Article pubs.acs.org/JPCA
Isotopic Dependence of Excited-State Proton-Tunneling Dynamics in Tropolone Probed by Polarization-Resolved Degenerate Four-Wave Mixing Spectroscopy Kathryn Chew, Deacon J. Nemchick, and Patrick H. Vaccaro* Department of Chemistry, Yale University, P.O. Box 208107, New Haven, Connecticut 06520-8107, United States ABSTRACT: The origin band of the à 1B2−X̃ 1A1 (π* ← π) absorption system in monodeuterated tropolone (TrOD) has been probed with near-rotational resolution by applying the frequency-domain techniques of polarization-resolved degenerate four-wave mixing (DFWM) spectroscopy under ambient, bulk-gas conditions. Judicious selection of polarization geometries for the incident and detected electromagnetic waves alleviated intrinsic spectral congestion and facilitated dissection of overlapping transitions, thereby enabling refined rotational-tunneling parameters to be extracted for the à 1B2(π*π) ̃ manifold. A tunneling-induced bifurcation of ΔA0 = 2.241(14) cm−1 was measured for the zero-point level of electronically excited TrOD, reflecting the presence of a substantial barrier along the O−D···O ↔ O···D−O reaction coordinate and representing nearly a 10fold decrease in magnitude over the analogous quantity in the parent (TrOH) isotopologue. Observed trends in hydron-migration rates are discussed in light of the changes in the potential-surface topology sustained from the π* ← π electron promotion and the dynamical effects incurred by selective isotopic modification of the nuclear framework, with similar considerations being applied to interpret rotational constants and inertial defects. Simultaneous analyses performed on an interloping sequence band built upon ν38(b1) gave an excited-state tunneling splitting of ΔAν38̃ = 1.217(61) cm−1, highlighting the ability of this symmetric, out-of-plane normal mode to inhibit the unimolecular tautomerization process.
1. INTRODUCTION The use of isotopes to unravel chemical mechanisms has a long and venerable history in molecular science,1−4 with measurements of kinetic isotope effects (KIEs) and their dependence on extrinsic parameters (e.g., temperature) serving to elaborate diverse phenomena of practical and fundamental importance.1,2,5,6 Of particular note are the ubiquitous processes that result in the transfer of a light hydron7 between discrete atomic centers,8 where the sizable shift in relative mass sustained from isotopic substitution can alter reaction rates markedly by strong perturbation of zero-point energies (which change effective barrier heights) and tunneling probabilities (which scale inversely as the exponential of transferring mass).3,4,9 Despite such auspicious circumstances, the interpretation of complex hydron-migration events, especially those of biochemical origin, often proves to be challenging. For example, the enzymatic cleavage of stable C−H bonds is believed to involve quantum-mechanical (proton) tunneling;10−12 however, the precise nature of the collective reaction coordinate, as mediated by random (thermal) motions and/or choreographed (vibrational) displacements of the protein matrix, remains controversial.13−22 Consequently, detailed investigations of model systems that are small enough to be tractable for potent electronic structure and nuclear dynamics calculations yet retain sufficient complexity to embody the multimode pathways that govern substantially larger complexes can afford unique insights regarding basic paradigms and dominant propensities. The present work focuses on a specific © 2013 American Chemical Society
excited electronic state (π*π) of monodeuterated tropolone (TrOD), in which the shuttling hydron responsible for a prototypical intramolecular proton-transfer reaction has been replaced by a more massive deuteron. These experiments have enlisted the versatile nonlinear optical platform of resonant four-wave mixing (RFWM) spectroscopy23−29 to probe the attendant π* ← π absorption feature, thereby elucidating primary KIEs incurred by the selective modification of bonds that break and form during the chemical transformation. Figure 1 schematically illustrates the tautomerization process responsible for the unimolecular dynamics of tropolone, highlighting preferential stabilization of the chelated enol form30 by the action of an internal hydrogen bond that adjoins the hydroxylic (proton-donating) and ketonic (proton-accepting) oxygen centers. Owing to the dual connectivity that exists for the central hydrogen atom, both the ground31 [X̃ 1A1] and the lowest-lying excited32 [Ã 1B2(π*π)] electronic states support symmetric double-minimum potential surfaces in which a barrier of finite height separates two stable and equivalent tautomers of planar (Cs) symmetry. Although classically hindered, rapid interconversion between these degenerate structures can take place by means of coherent quantum Special Issue: Prof. John C. Wright Festschrift Received: January 5, 2013 Revised: March 10, 2013 Published: March 18, 2013 6126
