A Molecular Theory for Doubly Resonant IR-UV-vis Sum-Frequency Generation�� M. Hayashi,��� S. H. Lin,*,�� M. B. Raschke,| and Y. R. Shen| Center for Condensed Matter Sciences, National Taiwan UniVersity, 1 RooseVelt Rd., Sec. 4, Taipei, Taiwan, 10617, ROC, Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei, Taiwan 106 ROC, and Department of Physics, UniVersity of California Material Sciences DiVision, Lawrence Berkeley National Laboratory, Berkeley, California 94720 ReceiVed: July 10, 2001 In Final Form: NoVember 28, 2001 The experiment for measuring doubly resonant infrared-visible (IR-vis) sum-frequency generation (SFG) has recently been developed by Shen and his co-worker and applied to Rhodamine 6G on silica surfaces. In this paper, based on the Born-Oppenheimer adiabatic approximation, a molecular theory for doubly resonant IR-vis SFG generation as a two-dimensional surface spectroscopy is presented. We shall show that this new nonlinear spectroscopy is closely related to IR and resonance Raman spectroscopy. In this preliminary theoretical derivation, the displaced harmonic potential energy surfaces model for the electronic ground and electronically excited states of the system are used to obtain the band shape functions for doubly resonant IR-vis SFG at a finite temperature. One of the unique and powerful abilities of this spectroscopy is that interference effects among IR-active modes can be observed. The calculated excitation profiles and IR spectra of the doubly resonant IR-UV SFG from Rhodamine 6G on fused silica are demonstrated. We shall show that this new nonlinear spectroscopy can provide experimentalists with access to microscopic properties of molecules on surfaces or at interfaces. 1. Introduction Optical signal for second-order nonlinear generation is dipole- forbidden in media with inversion symmetry. However, such signals have been generated at the interfaces of isotropic media since the early days of nonlinear optics.1,2 An important application of surface nonlinear optical measurements is the determination of adsorbate spectra via the resonant enhancement of the second-order nonlinear susceptibility l2. Early measure- ments exploited electronic resonance to record adsorbate electronic spectra.3,4 Recently, the emphasis has been placed on the application of infrared (IR) + visible or UV sum- frequency generation (SFG) to obtain adsorbate vibrational spectra.5-7 The vibrational spectra of a number of adsorbates have now been reported such as the methyl modes of alkanethiol self-assembled monolayers on gold 6 a number of alcohols on silica 2 Langmuir-Blodgett films on silica 10,11 methoxy on Ni (111) 11 acetonitrile on ZrO2,12 etc. (see, e.g., ref 13). As mentioned above, second-harmonic and sum-frequency generation (SHG/SFG) have recently found a wide rage of applications as probes in surface science due to their intrinsic surface and interface sensitivity and specificity.14-16 It is possible to have both ��IR and ��vis tunable in SFG for two-dimensional surface spectroscopy.17 With ��IR and ��vis near surface vibrational and electronic transitions, respectively, SFG could be doubly resonantly enhanced. In this respect, it is similar to resonant Raman spectroscopy (RRS), which is known for its powerful applications in condensed matter physics, chemistry, and biology however, SFG has the additional advantage of being surface specific and applicable to fluorescent molecules. As in RRS, doubly resonant (DR) enhancement in SFG occurs only when the probed vibrational and electronic transitions are coupled. This allows for more selective spectroscopic informa- tion and better assignment of the vibrational modes. Moreover, the coupling strengths between electronic and vibrational transitions can be deduced. The technique could also be valuable for studies of intermolecular interactions at surfaces and interfaces. However, to extract molecular properties such as potential energy surfaces of molecules at surface, a molecular theory for doubly resonant IR-UV SFG intensity is needed for multimode systems. It should be important to note here that molecular descriptions (or vibronic models) for RRS were developed and applied by many researchers.18-42 In particular, the time-correlator approach to the Raman polarizability tensor was originally presented by Hizhnyakov and Tehver. Later, transform technique was de- veloped based on the time-correlator approach for systems obeying the so-called standard assumptions (or the simplest vibronic model): (1) the adiabatic and Condon approximations, (2) a single electronic exited state, (3) the harmonic approxima- tion for the vibrations, (4) linear electron-vibration coupling (displaced harmonic potential surface model), and (5) a constant damping in each of the vibrational levels of the electronic excited state.19,22 Transform methods were subsequently ex- tended by Page and co-workers to systems beyond these assumptions for example, harmonic potential surfaces can be not only displaced but also distorted25 and/or rotated (Duschin- sky effect),39 i.e., the normal-mode frequencies change upon electronic excitation as a result of quadratic electron-vibration coupling. Another extension was to non-Condon active modes.39 Additional extensions of the transform technique included effects of inhomogeneous broadening,34 nonadiabatic corrections,31 anharmonicity,36 and formal temperature average.40,41 One major advantage of the transform technique allows one to calculate a RR profile of a Raman-active mode from the �� Part of the special issue ���Noboru Mataga Festschrift���. * Corresponding author. ��� National Taiwan University. �� Academia Sinica. | University of California. 2271 J. Phys. Chem. A 2002, 106, 2271-2282 10.1021/jp012633l CCC: $22.00 �� 2002 American Chemical Society Published on Web 02/14/2002

