Quantum Description of a Rotating and Vibrating Molecule

Quantum description of a rotating and vibrating molecule

Sylvain D. Brechet,∗ Francois A. Reuse, Klaus Maschke, and Jean-Philippe Ansermet Institute of Condensed Matter Physics, Station 3, Ecole Polytechnique F´ed´erale de Lausanne - EPFL, CH-1015 Lausanne, Switzerland

A rigorous quantum description of molecular dynamics with a particular emphasis on internal observables is developed accounting explicitly for kinetic couplings between nuclei and electrons. Rotational modes are treated in a genuinely quantum framework by defining a molecular orientation operator. Canonical rotational commutation relations are established explicitly. Moreover, physical constraints are imposed on the observables in order to define the state of a molecular system located in the neighborhood of the ground state defined by the equilibrium condition.

I. INTRODUCTION vere restrictions on the dynamics. Rotational states play an important role for low temperature spectroscopy [2], for THz spectroscopy [13], for molecular magnetism [14] The dynamics of quantum molecular systems has been and for molecular superrotors [15]. In order to establish studied analytically and numerically for decades. Molec- a genuine quantum description of rotational molecular ular rotations are usually characterised by Euler angles states, a molecular rotation operator has to be intro- defined with respect to a molecular reference frame [1–4] duced. and kinetic couplings between nuclei and electrons are neglected. However, in order to make a precise quantum Here, we develop a rigorous quantum description of description of molecular dynamics, these couplings have molecule with a particular emphasis on internal observ- to be taken explicitly into account, which leads to quan- ables. In this description, the vibrational and rotational tum deviations in the commutation relations. It is also modes are described by operators associated to the de- important to recognise that the notion of a molecular ref- formation and orientation of the molecule. The set of erence frame is inconsistent with quantum physics, due internal observables is related explicitly to the set of ob- to nonlocality. For the same reason, the orientation of a servables associated to nuclei and electrons through a molecule cannot be described simply using Euler angles rotation operator. The internal observables are chosen as in a classical framework. In order to treat rotational in order to satisfy canonical commutation relations in states of molecular systems in a genuine quantum frame- translation and rotation. Moreover, in order to define the work, the orientation and rotation of a molecule has to state of a molecular system, physical constraints need to be described using operators. This is done in this article. be imposed on the observables. In fact, the amplitudes Rotational states of molecular systems are currently of of the vibrational modes of a molecular system have to great interest. For example, in small molecular systems be sufficiently small. at low temperature, the rotational degrees of freedom The quantum molecular description presented in this play an important role since they can be distinguished publication is expected to be relevant for extremely fast experimentally from the vibrational degrees of freedom. rotating molecular systems exhibiting a large orbital an- Due to technological improvement, the distinction be- gular momentum. Such systems, called “superrotors”, tween these degrees of freedom became increasingly im- have been observed for the molecules listed in Table I. portant in the last decade. Rotating atoms [5], rotating The molecular orbital angular momentum depends in molecules [6, 7], rotating trapped Bose-Einstein conden- part on internal vibrations, as we will show below. Thus, sates [8] and even rotating microgyroscopes [9] are cur- this description is of importance for molecules with large rently studied experimentally and are attracting much vibration amplitudes. This can be realised for molecules attention. For example, physisorbed H2, HD and D2 on with weak bonds, such as van der Waals bonds. The a substrate at low temperature form a honeycomb lattice present quantum molecular formalism is expected also and rotational spectroscopy revealed a resonance width to be important for molecular systems, listed in Ta- of H2 twice as large as the resonance widths of HD and ble I, for which the “Coriolis interaction” leads to a large arXiv:1412.2119v2 [physics.chem-ph] 5 Jul 2015 D2 [5]. The theoretical explanation requires a rigorous rotational-vibrational coupling. quantum formalism with a genuine quantum treatment The structure of this publication is the following. In of molecular rotations. Sec. II, we formally describe the dynamics of a system A semi-classical approach is commonly used for the de- of N nuclei and n electrons. In Sec. III, we define the scription of molecular dynamics [10–12]. An important internal observables in order to obtain canonical commu- shortcoming of such an approach is that it requires the tations relations in translation and rotation. Sec. IV is rotational states to be in an eigenstate, thus imposing se- devoted to the description of the dynamics of the molecular system in terms of the internal observables. Finally, in Sec. V, we determine the equilibrium conditions that define the molecular ground state and we define explic- ∗Electronic address: sylvain.brechet@epfl.ch itly the angular frequency of the vibrational modes of the 2

associated to the electrons. The components of the op- TABLE I: Molecular systems erators Rµ and P µ acting on the Hilbert subspace HN satisfy the canonical commutation relations, i.e. Type Molecules References O , N [15] k k 2 2 ej · P µ, e · Rν = − i~ δµν ej · e 1N , (2) Superrotors CO2 [16] NO2 [17] and the components of the operator Sµ satisfy the canon-

Cl2 [18] ical commutation relation, i.e. PDT [19] [ ej · Sµ, ek · Sν ] = i δµν (ej × ek) · Sµ , (3) CH3CCl3 [20] ~

Ne − D2O [21] where ej are the units vectors of an orthonormal basis CH CNH [22] k Coriolis 2 and e are the units vectors of the dual orthonormal C2H2D2 [23] basis. The other commutation relations are trivial, i.e. C2H3F [24] j k FHF , FDF [25] e · Rµ, e · Rν = 0 ,

C5H8, C5H7D [26] [ ej · P µ, ek · P ν ] = 0 , (4) j e · Rµ, ek · Sν = 0 ,

[ ej · P µ, ek · Sν ] = 0 . molecular system. Similarly, the position, momentum and spin observables of the electron ν are characterised respectively by the II. QUANTUM DESCRIPTION OF A SYSTEM self-adjoint operators 1N ⊗ rν , 1N ⊗ pν and 1N ⊗ sν , OF N NUCLEI AND n ELECTRONS where ν = 1, .., n, acting trivially on the Hilbert subspace HN associated to the nuclei. The components of the The quantum dynamics of a molecular system consist- operators rν and pν acting on the Hilbert subspace He ing of N nuclei and n electrons is obtained from the clas- satisfy the canonical commutation relations, i.e. sical dynamics by applying the “correspondence princi- e · p , ek · r = − i δ e · ek 1 , (5) ple”. The Hilbert subspaces describing the nuclei and j µ ν ~ µν j e the electrons are denoted H and H respectively. The N e and the components of the operator s satisfy the canon- Hilbert space describing the whole system is expressed µ ical commutation relation, i.e. as,

[ ej · sµ, ek · sν ] = i~ δµν (ej × ek) · sµ . (6) H = HN ⊗ He . (1) The other commutation relations are trivial, i.e. To investigate the molecular kinematics, it is not restric- tive to assume that the nuclei are N discernible particles ej · r , ek · r = 0 , denoted by an index µ = 1, .., N. These particles have µ ν a mass M , an electric charges Z (− e) and a spin S . ej · p , ek · p = 0 , µ µ µ µ ν (7) The symbol e represents the electronic electric charge in- j e · rµ, ek · sν = 0 , cluding its sign and Zµ denotes the atomic number of the nucleus µ. ej · pµ, ek · sν = 0 . The Hilbert subspace He associated to the electrons is assumed to be isomorphic to the tensor product of n In order to discuss the dynamics of an electrically neu- one-electron Hilbert spaces, i.e. tral molecular system composed of N nuclei and n electrons, we implicitly assume that 2 3 3 2⊗n 2 3n 3n 2n He ∼ L (R , d x) ⊗ C ∼ L (R , d x) ⊗ C . N X Zµ = n . (8) In fact, the Hilbert spaces describing the electrons are µ=1 totally antisymmetric subspaces of He. For a molecular system, the contribution of the overlap integrals between In a non-relativistic framework, we restrict our anal- the nuclei is negligible. Thus, we do not need to take into ysis to instantaneous electromagnetic interactions be- account explicitly the fermionic nature of the nuclei. tween the particles, i.e. the electrons and the nuclei. In The position, momentum and spin observables of the this framework, the Hamiltonian governing the evolution nucleus µ are characterised respectively by the self- reads, adjoint operators Rµ ⊗ 1e, P µ ⊗ 1e and Sµ ⊗ 1e, where µ = 1, .., N, acting trivially on the Hilbert subspace He H = HN ⊗ 1e + 1N ⊗ He + HN −e , (9) 3 where the Hamiltonians HN and He associated to the in the absence of a centre of mass frame, the position and nuclei and electrons are defined respectively as, momentum observables of the centre of mass can be expressed mathematically as self-adjoint operators. This N 2 X P µ SO enables us to define other position and momentum ob- HN = + VN −N + VN , servables with respect to the centre of mass. We shall 2Mµ µ=1 (10) refer to them as “relative” position and momentum ob- n X p2 servables because they are the quantum equivalent of the H = ν + V + V SO , e 2m e−e e classical relative position and momentum variables de- ν=1 fined with respect to the center of mass frame. Then, using a rotation operator, we define the “rest” position and where m is the mass of an electron, V SO is the nuclear N momentum observables, which are the quantum equiva- spin-orbit coupling due to the interaction between the lent of the classical position and momentum variables spin and the orbital angular momentum of the nuclei defined in the molecular rest frame. Finally, the “rest” and V SO is the electronic spin-orbit coupling due to the e position and momentum observables are recast in terms interaction between the spin and the orbital angular mo- of internal observables characterizing the vibrational, ro- mentum of the electrons. The Coulomb potentials V N −N tational and electronic degrees of freedom. and V are respectively defined as, e−e Applying the correspondence principle, the position, 2 N momentum and angular momentum observables associ- e X ZµZν V = , ated to the center of mass are respectively given by the N −N 8πε kR − R k 0 µ,ν=1 µ ν self-adjoint operators, µ6=ν (11) n N n ! e2 X 1 1 X X Ve−e = , Q = Mµ Rµ ⊗ 1e + 1N ⊗ m rν , (14) 8πε0 krµ − rν k M µ,ν=1 µ=1 ν=1 µ6=ν N n X X where ε0 is the vacuum dielectric constant. The interac- P = P µ ⊗ 1e + 1N ⊗ pν , (15) tion Hamiltonian HN −e appearing in the definition (9) µ=1 ν=1 describes the interaction between the nuclei and the electrons. It is defined as, where M stands for the total mass of the molecule, i.e.

SO HN −e = VN −e + VN −e . (12) M = M + nm , (16)

The Coulomb potential VN −e between the nuclei and the and M represents the total mass of the nuclei, i.e. electrons is defined as, N 2 N n X e X X Zµ M = Mµ . (17) V = − , (13) N −e 4πε kR ⊗ 1 − 1 ⊗ r k µ=1 0 µ=1 ν=1 µ e N ν The commutation relations (2) and (5) imply that the SO and VN −e is the spin-orbit coupling due to the interaction operators P and Q satisfy the commutation relations, between the spin of the electrons and the orbital angular momentum of the nuclei and to the interaction between k k ej · P, e · Q = − i~ ej · e 1 . (18) the spin of the nuclei and the orbital angular momentum of the electrons. As usual in molecular physics, the ef- 0 Now we define the “relative” position operators Rµ fects of the magnetic field produced by the motion of the 0 0 particles are neglected. and rν , and the “relative” momentum operators P µ and 0 0 0 pν . The “relative” position operators Rµ and rν are related to the position operators Rµ and rν by, III. INTERNAL OBSERVABLES OF THE 0 MOLECULAR SYSTEM Rµ = Rµ ⊗ 1e − Q , 0 (19) rν = 1N ⊗ rν − Q . The description of molecular dynamics in a classical framework would be much simpler than in a quantum Similarly, the “relative” momentum operators P 0 and p0 framework since in the former a rest frame could be at- µ ν are related to the momentum operators P µ and p by, tached easily to the physical system. In quantum physics, ν the approach is slightly different because observables are 0 Mµ described mathematically by operators, which implies P µ = P µ ⊗ 1e − P , M (20) that there exists no rest frame and no centre of mass m p0 = 1 ⊗ p − P . frame associated to the molecular system. However, even ν N ν M 4

0 0 0 0 0 0 The operators Rµ, P µ, rν and pν commute with the P µ and pν by, operators Q and P. In addition, these operators satisfy the condition, 00 1 k 0 ej · P = e · R (ω) · ej, ek · P , µ 2 µ (26) N n 00 k 0 X 0 X 0 ej · pν = e · R (ω) · ej (ek · pν ) . Mµ Rµ + m rν = 0 , (21) µ=1 ν=1 where the brackets { , } denote an anticommutator accounting for the fact that the rotation operator R (ω) which is a direct consequence of the deﬁnitions (14) 0 does not commute with the position operator P µ of the and (19), and the condition, nuclei.

