Polarization-Angle-Scanning Two-Dimensional Spectroscopy

ARTICLE pubs.acs.org/JPCA

Polarization-Angle-Scanning Two-Dimensional Spectroscopy: Application to Dipeptide Structure Determination Jun-Ho Choi† and Minhaeng Cho*,†,‡ † ‡

Department of Chemistry and Research Institute for Natural Sciences, Korea University, Seoul 136-701, Korea Multidimensional Spectroscopy Laboratory, Korea Basic Science Institute, Seoul 136-713, Korea ABSTRACT: Coherent two-dimensional optical spectroscopy based on a heterodyne-detected stimulated photon echo measurement technique requires four ultrashort pulses whose pulse-to-pulse delay times, wavevectors, and frequencies are experimentally controllable variables. In addition, the polarization directions of the four radiations can also be arbitrarily adjusted. We show that the polarizationangle-scanning two-dimensional spectroscopy can be of effective use to selectively suppress either all the diagonal peaks or a cross-peak in a given two-dimensional spectrum. Theoretical relationships between the transition dipole vectors of a given pair of coupled modes or quantum transitions and the polarization angle configuration making the corresponding cross-peak vanish are established. Here, to shed light into the underlying principles of the polarization-angle-scanning two-dimensional spectroscopy, we considered the amide I vibrations of various isotope-labeled dipeptide conformers and show that one can selectively suppress a cross-peak by properly controlling the polarization angle of a chosen beam among them. Once the relative directions of the amide I transition dipole vectors are determined using the polarization-angle-scanning technique theoretically proposed here, they can serve as a set of constraints for determining structures of model peptides. The present work demonstrates that the polarization-controlled two-dimensional vibrational or electronic spectroscopy can provide invaluable information on intricate details of molecular structures.

I. INTRODUCTION A variety of two-dimensional spectroscopy techniques probing different nonlinear optical phenomena have been used to study molecular structures and dynamics in condensed phases. One of the most popular measurement methods is based on a heterodyne-detected stimulated photon echo technique, where three time-separated laser pulses are used to create two coherences that are separated by the so-called waiting time Tw.1-7 The time evolution of the last coherence generated by the third pulsematter interaction is responsible for producing the photon echo signal field.8 The relative phase and amplitude of the resultant echo field is then detected by employing an interferometric heterodyne detection method utilizing an additional reference (local oscillator) field.3,9-11 To carry out the heterodyne-detected photon echo measurement in general, one should have four laser pulses. In such an experiment, there are quite a number of experimentally controllable variables that are (i) center frequencies, (ii) pulse-to-pulse delay times, (iii) relative optical phases, (iv) propagation directions, (v) polarization states, e.g., linearly, circularly, or elliptically polarized radiations, and (vi) polarization directions of the four pulses. In the present paper, we shall focus on how the polarization directions of the incident pulse radiations affect the diagonal peak and cross-peak amplitudes in a given 2D spectrum. Typically, the crossing angles between incident beam propagation directions are fairly small so that they all can be assumed to be almost collinear. Now, without loss of generality, let us assume that the four radiations including the local oscillator field propagate along the X-axis in a space-fixed frame. One of the most convenient r 2010 American Chemical Society

beam configurations for a heterodyne-detected stimulated photon echo method is the case that the four field polarization directions are all parallel to the laboratory Z-axis. The thus obtained 2D spectrum has been referred to as the parallel polarization signal, i.e., S([0,0,0,0]). Also, the perpendicular polarization signal S([0,π/2,π/2,0]) is obtained by using the polarization configuration where the polarization directions of the k2 and k3 fields are perpendicular to those of the k1 and ks fields. Here, ks is the wavevector of the signal field whose phase and amplitude are detected by allowing its interference with a reference field at the detector. Then, it was shown that the diagonal peaks in the difference spectrum defined as S([0,0,0,0]) 3S([0,π/2,π/2,0]) all vanish.4,12,13 In general, the polarization directions of linearly polarized beams, which are on the Y-Z plane, can be continuously varied so that the relative angles of these directions with respect to the Z-axis specify the beam polarization configuration. These angles jj (j = 1, 2, 3, LO) can be varied by using properly chosen linear polarizers placed before the sample cell. Hereafter, the notation [jLO,j3,j2,j1] means that the corresponding polarization angles are jj (j = LO, 3, 2, 1) in the order. An early attempt made by Hochstrasser and co-workers is to control the polarization angles jj’s to be either 0 or π/2.13,14 Later, Zanni et al. showed that the Special Issue: Graham R. Fleming Festschrift Received: July 13, 2010 Revised: October 18, 2010 Published: November 16, 2010 3766

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signal obtained with either [0, -π/3, π/3,0] or [π/4, -π/4, π/2, 0] can produce 2D spectrum with vanished diagonal peaks.4 In addition, it was recently shown that one can selectively make each cross-peak vanish by controlling the polarization direction angles properly, where the specific set of polarization angles was found to be directly related to the angles between involved quantum transition dipole vectors.1 This suggests that the relative angles between any pair of transition dipoles can be determined by examining the polarization angles with which the corresponding cross-peak vanishes in the measured 2D spectrum. Here, we shall present theoretical descriptions on the relationships between the externally controlled polarization directions and the angles between transition dipoles of coupled chromophores in condensed phases. Then, we shall apply the theory to the structure determination of blocked dipeptide models having three amide bonds, where their amide I 2DIR (two-dimensional IR) spectra for varying polarization angles are numerically simulated to demonstrate the underlying principles.

II. POLARIZATION-ANGLE-SCANNING 2D SPECTROSCOPY In 2D optical spectroscopy, the interaction of the four classical electric fields with the molecular system can be theoretically analyzed in terms of the nonlinear response function of the system. II-A. Nonlinear Response Function Formalism. Spectroscopic signals are generated by the polarization (dipole density) P(r,t) of the molecular system that is induced by the radiationmatter interactions. In the electric dipole approximation, the system interacts with external electric field E(r,t) and the conjugate molecular property is the system dipole moment μ. Then, the interaction energy is approximately given as ^ 3 Eðr;tÞ ð1Þ Hint ðtÞ ¼ - μ where E(r,t) is the superposition of the three incident pulses E1, E2, and E3 in the four-wave-mixing spectroscopy The total Hamiltonian H of the system is given as H0 þ Hint with H0 the material Hamiltonian in the absence of radiation. One can start with the quantum Liouville equation for the density operator F(t) to obtain the nonlinear polarization that is defined as the expectation value of the electric dipole operator over the nthorder density operator. Using the time-dependent perturbation theory by treating Hint as the perturbation Hamiltonian to the reference system characterized by H0, the nth-order density operator F(n)(t) is given as Z Z τn i n t FðnÞ ðtÞ ¼ dτn dτn - 1 3 3 3 p t0 t0 Z τ2 dτ1 G0 ðt - τn Þ Lint ðτn Þ G0 ðτn - τn - 1 Þ Lint ðτn - 1 Þ 3 3 3 t0

