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A Concise Method for Kinetic Energy Quantisation

We present a straightforward method for obtaining exact classical and quantum molecular Hamiltonians in terms of arbitrary coordinates. As compared to other approaches the resulting expression are rather compact, the physical meaning of each quantity is quite transparent, and in some cases the calculation effort will be greatly reduced. We also investigate systems with constrains to find the suggested method to be applicable in contrast to most conventional approaches to kinetic energy operators which cannot directly be applied to constrained systems. Two examples are discussed in detail, the monohydrated hydroxide anion and the protonated ammonia dimer.


I. INTRODUCTION
Theoretical modeling of moelcular reaction dynamics requires to have at hand a Hamiltonian. For few-atom systems this may involve defining the potential energy surfaces (PES) explicitly in terms of properly chosen coordinates such as bond length and angles or distance vectors. In the latter case, for instance, it has been shown that PES in full-dimensionality including proper account of exchange symmetry can be generated for systems like the five-atomic H 3 O − 2 [1] or the seven-atomic H 5 O + 2 [2]. With increasing dimensionality this approach is no longer feasible due to the exponential scaling of the computational effort for explicit calculation of the PES on a numerical grid. Here, reaction surface concepts have been proven to be a versatile compromise, mapping out that PES region which is relevant for the considered reaction by combining suitable large amplitude reaction coordinates with orthogonal harmonic motions [3,4,5,6,7].
Besides the PES the kinetic energy operator (KEO) needs to be known in terms of the chosen coordinate system. In principle, kinetic energy quantisation had been discussed by Podolsky in terms of intricate tensor analysis right after the foundation of the Schrödinger equation had been laid [8]. Later studies are to some extent based on the same mathematical techniques [9,10]. Approximate KEOs using normal mode coordinates are widely adopted which go back to contributions by Eckart [11], Wilson [12], and Watson [13]. In particular, the Eckart equations [11] enable one to determine an orientation of molecule fixed axes suitable for using normal modes. The details of the derivation of a rovibrational Hamiltonian can be found in Ref. [12]. A significant simplification of this rovibrational Hamiltonian in terms of normal coordinates has been achieved by Watson [13,14]. In fact the Watson Hamiltonian served as a starting point for many later investigations and is implemented, e.g., in the MULTIMODE program of Bowman, Carter, and coworkers [15]. In principle normal mode coordinates provide a very compact representation of the PES which is taylored, however, to the stationary point for which they have been defined. For large amplitude motions away from this stationary point the above mentioned reaction surface approach or the use of general curvilinear coordinates are more suitable. A detailed discussion of advantages and disadvantages of various coordinates for rovibrational Hamiltonians has been presented, e.g., in Ref. [16].
In recent years Gatti and coworkers have developed a general scheme for obtaining a KEO using vector parameterization, i.e., the non-trivial coordinates are expressed with N − 1 real vectors for system of N atoms [17,18,19]. This scheme is quite successful and convenient for a full dimensional description [20]. However, often one needs a reduced dimensional description leading to a problem with imposed constraints [21,22]. Generally speaking in this case the coordinates can not be expressed as components of real vectors.
Finally, we should point out that a numerical alternative to the analytical determination of the KEO has been suggested in Ref. [23].
In the present work we introduce a simple physical method of kinetic energy quantisation which directly quantises the classical Lagrange/Hamilton functions. The formal theory is presented in Section II. Here it is also shown that our method can be directly applied to constrained systems. In Section III the approach is applied to two molecular systems, that is, H 3 O − 2 which is treated in full dimensionality and N 2 H + 7 for which certain constraints are applied.

