'main' term carries the dominant part of the physics and the correction is numerically smaller. However, this correction must be taken into account if one wants to reach high accuracy.

(3) It is advantageous to choose the active coordinates as a subset of polyspherical coordinates. Indeed, (i) the polyspherical coordinates straightforwardly provide the main term in a form that can be very easily implemented. (ii) If one chooses a judicious set of polyspherical coordinates, for instance a parametrization of Jacobi vectors combined with the separation into subsystems of Section IID, one strongly minimizes the coupling between the active and the inactive coordinates and, therefore, the effect of the 'correction' of the KEO on the dynamics.

The latter point is illustrated in the present Section: if one focuses on the dynamics within a small part of a very large molecular system, it is judicious to successively divide the whole molecule into smaller subsystems. In other words, one applies the 'separation into subsystems' of Section IID several times so that the subset of the active coordinates is almost decoupled from the rest of the molecule in the KEO. In Ref. [198], this strategy was adopted to calculate the KEO in order to undertake a first quantum dynamical study of migratory insertion and hydrogen elimination in a representative metal complex $[CpRh(PH_3)H(C_2H_4)]^+$. As for ethene, it was first necessary to identify a set of relevant coordinates. This is achieved by screening the relevant stationary points of the potential energy surface. The choice of the active coordinates therefore depends on the potential operator and not on the KEO, but it is always advantageous to translate them into the 'language of the polyspherical coordinates', i.e. to identify a subset of polyspherical coordinates as active coordinates.

The system is depicted in Figure 23 and separated into two subsystems '1' (HRhC₂H₄) and '2' (PH₃C₅H₅). $\overrightarrow{R12} = \overrightarrow{G_1G_2}$ is the Jacobi vector joining G₁ the center of mass of the subsystem '1' (HRhC₂H₄) to G₂ the center of mass of the subsystem '2' (PH₃C₅H₅).

For the whole system (HRhC₂H₄PH₃C₅H₅), the exact kinetic energy operator then reads
FIG. 23: Successive separation into subsystems of the metal complex $[CpRh(PH_3)H(C_2H_4)]^+$. G_2 is the center of mass of the subsystem (PH_3C_2H_5).

<figure><img src="image_24.png" /></figure>

$$2\hat{T} = \frac{\hat{P}_{R_{12}}^{\dagger}\hat{P}_{R_{12}}}{\mu_{12}} + 2\sum_{j=1}^{2}\hat{T}^{(j)} + \frac{(\hat{\vec{J}}^{\dagger} - (\sum_{j=1}^{2}\hat{\vec{J}}^{(j)})^{\dagger}) \cdot (\hat{\vec{J}} - (\sum_{j=1}^{2}\hat{\vec{J}}^{(j)}))}{\mu_{12}R_{12}^{2}}, \quad (244)$$

with

$$\mu_{12} = \frac{m_{PC_5H_8}m_{RhC_2H_5}}{m_{RhPC_7H_{13}}}. \quad (245)$$

The subsystem '1' can also be separated into two new subsystems: RhH and C_2H_4 that are linked by a Jacobi vector $\vec{R}$ joining G_{C2H4} the center of mass of C_2H_2 to G_{RhH} the center of mass of RhH, which can be approximated as G_{RhH} ≃ Rh (see Figure 23). The kinetic energy $\hat{T}^{(j)}$ reads

$$2\hat{T}^{(1)} = \frac{\hat{P}_{R}^{\dagger}\hat{P}_{R}}{\mu} + \frac{\hat{P}_{R_{H}}^{\dagger}\hat{P}_{R_{H}}}{\mu_{R_{H}}} + \frac{\hat{\vec{L}}^{\dagger}\hat{\vec{L}}}{\mu_{R_{H}}R_{R_{H}}^{2}} + 2\hat{T}_{C2H4} + \frac{(\hat{\vec{J}}^{\dagger} - (\hat{\vec{L}}^{\dagger} + \hat{\vec{J}}^{\dagger})) \cdot (\hat{\vec{J}} - (\hat{\vec{L}} + \hat{\vec{J}}^{\prime}))}{\mu R^{2}}. \quad (246)$$
$\vec{L}$ is the angular momentum associated with $\overrightarrow{RhH}$, $\vec{J}'$ is the total angular momentum of C₂H₄, the vector $RhH$ is described by three spherical coordinates: R<sub>H</sub>, θ, and φ. In addition, we have

$$ \mu_R = \frac{m_{RhH} m_{C_2H_4}}{m_{RhHC_2H_4}}. \qquad (247) $$

$\mu_{RH}$ is the mass of the H atom. The kinetic energy operator of $T_{C_2H_4}$ could be directly derived in terms of polyspherical coordinates from the general expression given by Eqs. (111,C4). If we keep only six internal polyspherical coordinates, as suggested in Ref. [198], R<sub>H</sub>, θ, φ, R (the length of $\overrightarrow{GG_{RhH}}$), $R_{CC}$ (the distance between the two C atoms in C₂H₂), and γ (the third Euler angle of the subsystem C₂H₂ that is depicted in Figure), the 'main' term of the final KEO for total J = 0 reads

$$ \begin{aligned} 2T = & -\frac{\hbar^2}{\mu_H} \frac{\partial^2}{\partial R_H^2} - \frac{\hbar^2}{\mu_R} \frac{\partial^2}{\partial R^2} - \frac{\hbar^2}{\mu_{CC}} \frac{\partial^2}{\partial R_{CC}^2} \\ & - \hbar^2 \left( \frac{1}{m_H R_H^2} + \frac{1}{\mu_R R^2} \right) \left( \frac{1}{\sin\theta} \frac{\partial}{\partial\theta} \sin\theta \frac{\partial}{\partial\theta} + \frac{1}{\sin\theta^2} \frac{\partial^2}{\partial\varphi^2} \right) \\ & + \frac{(2p_\gamma - i\hbar \cot\gamma) i\hbar \left( -\cos\varphi \frac{\partial}{\partial\theta} + \sin\varphi \cot\theta \frac{\partial}{\partial\varphi} \right)}{\mu_R R^2} \\ & - \hbar^2 \left( \frac{1}{\mu_{CC} R_{CC}^2} + \frac{1}{\mu_R R^2} \right) \left( \frac{1}{\sin\gamma} \frac{\partial}{\partial\gamma} \sin\gamma \frac{\partial}{\partial\gamma} \right) \\ & + \frac{p_\varphi^2}{\sin^2\gamma} \left( \frac{1}{\mu_{RCC} R_{CC}^2} + \frac{1}{\mu_R R^2} \right) - 2 \frac{p_\varphi^2}{\mu_R R^2} \\ & - \frac{\cot\gamma p_\varphi (\cos\varphi \cot\theta p_\varphi + \sin\varphi p_\theta) + (\cos\varphi \cot\theta p_\varphi + \sin\varphi p_\theta) \cot\gamma p_\varphi}{\mu_R R^2}, \end{aligned} \qquad (248) $$

with the following volume element:

$$ d\tau = dR_H dR dR_{CC} d\gamma d\varphi \sin\theta d\theta. \qquad (249) $$

In Ref. [198], the 'correction' and the extra-potential term were neglected. The successive separation into subsystems and the use of Jacobi vectors joining the different subsystems ($\vec{R}_{12}$, $\vec{R}$) allows the active coordinates to strongly decouple the active coordinates from the rest of the system in the KEO. The use of the polyspherical coordinates yields a final expression of the 'main' term that is not too complicated.
(ii) Large van der Waals clusters

In this last Section, we sketch how it is possible to combine the polyspherical coordinates with other families of coordinates when the system is separated into subsystems. Under particular conditions, it is indeed possible to directly make use of KEOs published in the literature within the polyspherical approach: the KEO in hyperspherical coordinates for a tri-atomic system and the Watson KEO in normal coordinates. In this Section, the separation into active and inactive coordinates does not play a major role. However, since we use almost all the results presented before (the properties of the angular momenta highlighted in Section IIB 2, the separation into subsystems, the rigid and flexible rotors, etc...), it is sensible that this Section appears at the end of the review.

(1) Introduction of hyperspherical coordinates in NH₃:

Before carrying on, it is important to be made clear that the polyspherical approach possesses properties that can be (and have been) exploited for other kinds of coordinates than polyspherical coordinates. One example is the calculation of the vibrational levels of NH₃ reported in Ref. [77]. For ammonia, the system had been separated into two subsystems: the N atom and the H₃ subsystem. If the system is parametrized by the set of three Jacobi vectors as shown in Figure 24, in view of Eqs. (125,126), the KEO reads

$$2\hat{T} = \frac{\hat{P}_{R_1}^{\dagger} \hat{P}_{R_1}}{\mu_1} + 2\hat{T}^{(1)} + \frac{(\hat{\vec{J}}^{\dagger} - \hat{\vec{J}}^{(1)})^{\dagger} \cdot (\hat{\vec{J}} - \hat{\vec{J}}^{(1)})}{\mu_1 R_1^2}, \quad (250)$$

with

$$2\hat{T}^{(1)} = \sum_{k=1}^{2} \frac{\hat{P}_{R_k^{(1)}}^{\dagger} \hat{P}_{R_k^{(1)}}}{\mu_k^{(1)}} + \sum_{k=1}^{2} \frac{\hat{\vec{L}}_k^{(1)}^{\dagger} \hat{\vec{L}}_k^{(1)}}{\mu_k^{(1)} R_k^{(1)}^2}. \quad (251)$$

($\mu_1 = \frac{3m_H m_N}{3m_H + m_N}$, $\mu_1^{(1)} = \frac{m_H}{2}$, $\mu_2^{(1)} = \frac{2m_H}{3}$.) However, in Ref. [77, 199], we preferred to parametrize the H₃ subsystem in terms of hyperspherical coordinates [200, 201] instead of polyspherical coordinates. The hyperspherical coordinates have the advantage that the three atoms are treated fully equivalently. The KEO in hyperspherical coordinates for a tri-atomic system is well-known [200, 201] and is given by:
FIG. 24: Jacobi description of NH$_3$: G$_{H2}$ is the center of mass of H$_2$, G$_{H3}$ is the center of mass of the H$_3$ subsystem.