N n The self-adjoint molecular orientation operator ω is X X fully determined by the position operators R and r . P 0 + p0 = 0 , (22) µ ν µ ν Thus, the components of ω satisfy the trivial commuta- µ=1 ν=1 tion relations, which is a direct consequence of the deﬁnitions (15) ej · ω, ek · ω = 0 . (27) and (20). Now, we can determine some further commutation relations. Clearly the components of the position 0 The orientation operator ω belongs to the rotation alge- operator Rµ commute and the components of the posi- 0 bra and it is related to the rotation operator R (ω) that tion operator rν commute as well. The components of 0 0 belongs to the rotation group by exponentiation, i.e. the momenta operators P µ and pν commute likewise, i.e. R (ω) = exp (ω · G) , (28) j 0 k 0 e · Rµ, e · Rν = 0 , 0 0 taking into account the commutation relation (27) of the ej · P , ek · P = 0 , µ ν (23) components of the orientation operator ω. The elements j 0 k 0 e · rµ, e · rν = 0 , of the rotation group have to satisfy the orthogonality 0 0 condition, i.e. ej · pµ, ek · pν = 0 . T Thus, the only non-trivial commutation relations read, R (ω) · R (ω) = 1 , (29) T −1 M which implies that R (ω) = R (ω) and in turn that, e · P 0 , ek · R0 = − i e · ek δ − µ 1 , j µ ν ~ j µν M G T = − G . (30) 0 k 0 k m ej · pµ, e · Rν = i~ ej · e 1 , M where the components of the vector G are rank-2 tensors 0 k 0 k Mµ that are generators of the rotation group acting on R3. ej · P µ, e · rν = i~ ej · e 1 , (24) M The action of the rotation group is locally deﬁned as, 0 k 0 k m ej · pµ, e · rν = − i~ ej · e δµν − 1 . M (ej · G) x = ej × x , (31)

00 00 3 Now we define the “rest” position operators Rµ and rν , which implies that the generators G of the rotation in R 00 00 and the “rest” momentum operators P µ and pν . These verify the well known commutation relations operators are related respectively to the operators “rel- 0 0 0 0 [ e · G, e · G ] = (e × e ) · G . (32) ative” Rµ, rν , P µ and pν by a rotation operator R (ω) j k j k that is a function of the operator ω describing the orientation of the molecular system. The rotation oper- The operators n(j) (ω) are Killing vectors [27] of the 00 rotation algebra that are defined in terms of the rotation ator commutes with the “rest” position operators Rµ, 00 00 operator R (ω) and the rotation generators as, rν and the “rest” momentum operator pν but not with 00 the “rest” momentum operator P µ as explained in Ap- −1 pendix C. The components of the “rest” position oper- R (ω) · (ej · ∂ω) R (ω) = n(j) (ω) · G . (33) ators R00 and r00, are related to the components of the µ ν The dual operator m(k) (ω) satisfies the duality condi- “relative” position operators R0 and r0 by, µ ν tion,

j 00 j −1 k 0 (k) k e · Rµ = e · R (ω) · ek e · Rµ , n(j) (ω) · m (ω) = ej · e . (34) (25) j 00 j −1 k 0 e · rν = e · R (ω) · ek e · rν . As shown in Appendix A, the Killing form [28] associated to the rotation group is given by, 00 The components of the “rest” momentum operators P µ 00 n (ω) · n (ω) = e · P + A (1 − P ) · e , (35) and pν are related to the “relative” position operators (j) (`) j ω ω ` 5

00 where the projector Pω and the scalar A are respectively Similarly, the “rest” momentum operator P µ is expressed defined as, in terms of the scalar operators Pα, the vectorial opera- ω ω tors p(ν0) and the angular velocity pseudo-vectorial op- P ≡ , ω kωk2 erator Ω as, (36) 2 kωk2 n A = sin . 00 (0) p α Mµ X P = Ω× M R + M P X − A 0 p 0 kωk 2 µ µ µ µ α µ M νν (ν ) ν,ν0=1 According to the definition (36), in the limit of an in- (43) finitesimal rotation, i.e. kωk −→ 0, the scalar A −→ 1, The relation (43) yields a kinetic coupling between the which implies that, “rest” momentum operators of the nuclei and electrons. 00 The “rest” momentum operator pν is expressed in terms lim n(j) (ω) = ej 1 , kωk→0 of the momentum operators p 0 as, (37) (ν ) lim m(k) (ω) = ek 1 . kωk→0 n 00 X p = Aνν0 p 0 . (44) (k) ν (ν ) The operators n(j) (ω) and m (ω) determine the struc- ν0=1 ture of the rotation algebra. Using the relations (25), the condition (21) is recast Note that the definition of the matrix elements Aνν0 is as, not unique. However, the particular choice made in relation (42) leads to canonical commutation relations be- N n X 00 X 00 tween the electronic position operator q(ν0) and the elec- Mµ Rµ + m rν = 0 . (38) tronic momentum operator p 0 . µ=1 ν=1 (ν ) The vector set {Xµα} is the orthonormal basis charac- Similarly, using the relations (26), the condition (22) is β terizing the vibrational modes and the vector set {Xµ} recast as, is the dual orthonormal basis, i.e. N n X X P 00 + p00 = 0 . (39) N µ ν X β β µ=1 ν=1 Xµα · Xµ = δα . (45) µ=1 Now, we can introduce operators characterizing the internal observables of the quantum molecular system. (0) α The vectors Rµ correspond to the equilibrium config- First, we introduce the scalar operators Q , where α = urations of the nuclei. In order for the identities (40) 1, .., 3N − 6, characterizing the amplitude of the vibra- and (41) to satisfy the condition (38) and for the identi- tional modes of the N nuclei. Second, we introduce the ties (43) and (44) to satisfy the condition (39), we need vectorial operators q 0 related to the relative position of (ν ) to impose conditions on the vectors R(0) and X . First, the electrons respectively. The “rest” position operator µ µα we choose the origin of the coordinate system such that R00 is expressed in terms of the scalar operators Qα and µ it coincides with the center of mass, i.e. the vectorial operators q(ν0) as, n N 00 (0) 1 α m X X (0) R = R 1 + Q Xµα − Aνν0 q 0 , Mµ Rµ = 0 . (46) µ µ pM M (ν ) µ ν,ν0=1 µ=1 (40) where we used Einstein’s implicit summation convention Then, we require the deformation modes of the molecule for the vibrational modes α. The relation (40) yields a to preserve the momentum, i.e. kinetic coupling between the “rest” position operators of 00 N the nuclei and electrons. The “rest” position operator rµ X p Mµ Xµα = 0 . (47) is expressed in terms of the position operators q(ν0) as, µ=1 n 00 X rν = Aνν0 q(ν0) , (41) We also require the deformation modes of the molecule ν0=1 to preserve the orbital angular momentum, i.e. −1 where the matrix elements Aνν0 and Aν0ν are defined as, N X p (0) r ! Mµ Rµ × Xµα = 0 . (48) 1 M A 0 ≡ δ 0 + − 1 , µ=1 νν νν n M (42) r ! The constraints (46)-(48) are known as the Eckart con- −1 1 M ditions [29]. Finally, we choose the orientation of the co- A 0 ≡ δν0ν + − 1 . ν ν n M ordinate system such that the inertia tensor of the equi- 6 librium position of the nuclei is diagonal, i.e. As shown in Appendix C, using the physical identity (50) defining the rotation operator, the mathemati- N cal identity (33) associated to the action of the rotation X (0) (0) Mµ ej · Rµ ek · Rµ = group, the commutation relation (23) and (24) and the µ=1 (49) transformation laws (25) and (26), the commutation re- N 00 00 00 00 2 lations between the operators Rµ, P µ, rν and pν are X (0) Mµ (ej · ek) ek · Rµ . found to be, µ=1 M e · P 00, ek · r00 = i e · ek µ 1 As shown in Appendix E, the first relation (25) and the j µ ν ~ j M physical constraints (46) and (48) determine the rotation 00 ` k 00 − ej · P , e · ω e · n (ω) × r , operator R (ω), i.e. µ (`) ν

N 00 00 X −1 ej · P µ, ek · pν = M R(0) × R (ω) · R0 = 0 . (50) µ µ µ 00 ` 00 µ=1 ej · P µ, e · ω ek · n(`) (ω) × pν , (52) To emphasize the physical motivation behind the pre- vious formal development, we consider the classical coun- 00 k 00 k Mµ ej · P µ, e · Rν = − i~ ej · e δµν − 1 terpart of a quantum molecular system. In a classical M framework, the classical counterpart of the operatorial 00 ` k 00 − ej · P µ, e · ω e · n(`) (ω) × Rν , relation (50) determines the rest frame of the molecular system. Moreover, the equilibrium conﬁguration of a (0) 00 00 molecule is given by a vector set {Rµ } describing the ej · P µ, ek · P ν = position of the nuclei. The condition (46) implies that the 1n o e · P 00, e · P 00, em · ω n (ω) × e · e` centre of mass of the molecule coincides with the origin 2 ` ν j µ (m) k of the coordinate system and the condition (49) requires 1 n o − e · P 00, e · P 00, em · ω n (ω) × e · e` the inertial tensor of this molecule to be diagonal with 2 ` µ k ν (m) j respect to the coordinate system. The set of orthonormal vectors {Xµα} characterize the The kinetic couplings (40) and (43) between the “rest” 3N − 6 normal deformation modes of the molecule and position and momentum operators of the nuclei and elec- thus account for the vibrations. The condition (47) im- trons lead to quantum deviation in the commutation rela- plies that the normal deformation modes preserve the tions (52), characterised by the mass ratio Mµ/M. These momentum of the molecule and the condition (48) re- deviations are larger for smaller molecules. For example, + quires that these modes also preserve the orbital angu- for a H3 molecule [30] : Mµ/M = 0.33. Moreover, the lar momentum of the molecule. Thus, the deformation fact that molecular rotations are treated in a genuine modes of the molecule do not generate translations or quantum framework leads to other quantum deviations rotations of the molecule. in the commutation relations (52). These deviations are The commutation relations (23) and (24) and the proportional to the commutator of the “rest” momentum 00 transformation laws (25) and (26) imply that the com- operator P ν and the molecular orientation operator ω. 00 00 00 mutation relations between the operators Rµ, rν and pν The internal observables are described by the scalar α are given by, operators Q , Pα, the vectorial operators q(ν), p(ν) and the pseudo-vectorial operators Ω and ω. As j 00 k 00 e · Rµ, e · Rν = 0 , shown in Appendix D, the inversion of the deﬁni- ej · r00, ek · r00 = 0 , tions (40), (41), (43), (44) yields explicit expressions for µ ν the internal observables Qα, P , q and p , i.e. 00 00 α (ν) (ν) ej · pµ, ek · pν = 0 , (51) ej · r00, ek · R00 = 0 , N µ ν α X p α 00 (0) m Q = Mµ Xµ · Rµ − Rµ 1 , e · p00, ek · R00 = i e · ek 1 , j µ ν ~ j M µ=1 N 00 k 00 k m 1 ej · pµ, e · rν = − i~ ej · e δµν − 1 . X 00 M Pα = p Xµα · P µ , Mµ µ=1 (53) The kinetic couplings (40) and (43) between the “rest” n r !! X 1 M 00 position and momentum operators of the nuclei and elec- q = δ 0 + − 1 r 0 , (ν) νν n M ν trons lead to quantum deviations in the commutation ν0=1 relations (51), characterised by the mass ratio m/M. n r !! These deviations are larger for smaller molecules. For X 1 M 00 p = δνν0 + − 1 p 0 . + −4 (ν) n M ν example, for a H3 molecule [30] : m/M = 2 · 10 . ν0=1 7

The components of the inertia tensorial operator I (Q .) well, i.e. are defined as, [ Pα,Pβ ] = 0 , h α i . α Q , ek · p 0 = 0 , ek · I (Q ) · e` = (ek · I0 · e`) 1 + Q (ek · Iα · e`) , (54) (ν ) h i Pα, ek · p(ν0) = 0 , where the dot in the argument of the operator I (Q .) h k i refers to all the vibrational modes. The first term on the Pα, e · q(ν0) = 0 , (60) RHS of the definition (54), i.e. k Pα, e · ω = 0 , h i N ej · p(ν), ek · p(ν0) = 0 , X (0) (0) ek · I0 · e` = Mµ ek × Rµ · e` × Rµ (55) h k i µ=1 ej · p(ν), e · ω = 0 . N X (0)2 (0) (0) The canonical commutation relations are given by, = Mµ Rµ (ek · e`) − ek · Rµ e` · Rµ , µ=1 β β Pα,Q = − i~ δα 1 , (61) h k i k ej · p(ν), e · q(ν0) = − i~ ej · e δνν0 1 . is required to be diagonal with respect to the rotating molecular system according to the constraint (49), i.e. The operator Ω does not commute with the operators α Q , Pα, q(ν), p(ν) and ω, i.e. e · I · e = (e · I · e )(e · e ) k 0 ` k 0 k k ` ej · Ω,Qα = N 2 2 (56) N X (0) (0) = Mµ Rµ − ek · Rµ (ek · e`) , X j . −1 α β − i~ e · I (Q ) · Xµ × Xµβ Q , µ=1 µ=1 and the term on the RHS, i.e. j e · Ω,Pα = N N 1 X h j . −1 β i X p (0) − i~ e · I (Q ) · Xµα,Pβ Xµ , ek · Iα · e` = Mµ (ek × Xµα) · e` × Rµ , (57) 2 × µ=1 µ=1

h i is shown in Appendix F to be symmetric, i.e. j k e · Ω, e · q(ν) =

j . −1 ` k − i~ e · I (Q ) · e e` × e · q(ν) , (62) ek · Iα · e` = e` · Iα · ek . (58)

h j i As shown in detail in Appendix F, the commutation e · Ω, ek · p(ν) = α relations between the operators Q , Pα, q(ν), p(ν), ω, j . −1 ` Ω accounting for internal degrees of freedom are deter- − i~ e · I (Q ) · e (e` × ek) · p(ν) , mined using the commutation relations (51) and (52), the deﬁnitions (53) and (42), and the constraints (45)-(48). α j k j . −1 (k) The operators Q , q(ν) and ω related to the molecular e · Ω, e · ω = − i~ e · I (Q ) · m (ω) , conﬁgurations commute, i.e. where we used the notation convention,

