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where <£ is the Lagrangian of the system. It is also possible to define a partial angular momentum corresponding to each Jacobi vector:

Figure 2 where <£ is the Lagrangian of the system. It is also possible to define a partial angular momentum corresponding to each Jacobi vector:

Related Figures (64)

Fig. 1. Definition of the three Jacobi vectors for an ABCD system. Here, g; is the center of mass of the (AB) subsystem, g> is the center of mass of the (CD) subsystem.
Fig. 1. Definition of the three Jacobi vectors for an ABCD system. Here, g; is the center of mass of the (AB) subsystem, g> is the center of mass of the (CD) subsystem.
Fig. 2. Definition of the three valence vectors for an ABCD system. since the expression of T*' is non-diagonal, the valence coordinates are said to be non-orthogonal. Again, to each vector R | = 1,2, 3), it is possible to associate a conjugate momentum (Eq. (40)) and to define a corresponding partial angular 1omentum (Eq. (41)). Finally, after having inserted Eq. (50) into Eq. (37) and used Eq. (40), we obtain a non-diagonal xpression of the kinetic energy in terms of the momenta
Fig. 2. Definition of the three valence vectors for an ABCD system. since the expression of T*' is non-diagonal, the valence coordinates are said to be non-orthogonal. Again, to each vector R | = 1,2, 3), it is possible to associate a conjugate momentum (Eq. (40)) and to define a corresponding partial angular 1omentum (Eq. (41)). Finally, after having inserted Eq. (50) into Eq. (37) and used Eq. (40), we obtain a non-diagonal xpression of the kinetic energy in terms of the momenta
Fig. 3. (AB)CB(EF) system parametrized by five valence vectors.
Fig. 3. (AB)CB(EF) system parametrized by five valence vectors.
In the general case, one should calculate the matrix A and derive the matrix M from it. ee ee ew Ss a ee ee ee eee eee aa eee ee In order to separate the overall rotation from the internal deformation, let us now introduce a Body-Fixed (BF) frame, {G; @,or, @,ar, €,sr}. Its orientation with respect to the SF frame is determined by three Euler angles: a, 6, y. After definition of the three Euler angles, the shape and the size of the molecules are described by 3N — 6 internal BF-coordinates. At this level, the exact definition of the BF frame is not yet specified (and not necessary), but the classical kinetic energy can
In the general case, one should calculate the matrix A and derive the matrix M from it. ee ee ew Ss a ee ee ee eee eee aa eee ee In order to separate the overall rotation from the internal deformation, let us now introduce a Body-Fixed (BF) frame, {G; @,or, @,ar, €,sr}. Its orientation with respect to the SF frame is determined by three Euler angles: a, 6, y. After definition of the three Euler angles, the shape and the size of the molecules are described by 3N — 6 internal BF-coordinates. At this level, the exact definition of the BF frame is not yet specified (and not necessary), but the classical kinetic energy can
Fig. 4. Definition of the EF, Frame and of the first two Euler angles.
Fig. 4. Definition of the EF, Frame and of the first two Euler angles.
Fig. 5. Definition of the BF frame and of the third angle.
Fig. 5. Definition of the BF frame and of the third angle.
Fig. 6. Definition of the BF frame and of the three Euler angles. The F2 Frame is the frame obtained after the first two Euler rotations
Fig. 6. Definition of the BF frame and of the three Euler angles. The F2 Frame is the frame obtained after the first two Euler rotations
Fig. 7. Definition of the standard polyspherical coordinates: the N — 1 vector lengths R; € [0, 00), N — 2 (BF) planar angles 6?" € [0, x], and N — 3 (BF dihedral angles y?" € [0, 27).
Fig. 7. Definition of the standard polyspherical coordinates: the N — 1 vector lengths R; € [0, 00), N — 2 (BF) planar angles 6?" € [0, x], and N — 3 (BF dihedral angles y?" € [0, 27).