A. Formal Development
In the following we focus on the most common cases where the kinetic energy includes purely quadratic terms and the potential energy does not depend on the velocity, although a generalisation would be straightforward. Further we adopt the laboratory reference frame (LRF). The classical Lagrange/Hamiltonian in terms of arbitrary variables can be written as where Q and P are single column vectors of the generalised coordinates and corresponding conjugate momenta, respectively. The generalised momentum vector is defined as The operator for each component of the momentum (P † X ) j = (P X ) j = −i ∂/∂X j is Hermitian. For non-Cartesian coordinates we define the momentum operator in the same way:P To obtain the quantum KEO in terms of non-Cartesian coordinates, we must exploit the coordinate transform relations, Q = Q(X). The following relations can be derived easilẏ where the elements of the transformation matrices are defined as (∂Q/∂X) ij = ∂Q i /∂X j and (∂X/∂Q) ij = ∂X i /∂Q j . The quantum KEO in terms of non-Cartesian coordinates can be obtained by the simple coordinate transform from Cartesian coordinateŝ where M X is the diagonal Cartesian mass matrix consisting of real mass associated with each Cartesian coordinate.
where the notation [Â,B] means the commutator of two operators. Eq. (5) shows the non-Hermiticity of generalised momenta associated with non-Cartesian coordinates. Additional terms appear due to the non-commutability of the momenta and the transformation matrix. Using the basic commutator [X j , (P X ) k ] = i δ jk , Eq. (5) can be simplified to (see also Ref. [24]) Here, the superscript • means a differential operator inside the bracket can not operate on functions outside, in other words, the result is just normal function of coordinates.
Eq. (6) clearly shows the relation between a momentum operator and its Hermitian conjugate. Apparently, any set of functions of coordinates, {f j (Q j )}, can be multiplied from the left to the derivative operator in Eq. (2) yielding different schemes of momenta quantisation, i.e., (P ) j = −i · f j (Q j ) · ∂/∂Q j constitutes also an acceptable scheme. Since these momentum operators are in general non-Hermitian, we cannot simply set f j (Q j ) = 1. For different schemes of momenta quantisation, the HCMOs will vary correspondingly to keep the KEO invariant. But, this enables us to optimise momenta quantisation schemes. In other words with proper functions one can obtain desired forms of momentum operators, e.g., symmetric forms.
With Eq. (4) and Eq. (5) we get the general scheme of constructing KEOs in terms of arbitrary coordinates. However, the final structure seems to be complicated and actually it can be simplified. To this end we compare Eq. (4) and Eq. (1) and take into account the invariance of the classical kinetic energy SinceQ can be any vector we must have the relation M = (∂X/∂Q) † M X ∂X/∂Q.
Calculating the inverse of both sides we immediately see that the complicated central (1). The final KEO is therefore simplified asT Therefore we can identify a concise and physically transparent scheme to construct the KEO: 1. Get the classical kinetic energy and make sure the mass matrix is symmetric, so that one has the same structure as Eq. (1), T (Q) = 1 2Q † MQ. The first step is quite familiar and one can choose arbitrary coordinates to get the classical kinetic energy. Actually we will introduce a useful partition method in Section II C which will significantly simplify this issue. The remaining three steps are quite straightforward to follow. Apparently, the major effort is in the second step, i.e. determining the HCMOs according to Eq. (6). However, for the familiar spherical coordinates the HCMOs are well-known which leads to a great simplification as shown in Section II D.
Apart from the KEO the volume element of integration is also of importance. Since we start from Cartesian coordinates and all that we have done is a coordinate transformation, the Euclidean normalisation remains correct:

B. Systems with Constraints
Reduced dimensional descriptions are always necessary for large systems. In the following we will give the general description for systems with constraints. Considering a system with some active coordinates Q 1 and some frozen coordinates Q 0 , the full dimensional coordinates and corresponding conjugate momenta read The constraining conditions are given byQ 0 = 0. Thus we can obtain the constrained classical kinetic energy where M ij are corresponding sub-matrices.