<figure><img src="image_25.png" /></figure>

$$ \hat{T}^{(1)} = -\frac{\hbar^2}{2\mu\varrho^5} \frac{\partial}{\partial\varrho} \varrho^5 \frac{\partial}{\partial\varrho} - \frac{\hbar^2}{2\mu\varrho^2} \left[ \frac{4}{\sin 2\theta} \frac{\partial}{\partial\theta} \sin 2\theta \frac{\partial}{\partial\theta} + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\varphi^2} \right] \\ + \frac{\hat{J}_{x\text{PAS}}^{(1)2}}{\mu\varrho^2(1-\sin\theta)} + \frac{\hat{J}_{y\text{PAS}}^{(1)2}}{\mu\varrho^2(1+\sin\theta)} + \frac{\hat{J}_{z\text{PAS}}^{(1)2}}{2\mu\varrho^2\sin^2\theta} - \frac{i\hbar \cos\theta}{\mu\varrho^2\sin^2\theta} \hat{J}_{z\text{PAS}}^{(1)} \frac{\partial}{\partial\varphi}, \qquad (252) $$

with

$$ 0 \le \varrho, $$

$$ 0 \le \theta \le \pi/2, $$

$$ 0 \le \varphi \le 2\pi, $$

$$ \mu = \frac{m_H}{\sqrt{3}}, $$

where PAS denotes the instantaneous principal axis system of H$_3$. ρ and {θ, ϕ} define the size and the shape of the H$_3$ triangle, respectively. They are defined such as

$$ R_{1x\text{PAS}}/d = \rho \cos(\theta/2 - \pi/4) \cos\varphi, $$

$$ R_{1y\text{PAS}}/d = -\rho \sin(\theta/2 - \pi/4) \sin\varphi, $$

$$ R_{2x\text{PAS}} * d = \rho \cos(\theta/2 - \pi/4) \sin\varphi, $$

$$ R_{2y\text{PAS}} * d = \rho \sin(\theta/2 - \pi/4) \cos\varphi, $$
with d a scaling factor that is equal to 1.07457 for an equal mass tri-atomic system [200, 201] (H₃ is a 'equal mass system' since the three atoms have the same mass). $\vec{J}^{(1)}$ is a function of three internal Euler angles $\alpha^{(1)}, \beta^{(1)}, \gamma^{(1)}$ allowing the localization of the H₃ PAS with respect to the E₂ frame. Since H₃ is not involved in the definition of the E₂ frame (the E₂ frame is specified by $\vec{R}_1$ only), the E₂-components of $\vec{J}^{(1)}$ have a regular expression and after applying the three Euler rotations (see Eqs. (69,70)), one obtains:

$$ \begin{bmatrix} \hat{J}_{x}^{(1)} \\ \hat{J}_{y}^{(1)} \\ \hat{J}_{z}^{(1)} \end{bmatrix} = \begin{bmatrix} -\frac{\cos \gamma^{(1)}}{\sin \beta^{(1)}} & \sin \gamma^{(1)} & \cot \beta^{(1)} \\ \frac{\sin \gamma^{(1)}}{\sin \beta^{(1)}} & \cos \gamma^{(1)} & -\cot \beta^{(1)} \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \frac{1}{i} \frac{\partial}{\partial \alpha^{(1)}} \\ \\ \frac{1}{i} \frac{\partial}{\partial \beta^{(1)}} \\ \\ \frac{1}{i} \frac{\partial}{\partial \gamma^{(1)}} \end{bmatrix} . \quad (253) $$

The third Euler angle is chosen such that $\gamma = \alpha^{(1)}$ (Note that, for the first time, we no longer use the definition of the BF frame as in Sec. IIC1 (i)). The KEO is given by Eqs. (250,252,253): the expression of the PAS-components of $\vec{J}^{(1)}$ (Eq. (253)) is unaffected by the choice $\gamma = \alpha^{(1)}$ and the BF-components of $\vec{J}^{(1)}$ are obtained from Eq. (253) along with Eqs. (69,70). The fact that we use the KEO in hyperspherical coordinates for a tri-atomic system for $\hat{T}^{(1)}$ and that we do not change the rest of the KEO comes from the fact that $\vec{R}_1$ originates from the center of mass of H₃. The correctness of the final KEO (Eqs. (250,252,253)) was checked by using the expression of Podolsky from Eq. (2) along with Mathematica. The hyperspherical coordinates were introduced in order to keep the full symmetry of the system and to work in each irreducible representation of the $D_{3h}(M)^2$ permutation inversion group of the molecule [202]. In addition, $R_1$ provides a good reaction coordinate for the umbrella inversion (see Ref. [77] for how to distinguish the two isomers), and $\rho$ a good coordinate to describe the symmetric stretching mode of vibration.

## (2) Introduction of hyperspherical coordinates for the water trimer:

Similarly, in Ref. [82], for the water trimer, we proposed to parametrize the three relative positions of the subsystems (what we called the 'remaining' vectors in Sec. IID (i)) by hyper-
spherical coordinates instead of polyspherical coordinates. Indeed, all the parametrizations proposed in Sec. IID for the water trimer (see Figures 8, 9, 10, 11) break the symmetry of permutation of the three monomers. If one wants to keep this symmetry, a solution is to start from the Jacobi vectors of Figure 8 (or Figure 9) and to parametrize the three 'remaining' vectors by hyperspherical instead of polyspherical coordinates. The KEO in Jacobi polyspherical coordinates is given by Eqs. (125,126) with n = 3. Now, it should be emphasized that the three 'remaining' vectors originate from or point towards the centers of mass of the three subsystems: there is no coupling in the matrix M between the n-1 'remaining' vectors and the vectors of the subsystems (like in Eqs. (130,131)). Consequently, $\hat{T} - \sum_{j=1}^{3} \hat{T}^{(j)}$ is, formally, identical to the KEO of a system of three particles except that $\hat{J}$ is replaced by $\hat{J} - \hat{L}$ with $\hat{L} = \sum_{j=1}^{3} \hat{J}^{(j)}$ (we recall that $\hat{J}^{(j)}$ is the total angular momentum of the j-th subsystem). The three 'particles' are not three atoms but three 'fictitious' particles, $G_1, G_2$ and $G_3$ (we recall that $G_j$ is the center of mass of the j-th subsystem). The mass associated with $G_j$ is equal to the mass of the j-th subsystem : $m_O + 2m_H$. Therefore, if we use hyperspherical (instead of polyspherical) coordinates to parametrize the three 'remaining' vectors, we simply have to rewrite Eq. (125) in terms of the hyperspherical coordinates associated with $G_1, G_2, G_3$. Therefore, the new KEO reads

$$ \begin{aligned} \hat{T} = & -\frac{\hbar^2}{2\mu\varrho^5} \frac{\partial}{\partial\varrho} \varrho^5 \frac{\partial}{\partial\varrho} - \frac{\hbar^2}{2\mu\varrho^2} \left[ \frac{4}{\sin 2\theta} \frac{\partial}{\partial\theta} \sin 2\theta \frac{\partial}{\partial\theta} + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\varphi^2} \right] \\ & + \frac{(\hat{J} - \hat{L})^2_{x^{PAS}}}{\mu\varrho^2(1 - \sin\theta)} + \frac{(\hat{J} - \hat{L})^2_{y^{PAS}}}{\mu\varrho^2(1 + \sin\theta)} + \frac{(\hat{J} - \hat{L})^2_{z^{PAS}}}{2\mu\varrho^2\sin^2\theta} - \frac{i\hbar \cos\theta}{\mu\varrho^2\sin^2\theta} (\hat{J} - \hat{L})_{z^{PAS}} \frac{\partial}{\partial\varphi} \\ & + \sum_{j=1}^{3} \hat{T}^{(j)}, \end{aligned} \quad (254) $$

with

$$ 0 \le \varrho, $$

$$ 0 \le \theta \le \pi/2, $$

$$ 0 \le \varphi \le 2\pi, $$

$$ \mu = \frac{2m_H + m_O}{\sqrt{3}}. $$

In Eq. (254), PAS denotes the instantaneous principal axis system of {$G_1, G_2, G_3$}, which is chosen as the BF frame of the system. $\rho$ and {$\theta, \varphi$} define the size and the shape of the
$\{G_1, G_2, G_3\}$ triangle, respectively. $\hat{T}^{(j)}$ is given by Eq. (126) or by Eqs. (B5,B6,B7) if each subsystem is parametrized by non-orthogonal vectors. $\hat{T}^{(j)}$ could also be a rigid (flexible) rotor of each monomer if the intramolecular motions are rigidly (adiabatically) constrained.

Again, we repeat that $\hat{T} - \sum_{j=1}^{3} \hat{T}^{(j)}$ in Eq. (254) is simply the KEO of a system of three particles in hyperspherical coordinates [200, 201] except that:

(1) the total angular momentum $\hat{J}$ is replaced by $\hat{J} - \hat{L} = \hat{J} - \sum_{j=1}^{3} \hat{J}^{(j)}$.

(2) the three 'particles' are not three atoms but three 'fictitious' particles, $G_1, G_2$ and $G_3$. The mass associated with $G_j$ is equal to the mass of the $j$-th subsystem : $m_O + 2m_H$.

The introduction of the hyperspherical coordinates that describe the relative positions of the three monomers allows the full symmetry of the system to be preserved. In particular, the definition of the BF frame for the whole system is different from the definition in the original formulation of the polyspherical approach (Sec. IIC1 (i)). The BF frame is now the instantaneous principal axis system of $\{G_1, G_2, G_3\}$: the symmetry of permutation of the three monomers is kept since no molecule plays a particular role in the definition of the BF frame. The overall rotation of each subsystem is parametrized by three Euler angles with respect to the new BF frame and, since the vectors in each subsystems play no role in the definition of the BF frame (as before), the expression of the projections of $\hat{J}^{(j)}$ onto the BF frame is given by the usual formulae (Eq. (128)). Finally, let us mention that it could be possible to use, instead of the three hyperspherical coordinates, Pekeris coordinates (corresponding to the three distances between the three 'fictitious' particles, $G_1, G_2$ and $G_3$) as proposed by Wang and Carrington for van der Waals trimers [203, 204].

(3) Generalization of the previous results:

The results presented above are of general character and can be used to combine the polyspherical coordinates with other families of coordinates. For instance, for large van der Waals clusters, it is possible to envision the use of normal coordinates to describe the relative motions of the centers of mass of the different subsystems, since the normal coordinates
simplify the final expression of the KEO. Let us separate the system into *n* subsystems as in Sec. IID.

In the formulation that follows two conditions must be satisfied (note that they were already satisfied for the water trimer in (2)):

(i) when the system is separated into subsystems, there is no coupling in the matrix **M** between the *n*-1 'remaining' vectors $\hat{\mathbf{R}}_i$ and all the vectors of the subsystems. It corresponds to the case (2) of Sec. IID. In other word, the *n*-1 'remaining' vectors and the vectors in each subsystem can be orthogonal or non-orthogonal vectors but the 'remaining' vectors must originate from or point towards the centers of mass of the subsystems. (The matrix **M** has a block-diagonal form as in Eq. (132) of Sec. IID.)