Inserting these relations into Eq. (79), we obtain
Inserting these relations into Eq. (79), we obtain
(We remind that the BF-projections of J have an ‘anomalous’ behavior [114], Le. Ja J, 2) =czJ, 2), 2 F1).) The symbol (angles | ...); denotes the current element of the angular basis set. Even though these results are not obvious they are not surprising as all angular momenta are calculated in the SF frame and several changes of coordinates and projections onto adequate axes is all that has been performed. Eqs. (92) and (99) lead straightforwardly, for orthogonal coordinates, to
(We remind that the BF-projections of J have an ‘anomalous’ behavior [114], Le. Ja J, 2) =czJ, 2), 2 F1).) The symbol (angles | ...); denotes the current element of the angular basis set. Even though these results are not obvious they are not surprising as all angular momenta are calculated in the SF frame and several changes of coordinates and projections onto adequate axes is all that has been performed. Eqs. (92) and (99) lead straightforwardly, for orthogonal coordinates, to
Fig. 8. Definition of the eight Jacobi vectors for the water trimer. Here, G, is the center of mass of the jth monomer. R? is the vector joining the two I atoms, RP is the vector joining the center of mass of H2 to the O atom. R2 joins G3 to G2 and R2 joins the center of mass of (G3G2) to Gy.
Fig. 8. Definition of the eight Jacobi vectors for the water trimer. Here, G, is the center of mass of the jth monomer. R? is the vector joining the two I atoms, RP is the vector joining the center of mass of H2 to the O atom. R2 joins G3 to G2 and R2 joins the center of mass of (G3G2) to Gy.
[n order to obtain the KEO in terms of the new set of polyspherical coordinates, let us first confine ourselves to studying the case of orthogonal vectors. Eq. (76) can be recast as
[n order to obtain the KEO in terms of the new set of polyspherical coordinates, let us first confine ourselves to studying the case of orthogonal vectors. Eq. (76) can be recast as
vhere j4; denotes the reduced mass associated with R; and ue ) the reduced mass associated with RY, Taking into account hat Ln-o has a particular behavior, we obtain from Eq. (125) (the key point is the fact that the BF-components of Ino must ‘emain on the right-hand side of all the scalar products):
vhere j4; denotes the reduced mass associated with R; and ue ) the reduced mass associated with RY, Taking into account hat Ln-o has a particular behavior, we obtain from Eq. (125) (the key point is the fact that the BF-components of Ino must ‘emain on the right-hand side of all the scalar products):
To be more specific, let us consider various parametrizations of the water trimer. In the case of the set of orthogonal vectors depicted in Fig. 8, the matrix M of Eq. (54) becomes Finally, Eq. (92) or Eq. (76) can be applied to express T”:
To be more specific, let us consider various parametrizations of the water trimer. In the case of the set of orthogonal vectors depicted in Fig. 8, the matrix M of Eq. (54) becomes Finally, Eq. (92) or Eq. (76) can be applied to express T”:
Let us give a new example: the mixed Jacobi/valence parametrization for the water trimer depicted in Fig. 9. As far as the matrix M is concerned, one obtains
Let us give a new example: the mixed Jacobi/valence parametrization for the water trimer depicted in Fig. 9. As far as the matrix M is concerned, one obtains
Fig. 9. Definition of the second set of vectors for the water trimer. R? and RP are the two valence vectors joining one H atom to the O atom.
Fig. 9. Definition of the second set of vectors for the water trimer. R? and RP are the two valence vectors joining one H atom to the O atom.
Fig. 10. Definition of the third set of vectors for the water trimer. Here, G; is the center of mass of the jth monomer.
Fig. 10. Definition of the third set of vectors for the water trimer. Here, G; is the center of mass of the jth monomer.
Fig. 11. Definition of the fourth set of vectors for the water trimer. Here, only valence vectors are used.
Fig. 11. Definition of the fourth set of vectors for the water trimer. Here, only valence vectors are used.
Fig. 12. Definition of the five Jacobi vector coordinates for the (H2)3 trimer. Here, g; is the center of mass of the ith monomer Hj, G; is the center of mass of the (H2)2 subsystem.
Fig. 12. Definition of the five Jacobi vector coordinates for the (H2)3 trimer. Here, g; is the center of mass of the ith monomer Hj, G; is the center of mass of the (H2)2 subsystem.
Fig. 13. Definition of the five Radau vector coordinates for the CHE system. Here, B is the canonical point of CH’, G is the center of mass of CHE, Gus is the center of mass of the H; subsystem. Isolated CHE is a stable but very floppy system, the protons continually hopping between 120 minima [140]. Very re- cently, Wang and Carrington [140] reported the first vibrational energy levels including all the 12 internal degrees of free- dom. They adopted the polyspherical approach based on a parametrization of five Radau vectors, as shown in Fig. 13. For the reader who is not familiar with Radau coordinates, let us add that these are often introduced to replace valence coordinates in describing the vibrational motions of molecules in which the central atom is much heavier than the end atoms [141]. For this type of molecules, both systems of coordinates are very similar, the former having the advantage of being orthogonal.
