Normal modes. The true story. Emil Zak∗ Department of Physics and Astronomy, University College London, London, WC1E 6BT, UK (Dated: January 20, 2016) I. INTRODUCTION KEO in arbitrary curvlinear coordinates was introduced by Yachmenev and Yurchenko [3] and successfully ap- The aim of this article is a comprehensive description plied to four-atomic systems by Chubb, Yachmenev and of normal modes of molecular vibrations. The starting Yurchenko [4]. Parallel analytic progress in the field has point is chosen to be a general molecular system with been made by Szalay [5], [6]. Several authors: Tennyson separated center of mass and an arbitrary embedding of et al. [7], [8], Jensen [9], Csaszar [10] and others also body-fixed axes. This allows to focus on internal de- succeeded to evaluate practical representations for many- grees of freedom only, leaving the problem of rotational body Hamiltonians in an arbitrary embedding of the motion behind. Nevertheless, for the sake of complete- body-fixed frame. In each of these cases one is theoreti- ness we first introduce a general quantum-mechanical cally able to linearize the internal coordinates and subse- Hamiltonian in Eckart-Watson representation, to make quently transfer into normal modes level. Nonetheless we a quick leap into a simplified harmonic description for shall follow the historically first attempt due to its strong internal motion and rigid rotor for rotational degrees of association with normal coordinates. After separating freedom. This regime constitutes a basis for more sophis- the nuclear center of mass, and moving into the Eckart ticated calculations. The first section introduces normal frame which guarantees minimal rotation-vibration cou- modes with emphasis on mathematical precision. Along- pling, some analytic manipulations yield the well known side we stress the points where approximations are made, Watson Hamiltonian: for the reader to be aware of limitations of the current model. Second section exemplifies presented methodol- ogy on a simple system of two masses connected with harmonic springs in 1D. Section V has supplementary 1 X 1 X character and underlines the formal route needed be H^ = M^ µ M^ + P^2 + U^ + V^ (1) 2 i ij j 2 i taken to derive the Watson Hamiltonian, which in turn i;j i is immersed in the framework of normal vibrations. The last section (VI) is a detailed discussion of rigid rotor model. Current review is largely based on classic books where U^ is mass dependent contribution to the potential by Bunker& Jensen [1] and Wilson,Decius&Cross [2] sup- V^ : plied with original papers, and a few author's own com- ments, derivations and theorems. 2 X 2 U^ = −~ µ ≡ −~ T rµ (2) 8 ii 8 II. GENERAL THEORY i Before proceeding to the main topic, a few words of explanation are needed. Section V presents a detailed and µij is the inverse of the dynamical moment of inertia insight into the formal procedure of derivation of the tensor[11]. M^ i = J^i − p^i − L^i stands for the i-th compo- arXiv:1601.04735v1 [quant-ph] 18 Jan 2016 Eckart-Watson form of quantum Hamiltonian for an ar- nent of the total angular momentum of the system and is bitrary many-body system. This is a popular but rather defined by the rovibronic angular momentum, vibrational specific approach, as is constrained to normal coordinates angular momentum and electronic angular momentum, only. Any set of coordinates which is not linearly related respectively. The potential operator V contains interac- to cartesian displacements brings significant complica- tions between all particles in the system, while P^ is the tions. Many efforts were made to evaluate an exact rep- linear momentum operator in normal coordinates repre- resentation of the kinetic energy operator (KEO) in curv- sentation. By imposing Born-Oppenheimer separation of linear coordinates and Eckart embedding. Most recently nuclear and electronic degrees of freedom the molecular an effective numerical procedure for expansion of nuclear Hamiltonian becomes a sum of an electronic and a rovi- brational part: ∗ [email protected]; http://www.homepages.ucl.ac.uk/~ucapejz/ H^ = H^el + H^rv (3) 2 The rovibrational Hamiltonian can be readily expanded later transfer to the rigid rotor approximation we are and arranged in the following fashion: left with internal nuclear motion problem. For a non- linear molecule, or non-linear system of atoms this leaves 1 X 1 X H^ = µe J^2 + P^2+ 3N − 6 internal degrees of freedom. Lets start from a rv 2 αα α 2 i α i fully classical picture. After formulating a suitable rep- 1 X resentation of the Hamilton function, quantization will + (µ − µ ) J^ − p^ J^ − J^ 2 αβ αβ α α β β be performed. From eq. 5 one can infer that the total αβ energy of the non-rotating system is a sum of kinetic and X e ^ X e 2 ^ ^ potential energy: − µααJαp^α + + µααp^α + U + Vnn α α E = T (x_ ) + V (x) (8) (4) Mixing terms appearing in above expression are re- where x = x1; y1; z1; :::; x3N−6; y3N−6; z3N−6 and x_ = sponsible for centrifugal distorsion and Coriolis cou- x_ 1; y_1; z_1; :::; x_ 3N−6; y_3N−6; z_3N−6 are position and veloc- plings. For sufficiently rigid molecules they can be ne- ity vectors of all 3N − 6 atoms, respectively. No net glected. What remains is a purely rotational part added external force is exerted on the system and all internal to internal motion energy operators decoupled from ro- forces are conservative, hence the total energy is equal to tations: the Hamilton function. We should be able to write ad hoc both components in cartesian coordinates: 3N−6 1 X e 2 1 X 2 1 H^rv = µ J^ + P^ + V^nn (5) X X 2 2 αα α 2 i Tcart = miα_i (9) α i 2 i=1 α=x;y;z The task now is to find normal coordinates for a general where α stands for cartesian coordinate in the molecule- many-body system, so that Watson approach may be re- i fixed (body-fixed) frame. It is very instructive to refor- alized. Hence it becomes clear that the choice of this par- mulate the above expression in a matrix form: ticular type of Hamiltonian representation was strongly dictated by its association to normal coordinates. How- 0 1 0 1 3N−6 m 0 0 x_ ever from practical point of view, one usually prefer start 1 X i i T = (_x ; y_ ; z_ ) 0 m 0 y_ (10) with a set of geometrically defined internal coordinates or cart 2 i i i @ i A @ i A i=1 0 0 m z_ cartesian coordinates. Thus, below we present a detailed i i instructions to obtain normal coordinates from arbitrar- and further: ily chosen internal coordinates. Assuming the knowledge of normal coordinates of the system we may expand inter- 1 T = x_ T Tx_ (11) nuclear interaction potential energy as: cart 2 where x is a 3N − 6 dimensional position vector, while T 3N−6 3N−6 1 X 1 X stands for a (3N − 6) × (3N − 6) matrix representation V = λ Q2 + Φ Q Q Q + ::: (6) nn 2 i i 6 ijk i j k of a diagonal metric tensor build from N − 2 diagonal i=1 i;j;k=1 blocks each containing three equal masses mi · 13. It and by truncating this expansion on quadratic terms we is clearly visible that the kinetic energy is a quadratic stay in harmonic approximation, in which normal coor- form in cartesian velocities. This observation will allow dinates were introduced. Now pick up the first two sums us to utilize some tools from quadratic forms theory in in eq. 5 and combine the second one with the truncated section IV. The internuclear potential energy is a func- potential energy. By neglecting all other terms we get: tion of 3N cartesian coordinates of all nuclei. However the functional form of potential energy is usually avail- able in a geometrically defined internal coordinates, i.e. 1 X 1 X H^ = µe J^2 + P^2 + λ Q2 (7) Vnn = Vnn(R1;R2; :::; R3N−6), where Ri is the i-th inter- rv 2 αα α 2 i i i nal coordinate. The reason for that is because the poten- α i tial energy function is in general a complicated function The first term in above Hamiltonian describes the energy of cartesian coordinates. Internal coordinates chosen in- of a rigid rotor - this model is solved and discussed in tuitively by geometrical considerations provide a more detail in section VI. The second term is the target of transparent perspective of types of internal motions and the present article. It is already shown in the final form, associated potential energy. Therefore internal coordi- however no recipe for finding λi and Qi was yet given. nate representation is more suitable at this stage. Nev- Until then this expression should be considered a black- ertheless, the potential energy function is typically also a box. fairly complicated function (but less complicated than in Having separated both center of mass motion and rota- cartesian representation) of internal coordinates. Thus, a tions via embedding in the Eckart body-fixed frame with for non-high-accuracy calculation purposes lowest terms 3 in the Taylor expansion are sufficient: where M is the total mass of the system, and 1 3N−6 1 3N−6 N X X eq X eq eq Vnn = fijRiRj + fijkRiRjRk + ::: (12) R = µ m (y ∆z − z ∆y ) (17) 2 6 x xx i i i i i i;j=1 i;j;k=1 i=1 Note that the minimum of V is set to 0 and it corre- while y and z rotational coordinates are obtained by sponds to equilibrium values of all internal coordinates: cyclic permutation of cartesian coordinates in above def- ~ ~ −1 R = (0; 0; :::; 0) = Req.