G0 ðτ2 - τ1 Þ Lint ðτ1 Þ G0 ðτ1 - t0 Þ Fðt0 Þ

ð2Þ

where G0(t) = exp(-iL0t/p) is the time evolution operator in the absence of the external radiation and the Liouville operators are defined as LaA = [Ha, A] for a = 0 or int. Equation 2 allows a clear interpretation of the density matrix evolution: the initial state F(t0) at time t0 propagates freely without perturbation for τ1 - t0, i.e., G0(τ1-t0), and at t = τ1, the first radiation-matter interaction takes place, which corresponds to the action of Lint(τ1). This is repeated n times until the final interaction at t = τn, which is represented by Lint(τn). Finally, the system evolves freely up to the observation time t, and its time-evolution is determined by

G0(t-τn). All the possible interaction times are allowed for by the multiple integrals over τ1, ..., τn under the time ordering condition t0 e τ1 e ... e τn e t. Each of the density matrices in eq 2 provides the corresponding nth-order nonlinear polarization P(n)(r,t) as PðnÞ μðr;tÞ

¼ Tr½^ μ FðnÞ ðtÞ R¥ R¥ ¼ 0 dtn 3 3 3 0 dt1 R ðnÞ ðtn ;:::;t1 ÞlEðr;t - tn Þ 3 3 3 ð3Þ Eðr;t - tn 3 3 3 - t1 Þ

Here, the nonlinear response function R(n)(tn,...,t1) given by R ðnÞ ðtn ;:::;t1 Þ ¼

n i θðtn Þ 3 3 3 θðt1 Þ p

Æμðtn þ ::: þ t1 Þ½μðtn - 1 þ ::: þ t1 Þ;½ 3 3 3 ½μðt1 Þ;½μð0Þ;Feq 3 3 3 æ

ð4Þ where μ(t) = exp(iH0t/p)μ̂ exp(-iH0t/p) is the dipole operator in the interaction picture and the angular bracket in eq 4 denotes the trace of a matrix. The causality condition is imposed in the response function in eq 4 by the Heaviside step function θ(t). To obtain a realistic response function, it is necessary to take into account the system-bath interaction that is responsible for various spectroscopic phenomena such as dephasing, relaxation, spectral diffusion, and population and coherence transfers. This can be achieved by employing the material Hamiltonian including the system-bath interaction, evaluating eq 4, and taking the ground state as the initial state. Since the three radiation-matter interactions can induce quantum transitions among up to four different states, the most general result for a four level system should be discussed. Denoting the four eigenstates of the material Hamiltonian as |gæ, |aæ, |bæ, and |cæ, the third-order response function can be expanded into eight terms 3 4 i R ð3Þ ðt3 ;t2 ;t1 Þ ¼ θðt3 Þ θðt2 Þ θðt1 Þ ∑ ½R R ðt3 ;t2 ;t1 Þ - ðccÞ p R¼1 ð5Þ where the components RR(t3,t2,t1) are given by R 1 ðt3 ;t2 ;t1 Þ ¼

∑ μgc μcb μba μag expð- iωab t3 - iωac t2 - iωag t1 Þ F1 ðt3 ;t2 ;t1 Þ abc R 2 ðt3 ;t2 ;t1 Þ

¼

∑ μgc μcb μba μag expð- iωab t3 - iωac t2 - iωgc t1 Þ F2 ðt3 ;t2 ;t1 Þ abc R 3 ðt3 ;t2 ;t1 Þ

¼

∑ μgc μcb μba μag expð- iωab t3 - iωgb t2 - iωgc t1 Þ F3 ðt3 ;t2 ;t1 Þ abc R 4 ðt3 ;t2 ;t1 Þ

¼

∑ μgc μcb μba μag expð- iωcg t3 - iωbg t2 - iωag t1 Þ F4 ðt3 ;t2 ;t1 Þ abc ð6Þ

Here, μab is the transition dipole moment between states a and b, and pωhab = ÆEa - Ebæ denotes the transition energy averaged over the bath degrees of freedom. We note that each component RR(t3,t2,t1) consists of three parts: (i) the product of four transition dipole moments that determines the amplitude of the transition (transition strength), (ii) three oscillatory terms representing coherences with corresponding transition frequencies determining peak positions in the multidimensional frequency domain (coherence oscillation), and (iii) F1-F4 that determine the 3767

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line shape and broadening (line-shape function). The cumulant expansion technique shows that the line shape functions F1-F4 are composed of the frequency-frequency correlation functions of the form Æδω(t1) δω(t2)æ where δω(t) denotes instantaneous fluctuation of the transition frequency due to the system-bath interaction. Explicit expressions for F1-F4 can be found in ref 1 for example. Individual terms RR(t3,t2,t1) of the total third order response function in eq 6 represent a series of distinct optical transition events that are called nonlinear optical transition pathways. Furthermore, they are fourth-rank tensors in a laboratory coordinate frame. Once the nonlinear polarization is obtained, the signal electric field E(n)(r,t) detected in experiments can be obtained by solving the Maxwell equation with the nonlinear polarization P(n)(r,t) as the source. Assuming that the temporal (n) amplitudes P(n) S (t) and ES (t) slowly vary in time compared to the optical period and that the phase-matching condition is satisfied, the signal electric field becomes linearly proportional to the nth-order polarization amplitude as E(n) S (t) (iωs/n(ωs)) P(n) S (t), where n is the frequency-dependent refractive index. II-B. Reorientational Contribution to the Nonlinear Response Function. To take into account the reorientational contribution to the nonlinear response function, we assume that the coupling between the vibronic and rotational degrees of freedom is negligible.15 Thus, the molecular Hamiltonian is partitioned as Hmol = Hvib þ Hrot. The transition dipole moment is then given by μ ¼ μ^ μ ð7Þ where μ is the vibronic part and μ̂ is the unit vector defining the orientation of the transition dipole moment of a given chromophore in the laboratory frame. As shown in ref 15, the contribution from each Liouville space pathway to the nonlinear response function can then be factorized into a product of rotational and vibrational terms, ð8Þ R R ðt3 ;t2 ;t1 Þ ¼ YR ðt3 ;t2 ;t1 Þ RR ðt3 ;t2 ;t1 Þ The fourth-rank tensor YR(t3,t2,t1) represents the rotational contribution to the nonlinear response function, which will be treated classically. Since the reorientational motion of molecules in solution can be described as a random walk over small angular orientations, one can generally use a Fokker-Planck equation for the conditional probability density in the rotational phase space (orientation and angular velocity). The initial distribution of angular velocities is given by the Maxwell-Boltzmann expression. By using Hubbard’s treatment16 and following the theoretical description of the rotational nonlinear response function,15,17,18 we can 3,t2,t1) in Z YR(tZ Z the form Z recast Y R ðt3 ;t2 ;t1 Þ ¼

dν3

dν2

dν1

^3 ðν3 Þ X dν0 μ

^2 ðν2 Þ Wðν2 ;t2 jν1 Þ X Wðν3 ;t3 jν2 Þ μ ^0 ðν0 Þ P0 ðν0 Þ ^1 ðν1 Þ Wðν1 ;t1 jν0 Þ μ μ