Fig. 13. Definition of the five Radau vector coordinates for the CHE system. Here, B is the canonical point of CH’, G is the center of mass of CHE, Gus is the center of mass of the H; subsystem. Isolated CHE is a stable but very floppy system, the protons continually hopping between 120 minima [140]. Very re- cently, Wang and Carrington [140] reported the first vibrational energy levels including all the 12 internal degrees of free- dom. They adopted the polyspherical approach based on a parametrization of five Radau vectors, as shown in Fig. 13. For the reader who is not familiar with Radau coordinates, let us add that these are often introduced to replace valence coordinates in describing the vibrational motions of molecules in which the central atom is much heavier than the end atoms [141]. For this type of molecules, both systems of coordinates are very similar, the former having the advantage of being orthogonal.
Fig. 14. Definition of the three Radau vectors for the NHD2 molecule. Here, B is the canonical point of NHDz, G is the center of mass of the molecule, Gyup2 is the center of mass of the HD» subsystem. By definition, we have (BGyp2)? = GGyp2 * NGup2 [141]. for HOOH in Jacobi coordinates and {4; = 2 = Mp, “43 = My} for NHDz in Radau coordinates. The three vectors are in turn parametrized by six internal polyspherical coordinates: Ry, R2, R3, U1, U2, yf. Eq. (C.7) yields (we use gy" = 0)
Fig. 14. Definition of the three Radau vectors for the NHD2 molecule. Here, B is the canonical point of NHDz, G is the center of mass of the molecule, Gyup2 is the center of mass of the HD» subsystem. By definition, we have (BGyp2)? = GGyp2 * NGup2 [141]. for HOOH in Jacobi coordinates and {4; = 2 = Mp, “43 = My} for NHDz in Radau coordinates. The three vectors are in turn parametrized by six internal polyspherical coordinates: Ry, R2, R3, U1, U2, yf. Eq. (C.7) yields (we use gy" = 0)
Fig. 15. Definition of the three valence vectors for the HONO molecule. Here, 07°" = 1 — off
Fig. 15. Definition of the three valence vectors for the HONO molecule. Here, 07°" = 1 — off
Fig. 16. Valence BF polyspherical coordinates for the HFCO system. Ro belongs to the xz plane.
Fig. 16. Valence BF polyspherical coordinates for the HFCO system. Ro belongs to the xz plane.
In order to be consistent with the conventions used for the potential energy surface calculated in Ref. [143], we also slightly changed the convention for the angle ORE > OM =n GH (see Fig. 15). Because of this, there appears the phase factor S; in Eq. (A.2) of Ref. [144]. The internal motions are described by six coordinates: R, Ro, R3, ui; = cos OFF, uy = cos 03", yPF, If the following volume element,
In order to be consistent with the conventions used for the potential energy surface calculated in Ref. [143], we also slightly changed the convention for the angle ORE > OM =n GH (see Fig. 15). Because of this, there appears the phase factor S; in Eq. (A.2) of Ref. [144]. The internal motions are described by six coordinates: R, Ro, R3, ui; = cos OFF, uy = cos 03", yPF, If the following volume element,
If we use the volume element of Eq. (156) in normalizing wave functions, the KEO can be obtained from Eqs. (111) and (C.4) and its explicit expression is given by Eq. (15) of Ref. [146]. The KEO of Eq. HFCO shows more terms than those of HONO and NHD3. The two last-mentioned operators can be derived from the KEO of HFCO just by replacing the matrix elements of M appearing in Eq. (157) by those of Eqs. (149) and (155), respectively. It is worth noting that the KEO used in Ref. [149] to study the selective population of the vibrational levels of H2CS with MCTDH is formally identical to the operator of HFCO. The only difference is that the matrix M, for H2CS, reads
If we use the volume element of Eq. (156) in normalizing wave functions, the KEO can be obtained from Eqs. (111) and (C.4) and its explicit expression is given by Eq. (15) of Ref. [146]. The KEO of Eq. HFCO shows more terms than those of HONO and NHD3. The two last-mentioned operators can be derived from the KEO of HFCO just by replacing the matrix elements of M appearing in Eq. (157) by those of Eqs. (149) and (155), respectively. It is worth noting that the KEO used in Ref. [149] to study the selective population of the vibrational levels of H2CS with MCTDH is formally identical to the operator of HFCO. The only difference is that the matrix M, for H2CS, reads