To get the quantum KEO we have to rewrite the constraint conditions in terms of momenta. According to the definition of momenta it is not difficult to find the following The final result is quite compact. Comparing Eq. (11) with Eq. (13) we observe that the same procedure mentioned in the full dimensional case can be followed provided we only consider the active coordinates and completely ignore the frozen ones from the very beginning when we generate the classical Lagrangian. This is an attractive point since when we need to deal with constrained systems the present scheme only requires the classical Lagrangian which can be obtained by traditional methods.

C. A General Method for Partitioning the Classical Kinetic Energy
So far we have given the general theory of kinetic energy quantization starting from the classical Lagrangian. In the following we will introduce a method for obtaining the classical kinetic energy with a partitioning technique which will greatly simplify the problem in most cases. In general it is quite convenient to divide a large system into small subsystems especially when a subsystems has certain symmetry. If we arbitrarily divide a system into N parts the kinetic energy is a sum of the N subsystems. According to the König theorem, where T B A is the kinetic energy of the part A with respect to the reference frame defined by B. Here 0 is the laboratory reference frame and (C) is the centre of mass reference frame.
In the special case when N = 2 the following relation can be obtained where (C i ) is the centre of mass of the ith part and T is the kinetic energy of the centre of mass of the first part with respect to the centre of mass of the second part. If the two parts are both mass points Eq. (15) can be simplified as follows which is a familiar result from two-body mechanics. By exploiting Eq. (15) repeatedly we can easily express the total kinetic energy in terms of kinetic energies of subsystems which are much easier to obtain as will be seen in the following.

D. Hermitian Conjugates of Momentum Operators
As mentioned in Section II A, the most tedious task for generating a KEO is to derive the expressions for the HCMOs. As an example let us consider a system described by the 3D spherical coordinates {R, θ, ϕ}. The coordinate transformation between 3D spherical coordinates and 3D Cartesian ones {x, y, z} is defined as Here, we only give the corresponding HCMOs leading to the well-known KEO (for details, So far our considerations have been concerned with the LRF. However, in order to describe a molecule in terms of its natural motions, e.g., bond lengths and bond angles, it is usually more convenient to use one or more molecular reference frames (MRFs). In the following we will discuss this point and derive results for HCMOs associated with MRF spherical coordinates. Consider the LRF and a MRF defined by sets of unit vectors {e x , e y , e z } and {e x ′ , e y ′ , e z ′ }, respectively. The relation between the LRF and the MRF is just an orthogonal transformation characterised by the three Euler angles {ϑ, φ, χ} where α = x, y, z and U α is a rotation around e α . (The expression for the rotational transformation matrix U α and more details on the following derivations can be found in the Appendix V A.) Now let us consider a vector R j characterised by three spherical coordinates {R j , θ j , ϕ j } in the MRF. According to Appendix V A, Eq. (46), we can express R j as In Appendix V A we also confirm that the momentum operators associated with MRF Cartesian coordinates are Hermitian (cf. Eq. (63)). This result is quite important as it shows that one does not need to take the effort to derive the expressions for HCMOs provided that one uses spherical coordinates, Cartesian coordinates or combinations of both, no matter whether they are defined in the LRF or MRFs. In general rotational excitation energies are quite small as compared with vibrational ones suggesting an approximate separation. For the following discussion it will be convenient to express the KEO as sum of rotational (three angles) and vibrational (3N − 6 other coordinates) parts as well as their coupling. Specifically, we exploit the 3D vector Q rot defined as

E. Angular Momentum and Rotation
containing the three Euler angles which connect the LRF and MRF according to Eq. (19).
The kinetic energy and total angular momentum J are defined as WithQ rot being the angular velocity of MRF the velocities can be re-expressed aṡ whereṘ ′ i is the velocity of R i measured in the MRF. With the help of Eq. (24) and some vector algebra we can rewrite the kinetic energy and angular momentum as Using Eqs. (25) it is straightforward to derive the following relation which is exactly the definition of the generalised momentum vector P rot associated with the three Euler angles.
