Rotations and vibrations of polyatomic molecules

Rotations and vibrations of polyatomic molecules When the potential energy surface V( R 1, R 2,..., R N ) is known we can compute the energy levels of the molecule. These levels can be an effect of: Rotation of a molecule as a whole (end-over-end rotation) Small vibrations around equilibrium configuration of the nuclei Internal rotation - free or hindered Tunneling Large amplitude vibrations (van der Waals vibrations) Transitions between these levels are are observed in spectroscopy: Infrared/Raman vibrations Terahertz (submillimeter) rotations, large amplitude vibrations Microwave rotations, tunneling These transitions occur between the eigenvalues of the Hamiltonian for the nuclear motion: Ĥ = 1 N 1 R 2 m i + V( R 1, R 2,..., R N ) i where m i is the mass of the i-th nucleus. The Born-Oppenheimer approximation is assumed here.

Rotations and vibrations of polyatomic molecules, continued For polyatomic molecules rotations and vibrations cannot be separated The separation is possible only approximately(often with good accuracy) for the so called rigid moleucles such as: water carbon dioxide methane ethylene benzene Rotation is then described as a rotation of a rigid body (a rotating top of the right kind). Vibrations are described as small amplitude oscillations around the (rotating) equilibrium positions of the nuclei. The total energy is a sum of the rotation energy and the vibration energy For nonrigid molecules (floppy molecules), such as: dimethylacetylene propane ammonia ethanol water dimer such an approximation does not work and each case must be treated separately. Theory of nuclear motion in floppy molecules is still in development and will not be presented in these lectures since: There are no commercial programs offering a possibility of computing rovibrational spectra of nonrigid molecules.

Rotations of polyatomic molecules Classically, the rotation of a rigid body is described by the vector of angular velocity ω such that the velocity v i of i-th particle (atomic nucleus) is given by: v i = ω R i The angular momentum of the rigid body is then given by N N L = R i (m i v i ) = m i R i ( ω R i ) = II ω, where II is the matrix (tensor) of the moment of inertia: II = i m i (Y 2 i + Z 2 i ) i m i X i Y i i m i X i, Z i i m i Y i X i i m i (X 2 i + Z2 i ) i m i Y i Z i i m i Z i X i i m i Z i Y i i m i (X 2 i + Y 2 i ) The matrix II is symemtric, e.g., II xy =II yx. In the coordinate system fixed to the rotating molecule the matrix II does not depend on time.

Rotations of polyatomic molecules, continued As each symmetric matrix the matrix II can be diagonalized. In the new coordinate frame (also fixed to the molecule) this matrix has the form: I A 0 0 II = 0 I B 0 0 0 I C where I A, I B, and I C are the so-called principal moments of inertia of the molecule. The normalized eigenvectors vectors A, B, and C, corresponding to the eigenvalues I A, I B, and I C determines the so-called principal exes of the top (a molecule). If the molecule has a symmetry then A, B, i C sa coincide with the symmetry axes of the molecule. The principal moments of inertia are given by the formulas: I A = i m i (B 2 i + C2 i ) I B = i m i (A 2 i + C2 i ) I C = i m i (A 2 i + B2 i ) where A i, B i and C i are the coordinates of the i-th nucleus in the coordinate frame determined by the principal axes A, B, and C, for instance Bi 2 + C2 i is the distance of the i-th nucleus form the A axis.

Rotations of polyatomic molecules, continued In the coordinate frame A, B, and C the matrix II is diagonal so the angular momentum vector L = II ω has the following coordinates in this frame: L A = I A ω A, L B = I B ω B, L C = I C ω C, One can easily show that the kinetic energy of rotations is given by E rot = 1 N m i v i v i = 1 N m i v i ( ω R i ) = 1 N m i ω ( R i v i ) = 1 2 2 2 2 ω L In the frame of principal axes ω L = ω A L A + ω B L B + ω C L C, hence E rot = 1 2 I A ω 2 A + 1 2 I B ω 2 B + 1 2 I C ω 2 C or E rot = 1 L 2 A + 1 2I B L 2 B + 1 2I C L 2 C To quantize and obtain the quantum mechanical Hamiltonian we add hats: Ĥ rot = 1 ˆL 2 A + 1 2I B ˆL 2 B + 1 2I C ˆL 2 C