If one uses the volume element of Eq. (159), the KEO is given by Eqs. (111) and (C.4), as usual. Its explicit expression can be found in Appendix D. Let us only point out that we have recently implemented this nine-dimensional operator in the Heidelberg MCTDH package [11-15,129]. In addition, we have used the program TNUM of Lauvergnat and Nauts [68] to check the correctness of the operator [127]. As already mentioned, the test does not only guarantee that the KEO is correct but also that the operator is correctly implemented in the code. es ee ee oe ee eee For the study of the vibrations of a molecule such as fluoroform, CHF3, the previous Radau parametrization is no longer adapted since the F atoms are rather heavy. Hence, the canonical point, used to define the Radau vectors, is located far from the C atom. The resulting Radau vectors are therefore very different from the valence vectors and cannot correctly describe the vibrations in the normal or local mode domains. It is more pertinent to parametrize CHF; by four valence vectors, which are shown in Fig. 17.
If one uses the volume element of Eq. (159), the KEO is given by Eqs. (111) and (C.4), as usual. Its explicit expression can be found in Appendix D. Let us only point out that we have recently implemented this nine-dimensional operator in the Heidelberg MCTDH package [11-15,129]. In addition, we have used the program TNUM of Lauvergnat and Nauts [68] to check the correctness of the operator [127]. As already mentioned, the test does not only guarantee that the KEO is correct but also that the operator is correctly implemented in the code. es ee ee oe ee eee For the study of the vibrations of a molecule such as fluoroform, CHF3, the previous Radau parametrization is no longer adapted since the F atoms are rather heavy. Hence, the canonical point, used to define the Radau vectors, is located far from the C atom. The resulting Radau vectors are therefore very different from the valence vectors and cannot correctly describe the vibrations in the normal or local mode domains. It is more pertinent to parametrize CHF; by four valence vectors, which are shown in Fig. 17.
Fig. 17. Definition of the four valence vectors for the CHF; molecule.
Fig. 17. Definition of the four valence vectors for the CHF; molecule.
Fig. 18. Jacobi description of the H;03 system. G, is the center of mass of the first monomer, G) is the center of mass of the second monomer. Gy,o, is the center of the dimer, H40>. roton. Each molecule of water is parametrized by two Jacobi vectors: R®, the vector connecting the center of mass ot 2 subsystem to the O atom, and RY, the vector joining one H atom to the other one (here, j = 1, 2). The two ‘remair ectors are the vector R2, which connects the center of mass of one monomer to the center of mass of the other mono! nd Ri, the Jacobi vector from the center of mass of the water dimer to the proton. Let J? be the total angular momentu! 5. 40) 40 1e jth molecule of water. This definition entails thatJ” = }°,_,5L, ,L, being the angular momentum associated with According to the general strategy adopted in Section 2.4, the z®* axis should be parallel to Ry and R; should be par ) the ((x®*, 25); x®F > 0) half-plane. However, for the definition of the ((x®", z®*); x8° > 0) half-plane in H503, itis n atural to choose a vector belonging to one monomer, let say R® (it is, indeed, more appropriate to link the overall rotatic 1e Zundel cation to the molecules of water than to the proton alone since the mass of the latter is much smaller). There! s already mentioned in Section 2.4(i), we have to face a particular case which is slightly more complicated than the ger ise thoroughly treated in Section 2.4. Using the properties of the angular momenta highlighted in Section 2.2.2(iii), we ready predict that the BF-projections of the total angular momenta of the second monomer have a non-regular beha' ee ye Oo Ms Eh lS, Cee Do i, pf pn ee ne eee Oe: oak ET ee eT ee ee TE fe Oe ie eS, Sh, oe crete
Fig. 18. Jacobi description of the H;03 system. G, is the center of mass of the first monomer, G) is the center of mass of the second monomer. Gy,o, is the center of the dimer, H40>. roton. Each molecule of water is parametrized by two Jacobi vectors: R®, the vector connecting the center of mass ot 2 subsystem to the O atom, and RY, the vector joining one H atom to the other one (here, j = 1, 2). The two ‘remair ectors are the vector R2, which connects the center of mass of one monomer to the center of mass of the other mono! nd Ri, the Jacobi vector from the center of mass of the water dimer to the proton. Let J? be the total angular momentu! 5. 40) 40 1e jth molecule of water. This definition entails thatJ” = }°,_,5L, ,L, being the angular momentum associated with According to the general strategy adopted in Section 2.4, the z®* axis should be parallel to Ry and R; should be par ) the ((x®*, 25); x®F > 0) half-plane. However, for the definition of the ((x®", z®*); x8° > 0) half-plane in H503, itis n atural to choose a vector belonging to one monomer, let say R® (it is, indeed, more appropriate to link the overall rotatic 1e Zundel cation to the molecules of water than to the proton alone since the mass of the latter is much smaller). There! s already mentioned in Section 2.4(i), we have to face a particular case which is slightly more complicated than the ger ise thoroughly treated in Section 2.4. Using the properties of the angular momenta highlighted in Section 2.2.2(iii), we ready predict that the BF-projections of the total angular momenta of the second monomer have a non-regular beha' ee ye Oo Ms Eh lS, Cee Do i, pf pn ee ne eee Oe: oak ET ee eT ee ee TE fe Oe ie eS, Sh, oe crete