Now we can draw the following important conclusion. If the set of coordinates contains the three Euler angles {ϑ, φ, χ}, the total angular momentum vector is just the generalised momentum vector associated with the three Euler angles However, since the direction of angular velocities {eθ, eφ, eχ} are complicated, it is better to transform the expressions to the Cartesian components in the LRF [12], i.e. be considered as being much smaller than the accuracy achievable by the quantum chemistry and quantum dynamics methods for obtaining the spectra. Therefore the following considerations will assume that rotational motion can be neglected, that is, we will set

III. APPLICATIONS
In the following we will give two applications of the present approach. First, we will consider the full-dimensional KEO for H 3 O − 2 employing polyspherical coordinates as introduced in more detail in Appendix V B. Second, we consider N 2 H + 7 under the assumption of certain constraints and using bond length and angle coordinates.
In this section we will give the full dimensional KEO for the monohydrated hydroxide ion, H 3 O − 2 , which had been used in the study of different isotopomers in Ref. [25]. The coordinates are defined in the MRF given by Eq. (19). The four Jacobi vectors shown in Fig. 1 are adopted. The three Euler angles are chosen in the same way as introduced in Section V B, i.e., the MRF spherical coordinates R 4 and R 1 are (R 4 , 0, 0) and (R 1 , θ 1 , 0), respectively. The other two vectors can be characterised by their spherical coordinates in the MRF (R j , θ j , ϕ j ) (j = 2, 3). We can calculate the G matrix according to Eq. (76) and the required matrix elements are shown in Eq. (78). Therefore, we can directly write down the exact 9D KEO for total angular momentum J = 0 as 2T vib =P † vib G vibP vib with the HCMOs given in Section II D. However, this KEO will contain a large number of terms describing the angular momentum coupling between the shared Hydrogen and the (OH) 2 fragments. In order to simplify the KEO we will derive an approximate KEO which has the advantage that these couplings do not appear.
The basic idea is to introduce a new MRF and express R 3 with Cartesian coordinates (x, y, z). The remaining six coordinates are spherical coordinates in the old MRF (R 1 , R 2 , θ 1 , θ 2 , R 4 , ϕ = ϕ 2 ). The new MRF is associated with the old MRF by a rotation of an angle ηϕ around the R 4 (the z-axis). The spherical coordinates for R 3 in the new is defined to minimise the Coriolis type couplings involving the central Hydrogen. Here, µ 1 (µ 2 ) is the reduced mass associated with the Jacobi vector R 1 (R 2 ). After transforming the spherical coordinates (R 3 , θ 3 ,φ 3 ) to the Cartesian ones (x, y, z) we can obtain the KEO for the central Hydrogen where µ 3 is the reduced mass associated with the Jacobi vector R 3 and Coriolis type terms are ignored. Since the e ′ z is defined along the direction of R 4 , the z coordinate corresponds to the shared Hydrogen stretching vibration.
