Alignment, Vibronic Level Splitting, and Coherent Coupling Effects on

ARTICLE pubs.acs.org/JPCA

Alignment, Vibronic Level Splitting, and Coherent Coupling Effects on the Pump
Probe Polarization Anisotropy Eric R. Smith† and David M. Jonas* Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309-0215, United States ABSTRACT: The pump
probe polarization anisotropy is computed for molecules with a nondegenerate ground state, two degenerate or nearly degenerate excited states with perpendicular transition dipoles, and no resonant excited-state absorption. Including finite pulse effects, the initial polarization anisotropy at zero pump
probe delay is predicted to be r(0) = 3/10 with coherent excitation. During pulse overlap, it is shown that the four-wave mixing classification of signal pathways as ground or excited state is not useful for pump
probe signals. Therefore, a reclassification useful for pump
probe experiments is proposed, and the coherent anisotropy is discussed in terms of a more general transition dipole and molecular axis alignment instead of experiment-dependent ground- versus excited-state pathways. Although coherent excitation enhances alignment of the transition dipole, the molecular axes are less aligned than for a single dipole transition, lowering the initial anisotropy. As the splitting between excited states increases beyond the laser bandwidth and absorption line width, the initial anisotropy increases from 3/10 to 4/10. Asymmetric vibrational coordinates that lift the degeneracy control the electronic energy gap and off-diagonal coupling between electronic states. These vibrations dephase coherence and equilibrate the populations of the (nearly) degenerate states, causing the anisotropy to decay (possibly with oscillations) to 1/10. Small amounts of asymmetric inhomogeneity (2 cm
1) cause rapid (130 fs) suppression of both vibrational and electronic anisotropy beats on the excited state, but not vibrational beats on the ground electronic state. Recent measurements of conical intersection dynamics in a silicon napthalocyanine revealed anisotropic quantum beats that had to be assigned to asymmetric vibrations on the ground electronic state only [Farrow, D. A.; et al. J. Chem. Phys. 2008, 128, 144510]. Small environmental asymmetries likely explain the observed absence of excited-state asymmetric vibrations in those experiments.

I. INTRODUCTION Integrated pump
probe and fluorescence polarization anisotropy measurements are a demonstrated method of observing ultrafast chemical behavior, such as molecular rotation,1 the dephasing of electronic levels,2 energy transfer,3 electronic relaxation,4 charge transfer,5 exciton delocalization,6 and nonadiabatic electronic dynamics.7 In these experiments, a linearly polarized pump pulse preferentially excites molecules with transition dipole moments aligned parallel to the optical electric field vector of the pump pulse (“photoselection”).8 If the pump pulse is faster than molecular rotation, the excited molecules are aligned and there is an aligned “hole” in the ground-state angular probability distribution. As a result of the alignment, the sample’s interaction with a subsequent linearly polarized probe may be anisotropic, dependent on the angle between the pump and probe optical electric fields. In studies of molecular rotation using nondegenerate transition dipoles and identical pump and probe pulses, the canonical ratio of pump
probe signals with parallel and perpendicularly polarized pulses is 3:1 before molecular rotation, reflecting the dipolar cos2(θ) excitation probability.1 Much related theoretical work has focused on dimers and systems with degenerate states.9
13 In these systems, the alignment is no longer achieved simply by angularly selective excitation, but can be created by coherent excitation of superposition states. The coherent dipole alignment can be greater than for a r 2011 American Chemical Society

weak excitation of a single, nondegenerate, transition dipole.9
13 Although an early experiment on a doubly degenerate molecular transition seemed to support the prediction of an enhanced parallel/perpendicular ratio in pump
probe signals without spectrally selective detection,2 other coherent experiments on chromophore aggregates observed only small enhancements,14 observed enhancements only before time zero15 (when small timing errors can cause large errors in the ratio), or observed a 3:1 ratio.3,16
18 The situation was clarified when Ferro and Jonas observed a 3:1 ratio on a doubly degenerate molecular transition and showed that, unlike the single dipole case, the coherent emission and pump
probe polarization ratios differ, and that the transition dipole alignment gives the stimulated emission ratio, not the pump
probe ratio.19 Further, when the coupling between chromophores is weak enough that all single excitations overlap the pulse spectrum, the usual 3:1 ratio is recovered in pump
probe.13,19 Departures from an initial 3:1 ratio have been explained by invoking additional excited-state absorption.5,18
20 The experimental results for the coherent fluorescence (spontaneous Special Issue: Graham R. Fleming Festschrift Received: February 28, 2011 Revised: March 4, 2011 Published: March 18, 2011 4101

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)

)

)