The BF-components of L, are given by Eq. (75) and BF" -components of L”’, with (j = 1, 2), and J by Eqs. (122) and (1: respectively. But what about the BF- and BF,-2)-components of J? As in Section 2.3.1(iii), we start from another frame the F, frame (we recall that the EF frame is the frame resulting from the first two Euler rotations only). Since the proto: and the two molecules of water are not involved in the definition of the Ey frame, the expressions of the E,-components i terms of the E> coordinates of all the angular momenta appearing in Eq. (165) have a regular form. In particular, one obtain
The BF-components of L, are given by Eq. (75) and BF" -components of L”’, with (j = 1, 2), and J by Eqs. (122) and (1: respectively. But what about the BF- and BF,-2)-components of J? As in Section 2.3.1(iii), we start from another frame the F, frame (we recall that the EF frame is the frame resulting from the first two Euler rotations only). Since the proto: and the two molecules of water are not involved in the definition of the Ey frame, the expressions of the E,-components i terms of the E> coordinates of all the angular momenta appearing in Eq. (165) have a regular form. In particular, one obtain
The BF-projections of J® are obtained by rotating the E>-projections of J? through the angle ag »
The BF-projections of J® are obtained by rotating the E>-projections of J? through the angle ag »
Fig. 19. Mixed Jacobi/valence description of the H;0} system. Go, is the center of mass of the Oz subsystem, or, in other words, the middle of Ro, the valence vector joining one O atom to the other one.
Fig. 19. Mixed Jacobi/valence description of the H;0} system. Go, is the center of mass of the Oz subsystem, or, in other words, the middle of Ro, the valence vector joining one O atom to the other one.
Consequently, we explicitly have all the components of the angular momenta appearing in Eq. (165). It results in a KEO with 95 terms whose explicit expression is given in the Appendix of Ref. [128]. As already explained in Section 2.5, the correctness of the derivation of the KEO was checked by comparing it with numerical results provided by the program TNUM of Lauvergnat and Nauts [68]. (ii) Mixed Jacobi/valence description:
Consequently, we explicitly have all the components of the angular momenta appearing in Eq. (165). It results in a KEO with 95 terms whose explicit expression is given in the Appendix of Ref. [128]. As already explained in Section 2.5, the correctness of the derivation of the KEO was checked by comparing it with numerical results provided by the program TNUM of Lauvergnat and Nauts [68]. (ii) Mixed Jacobi/valence description:
Let us recast only the first term. Since the conjugate momentum can be written as: with é; = R; /R; a unit vector along Ri; Pr,, the radial momentum and Li, the angular momentum; and its adjoint as
Let us recast only the first term. Since the conjugate momentum can be written as: with é; = R; /R; a unit vector along Ri; Pr,, the radial momentum and Li, the angular momentum; and its adjoint as
Fig. 20. Mixed Jacobi/Radau description of the Hg system (see the text)
Fig. 20. Mixed Jacobi/Radau description of the Hg system (see the text)
where 1 and 0 denote the identity matrix and the zero matrix, respectively. Moreover, is the Jacobian matrix of the transformation from {q’; q”; a, B, y} to {q'; f; w, B, y}. Ifthe motion of the inactive coordinates 8q" jlaa(@’) is decoupled from the active coordinates, all the terms — a; ogi must be very small. This is why, we proposed, in Ref. [185],
where 1 and 0 denote the identity matrix and the zero matrix, respectively. Moreover, is the Jacobian matrix of the transformation from {q’; q”; a, B, y} to {q'; f; w, B, y}. Ifthe motion of the inactive coordinates 8q" jlaa(@’) is decoupled from the active coordinates, all the terms — a; ogi must be very small. This is why, we proposed, in Ref. [185],
Fig. 21. Jacobi description of the water dimer. in the adiabatic model. This ‘main’ term is directly obtained from the operator without constraints Eq. (181) simply b' removing all terms involving conjugate momenta associated with the inactive coordinates. If polyspherical coordinates ar used, this term can always be written as a sum of products of mono-mode operators and is directly obtained from Eqs. (C.4)-(C.6 and (107).