For the other 6 DOFs of the (OH) 2 fragment, the 6D KEO is written in terms of the G vib matrix elements Eq. (78). Here we take the angular momentum of the (OH) 2 fragment as total angular momentum since the angular momentum of the shared Hydrogen is negligible. Combining both parts we finally obtain the 9D KEO where we introduce the new coordinates u i = cos θ i (i = 1, 2). For the simulation an additional normalisation transform was performed to reduce the numerical effort. This gives the following 9D KEO: with The non-Euclidean normalisation according to the volume element is d τ = dR 1 dR 2 dR 4 dxdydzdu 1 du 2 dϕ. The reduced masses for different isotopomers are defined as follows: In this section we give an example for a KEO for a system with constraints stressing again that our method is invariant for reduced dimensional descriptions. Specifically we focus on the protonated ammonia dimer, N 2 H + 7 , whose IR spectrum in the range < 1500 cm −1 had been discussed in Ref. [26] on the basis of a 5D model including only the asymmetric stretching vibration of the shared proton. Subsequently, the model was extended to account for the two degenerate bendings as well in Ref. [27]. Here we will derive the 5D model and comment on its extension. A reasonable reduced description for the spectrum which is influenced by the shared proton motion should take into account the following two conditions: (i) experimental data suggest that the relevant energy range is presumably below 1000 cm −1 , but might extend into the < 1500 cm −1 range due to combination bands and (ii) symmetry selection rules dominate the anharmonic couplings especially in this low-energy range. Therefore, we first assume that the C 3 symmetry of the N 2 H 6 fragment, i.e., excluding the central proton, will not be broken. Second, the length of the N-H covalent bonds shall be fixed. Notice that this neglects the bending motion of the shared proton perpendicular to the line connecting the two nitrogen atoms. This is justified by the different symmetry and the lower anharmonicity of these two modes as compared to the proton motion along the N-N line. These constraints leave five internal coordinates to describe the system as shown in Fig. 2, i.e., the shared proton stretching with respect to the center of mass of the rest N 2 H 6 fragment, z, the relative motion of the centres of mass of the two ammonia, R, the umbrella type motion of the two ammonia, θ 1 and θ 2 , and the rotation (torsion) of the NH 3 fragments with respect to each other, ϕ.
For these five active coordinates the classical kinetic energy with all other internal coordinates frozen can be obtained by exploiting Eq. (15) repeatedly where (AB) is the center of mass of AB. It is straightforward to obtain each term in above equation where . The orientation angle of each individual ammonia is denoted by ϕ 1,2 and only the difference between them is the torsion shown in Fig. 2 and R 0 is the free N-H covalent bond length. Thus the final kinetic energy is where I vib (θ) = I 0 (cos 2 θ + m N sin 2 θ)/(3m H + m N ), I rot (θ) = I 0 sin 2 θ and I 0 = 3m H R 2 0 . The last two terms in above equation can be rewritten and after separation of the global rotation we finally simplify the kinetic energy as where the reduced moment of inertia for the torsion is I tor (θ 1 , θ 2 ) = So far we did not consider the Coriolis type couplings with total angular velocity.
However, due to the C 3 symmetry, the total angular momentum equals to zero is equivalent to total angular velocity equals to zero for this specific case since none of the adopted coordinates contributes to total angular momentum. Therefore the Coriolis type couplings are zero.
According to the general procedure detailed in Section II A we can obtain the following Inspecting the adopted coordinates we notice that one of them is Cartesian and the other four are spherical coordinates defined in MRF. Therefore, according to Section II D the HCMOs are given byP † Hence the final KEO and the Euclidean normalisation condition are given as followŝ In Ref. [27] we have extended this model by including the shared proton bending coordinates x and y as shown in Fig. 2. In terms of the KEO this is straightforward since in Eq. (34) we simply have to set which is carried through the derivation to give a corresponding contribution to Eq. (39).
Compared to the experiment this gives a rather good description of the bending fundamental transitions as well as an improvement concerning the shared proton fundamental stretching transition.

IV. SUMMARY
In summary we have presented a new method for obtaining kinetic energy operators which involves a straightforward quantisation of the classical Lagrange/Hamilton function.
It is based on the notion of hermitian conjugate momentum operators whose derivation presents the major effort for practical applications. An important point of our approach is its validity for systems with constraints. This makes it particularly attractive for studying larger systems in reduced dimensionality.
We have applied our approach to the case of a general system described by polyspherical coordinates and showed the equivalence of the kinetic energy operator with that obtained by the method of Gatti and coworkers. Afterwards we discussed two specific applications to charged cluster systems having strong hydrogen bonds. The strong coupling between different coordinates as well their structural floppiness makes it necessary to perform high dimensional quantum dynamics simulations as done in Refs. [25,26,27] on the basis of the kinetic energy operators derived in Section III.