ð9Þ

where the molecular orientation is specified by the Euler angles ν (φ, θ, χ) and the conditional probability function in angular configurational space is denoted as W(νjþ1,tjþ1|νj). The initial probability distribution of the molecular orientations is denoted as P0(ν0), which is 1/8π2 for an isotropic system containing randomly oriented molecules, i.e., solutions. In the case of the general 2D optical spectroscopy, the involved transition dipoles associated with the Rth Liouville space pathway contribution to the total nonlinear response function could be different from one another, i.e., ^d ; ^3 ¼ μ μ

^2 ¼ μ ^d ; μ

^1 ¼ μ ^b ; μ

^0 ¼ μ â μ

ð10Þ

Hereafter, the [i,j,k,l]th element of the reorientational part of the nonlinear response function tensor is denoted as Ydcba ijkl (t3,t2,t1), where the indices i, j, k, and l represent the Cartesian coordinates. For the Rth Liouville space pathway involving a sequence of dipole transitions in the order given in eq 10, the resultant [i,j,k,l]th tensor element of the corresponding nonlinear response function component is given by Ydcba ijkl (t3,t2,t1)Rdcba(t3,t2,t1). Its vibrational part was extensively studied before, and also its rotational part Ydcba ijkl (t3,t2,t1) was discussed by Golonzka and Tokmakoff.18 II-C. Polarization-Angle-Scanning 2D Spectroscopy. Now, let us consider the 2D photon echo measurement method, where the incident electric field is composed of three pulses temporally separated by τ and T and the echo signal field emitted in the direction kS = -k1 þ k2 þ k3 is measured. The incident field is generally expressed as Eðr;tÞ ¼ e1 E1 ðt þ τ þ TÞ expðik1 3 r - iω1 tÞ þ e2 E2 ðt þ TÞ expðik 2 3 r - iω2 tÞ þ e3 E3 ðtÞ expðik3 3 r - iω3 tÞ þ ðccÞ

ð11Þ

where the unit vectors of linearly polarized beams are denoted as ej for j = 1, 2, 3. The resulting photon echo polarization under the rotating wave approximation is given as R¥ R¥ R¥ PPE ðr;tÞ ¼ eikS 3 r - iωt 0 dt1 0 dt2 0 dt3 ðR 2 þ R 3 Þle3 e2 e1 E3 ðt - t3 Þ E2 ðt þ T - t3 - t2 Þ E1 ðt þ T þ τ - t3 - t2 - t1 Þ expðiωt3 - iωt1 Þ ð12Þ Note that only the rephasing pathways R2 and R3 contribute significantly in the case of the well-separated pulse limit. The photon echo polarization in eq 12 produces the signal electric field, i.e., EPE(t) iPPE(t). This signal depends on three time variables τ, T, and t. The 2D photon echo spectrum is obtained by performing the 2D Fourier-Laplace transformation of EPE(t,T,τ) with respect to τ and t as ~ PE ðωt ;T;ωτ Þ ¼ E

Z

Z

¥

¥

dt 0

dτ EPE ðt;T;τÞ expðiωt t þ iωτ τÞ

0

ð13Þ In the present paper, we assume that the Z-component of the signal field is specifically measured, which can be achieved by using a linear polarizer placed in between the sample and the detector in a real experiment. To reduce the number of experimentally controlled polarization direction angles, we limit our consideration to the case that one of the three incident beam polarization directions, e.g., e1, is parallel to the Z-axis. Thus, the directions of linearly polarized beams are given as ^; e1 ¼ Z

^ cos j2 ; e2 ¼ Y^ sin j2 þ Z ^ cos j3 ; e3 ¼ Y^ sin j3 þ Z

^ ð14Þ es ¼ Z

The 2D photon echo spectrum measured by using this polarization configuration thus becomes a function of j2 and j3 as ~PE ðωτ ;T;ωt ;j2 ;j3 Þ ~PE ¼ E E

ð15Þ

The two frequencies ωτ and ωt correspond to the conjugate frequencies of the first and third delay times τ and t, respectively, 3768



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where the system is on coherences during these time evolution periods.19 Experimentally, one can carry out a series of 2D spectroscopic measurements for varying j2 and j3 at a fixed waiting time T, which have not been achieved before. II-D. Diagonal Peaks in the PAS 2D Spectrum. First of all, let us consider the limiting cases that all the diagonal peaks can be simultaneously vanished by properly choosing the external beam polarization configuration. For an arbitrary diagonal peak, the amplitude of the corresponding reorientational part of the nonlinear response function is determined by the rotationally averaged YZj2j3Z(τ,T,t) in the impulsive limit. Then, using the unit vectors given in eq 14, we find that mmmm YZj ðτ;T;tÞ 3 j2 Z mmmm mmmm ¼ YZZZZ ðτ;T;tÞ cos j2 cos j3 þ YZYYZ ðτ;T;tÞ sin j2 sin j3 þ mmmm mmmm ðτ;T;tÞ sin j2 cos j3 þ YZYZZ ðτ;T;tÞ cos j2 sin j3 YZZYZ

ð16Þ Here, the transition dipole moments involved in this groundstate bleach (GB) and stimulated emission (SE) contributions to the diagonal peak are all identical to μm. The excited-state absorption contribution to the diagonal peak involves the transition dipole between the first excited state and the overtone state. In general, due to the anharmonicity its direction can be different from that of the fundamental transition. However, we shall consider the polarization-angle dependencies of the GB þ SE diagonal peaks and cross-peaks in this work. In eq 16, the third and fourth terms vanish within the electric dipole approximation; note that they can be finite if the electric fieldelectric quadrupole and magnetic field-magnetic dipole interactions are taken into consideration though they are very small.1 Thus, eq 16 is simplified as mmmm mmmm ðτ;T;tÞ ¼ YZZZZ ðτ;T;tÞ cos j2 cos j3 YZj 3 j2 Z mmmm þ YZYYZ ðτ;T;tÞ sin j2 sin j3

ð17Þ

Here, the τ-, T-, and t-dependent ZZZZ and ZYYZ tensor elements of Ymmmm(τ,T,t) were already obtained and presented in ref 18 so that we have mmmm ðτ;T;tÞ ¼ c1 ðτÞ c1 ðtÞ ymmmm YZj Zj3 j2 Z ðTÞ 3 j2 Z

where the waiting time T-dependent part is given as 1 4 1 þ c ymmmm ðTÞ ¼ ðTÞ cos j2 cos j3 2 Zj3 j2 Z 9 5 1 þ c2 ðTÞ sin j2 sin j3 15

ð18Þ

compared to the vibrational relaxation or other relevant chemical or physical processes. Then, one can ignore the c2(T) term in eq 19 and the result is simplified as 1 c1 ðτÞ c1 ðtÞf3 cos j2 cos j3 15 þ sin j2 sin j3 g ð20Þ

mmmm YZj ðτ;T ¼ 0;tÞ ¼ 3 j2 Z

Thus, the diagonal peak amplitude is given as ~ diag ðωτ ¼ ωt ¼ ωm ;j2 ;j3 Þ 1 μm 4 f3 cos j2 cos j3 E 15 þ sin j2 sin j3 gΓðωτ ¼ ωt ¼ ωm Þ ð21Þ Here, μm is the vibrational transition dipole matrix element. The 2D line shape function is denoted as Γ(ωτ,ωt), and its precise functional form is not important here in the present work. Each individual diagonal peak has a different amplitude because the corresponding transition dipole matrix element depends on vibrational modes. However, as can be seen in eq 21, all the diagonal peak amplitudes commonly depend on the same factor that is determined by the two polarization direction angles, j2 and j3. This suggests that all the diagonal peaks vanish every time when we have 3 cos j2 cos j3 þ sin j2 sin j3 ¼ 0 or equivalently j3 ¼ tan-1 ð - 3 cot j2 Þ