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equated to a single sequence of pairwise nuclear exchanges that transform according to b2.35 The potential-surface topography in Figure 1 summarizes extensive quantum-chemical calculations performed on X̃ 1A1 and à 1B2 tropolone through use of complementary coupledcluster (CCSD)31 and equation-of-motion coupled-cluster (EOM-CCSD)32 levels of theory. For the ground state, the best estimates for the proton-transfer barrier height and the donor−acceptor (O···O) distance follow from fully relaxed CCSD(T)/aug-cc-pVDZ analyses that yield ΔEXpt̃ = 2557.0 ̃ ̃ cm−1 and rXO···O = 2.53 Å, with rXO···O decreasing to 2.31 Å at the symmetric transition-state geometry. As highlighted by the charge density-difference surface of Figure 1, the à 1B2−X̃ 1A1 transition is predicted to be a ring-centered π* ← π electron promotion that causes overall contraction of the five-membered reaction site (C−O−H···OC) and concomitant shortening of the intramolecular hydrogen bond, as reflected by optimized ̃ EOM-CCSD/aug-cc-pVDZ parameters of ΔEApt = 1270.6 cm−1 à and rO···O = 2.46 Å, where the latter decreases to 2.32 Å at the transition state and the former rises to 1316.1 cm−1 upon incorporation of a single-point triples correction. Given the exponential dependence of the hydron-migration probability on the barrier opacity (viz., the height and width), these results suggest a sizable enhancement in reaction rate to accompany the electronic excitation of tropolone, an assertion in keeping with the 20-fold difference in tunneling splitting (Δν) measured for TrOH between the vibrationless (υ = 0) or zero-point levels ̃ ̃ of the à 1B2 [ΔA0 = 19.846(25) cm−1] and X̃ 1A1 [ΔX0 = −1 0.973800245(37) cm ] manifolds by rotationally resolved degenerate four-wave mixing36 (DFWM) and high-resolution Fourier-transform microwave37,38 (FTMW) techniques. The present efforts rely on the analogous FTMW determination of ΔX0 ̃ = 0.05080938(17) cm−1 for the ground state of the monodeuterated isotopologue38 (TrOD) to make quantitative DFWM studies of isotopic tautomerization dynamics within the à 1B2(π*π) potential surface possible. The earliest quantitative information regarding the isotopic dependence of excited-state tropolone dynamics stems from seminal investigations of the near-ultraviolet absorption spectrum conducted under bulk-gas conditions by Alves and Hollas.39 Band-contour analyses published by these authors for the partially resolved à 1B2−X̃ 1A1 origin band revealed the (rotationless) spectral splitting of 18.93 ± 0.05 cm−1 for the parent isotopologue to be reduced to ∼2.2 cm−1 when the shuttling hydron was replaced by a deuteron, with the putatively small magnitude of ΔX0 ̃ (which had not been measured yet) suggesting these estimates of |ΔA0 ̃ − ΔX0 ̃ | to be ̃ dominated by ΔA0 . Building on the prior matrix-isolated infrared absorption work of Redington, et al.,40,41 Rossetti and Brus42 exploited laser-induced fluorescence (LIF) techniques to interrogate the π* ← π excitation of tropolone molecules entrained in a cryogenic neon matrix, finding bifurcation of the 000 resonance to be 21 ± 2 and 7 ± 1 cm−1 for TrOH and TrOD, respectively. Subsequent LIF measurements performed in a supersonic free-jet expansion by Tomioka and co-workers43 gave 19.0 ± 0.5 cm−1 (TrOH) and 2.0 ± 0.5 cm−1 (TrOD) for attendant values of |ΔA0 ̃ − ΔX0 ̃ |, which were interpreted to yield ΔA0 ̃ parameters of 25.2 and 2.4 cm−1. Extensions of such fluorescence-based probes have allowed the mode specificity of TrOD tunneling to be explored at vibronic resolution in both the ground44 and the lowest-lying (π*π) excited43,45,46 electronic states. A variety of other isotopic substitution studies