(ii) The BF frame is defined by the *n*-1 'remaining' vectors and the vectors in each subsystems play no role in its definition (before and after the coordinate transformation from the polyspherical coordinates to the new coordinates).

The condition (i) entails that $\hat{T} - \sum_{j=1}^{n} \hat{T}^{(j)}$ is, formally, identical to the KEO of an *n*-atomic system except that $\hat{\mathbf{J}}$ is replaced by $\hat{\mathbf{J}} - \hat{\mathbf{L}}$ with $\hat{\mathbf{L}} = \sum_{j=1}^{n} \hat{\mathbf{J}}^{(j)}$ (we recall that $\hat{\mathbf{J}}^{(j)}$ is the total angular momentum of the *j*-th subsystem). The centers of mass ($G_j$, $j = 1, \dots, n$) play the role of the 'atoms' and the mass associated with $G_j$ is $M_j$, the total mass of the *j*-th subsystem. A simple coordinate transformation from the polyspherical coordinates parametrizing the *n*-1 'remaining' vectors to the normal coordinates associated with the centers of mass ($G_j$, $j = 1, \dots, n$) leads to the new KEO

$$ \hat{T} = \frac{1}{2} \sum_{\alpha,\beta=x,y,z} \left( (\hat{\mathbf{J}} - \hat{\mathbf{L}})_\alpha - \hat{\pi}_\alpha \right) \mu_{\alpha\beta} \left( (\hat{\mathbf{J}} - \hat{\mathbf{L}})_\beta - \hat{\pi}_\beta \right) - \frac{1}{2} \sum_{k=1}^{3n-6} \frac{\partial^2}{\partial Q_k^2} - \frac{1}{8} \sum_{\alpha=x,y,z} \mu_{\alpha\alpha} \\ + \sum_{j=1}^{n} \hat{T}^{(j)}, \qquad (255) $$

where

$$ \mu_{\alpha\beta} = (\mathbf{I}'^{-1})_{\alpha\beta}, \quad \mathbf{I}'_{\alpha\beta} = \mathbf{I}_{\alpha\beta} + \sum_{k,l,m=1}^{3n-6} \zeta_{km}^{\alpha} \zeta_{lm}^{\beta} Q_k Q_l, \qquad (256) $$
and where $I_{\alpha\beta}$ is the inertia tensor and $\zeta_{km}^{\alpha}$ are Coriolis parameters defined for example in [88]. The vibrational angular momentum terms, $\pi_{\alpha}$, are

$$ \hat{\pi}_{\alpha} = -i \sum_{k,l=1}^{3n-6} \zeta_{kl}^{\alpha} Q_k \frac{\partial}{\partial Q_l}. \qquad (257) $$

The BF frame is specified by the positions of the centers of mass of the subsystems: in Eq. (255) its choice is free (it could be, for instance, the Eckart frame [111] of the $\{G_1, \dots, G_n\}$ system). Eq. (255) is the Watson KEO [88] except that (1) the total angular momentum $\hat{J}$ is replaced by $\hat{J} - \hat{L} = \hat{J} - \sum_{j=1}^{n} \hat{J}^{(j)}$. (2) The normal coordinates are the normal coordinates associated with $G_j$; $j = 1, \dots, n$ (the centers of mass of the subsystems).

In addition, since the BF frame is specified only by the position of the centers of mass of the subsystems and, therefore, since the vectors in each subsystem play no role in the definition of the BF frame (condition (ii) above), the expression of the projections of $\hat{J}^{(j)}$ onto the BF frame is given by the usual formulae (Eq. (128)).

$\hat{T}^{(j)}$ could be given by Eq. (126) or by Eqs. (B5,B6,B7). $\hat{T}^{(j)}$ could also be a rigid (flexible) rotor of each monomer if the intramolecular motions are rigidly (adiabatically) constrained.

Finally, for very large van der Waals clusters, it becomes possible to forget the separation of the overall rotation from the internal motions. The KEO simply reads

$$ 2\hat{T} = - \sum_{i=1,...,n-1} \sum_{\lambda=X,Y,Z} M_{ij} \frac{\partial}{\partial \lambda_i^{SF}} \frac{\partial}{\partial \lambda_j^{SF}} + 2 \sum_{j=1}^{n} \hat{T}^{(j)}, \qquad (258) $$

where $\{X_i^{SF}, Y_i^{SF}, Z_i^{SF}\}$ are the Cartesian coordinates of the $i$-th 'remaining' vector in the SF frame. $M_{ij}$ are the elements of the mass matrix $M$ of Eq. (54). $\hat{T}^{(j)}$ could be given by Eq. (126) or by Eqs. (B5,B6,B7). As above, $\hat{T}^{(j)}$ could be a rigid (flexible) rotor of each monomer if the intramolecular motions are rigidly (adiabatically) constrained.
V. CONCLUSION AND OUTLOOK

In the last decades, important progress has been made in the development of quantum
dynamics simulations. This has been made possible by the increase in computer power and,
as importantly, the development of new efficient algorithms. In this context, the growing
interest in systems presenting large amplitude motions calls for correct quantum-mechanical
expressions of KEOs in curvilinear coordinates. The polyspherical approach recently
developed and reviewed in the present paper was devised in order to obtain these operators
in a systematic way. Numerous applications have already been performed using KEOs
obtained from this approach (see Part III for an non-exhaustive list). These applications
demonstrate conclusively that this approach can live up to its early promise of being
capable of providing compact expressions of KEOs in a form well-adapted to the numerical
approaches used in dynamics.

Looking ahead, we can say that the next step in the development of the polyspherical
approach will probably be the implementation of the results presented in Part II of the
present review, by way of an automatic procedure. Indeed, it should be possible to write
a program that uses the polyspherical approach combined with the (successive) separation
into subsystems. This program could give the explicit analytical expression of the KEO for
any particular case and could even directly implement the operator in a code used to solve
numerically the Schrödinger equation.

The polyspherical approach has given rise to many applications chiefly in the field of
Infra-red spectroscopy (see Part III). Another goal will be the study of chemical processes
for instance in photochemical reactivity involving conical intersections. As explained in
the introduction of Part IV, the number of quantum mechanical simulations with curvi-
linear coordinates for these systems is rather limited because it is difficult to obtain the
corresponding correct KEO. The results presented in Part IV could bring an important
contribution in order to generalize the use of curvilinear coordinates in this domain.
Acknowledgements

F.G. thanks Prof. H.-D. Meyer for his strong support of this work and for having sug-
gested to write the present review. The authors thank Prof. A. Nauts, Dr. B. Lasorne,
Dr. M. Sanrey, Dr. G. Barg, Dr. F. Richter, and Dr. D. Lauvergnat for having read the
manuscript very carefully and for their numerous corrections.
# APPENDIX A: PROPERTIES OF THE PARTIAL ANGULAR MOMENTA

When defining the conformation of an N-particle system by N-1 vectors, one unavoidably comes across several angular momenta: not only the total angular momentum of the system but also the various partial angular momenta corresponding to the motion of the various vectors. All these momenta can, in addition, be referred to a variety of reference frames such as the so-called Body-Fixed (BF) frame whose origin coincides with the center of mass and whose axes rotate in a conventional manner when the particles move. The introduction of the projections onto the BF-axes is necessary when using the BF coordinate (see Sec. IIB 2), but raises a new technical problem: while the projections of the angular momenta onto the SF-axes satisfy the usual commutation relations, and their action onto a basis set of spherical harmonics in terms of the SF spherical coordinates is well known (see for instance Ref. [109]), the projections of the same angular momenta onto the axes of a moving frame may satisfy anomalous, non-definite commutation relations [81].

We have shown elsewhere [81] that two cases must be distinguished: whether the vector is involved in the definition of the BF or not. Indeed, if a vector is not involved in the definition of a frame F, the expression for the projection of the corresponding angular momentum onto the F-axes expressed in the coordinates in this frame is identical to the usual one in a SF frame [58] (it should be stressed that this result is not trivial at all, see below). To be more specific, if $\vec{R}_j$ is not involved in the definition of the BF frame, the projections of $\vec{L}_j$ onto the BF-axes and expressed in terms of the BF-Cartesian coordinates are given by:

$$ \begin{bmatrix} \hat{L}_{j x^{BF}} \\ \hat{L}_{j y^{BF}} \\ \hat{L}_{j z^{BF}} \end{bmatrix} = \begin{bmatrix} R_{j x^{BF}} \\ R_{j y^{BF}} \\ R_{j z^{BF}} \end{bmatrix} \times \begin{bmatrix} \frac{1}{i} \frac{\partial}{\partial R_{j x^{BF}}} \\ \\ \frac{1}{i} \frac{\partial}{\partial R_{j y^{BF}}} \\ \\ \frac{1}{i} \frac{\partial}{\partial R_{j z^{BF}}} \end{bmatrix} . \quad (A1) $$

Consequently, the usual commutation relations are verified.

If the vector is involved in the definition of the BF frame, the previous results are no longer true. The projections have to be calculated explicitly and the commutation relations are non-definite. Several examples can be found in Sec. IIC 1 (Eq. (84)) and Ref. [81].
Let us return to the first case, namely when the vector is not involved in the definition of the BF. It would be altogether mistaken to suppose that Eq. (A1) refers to an angular momentum computed in the BF frame. Indeed, Eq. (A1) is nothing but the exact quantum mechanical counterpart of the classical expression [58]

$$ \begin{bmatrix} L_{j x^{BF}} \\ L_{j y^{BF}} \\ L_{j z^{BF}} \end{bmatrix} = \begin{bmatrix} R_{j x^{BF}} \\ R_{j y^{BF}} \\ R_{j z^{BF}} \end{bmatrix} \times \begin{bmatrix} \frac{\partial T^{SF}}{\partial \dot{R}_{j x^{BF}}} \\ \frac{\partial T^{SF}}{\partial \dot{R}_{j y^{BF}}} \\ \frac{\partial T^{SF}}{\partial \dot{R}_{j z^{BF}}} \end{bmatrix}. \qquad (A2) $$

