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 molec- ular 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. P DT [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 , F DF [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 elec- trons, 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 ex- pressed 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.