Chapter 10
Chirality: Physical Chemistry Downloaded from pubs.acs.org by AUBURN UNIV on 03/17/19. For personal use only.
Sum-Frequency Generation at Second Order in Isotropic Chiral Systems: The Microscopic View and the Surprising Fragility of the Signal Jason Kirkwood , A. C. Albreeht1, , Peer Fischer , and A. D. Buckingham 1,3
*
2
2
Department of Chemistry, Cornell University, Ithaca, N Y 14853 Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge C B 2 1EW, United Kingdom Department of Chemistry, University of Rochester, Rochester, N Y 14627-0216 1
2
3
The rotationally invariant component o f the second order hyperpolarizability tensor is examined. Sum-frequency generation (SFG) arises in a chiral system only when at least two excited states are involved. N o D C component and exceptionally sensitive resonance effects appear in this 3-wave mixing (3-WM) process. These are entirely unlike those seen in the more familiar even-WM spectroscopies. Both electronic (UV) and vibrational (IR) resonances are explored. Causes for the apparent fragility of SFG in optically active solutions are discussed and ways to circumvent them are suggested.
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© 2002 American Chemical Society
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Introduction The many nonlinear electric-field based spectroscopies, once treated perturbatively, are naturally classified by order. The sth order spectroscopies involve s incident field actions upon a material. The outcome of this action is a macroscopic, sth order polarization (the induced dipole density). The scalar product of its time derivative and the total field serves as the integrand in the calculation of the cycle-averaged rate of energy exchange between the incident fields and the material. Such net energy exchange must take place in all (onephoton or multiphoton) absorption and emission spectroscopies and, provided all incident fields are oscillating (none are DC), these so-called "Class Γ spectroscopies can appear only when s is odd (J). The induced, oscillating, sth order polarization also serves as a source term in Maxwell's equation for the electric field to produce a new field — the ( 5 · + l)th field in an overall process called (y+ l)-wave mixing ((s+ 1)-WM). The intensity of the new field may be detected in self-quadrature (homodyned) or placed into quadrature with an idler field (heterodyned). Unlike the Class I spectroscopies, here phase-matching becomes an important feature since the signal requires the successful macroscopic build-up of the new field. These "coherent" nonlinear spectroscopies need not have light/matter energy exchange that survives cycle averaging. They have been termed the "Class II" nonlinear spectroscopies and, unlike those of Class I, may appear at all orders in s. However the macroscopic material response to the incident fields is built of contributions from each of the constituent molecules of the sample. This entails an ensemble average of the microscopic (molecular) induced polarization. At 5th order such an average involves (s+1) direction cosines that project the s actions of the incident field and the consequent induced dipole within the molecular frame to the macroscopic polarization of the sample in the laboratory frame. The ensemble average amounts to an average of the product of the (s+ 1) direction cosines over the orientational distribution of the molecules. Normally samples consisting of microscopically randomly oriented molecules such as gases, liquids, and microscopically disordered solids, are symmetric with respect to inversion in the macroscopic frame. However each direction cosine is odd with respect to such inversion. Thus Class II spectroscopies for such randomly oriented systems should not exist at even s (an odd number of direction cosines). Indeed Class II spectroscopies at even order are unknown for such randomly oriented systems. However there is one exception ! The formal requirement for the microscopic susceptibility at any order to survive unweighted rotational averaging is that the tensor possess a rotationally invariant component. It was pointed out long ago (2) how at second order the microscopic polarizability tensor, beta, does in fact have a rotationally invariant component — it is a pseudoscalar. Though clearly vanishing for ordinary systems, it was noted how for any material lacking inversion symmetry on the macroscopic scale, this special component of the beta tensor should survive. A n optically active system satisfies this requirement. So a solution of randomly oriented, optically resolved,
132 chiral centers may give rise to sum-frequency generation (SFG). This suggested an exciting new nonlinear spectroscopy that would promise a background-free way to examine only the chiral centers in any complex system. Chiral centers are of general interest, but since they are ubiquitous in both proteins and nucleic acids such a new method to study them in the absence of any background might be of considerable use in biophysics. Two reports of SFG from optically resolved arabinose have appeared (3, 4). However just last year an extensive and careful re-examination of SFG from these systems suggests a likely artifactual nature of the reported signals and that in fact SFG from such optically active systems is yet to be demonstrated (5, 6). Nevertheless an encouraging positive experiment first reported at this Symposium (7) shows an SFG signal from randomly oriented limonene in which an infrared resonance plays an enhancing role. The present contribution reports results from a detailed study (8) of the rotationally invariant microscopic beta tensor element. The aim is to search for reasons behind the evident fragility of the S F G signal and thereby to suggest ways for its optimization. The exploration is cast in the frequency domain so it is the dispersive properties of SFG that are examined. To our surprise we have discovered that the dispersion of such o # W M from randomly oriented systems is qualitatively different from that seen in the well-known, ubiquitous, everc-WM spectroscopies. We begin by presenting the analytic form for the rotationally invariant part of the beta tensor. Then a minimal three-state model is introduced for exploring the dispersive properties of SFG. Both electronic and vibrational resonances are included. These properties are displayed in the form of six figures in which both the amplitude (the modulus) and the phase of the response are shown, with the two incident frequencies and the energy level spacings as parameters. The dispersive behaviour of the SFG intensity (if homodyned) is not shown. But this is proportional to the square of the plotted amplitude. A brief concluding section with some speculation ends this contribution.
Theory General Let the sth order microscopic electrical susceptibility be called in more familiar terms
s a , the linear polarizability; φ
{2)
hyperpolarizability, and φ
{3)
rank tensor having 3
( i + l )
(where
ξ β , the first
= y , the second hyperpolarizability). It is an (s +1)-
Cartesian components. In the density matrix formalism
the analytic expression for φ^ is built from contributions from all possible "Liouville paths" that trace the evolution of states of the molecule as it becomes
133 (nonlinearly) polarized. Each path defines a particular time ordering of the actions on the molecule of the incident fields and whether a given action is on the bra side or the ket side of the basis set used to describe the molecule. A dual Feynman diagram, or equivalently, a wave-mixing energy level diagram (WMEL), is used to depict uniquely any given path. Provided no incident field acts more than once (this is the case of fully nondegenerate wavemixing) there are altogether s!2 paths to consider at 5th order (2* for the choice of bra or ket action at each perturbing event and s\ for the ways of time ordering the actions of s distinct fields). The contribution of each such path to the analytic expression for