Figure 1. Unimolecular dynamics in tropolone. The coordinate that mediates the proton-transfer reaction in tropolone is shown, highlighting the shuttling hydron that interconnects two equivalent and asymmetric (Cs) tautomers by way of a symmetric (C2v) transition state. Schematic potential curves for the ground (X̃ 1A1) and excited (à 1B2) electronic manifolds are drawn on the basis of barrier heights (ΔEpt) and widths (rO···O) predicted from coupled-cluster calculations,31,32 with the magnitude of tunneling-induced bifurcation for representative vibronic features indicated by Δν. The origin band for the à −X̃ transition manifests two symmetry-allowed components, 0++ and 0−−, that are separated in energy by the absolute value of the change in tunneling splitting for the pertinent (vibrationless) levels, ̃ ̃ |ΔA0 − ΔX0 |. The redistribution of electronic charge accompanying this ring-centered π* ← π excitation is depicted in the density-difference plot on the right, where light and dark colors signify regions of gain and loss, respectively.
mechanical (proton) tunneling, thereby producing a characteristic bifurcation of all rovibronic features into components that are symmetric (+) and antisymmetric (−) with respect to the reaction coordinate. The magnitude of such tunneling-induced splitting scales in direct proportion to the rate of proton transfer and reflects not only motion of the shuttling hydron but also the concomitant displacement of other (heavy) atoms in the molecular framework and the redistribution of charge density about the constituent nuclei. Consequently, selective vibronic excitation and/or isotopic modification of tropolone can be expected to influence hydron-migration markedly,33 strongly affecting the pathway and the efficacy of intramolecular dynamics. Indeed, the nonrigidity conferred by this largeamplitude degree of freedom demands that eigenstates be classified under the encompassing G4 molecular symmetry group,34 where an isomorphism with the C2v point group used to describe the static transition-state configuration (at the barrier crest) allows irreducible representations to be designated as a1, b2, b1, and a2. From this permutationinversion perspective, the proton-transfer reaction can be 6127
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to be plane-polarized and to propagate collinearly along the Zaxis, kj ∝ ⇀ e Z, the polarization properties for each optical field can be described fully by angle ϕj56,57
have been reported for the tropolone system, including spectroscopic efforts focusing on selective deuteration of the seven-membered aromatic ring47,48 as well as replacement of naturally abundant carbon (12C) and oxygen (16O) atoms by their more massive nuclides (13C and 18O).38,49−51 The subtle changes in behavior uncovered by these endeavors have served to highlight the multidimensional nature of the underlying reaction coordinate and the important roles that choreographed vibrational motion of the molecular framework can play for the mechanism of tautomerization.33 In the present work, the lowest-lying portion of the TrOD π* ← π absorption system has been interrogated under ambient, bulk-gas conditions by exploiting the frequencydomain techniques of DFWM spectroscopy.28,29 Of particular importance is the inherent ability of this nonlinear optical probe to alleviate spectral congestion and minimize rovibronic complexity, thus permitting the tunneling-doubled resonances that constitute the à 1B2−X̃ 1A1 origin band to be examined with near-rotational resolution. The definitive assignment of observed features was facilitated by polarization-resolved detection schemes,52,53 which allowed rotational lines to be discriminated according to their attendant changes in angular momentum quantum numbers. The theoretical principles underlying such branch-suppression methods are the topic of section 2, while section 3 discusses practical considerations required for their experimental realization. Detailed spectroscopic results obtained for monodeuterated tropolone will be highlighted in section 4, with quantitative simulations of DFWM data sets acquired in the weak-field (optically unsaturated) regime enabling refined rotation-tunneling information to be extracted for the electronically excited à 1B2(π*π) manifold. The structural and dynamical parameters determined for TrOD will be contrasted with analogous quantities reported for the parent isotopologue and interpreted by reference to high-level ab initio calculations, before concluding in section 5 with a summary of key findings as well as suggestions for future studies.