Fig. 21. Jacobi description of the water dimer. in the adiabatic model. This ‘main’ term is directly obtained from the operator without constraints Eq. (181) simply b' removing all terms involving conjugate momenta associated with the inactive coordinates. If polyspherical coordinates ar used, this term can always be written as a sum of products of mono-mode operators and is directly obtained from Eqs. (C.4)-(C.6 and (107).
The dimer is split into two subsystems that are simply the two molecules of water. As far as the exact KEO is concerned, ne obtains in view of Eqs. (125) and (126) of Section 2.4 where 4; denotes the reduced mass associated with Ri and Be’, the reduced mass associated with Re. They reac Furthermore, we have (see Eq. (124) of Section 2.4)
The dimer is split into two subsystems that are simply the two molecules of water. As far as the exact KEO is concerned, ne obtains in view of Eqs. (125) and (126) of Section 2.4 where 4; denotes the reduced mass associated with Ri and Be’, the reduced mass associated with Re. They reac Furthermore, we have (see Eq. (124) of Section 2.4)
If we now subject each molecule of water (j = 1, 2) to the following rigid constraints: where |e, means that the coordinate is fixed at its value at the equilibrium geometry. The ‘main’ term of the rigid] constrained operator reads (see Eqs. (218) and (230))
If we now subject each molecule of water (j = 1, 2) to the following rigid constraints: where |e, means that the coordinate is fixed at its value at the equilibrium geometry. The ‘main’ term of the rigid] constrained operator reads (see Eqs. (218) and (230))
and it is worth noting that the resulting coefficients are not the principal values of the inertia tensor of the water molecule. The ‘correction’ reads (see Eqs. (220), (227) and (228))
and it is worth noting that the resulting coefficients are not the principal values of the inertia tensor of the water molecule. The ‘correction’ reads (see Eqs. (220), (227) and (228))
The operator in Eq. (234) is the rigid rotor of the water molecule, i.e. the resulting coefficients are the principal values of the inertia tensor of the water molecule. The result given by Eqs. (233) and (234) is a particular case of the general formulation of Brocks et al. [60] for van der Waals dimers.
The operator in Eq. (234) is the rigid rotor of the water molecule, i.e. the resulting coefficients are the principal values of the inertia tensor of the water molecule. The result given by Eqs. (233) and (234) is a particular case of the general formulation of Brocks et al. [60] for van der Waals dimers.