α β Q ,Q = 0 , [ A, B ]× = A × B − B × A . (63) h α k i According to the commutation relations (62), the an- Q , e · q(ν0) = 0 , gular velocity operator Ω does not commute with the α k Q , e · ω = 0 , (59) other internal observables. Thus, it is not a suitable ob- h j k i servable for a quantum description of molecular rotation. e · q(ν), e · q(ν0) = 0 , Therefore, we introduce the angular momentum operator 0 h j k i L deﬁned as, e · q(ν), e · ω = 0 . N n 0 X 0 0 X 0 0 L = Rµ × P µ + rν × pν , (64) The other non-canonical commutation relations vanish as µ=1 ν=1 8 and the angular momentum operator L deﬁned as, The orbital angular momentum operator L does not com-

N n mute with the operators ω, Ω and L, i.e. 1 X 1 X L = R00, P 00 + [ r00, p00 ] , (65) j (j) 2 µ µ × 2 ν ν × L, e · ω = − i~ m (ω) , µ=1 ν=1 j j . −1 k L, e · Ω = i~ e · I (Q ) · e (ek × L) , where we used the notation convention (63). As shown in Appendix G, using the definitions (40)- [ L, ej · L ] = i~ (ej × L) . (72) (44), (54)-(58) and the notation conventions (63) and, The third commutation relation (72) implies that, { A, B } = A · B + B · A , (66) • 2 L , ej · L = 0 , the orbital angular momentum (65) is recast as, (73) [ ej · L, ek · L ] = i~ δµν (ej × ek) · L , N 1 n . o 1 X h α β i as expected. Finally, the property (34) and the commuta- L = I (Q ) , Ω + Q Xµα,Pβ Xµ 2 • 2 × µ=1 tion relation (72) imply that the canonical commutation n relations for a quantum rotation are given by, 1 X h i + q(ν), p(ν) , (67) k k 2 × n(j) (ω) · L, e · ω = − i~ ej · e . (74) ν=1 which is a self-adjoint operator. Since the orbital an- The internal spin observables are the nuclear spin op- gular momentum operator commutes with the position erator S(µ) and the electronic spin operator s(ν) that are and momentum operators, as shown explicitly below, it respectively defined as, is convenient to recast the angular rotation rate Ω in k ej · S(µ) = e · R (ω) · ej ek · Sµ , terms of L. In order to do so, we define the molecular (75) k orbital angular momentum L, the deformation orbital ej · s(ν) = e · R (ω) · ej ek · sν . angular momentum L and the electronic orbital angular As shown in Appendix I the definitions (75), the commu- momentum ` respectively as, tation relations (3) and (6), and the fact that the spins 1 n o commute with the molecular orientation observable ω im- L = I (Q .) , Ω , 2 • ply that, N 1 X h α β i ej · S(µ), ek · S(ν) = i~ δµν (ej × ek) · S(µ) , L = Q Xµα,Pβ Xµ , (76) 2 × (68) µ=1 ej · s(µ), ek · s(ν) = i~ δµν (ej × ek) · s(µ) . n α 1 X h i The operators Q , q(ν) and ω commute with the nuclear ` = q(ν), p(ν) . 2 × spin operators S(µ), i.e. ν=1 h j i Using the definitions (68) and the fact that the iner- S(µ), e · q(ν) = 0 , (77) tia tensor I (Q .) commutes with the operators L, q (ν) α S(µ),Q = 0 , (78) and p(ν), the inversion of the expression (67) yields the j rotation rate, i.e. S(µ), e · ω = 0 , (79)

1 n −1 o Ω = I (Q .) , L , (69) and also with the electronic spin operators s(µ), i.e. 2 • h j i where s(µ), e · q(ν) = 0 , (80) α L = L − L − ` . (70) s(µ),Q = 0 , (81) j As shown in Appendix H, the commutation relations s(µ), e · ω = 0 . (82) involving the orbital angular momentum operator L are Moreover, as demonstrated in the Appendix I, the opera- obtained using the expression (67) and the commutation tors P and p commute with the nuclear spin operators relations (59). α (ν) S , i.e. The orbital angular momentum operator L commutes (µ) α with the conﬁguration and momentum operators Q , Pα, h i S(µ), ej · p(ν) = 0 , (83) q(ν) and p(ν), i.e. S ,P = 0 , (84) [ L,Qα ] = 0 , (µ) α

[ L,Pα ] = 0 , and also with the electronic spin operators s(µ), i.e. h i h i L, ek · q = 0 , (71) (ν) s(µ), ej · p(ν) = 0 , (85) h i L, ek · p(ν) = 0 . s(µ),Pα = 0 . (86) 9

Finally, as shown in the Appendix I, the commutation angular momentum, i.e. relations between the orbital angular momentum L and 2 n o n o the spin operators S(µ) and s(µ) are respectively given P 1 . −1 . −1 T = + L, I (Q ) · I0 · I (Q ) , L by, 2 M 8 • • n p2 2 1 α X (ν) ~ . S , e · L = i e × S , + Pα P + + Φres (Q ) . (94) (µ) j ~ j (µ) 2 2 m 8 (87) ν=1 s(µ), ej · L = i~ ej × s(µ) . The expression (94) of the kinetic energy T separates the rotational and vibrational degrees of freedom. It is a IV. DYNAMICAL DESCRIPTION OF A consequence of the Eckart conditions (46), (47) and (48) ROTATING AND VIBRATING MOLECULE and of the physical definition (50) of the rotation operator R (ω). It is the quantum counterpart of the expres- The observable corresponding to the kinetic energy is sion derived by Jellinek and Li [31, 32] and extended by defined as, Essen [33] in a classical framework. The Hamiltonian (9) is expressed in terms of the ki- N 2 n 2 netic energy as, X P µ X pν T = ⊗ 1e + 1N ⊗ , (88) 2 Mµ 2 m . . µ=1 ν=1 H = T + VN −N (Q ) + Ve−e (q(.)) + VN −e Q , q(.) SO . SO As shown in Appendix J, it is recast as, + VN (Q ,P ., L, S(.)) + Ve (q(.), p(.), s(.)) SO . + VN −e(Q ,P ., L, S(.), q(.), p(.), s(.)) (95) 2 N 00 2 n 00 2 2 P X P µ X pν ~ . T = + + + Φres (Q ) , (89) Taking into account the fact that a norm is invariant 2 M 2 M 2 m 8 µ=1 µ ν=1 under rotation and using the definitions (10), (13) and (40), the nuclear, electronic and interaction Coulomb po- . where Φres (Q ) is a residual operator resulting from the tentials are recast in terms of the internal observables non-commutation of the rotation operator R (ω) and the respectively as, momentum operator P µ that is given by, . VN −N (Q ) . . −2 N Φres (Q ) = Tr (I00) Tr I (Q ) 2 e X Zµ Zν = , . −1 . −1 8πε0 (0) (0) α − 2 Tr I00 · I (Q ) Tr I (Q ) µ,ν=1 k Rµ − Rν 1 + Q (Y µα − Y να) k µ6=ν 2 (90) n + Tr I · I (Q .)−2 − 2 I · I (Q .)−1 e2 X 1 00 α V (q ) = , (96) e−e (.) 8πε k q¯ − q¯ k 0 µ,ν=1 (µ) (ν) . −1 . −2 α β + 2 Tr I (Q ) · Iα · I (Q ) · I0β Q , µ6=ν . VN −e(Q , q(.)) α and the rank-2 tensors I00 and I0β are defined respectively 2 N n e X X Zµ as, = − , 4πε (0) α 0 µ=1 ν=1 k Rµ 1 + Q Y µα − q¯(ν) k N X (0) (0) I00 = Mµ Rµ Rµ , (91) where the orthogonal basis vector Y µα is related to the µ=1 orthonormal basis vector Xµα by, N α X (0) (0) α (0) α p I0β = Rµ Rµ · Xν Xνβ − Rµ · Xνβ Xν Xµα = Mµ Y µα , (97) µ,ν=1 00 (92) and the operator q¯(ν) ≡ r(ν) is a function of the operators q(.) according to the relation (41). As shown in Appendix J, using the appropriate commu- The potential energy operator associated to the spin- tation relations, the kinetic energy (89) is recast in terms orbit coupling between the nuclei is given by, of the internal observables as, N SO . X N . 2 n 2 2 VN (Q ,P ., L, S(.)) = − γµ S(µ) · B(µ)(Q ,P ., L) , P 1 1 X p(ν) T = + Ω·I ·Ω+ P P α+ + ~ Φ (Q .) . µ=1 2 M 2 0 2 α 2 m 8 res ν=1 (98) (93) where γµ > 0 is the gyromagnetic ratio of the nuclei µ N . Finally, using the definition (69), it is useful to recast and B(µ)(Q ,P ., L) is a pseudo-vectorial operator cor- the kinetic energy (93) explicitly in terms of the orbital responding to the internal magnetic field exerted on the 10 nucleus µ and generated by the relative motion of the the molecular inertia tensor I (Q .) are second-order con- other nuclei. Similarly, the potential energy operator as- tributions that we neglect as well. Thus, to zeroth-order sociated to the spin-orbit coupling between the electrons in Qα the inertia tensor I (Q .) reduces, yields, . . I (Q ) = I0 + O (Q ) . (103) n SO X e Using the definitions (68) and (70), and the rela- Ve (q(.), p(.), s(.)) = − γe s(ν) · B(ν)(q(.), p(.)) , ν=1 tion (103), the rotational part of the Hamiltonian H in (99) equation (94) is explicitly recast to zeroth-order in Qα where γe < 0 is the gyromagnetic ratio of the electron as, e and B(ν)(q(.), p(.)) is a pseudo-vectorial operator corre- 1 n . −1 o n . −1 o sponding to the internal magnetic field exerted on the L, I (Q ) · I0 · I (Q ) , L (104) electron ν and generated by the relative motion of the 8 • • other electrons. Finally, the potential energy operator 1 −1 1 −1 −1 . = L · I0 · L + ` · I0 · ` − L · I0 · ` + O (Q ) . associated to the spin-orbit coupling between the nuclei 2 2 and the electrons is given by, According to the relations (93), (95) (101), (102) SO . and (103), the Hamiltonian H is expanded in terms of VN −e(Q ,P ., L, S(.), q(.), p(.), s(.)) (100) the deformation operators Qα as, n X N −e . 1 = − γe s(ν) · B (Q ,P ., L) α α β .3 (ν) H = H(0) + H(α) Q + H(αβ) Q Q + O Q , (105) ν=1 2 N X e−N where − γµ S(µ) · B(µ) (q(.), p(.)) , P2 1 1 µ=1 H = + P P α + L · I−1 · L + V 1 , (0) 2 M 2 α 2 0 N −N (0) where B N −e(Q .,P , L) is a pseudo-vectorial operator n 2 (ν) . X p(ν) 1 + + ` · I−1 · ` + V (q ) , corresponding to the internal magnetic field exerted on 2 m 2 0 e−e (.) the electron ν and generated by the relative motion of the ν=1 e−N −1 nuclei and B(µ) (q(.), p(.)) is a pseudo-vectorial operator − L · I0 · ` + VN −e (0) (q(.)) , (106) corresponding to the internal magnetic field exerted on the nucleus µ and generated by the relative motion of the H(α) = VN −N (α) + VN −e (α) (q(.)) , electrons. H(αβ) = VN −N (αβ) + VN −e (αβ) (q(.)) .

V. MOLECULAR GROUND STATE AND In order to ensure that the molecular dynamics occurs VIBRATIONAL MODES in the neighbourhood the equilibrium ground state, we vary the energy E = h H i of the system with respect to A molecular system is described by bound states de- the deformation modes Qα, where the brackets denote fined in the neighbourhood of the ground state in the the expectation value taken on the Hilbert space (1) of Hilbert space (1). For such states, the vibrational de- the molecular system. At equilibrium, the density matrix grees of freedom are sufficiently small to allow a series commutes with the Hamiltonian (95). We assume that expansion in terms of the deformation operators Qα [1]. this is also the case in the neighbourhood of the equi- To second-order, the series expansion of the Coulomb po- librium state. Thus, to first-order, the variation of the tentials read, energy yields the equilibrium condition, i.e.