εj = cos ϕj ⇀ eX + sin ϕj ⇀ eY
(3)
e X and ⇀ e Y span the transverse plane. where the unit vectors ⇀ The branch-suppression schemes exploited in the present work rely on judiciously selected polarization geometries, as specified compactly by the ordered list of parameters ε*4 ε1ε*2 ε3 ≡ ϕ4ϕ1ϕ2ϕ3 to dissect and analyze the rovibronically congested spectrum of the TrOD Ã 1B2−X̃ 1A1 origin band. To elaborate polarization-resolved spectral patterns, it proves convenient to focus on an isolated one-photon (rovibronic) transition taking place between energy levels designated by labels “g” and “e” for the ground and excited states, respectively, where only the former is presumed to be populated prior to the four-wave mixing interaction. The attendant eigenbasis for the unperturbed molecular Hamiltonian can be specified by |α⟩ ≡ |ηαJαMα⟩ (α = g or e), where Mα signifies projection of the total angular momentum Jα on the space-fixed quantization axis while ηα denotes all other rovibronic quantum numbers. When necessary, primes will be used to distinguish the magnetic sublevels composing each (2Jα+1)-fold degenerate manifold of eigenstates such that |g′⟩ ≡ |ηgJgMg′⟩. The angular frequency, ωαβ, and the dephasing rate, Γαβ, for the |α⟩ ↔ |β⟩ resonance are defined by ωαβ =
1 (Eα − Eβ ) ℏ
(4)
Γαβ =
1 φ (Γαα + Γββ) + Γ αβ 2
(5)
where Eα and Γαα denote the unperturbed energy and the depopulation rate for quantum state |α⟩. The additional factor of Γφαβ in the latter expression accounts for the action of phasedisrupting processes that destroy molecular coherence without affecting the attendant molecular populations.23,24 For convenience, the molecular properties embodied in eqs 4 and 5 often are combined into a single complex quantity to yield Ωαβ = ωαβ − iΓαβ.28,29,54 Building upon the canonical (weak-field) framework of timedependent perturbation theory,23,24,28,29 the instantaneous response evoked during a resonant four-wave mixing interaction can be shown to scale in proportion to the square modulus of the third-order electric polarization vector, ε*4 ·P(3)(t), as induced by successive coupling of the three impinging electromagnetic waves. For frequency-domain DFWM spectroscopy performed on nonstationary (gasphase) target molecules without spectral discrimination of the emerging signal field,54 the intensity observed at incident frequency ω, I4(ω), can be related to the Fourier transform of this pivotal quantity, ε4*·P(3)(ωs), such that
2. THEORETICAL BACKGROUND The ensuing discussion of rotationally resolved DFWM spectroscopy builds upon our previous tensor-algebraic analyses of weak-field response,52 as generalized to encompass multiple excitation paths.53 Each electromagnetic wave is assumed to be monochromatic,54 with a numerical index used to distinguish among the incident (j = 1, 2, or 3) and generated (j = 4) beams. The electric field vector, Ej(r,t), for a monochromatic plane wave having angular frequency ωj and propagation wavevector kj is defined by 1 Ej(r, t ) = [E ωje i(k j·r − ωjt ) + E*ωje−i(k j·r − ωjt )] (1) 2 where the vector amplitude Eωj embodies a constant scalar amplitude, Eωj = |Eωj|, and a polarization unit vector, εj = Eωj/|Eωj|, such that Eωj = Eωjεj. For an isotropic target medium characterized by a frequency-dependent index of refraction n(ωj), the corresponding cycle-averaged intensity follows from55 cε Ij = 0 n(ωj)|E ωj|2 (2) 2
+∞
I4(ω) ∝
∫−∞
|⟨⟨ε4*·P(3)(ωs)⟩⟩v |2 dωs
(6)
where the angular brackets signify an average over translational degrees of freedom as specified by the normalized velocity distribution f(v), as defined below.