ð22Þ

The case that j2 = -j3 = π/3 satisfies the above relationship so that the 2D spectrum S([0,-π/3,π/3,0]) will have no diagonal peaks. II-E. Cross-Peaks in the PAS 2D Spectrum. We next consider the relationship between the beam polarization direction angles and the cross-peak amplitudes. Let us consider the crosspeak at ωτ = ωm and ωt = ωn, where m and n (+m) represent two different transitions. We now assume that the direction of μ̂m is parallel to the z-axis in a molecule-fixed frame i.e., μ̂m = ^z, and that the angle between μ̂m and μ̂n will be denoted as θn,m. Then, it is always possible to write the transition dipole vector μ̂n as ^n ¼ ^z cos θn:m þ ^x sin θn;m μ

ð23Þ

Then, the corresponding cross-peak spectra originating from the GB and SE contributions can be written as ð19Þ

For a spherical rotor, cJ(t) = exp[-J(J þ 1)Dort] and Dor is the orientational diffusion coefficient. The expressions for cJ(t) for J = 1, 2 were obtained by using the small angle diffusion equation, but one can use the more general expressions for these by solving the Fokker-Planck equation. Here, we shall not pursue this here, because detailed derivations of the reorientational relaxation functions are not of main interest in the present work. From mmmm (τ,T,t) eq 18, one can find that the τ and t dependencies of YZj 3j2Z are determined by c1(τ) c1(t). Therefore, the 2D spectrum is additionally broadened by the reorientational relaxations during τ and t periods. In the present work, we focus on the case that the waiting time T is zero or that the reorientational motion is sufficiently slow

GB 2 2 nnmm ~GB E cross ðωτ ;T;ωt ; j2 ;j3 Þ μn μm yZj3 j2 Z ðTÞ Γcross ðωτ ;T;ωt Þ

ð24Þ SE 2 2 nmnm ~SE E cross ðωτ ;T;ωt ; j2 ;j3 Þ μn μm yZj3 j2 Z ðTÞ Γcross ðωτ ;T;ωt Þ

ð25Þ Again, the T-dependent reorientational parts in eqs 24 and 25 depend on the two polarization direction angles and they are found to be 1 4 1 2 nnmm 2 1 þ c2 ðTÞ cos θn:m - sin θn;m yZj3 j2 Z ðTÞ ¼ 9 5 2 1 1 cos j2 cos j3 þ c2 ðTÞ cos2 θn;m - sin2 θn;m 15 2 ð26Þ sin j2 sin j3 3769



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The above result shows that the cross-peak amplitude is determined by not only the polarization direction angles j2 and j3 but also the corresponding transition dipole angle θn,m. From eq 30, one can find that the GB þ SE (m,n) cross-peak at ω1 = ω hm and ω3 = ω hn vanishes when the following equality is satisfied, cos j2 cos j3 ð4 cos2 θn;m þ 2Þ þ sin j2 sin j3 ð3 cos2 θn;m - 1Þ ¼ 0 or equivalently cos2 θn;m ¼

tan j2 tan j3 - 2 4 þ 3 tan j2 tan j3

ð31Þ

Again, there exist infinite possible pairs of j2 and j3 angles that satisfy the above relationship for a given θn,m value. This is an interesting relationship showing that the macroscopically controlled polarization directions of the incidence beams are directly related to the transition dipole angle of a given pair of modes in a randomly oriented molecule in a space-fixed frame. For any θn,m, the cos2 θn,m value is in the range from 0 to 1. Therefore, it is not necessary to scan the entire range of j2 and j3 angles, i.e., 0e

Figure 1. Two-dimensional contour plot of (tan j2 tan j3 - 2)/(4 þ 3 tan j2 tan j3) (see eq 9) with respect to the two polarization direction angles j2 and j3 from the laboratory Z-axis (a). If j1 = jLO = 0 and j2 = π/4, the diagonal peak amplitudes, which are functions of j3, are commonly dependent on (3 3 21/2 cos j3 þ 21/2 sin j3)/2 (see the blue line in (b)). The function (tan j2 - 2)/(4 þ 3 tan j2) on the right-hand side of eq 13 is plotted for varying j3 from 63.4 to 108.4 (see the red line in (b)).

1 1 2 2 2 cos θ þ c2 ðTÞ½4 cos θn;m þ 3 sin θn;m ¼ 9 5 1 1 c2 ðTÞ½4 cos2 θn;m þ 3 sin2 θn;m cos j2 cos j3 þ 12 5 - c1 ðTÞ sin2 θn;m sin j2 sin j3 ð27Þ

ynmnm Zj3 j2 Z ðTÞ

In the limiting case that the reorientation relaxation is sufficiently slow compared to the other dynamic time scales or at T = 0, we found nnmm nmnm yZj (T=0) = yZj (T=0) regardless of the sequence of involved 3j2Z 3j2Z dipolar transitions, either m f m f n f n or m f n f m f n. Since the rotationally averaged transition strengths of the (m,n) crosspeak originating from the GB þ SE contribution are the same, we have GB 2 2 nnmm ~GB E cross ðωτ ;ωt ;j2 ;j3 Þ μn μm yZj3 j2 Z Γcross ðωτ ;ωt Þ

ð28Þ

Due to the periodicity of the tan-1(-3/tan j2*) function, a proper value of tan-1(-3/tan j*) 2 that should be chosen is the one in the range from π/2 to 3π/2. Now, in general, one can carry out a full 2D polarization angle scanning (PAS) to find the specific beam polarization configuration at which the GB þ SE (m, n) cross-peak amplitude vanishes. However, without the loss of any generality, one can fix one of the two angles j2 and j3 and varies the other. To demonstrate such a one-dimensional polarization-angle-scanning 2D spectroscopy, let us consider the particular case that j2 = π/4 and the polarization angle j3 is scanned experimentally. The (m, n) cross-peak amplitude given in eqs 28 and 29 is in this case simplified as ~ cross ðωτ ;ωt ; j2 ;j3 Þ E

ð30Þ

μ2n μ2m pffiffiffi fcos j3 ð4 cos2 θn;m þ 2Þ 30 2

þ sin j3 ð3 cos2 θn;m - 1Þg

ð34Þ

Consequently, if the scanning polarization angle j3 obeys the following equality, cos2 θn;m ¼

where 1 fcos j2 cos j3 ð4 cos2 θn;m þ 2Þ 30 þ sin j2 sin j3 ð3 cos2 θn;m - 1Þg

ð32Þ

In Figure 1a, this j2- and j3-dependent function is plotted, which is symmetric with respect to the interchange between j2 and j3 variables. If the j2 angle is additionally fixed at j*2 in the range from 0 to π/2, the scanning range of j3 (or j2) should be limited as ! ! 2 -3 tan-1 e j3 e tan-1 ð33Þ tan j2 tan j2