. α D E VN −N (Q ) = VN −N (0) 1 + VN −N (α) Q VN −N (α) + VN −e (α) (q(.)) = 0 , (107) 1 + V Qα Qβ + O Q .3 , (101) 2 N −N (αβ) which deﬁnes the ground state of the molecular system. Using the deﬁnitions (96), the series expansions (101) . α VN −e Q , q(.) = VN −e (0) (q(.)) 1 + VN −e (α) (q(.)) Q and (102), and the fact that molecular vibration modes 1 Xµα are linearly independent, the condition (107) im- + V (q ) Qα Qβ + O Q .3 . (102) 2 N −e (αβ) (.) plies that,

2 N (0) (0) To establish the molecular equilibrium conditions deﬁn- e X Rµ − Rν Zν ing the ground state, we neglect the contributions due to 4πε (0) (0) 3 0 ν=1 kRµ − Rν k the spin-orbit couplings. Moreover, the contribution due µ6=ν (108) to the residual term Φ (Q .) in the kinetic energy oper- res 2 n (0) 2 e X Rµ 1 − q¯(ν) ator (94) is proportional to ~ and can be neglected also. − = 0 . α 4πε (0) 3 Furthermore, the eﬀects of the deformation modes Q on 0 ν=1 kRµ 1 − q¯(ν)k 11

The ﬁrst term in the condition (108) represents the clas- the molecule as well as a the position and the momen- sical Coulomb force exerted by the nuclei on the nucleus tum operators of the electrons. µ. The second term in the condition (108) represents the In a rigorous quantum description of a molecule, the expectation value of the quantum Coulomb force exerted molecular orientation and rotation are described by op- by the electrons on the nucleus µ. The equilibrium con- erators. As a result the “rest” momentum operators do dition (108) implies that the resulting force exerted on not commute with the molecular orientation and rota- the nucleus µ vanishes at the equilibrium. tion operators. This generates an additional contribution . The 3N − 6 equilibrium conditions (108) imposed on proportional to the residual term Φres (Q ) in the expres- all the vibrations modes Xµα, the 3 conditions (46) im- sion of the molecular Hamiltonian and leads to canonical posed on the origin of the coordinate system and the 3 rotational commutation relations (74) between the total conditions (49) imposed on the orientation of the coordi- angular momentum operator L and the orientation op- nate system fully determine the 3N degrees of freedom erator ω. The molecular Hamiltonian satisﬁes a “molec- corresponding to the positions of the nuclei. ular correspondence principle” : replacing the operators The molecular deformation modes Xµα and Xνβ are describing physical observables by variables leads to a orthogonal if α 6= β according to the condition (45). classical molecular Hamiltonian where the residual term . Moreover, to second-order with respect to the deforma- Φres (Q ) vanishes. tion Qα, the deformation modes are decoupled. Thus, The rigorous quantum description of the dynamics of the variation of the energy E = h H i with respect to the a rotating and vibrating molecule presented in this pub- deformation Qα yields, lication, is a prelude to the study of quantum dissipation D E at the molecular level [34]. In order to describe molecular VN −N (αβ) + VN −e (αβ) (q(.)) = 0 , ∀ α 6= β . (109) dissipation, the quantum statistical framework provided by the quantum master equations needs to be introduced. As shown in Appendix K, the diagonal components, i.e. In such a framework, where certain internal observables α = β, of the Hessian matrix of the Hamiltonian (95) are treated as a statistical bath that is weakly coupled yield the square of the angular frequency of the molecular and weakly correlated to the other internal observables vibration eigenmodes, i.e. representing the system of interest [35]. The quantum master equations of the molecular system are expected to 2 D E ωα = VN −N (αα) + VN −e (αα) (q(.)) > 0 , (110) lead to dissipative couplings between the rotational, vibrational and magnetic quantum modes and to describe molecular dissipative phenomena such as molecular mag- which ensures that ωα > 0 in the neighbourhood of the molecular ground state. netism [36, 37].

VI. CONCLUSION Appendix A: Killing form of the rotation algebra

A rigorous quantum treatment of a molecular system In this appendix, we determine the explicit expression includes kinetic couplings between nuclei and electrons of the Killing form n(j) (ω) · n(`) (ω) of the rotation al- and leads to quantum deviations in the commutation re- gebra. lations between the position and momentum operators. The rotation group action (31) implies that, These deviations are proportional to the ratio of the electron mass to the molecular mass and to the ratio of a T i T specific nuclear mass to the molecular mass. Thus, these Tr ej · G (ek · G) = e · ej · G (ek · G) ei deviations are larger for smaller molecules. i i In this quantum description, the vibrational and rota- = − e · (ej · G) (ek · G) ei = − e · ej × (ek × ei) tional degrees of freedom are treated in a genuine quan- = (e · e ) · ei · e − (e · e ) ei · e = 2 (e · e ) . tum framework. Since quantum nonlocality forbids the j k i j i k j k (A1) existence of a center of mass frame and a molecular rest frame, we defined “relative” position and momentum observables, which are the quantum equivalent of the rel- Thus, using the identity (A1) and the definitions (30) ative position and momentum variables expressed with and (33), the Killing form is expressed as, respect to the centre of mass frame in a classical framework. In order to define “rest” position and momentum n(j) (ω) · n(`) (ω) observables expressed with respect to a rotating molecule 1 = Tr n (ω) · G T · n (ω) · G we defined a molecular rotation operator that is function 2 (j) (`) (A2) of a molecular orientation operator. Then, we defined 1 = Tr (e · ∂ ) R (ω)−1 · (e · ∂ ) R (ω) . internal observables that account for the quantum de- 2 j ω ` ω grees of freedom of the molecular system. These observables are the deformation and orientation operators of Moreover, the rotation group action (31) implies that 12

∀ x ∈ R3, Finally, the bilinear form (A9) correspond to the Killing form (35) that is expressed in components as (ω · G)2 x = ω × (ω × x) = − ω2 x + ω (ω · x) (A3) e · K (ω) · e ≡ n (ω) · n (ω) = − ω2 (1 − P ) x , j ` (j) (`) ω (A10) = ej · Pω + A (1 − Pω) · e` . where the self-adjoint projection operator Pω satisfies the following identity ∀ n ∈ N?, The components of the symmetric rank-2 tensor K (ω)−1 2 n Pω = Pω ⇒ (1 − Pω) = 1 − Pω . (A4) are given by The projection operator (36) is orthogonal to the rotation j −1 ` j −1 ` e · K (ω) · e = e · Pω + A (1 − Pω) · e . (A11) group action (31), i.e. ω The expressions (A10) and (A11) imply that, Pω ·(ω · G) x = Pω ·(ω × x) = ω ·(ω × x) = 0 . kωk2 K (ω) · ω = ω , (A5) −1 (A12) Using the properties (A3), (A4) and (A5), the rotation K (ω) · ω = ω . operator (28) is recast as, The operator m(j) (ω) is defined as the dual of the ∞ 2n ∞ 2n+1 X (ω · G) X (ω · G) operator n(`) (ω), i.e. R (ω) = 1 + + (2n)! (2n + 1)! (j) j −1 ` n=1 n=0 m (ω) ≡ e · K (ω) · e n(`) (ω) . (A13) ∞ n X (−1) ω2n = 1 + (1 − Pω) The expressions (A10) and (A13) yield the duality con- (2n)! n=1 (A6) dition (34), i.e. ∞ n 1 X (−1) ω2n+1 + (ω · G) m(j) (ω) · n (ω) ω (2n + 1)! (`) n=0 j −1 k sin ω = e · K (ω) · e n(k) (ω) · n(`) (ω) (A14) = cos ω + (1 − cos ω) Pω + (ω · G) . ω j −1 k j = e · K (ω) · e (ek · K (ω) · e`) = e · e` . The definition (A6), the trace properties The definitions (28) and (33) imply that Tr (Pω) = 1 , j e · ω n(j) (ω) · G Tr (ω · G) = 0 , j −1 = e · ω R (ω) · (ej · ∂ω) R (ω) (A15) (ω0 · ω)2 −1 −1 Tr (Pω0 · Pω) = , (A7) ω02 ω2 = R (ω) · (ω · ∂ω) R (ω) = R (ω) · (ω · G) · R (ω) Tr (ω0 · G)(ω · G) = − 2 ω0 · ω , = ω · G , Tr Pω0 · (ω · G) = 0 , which yields the identity, j and trigonometric identities imply that, e · ω n(j) (ω) = ω . (A16)

0 −1 Moreover, the Killing form (A10) and the identity (A16) Tr R (ω ) · R (ω) (A8) then imply that, 0 0 0 2 j ω ω ω ω ω ω ω · n(`) (ω) = e · ω n(j) (ω) · n(`) (ω) = 4 cos cos + 0 sin sin − 1 . 2 2 ω · ω 2 2 j = e · ω (ej · K (ω) · e`) = ω · e` . (A17) The relations (A2) and (A8) with the help of trigonomet- Using the deﬁnition (A13), the ﬁrst property (A12) ric identities then imply in turn that, yields,

n(j) (ω) · n(`) (ω) (j) j −1 ` (ej · ω) m (ω) = (ej · ω) e · K (ω) · e n(`) (ω) 1 0 −1 = (ej · ∂ω)(e` · ∂ω0 ) Tr R (ω ) · R (ω) δωω0 ` 2 = e · ω n(`) (ω) = ω . (A18) = 2 (e · ∂ )(e · ∂ 0 ) (A9) j ω ` ω Moreover, the identity (A18) and the Killing form (A10) ω0 ω ω0 ω ω0 ω 2 then imply that, cos cos + 0 sin sin δωω0 2 2 ω · ω 2 2 (j) j −1 ` ! ω · m (ω) = ω · e · K (ω) · e n(`) (ω) (A19) ω ω 2 kωk2 ω ω = ej · 2 + sin 1 − 2 · e` . j −1 ` j kωk kωk 2 kωk = e · K (ω) · e (ω · e`) = e · ω . 13

Appendix B: Commutation relations of the Performing a similar calculation using the properties (30) momentum and orientation operators and (33) yields,

In this appendix, we determine the commutations re- h 0 −1 i e` · P , R (ω) (B6) 00 00 ν lations of the momentum operators P µ and pν with the 0 j −1 orientation ω. = − e` · P ν , e · ω n(j) (ω) · G · R (ω) , In order to determine these relations, we use the Baker- Campbell-Hausdorﬀ formulas, i.e. 00 e` · P , R (ω) (B7) ∞ ν X − X X 1 00 j e Y e = [ X, Y ] , = e` · P ν , e · ω R (ω) · n(j) (ω) · G , k! k k=0 (B1) ∞ k − X X X (− 1) h 00 −1 i e Y e = [ X, Y ] , e` · P , R (ω) (B8) k! k ν k=0 00 j −1 = − e` · P , e · ω n(j) (ω) · G · R (ω) . where ν [ X, Y ] = [ X, [ X, Y ]] , The condition (50) implies that, k k−1 (B2) [ X, Y ] = Y . 0 " N # 0 X (0) 0 According to the properties (28), (30) and the formu- e` · P ν , Mµ R (ω) · Rµ × Rµ = 0 , (B9) las (B1), µ=1

0 e` · P ν , R (ω) which is expanded as,

0 0 = R (ω) · exp (− ω · G) e` · P ν exp (ω · G) − e` · P ν N X 0 (0) 0 ∞ Mµ e` · P ν , R (ω) · Rµ × Rµ X (− 1)k = R (ω) · ω · G, e · P 0 (B3) µ=1 k! ` ν k (B10) k=1 N X (0) 0 0 ∞ k + Mµ R (ω) · Rµ × e` · P ν , Rµ = 0 . j 0 X (− 1) = R (ω) · e · ω, e · P [ ω · G, e · G ] µ=1 ` ν k! j k−1 k=1 where the k = 0 term cancels out on the third line and the The commutator (B5) and the rotation group action (31) orientation operator is a function of the rest positions of imply that, the nuclei R(0) and the deformation operators R0 . Using µ µ N the relation, X 0 (0) 0 Mµ e` · P ν , R (ω) · Rµ × Rµ [ ej · ∂ω, ω · G ] x µ=1 0 j = (ej · ∂ω)(ω × x) − ω × (ej · ∂ω) x (B4) = e` · P ν , e · ω = (ej · ∂ω) ω × x = ej × x = (ej · G) x , N X (0) 0 · Mµ R (ω) · n(j) (ω) · G Rµ × Rµ (B11) the Baker-Campbell-Hausdorﬀ formula (B1) and the µ=1 group identity (33), the commutator (B3) is recast as, 0 j = e` · P ν , e · ω 0 e` · P , R (ω) ν N ∞ k X (0) 0 X (−1) · Mµ R (ω) · n(j) (ω) × Rµ × Rµ , = − R (ω) · ej · ω, e · P 0 [ ω · G, e · ∂ ] ` ν k! j ω k µ=1 k=0

0 j the commutation relations (24) and the constraint (46) = R (ω) · e` · P ν , e · ω imply that,