where c and ε0 denote the speed of light and electrical permittivity for a vacuum, respectively. By assuming all beams
⟨⟨ε4*·P(3)(ωs)⟩⟩v = 6128
∫v [ε4*·P(3)(ωs)]f (v) dv
(7)
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Table 1. Expansion Coefficients for Rank-Zero DFWM Response Tensorsa R(0) 0 (ϕ4ϕ3ϕ2ϕ1;Jg,Je) tensor elements
R(0) 0 (ϕ4ϕ1ϕ2ϕ3;Jg,Je) tensor elements
quantity
P(Jg)
Q(Jg)
R(Jg)
P(Jg)
Q(Jg)
R(Jg)
c00(Jg) c12(Jg) c13(Jg) c14(Jg) A B C restriction
Jg(2Jg + 1) (Jg + 1)(2Jg + 3) 2(6J2g − 1) (Jg − 1)(2Jg − 3) 1 6 1 Jg ≥ 1
Jg(Jg + 1) 4Jg(Jg + 1)− 3 4J2g + 4Jg + 2 4J2g + 4Jg − 3 4 4 4 Jg ≥ 1
(Jg + 1)(2Jg + 3) Jg(2Jg − 1) 12J2g + 24Jg + 10 2J2g + 9Jg + 10 1 6 1
Jg(2Jg − 1) (Jg − 1)(2Jg − 3) 2(6J2g − 1) (Jg + 1)(2J + 3) 1 6 1 Jg ≥ 1
Jg(Jg + 1) 4Jg(Jg + 1)− 3 4J2g + 4Jg + 2 4J2g + 4Jg + − 3 4 4 4 Jg ≥ 1
(Jg + 1)(2Jg + 3) (Jg + 2)(2Jg + 5) 12J2g + 24Jg + 10 Jg(2Jg − 1) 1 6 1
a The dimensionless coefficients used to expand the scaled isotropic (rank-zero) DFWM response tensor in eq 13 are tabulated as a function of initial rotational quantum number (Jg), with results being partitioned according to the angular momentum change (ΔJeg = Je − Jg) for each one-photon allowed rotational branch. Also shown are the high-Jg limits of these values, as introduced by eq 17 and defined by eqs 18−20.
By assuming the ground state of the isolated |e⟩ ↔ |g⟩ transition to be populated uniformly (without angular momentum anisotropy)58 and to support a target number density of NηgJg, detailed analyses yield the following compact expression for the pertinent component of the third-order polarization vector56,57 NηgJ g Eω Eω* Eω e ik σ·r ⟨⟨ε4*· P(3)(ωs)⟩⟩v = 8ℏ3 1 2 3 ̂ ⟩W4321(ωs) + ⟨O4123 ̂ ⟩W4123(ωs)] × [⟨O4321
evaluated analytically by means of the plasma dispersion function.54 Of special importance for the present studies are the two expectation values of the four-fold transition-dipole operators, ⟨Ô 4321⟩ and ⟨Ô 4123⟩, that appear in the DFWM signal expression of eq 8. These quantities follow from an implicit assumption that relaxation processes take place in an isotropic fashion (viz., Γαα = Γα′α′ and Γαβ = Γα′β′ = Γβα),56,57 with use of a tensor-recoupling formalism allowing for effective separation of intrinsic (e.g., molecular populations and transition moments) and geometric (e.g., incident/generated field polarizations) contributions to the induced third-order response. For an isolated |e⟩ ↔ |g⟩ resonance, one finds56,57
(8)
where the complex exponential eikσ·r, with kσ = k1 − k2 + k3, reflects the constraint of linear-momentum conservation imposed on the four-wave mixing process. The intrinsic frequency dependence of eq 8 is encoded in two linear combinations of velocity-averaged quantities
̂ ⟩ = (εi*·μ̂ )Pe ′(εj*·μ̂ )Pg ′(εk*·μ̂ )Pe(εl ·μ̂ ) ⟨Oijkl =
W4321(ωs) = ⟨⟨D12g ⟩⟩v + ⟨⟨D21g ⟩⟩v + ⟨⟨D32e⟩⟩v + ⟨⟨D23e⟩⟩v
(10)
where each factor stems from one of the eight possible time orderings of matter−field interactions contributing to the frequency-domain DFWM response (in the absence of temporal discrimination).28,29,56,57 More specifically, Dijα signifies a resonant denominator term arising from the successive coupling of Ei(r,t) and Ej(r,t) to produce an intermediate (diagonal) coherence in state |α⟩. The definition of these quantities depends on the precise nature of the impinging electromagnetic waves,54 with the monochromatic form of eq 1 giving in the case of D12g D12g =