)

emission) anisotropy are often less clear because of the slower instrument response.21 More recently, the question of whether “improperly ordered” (coherent coupling)22 interactions with the pump and probe can significantly alter the anisotropy during pulse overlap has been raised.23 Models incorporating impulsive excitation have provided zero-order explanations of the coherent anisotropy recorded with nonoverlapping finite duration pulses;12,13,24 this work examines the coherent contributions to the anisotropy when the pump and the probe pulses overlap in time and there is no excited-state absorption. To construct the anisotropy, one records the pump
probe signals as a function of the pump
probe delay T using linearly polarized pump and probe fields aligned parallel [S (T)] and perpendicular [S^(T)] to each other. Macroscopically, the anisotropy is given by r(T) = [S (T)
S^(T)]/[S (T) þ 2S^(T)]. The isotropic dynamics can be reconstructed via SMA(T) = [S (T) þ 2S^(T)]/3 or measured directly by setting the angle between the pump and probe polarizations at the “magic angle” of arccos(1/(3)1/2) ≈ 54.7. On a molecular level, the anisotropy r = [3Æcos2θæ
1]/2 measures the “degree of alignment”, Æcos2θæ, where θ is the angle between the probed transition dipole moment in the molecular frame and the pump polarization axis in the laboratory frame and the brackets denote an average over the angular probability distribution. It is common practice25 to separate pump
probe signals into three contributions: (1) a probe absorption decrease due to pump pulse depletion of the ground-state population [increased probe transmission from ground-state bleaching (GSB)], (2) probe gain due to stimulated emission from the excited state [increased probe transmission from excited-state emission (ESE)], and (3) absorption from the excited state to higher optically accessible levels [decreased probe transmission from excited-state absorption (ESA)]. Since the work of Ferro and Jonas,19 discussions of the pump
probe anisotropy have focused on the anisotropy of these three contributions to the signal, which are different.7,12,13,26
28 Such an approach is useful because the relative signs of the three signal contributions (and hence the total anisotropy) depend on the pump
probe detection method (e.g., transient absorption vs fluorescence or ionization detection).29 It will be shown that the standard GSB/ESE classification becomes problematic during pulse overlap, necessitating at least a reclassification specific to pump
probe experiments. To discuss the anisotropy without such GSB/ESE classifications, we use transition dipole and molecular axis alignment. We point out here that, although the transition dipole is more highly aligned with coherent excitation, the molecules are less aligned. For weakly coupled excitations, the effects of more aligned transition dipoles and less aligned molecules initially cancel when all three contributions to the signal are added, as found previously for the sum of GSB, ESE, and ESA signals.13,19 However, when a degenerate state of a molecule (e.g., E, Π, or T symmetry) can be resonantly excited from a nondegenerate state without resonant excited-state absorption, the parallel/perpendicular signal ratio when the probe just follows the pump is predicted to be symmetrydependent (16:7 for E and Π states, 2:1 for T states).13 In perturbation theory, pump
probe signals arise from three field-matter interactions (two with the pump, one with the probe), radiation from the oscillating polarization, and interference of the radiated field with the probe. For overlapping pulses, “improperly ordered” coherence pathways, terms in the nonlinear optical response function in which the first or second

Figure 1. Parameters of the three electronic state model system. The system consists of a ground state |gæ with potential energy V(q)/p = (1/ 2)ωq2 and two excited states, |Ræ with V(q)/p = ωeg
δ/2 þ (1/2)ω (q
d)2 and |βæ with V(q)/p = ωeg þ δ/2 þ (1/2)ω(q þ d)2. The excited-state stabilization energy is (Dω) = (1/2)ωd2. In the limit that the asymmetric vibrational coordinate q is overdamped, the absorption spectrum is a sum of two Gaussians centered at ωeg þ (Dω) ( δ/2, each with variance Δ2 = (Dω)ω coth(βpω/2), where β is the inverse temperature. For δ g Δ, preferential excitation of one excited state is possible. The transition dipoles from |gæ to |Ræ and |gæ to |βæ are polarized along the x and y axes, respectively, in the molecular frame and are equal in magnitude. In contrast to the figure, the parameters used in this paper for asymmetric vibrations all involve vibrational displacements d < 1, hence stabilization energies less than zero point energies, (Dω) < ω/2.

perturbation theoretic interaction between the sample and the field may arise from the probe, contribute to the generation of the signal by “coherent coupling” of the pump and probe.22,30 These “improperly ordered” pathways are responsible for the so-called “coherence spike” at time zero,31
36 which can alter the parallel/ perpendicular signal ratio for an electronic two-level system,33 but vanishes when the ratio is 3:133 or when the pulses become much shorter than the dephasing time for a two-level system.35 If the probe pulse has a different laboratory-fixed electric field polarization than the pump, the orientational average of the projection of the field onto the molecular dipole can be different for the improperly ordered pathways. Because this may change the anisotropy in the pulse overlap region, an understanding of these effects is crucial to interpreting changes in the anisotropy observed as the temporal pulse overlap changes. Numerical calculations of the signal for pulses with finite durations will be presented as a function of pump
probe delay for model systems with the ground state and two excited-state potential energy surfaces shown in Figure 1. The model reduces to that in Figure 4 of ref 12 in the absence of a splitting between the two excited states (δ = 0). This limit describes the doubly degenerate excited states of D4h symmetry metalloporphyrins, which are Jahn
Teller distorted, in the fashion shown in Figure 1, by asymmetric vibrations.37 Finally, we discuss the effect that an excited-state electronic splitting (δ in Figure 1) has on the ground-state bleach and excited-state emission anisotropy. This is relevant to the pump
probe anisotropy of strongly coupled heterodimers and free-base porphyrins (D2h molecules closely related to metalloporphyrins in which the first singlet excited states are not energetically degenerate by symmetry). Numerical calculations on model systems illustrate the effect of laser detuning on the ground-state bleach and excited-state emission anisotropies for an electronic system with a splitting.