2 2 nnmm SE ~SE E cross ðωτ ;ωt ;j2 ;j3 Þ μn μm yZj3 j2 Z ðTÞ Γcross ðωτ ;ωt Þ ð29Þ

ynnmm Zj3 j2 Z ¼

tan j2 tan j3 - 2 e1 4 þ 3 tan j2 tan j3

tan j3 - 2 4 þ 3 tan j3

ð35Þ

the corresponding cross-peak amplitude would vanish. Since the value of cos2 θn,m is again in the range from 0 to 1, the scanning angle j3 should vary from 63.4 to 108.4 or equally from -116.6 to -71.6. Unfortunately, one cannot distinguish whether 3770



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Table 1. Amide I Vibrational Properties and Angles θm,n between Transition Dipole Vectors of Two Amide I Normal Modesa

Figure 2. Molecular structures of isotope-labeled blocked dialanine. The N- and C-termini are blocked by acetylation and N-methyl amidation, respectively. The dihedral angles φ and ψ determine the dipeptide backbone conformation of this model peptide. The amide group on the peptide 1 is isotope-labeled with 13Cd18O so that its amide I vibrational frequency is red-shifted by about 75 cm-1 from that of unlabeled peptide bond. The second (middle) amide group on the peptide 2 is isotope-labeled with 13CdO and its amide I frequency is red-shifted by about 49 cm-1. Due to the isotope-labeling effects, the three amide I local and normal-mode frequencies are separated from one another, and consequently the resultant IR peaks and 2DIR diagonal peaks would appear to be frequency-resolved in the corresponding spectra.

the relative angle between two transition dipole vector θn,m lies in the range from 0 to 90 or in the range from 90 and 180, because the square of cos θn,m not cos θn,m is determined. Nevertheless, the transition dipole angles determined in this way can serve as a good set of constraints used to determine the 3D structure of complex molecules. In Figure 1b, we plot the function (tan j3 - 2)/(4 þ 3 tan j3) for varying j3 from 63.4 to 108.4. One can use this relationship to determine the value of cos2 θn,m from the PAS 2D experiments. In the [0, j3, π/4, 0] PAS 2D measurements, the cross-peak amplitudes are differently dependent on the j3 angle because the transition dipole angles θn,m are not the same for all the (m, n) cross-peaks. However, all the diagonal peak amplitudes in these specific [0, j3, π/4, 0] PAS 2D measurements are commonly dependent on the same factor that is (3 3 21/2 cos j3 þ 21/2 sin j3)/2, which is obtained from eq 21 with j2 = π/4 (see the blue line in Figure 1b). As j3 increases from 63.4 to 108.4, this j3dependent amplitude factor varies from 1.58 to zero. This is an important piece of information in practice because the absolute phases (signs) of diagonal peaks cannot be completely characterized even in heterodyne-detected 2D photon echo measurements. Since the signs of the diagonal peaks for varying j3 angle are predetermined by eq 21, the relative signs of all the measured cross-peaks compared to the signs of diagonal peaks can be absolutely determined without ambiguity.

III. DENSITY FUNCTIONAL THEORY CALCULATIONS OF MODEL DIPEPTIDES In the previous section, we presented a discussion on the general principles of PAS 2D spectroscopy. To show the use of this technique, we apply it to peptide structure determinations. Particularly, a few different conformers of blocked dialanine (AcAla-Ala-NHMe) containing three peptide bonds (see Figure 2) are considered. Among the peptide vibrational degrees of freedom, we shall focus on the amide I vibrations not only because they have been extensively studied by using a variety of linear and nonlinear IR spectroscopic methods but also because their vibrational frequencies and the coupling constants between amide I local modes were found to be highly sensitive to peptide backbone conformations.4,7,20-29 To make the three amide I normal modes frequency-resolved, the first (N-terminal) peptide bond is assumed to be 13Cd18O isotope-labeled and the second (middle) peptide is 13CdO isotope-labeled (see Figure 2).

RR-RR

β-β

PII-PII

PII-β

β-PII

ν~1

1611.8

1602.1

1617.9

1617.9

1601.6

ν~2

1658.1

1621.3

1639.9

1629.0

1633.5

ν~3

1690.5

1680.3

1686.5

1679.2

1688.6

J12

4.8

7.5

3.0

2.1

7.7

J23 J13

5.9 -3.0

7.4 1.0

3.0 -0.4

7.1 0.6

2.9 0.2

θ1,2

118.2

41.9

53.8

42.7

41.0

θ2,3

52.2

123.3

132.8

138.0

123.4

θ1,3

131.5

163.2

93.2

133.8

140.8

j(12) 3

83.5

98.9

90.8

98.5

99.4

j(23) 3

92.1

88.3

95.7

98.9

88.4

j(13) 3

94.8

107.2

63.8

96.4

100.3

The amide I local mode frequencies (in cm-1) of peptides 1, 2, and 3 are denoted as ν ~1, ν ~2, and ν ~3, respectively. Jjk is the vibrational coupling constant (in cm-1) between the jth and kth amide I local modes. θm,n (in degrees) is the angle between the transition dipole vectors of the mth and nth amide I normal (not local) modes. j(mn) (in degrees) is the 3 polarization angle of the third pulse with respect to the laboratory Z-axis. a

Thus, due to these isotope-labeling effects on the amide I frequencies,30-36 the three amide I IR bands are frequencyresolved, regardless of the peptide backbone conformations. A. Model Blocked Dipeptide Conformations. The detailed peptide backbone conformation of the blocked dialanine in Figure 2 is completely specified by four dihedral angles (φ1, ψ1) and (φ2, ψ2). For each pair of Ramachandran dihedral angles, we consider three representative conformations that are right-handed R-helix (RR: φ = -57 and ψ = -47), antiparallel β-strand (β: φ = -139 and ψ = 135), and polyproline II (PII: φ = -78 and ψ = 149) structures. These three conformers will be denoted as RRRR, β-β, and PII-PII. In addition, we considered two more extended structures denoted as PII-β and β-PII, since it was recently shown that these two conformers also populate in water at room temperature.37 To obtain amide I vibrational properties needed to calculate both IR and 2DIR response functions, we carried out quantum chemistry calculations using B3LYP/6-31þG(d) method implemented in the Gaussian03 program suite.38 B. Local Amide I Mode Frequency and Coupling Constants. First of all, for each conformer we performed geometry optimization and vibrational analysis with fixed four dihedral angles, φ1 - ψ2. The latter provides the amide I normal-mode frequencies, eigenvectors, and transition dipole moments. Here, the frequency scaling factor used is 0.9613.39 It is well-known that a given amide I normal mode can be described as a linear combination of amide I local oscillators that are coupled to each other. By using the generalized Hessian matrix reconstruction (GHMR) method,40-42 we could retrieve the corresponding amide I local mode frequencies and coupling constants from the amide I normal-mode frequencies and eigenvectors. The GHMR is based on the assumption that the amide I local modes are mainly delocalized over the six atoms constituting a peptide bond, which are O(=C), C(dO), N(-H), H(-N), CR, and H(-CR). More specifically, the Hessian matrix obtained from quantum chemistry vibrational analyses are divided into submatrices describing amide I local oscillators on each peptide bond. Detailed computational procedure was discussed in ref 42 so we 3771



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To numerically simulate the amide I IR spectra, the transition dipole moments of the three amide I normal modes are needed to be calculated. Since each normal mode is described as a linear combination of the three amide I local modes, its transition dipole vector is also expressed by the same linear combination of those of amide I local modes.7,46 In Figure 3, all the amide I IR spectra of five conformers are plotted, where the line shape of each band was assumed to be the Gaussian function of exp{-(ν ~ν~0)2/2σ2} with σ = 6 cm-1. Except for the PII-β conformer, the amide I IR spectra exhibit three peaks. Furthermore, it is noted that the IR peak intensities strongly depend on the conformation because each normal mode transition dipole moment is determined by finite vibrational couplings between amide I local modes.