A. Hermitian Conjugate Momentum Operators for Spherical Coordinates
The derivation of the expressions for the HCMOs in LRF is the most tedious part of the present method and it becomes even more involved if coordinates defined in MRFs are used. In this Appendix we will first summarise the relevant relations between the LRF and MRFs and subsequently turn to the derivation of HCMOs.
The new reference frame generated by applying U y (ϑ) and U z (φ) to the LRF has the same z axis as the MRF, i.e., the only difference between the two reference frames is just a rotation by χ around e z ′ . Therefore, the MRF can be obtained by applying three successive rotations U y (ϑ), U z (φ) and U z ′ (χ) to the LRF, namely where U z ′ (χ) means rotating χ around e z ′ . The matrix representation for U z ′ (χ) in the MRF is the same with U z (χ) in the LRF. Since the third rotation U z ′ (χ) in Eq. (42) has no effects on e z ′ we actually have e z ′ = U z (φ)U y (ϑ)e z . Exploiting the rules between vector and operator transformations we can obtain the expression for U z ′ (χ) in the LRF Consequently Eq. (42) can be rewritten as Eq. (44) tells us that an equivalent way to obtain the MRF is to apply three successive rotations U z (χ), U y (ϑ) and U z (φ) to the LRF. Now let us consider vectors R j characterised by spherical coordinates (R j , θ j , ϕ j ) in the MRF (j = 1 − N). With Eq. (44) and the rule between basis vectors and components transformations we can obtain the Cartesian components of these vectors in LRF where R jα and R jα ′ are Cartesian components of R j in the LRF and the MRF, respectively. We can rewrite Eq. (45) in more formally as Eq. (46) is a vector equation therefore it is also valid for an arbitrary reference frame.
Now we turn to the main task, that is, deriving the Hermitian conjugate of momentum operators. This can be done step by step according to Eq. (6) with the help of the coordinate transformation Eq. (45). From Eq. (45) we can see that (θ j , ϕ j ) only appear in the Cartesian components of R j . This greatly simplifies the expression for the Hermitian conjugates of momentum operators as followŝ Here the three Euler angles in Eq. (45) are purely parameters. To calculate the partial derivatives we first give some useful relations following from the orthogonality of the transformation Eq. (45) where α = x, y, z and β ′ = x ′ , y ′ , z ′ . Note here all the derivatives {∂R jβ ′ /∂R jα } are parameters only depending on the three Euler angles.
Let us first derive the expression forP † R j . It is straightforward to get the following derivatives of MRF Cartesian coordinates where R 2 j = R 2 jx + R 2 jy + R 2 jz = R 2 jx ′ + R 2 jy ′ + R 2 jz ′ . Based on Eq. (49) we can obtain the derivatives of LRF Cartesian coordinates with respect to R j and derive the final result where Eq. (48) has been used in the fourth step. Finally we can get the HCMO associated with R j as followsP † Next we will derive the expression forP † θ j following the same procedure. The derivatives of MRF Cartesian coordinates read Based on Eq. (52) we can obtain the derivatives of Cartesian coordinates in LRF with Furthermore, we obtain Since cot θ j = R jz ′ / R 2 jx ′ + R 2 jy ′ we can get the following derivatives Based on Eq. (55) we have Combining Eq. (54) and Eq. (56) we can now derive The Hermitian conjugate of eachP θ j associated with the corresponding MRF polar angle θ j can be finally expressed aŝ Finally we come toP † ϕ j . The procedure is the same, however, it is much simpler as compared toP † θ j . Again we start with the derivatives of MRF Cartesian coordinates The momentum operator associated with each orientation angle ϕ j is Hermitian according Let us recall the above detailed procedure. The only condition we need is that the Finally, we can obtain the HCMOsP † jα ′ associated with MRF Cartesian coordinates following the same procedure. Since ∂R jα /∂R jα ′ is just a parameter which does not depends on R jx , R jy or R jz we can easily get where α ′ = x ′ , y ′ , z ′ . Apparently, the momenta associated with MRF Cartesian coordinates are Hermitian.

