- Email: [email protected]
Linear and nonlinear Herzberg−Teller vibronic coupling effects. II: Hole-burning and fluorescence line narrowing spectroscopy
Chemical Physics 525 (2019) 110402
Contents lists available at ScienceDirect
Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Linear and nonlinear Herzberg−Teller vibronic coupling effects. II: Holeburning and fluorescence line narrowing spectroscopy
T
Mohamad Toutounji Department of Chemistry, P. O. Box 15551, UAE University, Al Ain, United Arab Emirates
ABSTRACT
Part I of this series showed that Herzberg-Teller vibronic coupling manifests itself in linear spectroscopy through the breakdown of the mirror image symmetry between the absorption and emission spectra, whereby a nuclear exponential function was used to represent the molecular electronic transition dipole moment. This part tests the applicability of the framework developed in Paper I to absorption-emission asymmetry in nonlinear signals such as hole-burned and fluorescence line narrowed spectra. Model calculations of nonlinear absorption and emission spectra showing asymmetry is provided. Hole-burning and fluorescence line narrowing signals are to exemplify nonlinear absorption and emission spectra so as to reveal the anticipated asymmetry in 4-wave mixing experiments, and probe the consequent zero-phonon hole and the phonon-sideband hole. These calculations prove to be efficient and fast, exhibiting remarkable numerical stability. The observation of richer phonon-side band (PSB) in linear spectra, phono-side band hole (PSBH) in Hole-burning and fluorescence line narrowing signals, and emergence of new quantum beats over longer time scale (despite the dissipative medium) under the non-Condon regime are linked to the long quantum coherence phenomenon and stimulated photon peak shift in photosynthetic systems.
1. Introduction
2. Linear spectroscopy in Hertzberg-Teller coupled systems
Non-Condon effects play an important role in spectroscopy, kinetics, and dynamics [1–9], particularly in media where Born-Oppenheimer approximation (BOA) fails. It was shown in [6] that one ramification of BOA failure is the emergence of Hertzberg-Teller coupling through the dependence of the transition dipole moment on nuclear coordinates. The model employed in [6] to treat the transition dipole moment as a function of molecular position [2,3] offers several advantages, besides obtaining a closed-form expression for the linear dipole moment time correlation function. As pointed out earlier, the model adopted in [1,6,7] features a 4-fold advantage. i) It precludes eigenstate expansion, (ii) it eliminates perturbation theory when treating non-Condon effects, (iii) starting with one-photon absorption in the time domain, thereby allowing calculating nonlinear spectra in both time- and frequency domains, and finally (iv) it is computationally fast and stable. This paper is a continuation of Paper I, for it tests the efficacy and applicability of the model presented in [1,6] upon extending it to probe nonlinear absorption-emission asymmetry in spectra, e.g. hole burned and fluorescence line narrowing. [10–15] One may readily see that this is a first time-report showing hole-burned and fluorescence line narrowed spectra taking account Hertzberg-Teller vibronic coupling (HTVC); [6,16–18] hence unique results presented herein. These siteselective experiments contribute significantly to our understanding of optical dephasing in condensed media.
Consider a two-level system making an electronic transition from a ground state g to an excited state e whereby the electronic transition is coupled to some vibrational mode j with frequency ωj, momentum Pj, and coordinate Qj. The adiabatic electronic Hamiltonian of the twolevel system is
E-mail address: [email protected]. https://doi.org/10.1016/j.chemphys.2019.110402 Received 5 June 2019; Accepted 6 June 2019 Available online 08 June 2019 0301-0104/ © 2019 Elsevier B.V. All rights reserved.