II. THEORY Without the pump, probe absorption may be regarded as caused by destructive interference between the probe and the 4102

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The Journal of Physical Chemistry A radiation from the oscillating electric polarization that it excites (the linear optical free induction decay).29,38 A pump pulse can modify the polarization excited by the probe (most obviously, by changing level populations before the probe arrives). More generally, the oscillating macroscopic electric polarization excited by the pump and probe fields radiates a field that interferes with the probe field in a pump
probe experiment.25 For weak pump and probe pulses, this radiated field can be calculated from the third-order nonlinear polarization. The third-order polarization derives from the third-order nonlinear optical response, which is calculated using density matrix perturbation theory.39 The double-sided Feynman diagrams22,39
41 that represent individual perturbation theory terms that add together to form the third-order response functions for a four-level system are presented in a review of 2D spectroscopy.29 The conventions used for labeling individual terms in the nonlinear response and electronic levels used in that review will be applied throughout this work. To clarify the states involved, the wave-mixing energy level diagrams of Lee and Albrecht, which employ dashed and solid arrows to indicate bra and ket index changes, are used.41 Briefly, a coherence pathway represented by a Feynman diagram in Figure 2 contains five factors multiplied together: a sign representing its contribution to the radiated signal (absorption/ emission factor determined by the number of dashed vs solid arrows),29,41 an orientational factor based on the average of the projection of the laboratory-frame electric fields onto the transition dipoles in the isotropically oriented molecular frame, an oscillatory term from the evolution of coherence during each interaction period, and electronic dephasing functions for symmetric and asymmetric vibrations present in the system. The first three factors can be derived directly from the diagrams in Figure 2 using the optical transition frequencies and the transition dipoles between each pair of levels. The discussion here will be concerned with transitions between the symmetric electronic ground state |gæ and levels of the quasi-degenerate excited states |Ræ (designated an x-polarized transition) and |βæ (y-polarized) with one excitation shown in Figure 1. Here, we treat the predicted strengths of each contribution to the signal individually. To begin, we neglect relaxation processes, such as vibrational dephasing, electronic dephasing, or population transfer. This is a useful approximation at very short delays for short pulses. There are six possible orders in which the fields can interact,29 but only the middle four shown across the top of Figure 2 occur without excited-state absorption. Only diagrams for the nonlinear polarization with positive wavevector kc þ kb
ka and a positive radiated frequency are shown (changing the sign of all wavevectors and frequencies gives the complex conjugate contribution to the real time-domain polarization). For pump
probe signals, the field interactions labeled a and b both derive from the pump pulse, the interaction labeled c derives from the probe, and the signal field is detected through interference only when it overlaps the probe pulse in space and time (thus, the nonlinear polarization with wavevector kc þ kb
ka also contributes to the signal with complex conjugate roles for pulses a and b). This field description is more general than the description in terms of pump-induced changes in probe photon absorption and is absolutely necessary for describing the pulse overlap region,22,25 where finite pulse durations not only blur the intervals between pulses but also their ordering. The total pump
probe signal can be constructed by adding together the contributions from each diagram individually.42 In discussing the diagrams, it is often convenient to use the positive

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Figure 2. Sixteen density matrix perturbation theoretic diagrams that describe the optical response for a pump
probe experiment. Pulse labels for four-wave mixing are used for generality. In pump
probe, fields a and b are both the pump field, and field c is the probe field. The time intervals across the top define the order of interaction (arrows) with the fields a, b, and c, with the sample, each referenced to the time of signal radiation (wavy line) (t). Time increases from left to right. Interactions with pulses a, b, and c occur at t
τa, t
τb, and t
τc, respectively. Thus, the eight diagrams on the right (D1
D8) contribute when the probe strictly follows the pump (positive pump
probe delay T), five diagrams on the left (D9, D12, D14
D16) contribute to the spectrally resolved pump
probe signal when the probe strictly precedes the pump (negative T), and all 16 contribute when the pump and probe overlap in time (near T = 0). The solid and dashed arrows correspond to field-induced changes in the density matrix ket and bra indices, respectively. The final wavy line for signal radiation represents both the upward and the downward arrows (one dashed and one solid) that make the bra and ket indices the same. The numbers 0, 1, and 2 at the left of the leftmost diagrams indicate the number of excitations in states of the same approximate energy. The boxes classify the diagrams as ground-state bleach (GSB), excited-state emission (ESE), excited-state absorption (ESA), or double quantum coherence (DQC) according to the density matrix element change probed after the first two field interactions (t2 dependence).25 DQC diagrams are further divided according to their absorptive/emissive sign. Diagrams and labeling indicate a proposed reclassification of the improperly time-ordered diagrams, specific to pump
probe experiments, with red for GSB, black for ESE, green for ESA, purple for negative DQC, and blue for positive DQC.

time intervals between the first and second interactions (t1), second and third interactions (t2), and third interaction and radiation (t3).39 In the limit of δ function pulses, the probe always follows the pump for T > 0, and (assuming no ESA) diagrams D1
D4 are sufficient to describe the signal with t1 = t3 = 0 and t2 = T. For positive pump probe delays with finite pulses of duration tp, 0 e t1 j tp, T
tp j t2 j T þ tp, and 0 e t3 j tp. Conventionally, each diagram is described as either ground-state bleach (GSB) or excited-state emission (ESE) based on the density matrix element probed during t2,22,25,29 as labeled in Figure 2. Both increase probe transmission through the sample and thus have an absorption/emission factor with a positive sign.