N · exp (− ω · G)(ej · ∂ω) exp (ω · G) X (0) 0 0 Mµ R (ω) · Rµ × e` · P ν , Rµ µ=1 0 j −1 N = R (ω) · e` · P , e · ω R (ω) · (ej · ∂ω) R (ω) X Mν ν = − i M R (ω) · R(0) × δ − e ~ µ µ νµ M ` µ=1 0 j = e` · P , e · ω R (ω) · n(j) (ω) · G . (B5) (0) ν = − i~ Mν R (ω) · Rν × e` . (B12) 14

The identities (B10), (B11) and (B12) yield, tion (31) imply that,

N N X X 0 (0) 0 0 j (0) 0 Mµ [ e` · pν , R (ω)] · Rµ × Rµ e` · P ν , e · ω Mµ R (ω)· n(j) (ω)×Rµ ×Rµ µ=1 µ=1 0 j (0) = e` · pν , e · ω = i~ Mν R (ω) · Rν × e` . (B13) N X (0) 0 · Mµ R (ω) · n(j) (ω) · G Rµ × Rµ (B20) ` Multiplying the commutation relation (B13) by e ·R (ω)· µ=1 0 00 ek and using the relation (26) between P µ and P µ yields, 0 j = e` · pν , e · ω N X N · M R (ω) · n (ω) × R(0) × R0 , 00 j X (0) 0 µ (j) µ µ e` · P ν , e · ω Mµ R (ω)· n(j) (ω)×Rµ ×Rµ µ=1 µ=1 = i M R (ω) · R(0) × (R (ω) · e ) (B14) the commutations relations (24) and the constraint (46) ~ ν ν ` imply that, (0) = i Mν R (ω) · R × e` . ~ ν N X (0) 0 0 Mµ R (ω) · Rµ × e` · pν , Rµ 0 00 µ=1 The relation (25) between Rµ and Rµ implies that, N X (0) m = − i Mµ R (ω) · R × e` = 0 . (B21) (0) 0 ~ µ M R (ω) · n(j) (ω) × Rµ × Rµ µ=1 (0) 00 = R (ω) · n(j) (ω) × Rµ × R (ω) · Rµ (B15) The identities (B19), (B20) and (B21) yield, (0) 00 = R (ω) · n(j) (ω) × Rµ × Rµ . 0 j e` · pν , e · ω N X (0) 0 (B22) The identities (B14) and (B15) imply in turn that, · Mµ R (ω) · n(j) (ω) × Rµ × Rµ = 0 . µ=1 N 00 j X (0) 00 Finally, by analogy with the identities (B13)-(B16), we e` · P ν , e · ω Mµ n(j) (ω) × Rµ × Rµ µ=1 obtain the last commutation relation (60), i.e. (0) = i~ Mν Rν × e` . (B16) 00 j e` · pν , e · ω = 0 . (B23)

Now, by analogy with the commutation relation (B5), Appendix C: Commutation relations of the position 0 0 j and momentum operators [ e` · pν , R (ω) ] = e` · pν , e · ω R (ω)· n(j) (ω)·G (B17) The condition (50) implies that, In this appendix, we determine the commutations re- 00 lations of the momentum operator P µ with the position 00 00 00 " N # operators Rν and rν , and the momentum operators P ν 0 X (0) 0 00 e` · pν , Mµ R (ω) · Rµ × Rµ = 0 , (B18) and pν respectively. µ=1 Using the relations (25) and (26), the commutation relation yields, which is expanded as, 00 k 00 ej · P µ, e · rν N h 0 k −1 0 i X = (R (ω) · ej) · P µ, e · R (ω) · rν (C1) M [ e · p0 , R (ω)] · R(0) × R0 µ ` ν µ µ h i µ=1 ` 0 k −1 m 0 (B19) = e · R (ω) · ej e` · P µ, e · R (ω) · em e · rν N X (0) 0 0 ` k −1 0 m 0 + Mµ R (ω) · Rµ × e` · pν , Rµ = 0 . + e · R (ω) · ej e · R (ω) · em e` · P µ, e · rν µ=1 Using the commutation relations (24) and (B6), the def- The commutator (B17) and the rotation group ac- inition of the group action (31) and the relations (25) 15 and (26), the commutation relation (C1) yields the ﬁrst According to equation (B7), commutation relation (52), i.e. 00 ` 00 k 00 ej · P µ, e · R (ω) · ek (C8) ej · P µ, e · rν 00 m ` ` 0 s = ej · P µ, e · ω e · R (ω) · n(m) (ω) · G · ek . = − e · R (ω) · ej e` · P µ, e · ω k −1 m 00 Introducing the rank-2 tensorial operator, · e · n(s) (ω) · G · R (ω) · em (e · R (ω) · rν )

−1 Mµ 00 m + e` · R (ω) · e ek · R (ω) · e i (e · em) 1 Aµ(j) = ej · P µ, e · ω n(m) (ω) · G , (C9) j m ~ ` M 00 s k 00 = − ej · P µ, e · ω e · n(s) (ω) × rν (C2) and using the relation (C8), the commutator (C7) is recast as, k Mµ + i~ ej · e 1 . M 00 00 ej · P , ek · P = (C10) Similarly, using the relations (25) and (26), the commu- µ ν 1 n o tation relation yields, e · P 0 , e` · R (ω) · e en · A · e 2 ` ν n µ(j) k 00 00 ej · P µ, ek · pν 1 n 0 ` n o − e` · P , e · R (ω) · en e · A · ej . 0 0 2 µ ν(k) = (R (ω) · ej) · P µ, (R (ω) · ek) · pν (C3) ` 0 m 0 = e · R (ω) · ej e` · P µ, e · R (ω) · ek em · pν The relation (26) implies that the commutation rela- ` m 0 0 tion (C10) reduces to, + e · R (ω) · ej (e · R (ω) · ek) e` · P µ, em · pν . Using the commutation relations (24) and (B5), the deﬁ- 1 n o e · P 00, e · P 00 = e · P 00, e` · A · e nition of the group action (31) and the relation (26), the j µ k ν 2 ` ν µ(j) k commutation relation (C3) yields the second commuta- 1 n o − e · P 00, e` · A · e . (C11) tion relation (52), i.e. 2 ` µ ν(k) j 00 00 ej · P µ, ek · pν Finally, using the expression (C9) of the rank-2 tensor ` 0 s = e · R (ω) · ej e` · P µ, e · ω (C4) Aµ(j) and the deﬁnition of the rotation group action (31) the commutation relation (C11) yields the last commu- m −1 00 · ek · n(s) (ω) · G · (e · R (ω)) R (ω) · em · pν tation relation (52), i.e. 00 s 00 = ej · P µ, e · ω ek · n(s) (ω) × pν . 00 00 ej · P µ, ek · P ν = (C12) By analogy with the commutation relation (C1), using 1 n 00 00 m ` o the relations (25) and (26), the commutation relation e` · P , ej · P , e · ω n (ω) × ek · e 2 ν µ (m) yields, 1 n 00 00 m ` o 00 k 00 − e` · P µ, ek · P ν , e · ω n(m) (ω) × ej · e ej · P µ, e · Rν (C5) 2 ` h 0 k −1 i m 0 = e · R (ω) · ej e` · P µ, e · R (ω) · em e · Rν

` k −1 0 m 0 Appendix D: Internal observables + e · R (ω) · ej e · R (ω) · em e` · P µ, e · Rν By analogy with the commutation relation (C2), using In this appendix, we determine the expressions for the α the commutation relations (24) and (B6), the deﬁnition internal observables Q , Pα and Ω. The deﬁnition (40) of the group action (31) and the relations (25) and (26) and the constraint (45) imply that yield the third commutation relation (52), i.e. N N Mµ X p α 00 X p α (0) α 00 k 00 k Mµ Xµ · Rµ = Mµ Xµ · Rµ 1 + Q ej · P µ, e · Rν = − i~ ej · e δµν − 1 M µ=1 µ=1 00 s k 00 N n − ej · P , e · ω e · n(s) (ω) × R . (C6) µ ν m X X p α − M A 0 X · q 0 . (D1) M µ νν µ (ν ) Using the relation (26), the commutation relation of the µ=1 ν,ν0=1 momentum operators is expressed as, 00 00 Using the constraint (47), the identity (D1) yields the ej · P , ek · P = µ ν expression (53) for the internal observable Qα, i.e. 1 n 0 00 ` o e` · P ν , ej · P µ, e · R (ω) · ek (C7) 2 N 1 n o α X p α 00 (0) 0 00 ` Q = Mµ X · R − R 1 . (D2) − e` · P µ, ek · P ν , e · R (ω) · ej . µ µ µ 2 µ=1 16

α Similarly, the deﬁnition (43) and the constraint (45) im- Q and q(ν0) implies that, ply that, N X M R(0) × R00 N 1 µ µ µ X 00 µ=1 p Xµα · P µ = µ=1 Mµ N X (0) (0) N = Mµ R × R 1 X 1 µ µ X · Ω × M R(0) 1 + P (D3) µ=1 p µα µ µ α (E1) µ=1 Mµ N ! α X p (0) N n + Q Mµ Rµ × Xµα m X X p − M A 0 X · p 0 . µ=1 M µ νν µα (ν ) µ=1 ν,ν0=1 N !  n  X (0) m X − M R × A 0 q 0 . µ µ M νν (ν ) Using the constraints (47) and (48) the identity (D3) µ=1 ν,ν0=1 yields the expression (53) for the internal observable Pα, i.e. Using the Eckart conditions (46) and (48), the RHS of the relation (E1) vanishes, i.e. N X 1 P = X · P 00 . (D4) N α p µα µ X (0) 00 µ=1 Mµ Mµ Rµ × Rµ = 0 . (E2) µ=1 The relation (26), the constraints (46)-(48) and the Finally, using the relation (25) the condition (E2) is re- deﬁnition (55) imply that, cast as,

N N N X −1 X (0) 00 X (0) (0) M R(0) × R (ω) · R0 = 0 , (E3) Rµ × P µ = Rµ × Ω × Mµ Rµ µ µ µ µ=1 µ=1 µ=1 N which is equivalent to the condition k X (0) (0) ` = e · Ω Mµ e` · Rµ × ek × Rµ e N µ=1 X M R (ω) · R(0) × R0 = 0 . (E4) N µ µ µ k X (0) (0) ` µ=1 = e · Ω Mµ e` × Rµ · ek × Rµ e µ=1 The condition (E3) is the physical deﬁnition (50) of the rotation operator R (ω). k ` = e · Ω (e` · I0 · ek) e , (D5)

Appendix F: Commutation relations of the internal which implies in turn that, observables

N X In this appendix, we determine the commutation rela- e × R(0) · P 00 = (e · I · e ) ej · Ω . (D6) α k µ µ k 0 j tions between the internal observables Q , Pα, q(ν), p(ν), µ=1 ω and Ω. Using the definition (D4) of the operator Pα, the con- Using the property (56) of the inertia tensor I0, the iden- straint (48) and the identity (B16) imply that, tity (D6) yields the internal observable ek · Ω, i.e. N j X (0) 00 N (0) ! Pα, e · ω Mµ n(j) (ω) × Rµ × Rµ X ek × Rµ ek · Ω = · P 00 . (D7) µ=1 e · I · e µ (F1) µ=1 k 0 k N X p (0) = i~ Mν Rν × Xνα = 0 , ν=1 Appendix E: Rotation operator which yields the fifth commutation relation (60), i.e. j In this appendix, we show that the Eckart condi- Pα, e · ω = 0 . (F2) tions (46) and (48), and the relations (25) and (40) imply Moreover, the definition (44) and the commutation rela- the physical definition (50) of the rotation operator R (ω). tion (B23) imply that, 00 The expression (40) of the rest momentum Rµ in terms h i (0) ek · p , ej · ω = 0 . (F3) of the equilibrium position Rµ and the rest observables (ν) 17

Using the expression (D7) for the operator ek · Ω, the properties (48), (54)-(56), we obtain, constraint (46) and the diagonality condition (56), the N identity (B16) is recast as, X (0) 00 Mµ n(j) (ω) × Rµ × Rµ · e` µ=1 N X N ek · Ω, ej · ω M n (ω) × R(0) × R00 X (0) 00 µ (j) µ µ = − Mµ n(j) (ω) × Rµ · e` × Rµ (F8) µ=1 µ=1 N (0) !! N X ek × Rµ = i M R(0) × (F4) k X (0) (0) ~ ν ν = − e · n(j) (ω) Mµ ek × Rµ ek × Rµ ek · I0 · ek ν=1 µ=1 ek · I0 · e` ` k N = i~ e = i~ e . k X p (0) α ek · I0 · ek − e · n(j) (ω) Mµ ek × Rµ (ek × Q Xµα) µ=1 k = − e · n(j) (ω) The condition (48) implies that, k α · (ek · I0 · ek) e · e` + Q (ek · Iα · e`) N X p (0) M e · e × R × X = 0 , (F5) . µ k ` µ µα = − n(j) (ω) · I (Q ) · e` . µ=1 Using the identity (F8), the identity (F4) is recast as, k j . k which is recast as, e · Ω, e · ω n(j) (ω) · I (Q ) · e` = − i~ e · e` . (F9) Using the property (A14), which implies that, N X p (0) −1 Mµ ek · Rµ (e` · Xµα) . . −1 (j) n(j) (ω) · I (Q ) = I (Q ) · m (ω) , (F10) µ=1 (F6) N the identity (F9) yields the ﬁfth commutation rela- X p (0) = Mµ e` · Rµ (ek · Xµα) . tion (62), i.e. µ=1 k j k . −1 (j) e · Ω, e · ω = − i~ e · I (Q ) · m (ω) . Using the identity (F6), the expression (57) yields the (F11) property (58), i.e. The relation (43), the commutation relations (B23), (F2) and (F11) imply that, 00 j N e` · P µ, e · ω X p (0) ek · Iα · e` = Mµ ek × Rµ · (e` × Xµα) k j (0) = e · Ω, e · ω ek × Mµ Rµ · e` µ=1 j p α N + Pα, e · ω Mµ e` · Xµ (F12) X p (0) = Mµ Rµ · Xµα (ek · e`) n X j Mµ µ=1 − A 0 e · p 0 , e · ω νν ` ν M ν,ν0=1 (0) − (ek · Xµα) e` · Rµ k . −1 (j) (0) = − i~ e · I (Q ) · m (ω) ek × Mµ Rµ · e`