Let us point out that the functions q”;|aq(q’), and hence the matrix elements of Eqs. (237), (238) and (240) are not known analytically but are numerically defined on a grid. Interestingly enough, in Ref. [192], the extra-potential term of Eq. (217) (iL) PAA DAU aLyY CULISUIAIICU UPCLAalUl LIC AIDIE LULUL ), In Ref. [192], it was shown that the flexibility of each water molecule plays a crucial role in the description of the hydrogen bond between the two molecules. A very relevant adiabatic formulation was adopted for the water dimer. Indeed, for the water dimer, a single quantum of excitation energy in one intramolecular mode of vibration is sufficient to break the hydrogen bond. This implies that only the ground vibrational state of each monomer has to be considered to study the rovibrational bound states of the dimer. Another possibility is to take into account the first excited vibrational state of each monomer alone if the dissociation of the dimer is to be considered. Such a situation is reminiscent of the Born—Oppenheimer approximation that allows one to confine the description of a bound molecule to one electronic state. Consequently, it turns out to be very relevant to define the adiabatic dependence of the inactive coordinates as follows. We first work out the vibrational equation for the six inactive coordinates (see Eq. (181) for the definition of 2”):
Let us point out that the functions q”;|aq(q’), and hence the matrix elements of Eqs. (237), (238) and (240) are not known analytically but are numerically defined on a grid. Interestingly enough, in Ref. [192], the extra-potential term of Eq. (217) (iL) PAA DAU aLyY CULISUIAIICU UPCLAalUl LIC AIDIE LULUL ), In Ref. [192], it was shown that the flexibility of each water molecule plays a crucial role in the description of the hydrogen bond between the two molecules. A very relevant adiabatic formulation was adopted for the water dimer. Indeed, for the water dimer, a single quantum of excitation energy in one intramolecular mode of vibration is sufficient to break the hydrogen bond. This implies that only the ground vibrational state of each monomer has to be considered to study the rovibrational bound states of the dimer. Another possibility is to take into account the first excited vibrational state of each monomer alone if the dissociation of the dimer is to be considered. Such a situation is reminiscent of the Born—Oppenheimer approximation that allows one to confine the description of a bound molecule to one electronic state. Consequently, it turns out to be very relevant to define the adiabatic dependence of the inactive coordinates as follows. We first work out the vibrational equation for the six inactive coordinates (see Eq. (181) for the definition of 2”):
Fig. 22. Mixed Jacobi/valence description of the CH, system (see the text).
Fig. 22. Mixed Jacobi/valence description of the CH, system (see the text).
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] (y, r, ,, a), 3, 3). Indeed, since the potential energy surface is given only in terms of six degrees of freedom, we subject the system to the following constraints: 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: (Ri, R2, R3, Ra, Rs, 07", 07", 08", ORF, pBF, BF, o8"). The corresponding matrix M reads In Ref. [196], Krawczyk et al. have computed the three lowest potential energy surfaces of ethene. Ethene has si) 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 tha 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, lef and right scissoring and left and right wagging (pyramidalization). The left and right rockings and the four C-H stretche: were ignored and their coordinates were fixed to their equilibrium positions, i.e. we have the case of a rigid constrain studied in Section 4.1.1. The six active modes are parametrized by six coordinates, which are: g, the torsional angle; r the C-C distance; a@,, the right scissor angle; a, the left scissor angle; v,, the right pyramidalization angle; 1, the lef pyramidalization angle. The PES were transformed to a diabatic representation and dynamical calculations were performec in Ref. [197] using the Multi-Configuration Time-Dependent Hartree (MCTDH) method. Unfortunately the KEO, obtainec by inversion of the 6 x 6 matrix: an a oe where the x, are mass-weighted Cartesian coordinates of the six atoms o the molecule, and q; € {¢, Tr, a, 3, 0} (see Eqs. (2), (5) and (6) of Section 2.1), is unsuitable for the MCTDH propagation J 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
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] (y, r, ,, a), 3, 3). Indeed, since the potential energy surface is given only in terms of six degrees of freedom, we subject the system to the following constraints: 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: (Ri, R2, R3, Ra, Rs, 07", 07", 08", ORF, pBF, BF, o8"). The corresponding matrix M reads In Ref. [196], Krawczyk et al. have computed the three lowest potential energy surfaces of ethene. Ethene has si) 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 tha 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, lef and right scissoring and left and right wagging (pyramidalization). The left and right rockings and the four C-H stretche: were ignored and their coordinates were fixed to their equilibrium positions, i.e. we have the case of a rigid constrain studied in Section 4.1.1. The six active modes are parametrized by six coordinates, which are: g, the torsional angle; r the C-C distance; a@,, the right scissor angle; a, the left scissor angle; v,, the right pyramidalization angle; 1, the lef pyramidalization angle. The PES were transformed to a diabatic representation and dynamical calculations were performec in Ref. [197] using the Multi-Configuration Time-Dependent Hartree (MCTDH) method. Unfortunately the KEO, obtainec by inversion of the 6 x 6 matrix: an a oe where the x, are mass-weighted Cartesian coordinates of the six atoms o the molecule, and q; € {¢, Tr, a, 3, 0} (see Eqs. (2), (5) and (6) of Section 2.1), is unsuitable for the MCTDH propagation J 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