B. KEO in Terms of Polyspherical Coordinates
In order to connect our approach to existing one we consider the KEO in terms of so called polyspherical coordinates defined in the LRF and a MRF. This will lead us to expression which have been reported in Ref. [17]. Consider a molecular system composed of N + 1 atoms. After separating the total centre of mass motion we can describe it with N vectors R 1 , R 2 , · · · , R N . The three Euler angles (ϑ, φ, χ) are chosen in such a way that the spherical coordinates of R N in the LRF are (R N , ϑ, φ) and the spherical coordinates of R 1 in the MRF are (R 1 , θ 1 , ϕ 1 = 0). That is to say the e z ′ axis of the MRF is defined to be along the direction of R N . The rest N − 2 vectors are characterised by spherical coordinates {R j , θ j , ϕ j } in the MRF (j = 2, · · · , N − 1).
Using Eq. (20) with j = 1, · · · , N and θ N = ϕ N = ϕ 1 = 0, we can derive the LRF components of the N velocity vectorṡ where U ′ is the first order derivative of the corresponding rotational transformation matrix. The above equations are still vector equations, however, they are only valid in the LRF since we have used the relationė z = 0. To make them valid in an arbitrary reference frame we only need to add the corresponding terms containingė z to the right hand side.
Before proceeding we will discuss the orthogonality of the terms appearing in Eq. (64) .
where eθ is the direction of the angular velocityθ, and similar for eφ, eχ, eθ j , and eφ j . With the help of Eq. (65) one can immediately see that the fourth term is parallel to R j while the other terms are perpendicular to R j . Recalling the velocity in spherical coordinates we know that the last three terms are orthogonal to each other. Therefore, in Eq. (64) (or Eq. (65) ), the fourth term (namely theṘ j term) is orthogonal to the other terms and the last three terms are orthogonal to each other.
Having at hand all the velocities it is quite straightforward to write the classical kinetic energy [28] according to where the matrix {µ jk } combines the reduced masses and the transformation matrix between adopted and Jacobi vectors [28]. The final results in LRF can be written in a symmetric form as in Eq. (1) where the coordinates are separated as rotational, Eq. (22), and vibrational DOFs The next step is to calculate the inverse matrix of M to generate the quantum KEO.
Aiming at a separation of the rotational and vibrational DOFs, we divide the matrix M into the following four blocks where the subscripts " rot ", " Cor ", and " vib " correspond to rotational, Coriolis, and vibra- According to Section II E, the generalised momentum vectorP rot associated with the rotational DOFs is just the total angular momentum vector J. If we are interested in the rotational DOFs only, we obtain upon setting the vibrational momenta equal to zero: where the components of J should be along the directions of eθ, eφ and eχ according to Eq. (27). In passing we note that one can also transform the components to Cartesian ones according to Eq. (28). On the other hand, if we are interested in the vibrational DOFs, the KEO for the total angular momentum J = 0 readŝ Calculating the matrix product in above equation leads to the final results The matrix G is essentially the same as the one reported in Ref. [17]. There, however, the essential point has been to express the MRF Cartesian components of the angular momentum associated with each vector {R i } in terms of J and {P θ i , P ϕ i }.
For the special case of Jacobi vectors, M vib is diagonal which greatly simplifies the calculation of Eq. (76). It is straightforward to obtain the diagonal elements according to Eq. (64) The inverse matrix of M vib can be obtained quite easily. We only need to calculate the inverse of a matrix of dimension 3×3 as well as some matrix products to get the final results. Here we give the matrix elements for G vib (Hermitian) which will be used in Section III: j sin 2 θ j + 1 µ 1 R 2 1 sin 2 θ 1 + cot θ i cot θ j cos(ϕ i − ϕ j ) + cot 2 θ 1 − cot θ 1 (cos ϕ i cot θ i + cos ϕ j cot θ j ) µ N R 2 N . (78)