It is important then to notice that ($\mathcal{L}$ is the Lagrangian of the system)

$$ \frac{\partial T^{SF}}{\partial \dot{R}_{j \lambda^{BF}}} = \frac{\partial \mathcal{L}}{\partial \dot{R}_{j \lambda^{BF}}} \neq \frac{\partial T^{BF}}{\partial \dot{R}_{j \lambda^{BF}}}, \qquad (A3) $$

i.e. that $\frac{\partial T^{SF}}{\partial \dot{R}_{j \lambda^{BF}}}$ are not the components of a momentum computed in the BF frame, i.e. they are not computed from velocities relative to the BF frame. Consequently, the angular momentum operator in Eq. (A1) turns out to be the angular momentum computed in the SF frame, projected onto the BF-axes and expressed in terms of BF coordinates. For the sake of completeness, let us add that

$$ \frac{\partial T^{SF}}{\partial \dot{R}_{j \lambda^{BF}}} = \frac{\partial \mathcal{L}}{\partial \dot{R}_{j \lambda^{BF}}} \rightarrow \frac{1}{i} \frac{\partial}{\partial R_{j \lambda^{BF}}} \qquad (A4) $$

is the only correct quantization rule [57] and not

$$ \frac{\partial T^{BF}}{\partial \dot{R}_{j \lambda^{BF}}} \rightarrow \frac{1}{i} \frac{\partial}{\partial R_{j \lambda^{BF}}}. \qquad (A5) $$

To explain the origin of these results, let us point out that Eq. (A1) and Eq. (A2) result from the vector $\vec{R}_j$ being not involved in the definition of the BF frame, which can also be mathematically expressed as follows (a systematic parallel between classical and quantum mechanics is made to highlight the physical meaning of the operators):

(i) by means of Poisson brackets:

$$ \{ \mathcal{R}_{\lambda\lambda'}, L_{j\rho}^{SF} \} = 0, \qquad (A6) $$
(ii) or commutation relations:

$$[\mathcal{R}_{\lambda\lambda'}, \hat{L}_{j\rho^{\text{SF}}}] = 0, \qquad (A7)$$

(with $(\lambda, \lambda', \rho = x, y, z)$ and $\mathcal{R}(\alpha, \beta, \gamma)$ the Euler rotation matrix) We then have in classical mechanics

$$\begin{bmatrix} L_{j x^{\text{BF}}} \\ L_{j y^{\text{BF}}} \\ L_{j z^{\text{BF}}} \end{bmatrix} = \mathcal{R}^T(\alpha, \beta, \gamma) \begin{bmatrix} L_{j x^{\text{SF}}} \\ L_{j y^{\text{SF}}} \\ L_{j z^{\text{SF}}} \end{bmatrix} \qquad (A8)$$

$$= (\mathcal{R}^T(\alpha, \beta, \gamma) \begin{bmatrix} R_{i x^{\text{SF}}} \\ R_{i y^{\text{SF}}} \\ R_{i z^{\text{SF}}} \end{bmatrix}) \times (\mathcal{R}^T(\alpha, \beta, \gamma) \begin{bmatrix} \frac{\partial T^{\text{SF}}}{\partial R_{i x^{\text{SF}}}} \\ \frac{\partial T^{\text{SF}}}{\partial R_{i y^{\text{SF}}}} \\ \frac{\partial T^{\text{SF}}}{\partial R_{i z^{\text{SF}}}} \end{bmatrix})$$

$$= \begin{bmatrix} R_{j x^{\text{BF}}} \\ R_{j y^{\text{BF}}} \\ R_{j z^{\text{BF}}} \end{bmatrix} \times (\mathcal{R}^T(\alpha, \beta, \gamma) \begin{bmatrix} \frac{\partial T^{\text{SF}}}{\partial R_{j x^{\text{SF}}}} \\ \frac{\partial T^{\text{SF}}}{\partial R_{j y^{\text{SF}}}} \\ \frac{\partial T^{\text{SF}}}{\partial R_{j z^{\text{SF}}}} \end{bmatrix}), \qquad (A9)$$

and in quantum mechanics
$$
\begin{align}
\begin{bmatrix} \hat{L}_{j x^{BF}} \\ \hat{L}_{j y^{BF}} \\ \hat{L}_{j z^{BF}} \end{bmatrix} &= \mathcal{R}^T(\alpha, \beta, \gamma) \begin{bmatrix} \hat{L}_{j x^{SF}} \\ \hat{L}_{j y^{SF}} \\ \hat{L}_{j z^{SF}} \end{bmatrix} \nonumber \\
&= (\mathcal{R}^T(\alpha, \beta, \gamma) \begin{bmatrix} R_{i x^{SF}} \\ R_{i y^{SF}} \\ R_{i z^{SF}} \end{bmatrix}) \times (\mathcal{R}^T(\alpha, \beta, \gamma) \begin{bmatrix} \frac{1}{i} \frac{\partial}{\partial R_{j x^{SF}}} \\ \frac{1}{i} \frac{\partial}{\partial R_{j y^{SF}}} \\ \frac{1}{i} \frac{\partial}{\partial R_{j z^{SF}}} \end{bmatrix}) \tag{A10} \\
&= \begin{bmatrix} R_{j x^{BF}} \\ R_{j y^{BF}} \\ R_{j z^{BF}} \end{bmatrix} \times (\mathcal{R}^T(\alpha, \beta, \gamma) \begin{bmatrix} \frac{1}{i} \frac{\partial}{\partial R_{j x^{SF}}} \\ \frac{1}{i} \frac{\partial}{\partial R_{j y^{SF}}} \\ \frac{1}{i} \frac{\partial}{\partial R_{j z^{SF}}} \end{bmatrix}). \tag{A11}
\end{align}
$$

An important question to ask is the following: is it true that

$$ (\mathcal{R}^T \begin{bmatrix} \frac{\partial T^{SF}}{\partial \hat{R}_{i x^{SF}}} \\ \frac{\partial T^{SF}}{\partial \hat{R}_{i y^{SF}}} \\ \frac{\partial T^{SF}}{\partial \hat{R}_{i z^{SF}}} \end{bmatrix}) = \begin{bmatrix} \frac{\partial T^{SF}}{\partial \hat{R}_{i x^{SF}}} \\ \frac{\partial T^{SF}}{\partial \hat{R}_{i y^{SF}}} \\ \frac{\partial T^{SF}}{\partial \hat{R}_{i z^{SF}}} \end{bmatrix}, \quad (A12) $$

or

$$ (\mathcal{R}^T \begin{bmatrix} \frac{1}{i} \frac{\partial}{\partial R_{j x^{SF}}} \\ \frac{1}{i} \frac{\partial}{\partial R_{j y^{SF}}} \\ \frac{1}{i} \frac{\partial}{\partial R_{j z^{SF}}} \end{bmatrix}) = \begin{bmatrix} \frac{1}{i} \frac{\partial}{\partial R_{j x^{BF}}} \\ \frac{1}{i} \frac{\partial}{\partial R_{j y^{BF}}} \\ \frac{1}{i} \frac{\partial}{\partial R_{j z^{BF}}} \end{bmatrix}, \quad (A13) $$

or, in other words, that the conjugate momentum associated with the BF coordinates (which is not a 'BF conjugate momentum' in the sense of a momentum computed in the BF frame)
can be identified with the projection of the SF conjugate momentum? The explicit calculation of the projections shows that the answer is yes and comes directly from Eq. (A6) and (A7), i.e. from the fact that the vector is not involved in the definition of BF.

We thus get

$$ \begin{bmatrix} L_{j x^{BF}} \\ L_{j y^{BF}} \\ L_{j z^{BF}} \end{bmatrix} = \begin{bmatrix} R_{j x^{BF}} \\ R_{j y^{BF}} \\ R_{j z^{BF}} \end{bmatrix} \times \begin{bmatrix} \frac{\partial T^{SF}}{\partial \tilde{R}_{i x^{BF}}} \\ \frac{\partial T^{SF}}{\partial \tilde{R}_{i y^{BF}}} \\ \frac{\partial T^{SF}}{\partial \tilde{R}_{i z^{BF}}} \end{bmatrix}, \qquad (A14) $$

and

$$ \begin{bmatrix} \hat{L}_{j x^{BF}} \\ \hat{L}_{j y^{BF}} \\ \hat{L}_{j z^{BF}} \end{bmatrix} = \begin{bmatrix} R_{j x^{BF}} \\ R_{j y^{BF}} \\ R_{j z^{BF}} \end{bmatrix} \times \begin{bmatrix} \frac{1}{i} \frac{\partial}{\partial R_{j x^{BF}}} \\ \frac{1}{i} \frac{\partial}{\partial R_{j y^{BF}}} \\ \frac{1}{i} \frac{\partial}{\partial R_{j z^{BF}}} \end{bmatrix}, \qquad (A15) $$

if the vector $\vec{R}_j$ is not involved in the definition of the BF frame.
APPENDIX B: GENERAL EXPRESSION OF THE KEO IN ANGULAR MOMENTA FOR NON-ORTHOGONAL COORDINATES

In Section IIC1, we have derived the expression of the KEO in angular momenta for orthogonal vectors (Eq. (92)). Now, what happens if non-orthogonal vectors are used?

Whereas orthogonal vectors diagonalize the kinetic energy in Eq. (63), non-orthogonal vectors give rise to off-diagonal terms, also called 'mass polarization' terms: $\hat{P}_i^\dagger M_{ij} \hat{P}_j$ with $i \neq j$. Since the conjugate momentum can be separated into a radial and angular term as:

$$ \hat{P}_i = \hat{P}_{R_i} \vec{e}_i - \frac{\vec{e}_i \times \hat{L}_i}{R_i}, \qquad (B1) $$

with $\vec{e}_i = \vec{R}_i/R_i$ a unit vector along $\vec{R}_i$; $\hat{P}_{R_i}$, the radial momentum and $\hat{L}_i$, the angular momentum; and its adjoint as

$$ (\vec{P}_i)^\dagger = \hat{P}_{R_i}^\dagger \vec{e}_i + \frac{\hat{L}_i^\dagger \times \vec{e}_i}{R_i}, \qquad (B2) $$

in view of Eqs. (63), (B1,B2), $\hat{P}_i^\dagger M_{ij} \hat{P}_j + \hat{P}_j^\dagger M_{ij} \hat{P}_i$ can be rewritten as

$$ \begin{aligned} 2\hat{T}_{ij} = M_{ij} & [\vec{e}_i \cdot \vec{e}_j (\hat{P}_{R_i}^\dagger \hat{P}_{R_j}^r + \hat{P}_{R_j}^\dagger \hat{P}_{R_i}) \\ & - \frac{\hat{L}_i^\dagger \times \vec{e}_i}{R_i} \cdot \frac{\vec{e}_j \times \hat{L}_j}{R_j} - \frac{\hat{L}_j^\dagger \times \vec{e}_j}{R_j} \cdot \frac{\vec{e}_i \times \hat{L}_i}{R_i} \\ & + \hat{P}_{R_j} \frac{\hat{L}_i^\dagger \times \vec{e}_i}{R_i} \cdot \vec{e}_j - \hat{P}_{R_j}^\dagger \vec{e}_j \cdot \frac{\vec{e}_i \times \hat{L}_i}{R_i} \\ & - \hat{P}_{R_i}^\dagger \vec{e}_i \cdot \frac{\vec{e}_j \times \hat{L}_j}{R_j} + \hat{P}_{R_i} \frac{\hat{L}_j^\dagger \times \vec{e}_j}{R_j} \cdot \vec{e}_i] \end{aligned} \qquad (B3) $$