IV. NUMERICAL SIMULATIONS OF PAS 2DIR SPECTRA A. Parallel-Polarization 2DIR Spectroscopy and 2D LineShape Function. Once the amide I normal-mode frequencies

and transition dipole moments are obtained, it is now possible to numerically simulate PAS 2DIR spectra of the five conformers considered in the previous section. They will be directly compared to the parallel polarization 2DIR spectrum S([0,0,0,0]). The rephasing 2D photon echo spectrum obtained with the parallel polarization beam configuration can be written as47 1 ð0 T 1Þ T 1Þ ð1 T 2Þ Re½Sdiag þ Sð0 cross þ S 5

Sð½0;0;0;0Þ Figure 3. Numerically simulated IR absorption spectra of conformers RR-RR, β-β, and PII-PII (a) and PII-β and β-PII (b). For the sake of simplicity, the line shape of each band is assumed to be a Gaussian function of exp{-(ν~ - ν~0)2/2σ2} with σ = 6 cm-1.

shall not repeat it here in this paper. The resultant amide I local mode frequencies and coupling constants for the five conformers are summarized in Table 1. It was already found that the coupling constant between two neighboring amide I local modes strongly depends on the dihedral angles,43,44 whereas the side groups of amino acids or isotope labels used here do not make any notable differences in the resultant coupling constants.33,45 Thus, it is not surprising to find that the calculated vibrational coupling constants are quantitatively similar to those obtained from the previous Hessian matrix reconstruction analyses of blocked dipeptides without any isotope labels. For the RR-conformation, the average coupling constant between neighboring amide I local modes is about 5.4 cm-1, whereas those of β and PII conformations are 7.4 and 2.8 cm-1, respectively. The coupling constant J13’s are comparatively small due to the long distance R between the two peptide groups; note that the transition dipole coupling theory suggests that the coupling constant is inversely proportional to R3.21 Unlike the coupling constants, the isotope-labeling effects on the amide I local mode frequencies are very large as expected. First, the frequency (ν ~1) of the amide I local mode on the peptide 1, which is 13Cd18O isotope-labeled, is about 75 cm-1 red-shifted from the peptide 3 amide I local mode frequency (ν ~3). That of the peptide 2 (ν ~2) is red-shifted by about 49 cm-1 due to the 13 C-isotope labeling. Note that these frequency red shifts result from both isotope-labeling and interpeptide interaction effects. Consequently, even with finite couplings among the three amide I local modes the three amide I normal modes are relatively decoupled and localized, and their frequencies are sufficiently different from one another so that the amide I IR spectrum usually exhibits three frequency-resolved peaks.

ð36Þ

where the three contributions are ð0 T 1Þ

Sdiag

¼

∑m jμe g j4ðR~eð2Þe ðωτ ;T;ωt Þ þ R~eð3Þe ðωτ ;T;ωt ÞÞ m

m m

m m

∑ n6∑¼m jμe g j2 jμe g j2ð2 cos2 θm;n þ 1Þ

1 T 1Þ Sð0 ¼ f cross 3 m

m

n

~eð2Þe ðωτ ;T;ωt Þ þ R ~eð3Þe ðωτ ;T;ωt Þg ½R n m n m

∑l ∑m ∑m

1 fðμ μ Þðμ μ Þ 3 fl em 3 fl en en g 3 em g þ ðμfl em 3 μen g Þðμfl en 3 μem g Þ ~fð5Þ þ ðμfl ej 3 μej g Þðμfl ek 3 μek g ÞgR ðωτ ;T;ωt Þ l ek ej

Sð1 T 2Þ ¼ -

ð37Þ

~ e em(ωτ,T,ωt) are the response function components Here, R m contributing to the rephasing photon echo signal, and their detailed expressions can be found in ref 41. The ground state is denoted as |gæ and the one- and two-quantum excited states are as |ejæ and |fjæ, respectively. Since there are three amide I normal modes, we need to consider three one-quantum excited states and six two-quantum excited states. The latter states approximately correspond to the three overtone and three combination states. The angle between the two transition dipole vectors, μemg and μeng, is denoted as θm,n. The three angles θ1,2, θ2,3, and θ1,3 for each conformer were obtained using the DFT calculation methods and are given in Table 1. The parallel polarization 2DIR spectrum results from three groups of contributions. The first term in eq 36, S(0T1) diag , is given by the sum of ground-state bleach (GB) and stimulated emission (SE) contributions that produce diagonal peaks in a given 2D spectrum. Note that the amplitude of the diagonal peak at (ωτ = ωemg, ωt = ωemg) is determined by μemg4. Second, the term S(0T1) cross 3772



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Figure 4. Numerically simulated 2DIR spectra of RR-RR, β-β, and PII-PII conformers. The parallel polarization 2DIR spectra are plotted in the first (23) (13) column of this figure. The remaining spectra correspond to the PAS 2DIR spectra when the polarization direction angle j3 equals j(12) 3 , j3 , or j3 (see Table 1). In the parallel polarization 2DIR spectra, the three cross-peaks in the upper-left diagonal part of the spectrum are pointed by blue arrows, whereas the position of the vanished cross-peak is pointed by a red arrow in each PAS 2DIR spectrum.

is the sum of GB and SE contributions producing cross-peaks. The amplitude of the (m,n) cross-peak at (ωτ = ωemg, ωt = ωeng) is determined by both transition dipole moments and the angle between the two transition dipole vectors, i.e., μemg2μeng2 (2 cos 2θm,n þ 1). Finally, the third term S(1T2) represents the excited-state absorption (EA) contributions. The resultant peaks produced by this term are, however, complicated because their amplitudes are determined by dot products of different transition dipoles. The excited-state absorption contribution has opposite sign compared to the ground-state bleach and stimulated emission contributions. However, due to the intrinsic diagonal and offdiagonal vibrational anharmonicities of the amide I vibrational potential surface, the destructive interference between EA and GB þ SE is not complete. Consequently, the diagonal peak or crosspeak in a given 2DIR spectrum usually appears as a pair of positive and negative peaks. Here, the diagonal and off-diagonal anharmonicities will be assumed to be 16 and 4 cm-1; note that the offdiagonal anharmonicity is in fact dependent on detailed peptide backbone conformation but for the sake of simplicity it is assumed to be 4 cm-1 for all the conformers.48 Although the general expressions for the 2D line shape functions were studied and reported before,1,7 we shall use a highly simplified 2D Gaussian line shape function, i.e., ( ) π ðωτ - ωm Þ2 exp Γðωn ;σ n ;ωm ;σ m Þ ¼ σn σm 2σ2m ( ) ðωt - ωn Þ2 exp 2σ 2n 2