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Table 1. Field Polarization Sequences, Transition Dipole Sequences, and Orientational Factors, Conventionally Classified as ESE or GSB According to Their t2 Dependence25 (Columns) and with the Proposed Reclassification Indicated by Roman vs Boldface Type GSB

ESE Dn ^

a

D9

R 2d3 ðτc , τb
τc , τa
τb Þ ¼

i ! ! ! ! Æð μ 0R • ε d Þð μ β0 • ε c Þ p3 ð! μ R0 •! ε b Þð! μ 0β •! ε a Þæorientational

ÆG R0 ðτc ÞG Rβ ðτb
τc ÞG 0β ðτa
τb ÞF00 ævibrational

ð1Þ

The brackets indicate averages. Note that time ordering in the response conventionally runs from right to left (opposite the diagrams). Gjk(τ) is the Green function that describes the time evolution of density matrix element Fjk during the interval between field interations τ. F00 is the diagonal matrix element for the starting vibrational level on the ground electronic state. In this section, we are considering time delays before significant dynamics has taken place; therefore, the average Green function product will remain unity. The contribution of a subdiagram to the optical response for small τc, τb, and τa is thus proportional to the orientationally averaged term. Factoring out the scalar amplitudes of the fields and dipoles in eq 1, the first bracketed quantity becomes the orientational average of the dot products between laboratory frame fields and molecular frame dipoles. Table 1 summarizes the factors that determine the orientational averages for each subdiagram. As in the response function, time runs from right to left. The field polarization sequences are indicated by uppercase letters: the pump interactions always have laboratory frame polarization εBa = εBb = Z; for parallel polarization, the probe interaction and detected nonlinear field have εBc = εBd = Z; and for perpendicular polarization, εBc = εBd = Y. The order of field interactions depends only on the column in which the diagram appears in Figure 2. For parallel polarization, the sequence is always ZZZZ. With perpendicular polarization, the sequence YYZZ for properly ordered diagrams becomes YZYZ when a probe interaction is interlaced between two pump

yxyx 2/15

D1

D12

D11

YYZZ

YZZY

YZYZ

D3

D4

YYZZ YYZZ

yxyx yxxy xxyy xxyy xxyy
1/30
1/30
1/30
1/30 2/15

xxyy 2/15

)

)

)

)

a The parallel field polarization sequence is always ZZZZ. b The transition dipole sequences for nd1 are all xxxx, those for nd2 are all yyyy, and those for nd4 are all obtained from those for nd3 by interchanging all x and y. c The remaining orientational factors are Ond1 = Ond2 = 1/5, nd2 1 Ond3 = Ond4 = 1/15, and Ond ^ = O^ = 1/15.

interactions (D10 and D11) and YZZY when the probe interacts first (D9 and D12). The transition dipole directions from the totally symmetric ground state to the |βæ and |Ræ excited levels are μBβ0 = μB0β = y and μBR0 = μB0R = x (we have suppressed vector notation in this discussion for convenience). The dipole sequences are indicated in lowercase (for example, xyxy for 2d3 and 10d3). The molecular frame transition dipole pattern depends on both the diagram subscript n and the subdiagram subscript i. The orientational averages are labeled by a superscript for the subdiagram and a subscript for the polarization; for example, 3 O2d ^ = Æ(Y 3 x)(Y 3 y)(Z 3 x)(Z 3 y)æ for subdiagram 2d3 with perpendicular polarization. Elements such as (Z 3 x) are direction cosines. The temporal order of the dot products does not affect the averages, which may be found in ref 13. Even though the order of molecular transition dipoles is the same, the orientational averages for 10d3 and 2d3 differ from each other in perpendicular polarization because the field interaction order is 2d4 10d3 10d4 3 different (O2d YYZZ = OYYZZ =
1/30 vs OYZYZ = OYZYZ = 2/15). The orientational factor for the polarized pump
probe response of the subdiagram set is ODP n ¼

∑ OndP i

i

ð2Þ

where n is the diagram subscript on Dn, i = 1
4 is the subdiagram subscript, and P specifies the field polarization sequence (see n Table 1). OD P is proportional to the T = 0 signal for δ function pulses. For all eight diagrams, ODn = 4/15. Using the conventional classification based on t2 dependence, the perpendicular n orientational factor OD ^ is different for GSB versus ESE and for properly ordered versus improperly ordered diagrams: for ESE, D2 1 the properly ordered diagrams have OD ^ = O^ = 1/30 and the D10 9 = O improperly ordered diagrams have OD ^ ^ = 3/15; for GSB, 3 4 = OD the properly ordered diagrams have OD ^ ^ = 3/15 and the D11 D12 improperly ordered diagrams have O^ = O^ = 1/30. The consequences of this reversal for the anisotropy and the isotropic strength of ESE versus GSB in the pulse overlap region suggest a proposed reclassification that is indicated in both Table 1 and Figure 2. This proposed reclassification of the improperly ordered diagrams will be examined in section IV. All of the discussion here in this section, in section III, and Figures 4, 5, and 7 is based on the conventional t2 dependent classification. The relative contribution from each diagram to the pump
probe signal depends on the pump
probe delay. For pump
probe delays very much greater than the pulse duration (T . tp), the )