(1)
H = Hg |g g| + He |e e| where the nuclear Hamiltonians Hg andHe are
Hg =
He =
P j2 2mj
P j2 2mj
+
1 m 2
2 2 j Qj
+
1 m 2
2 j (Qj
(2a)
Dj )
2
(2b)
Dj is the linear displacement of the upper energy curve with respect to the ground state of the vibrational mode j. Expressing the nuclear Hamiltonians in dimensionless quantities
Pj = pj mj
j
, Qj = qj
mj
j
, Dj = dj
mj
j
(3)
The linear electronic transition dipole moment correlation function J (t )
Chemical Physics 525 (2019) 110402
M. Toutounji
Trel Trn (eiH t / µ e Trel Trn (e
J (t ) =
iH t /
lineshape function for mode j
H)
µe
(4)
H)
a(
where β = 1/kT, and Trel and Trn denote the trace over the electronic and nuclear degrees of freedom, respectively. The electronic transition dipole moment of mode j accounting for Herzberg-Teller (HT) is given by [1–3,6,7]
µeg (qj ) = µge (qj ) = µge0 e kj qj
(µge0 ) 2
J (t ) =
1 z|z
z |eiHg
t/
e kj
qj e iHe t /
e kj
qj
Hg |
e
j
j
f
1 2 (kj 2
b1 = e
i jt
b2 =
ie
j
( k j + dj + ( k j
i jt
= csc (
dj2 + (kj 2 + 2kj dj + dj2 )e
2k j d j
j
j
(
dj )ei
jt
)(1 + ei
dj )ei
k j + dj + ( k j
jt
)(
ei
j t+
j t+
j
j
)/
2
(8b)
1 / 2
(8c)
)
and
=1
e
(8e)
j
The linear emission time correlation function may be cast as [6]
F (t ) =
2sin ( (1
j /2)
e
j)
d2
Exp
2
+ 2dj
+
0
2(e
j
1
+
j (1
1)
e
j)
(9)
where 2
= (1
1
= kj e
ei i jt
jt
+e
e
j
(1 + ei t)(ei j
i j t+ jt
(9a)
j)
e
j
)
(9b)
and 0
= kj (1 + ei
jt
+e
j
+e
i j t+
(9c)
j)
The linear absorption and fluorescence lineshape functions are, respectively, given by
)=
1 2
dtJ (t ) e i
t
( )=
1 2
dtF (t ) e
i t
a(
j | t|/2
(10)
and f
j | t|/2
(11)
where j is a damping constant of mode j. An interesting result may be obtained from Eqs. (7) and (8) in the low T limit, i.e. T 0, leading to a tremendous simplification of the dipole moment time correlation functions
J (t ) = exp
1 ((kj 2 2
2kj dj
dj2) + (kj + dj ) 2e
i j t)
=
( )=e
=
2kj dj
dj2 ) + (kj
d j ) 2e
i j t)
(14b)
( j)
j
1 (kj 2
j /2 2 j) +
! ( +
( j /2) 2
(15)
dj )2
(15a)
d N(
m) a (
) exp [
a( B
)]
(16)
where B is the burning frequency. The inhomogeneous broadening due to the environmental heterogeneity may be assigned a Gaussian distribution, encompassing all zero-phonon line (ZPL) frequencies,
(12)
N( ) = 1 ((kj 2 2
(14a)
1 (kj + dj )2 2
HB ( ) =
and
F (t ) = exp
dj2)
k j dj
Nonlinear spectra are indispensable tool that may be used to unmask the hidden structural and dynamical information underneath the inhomogeneous broadening. Hole-burning is one of the selective spectroscopy techniques to probe electronic dephasing of a guest molecule embedded in a solid or amorphus host. The idea of hole-burning is basically exciting a group of molecules within an inhomogeneously broadened band, leaving a hole in the ground state. The hole structure is very important experimentally for measuring pure electronic dephasing time [10]. This Section provides calculations of hole-burned (HB) absorption and fluorescence line narrowed (FLN) spectra of model systems using the lineshape functions in Eqs. (10) and (11), respectively, to test the applicability of the above developed theory in revealing the associated non-Condon effects (HT coupling) in nonlinear spectral signals, i.e., nonlinear absorption-emission asymmetry. Two key components of HB and FLN spectra, namely zero-phonon hole (ZPH), with a Franck-Condon factor (FCF) equal e 2 j (as opposed to corresponding ZPL FCF which is equal e j ) and twice the respective ZPL width, and the phonon-side band hole (PSBH) are important to probing electronic dephasing and the strength of linear elctron-phonon coupling in condensed systems. The effect of HT coupling manifests itself through HB-FLN asymmetry and the enhancement of the PSBH richer (richer vibronic structure) through enclosing more phonon profiles as opposed to that of Condon PSBH, as has clearly been pointed out in [6] and rigorously proved in [7]. (In photon echo signals, this would show longer lasting quantum beats (vibronic structure) as shown in [6].) The theory of Hays-Small [19] and Jankowiak [11,13–15] will be utilized for calculating HB and FLN spectra with the aid of Eqs. (7)–(11) or by directly using Eqs. (14) and (15). The hole-burning signal after burning time is given by [19]