N Now, we establish a useful commutation relation for the X p (0) = Mµ Rµ · Xµα (ek · e`) (F7) dynamics. The identities (A14), (B8) and (F12) imply µ=1 that,

(0) h 00 −1 i − (e` · Xµα) ek · Rµ e` · P µ, R (ω) R (ω) 00 j N = − e` · P µ, e · ω n(j) (ω) · G (F13) X p (0) = M e × R · (e × X ) = e · I · e , k . −1 (j) µ ` µ k µα ` α k = i~ e · I (Q ) · m (ω) µ=1 (0) · ek × Mµ Rµ · e` n(j) (ω) · G . which shows that the tensors Iα and I (Q ) are symmet- = i ek · I (Q .)−1 · ej e × M R(0) · e (e · G) ric. Using the relation (40), the constraint (46) and the ~ k µ µ ` j 18

The orthogonality condition (30) and the commutation commutation relation (60), i.e. relation (F13) imply that, h i ` 00 k m −1 k P , e · q 0 e · P µ, e · R (ω) · e` e · R (ω) · ek (F14) α (ν ) h i N 00 ` −1 k n m 1 = e` · P µ, e · R (ω) · ek e · R (ω) · en (e · e ) X 00 s = − p Xµα · P µ, e · ω Mµ k . −1 j (0) m µ=1 = i e · I (Q ) · e ek × Mµ R · (ej × e ) ~ µ n X −1 k 00 k . −1 j (0) m · Aν0ν e · n(s) (ω) × rν (F18) = − i~ e · I (Q ) · e ej × ek × Mµ Rµ · e ν=1 N n p The deﬁnitions (42) and (53), the commutation rela- X X Mµ −1 k + i A 0 X · e 1 tion (51) and the condition (47) yield the commutation ~ M ν ν µα µ=1 ν=1 relations (59), i.e. n s X −1 k 00 = − [ P , e · ω ] A 0 e · n (ω) × r = 0 . Qα,Qβ = 0 , α ν ν (s) ν ν=1 h j k i e · q(ν), e · q(ν0) = 0 , h i The deﬁnitions (44) and (D4) and the commutation re- e · p , e · p 0 = 0 , j (ν) k (ν ) lations (C4) and (F2) yield the third commutation rela- h j β i tion (60), i.e. e · q(ν),Q = 0 , (F15)

h β i ej · p(ν),Q h i Pα, ek · p(ν0) n N mpM X X µ −1 β N = i A 0 e · X 1 = 0 , ~ M νν j µ X 1 00 s ν0=1 µ=1 = p Xµα · P µ, e · ω (F19) µ=1 Mµ h k i ej · p(ν), e · q(ν0) n X −1 00 n · A 0 ek · n (ω) × p X m ν ν (s) ν −1 −1 k ν=1 = − i~ Aνν δνν0 − Aν0ν0 ej · e 1 . M n ν,ν0=1 s X −1 00 = [ Pα, e · ω ] Aν0ν ek · n(s) (ω) × pν = 0 . The deﬁnitions (16) and (42) yield the relation, ν=1

n X −1 m −1 The definitions (D2) and (D4), the constraint (47) and A δ 0 − A 0 νν νν M ν0ν ν,ν0=1 the commutation relation (C6) yield the first canonical n r !! commutation relation (61), i.e. X 1 M m = δ + − 1 δ 0 − νν n M νν M ν,ν0=1 β Pα,Q r !! 1 M N · δν0ν0 + − 1 (F16) X 1 00 s n M = − p Xµα · P µ, e · ω µ=1 Mµ  r ! r !2 1 m M M N = − 2 − 1 + − 1  X p β 00 n M M M · Mν Xν · n(s) (ω) × Rν (F20) ν=1 m N √ − + δνν0 = δνν0 , X Mν Mµ M − i X · Xβ δ − 1 ~ p µα ν µν M µ, ν=1 Mµ which implies that the last commutation relation (F15) N yields the second canonical commutation relation (61), X = − i X · Xβ 1 = − i δβ . i.e. ~ µα µ ~ α µ=1 h k i k ej · p(ν), e · q(ν0) = − i~ ej · e δνν0 . (F17) The definitions (41) and (D3), the properties (46) The definitions (41) and (D4), the constraint (47), and and (D7), the commutation relations (C2) and (F11) the commutation relations (C2) and (F2) yield the fourth and the property (A14) yield the third commutation re- 19 lation (62), i.e. yield the first commutation relation (62), i.e.

ek · Ω,Qα " N (0) # X ek × Rµ h i = − · P 00, es · ω k j e · I · e µ e · Ω, e · q(ν0) µ=1 k 0 k " N (0) # N X ek × Rµ 00 s X p α 00 = − · P , e · ω · Mν X · n (ω) × R (F23) e · I · e µ ν (s) ν µ=1 k 0 k ν=1 n ! N N (0) ! X X p Mµ ek × Rµ j X −1 00 − i M δ − · Xα 1 · e · n(s) (ω) × Aν0ν rν (F21) ~ ν µν ν M ek · I0 · ek ν=1 µ=1 ν=1   N N e × M R(0) n ! X k µ µ X k s X p 00 α j −1 = − i~ e · Ω, e · ω Mν n(s) (ω) × Rν · Xν + i~   · e Aν0ν 1 M (e · I · e ) ν=1 µ=1 k 0 k ν=1 k . −1 (s) k s j = i~ e · I (Q ) · m (ω) = e · Ω, e · ω n(s) (ω) · e × q(ν0) N k . −1 (s) j X p 00 α = − i~ e · I (Q ) · m (ω) n(s) (ω) · e ×q(ν0) · Mν Rν × Xν · n(s) (ω) ν=1 k . −1 ` j = − i~ e · I (Q ) · e e` × e · q(ν0) . N β k . −1 ` X α = i~ Q e · I (Q ) · e (Xνβ × Xν ) · e` ν=1 N X k . −1 α β = − i~ e · I (Q ) · Xµ × Xµβ Q . µ=1 Similarly, the definition (44), the properties (A14) and (D7), and the commutation relations (C4) and (F11) yield the fourth commutation relation (62), i.e. The definition (D4) and the commutation relations (C12) and (F2) yield the first commutation relation (60), i.e.

[ Pα,Pβ ] h k i N e · Ω, e · p 0 X 1 1 j (ν ) = √ X · P 00, X · P 00 p M µα µ νβ ν " N (0) # µ, ν=1 Mµ ν X ek × Rµ 00 s = − · P , e · ω N e · I · e µ 1 X 1 n µ=1 k 0 k = √ e · P 00, [ P , em · ω ] 2 M ` ν α n ! ν=1 ν X −1 00 · ej · n(s) (ω) × Aν0ν pν (F22) ` o ν=1 n(m) (ω) × Xνβ · e (F24) k s = e · Ω, e · ω n(s) (ω) · ej × p(ν0) 1 N 1 n −1 X 00 m k . (s) − e` · P , [ Pβ, e · ω ] = − i~ e · I (Q ) · m (ω) n(s) (ω) · ej ×p(ν0) 2 p ν µ=1 Mµ k . −1 ` = − i~ e · I (Q ) · e (e` × ej) · p(ν0) . ` o n(m) (ω) × Xµα · e = 0 .

The definition (D4), the commutation rela- The definition (D2), the properties (47), (48), (A14) tion (C12), (F2) and (F11), and the properties (45) and (D7), and the commutation relations (C6) and (F11) and (A14), the expressions (D6)-(D7) yield the second 20 commutation relation (60), i.e. Using the relation (G3) and the definition (G2) of the orbital angular momentum L0 yields the expression (65) k e · Ω,Pα for the orbital angular momentum L, i.e. N 1 X 1 = √ e · P 00, ek · Ω, em · ω 2 M ` ν ν=1 ν

N n · n (ω) × X · e` 1 X 1 X (m) να L = R00, P 00 + [ r00, p00 ] . (G4) 2 µ µ × 2 ν ν × µ=1 ν=1 N 1 X 1 n 00 k . −1 (m) = − i~ √ e` · P ν , e · I (Q ) · m (ω) 2 M ν=1 ν

` · n(m) (ω) × Xνα · e Using the relations (40)-(44) the expression (G4) is recast as, N 1 X 1 00 ` = − i~ √ e` · P ν (em × Xνα) · e , 2 M ν=1 ν

−1 o N  n  · ek · I (Q .) · em 1 X 1 m X L = R(0) 1 + QαX − A q 2  µ p µα M νν (ν) µ=1 Mµ N ν, ν=1 1 X 1 n 00 β   = − i~ √ Xµβ · P ν Xνα × Xµ · em, n 2 Mν (0) p β Mµ X µ,ν=1 × Ω× M R + M P X − A 0 0 p 0  µ µ µ β µ M ν ν (ν ) ν0, ν0=1 k . −1 m o · e · I (Q ) · e  n  (0) p β Mµ X − Ω× Mµ Rµ + Mµ PβXµ − Aν0ν0 p(ν0) N M ν0, ν0=1 1 X h k . −1 β i = − i~ e · I (Q ) · Xµα,Pβ Xµ . (F25) 2 ×  n  µ=1 (0) 1 α m X ×Rµ 1 + Q Xµα − Aνν q(ν) pM M µ ν, ν=1 Appendix G: Orbital angular momentum n n ! n !! 1 X X X + A q × A 0 p 0 2 νν (ν) νν (ν ) In this appendix, we establish an explicit expression of ν=1 ν=1 ν0=1 n n ! n !! the orbital angular momentum in terms of the internal 1 X X X observables. − Aνν0 p 0 × Aνν q . 2 (ν ) (ν) According to the first and last relations (24), the com- ν=1 ν0=1 ν=1 j 0 (G5) mutation relations between the operators e · Rµ and 0 j 0 0 ek · P µ and the operators e · rν and ek · pν are symmetric with respect to the permutation of the basis vectors j e and ek, which implies that the vector product of these operators satisfies the identities, Using the definitions (56)-(57) and the proper- 0 0 0 0 ties (17), (46)-(48) and (F7), L is recast as, Rµ × P µ = − P µ × Rµ , 0 0 0 0 (G1) rν × pν = − pν × rν . Thus, using the relation (G1) the angular momen- ` k tum (64) is recast as, L = e (e` · I0 · ek) e · Ω

N n 1 ` α k + e (e` · Iα · ek) Q , e · Ω 0 1 X 0 0 1 X 0 0 2 L = Rµ, P µ + [ rν , pν ]× . (G2) 2 × 2 N µ=1 ν=1 1 X h α β i + Q Xµα,Pβ Xµ (G6) 2 × The angular momentum L is a pseudo-vectorial operator µ=1 0 that is related to the angular momentum L by, n  n  1 X X m h i 1 +  Aνν δνν0 + Aν0ν0 q(ν), p(ν0) 0 0 × e · L = (R (ω) · e ) · L + L · (R (ω) · e ) . (G3) 2 0 0 M ` 2 ` ` ν, ν =1 ν, ν =1 21