measured absorption line shape using the model parameters of only this active mode while the information on the remaining inactive modes can be included via the use of the measured absorption. In this fashion, many of the complications which appear when dealing with molecular transitions within the full adiabatic approximation can be bypassed.28 Recent development of ab initio molecular orbital calculation methods has made it possible to provide potential surface properties of various molecules at a specific calculation level. Thus, information about part of inactive mode space can be made available, especially for high-frequency intramolecular modes. This allows one to extract ���pure��� information on low- frequency modes resulting from both intramolecular modes and environmental modes, which, in turn, provide molecular dynam- ics and/or ab initio developers with useful information for their challenging tasks to construct these low-frequency modes theoretically. In this case, a full vibronic description of inactive modes is still meaningful. A main purpose of this paper is to provide a theoretical treatment for SFG with the doubly resonant case, i.e., IR-vis or UV. We shall also show how to apply the Born-Oppenheimer (B-O) adiabatic approximation to obtain the expressions for the doubly resonant SFG for molecular systems. In this case, as has been discussed above, we will apply the B-O adiabatic approximation to all the possible vibrational modes. It will be shown that the recasting method cannot be applied to obtain the general expression of IR-UV SFG susceptibility.43 We shall also perform numerical simulation for model systems. In particular, we will focus on how potential energy surface properties affect doubly resonant IR-UV or vis SFG spectra. We will demonstrate how to theoretically construct doubly resonant IR-vis SFG spectra of Rhd6G on fused silica we shall present the IR spectra and excitation profiles of doubly resonant IR-vis SFG signal of this system. 2. General Theory We shall consider a model system shown in Figure 1. In Figure 1, g, m, k denote the initial, intermediate, final state manifolds, and ��I, ��II represent the frequencies of the two lasers used in SFG experiments. According to the definition of the second-order SFG suscep- tibility lR��(��I (2) + ��II), i.e., we find, for the doubly resonant case,44-46 In eq 2, for example, p��mg ) p(��m - ��g) ) Em - Eg, ��gg(T) is the initial population distribution function, ��mg represents the dephasing rate constant, and ��gk(R) denotes the R-component transition moment. In this section, we shall derive theoretical expressions for second-order susceptibility of IR-UV(vis) and UV(vis)-IR SFG in terms of a molecular description. For this purpose, we shall employ the Born-Oppenheimer adiabatic approximation and the harmonic potential surfaces for the excited and ground electronic states. It should be noted that the term ��mm(T) in eq 2 is often ignored for the case of doubly resonant excitations and recast the dummy indices, for example, g, and m to m and g in eq 2. For the IR-UV(vis) SFG in a molecular description, ��mm(T) cannot be ignored. Herein, for simplicity we will omit notation of (vis), that is, simply using IR-UV SFG. 2.1. IR-UV SFG. In the adiabatic approximation, IR-UV SFG can be described by noting gfg{V}, mfg{V���}, kfe{u} in eq 2, where g and e denote the electronic states while {V} and {u} represent the vibrational states. Figure 1 shows a schematic representation of IR-UV and UV-IR SFG. For the IR-UV case, setting ��I ) ��IR and ��II ) ��UV, we find As can be seen from eq 3, this type of SFG case is very similar to resonance Raman (RR) scattering. The square of the last term (involving the summation over {u}) in eq 3 corre- sponds to the band-shape function of the RR excitation pro- file.47,48 Equation 3 shows that the selection rule for this IR- UV SFG is that the vibrational mode should be both IR active and the resonance Raman active. (i.e., usually totally symmetric). Here, for example, ��gV,ku(R) in eq 3 can be written as where ��gV and ��eu denote the vibrational wave function for the normal modes. In this case, we have where, for example, leui(Q���) i is the vibrational wave function for the mode i of the electronically excited state e and it is given by Here Hul(x�����/pQ���) l l represents the Hermite polynomials. We assume that the Condon approximation49 can be applied to eq 4. It follows that Figure 1. A schematic representation of doubly resonant IR-UV and UV-IR SFG. PR (2) (��I + ��II) ) ������lR��(��I �� (2) + ��II)E1 (��I)E2��(��II) (1) lR��(��I (2) + ��II) ) 1 p2 ���������{��gg(T) g m k - ��mm(T)} �� ��gk(R)��mg( )��km(��) [(��I - ��mg) + i��mg][(��I + ��II - ��kg) + i��kg] (2) lR�� (2)(IR-UV) (��IR + ��UV) ) 1 p2{V} ������{��gV,gV(T) {V���} - ��gV���,gV���(T)} �� ��gV���,gV( ) [(��IR - ��gV���,gV) + i��gV���,gV] �� ��� {u} ��gV,eu(R)��eu,gV���(��) [(��IR + ��UV - ��eu,gV) + i��eu,gV] (3) ��gV,eu(R) ) �����gV|��ge(R)|��eu��� (4) ��gV(Q) ) ���lgVi(Qi) i)1 N and ��eu(Q���) ) ���leui(Q���)i i)1 N leul(Q���) l ) NulHul(x�����/pQ���)e-��l���Ql���2/(2p) l l (5) 2272 J. Phys. Chem. A, Vol. 106, No. 10, 2002 Hayashi et al.