(8d)
j /2)/2
1 2 (kj 2
3. Nonlinear Absorption-Emission asymmetry in Hertzberg-Teller coupled systems
(8a)
2
(14)
where
j
i j t)
=
=0
where
A = exp
+ ( j/2)2
Similarly, the linear fluorescence lineshape function in the low T limit is
(6)
(7)
)
m
and
= Tr (e Hg ) and where the canonical partition function is z|z = exp( |z|2 ) . Without loss of generality, µge0 in Eq. (5) has been set equal to unity for simplicity and tractability. Evaluating the integral in Eq. (6) yields J (t ) = Aexp [(b12 + b22 )/4 ]/(
m! (
j/2 2 j)
where
mj j / .
z d 2z
( j )m
j
m=0
(5)
where kj is an HT coupling coefficient measured in units of Using harmonic coherent states {|z } in Eq. (4) yields [6]
)=e
1 2
2
exp
m
2
2
(17)
2
Equation (17) represents a Gaussian profile with variance of ZPL frequencies ( ) centered at frequency m . While signifies the burning time, ρ is the product of the absorption cross-section, the laser burn 2
(13)
Inserting Eq. (12) in Eq. (10) produces the linear absorption 2
Chemical Physics 525 (2019) 110402
M. Toutounji
Fig. 2. Condon homogeneous absorption (green) and fluorescence (red) spectra of the insets in Fig. 1 superimposed to ratify their mirror image symmetry when there is no HT coupling.
Fig. 1. An inhomogeneously broadened absorption spectrum of a linearly coupled system with one vibrational mode whose ωj = 100 cm−1, dj = 1.1, T = 0.10 K, Δ = 120 cm−1, and γj = 20 cm−1. The green and the red spectra in the insets of the broad band are, respectively, the corresponding the Condon single-site linear absorption and linear fluorescence spectra. The inhomogeneous structureless band is calculated by convolving Eq. (17) and Eq. (10).
never accounted for non-Condon effects and their had always been in the Condon regime. Therefore, the reported results herein should be noteworthy.) Fig. 1 shows an inhomogeneously broadened absorption spectrum of a linearly coupled system with one vibrational mode whose dj = 1.1, T = 0.10 K, Δ = 120 cm−1, and ωj = 100 cm−1, −1 γj = 20 cm . HT coupling is nil by setting kj = 0 in Eq. (12) or (13). The green and the red spectra in the upper corners of the inhomogeneous broad band are, respectively, the corresponding homogeneous (single-site) linear absorption and linear fluorescence spectra. These are Condon spectra whereby BOA is followed. The inhomogeneous structureless band is calculated by convolving Eq. (17) and Eq. (10) (or 14) using Δ = 120 cm−1. The absorption and emission spectra in the insets are obtained by inserting Eqs. (7) and (9) in Eqs. (10) and (11), respectively, after setting kj = 0 . Fig. 2 ratifies that the Condon homogeneous absorption and fluorescence spectra of Fig. 1 are mirror image of each other in case of negligible HTVC where kj = 0. Carrying out Condon (kj = 0 ) HB (blue) and FLN (orange) calculations using Eqs. (16) and (21) produces Fig. 3 in which HB (blue) and FLN (brown) spectra exhibit perfect mirror image symmetry, thereby successfully eliminating the inhomogeneous broadening, due to the surrounding environment heterogeneity, and thus recovering the homogeneous structure shown in Fig. 2. Figs. 4 and 6 are the same as in Fig. 2 but with finite HT