The definitions (16) and (42) yield the identity and the expression (G8) yield the first commutation relation (71), i.e. n X m A δ 0 + A 0 0 νν νν M ν ν 1 ν, ν0=1 α . k α [ L,Q ] = I (Q ) · ek, e · Ω,Q n r !! 2 X 1 M m N = δνν + − 1 δνν0 + 1 X n M M + X × Xβ { Qγ , [ P ,Qα ] } (H3) ν, ν0=1 2 γµ µ β r !! µ=1 1 M ( N ) · δν0ν0 + − 1 (G7) 1 −1 X n M = I (Q .) , − i I (Q .) · Xα × X Qβ 2 ~ µ µβ µ=1  r ! r !2 • 1 m M M N = + 2 − 1 + − 1  X α β n M M M − i~ Xµβ × Xµ Q µ=1 m + + δνν0 = δνν0 . N M X α α β = − i~ Xµ × Xµβ + Xµβ × Xµ Q = 0 . Using the identity (G7) and the relation (54), the ex- µ=1 pression (G6) yields the expression (67) for the orbital α angular momentum, i.e. Since the operator q(ν) commutes with the operators Q and Pβ, the commutation relations (59)-(57) and the ex- N 1 n . o 1 X h α β i pression (G8) yield the third commutation relation (71), L = I (Q ) , Ω + Q Xµα,Pβ Xµ i.e. 2 • 2 × µ=1 n h i 1 X h j i 1 n . h k j i o + q(ν), p(ν) . (G8) L, e · q 0 = I (Q ) · ek, e · Ω, e · q 0 2 × (ν ) (ν ) ν=1 2 n 1 X h h j i i + q(ν), p(ν), e · q(ν0) (H4) 2 × ν=1 Appendix H: Commutation relations of orbital 1 n . k . −1 j o angular momentum and the internal observables = I (Q ) · e , − i e · I (Q ) · e × q 0 2 k ~ (ν ) n 1 X h j i In this appendix, we determine explicitly the commu- + q(ν), − i~ δνν0 e 2 × tation relations between the orbital angular momentum ν=1 α L and the internal observables Q , Pα, q(ν), p(ν), ω and j j = − i e × q 0 + q 0 × e = 0 . Ω. ~ (ν ) (ν ) Since the operator Ω commutes with the operators Qα, Pβ, q and p , the commutation relations (59)-(57) α (ν) (ν) Since the operator p(ν) commutes with the operators Q and the expression (G8) yield the first commutation re- and Pβ, the commutation relations (60)-(57) and the ex- lation (72), i.e. pression (G8) yield the fourth commutation relation (71), i.e. 1 L, e` · ω = I (Q .) · e , ek · Ω, e` · ω 2 k h i 1 n . h k i o 1 n . k . −1 (`) o L, ej · p(ν0) = I (Q ) · ek, e · Ω, ej · p(ν0) = I (Q ) · ek, − i~ e · I (Q ) · m (ω) 2 2 n (`) 1 X h h i i = − i~ m (ω) . (H1) + q(ν), ej · p(ν0) , p(ν) (H5) 2 × ν=1

The commutation relation (H1) and the property (A14) 1 n . k . −1 j o = I (Q ) · e , − i e · I (Q ) · e × p 0 yield the canonical commutation relation in rotation (74), 2 k ~ (ν ) i.e. n 1 X h i + i~ δνν0 ej, p(ν) ` ` 2 × n(k) (ω) · L, e · ω = − i~ ek · e , (H2) ν=1 = i p 0 × e + e × p 0 = 0 . ` ~ (ν ) j j (ν ) which means that n(k) (ω) · L and e · ω are canonically conjugated operators. α α Since the operator Q commutes with the operators Since the operator Pα commutes with the operators Q γ Q , q(ν) and p(ν), the commutation relations (59)-(57) and p(ν), the commutation relations (60)-(57) and the 22 expression (G8) imply that which is recast as,

1 . k [ L,Pα ] = I (Q ) · ek, e · Ω,Pα [ L, ek · L ] = i~ (ek × L) . (H13) 2 N α 1 X β γ Since the operator L commutes with the operators Q , + Xγµ × Xµ { [ Q ,Pα ] ,Pβ } (H6) α 2 Pα, q , pα, the commutation relations (H3)-(H5) and µ=1 (H8), and the expression (G8) imply that ( N ) 1 . 1 X h . −1 β i = I (Q ) , − i~ I (Q ) · Xµα,Pβ Xµ 2 2 × 1 . µ=1 [ L, ek · L ] = [ { I (Q ) , Ω } , ek · L ] • 2 • N X 1 . ` β = I (Q ) · e`, e · Ω , ek · L + i~ Xµα × Xµ Pβ 2 • µ=1 1 = I (Q .) · e , e` · Ω, e · L (H14) N ! 2 ` k 1 . . −1 X β = − i~ [ I (Q ) ,Pβ ] · I (Q ) · Xµα × Xµ 1 4 + e` · Ω, [ I (Q .) · e , e · L ] . µ=1 2 ` k N ! 1 −1 X − i I (Q .) · X × Xβ · [ P , I (Q .)] . The deﬁnition (54) and the commutation relation (H3) 4 ~ µα µ β µ=1 imply that,

Now, the deﬁnition (54) implies that, . α [ I (Q ) · e`, ek · L ] = [ I0 · e` + Q (Iα · e`) , ek · L ] . γ α [ I (Q ) ,Pβ ] = Iγ [ Q ,Pβ ] = i~ Iβ . (H7) = Iα · e` [ Q , ek · L ] = 0 , (H15) Substituting the commutation relation (H7) into the commutation relation (H6), the latter yields the second which implies in turn that, commutation relation (71), i.e.

h ` . −1 i [ L,Pα ] = 0 . (H8) e · I (Q ) , ek · L = 0 . (H16) The commutation relation (B7) applies for the operator 00 The commutation relation (H15) implies that the com- L as well as for the operator P ν . The commutation relations (B7) and (H1) and the property (A14) imply mutation relation (H14) is recast as, that, 1 . ` −1 j [ L, ek · L ] = I (Q ) · e`, e · Ω, ek · L . R (ω) [ L, R (ω) ] = L, e · ω n(j) (ω) · G 2 (H9) (H17) (j) = − i~ m (ω) · n(j) (ω) G = − i~ G . The commutation relations (H13) and (H16)-(H17) imply that, The commutation relation (H9) implies that, ` ` . −1 ek · L, e · Ω = − e · I (Q ) · [ L, ek · L ] [ ej · L, R (ω) ] = − i~ R (ω)(ej · G) . (H10) ` . −1 j Now, the commutation relations (32) and (H10) imply = − i~ e · I (Q ) · e (ej · (ek × L)) (H18) that, ` . −1 j = i~ e · I (Q ) · e (ek · (ej × L)) , [[ ej · L, ek · L ] , R (ω)] which yields the second commutation relation (72), i.e. = [ ej · L, [ ek · L, R (ω)]] − [ ek · L, [ ej · L, R (ω)]] = − i (ek ·G)[ ej · L, R (ω)] − (ej ·G)[ ek · L, R (ω)] ` ` . −1 j ~ L, e · Ω = i~ e · I (Q ) · e (ej × L) . (H19) 2 = ~ R (ω)[ ej · G, ek · G ] (H11) = i~ − i~ R (ω)(ej × ek) · G Appendix I: Commutation relations of spin and the = [ i~ (ej × ek) · L, R (ω)] . internal observables Identifying the terms on the LHS of the commutation relation (H11) yields the second commutation relation (72), In this appendix, we determine explicitly the commu- i.e. tation relations between the nuclear spin S(µ), the electronic spin s(µ) and the internal observables Pα, p(ν) and [ ej · L, ek · L ] = i~ (ej × ek) · L , (H12) L. 23

The deﬁnitions (75) and the commutations rela- relations, tions (3) and (6) imply that ei · S(µ), ej · S(ν) (I1) Pα, S(µ) k ` N = e · R (ω) · ei e · R (ω) · ej [ ek · Sµ, e` · Sν ] X 1 = √ ej · X e · P 00, ek · R (ω) (e · S ) k ` M αν j ν k µ = i~ δµν e · R (ω) · ei e · R (ω) · ej ν=1 ν · (e × e ) · em (e · S ) N k ` m µ X 1 = √ ej · X e · P 00, e` · ω (I8) M αν j ν ν=1 ν ei · s(µ), ej · s(ν) (I2) k ` = e · R (ω) · ei e · R (ω) · ej [ ek · sµ, e` · sν ] · n(`) (ω) · G S(µ) = i δ ek · R (ω) · e e` · R (ω) · e ~ µν i j ` m = Pα, e · ω n(`) (ω) · G S(µ) = 0 · (ek × e`) · e (em · sµ)

m The triple product (ek × e`) · e is invariant under rota- P , s tion, which implies that, α (µ) N 1 k ` m X j 00 k e · R (ω) · ei e · R (ω) · ej (ek × e`) · e = √ e · Xαν ej · P ν , e · R (ω) (ek · sµ) Mν n m (I3) ν=1 = (ei × ej) · e (e · R (ω) · en) N X 1 = √ ej · X e · P 00, e` · ω (I9) Using the vectorial identity (I3) and the definitions (75), M αν j ν the commutation relations (I1) and (I2) yield the com- ν=1 ν mutation relations (76), i.e. · n (ω) · G s (`) (µ) ei · S(µ), ej · S(ν) n m ` = i~ δµν (ei × ej) · e (e · R (ω) · en)(em · Sµ) (I4) = Pα, e · ω n(`) (ω) · G s(µ) = 0 = i~ δµν (ei × ej) · S(µ) Using the definitions (75), the commutation relations (B7) and (F3), we compute the commutation re- ei · s(µ), ej · s(ν) n m lations, = i~ δµν (ei × ej) · e (e · R (ω) · en)(em · sµ) (I5) = i~ δµν (ei × ej) · s(µ) h i ej · p(ν), S(µ) Using the group action (31), the identity (A14), the defi- h i nitions (75), the commutation relation (B7) where e ·P 00 k ` ν = ej · p(ν), e · R (ω) (ek · Sµ) (I10) is replaced by ej · L and the commutation relation (H1), h ` i we compute the commutation relations, = ej · p(ν), e · ω n(`) (ω) · G S(µ) = 0 ej · L, S(µ) h i = e · L, ek · R (ω) (e · S ) j k µ ej · p(ν), s(µ) ` = ej · L, e · ω n(`) (ω) · G S(µ) (I6) h k i = ej · p(ν), e · R (ω) (ek · sµ) (I11) (`) = − i~ ej · m (ω) n(`) (ω) · G S(µ) h ` i = ej · p(ν), e · ω n(`) (ω) · G s(µ) = 0 = − i~ (ej · G) S(µ) = − i~ ej × S(µ)

ej · L, s(µ) = e · L, ek · R (ω) (e · s ) j k µ Appendix J: Kinetic energy operator ` = ej · L, e · ω n(`) (ω) · G s(µ) (I7) (`) = − i~ ej · m (ω) n(`) (ω) · G s(µ) In this appendix, we determine the expression of the kinetic energy operator T in terms of the internal observ- = − i~ (ej · G) s(µ) = − i~ ej × s(µ) ables. Using the definitions (75) and (D4), the commutation Using the relation (17)and the definition (20), the ex- relations (B7) and (F2), we compute the commutation pression (88) of the kinetic energy operator T is recast 24 as, where the operator accounting for the residual terms is given by, N 2 n 2 X 1 0 Mµ X 1 0 m . T = P + P + p + P Φres (Q ) (J8) 2 M µ M 2 m ν M µ=1 µ ν=1 N N 2i X 1 00 . X 1 . 2 N n ! 2 N 0 2 n 0 2 = − P µ, Aµ (Q ) + Aµ (Q ) . X X P X P µ X p M • M ν ~ µ=1 µ µ=1 µ = Mµ + m 2 + + 2 M 2 Mµ 2 m µ=1 ν=1 µ=1 ν=1 Now, we recast the kinetic energy operator T in terms N n ! X X P of the internal observables. The relation (43), the con- + P 0 + p0 · . (J1) µ ν M ditions (46)-(48) and the definitions (17) and (55) imply µ=1 ν=1 that the second term on the RHS of the expression (J7) is recast as, Using the relations (16)-(17) and (22), the kinetic energy operator (J1) reduces to, N 00 2 X P µ 1 1 α = Ω · I0 · Ω + Pα P 2 Mµ 2 2 2 N 0 2 n 0 2 µ=1 P X P µ X p (J9) T = + + ν . (J2) n n 2 M 2 Mµ 2 m 1 X X m µ=1 ν=1 + A A 0 0 p · p 0 . 2 m M νν ν ν (ν) (ν ) ν, ν=1 ν0, ν0=1 Using the relation (26), the first term on the RHS of the expression (J2) is recast as, Similarly, the relation (44) implies that the third term on the RHS of the expression (J7) is recast as, N 0 2 N 00 2 N P P 2 n 00 2 n n X µ X µ X ~ . 2 X p 1 X X = + Aµ (Q ) ν 2 M 2 M 8 M = δνν0 Aνν Aν0ν0 p(ν) · p(ν0) . µ=1 µ µ=1 µ µ=1 µ 2 m 2 m (J3) ν=1 ν, ν=1 ν0, ν0=1 N (J10) X i~ 00 ` . − e` · P , e · Aµ (Q ) , Using the identity (G7), the sum of the relations (J9) 4 M µ µ=1 µ and (J10) reduces to,

. N 00 2 n 00 2 n 2 where the operator Aµ (Q ) is deﬁned as, X P µ X p 1 1 X p(ν) + ν = Ω · I · Ω + P P α + 2 M 2 m 2 0 2 α 2 m ` . µ=1 µ ν=1 ν=1 − i~ e · Aµ (Q ) (J4) (J11) k 00 j ` −1 = e · P µ, e · R (ω) · ek e · R (ω) · ej . which implies that the kinetic energy yields the expression (93), i.e.