All the different terms appearing in Eq. (B3) have to be explicitly calculated. Taking into account that

$$ \hat{L}_{N-1} = \hat{J} - \sum_{i=1}^{N-2} \hat{L}_i \qquad (B4) $$

and that $\hat{L}_{N-2}$ has a particular behavior, it is straightforward, but tedious to show that [78]
$$ \hat{T} = \hat{T}_1 + \hat{T}_2, \qquad (B5) $$

with

$$ \begin{aligned} \hat{T}_1 = & \sum_{i=1}^{N-2} \left( \frac{M_{iN-1} P_{R_i} \sin \theta_i^{BF}}{R_{N-1}} \left( \frac{e^{-i\varphi_i^{BF}} \hat{J}_{+BF} - e^{i\varphi_i^{BF}} \hat{J}_{-BF}}{2i} \right) \right. \\ & \left. - \frac{M_{iN-1} \sin \theta_i^{BF}}{R_i R_{N-1}} \left( \frac{e^{-i\varphi_i^{BF}} \hat{J}_{+BF} + e^{i\varphi_i^{BF}} \hat{J}_{-BF}}{2} \right) \hat{L}_{iz^{BF}} \right. \\ & \left. + \left( \frac{M_{iN-1} \cos \theta_i^{BF}}{R_i R_{N-1}} - \frac{M_{N-1N-1}}{R_{N-1}^2} \right) \left( \frac{\hat{J}_{+BF} \hat{L}_{i-BF} + \hat{J}_{-BF} \hat{L}_{i+BF}}{2} \right) \right) \\ & - \sum_{i=1}^{N-2} \left( \frac{M_{N-1N-1} \hat{J}_{z^{BF}} \hat{L}_{iz^{BF}}}{R_{N-1}^2} \right) + \frac{M_{N-1N-1} (\hat{J})_{BF}^2}{2R_{N-1}^2}, \end{aligned} \qquad (B6) $$

and with
$$
\begin{align}
\hat{T}_2 = & \sum_{i=1}^{N-1} M_{ii} \frac{\hat{P}_{R_i}^\dagger \hat{P}_{R_i}}{2} + \sum_{i,j=1; i<j}^{N-1} M_{ij} (\sin \theta_i^{BF} \sin \theta_j^{BF} \cos(\varphi_i^{BF} - \varphi_j^{BF}) + \cos \theta_i^{BF} \cos \theta_j^{BF}) \hat{P}_{R_i} \hat{P}_{R_j} \nonumber \\
& + \sum_{i,j=1; i \neq j}^{N-2} \{ M_{ij} \sin \theta_i^{BF} \sin \theta_j^{BF} \sin(\varphi_i^{BF} - \varphi_j^{BF}) \frac{\hat{P}_{R_i} \hat{L}_{jz^{BF}}}{R_j} \nonumber \\
& + \hat{P}_{R_i} \sin \theta_i^{BF} \left( \frac{M_{ij} \cos \theta_j^{BF}}{R_j} - \frac{M_{iN-1}}{R_{N-1}} \right) \left( \frac{e^{-i\varphi_i^{BF}} \hat{L}_{j+^{BF}} - e^{i\varphi_i^{BF}} \hat{L}_{j-^{BF}}}{2i} \right) \} \nonumber \\
& - \sum_{i=1}^{N-2} \frac{M_{iN-1}}{R_{N-1}} \hat{P}_{R_i} \sin \theta_i^{BF} \left( \frac{e^{-i\varphi_i^{BF}} \hat{L}_{i+^{BF}} - e^{i\varphi_i^{BF}} \hat{L}_{i-^{BF}}}{2i} \right) \nonumber \\
& - \sum_{i,j=1; i \neq j}^{N-1} \frac{M_{ij}}{R_i} \hat{P}_{R_j} \cos \theta_j^{BF} \sin \theta_i^{BF} \left( \frac{e^{-i\varphi_i^{BF}} \hat{L}_{i+^{BF}} - e^{i\varphi_i^{BF}} \hat{L}_{i-^{BF}}}{2i} \right) \nonumber \\
& + \sum_{i,j=1; i<j}^{N-2} \left\{ \frac{M_{ij}}{R_i R_j} \left( \cos \theta_i^{BF} \cos \theta_j^{BF} + \frac{1}{2} e^{-i\varphi_i^{BF}} \sin \theta_i^{BF} e^{i\varphi_j^{BF}} \sin \theta_j^{BF} \right) \right. \nonumber \\
& - \frac{M_{iN-1}}{R_i R_{N-1}} \cos \theta_i^{BF} - \frac{M_{jN-1}}{R_j R_{N-1}} \cos \theta_j^{BF} + \frac{M_{N-1N-1}}{R_{N-1}^2} \left\{ \frac{\hat{L}_{i+^{BF}} \hat{L}_{j-^{BF}}}{2} \right\} \nonumber \\
& + \sum_{i,j=1; i<j}^{N-2} \left\{ \frac{M_{ij}}{R_i R_j} \left( \cos \theta_i^{BF} \cos \theta_j^{BF} + \frac{1}{2} e^{i\varphi_i^{BF}} \sin \theta_i^{BF} e^{-i\varphi_j^{BF}} \sin \theta_j^{BF} \right) \right. \nonumber \\
& - \frac{M_{iN-1}}{R_i R_{N-1}} \cos \theta_i^{BF} - \frac{M_{jN-1}}{R_j R_{N-1}} \cos \theta_j^{BF} + \frac{M_{N-1N-1}}{R_{N-1}^2} \left\{ \frac{\hat{L}_{i-^{BF}} \hat{L}_{j+^{BF}}}{2} \right\} \nonumber \\
& - \sum_{i,j=1; i < j}^{N-2} \frac{M_{ij}}{4 R_i R_j} \sin \theta_i^{BF} \sin \theta_j^{BF} \left( e^{-i(\varphi_i^{BF} + \varphi_j^{BF})} \hat{L}_{i+^{BF}} \hat{L}_{j+^{BF}} + e^{i(\varphi_i^{BF} + \varphi_j^{BF})} \hat{L}_{i-^{BF}} \hat{L}_{j-^{BF}} \right) \nonumber \\
& + \sum_{i=1}^{N-2} \left\{ \frac{M_{N-1N-1}}{2 R_{N-1}^2} + \frac{M_{ii}}{2 R_i^2} - \frac{M_{iN-1} \cos \theta_i^{BF}}{R_i R_{N-1}} \right\} (\hat{\vec{L}}_i^\dagger \hat{\vec{L}}_i)_{BF} \nonumber \\
& + \sum_{i,j=1; i < j}^{N-2} \left\{ \frac{M_{ij}}{R_i R_j} \sin \theta_i^{BF} \sin \theta_j^{BF} \cos(\varphi_i^{BF} - \varphi_j^{BF}) + \frac{M_{N-1N-1}}{R_{N-1}^2} \right\} \hat{L}_{iz^{BF}} \hat{L}_{jz^{BF}} \nonumber \\
& + \sum_{i,j=1; i < j}^{N-2} \sin \theta_j^{BF} \left( -\frac{M_{ij}}{R_i R_j} \cos \theta_i^{BF} + \frac{M_{jN-1}}{R_j R_{N-1}} \right) \left( \frac{e^{-i\varphi_j^{BF}} \hat{L}_{i+^{BF}} + e^{i\varphi_j^{BF}} \hat{L}_{i-^{BF}}}{2} \right) \hat{L}_{jz^{BF}} \nonumber \\
& + \sum_{i,j=1; i < j}^{N-2} \sin \theta_i^{BF} \left( -\frac{M_{ij}}{R_i R_j} \cos \theta_j^{BF} + \frac{M_{iN-1}}{R_i R_{N-1}} \right) \left( \frac{e^{-i\varphi_i^{BF}} \hat{L}_{iz^{BF}} \hat{L}_{j+^{BF}} + e^{i\varphi_i^{BF}} \hat{L}_{iz^{BF}} \hat{L}_{j-^{BF}}}{2} \right) \nonumber \\
& + \sum_{i=1}^{N-2} \frac{M_{iN-1}}{R_i R_{N-1}} \sin \theta_i^{BF} \left( \frac{e^{-i\varphi_i^{BF}} \hat{L}_{i+^{BF}} - e^{i\varphi_i^{BF}} \hat{L}_{i-^{BF}}}{2} \right). \tag{B7}
\end{align}
$$

The previous equations should be used along with

$$
\theta_{N-1}^{BF} = \varphi_{N-1}^{BF} = \varphi_{N-2}^{BF} = 0 \qquad (B8)
$$
since $\vec{R}_{N-1}$ is parallel to the $z^{BF}$ axis and $\vec{R}_{N-2}$ is parallel to the $((x^{BF}, z^{BF}); x^{BF} > 0)$ half-plane. Even if this does not appear explicitly for each term, it is emphasized that the operator in Eq. (B5) is Hermitian. This operator is correct for the case in which the wave functions are normalized with an Euclidean convention of normalization, i.e. with the volume element given by Eq. (73). All the equations (75), (77), (83), and (99) given in Sec. II C 1 remain true.
APPENDIX C: GENERAL EXPRESSION OF THE KEO IN CONJUGATE MO- MENTA