Fig. 23. Successive separation into subsystems of the metal complex [CpRh(PH3)H(C2H,4)]”. G2 is the center of mass of the subsystem (PH3Cs5Hs). For the whole system (HRhC2H,PH3C5Hs), the exact kinetic energy operator then reads
Fig. 23. Successive separation into subsystems of the metal complex [CpRh(PH3)H(C2H,4)]”. G2 is the center of mass of the subsystem (PH3Cs5Hs). For the whole system (HRhC2H,PH3C5Hs), the exact kinetic energy operator then reads
Fig. 24. Jacobi description of NH3: Gyo is the center of mass of H2, Gu3 is the center of mass of the H3 subsystem. in Refs. [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 (Rj 2, 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 Se ee Ee ee ee ee ee 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 2.2.2, the separation into subsystems, the rigid and flexible rotors, etc...), it is sensible that this section appears at the end of the review. 44 N Wen tien dlee mk men, mflleccenmenwmilancninadnl mmnandiieaanéimn xen RITT «
Fig. 24. Jacobi description of NH3: Gyo is the center of mass of H2, Gu3 is the center of mass of the H3 subsystem. in Refs. [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 (Rj 2, 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 Se ee Ee ee ee ee ee 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 2.2.2, the separation into subsystems, the rigid and flexible rotors, etc...), it is sensible that this section appears at the end of the review. 44 N Wen tien dlee mk men, mflleccenmenwmilancninadnl mmnandiieaanéimn xen RITT «
We have shown elsewhere [81] that two cases must be distinguished: whether the vector is involved in the definitior 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 ir a SF frame [58] (it should be stressed that this result is not trivial at all, see below). To be more specific, if Rj is not involved ir the definition of the BF frame, the projections of Lj onto the BF-axes and expressed in terms of the BF-Cartesian coordinate: are given by:
We have shown elsewhere [81] that two cases must be distinguished: whether the vector is involved in the definitior 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 ir a SF frame [58] (it should be stressed that this result is not trivial at all, see below). To be more specific, if Rj is not involved ir the definition of the BF frame, the projections of Lj onto the BF-axes and expressed in terms of the BF-Cartesian coordinate: are given by:
It is important then to notice that (£ is the Lagrangian of the system) 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. (A.1) refers to an angular momentum computed in the BF frame. Indeed, Eq. (A.1) is nothing but the exact quantum mechanical counterpart of the classical expression [58]
It is important then to notice that (£ is the Lagrangian of the system) 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. (A.1) refers to an angular momentum computed in the BF frame. Indeed, Eq. (A.1) is nothing but the exact quantum mechanical counterpart of the classical expression [58]
(with (A, 4’, 0 = x, y,z) and R(a@, B, y) the Euler rotation matrix) We then have in classical mechanics
(with (A, 4’, 0 = x, y,z) and R(a@, B, y) the Euler rotation matrix) We then have in classical mechanics
An important question to be asked is the following: is it true that or, in other words, that the conjugate momentum associated with the BF-coordinates (which is not a ‘BF conjugate momentum’ in the sense of amomentum 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 Eqs. (A.6) and (A.7), i.e. from the fact that the vector is not involved in the definition of BF. Wile thiie cet
An important question to be asked is the following: is it true that or, in other words, that the conjugate momentum associated with the BF-coordinates (which is not a ‘BF conjugate momentum’ in the sense of amomentum 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 Eqs. (A.6) and (A.7), i.e. from the fact that the vector is not involved in the definition of BF. Wile thiie cet
if the vector R is not involved in the definition of the BF frame.
if the vector R is not involved in the definition of the BF frame.
The previous equations should be used along with
The previous equations should be used along with
and if the following volume element is used in normalizing the wave functions Let us parametrize a penta-atomic system such as fluoroform, CHF3, by four valence vectors as shown in Fig. 17. The corresponding matrix M reads
and if the following volume element is used in normalizing the wave functions Let us parametrize a penta-atomic system such as fluoroform, CHF3, by four valence vectors as shown in Fig. 17. The corresponding matrix M reads
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