To calculate the coherent anisotropy, each diagram must be expanded into four subdiagrams.13 The conventions we follow for dividing the diagrams into subdiagrams are summarized in Table 2 of Smith et al.12 The pattern of molecular-frame transition dipoles, and hence the orientational average for a particular subdiagram, is dictated by the order of interactions with the fields a, b, and c. Diagrams D2 (properly ordered) and D10 (improperly ordered) are expanded into their subdiagrams in Figure 3. Using the rules in refs 43 and 29, the response from subdiagram 2d3 is

D2

YZZY YZYZ YYZZ

nd3b yxxy 3c 2/15 Ond ^

Figure 3. Expansion of diagrams D2 and D10 into subdiagrams. Time conventionally runs from left to right in diagrams. The molecular transition dipole pattern and sequence of laboratory electric field polarizations that give the orientational average are indicated by the subdiagrams. The molecular transition dipole patterns are the same in each column while the sequence of laboratory electric fields is the same in each row. As a result, the D2 and D10 subdiagrams have different orientational averages.

D10

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)

r ESE

SESE
SESE 7 ^ ¼ ¼ ESE 10 3SMA

ðT >> tp Þ

ð3Þ

ðT >> tp Þ

ð4Þ

the ground-state bleach anisotropy SGSB
SGSB 1 ^ ¼ GSB 10 3SMA )

r GSB ¼

superposition state that is maximally aligned to the pump polarization in the x
y plane, with an anisotropy of r = 7/10 before any loss of alignment. The total pump
probe polarization anisotropy is given by adding the GSB and ESE contributions to each polarized signal, yielding r TOT ¼ 3=10

It will be seen below that this result is a general one for impulsive excitation. At zero pump
probe delay, all eight diagrams contribute with D1 D2 D9 D10 equal weights, so that SESE P = β[OP þ OP þ OP þ OP ], where GSB β is a proportionality constant, and similarly for SP . At time zero, the excited-state emission anisotropy is r ESE ð0Þ ¼

and the ratio of isotropic strengths ESE SGSB MA ¼ 2SMA

ðT . tp Þ

ð5Þ

that hold before excited-state evolution and relaxation dynamics have set in. This factor of 2 greater strength for GSB over ESE arises because population of one of the doubly degenerate excited states must depopulate the ground state, thus bleaching the absorption transitions from the ground state to both degenerate excited states. This bleach of both x- and y-polarized transitions is delocalized over the x
y plane, which explains the anisotropy of r = 1/10. In contrast, the excited state emits from a coherent

ð6Þ

SESE ð0Þ
SESE 3 ^ ð0Þ ¼ 3SESE 10 ð0Þ MA )

probe pulse follows and does not overlap the pump, and only the properly ordered diagrams contribute. For each process (ESE vs GSB), the weighting of the two diagrams (interaction with a vs b D1 D2 GSB 3 = R[OD first) is equal, so that SESE P = R[OP þ OP ] and SP P þ D4 OP ] where R is some proportionality constant. This leads to the excited-state emission anisotropy

Figure 5. Calculated ground-state bleach anisotropy (top), excitedstate emission anisotropy (middle), and total anisotropy (bottom) for two model three-level systems. The inset at the top shows the absorption (dashed line) and emission (dotted line) line shapes of the model compared to the laser spectrum of a 30 fs pulse centered at ωeg. The model system has a δ/2πc = 100 cm
1 splitting between orthogonally polarized transitions to the excited levels hidden under line broadening by a slow totally symmetric mode (see Figure 4, caption). Solid lines show a calculation with no asymmetric dephasing of the electronic levels and dotted lines include an asymmetric critically damped mode with
1 ωasymm and stabilization energy (Dω)asymm CDO /2πc = 50 cm CDO /2πc = 5 cm
1 at a temperature of 300 K (conventional t2-dependent distinction between GSB and ESE).

the ground-state bleach anisotropy is r GSB ð0Þ ¼

SGSB ð0Þ
SGSB 3 ^ ð0Þ ¼ GSB 10 3SMA ð0Þ )

Figure 4. Calculated pump
probe signal and anisotropy in the pulse overlap region for a three-level system with no anisotropic dynamics (d = δ = 0 in Figure 1). Line broadening is provided by a totally symmetric overdamped Brownian oscillator with frequency ωODO/(2πc) = 5 cm
1, damping γODO/(2πc) = 276 cm
1, and reorganization energy λODO/(2πc) = 55 cm
1 at a temperature of 300 K. The 30 fs (field amplitude full width at half-maximum) duration pump and probe pulses are tuned to ωeg/(2πc) = 12685 cm
1. (Top) The solid black line is the total anisotropy, including contributions from ground-state bleach (dotted) and excited-state emission (dashed). The pulse envelope is shown for comparison. (Bottom) The solid line shows the ratio of calculated isotropic strengths of the ground-state bleach and excitedstate emission. The long-dashed line shows evolution of total isotropic TOT strength, STOT MA (T)/SMA (0) (conventional t2-based distinction between GSB and ESE).

and the ratio of isotropic strengths is ESE SGSB MA ð0Þ ¼ SMA ð0Þ

The total anisotropy is again rTOT = 3/10 for fully overlapping pulses at time zero. 4105

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Figure 6. Comparison of electronic anisotropy quantum beat decay for different amplitudes of asymmetric stabilization energy. All use the same model as in Figure 5, but the amount of stabilization energy in the
1 critically damped mode is changed. Dashed line: (Dω)asymm CDO = 2 cm . asymm asymm
1
1 Dotted line: (Dω)CDO = 5 cm . Solid line: (Dω)CDO = 10 cm .