This Appendix is devoted to the explicit expression of the coefficients $G_{q_l q_m}$, $\Gamma_{\lambda\lambda'}$, and $C_{\lambda q_l}$ appearing in the KEO in Eqs. (106) and (111). Here, $\{q_i; i = 1, \dots, 3N - 6\} = \{(R_i; i = 1, \dots, N - 1), (u_i; i = 1, \dots, N - 2), (\varphi_i^{BF}; i = 1, \dots, N - 3)\}$ and $\lambda, \lambda' = x, y, z$. The sub-matrices are all symmetrical, i.e. $G_{q_l q_m} = G_{q_m q_l}$, $\Gamma_{\lambda\lambda'} = \Gamma_{\lambda'\lambda}$, and $C_{\lambda q_l} = C_{q_l \lambda}$. The following equations should be used along with $\theta_{N-1}^{BF} = \varphi_{N-1}^{BF} = \varphi_{N-2}^{BF} = 0$ since $\vec{R}_{N-1}$ is parallel to the $z^{BF}$ axis and $\vec{R}_{N-2}$ is parallel to the $((x^{BF}, z^{BF}); x^{BF} > 0)$ half-plane. We recall that

$$u_i = \cos \theta_i^{BF} \qquad (C1)$$

and that

$$\cot \theta_i^{BF} = \frac{u_i}{v_i} \qquad (C2)$$

with

$$v_i = \sqrt{1 - u_i^2} = \sin \theta_i^{BF}. \qquad (C3)$$

(i) General case:
Vibrational terms:

$$
G_{R_i R_j} = M_{ij} (\sin \theta_i^{BF} \sin \theta_j^{BF} \cos(\varphi_i^{BF} - \varphi_j^{BF}) + \cos \theta_i^{BF} \cos \theta_j^{BF}),
$$

$$
G_{R_i u_j} = - \sin \theta_j^{BF} M_{ij} \frac{\sin \theta_i^{BF} \cos \theta_j^{BF} \cos(\varphi_i^{BF} - \varphi_j^{BF}) - \cos \theta_i^{BF} \sin \theta_j^{BF}}{R_j} \\
-M_{iN-1} \frac{\sin \theta_i^{BF} \cos(\varphi_i^{BF} - \varphi_j^{BF})}{R_{N-1}},
$$

$$
G_{R_i \varphi_j^{BF}} = M_{ij} \frac{\sin \theta_i^{BF} \sin(\varphi_i^{BF} - \varphi_j^{BF})}{R_j \sin \theta_j^{BF}} - M_{iN-2} \frac{\sin \theta_i^{BF} \sin \varphi_i^{BF}}{R_{N-2} \sin \theta_{N-2}^{BF}} \\
+ M_{iN-1} \frac{\sin \theta_i^{BF} (\sin \varphi_i^{BF} \cot \theta_{N-2}^{BF} + \cot \theta_j^{BF} \sin(\varphi_j^{BF} - \varphi_i^{BF}))}{R_{N-1}},
$$

$$
G_{u_i u_j} = \sin \theta_i^{BF} \sin \theta_j^{BF} M_{ij} \frac{\cos \theta_i^{BF} \cos \theta_j^{BF} \cos(\varphi_i^{BF} - \varphi_j^{BF}) + \sin \theta_i^{BF} \sin \theta_j^{BF}}{R_i R_j} \\
- M_{iN-1} \frac{\cos \theta_i^{BF} \cos(\varphi_i^{BF} - \varphi_j^{BF})}{R_i R_{N-1}} \\
- M_{jN-1} \frac{\cos \theta_j^{BF} \cos(\varphi_i^{BF} - \varphi_j^{BF})}{R_j R_{N-1}} + M_{N-1N-1} \frac{\cos(\varphi_i^{BF} - \varphi_j^{BF})}{R_{N-1}^2},
$$

$$
G_{u_i \varphi_j^{BF}} = - \sin \theta_i^{BF} M_{ij} \frac{\cos \theta_i^{BF} \sin(\varphi_i^{BF} - \varphi_j^{BF})}{R_i R_j \sin \theta_j^{BF}} - M_{iN-2} \frac{\cos \theta_i^{BF} \sin \varphi_i^{BF}}{R_i R_{N-2} \sin \theta_{N-2}^{BF}} \\
+ M_{iN-1} \frac{\cos \theta_i^{BF} (\cot \theta_j^{BF} \sin(\varphi_j^{BF} - \varphi_i^{BF}) + \cot \theta_{N-2}^{BF} \sin \varphi_i^{BF})}{R_i R_{N-1}} \\
+ M_{jN-1} \frac{\sin(\varphi_j^{BF} - \varphi_i^{BF})}{R_j R_{N-1} \sin \theta_j^{BF}} + M_{N-2N-1} \frac{\sin \varphi_i^{BF}}{R_{N-2} R_{N-1} \sin \theta_{N-2}^{BF}} \\
+ M_{N-1N-1} \frac{\cot \theta_j^{BF} \sin(\varphi_i^{BF} - \varphi_j^{BF}) - \sin \varphi_i^{BF} \cot \theta_{N-2}^{BF}}{R_{N-1}^2},
$$

$$
G_{\varphi_i^{BF} \varphi_j^{BF}} = M_{ij} \frac{\cos(\varphi_i^{BF} - \varphi_j^{BF})}{R_i R_j \sin \theta_i^{BF} \sin \theta_j^{BF}} - M_{iN-2} \frac{\cos \varphi_i^{BF}}{R_i R_{N-2} \sin \theta_i^{BF} \sin \theta_{N-2}^{BF}} \\
- M_{jN-2} \frac{\cos \varphi_j^{BF}}{R_j R_{N-2} \sin \theta_j^{BF} \sin \theta_{N-2}^{BF}} \\
+ M_{iN-1} \frac{\cos \varphi_i^{BF} \cot \theta_{N-2}^{BF} - \cos(\varphi_i^{BF} - \varphi_j^{BF}) \cot \theta_j^{BF}}{R_i R_{N-1} \sin \theta_i^{BF}} \\
+ M_{jN-1} \frac{\cos \varphi_j^{BF} \cot \theta_{N-2}^{BF} - \cos(\varphi_i^{BF} - \varphi_j^{BF}) \cot \theta_i^{BF}}{R_j R_{N-1} \sin \theta_j^{BF}} \\
+ M_{N-2N-2} \frac{1}{(R_{N-2} \sin \theta_{N-2}^{BF})^2} + M_{N-2N-1} \frac{\cos \varphi_i^{BF} \cot \theta_i^{BF} + \cos \varphi_j^{BF} \cot \theta_j^{BF} - 2 \cot \theta_{N-2}^{BF}}{R_{N-1} R_{N-2} \sin \theta_{N-2}^{BF}} \\
+ M_{N-1N-1} \frac{\cot \theta_i^{BF} \cot \theta_j^{BF} \cos(\varphi_i^{BF} - \varphi_j^{BF}) + (\cot \theta_{N-2}^{BF})^2}{R_{N-1}^2} \\
- M_{N-1N-1} \frac{\cot \theta_{N-2}^{BF} (\cos \varphi_i^{BF} \cot \theta_i^{BF} + \cos \varphi_j^{BF} \cot \theta_j^{BF})}{R_{N-1}^2}.
$$
Rotational terms:

$$ \Gamma_{zz} = M_{N-2N-2} \frac{1}{(R_{N-2} \sin \theta_{N-2}^{BF})^2} - M_{N-2N-1} \frac{2 \cot \theta_{N-2}^{BF}}{R_{N-2} R_{N-1} \sin \theta_{N-2}^{BF}} + M_{N-1N-1} \frac{(\cot \theta_{N-2}^{BF})^2}{R_{N-1}^2}, $$

$$ \Gamma_{yy} = \Gamma_{xx} = M_{N-1N-1} \frac{1}{R_{N-1}^2}, $$

$$ \Gamma_{xz} = -M_{N-2N-1} \frac{1}{R_{N-2} R_{N-1} \sin \theta_{N-2}^{BF}} + M_{N-1N-1} \frac{\cot \theta_{N-2}^{BF}}{R_{N-1}^2}, $$

$$ \Gamma_{xy} = \Gamma_{yz} = 0. $$

(C5)

Coriolis terms:

$$ C_{xR_j} = -M_{jN-1} \frac{\sin \theta_j^{BF} \sin \varphi_j^{BF}}{R_{N-1}}, $$

$$ C_{yR_j} = M_{jN-1} \frac{\sin \theta_j^{BF} \cos \varphi_j^{BF}}{R_{N-1}}, $$

$$ C_{zR_j} = M_{jN-2} \frac{\sin \theta_j^{BF} \sin \varphi_j^{BF}}{R_{N-2} \sin \theta_{N-2}^{BF}} - M_{jN-1} \frac{\sin \theta_j^{BF} \sin \varphi_j^{BF} \cot \theta_{N-2}^{BF}}{R_{N-1}}, $$

$$ C_{xu_j} = -\sin \theta_j^{BF} \left( -M_{jN-1} \frac{\cos \theta_j^{BF} \sin \varphi_j^{BF}}{R_{N-1} R_j} + M_{N-1N-1} \frac{\sin \varphi_j^{BF}}{R_{N-1}^2} \right), $$

$$ C_{yu_j} = -\sin \theta_j^{BF} \left( M_{jN-1} \frac{\cos \theta_j^{BF} \cos \varphi_j^{BF}}{R_{N-1} R_j} - M_{N-1N-1} \frac{\cos \varphi_j^{BF}}{R_{N-1}^2} \right), $$

$$ C_{zu_j} = -\sin \theta_j^{BF} \left( M_{jN-2} \frac{\cos \theta_j^{BF} \sin \varphi_j^{BF}}{R_{N-2} R_j \sin \theta_{N-2}^{BF}} - M_{jN-1} \frac{\cot \theta_{N-2}^{BF} \cos \theta_j^{BF} \sin \varphi_j^{BF}}{R_{N-1} R_j} \right. \\ \left. - M_{N-2N-1} \frac{\sin \varphi_j^{BF}}{R_{N-2} R_{N-1} \sin \theta_{N-2}^{BF}} + M_{N-1N-1} \frac{\sin \varphi_j^{BF} \cot \theta_{N-2}^{BF}}{R_{N-1}^2} \right), $$

$$ C_{x\varphi_j^{BF}} = -M_{jN-1} \frac{\cos \varphi_j^{BF}}{R_{N-1} R_j \sin \theta_j^{BF}} + M_{N-2N-1} \frac{1}{R_{N-1} R_{N-2} \sin \theta_{N-2}^{BF}} \\ + M_{N-1N-1} \frac{\cos \varphi_j^{BF} \cot \theta_j^{BF} - \cot \theta_{N-2}^{BF}}{R_{N-1}^2}, $$

$$ C_{y\varphi_j^{BF}} = -M_{jN-1} \frac{\sin \varphi_j^{BF}}{R_{N-1} R_j \sin \theta_j^{BF}} + M_{N-1N-1} \frac{\sin \varphi_j^{BF} \cot \theta_j^{BF}}{R_{N-1}^2}, $$

$$ C_{z\varphi_j^{BF}} = M_{jN-2} \frac{\cos \varphi_j^{BF}}{R_{N-2} R_j \sin \theta_{N-2}^{BF} \sin \theta_j^{BF}} - M_{jN-1} \frac{\cot \theta_{N-2}^{BF} \cos \varphi_j^{BF}}{R_{N-1} R_j \sin \theta_j^{BF}} \\ + M_{N-2N-1} \frac{2 \cot \theta_{N-2}^{BF} - \cos \varphi_j^{BF} \cot \theta_j^{BF}}{R_{N-2} R_{N-1} \sin \theta_{N-2}^{BF}} \\ - M_{N-2N-2} \frac{1}{(R_{N-2} \sin \theta_{N-2}^{BF})^2} + M_{N-1N-1} \frac{\cot \theta_{N-2}^{BF} (-\cot \theta_{N-2}^{BF} + \cos \varphi_j^{BF} \cot \theta_j^{BF})}{R_{N-1}^2}. \quad (C6) $$
(ii) Orthogonal coordinates :