2

ð38Þ

where σm (σn) is the standard deviation of the Gaussian function along the ωτ (ωt) axis. For numerical simulations of the 2DIR spectra, we shall also assume that all the standard deviations determining spectral band widths are 6 cm-1. Another simplification made to the 2D line shape function is that the inhomogeneous distributions of vibrational transition frequencies are ignored and the waiting time is zero. Consequently, the 2D shape of each peak in a given 2D spectrum would appear to be symmetric without the well-known diagonal elongation of each peak at a short time.49,50 The parallel polarization 2DIR spectra of five conformers are plotted on the first column in Figures 4 and 5. Here, the horizontal and vertical lines are drawn at the three amide I normal-mode frequencies for each conformer. In addition to the three diagonal peaks, where each diagonal peak appears as a couplet consisting of positive (red) and negative (blue) peaks, there appear three cross-peaks in the upper and lower diagonal regions. The overall 2D lineshapes and amplitude distribution of cross-peaks depend on the dipeptide conformation. Again, due to the off-diagonal anharmonicities, the corresponding cross-peak originating from the EA contribution is slightly red-shifted along the ωt-axis in comparison to the GB þ SE cross-peak located just above the EA cross-peak. For comparisons with the PAS 2DIR spectra later, the three cross-peak positions in the upper-left region in each parallel polarization 2DIR spectrum are highlighted using blue arrows in these figures. B. Polarization-Angle-Scanning 2DIR Spectroscopy. We next carried out numerical simulations of polarization-anglescanning 2DIR spectra for varying polarization angle j3. From the full j3-dependent 2DIR spectra, we chose three cases for each (23) conformer, which are [0, j3 = j(12) 3 , π/4, 0], [0, j3 = j3 , π/4, 0], 3773



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Figure 5. Numerically simulated 2DIR spectra of PII-β and β-PII conformers. (mn) and [0, j3 = j(13) is the polarization 3 , π/4, 0]. Here, j3 angle of the third pulse, when the cross-peak at (ωτ = ωm, ωt = ωn) vanishes. This means that the angle θm,n between the transition dipole vectors of the mth and nth amide I normal modes obeys the equality in eq 35. Since we have already determined θm,n values in the present cases, the corresponding can be calculated using eq 35 (see Table 1). In real angles j(mn) 3 experiments, one should, however, scan the polarization angle values first, and then they are used to j3 to determine j(mn) 3 extract quantitative information on θm,n values. In Figure 4, the PAS 2DIR spectra of RR-RR, β-β, and PII-PII conformers, when the polarization direction configurations are (23) (13) [0, j3 = j(12) 3 , π/4, 0], [0, j3 = j3 , π/4, 0], and [0, j3 = j3 , π/4, 0], are plotted, in addition to the parallel-polarization 2DIR spectra on the first column. As j3 increases from 63.4, both cross-peak and diagonal peak amplitudes and also cross-peak signs change. However, it should be emphasized that all the three diagonal peak amplitudes change in the same manner because their j3 dependencies are completely governed by the same function of j3, i.e., (3 3 21/2 cos j3 þ 21/2 sin j3)/2. In contrast, the changes to each cross-peak amplitude and sign are differently dependent on j3, because the m-n cross-peak amplitude is a function of both j3 and θm,n. For the RR-RR conformer, when the j3 angle equals 83.5 (=j(12) 3 ), the (1,2) cross-peak originating from the GB þ SE contribution vanishes; note that the (1,2) cross-peak is induced by the coupling between the amide I normal modes 1 and 2. The position of the vanished 1-2 cross-peak is noted by a red arrow in the [1,2] panel in Figure 4. Here, the [m, n] panel in Figures 4 and 5 corresponds to that in the mth row and nth column in the figure. Although the GB þ SE (1,2) cross-peak disappears, still there appears a trace of weak cross-peaks, which in fact originates from the EA contributions involving transitions between |e1æ and |fjæ (for j = 1 - 6) states. It should be emphasized that the EA cross-peak amplitude is a complicated function of j3 as well as of more than one transition dipole angles. Consequently, even at j3= j(12) 3 , the EA (1,2) cross-peak could have a finite amplitude. Thus, if a cross-peak couplet undergoes a change to a slightly blue-shifted singlet along the ωt-axis, it can be an important spectral signature of vanished cross-peak. Again, it is the fact that that is the GB þ SE (1,2) cross-peak vanishes at j3 = j(12) 3

important. For the (2,3) and (1,3) cross-peaks, such spectral signatures can be easily found in the spectra shown in the [1,3] and [1,4] panels in Figure 4. In the cases of the two extended conformers, β-β and PII-PII, the two low frequency amide I diagonal peaks are close to each other so that the (1,2) cross-peak is spectrally congested due to its overlaps with the diagonal peaks. However, the (2,3) and (1,3) cross-peaks are well separated from the diagonal peaks. The corresponding cross-peak vanishes when the scanning angle j3 becomes identical to j(mn) . Furthermore, the sign of each 3 different cross-peak changes for varying j3, which will be discussed later. In Figure 5, the 2DIR spectra of PII-β and β-PII conformers are plotted. It was recently shown that the trialanine in water adopts four different conformers that are PII-PII, PII-β, β-PII, and β-β.37,51-53 To quantitatively determine the relative populations of these four conformers in an aqueous solution, we carried out extensive circular dichroism (CD) and NMR experiments.37 The temperature-dependent CD spectra and the NMR 3J(HN,HR) coupling constants were measured. The former provided information on temperature-dependent conformational change from PII to β-strand. The NMR results allowed us to determine the dihedral angle φ of the chemical bond connecting the CR atom and the amide N atom.54 Since there are two φ angles in a given trialanine, two different sets of experimentally measured 3J(HN, HR) coupling constants were obtained and used to determine the detailed peptide backbone conformation. Developing a modified singular value decomposition analysis method37,55 and using a self-consistent procedure to take in account the CD and NMR experimental data, we could obtain the molar CD spectra of PII and β-strand conformers of the peptide separately; note that it has not been possible to directly measure the PII and β-strand CD spectra because both conformers are present at a thermal equilibrium state. In addition, it was possible to determine the relative populations of these four possible conformers, PII-PII, PII-β, β-PII, and β-β, in an aqueous solution, which were found to be 24, 42, 12, and 22%, respectively.37 Thus, not only the PAS 2DIR spectra of PII-PII and β-β conformers but also those of mixed conformers PII-β and β-PII will be needed in the future to properly interpret experimentally measured PAS 2DIR spectra of such alanine-based peptides in solutions. 3774



Figure 6. Numerically simulated PAS 2DIR spectra of the PII-PII conformer in the frequency region near the (1,3) and (2,3) cross-peaks that are the left and right couplets in this figure. From (a) to (e), the j3 angle varies from 64 to 104 with 10 intervals. Each 2D spectrum is normalized so that the largest positive peak amplitude is set to be one.