Well before time zero, at negative delay, the improperly time ordered diagrams dominate. It is not a priori clear how the two improper pulse orderings are weighted relative to each other, but their orientational factors are the same, so it does not matter. For simplicity, we assume the pulse interaction order cab dominates, GSB 9 12 = χOD = χOD so that SESE P P and SP P , where χ is some proportionality constant. This leads to the excited-state emission anisotropy SESE
SESE 1 ^ ¼ ESE 10 3SMA )

r ESE ¼

ðT , 0Þ

the ground-state bleach anisotropy S

)

r GSB ¼


SGSB ^ 3SGSB MA

GSB

¼

7 10

ðT , 0Þ

and the ratio of isotropic strengths ESE 2SGSB MA ¼ SMA

ðT , 0Þ

for large negative delays. Although the pump
probe signal disappears when the probe fully precedes the pump, the above reversal of GSB versus ESE anisotropies and isotropic strengths relative to positive pump
probe delays underlies their equality at time zero. The total anisotropy is still 3/10. As the pump
probe delay increases from negative to positive, the excited-state population continues to grow until the pulses cease to overlap, as do the isotropic strength of both ground-state bleaching and excited-state emission. The ESE versus GSB isotropic strength ratios and anisotropies are interchanged as the pulse ordering reverses, with equality at T = 0. As total anisotropies of r = 3/10 are calculated throughout the pulse overlap region, this suggests that r = 3/10 for coherent excitation so long as the coherent superposition of the two excited states remains aligned. The above results approximate the pulses as δ functions and thus miss the effects of finite frequency bandwidth and finite pulse duration that are needed to select one of two nondegenerate transitions and recover the anisotropy (4/10) and isotropic strength ratio (1:1) for a single dipole transition. Numerical calculations in the next section include these effects and relaxation dynamics.

Figure 7. Effect of an asymmetric mode with a small stabilization energy on asymmetric vibrational quantum beats in the anisotropy. The electronic splitting δ is zero. The vibrational quantum beats arise from an underdamped Brownian oscillator with ω1/(2πc) = 176 cm
1, stabilization energy (Dω)1 = 9 cm
1, and damping constant γ/(2πc) = 2.5 cm
1 at 300 K. The 30 fs laser pulse is tuned to 12 485 cm
1, below the vibronic origin at 12 685 cm
1. (Top) The solid line is the groundstate bleach anisotropy for the underdamped asymmetric vibration with only totally symmetric line broadening; the dotted line is the groundstate bleach anisotropy for the underdamped asymmetric vibration and the critically damped “asymmetric solvation” mode (as in Figure 5), but
1 with (Dω)asymm stabilization energy. (Bottom) The solid line CDO = 2 cm is the excited-state emission anisotropy with vibration and only symmetric inhomogeneous dephasing; the dotted line includes the asymmetric solvation mode (conventional t2-based distinction between GSB and ESE).

III. CALCULATIONS The model system shown in Figure 1 incorporates both a static excited-state splitting and an asymmetric vibrational coordinate that causes Jahn
Teller distortion by vibronic activity in 4-fold symmetric molecules. More generally, asymmetric vibrations cause both a distribution of the splitting around its mean and a vibrational modulation of the splitting (totally symmetric vibrations of 4-fold symmetric molecules do neither).12 Damping and thermal equilibration (at 300 K) of the asymmetric harmonic oscillator vibrations on each electronic state are incorporated by means of Brownian oscillator line-shape functions. The quantum line-shape functions in Gu et al.44 were inserted into nonlinear response functions given by eqs 2
4 of ref 12 to calculate the nonlinear polarization, from which the pump
probe signal was calculated using eqs 4
6 of ref 45. A. Pulse Overlap Effects. The calculations were performed using the code described in Farrow et al.7,45 (see appendix of ref 46). Figure 4 shows the calculated pump
probe signals for a model with no asymmetric vibrational distortions (d = 0 in Figure 1) and no splitting (δ = 0 in Figure 1). For computational reasons, a totally symmetric vibration is coupled to the electronic excitation in all calculations reported here. This mode dominates the absorption and emission line shapes, broadening them to about the width seen in naphthalocyanines. The line-shape function for this mode consists of a single totally symmetric overdamped Brownian oscillator with an ∼60 ps decay chosen so that electronic dynamics could be neglected in the early time anisotropy; this decay time is consistent with the slowest decays seen in naphthalocyanine molecules.7 Pump and probe pulses with a duration of 30 fs resonantly excite the electronic origin. 4106