In the particular case of orthogonal coordinates, Eqs. (C4, C5, C6) reduce to $\mu_i = \frac{1}{M_{ii}}$, $\delta_{ij} = 0$ if $i \neq j$, 1 otherwise)

Vibrational terms:

$$
\begin{align*}
G_{R_i R_j} &= \frac{\delta_{ij}}{\mu_i}, \quad G_{R_i u_j} = 0, \quad G_{R_i \varphi_j^{BF}} = 0, \\
G_{u_i u_j} &= \frac{1}{\mu_{N-1} R_{N-1}^2} \sin \theta_i^{BF} \sin \theta_j^{BF} \cos(\varphi_i^{BF} - \varphi_j^{BF}) + \frac{\delta_{ij}}{\mu_i R_i^2} (\sin \theta_i^{BF})^2, \\
G_{u_i \varphi_j^{BF}} &= -\frac{1}{\mu_{N-1} R_{N-1}^2} \sin \theta_i^{BF} (\cot \theta_j^{BF} \sin(\varphi_i^{BF} - \varphi_j^{BF}) - \sin \varphi_i^{BF} \cot \theta_{N-2}^{BF}), \\
G_{\varphi_i^{BF} \varphi_j^{BF}} &= \frac{1}{\mu_{N-1} R_{N-1}^2} (\cot \theta_i^{BF} \cot \theta_j^{BF} \cos(\varphi_i^{BF} - \varphi_j^{BF}) + \cot^2 \theta_{N-2}^{BF} \\
&\quad - \cot \theta_{N-2}^{BF} (\cos \varphi_i^{BF} \cot \theta_i^{BF} + \cos \varphi_j^{BF} \cot \theta_j^{BF})) \\
&\quad + \frac{\delta_{ij}}{\mu_i R_i^2 \sin^2 \theta_i^{BF}} + \frac{1}{\mu_{N-2} R_{N-2}^2 \sin^2 \theta_{N-2}^{BF}}. \tag{C7}
\end{align*}
$$

Rotational terms:

$$
\begin{align*}
\Gamma_{zz} &= \frac{1}{\mu_{N-2} R_{N-2}^2 \sin^2 \theta_{N-2}^{BF}} + \frac{\cot^2 \theta_{N-2}^{BF}}{\mu_{N-1} R_{N-1}^2}, \\
\Gamma_{xx} &= \Gamma_{yy} = \frac{1}{\mu_{N-1} R_{N-1}^2}, \\
\Gamma_{xy} &= \Gamma_{yz} = 0, \\
\Gamma_{xz} &= \frac{\cot \theta_{N-2}^{BF}}{\mu_{N-1} R_{N-1}^2}. \tag{C8}
\end{align*}
$$
Coriolis terms:

$$C_{xR_j} = C_{yR_j} = C_{zR_j} = 0,$$

$$C_{xu_j} = -\frac{\sin \theta_j^{BF} \sin \varphi_j^{BF}}{\mu_{N-1} R_{N-1}^2},$$

$$C_{yu_j} = \frac{\sin \theta_j^{BF} \cos \varphi_j^{BF}}{\mu_{N-1} R_{N-1}^2},$$

$$C_{zu_j} = -\frac{\sin \theta_j^{BF} \sin \varphi_j^{BF} \cot \theta_{N-2}^{BF}}{\mu_{N-1} R_{N-1}^2},$$

$$C_{x\varphi_j^{BF}} = \frac{\cos \varphi_j^{BF} \cot \theta_j^{BF} - \cot \theta_{N-2}^{BF}}{\mu_{N-1} R_{N-1}^2},$$

$$C_{y\varphi_j^{BF}} = \frac{\sin \varphi_j^{BF} \cot \theta_j^{BF}}{\mu_{N-1} R_{N-1}^2},$$

$$C_{z\varphi_j^{BF}} = -\frac{1}{\mu_{N-2}(R_{N-2} \sin \theta_{N-2}^{BF})^2} + \frac{\cot \theta_{N-2}^{BF}(-\cot \theta_{N-2}^{BF} + \cos \varphi_j^{BF} \cot \theta_j^{BF})}{\mu_{N-1} R_{N-1}^2}. \quad (C9)$$
APPENDIX D: KEO FOR HCF₃ IN VALENCE COORDINATES

Let us parametrize a penta-atomic system such as fluoroform, CHF₃, by four valence vectors as shown in Figure 17. The corresponding matrix **M** reads

$$ M = \begin{bmatrix} \frac{1}{m_C} + \frac{1}{m_H} & \frac{1}{m_C} & \frac{1}{m_C} & \frac{1}{m_C} \\ \frac{1}{m_C} & \frac{1}{m_C} + \frac{1}{m_F} & \frac{1}{m_C} & \frac{1}{m_C} \\ \frac{1}{m_C} & \frac{1}{m_C} & \frac{1}{m_C} + \frac{1}{m_F} & \frac{1}{m_C} \\ \frac{1}{m_C} & \frac{1}{m_C} & \frac{1}{m_C} & \frac{1}{m_C} + \frac{1}{m_F} \end{bmatrix}, \qquad (D1) $$

and if the following volume element is used in normalizing the wave functions

$$ d\tau' = dR_4 dR_3 dR_2 dR_1 du_3 du_2 du_1 d\varphi_2^{BF} d\varphi_1^{BF}, \qquad (D2) $$

the KEO of Eqs. (111,C4) yields

$$ \begin{aligned} \hat{T} = \hat{T}_{R_i R_j} + V_{extra} + \hat{T}_{\varphi_i \varphi_j} & \\ + \hat{T}_{R_i u_j} + \hat{T}_{R_i \varphi_j} + \hat{T}_{u_i u_j} + \hat{T}_{u_i \varphi_j}, & \end{aligned} \qquad (D3) $$

with

$$ \begin{aligned} \hat{T}_{R_i R_j} + V_{extra} = & - \sum_{i=1}^{4} \frac{1}{2\mu_i} \frac{\partial^2}{\partial R_i^2} - \sum_{i=1}^{3} \frac{u_i}{m_C} \left( \frac{\partial^2}{\partial R_i \partial R_4} + \frac{1}{R_i R_4} \right) \\ & - \frac{v_1 v_2 \cos(\varphi_1 - \varphi_2) + u_1 u_2}{m_C} \left( \frac{\partial^2}{\partial R_1 \partial R_2} + \frac{1}{R_1 R_2} \right) \\ & - \sum_{i=1}^{2} \frac{v_i v_3 \cos \varphi_i + u_i u_3}{m_C} \left( \frac{\partial^2}{\partial R_i \partial R_3} + \frac{1}{R_i R_3} \right), \end{aligned} \qquad (D4) $$