In the case of the PII-β conformer, the first and second amide I normal-mode frequencies are very close to each other so that the (1,2) cross-peak is spectrally congested due to its overlap with the two diagonal peaks. However, in the PAS 2DIR = 98.5, the (1,2) cross-peak appearing spectrum with j3 = j(12) 3 as a bulged part in the parallel polarization 2DIR spectrum disappears. Interestingly, in this specific case of the PII-β value (=98.9) is very close to j(12) conformer, the j(23) 3 3 , even though the transition dipole angles θ1,2 and θ2,3 are significantly different from each other. Consequently, the (2,3) cross-peak is already very weak in the PAS 2DIR spectrum with [0, j3 = j(12) 3 , (=96.4), π/4, 0] geometry. As the j3 angle approaches j(13) 3 the (1,3) cross-peak clearly vanishes. In the case of the β-PII conformer, the three diagonal peaks are well-resolved in the 2D frequency space. Also, the three PAS 2DIR spectra show that the corresponding cross-peak vanishes for each case, as expected.

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Although the overall spectral changes to the PAS 2DIR spectrum for varying j3 angle were discussed on the basis of numerically simulated spectra in Figures 4 and 5. We next focus on a narrow spectral region around some cross-peaks and examine how the cross-peak amplitudes and signs change upon the polarization angle scanning. In Figure 6, a series of PAS 2DIR spectra of the PII-PII conformer, where the (1,3) and (2,3) cross-peak region (see the small dashed box in the [3,1] panel of Figure 4) is chosen for detailed investigation for instance, are plotted for varying j3 angle from 64 to 104 for every 10. Here, it should be noted that the θ1,3 and θ2,3 angles are 93.2 and 132.8 degrees (see Table 1). Therefore, from eq 35, we have that = 63.9 and j(23) = 95.7. Therefore, it is expected that the j(13) 3 3 (1,3) and (2,3) cross-peaks vanish at j3 = 63.9 and j3 = 95.7, respectively. Indeed, Figure 6a shows just a single couplet representing the (2,3) cross-peak without the (1,3) cross-peak. As j3 increases, the (1,3) cross-peak amplitude increases. Furthermore, the (1,3) cross-peaks in Figure 6b-e exhibit a negative-positive couplet, where the upper peak in the couplet is negative and the lower peak is positive. Now, let us examine the (2,3) cross-peaks in this series of PAS 2DIR spectra. As the j3 angle increases and reaches to 94 (see Figure 6d), the relative (2,3) cross-peak amplitude is significantly weaker than those in the other spectra. Furthermore, the (2,3) cross-peak couplet in Figure 6a is normal, meaning that the upper peak in this 2-3 cross-peak couplet is positive and the lower peak is negative. However, as j3 increases and becomes larger than 94 (=j(23) 3 ), its sign changes. This is an important observation angle in and will be of help to accurately determine the j(23) 3 practice. Experimentally, one can plot the cross-peak amplitude as a function of j3. Then, using an interpolated curve connecting the experimentally measured data on the target cross-peak value with high amplitude, it will be possible to estimate j(23) 3 precision. Here, we theoretically described the PAS 2D spectroscopy. Experimentally, for varying j3 angle, one can record a series of 2DIR spectra and examine the changes to the 2DIR cross-peak shapes and amplitudes with respect to the scanning j3 angle to values, which in turn provide information on find a clue on j(mn) 3 θm,n angles. Perhaps, a practically useful method for estimating θm,n angles is to directly compare theoretically simulated 2DIR spectra with experimental results. For an initially guessed conformation, one could simulate the j3-dependent PAS 2DIR spectra. If the simulation results agree with the experiments, the initial guess for the conformation is accepted to be valid. However, if not, different conformations should be chosen for subsequent comparative investigations. Then, iterations of this procedure particularly focusing on the amplitudes and signs of the diagonal peaks and cross-peaks with respect to j3 would provide intricate details on the solution structure. In the case that the solution conformation has a distribution, it would be necessary to make significant simulation efforts to ultimately achieve the goal. Nevertheless, only when the theoretically predicted conformation distribution is correct, the series of simulated PAS 2DIR spectra would bear resemblance to experimentally measured spectra.

V. SUMMARY In the present paper, a theoretical description on the PAS 2D spectroscopy that can be of use to obtain detailed information on directions of transition dipole vectors and eventually on 3775


The Journal of Physical Chemistry A molecular conformation was presented. Among a number of experimentally controllable variables, the polarization directions of incident beams were shown to be important factors determining the diagonal peak and cross-peak amplitudes. A properly chosen set of polarization direction angles can make the entire diagonal peak amplitude vanish, which would eventually enhance frequency-resolution of the cross-peaks in a given 2D spectrum. One of the most interesting findings here is that one can selectively suppress a specific cross-peak at a time. This is because each cross-peak amplitude is determined by both incident beam polarization directions and the angle between the two transitions associated with the cross-peak. From the general expression for the GB þ SE cross-peak amplitude with respect to the polarization direction angles and transition dipole angles, it was shown that the angle between two transition dipole vectors can be determined by carrying out the PAS 2D spectroscopy experiments. It should be mentioned that the PAS 2D spectrum is given as a linear combination of the ZZZZ and ZYYZ 2D spectra, where the weighting factors are determined by the polarization direction angles. This is theoretically correct in an ideal case, but in practice it is not easy to measure the absolute intensities of these two 2D spectra due to laser intensity fluctuations. Consequently, the interpolation with only two 2D spectra to obtain any arbitrary PAS 2D spectrum should be of limited success. Despite the fact that the PAS 2D spectra also have the same limitation, from the series of multiple PAS 2D spectra one can obtain quantitatively reliable results. To illustrate the basic principle of the PAS 2D spectroscopy, we considered isotope-labeled blocked dialanine conformers and calculated vibrational frequencies, eigenvectors, and transition dipoles of the three amide I normal modes. Then, the amide I PAS 2DIR spectra simulated with the parameters determined by the density function theory methods were presented and directly compared with the parallel-polarization 2DIR spectra. It was shown that the spectral signatures of vanished cross-peaks in a given PAS 2DIR spectrum are manifest when the polarization direction angles are properly chosen. Currently, we are carrying out such PAS 2D vibrational spectroscopic investigations for a few molecular systems and will present the experimental results elsewhere. We anticipate that the present PAS 2D vibrational or electronic spectroscopy will be of critical use in not just determining molecular structure but also studying conformational changes to proteins and other related biomolecules in condensed phases in real time.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work was supported by NRF grant (20090078897) and KBSI grant (T30401) to M.C. Also, we thank the support by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (NRF20100020209). Finally, we are grateful to an anonymous reviewer who pointed out the reorientational relaxation contributions to the PAS 2D spectroscopy.

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Polarization-Angle-Scanning Two-Dimensional Spectroscopy

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