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The Journal of Physical Chemistry A The numerical calculation of the triple field convolution with the optical response functions for the third-order polarization ranged over (5 times the pulsewidth, using 24 Gaussian quadrature points and a 1 fs step size for the integration over t. Based on the comparison tests in ref 45, where analytic results are available for reference, convergence with an accuracy of 0.01
0.05% is expected for these numerical integration parameters. Figure 4 shows a continuous change of the ground-state bleach and excited-state emission anisotropies during pulse overlap. The analytic results regarding the anisotropies of GSB and ESE are reproduced from these calculations. The top half of Figure 4 shows that the ground-state bleach anisotropy changes smoothly from 7/10 (resulting from diagram D12 contributing to the response) before zero delay to 1/10 when the pulses are separated at positive delay (when only diagrams D3 and D4 contribute). The excited-state emission anisotropy changes smoothly from 1/10 before zero delay (D9 contributes) to 7/ 10 after; thus, the overall anisotropy remains constant at 3/10. The anisotropy and strength of each component approximately reach their T . tp value by 2*tp. The bottom half of the figure shows the change in the ratio of ground-state bleach isotropic strength to excited-state emission isotropic strength, reproducing the analytic predictions of the isotropic sigESE GSB ESE nal strength ratios SGSB MA /SMA = 1 for T = 0 and SMA /SMA = 2 for T . tp. Figure 4 also shows that the interchange of anisotropies between GSB and ESE across time zero is accompanied by an interchange of the ratio of their isotropic strengths; GSB before time zero, SESE MA = 2SMA . The evolution of the absolute ESE isotropic strength in time, SGSB MA (T) þ SMA (T), overplotted on the bottom of Figure 4, reveals a ratio of STOT MA (T . tp) = 3STOT MA (T = 0) caused by excited-state population buildup during the pulse. The model includes no fast processes that could begin to take effect during the pulse. We additionally checked the TOT ratio STOT MA (T . tp)/SMA (T = 0) for a two-level system, and the 3:1 ratio also obtains there. Next, the code is used to calculate the signals expected from a three-level system in which the pair of singlet excited states are split in energy. B. Vibronic Level Splitting Effects. Expanding the time window of observations slightly can show the effect of an excited-state electronic level splitting on the anisotropy dynamics and reveal the effect of an asymmetric dephasing coordinate. For impulsive excitation of two electronic levels with perpendicular, equal amplitude transition dipoles, the total anisotropy becomes (in the absence of excited-state absorption) r(T) = (1/10)[1 þ cþ(T) þ d
(T)],7 where the coherence dephasing function cþ(T) and population difference function d
(T) both have initial values of 1 [cþ(0) = d
(0) = 1], are bounded by (1, and ultimately decay to equilibrium [zero for cþ(T), a temperature- and splitting-dependent value for d
(T)]. In D4h symmetry Jahn
Teller systems with simultaneous vibrational distortions along b1g and b2g coordinates and no static level splitting, cþ(T) and d
(T) involve coupled nonadiabatic dynamics and are not simply connected to line-shape functions. Similarly, a static splitting and vibration of opposite symmetries (b1g and b2g in D4h) lead to nonadiabatic dynamics not treated in this paper. All models used in this paper correspond to the “accidental Born
Oppenheimer case”47 of vibronic coupling that does not permit population transfer so that d
(T) = 1. In this case, cþ(T) represents the coherence dephasing between x- and y-polarized electronic states caused by b1g vibrations, and analytic response functions can be obtained for damped motion on identical, but displaced, harmonic potentials. Using eqs 6a and 6b of ref 12 and

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neglecting excited-state absorption yields the functional form of the impulsive anisotropy, considering electronic dephasing as the sole driver of electronic relaxation rðTÞ ¼ ð1=10Þ½2 þ cosðωβR TÞexpð
4Re½g1 ðTÞÞ

ð7Þ

where ωβR = δ is the b1g splitting between electronic levels |βæ = |yæ and |Ræ = |xæ in Figure 1 and g1(T) is the line-shape function for the asymmetric b1g vibrational coordinate.43,48,49 The derivation of eq 7 assumes that the symmetry of the splitting and the asymmetric vibration are equal in the D4h point group so that both became ag symmetry in the D2h point group (the accidental Franck
Condon case). Equation 7 reproduces that for coherently excited perpendicular quasi-degenerate dimer transitions.6 Vibrations and splitting of b2g symmetry in D4h cause only population transfer and lead to a formula just like eq 7, except |Ræ = |x0 æ and |βæ = |y0 æ [where |x0 æ and |y0 æ are polarized along (^x ( ^y)/(2)1/2]. The subscript 1 is replaced by 2 everywhere, and the time-dependent part corresponds to the coherence dephasing between |x0 æ and |y0 æ, which is equal to the population transfer function d
(T) in the |xæ, |yæ basis. The cosine term in eq 7 manifests the evolution of the transition dipole direction of a coherent superposition of two wave functions with orthogonal polarizations. For the system discussed by Farrow et al.,7 an asymmetric vibrational mode produced a dynamic splitting, which drove electronic wavepacket dynamics (and eventually dephased the electronic states). We begin the current analysis by considering a static splitting with only the totally symmetric dephasing (ag symmetry in D2h) used for computational reasons in Figure 4. The excitation pulse in the calculations is again centered at ωeg with a 30 fs duration. The anisotropies for this situation are shown as solid lines in Figure 5. After pump
probe overlap, the GSB anisotropy becomes constant and the ESE anisotropy oscillates cosinusoidally. Starting at time zero, the total anisotropy oscillates cosinusoidally without damping. The amplitude-normalized absorption and emission line shapes for the model are compared against the laser spectrum in the inset at the top of the figure. Comparison of Figures 4 and 5 shows that the initial decrease of the ground-state bleach anisotropy from ∼3/10 to ∼1/10 during pulse overlap is largely unchanged by the splitting, as the excited state does not evolve significantly during the pulse. However, the GSB anisotropy after pulse overlap is slightly changed, reaching an asymptotic value of r(¥) = 0.105. Because of its finite bandwidth, a pulse centered at ωeg will excite a slightly greater fraction of molecules to the lower (in this case, xpolarized) electronic level, as can be seen from Figure 1. This is also the reason why the total anisotropy starts slightly higher than 3/10. Effects of pulse-tuning on the anisotropy will be discussed later, in the context of a large electronic splitting. In contrast to Figure 4, the excited-state emission anisotropy and total anisotropy in Figure 5 show large oscillations of (2π/δ) = 333 fs period due to quantum beating of the two electronic levels. The excited-state emission anisotropy starts at about 3/10 at T = 0 but is modulated by coherent electronic wavepacket motion, oscillating between ∼7/10 and ∼1/10. The quantum beats are present if both perpendicularly polarized transitions fall within the bandwidth of the laser. With only totally symmetric dephasing processes and no inhomogeneity in the electronic splitting, the electronic quantum beats do not dephase. The quantum beats are not detected in the magic-angle 4107

