Aprendizaje de las estudiantes Sordas

The rotational spectrum of nonlinear molecules can be treated using an effective Hamilto- nian that describes the rotations along the principal axes (a, b, c) of the molecule. In the rigid rotor approximation, this leads to the following form of the Hamiltonian:

H=AP_a2+BP_b2+CP_c2 (1.1)

where Pa, Pb and Pc are the angular momentum operators along the principal axes. A, B

and C are the rotational constants, which are simply the inverse of the moment of inertia along the principle axes according to:

A= ~ 4πcIA

(1.2)

where IA is the moment of inertia along the a axis of the molecule. When the moments

of inertia along all three axes are different, which is the most common class of molecules, a species is classified as an ‘asymmetric top’, and this requires a more complex treatment. The matrix of the effective Hamiltonian in a given basis set must be diagonalized to find the energy eigenvalues and eigenstates. This leads to a Hamiltonian of the following form:

H = B+C 2 P2+ A−B+C 2 P_a2+ B−C 4 (P_b2−P_c2)−∆JP4 −∆J KP2Pa2−∆KPa2−2δJP2(Pb2−Pc2)−δK{Pa2,(Pb2−Pc2)} (1.3)

Figure 1.3: Rotational potential energy curve of a hindered internal rotor. The angle of rotation, α, of the methyl top is shown along the x-axis, and the resulting potential energy lies along the y-axis, with maxima at the barrier to rotation, V3, every 2π/3. Tunneling of the hydrogen atoms

through the barrier results in splitting of the torsional levels,ν, into A- andE- sublevels.

where P2 = P2_a + P2_b + P2_c, and ∆J, ∆J K, ∆K, δJ and δK are the first order centrifugal

distortion constants that follow from a perturbation treatment on the zeroth order (rigid rotor) Hamiltonian. Higher order centrifugal distortion corrections also exist, but their con- tribution is decreasingly significant in magnitude. Nonetheless, for weaker, high quantum number states at high frequencies these corrections can lead to shifts on the order of MHz or more – large enough to lead to errors in assignment given the complex nature of dense cloud spectra. This indicates the need to measure or accurately predict the rotational spectrum of an observational target across the observational windows to be studied astronomically and describing the laboratory spectrum in detail. For large frequency jumps from millimeter wavelengths into the THz regime (such as will occur with spectral line surveys undertaken by Herschel) this is of great importance.

In addition to the basic rotational treatment of asymmetric molecules above, further complications to the spectrum can arise due to the contribution of low-energy, large- amplitude vibrational motions, in particular the internal rotation of a methyl (CH3) group.

group with the rest of the framework of the molecule leads to a periodic contribution in the potential energy, which is shown in Fig. 1.3. In a barrier of finite size, the hydrogen atoms are capable of tunneling through the torsional barrier, which leads to a splitting of the rotational levels intoA- andE-state sublevels. The magnitude of the splitting depends on the size of the barrier, with a large splitting for low barriers and a small splitting for high barriers. In the case of very high barriers theA-E splitting in the spectrum can essentially be treated as a perturbation correction to the rotational Hamiltonian; whereas at negligi- ble barriers, the splitting can be treated as a free-rotor problem [29]. Both of these limits simplify the treatment of the problem. At low to intermediate levels, however, significant coupling between the torsional motion of the top and the rotational motion of the molecule can occur, leading to significant perturbations. This leads to a zeroth-order Hamiltonian of the general form:

H=Hrot+ (1/2)V3(1−cos3γ) +F(Pγ+ρPa)2 (1.4)

where the first term is the pure rotational Hamiltonian of the form in equation (1.3), the second term accounts for the periodic contribution to the potential energy, and the third term accounts for the coupling between the torsional and rotational motions of the molecule. V3 is the barrier to internal rotation, γ the angle of internal rotation, Pγ is the angular

momentum operator of the methyl top, ρ is the projection of the moment of inertia of the top onto that of the molecule, and F is the reduced rotational constant for the methyl top. Higher order terms in the Hamiltonian, i.e. centrifugal distortion constants, higher order torsional constants, and their cross-products, arise in treatments where a power se- ries expansion of the various angular momenta are used, since there are products of these

83558A HP8341b SR830 M1611/2D HP FG AM/FM/TM sync InSb or Si gas pre-amp vacuum pump sample sweep

synthesizer _filterYIG mm-wave

module MMIC

amps multiplierssubmm

waveguide feedhorns sample cell He cooled detector waveform generator lock-in-amplifier PC GPIB modulation DAQ tuning voltage

Figure 1.4: Block diagram of the JPL frequency multiplier flow cell spectrometer, taken from [2]. Details on the system can be found in the text.

terms with the torsional potential term in the effective Hamiltonian. For further details on rotational spectroscopy and the effects of internal rotation, see [30, 29]. The next two Chapters of this thesis deal with rotational spectra that show the effects of internal rota- tion. In the case of methanol (Chapter 2), the barrier is of intermediate value, and only represents a challenging but fairly well understood problem due to its long history of study. Hydroxyacetone (Chapter 3) has a low barrier to internal rotation, which thus represents a challenging spectroscopic problem, and one that can now be solved with modern numerical approaches to the spectroscopy of internal rotation.