Using the commutation relation (F14), the operator 2 n p2 2 . P 1 1 α X (ν) ~ . Aµ (Q ) is recast as, T = + Ω·I ·Ω+ P P + + Φ (Q ) 2 M 2 0 2 α 2 m 8 res ν=1 . j . −1 k (0) (J12) Aµ (Q ) = e · I (Q ) · e ej × ek × MµRµ Now, we determine the expression of the residual terms = ej · I (Q .)−1 · ek in the expression (J12). Using the commutation relation (60) and the relation (43), the ﬁrst residual term (0) (0) · Mµ ej · Rµ ek − (ej · ek) Mµ Rµ appearing in the expression (J3) is recast as,

(0) . −1 (0) . −1 N = Mµ Rµ · I (Q ) − Mµ Rµ Tr I (Q ) . 2i X 1 00 . − P , Aµ (Q ) M µ • (J5) ~ µ=1 µ N Using the deﬁnitions (30) and (26), the third term on the 2i X h (0) ` . i = − e` · Ω × R , e · Aµ (Q ) RHS of the relation (J2) is recast as, µ ~ µ=1 N " # n 0 2 n 00 2 X pν X pν 2i X 1 α ` . = . (J6) − p Pα e` · Xµ , e · Aµ (Q ) 2 m 2 m Mµ ν=1 ν=1 ~ µ=1 " N !# The identities (J3) and (J6) imply that the kinetic en- 2i k X (0) . = − e · Ω, ek · Rµ × Aµ (Q ) ergy (J7) is recast as, ~ µ=1 " N # N 00 2 n P2 P p00 2 2 2i X 1 α . X µ X ν ~ . − Pα, X · Aµ (Q ) (J13) T = + + + Φres (Q ) , (J7) p µ 2 M 2 M 2 m 8 ~ µ=1 Mµ µ=1 µ ν=1 25

We determine in turn the two terms in the commutation as, relation (J13). The deﬁnition (J5) implies that, " N !# 2i X − ek · Ω, e · R(0) × A (Q .) N k µ µ X (0) . ~ µ=1 Rµ × Aµ(Q ) m . −1 . −1 ` µ=1 = 2 e · I (Q ) · Iα · I (Q ) · I00 · e N X −1 . −1 X α β (0) (0) . (e` × em) · I (Q ) · X × Xµβ Q = Rµ × Mµ Rµ · I (Q ) µ µ=1 (J14) µ=1 −1 −1 m . −1 j m . . α β = e · I (Q ) · e = 2 e · I (Q ) · Iα · I (Q ) · I0β Q · em N . −1 . −1 α β X = 2 Tr I (Q ) · Iα · I (Q ) · I0β Q · M e · R(0) e · R(0) e` × e µ j µ ` µ m (J19) µ=1

Similarly, using the deﬁnition (J5), the second term in Using the deﬁnition (91), equation (J14) is recast as, the commutation relation (J13) is recast as,

N " N # 2i X 1 α . X (0) . m . −1 ` − Pα, X · Aµ (Q ) Rµ × Aµ(Q ) = e · I (Q ) · I00 · e (e` × em) p µ ~ Mµ µ=1 µ=1 (J15) 2i h j . −1 ` = − Pα, e · I (Q ) · e (J20) which implies in turn that the ﬁrst term in the commu- ~ tation relation (J13) becomes N # X p α (0) Mµ Xµ · ej × e` × Rµ . µ=1 " N !# 2i k X (0) . − e · Ω, ek · Rµ × Aµ(Q ) ~ µ=1 The deﬁnition (57) implies that,

2i m h k . −1 i ` = − e · e · Ω, I (Q ) · I00 · e ek · (e` × em) N ~ X p α (0) (J16) Mµ Xµ · ej × e` × Rµ µ=1 N (J21) X p α (0) The definition (54) and the first commutation rela- = − Mµ ej × Xµ · e` × Rµ tion (62) and imply that, µ=1 α = − (ej · I · e`) . k . k α e · Ω, I (Q ) = Iα e · Ω,Q ! Using the relation (J21), the commutation relation (J20) k . −1 X α β is recast as, = − i~ Iα e · I (Q ) · Xµ × Xµβ Q µ=1 (J17) " N # 2i X 1 α . − Pα, X · Aµ (Q ) pM µ ~ µ=1 µ (J22) which implies that, 2i α h j . −1 ` i = (ej · I · e`) Pα, e · I (Q ) · e . ~ h i ek · Ω, I (Q .)−1 Using the definition (54) and the canonical commutation = − I (Q .)−1 · ek · Ω, I (Q .) · I (Q .)−1 relation (62), we commute the commutation relation on the RHS of the expression (J22), i.e. . −1 . −1 = i~ I (Q ) · Iα · I (Q ) (J18) ! h j . −1 ` i Pα, e · I (Q ) · e k . −1 X α β · e · I (Q ) · Xµ × Xµβ Q (J23) j . −1 . −1 ` µ=1 = i~ e · I (Q ) · Iα · I (Q ) · e .

Using the commutation relation (J18) and the deﬁni- Using the commutation relation (J23) and the fact tion (91), the commutation relation (J16) can be recast that the tensor Iα is symmetric, the commutation rela- 26 tion (J23) is recast as, order terms of the series expansion (101), i.e.

2 N " N # e X Zµ Zν 2i X 1 α . VN −N (0) = , − Pα, X · Aµ (Q ) 8πε (0) p µ 0 k∆ Rµν k ~ Mµ µ,ν=1 µ=1 µ6=ν α . −1 . −1 ` (J24) 2 N (0) ! = − 2 e` · I · I (Q ) · Iα · I (Q ) · e e X ∆ Y µνα · ∆ Rµν V = Z Z − , N −N (α) µ ν (0) 2 8πε0 3 . −1 µ,ν=1 k∆ Rµν k = − 2 Tr Iα · I (Q ) . µ6=ν

2 N e X ∆ Y µνα · ∆ Y µνβ VN −N (αβ) = Zµ Zν − The commutation relations (J13), (J19) and (J24) imply 8πε (0) 3 0 µ,ν=1 k∆ Rµν k that the ﬁrst term of the operator (J8) is recast as, µ6=ν  ∆ Y · ∆ R(0) ∆ R(0) · ∆ Y 2i N 1 µνα µν µν µνβ X 00 . + 3  , (K1) − P µ, Aµ (Q ) (0) 5 M • k∆ Rµν k ~ µ=1 µ

. −1 . −1 α β (J25) (0) (0) (0) = 2 Tr I (Q ) · Iα · I (Q ) · I0β Q where ∆ Rµν ≡ Rµ − Rν and ∆ Y µνα ≡ Y µα − Y να . Similarly, using the expression (96) of the Coulomb po- 2 − 2 Tr Iα · I (Q .)−1 . tential between the nuclei and the electrons, we deduce the expressions for the zeroth, first and second order terms of the series expansion (102), i.e. Using the definitions (91) and (J5) the last term of the 2 N n the RHS of the expression (J8) is recast as, e X X Zµ V (q ) = − , N −e (0) (.) (0) 4πε0 k ∆ Q k N µ=1 ν=1 µν X 1 . 2 A (Q ) 2 N n (0) ! M µ e X X Y µα · ∆ Qµν µ=1 µ VN −e (α) (q ) = − Zµ − , (.) 4πε (0) 3 0 µ=1 ν=1 k ∆ Qµν k N 2 X (0) . −1 (0) . −1 = M R · I (Q ) − R Tr I (Q ) 2 N n µ µ µ e X X Y µα · Y µβ µ=1 VN −e (αβ) (q ) = − Zµ − (.) 4πε (0) 3 0 µ=1 ν=1 k ∆ Qµν k . −2 . −1 . −1 = Tr I00 · I (Q ) − 2 Tr I00 · I (Q ) Tr I (Q ) (0) (0)  Y µα · ∆ Qµν ∆ Qµν · Y µβ . −2 + 3 . (K2) + Tr (I00) Tr I (Q ) . (0) 5  k ∆ Qµν k (J26) (0) (0) where ∆ Qµν ≡ Rµ 1− q¯(ν) . We define three symmetric The relations (J25) and (J26) imply that the expres- and trace-free tensors, i.e. sion (J8) yields the residual operator (90), i.e. 2 (0) (0) ! e Zµ Zν 1 ∆ Rµν ∆ Rµν Bµν = − + 3 , . . −2 4πε (0) 3 (0) 5 Φres (Q ) = Tr (I00) Tr I (Q ) 0 k∆ Rµν k k∆ Rµν k N ! . −1 . −1 2 ∆ R(0) ∆ R(0) − 2 Tr I00 · I (Q ) Tr I (Q ) e Zµ X 1 µν µν Aµ = Zν − + 3 , 4πε (0) 3 (0) 5 2 (J27) 0 ν=1 k∆ Rµν k k∆ Rµν k . −2 . −1 + Tr I00 · I (Q ) − 2 Iα · I (Q ) 2 * n (0) (0) !+ e Zµ X 1 ∆ Qµν ∆ Qµν E = − − + 3 µ (0) (0) . −1 . −1 α β 4πε0 k∆ Q k3 k∆ Q k5 + 2 Tr I (Q ) · Iα · I (Q ) · I0β Q . ν=1 µν µν (K3) where

Appendix K: Vibrational modes Bµν = Bνµ , N N X X (K4) In this appendix, we determine the expression of the Aµ = Bµν = Bµν (1 − δµν ) . angular frequency of the vibrational modes and show that ν=1 ν=1 it is positively defined. µ6=ν Using the expression (96) of the Coulomb potential Using the definitions (96) and (K3), and the proper- between the nuclei, we deduce the zeroth, first and second ties (K4), the second-order term of the Coulomb potential 27

VN −N (αβ) is expressed as, Using the conditions (46) and (47) and taking the sum over all the nuclei in the relation (K10) after multiplying N (0) 1 X by bβ × R yields, V = ∆ Y · B (1 − δ ) · ∆ Y µ N −N (αβ) 2 µνα µν µν µνβ µ,ν=1 N N X X (0) = Y µα · Aµ δµν · Y νβ (K5) bβ × Rµ · Dµν · Y νβ µ,ν=1 µ,ν=1 (K13) N N X X (0) (0) − Y µα · Bµν (1 − δµν ) · Y νβ . = Mµ bβ × Rµ · bβ × Rµ . µ,ν=1 µ=1

Similarly, using the deﬁnitions (96) and (K3), the second- The deﬁnitions (55) implies that the relation (K13) re- order term of the Coulomb potential VN −e (αβ) is expressed as, duces to,

N X N VN −e (αβ) = Y µα · Eµ δµν · Y νβ . (K6) X (0) ej × Rµ · Dµν · Y νβ = ej · I0 · bβ , (K14) µ,ν=1 µ,ν=1 According to the expressions (K5) and (K6) of the second-order Coulomb potentials, the condition (109) is which is recast as, recast in terms of the tensors (K3) as,

N N N X X −1 j X (0) Y µα· (Aµ + Eµ) δµν − Bµν (1 − δµν ) ·Y νβ = 0 bβ = I0 · e ej × Rρ · Dρν · Y νβ . (K15) µ=1 ν=1 ρ,ν=1 (K7) To simplify the notation, we deﬁne a symmetric and Substituting the relations (K12), (K8) and (K15) into the trace-free tensor Dµν that is a linear combination of the condition (K10) yields the eigenvalue equation, tensors Aµ, Bµν and Eµ, i.e.

Dµν = (Aµ + Eµ) δµν − Bµν (1 − δµν ) , (K8) N N X Mµ X D − E − M e × R(0) ek · I−1 · ej µν M ν µ k µ 0 which implies that the relation (K7) is recast as, ν=1 ρ,ν=1 N N ! (0) p X X · ej × Rρ · Dρν · Y νβ = cβ Mµ Xµβ . (K16) Y µα · Dµν · Y νβ = 0 , (K9) µ=1 ν=1 where α 6= β . The conditions (45), (47) and (48) im- Using the conditions (45), (46) and (47) and taking the ply that the condition (K9) is satisﬁed provided all the sum over all the nuclei in the relation (K16) after multi- vibration modes Y νβ satisfy the condition, plying by Y µβ yields,

N X p (0) N N ! Dµν ·Y νβ = cβ Mµ Xµβ +Mµ aβ +Mµ bβ × Rµ X X Y · D · Y = c . (K17) ν=1 µβ µν νβ β (K10) µ=1 ν=1 where cβ is a scalar parameter, aβ and bβ are vectorial parameters. At equilibrium, the energy of the stable molecular system Using the deﬁnition (K8), the conditions (46), (47) is minimal. Thus, in the neighbourhood of the equilib- and (K4) and taking the sum over all the nuclei in the rium, the relation (K17) is a positive deﬁnite quadratic relation (K10) yields, form. Identifying the relations (109) and (K9) and com-

N N N paring them to the relation (K17), we conclude that, i.e. X X X Dµν · Y νβ = Eµ · Y µβ = Mµ aβ . (K11) µ,ν=1 µ=1 µ=1 2 D E ωβ = VN −N (ββ) + VN −e (ββ) (q(.)) , (K18) The deﬁnition (17) and the relation (K11) yield,

N where ω2 ≡ c > 0 is identiﬁed physically as the square 1 X β β a = E · Y . (K12) of the angular frequency of the vibration eigenmodes. β M ν νβ ν=1 28

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