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The Journal of Physical Chemistry A transient, indicating they are completely anisotropic within the 0.05% estimated convergence error. The dephasing process that destroys the anisotropic quantum beats need not involve homogeneous damping. The dotted line in Figure 5 shows the calculated results when an asymmetric
1 and critically damped oscillator of ωasymm CDO /2πc = 50 cm
1 asymm stabilization energy (Dω)CDO /2πc = 5 cm is added to the model. This asymmetric dephasing mode can be attributed to an asymmetric solvation process,7 or an asymmetric inhomogeneous broadening that produces a distribution of nearly static splittings on a ∼200 fs time scale.50,51 The obvious effect on the anisotropy is a very fast apparent damping of the beats. The “death” of the coherent superposition of the orthogonally polarized states reduces the anisotropy to the fully dephased value of 2/10. The dephasing of the coherent superposition depends almost entirely on the stabilization energy of the asymmetric mode and is not very sensitive to its damping time scale. This was determined by recalculating the anisotropy decay for an asymmetric overdamped oscillator with a decay time of ∼1000 fs. The shape of the anisotropy agreed within ∼1
2% of the dotted trace shown in Figure 5. Though perhaps surprising, this follows from the discussion in ref 12, where the amount of stabilization energy was found to be the main factor in the early anisotropy decay, and not the speed of the vibrational stabilization. Because the main process driving the initial drop of the anisotropy toward its asymptotic value is actually coherent electronic motion, not vibrational motion, it seems reasonable to conclude that the form of the dephasing process plays a minor role in the decay. The importance of the stabilization energy is shown in Figure 6. The anisotropy with a stabilization energy of 2 cm
1 goes through more than a full oscillation before reaching its asymptotic value, whereas increasing the asymmetric stabiliza
1 yields an anisotropy tion energy to (Dω)asymm CDO /2πc = 10 cm that just barely oscillates around 2/10. These calculations show that an asymmetric dephasing process is sufficient to dephase the two electronic states within a few hundred femtoseconds. Cho et al. came to similar conclusions about the strong damping effect of asymmetric broadening on electronic coherences in transient grating and three pulse photon echo peak shift signals.50 Figure 7 shows that underdamped asymmetric vibrations can produce large amplitude anisotropy beats in the absence of an electronic splitting. The solid line in Figure 7 shows the groundstate bleach and excited-state emission anisotropy for an underdamped b1g symmetry vibration with frequency ω1/(2πc) = 176 cm
1 that is damped on a time scale of τ ∼ 2 ps. The amplitude of the beats depends on the stabilization energy [(Dω)1 = 9 cm
1] and laser tuning. The 30 fs laser pulse was tuned to the red of the vibronic origin to make the vibrational beats stronger, an effect caused by poorer overlap of the laser spectrum with vibrational wavepackets as they evolve to the other side of the potential well.52 The ground-state bleach anisotropy oscillates around 1/10 at the vibrational frequency. The excitedstate emission anisotropy starts at 3/10 (a result of pulse overlap) and proceeds to evolve in a way consistent with the impulsive analytical prediction10,12 r ESE ðTÞ ¼ ð1=10Þ½4 þ 3cosðωβR TÞexpð
4Re½g1 ðTÞÞ ð8Þ where g1(T) is the vibrational line-shape function for the underdamped oscillator44 and ωβR = 0. In this case, the degenerate electronic states dephase at the dephasing rate of the asymmetric

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Figure 8. Change in electronic anisotropy dynamics as the splitting is increased and the laser is tuned to the higher-frequency (y-polarized) transition, ωlaser = ωeg þ δ/2. (Top) δ/2πc = 500 cm
1. (Bottom) δ/ 2πc = 1000 cm
1. Line broadening is dominated by the totally symmetric mode. Solid lines have no coupling to asymmetric vibrations, whereas dotted lines show the effect of the critically damped asymmetric solvation mode (introduced in Figure 5) with a stabilization energy of 2 cm
1.

vibration. Unlike the electronic quantum beats in Figure 5, the magic-angle signal has a very weak (but computationally significant) quantum beat amplitude (about 0.27% peak-to-peak oscillations between 400 and 800 fs). The quantum beats in the GSB signals oscillate 0.02% peak to peak, within the range of estimated convergence errors. In contrast, the quantum beat strengths in the ESE signals are significant, about 0.77% peak to peak. Both of these results are consistent with the analytic results obtained in ref 28, which showed that asymmetric vibrational quantum beats on the excited state can be less than completely anisotropic (|rvib|