Rotations of polyatomic molecules, continued Eigenvalues and eigenfunctions of Ĥ rot depend significantly on the relative values of the principal moments of inertia. We distinguish four kinds of tops: spherical tops, I A = I B = I C (methane, hexafluoride of uranium) prolate symmetric tops, I A < I B = I C (ammonia, CH 3 Cl) oblate symmetric tops, I A = I B < I C (benzene, SO 3, CHCl 3 ) asymmetric tops, I A I B I C (water, ethylene, naphthalene) For the oblate symmetric top I A = I B so we can write: Ĥ rot = 1 ˆL 2 A + 1 2I B ˆL 2 B + 1 2I C ˆL 2 C = 1 ˆL 2 A + 1 ˆL 2 B + 1 ˆL 2 C + 1 2I C ˆL 2 C 1 hence Ĥ rot = 1 ˆL 2 + ( 1 1 ) 2I C ˆL 2 C ˆL 2 C Since the eigenvalues of ˆL 2 and ˆL C are given by J(J + 1) 2 i K, where J K J, the eigenvalues of Ĥ rot must be given by (in atomic units) E JK = 1 ( 1 J(J + 1) + 1 ) K 2 2I C

Rotations of polyatomic molecules, continued For the prolate symmetric top we obtain a similar formula only the smallest and largest moments of inertia I A and I C are interchanged: E JK = 1 ( 1 J(J + 1) + 1 ) K 2 2I C 2I C For symmetric tops the degeneracy of the level E JK is 2(2J +1), 2 with respect to K K and 2J + 1 with respect to M. For the spherical top I A =I B =I C thus: E J = 1 J(J + 1) The degeneracy in this case is (2J + 1) 2. For the asymetric top (water, ethylene,) there is no analytical formula for the rotational levels. The degeneracy with respect to K is lifted. K ceases ti be a god quantum number. Linear molecules in states with Λ 0, i.e. Π,, etc. states are also symmetric tops (prolate ones) with K = Λ where Λ = 1 for the Π, Λ = 2 for the, states, etc. The rotational energy of this states is given by: E JΛ = 1 [J(J + 1) Λ 2 ] + 1 Λ 2 2I C where I C =µre 2 and Λ2 /( ) is a constant included in the electronic energy.

Rotational energies pattern for symmetric rotors

Rotational energies pattern for asymmetric rotors

Vibrations of polyatomic molecules Consider N atomic nuclei. Position of the i-th nucleus will be specified by the Cartesian coordinates x i, y i and z i of its displacement from the equilibrium position (assumed fixed in space). The Hamiltonian in this coordinates is: Ĥ vib = 1 2m i 2 u 2 i + V (u 1, u 2,..., u 3N ) where u i,,...,3n denote collectively all Cartesian coordinates x 1, y 1, z 1, x 2, y 2, z 2,... and masses are defined such that m 1 =m 2 =m 3, m 4 = m 5 =m 6,... The potential V (u 1, u 2,..., u 3N ) can be expanded in the Taylor series around u 1 =u 2 =u 3 = =0 ( ) V V (u 1, u 2,..., u 3N ) = V 0 + u i + 1 ( 2 ) V u i u j + u i 2 u i u j The first derivatives vanish at the minimum, the third ones are neglected (harmonic approximations). The constant V 0 is included in the electronic energy, hence: Ĥ vib = 1 2m i 2 u 2 i + + 1 2 j=1 j=1 V ij u i u j where V ij is the matrix of second derivatives ( the Hessian) at the minimum.

Vibrations of polyatomic molecules, continued To find eigenvalues of this Hamiltonian it is convenient to introduce the so-called mass scaled coordinates : w i = m i u i In these coordinates the Hamiltonian has the form : Ĥ vib = 1 2 2 w 2 i + 1 2 j=1 V ij mi m j w i w j The matrix V ij / m i m j is symmetric so it can be diagonalized with a matrix U ij and one can introduce the normal coordinates q k using this U ij matrix: q k = U ik w i henceczyli u i = 1 mi 3N k=1 U ik q k e.g., u (k) i = 1 mi U ik q k In this normal coordinates the Hamiltonian has a very simple from: ( Ĥ vib = 1 2 + 1 ) 2 2 λ k q 2 k k=1 where λ k are eigenvalues of the V ij / m i m j matrix. The eigenvalules of Ĥ vib are: ( E v1,v 2,... = ω k v k + 1 ) gdzie ω k = λ k 2 k=1 q 2 k

Translations, rotations, and vibrations of a triatomic molecule

Harmonic vibration spectrum of water

Resonance tunneling in the vibration spectrum of ammonia

Fermie resonance of the vibrational excitations ω 1 = 1333 cm 1 and 2 ω 2 = 1338 cm 1 in the CO 2 molecule

Coriolis force effects on the normal vibrations of the CO 2 molecule