2Jg + 1
R 0(0)(εi*εj εk*εl ; Jg , Je ) (12)
where Pα = ∑ Mα|ηαJαMα⟩⟨ηαJαMα| signifies the normalized projection operator onto the subspace spanned by eigenvectors |α⟩ ≡ |ηαJαMα⟩ and μeg = ⟨ηeJe∥μ̂∥ηgJg⟩ denotes the reduced matrix element for the electric dipole moment operator, μ̂. The final equality in eq 12 holds true only in the case of an isotropic target medium (without angular momentum anisotropy),58 with the rank-zero element of the DFWM response tensor, R(0) 0 (εi*εjεk*εl; Jg, Je), representing a fully calculable quantity that embodies the angular momentum properties of the selected transition (Jg and Je) as well as the transverse characteristics of the incident/generated electromagnetic waves (εi*εjεk*εl).56,57 In the case of linearly polarized fields propagating along a common axis such that ε*i εjε*k εl ≡ ϕiϕjϕkϕl [cf., eq 3], a straightforward, albeit tedious, analysis yields
(9)
W4123(ωs) = ⟨⟨D12e⟩⟩v + ⟨⟨D21e⟩⟩v + ⟨⟨D32g ⟩⟩v + ⟨⟨D23g ⟩⟩v
|μeg |4
(2Jg + 1)3/2 R 0(0)(ϕϕϕ ϕ; J , J ) = i j k l g e
δ(ωs − ωσ ) (Ωeg ″ − ω1)(Ωg ′ g ″ − ω1 + ω2)(Ωe ′ g ″ − ω1 + ω2 − ω3)
1 60c00(Jg )
[c12(Jg ) cos(ϕ1 + ϕ2 − ϕ3 − ϕ4) + c13(Jg )
(11)
cos(ϕ1 − ϕ2 + ϕ3 − ϕ4) + c14(Jg )
where ωσ = ω1 − ω2 + ω3 and the Dirac delta function, δ(ωs − ωσ), reflects energy conservation. Wasserman et al.54 have reported a substitution table that allows all other Dijα factors to be obtained from this expression for D12g. The velocity averaging in eqs 9 and 10 demands that the common rest frequency for the incident (degenerate) fields be replaced by their Doppler-shifted counterparts, ωj → ωj − kj·v, with a collinear approximation permitting requisite integrals to be
cos(ϕ1 − ϕ2 − ϕ3 + ϕ4)]
(13)
where the coefficients c00(Jg), c12(Jg), c13(Jg), and c14(Jg) are polynomials in Jg that reflect the change in angular momentum quantum number (ΔJeg = Je − Jg) taking place during the fourwave mixing process. Table 1 contains a compilation of expansion parameters for the pertinent R(0) 0 (ϕ4ϕ3ϕ2ϕ1;Jg,Je) 6129
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Figure 2. Experimental configuration for DFWM spectroscopy. The apparatus employed for studies of ambient TrOD vapor is depicted schematically. Tunable ultraviolet radiation produced by frequency doubling the narrow-bandwidth output of a Nd:YAG-pumped dye laser was partitioned into the three input waves, E1, E2, and E3, needed to implement the forward-box configuration for DFWM spectroscopy, with the signal photons, E4 (which emerge in the direction specified by wavevector k4 = k1 − k2 + k3), being isolated through use of spectral and spatial filters. Optical components inserted into the incident and detected fields permitted various linear-polarization geometries to be established, as required for the rovibronic branch-suppression methodology discussed in the text. Analogous linear-absorption data were recorded by directing a single ultraviolet beam through the sample chamber and monitoring the transmitted intensity as a function of excitation frequency.