and with (we use $\varphi_i$ instead of $\varphi_i^{BF}$ and $\theta_i$ instead of $\theta_i^{BF}$ to lighten the notations)
$$
\begin{equation}
\begin{aligned}
\hat{T}_{\varphi_i \varphi_j} = & - \sum_{i=1}^{2} \frac{\partial}{\partial \varphi_i} \left[ \frac{M_{ii}}{2R_i^2 v_i^2} - \frac{\cos \varphi_i}{m_C v_i R_i R_3 v_3} + \frac{\cos \varphi_i \cot \theta_3 - \cot \theta_i}{m_C v_i R_i R_4} \right. \\
& \left. \qquad + \frac{M_{33}}{2R_3^2 v_3^2} + \frac{\cos \varphi_i \cot \theta_i - \cot \theta_3}{m_C v_3 R_3 R_4} + \frac{\cot^2 \theta_i + \cot^2 \theta_3}{2\mu_4 R_4^2} - \frac{\cot \theta_3 \cos \varphi_i \cot \theta_i}{\mu_4 R_4^2} \right] \frac{\partial}{\partial \varphi_i} \\
& - \frac{\partial}{\partial \varphi_1} \left[ \frac{\cos (\varphi_2 - \varphi_1)}{2m_C R_1 R_2 v_1 v_2} + \frac{M_{33}}{2R_3^2 v_3^2} + \frac{\sum_{i=1,2} \cos \varphi_i \cot \theta_i - 2 \cot \theta_3}{2m_C R_4 R_3 v_3} \right. \\
& \left. \qquad + \sum_{i,j=1,2}^{i \neq j} \left( \frac{\cos \varphi_i \cot \theta_3 - \cos (\varphi_2 - \varphi_1) \cot \theta_j}{2m_C R_i R_4 v_i} - \frac{\cos \varphi_i}{2m_C R_i R_3 v_i v_3} \right) \right. \\
& \left. \qquad + \frac{\cot \theta_2 \cot \theta_1 \cos (\varphi_2 - \varphi_1) + \cot^2 \theta_3}{2\mu_4 R_4^2} - \frac{\cot \theta_3 (\sum_{i=1,2} \cos \varphi_i \cot \theta_i)}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial \varphi_2} \\
& - \frac{\partial}{\partial \varphi_2} \left[ \frac{\cos (\varphi_2 - \varphi_1)}{2m_C R_1 R_2 v_1 v_2} + \frac{M_{33}}{2R_3^2 v_3^2} + \frac{\sum_{i=1,2} \cos \varphi_i \cot \theta_i - 2 \cot \theta_3}{2m_C R_4 R_3 v_3} \right] \\
& + \sum_{i,j=1,2}^{i \neq j} \left( \frac{\cos \varphi_i \cot \theta_3 - \cos (\varphi_2 - \varphi_1) \cot \theta_j}{2m_C R_i R_4 v_i} - \frac{\cos \varphi_i}{2m_C R_i R_3 v_i v_3} \right) \\
& \left. \qquad + \frac{\cot \theta_2 \cot \theta_1 \cos (\varphi_2 - \varphi_1) + \cot^2 \theta_3}{2\mu_4 R_4^2} - \frac{\cot \theta_3 (\sum_{i=1,2} \cos \varphi_i \cot \theta_i)}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial \varphi_1},
\end{aligned}
\tag{D5}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\hat{T}_{R_i u_j} = & - \sum_{i=1}^{3} \left( \frac{\partial}{\partial R_i} \frac{v_i^2}{2m_C R_4} \frac{\partial}{\partial u_i} + \frac{\partial}{\partial u_i} \frac{v_i^2}{2m_C R_4} \frac{\partial}{\partial R_i} \right) - \sum_{i,j=1,2; i \neq j} \frac{\partial}{\partial R_i} \left[ \right. \\
& \left. \frac{v_j (-v_i u_j \cos(\varphi_i - \varphi_j) + u_i v_j)}{2m_C R_j} \frac{\partial}{\partial u_j} + \frac{\partial}{\partial u_j} \frac{v_j (-v_i u_j \cos(\varphi_i - \varphi_j) + u_i v_j)}{2m_C R_j} + \frac{v_i v_j \cos(\varphi_i - \varphi_j)}{2m_C R_4} \frac{\partial}{\partial u_j} \right. \\
& + \frac{\partial}{\partial u_j} \frac{v_i v_j \cos(\varphi_i - \varphi_j)}{2m_C R_4} + \frac{v_j (-v_i u_j \cos(\varphi_i + u_i v_3))}{2m_C R_3} \frac{\partial}{\partial u_j} + \frac{\partial}{\partial u_j} \frac{v_j (-v_i u_j \cos(\varphi_i + u_i v_3))}{2m_C R_3} \\
& + \frac{\partial}{\partial u_j} \frac{v_i v_3 \cos(\varphi_i - \varphi_j)}{2m_C R_4} \frac{\partial}{\partial u_3} + \frac{\partial}{\partial u_3} \frac{v_i v_3 \cos(\varphi_i - \varphi_j)}{2m_C R_4} \left. - \frac{\partial}{\partial R_3} \left[ \frac{v_i (-v_3 u_i \cos(\varphi_i + u_3 v_i))}{2m_C R_i} \frac{\partial}{\partial u_i} + \frac{\partial}{\partial u_i} \frac{v_i (-v_3 u_i \cos(\varphi_i + u_3 v_i))}{2m_C R_i} \right] \right. \\
& \left. + \frac{\partial}{\partial u_i} \frac{v_3 v_i \cos(\varphi_i - \varphi_j)}{2m_C R_4} \frac{\partial}{\partial u_i} + \frac{\partial}{\partial u_i} \frac{v_3 v_i \cos(\varphi_i - \varphi_j)}{2m_C R_4} \right] - \sum_{i=1}^{3} \frac{\partial}{\partial R_4} \left( \frac{v_i^2}{2m_C R_i} \frac{\partial}{\partial u_i} + \frac{\partial}{\partial u_i} \frac{v_i^2}{2m_C R_i} \right),
\end{aligned}
\tag{D6}
\end{equation}
$$
$$
\begin{aligned}
\hat{T}_{R_i \varphi_j} = & - \sum_{i=1}^{2} \frac{\partial}{\partial R_i} \left[ \left( \frac{v_i \sin \varphi_i \cot \theta_3}{2m_C R_4} - \frac{v_i \sin \varphi_i}{2m_C R_3 v_3} \right) \frac{\partial}{\partial \varphi_i} + \frac{\partial}{\partial \varphi_i} \left( \frac{v_i \sin \varphi_i \cot \theta_3}{2m_C R_4} - \frac{v_i \sin \varphi_i}{2m_C R_3 v_3} \right) \right] \\
& - \sum_{i,j=1,2}^{i \neq j} \frac{\partial}{\partial R_i} \left[ \left( \frac{v_i \sin (\varphi_i - \varphi_j)}{2m_C R_j v_j} - \frac{v_i \sin \varphi_i}{2m_C R_3 v_3} + \frac{v_i (\sin \varphi_i \cot \theta_3 + \cot \theta_j \sin (\varphi_j - \varphi_i))}{2m_C R_4} \right) \frac{\partial}{\partial \varphi_j} \right. \\
& \left. + \frac{\partial}{\partial \varphi_j} \left( \frac{v_i \sin (\varphi_i - \varphi_j)}{2m_C R_j v_j} - \frac{v_i \sin \varphi_i}{2m_C R_3 v_3} + \frac{v_i (\sin \varphi_i \cot \theta_3 + \cot \theta_j \sin (\varphi_j - \varphi_i))}{2m_C R_4} \right) \right] \\
& - \sum_{i=1}^{2} \frac{\partial}{\partial R_3} \left[ \left( \frac{v_3 \sin \varphi_i \cot \theta_i}{2m_C R_4} - \frac{v_3 \sin \varphi_i}{2m_C R_i v_i} \right) \frac{\partial}{\partial \varphi_i} + \frac{\partial}{\partial \varphi_i} \left( \frac{v_3 \sin \varphi_i \cot \theta_i}{2m_C R_4} - \frac{v_3 \sin \varphi_i}{2m_C R_i v_i} \right) \right],
\end{aligned}
\tag{D7}
$$

$$
\begin{aligned}
\hat{T}_{u_i u_j} = & - \sum_{i=1}^{3} \frac{\partial}{\partial u_i} v_i^2 \left( \frac{M_{ii}}{2R_i^2} + \frac{M_{44}}{2R_4^2} - \frac{u_i}{m_C R_i R_4} \right) \frac{\partial}{\partial u_i} \\
& - \frac{\partial}{\partial u_1} v_1 v_2 \left[ \frac{u_1 u_2 \cos (\varphi_1 - \varphi_2)}{2m_C R_1 R_2} + \frac{v_1 v_2}{2m_C R_1 R_2} - \sum_{i=1,2} \frac{u_i \cos (\varphi_1 - \varphi_2)}{2m_C R_i R_4} + \frac{\cos (\varphi_1 - \varphi_2)}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial u_2} \\
& - \frac{\partial}{\partial u_2} v_1 v_2 \left[ \frac{u_1 u_2 \cos (\varphi_1 - \varphi_2)}{2m_C R_1 R_2} + \frac{v_1 v_2}{2m_C R_1 R_2} - \sum_{i=1,2} \frac{u_i \cos (\varphi_1 - \varphi_2)}{2m_C R_i R_4} + \frac{\cos (\varphi_1 - \varphi_2)}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial u_1} \\
& - \sum_{i=1}^{2} \left\{ \begin{aligned}
& \frac{\partial}{\partial u_i} v_i v_3 \left[ \frac{u_i u_3 \cos \varphi_i + v_i v_3}{2m_C R_i R_3} - \frac{u_i \cos \varphi_i}{2m_C R_i R_4} - \frac{u_3 \cos \varphi_i}{2m_C R_3 R_4} + \frac{\cos \varphi_i}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial u_3} \\
& + \frac{\partial}{\partial u_3} v_i v_3 \left[ \frac{u_i u_3 \cos \varphi_i + v_i v_3}{2m_C R_i R_3} - \frac{u_i \cos \varphi_i}{2m_C R_i R_4} - \frac{u_3 \cos \varphi_i}{2m_C R_3 R_4} + \frac{\cos \varphi_i}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial u_i}
\end{aligned} \right\},
\end{aligned}
\tag{D8}
$$

and
$$
\begin{equation}
\begin{aligned}
\hat{T}_{u_i \varphi_j} = & - \sum_{i,j=1,2; i \neq j} \left\{ \frac{\partial}{\partial u_i} v_i \left[ \frac{u_i \sin (\varphi_j - \varphi_i)}{2m_C R_i R_j v_j} + \frac{u_i \sin \varphi_i}{2m_C R_i R_3 v_3} - \frac{u_i (\cot \theta_j \sin (\varphi_j - \varphi_i) + \cot \theta_3 \sin \varphi_i)}{2m_C R_i R_4} \right. \right. \\
& \left. \left. + \frac{\sin (\varphi_i - \varphi_j)}{2m_C R_j R_4 v_j} - \frac{\sin \varphi_i}{2m_C R_3 R_4 v_3} - \frac{\cot \theta_j \sin (\varphi_i - \varphi_j) - \sin \varphi_i \cot \theta_3}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial \varphi_j} \right. \\
& + \frac{\partial}{\partial \varphi_j} v_i \left[ \frac{u_i \sin (\varphi_j - \varphi_i)}{2m_C R_i R_j v_j} + \frac{u_i \sin \varphi_i}{2m_C R_i R_3 v_3} - \frac{u_i (\cot \theta_j \sin (\varphi_j - \varphi_i) + \cot \theta_3 \sin \varphi_i)}{2m_C R_i R_4} \right. \\
& \left. + \frac{\sin (\varphi_i - \varphi_j)}{2m_C R_j R_4 v_j} - \frac{\sin \varphi_i}{2m_C R_3 R_4 v_3} - \frac{\cot \theta_j \sin (\varphi_i - \varphi_j) - \sin \varphi_i \cot \theta_3}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial u_i} \right\} \\
& - \sum_{i=1}^2 \left\{ \frac{\partial}{\partial u_3} v_3 \left[ \frac{u_3 \sin \varphi_i}{2m_C R_i R_3 v_i} - \frac{u_3 \cot \theta_i \sin \varphi_i}{2m_C R_3 R_4} - \frac{\sin \varphi_i}{2m_C R_i R_4 v_i} + \frac{\sin \varphi_i \cot \theta_i}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial \varphi_i} \right. \\
& \left. + \frac{\partial}{\partial \varphi_i} v_3 \left[ \frac{u_3 \sin \varphi_i}{2m_C R_i R_3 v_i} - \frac{u_3 \cot \theta_i \sin \varphi_i}{2m_C R_3 R_4} - \frac{\sin \varphi_i}{2m_C R_i R_4 v_i} + \frac{\sin \varphi_i \cot \theta_i}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial u_3} \right\} \\
& - \sum_{i=1}^2 \left\{ \frac{\partial}{\partial u_i} v_i \left[ \frac{u_i \sin \varphi_i}{2m_C R_i R_3 v_3} - \frac{u_i \cot \theta_3 \sin \varphi_i}{2m_C R_i R_4} - \frac{\sin \varphi_i}{2m_C R_3 R_4 v_3} + \frac{\sin \varphi_i \cot \theta_3}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial \varphi_i} \right. \\
& \left. + \frac{\partial}{\partial \varphi_i} v_i \left[ \frac{u_i \sin \varphi_i}{2m_C R_i R_3 v_3} - \frac{u_i \cot \theta_3 \sin \varphi_i}{2m_C R_i R_4} - \frac{\sin \varphi_i}{2m_C R_3 R_4 v_3} + \frac{\sin \varphi_i \cot \theta_3}{2\mu_4 R_4^2} \right] \frac{\partial}{\partial u_i} \right\}.
\end{aligned}
\tag{D9}
\end{equation}
$$

We recall that

$$ u_i = \cos \theta_i \tag{D10} $$

and that

$$ \cot \theta_i = \frac{u_i}{v_i} \tag{D11} $$

with

$$ v_i = \sqrt{1 - u_i^2} = \sin \theta_i. \tag{D12} $$
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