and R(0) 0 (ϕ4ϕ1ϕ2ϕ3;Jg,Je) tensor elements, highlighting the subtle differences that exist among these quantities for the one-photon accessible P-branch (ΔJeg = −1), Q-branch (ΔJeg = 0), and R-branch (ΔJeg = +1) rovibronic lines. By combining eqs 6, 8, and 12 under the tacit assumption of linearly polarized electromagnetic waves, the DFWM signal expression can be reformulated as I4(ω) ∝
g
2Jg + 1
×
∫−∞
g
2Jg + 1
|μeg |8 I1I2I3 |R 0(0)(ϕ4 ϑϕ2 ϑ; Jg , Je )|2 3(ω)
where the velocity-averaged line shape function defined below, 3 (ω), encodes the spectral behavior inherent to eqs 9 and 10.
8
|μeg | I1I2I3
+∞
∫−∞
|W4321(ωs) + W4123(ωs)|2 dωs
(16)
The requisite conditions for achieving rovibronic branchsuppression in one-photon resonant DFWM spectroscopy follow from consideration of the pertinent rank-zero response tensor elements in the limit of high rotational quantum numbers (i.e., Jg → ∞)52,53
|R 0(0)(ϕ4ϕ3ϕ2ϕ1; Jg , Je )W4321(ωs)
+ R 0(0)(ϕ4ϕ1ϕ2ϕ3; Jg , Je )W4123(ωs)|2 dωs
Nη2gJ
(15)
3(ω) =
Nη2gJ
+∞
I4(ω) ∝
(14)
ϕ; J , J ) lim (2Jg + 1)3/2 R 0(0)(ϕ4ϕϕ i 2 j g e
where the product of incident cycle-averaged intensities, Ij ∝ |Eωj|2, reflects the expected intensity-cubed dependence of the weak-field response.28,29 This expression shows I4(ω) to scale in proportion to the square of the ground-state population and the eighth power of the |e⟩ ↔ |g⟩ transition moment, both of which will tend to alleviate spectral congestion by minimizing the contributions made by weaker features. The branchsuppression schemes exploited for the present study rely on manipulation of the rank-zero DFWM response tensors in eq 14, with proper selection of incident and detected polarization properties enabling these quantities to vanish simultaneously for a preselected subset of rovibronic transitions.52,53 Because (0) the pertinent R(0) 0 (ϕ4ϕ3ϕ2ϕ1;Jg,Je) and R0 (ϕ4ϕ1ϕ2ϕ3;Jg,Je) elements can be interconverted by exchanging ϕ1 and ϕ3, it proves convenient to exploit a mutual polarization geometry where ϕ1 = ϕ3 = ϑ such that ε*4 ε1ε*2 ε3 ≡ ϕ4ϑϕ2ϑ ≡ ε*4 ε3ε*2 ε1. In this manner, the DFWM signal expression can be simplified to
Jg →∞
=
1 [A cos(ϕ1 + ϕ2 − ϕ3 − ϕ4) 60
+ B cos(ϕ1 − ϕ2 + ϕ3 − ϕ4) + C cos(ϕ1 − ϕ2 − ϕ3 + ϕ4)]
(17)
where i,j = 1 or 3 and the coefficients A, B, and C now represent positive integers (cf., Table 1) that depend solely on the absolute value of ΔJeg as shown below.
6130
A = 4 − 3|ΔJeg |
(18)
B = 4 + 2|ΔJeg |
(19)
C = 4 − 3|ΔJeg |
(20)
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Figure 3. Linear and nonlinear spectroscopy of TrOD. The region proximate to the origin of the TrOD Ã 1B2−X̃ 1A1 (π* ←π) system was probed under ambient, bulk-gas conditions by exploiting linear-absorption (uppermost trace in red) and nonlinear-DFWM (lowermost trace in blue) techniques based upon a high-resolution excitation source (