coupling (kj = 0.1and0.30 ), thereby causing a breakdown of the mirror image symmetry that appears in both Figs. 2 and 3. Figs. 5 and 7, however, use Eqs. (14) and (15) in Eqs. (16) and (21) to calculate the non-Condon HB and FLN spectra with kj = 0.1and0.30 . Note that the HB-FLN asymmetry is caused by the HTVC coupling in Figs. 5 and 7. The non-Condon HB-FLN spectra in Figs. 5 and 7 remarkably recover the corresponding homogenous absorptionemission in Figs. 4 and 6 by eliminating the inhomogeneous broadening. Because the HTVC magnitude in Figs. 6 and 7 is greater than that in Figs. 4 and 5, the asymmetry is more pronounced. The ZPH in both Figs. 5 and 7 has lost intensity and broadened in the hole-burned spectra as opposed to their respective ZPL appearing in Figs. 4 and 6 absorption spectra, whereas the PSBH has gotten richer [1,2] due to
flux, and the quantum yield. One can get an informative hole expression under short burning time and infinite inhomogeneous broadening that the hole signal in Eq. (16) may be recast as [20]
HB ( )
d
a(
)
a( B
)
(18)
The integral in Eq. (16) is not trivial. One can however resort to Fourier analysis theory which will reveal that the integral in Eq. (16) is just a convolution and may be recast using Parseval’s identity as [20]
HB ( )
d t |J (t )|2 e
2
i t ei B t
(19)
Carrying out the evaluation of Eq. (19), and guided by the work in Ref. [20], one can arrive at the hole-burned spectrum, from which the ZPH profile ( B = 0 ) may be drawn as
ZPH ( )
e2
j
j 2
+
j
2
(20)
The FCF and fwhm of ZPH are and 2 j , respectively, as expected. An additional important spectral feature of a hole-burned spectrum besides ZPH besides the real PSBH, should be pointed out is the pseudo phonon-sideband hole (PSBH) profiles which appear on left of the ZPH, respectively. More information on experimental hole-burning spectroscopy may be found in [10]. The FLN spectrum equation which will be utilized in this work is [11]
e2 j
FLN ( ) =
d N(
m) f
(
) exp [
f
(
B
)]
(21)
The same parameters used in [6] for linear spectroscopy will be used in the forthcoming model calculations. (Note that Refs. [11,19] 3
Chemical Physics 525 (2019) 110402
M. Toutounji
Fig. 3. Condon hole-burned (blue) absorption and fluorescence line narrowed (brown) spectra of model systems exhibit perfect mirror image symmetry, thereby successfully eliminating the inhomogeneous broadening in Fig. 1 and thus recovering the homogeneous structure shown in Fig. 2.
Fig. 5. Non-Condon hole-burned (blue) absorption and fluorescence line narrowed (brown) spectra of model systems fail to exhibit mirror image symmetry, thereby successfully eliminating the inhomogeneous broadening in Fig. 1 but with very small HT coupling (kj = 0.10) and thus recovering the homogeneous structure shown in Fig. 4. The slight HB-FLN asymmetry is caused by HTVC. Note that ZPH is broader than that in Fig. 4.
Fig. 4. Non-Condon homogeneous absorption (green) and fluorescence (red) spectra with very small HT coupling (kj = 0.10) where homogeneous absorption- fluorescence slight asymmetry is starting to emerge. Notably, more vibrational structure is also starting to appear.
Fig. 6. Non-Condon homogeneous absorption (green) and fluorescence (red) spectra with kj = 0.30 where homogeneous absorption- fluorescence mild asymmetry is more pronounced that that in Fig. 4. Notably, richer vibrational structure in the PSB than both Figs. 2 and 4.
HTVC than the corresponding PSB in Figs. 4 and 6. The transactions of the ZPH and PSBH in HB and FLN spectra reveal a wealth of information of electronic dephasing and linear electron-phonon coupling due to the phonon wing profile. The burning frequency in Figs. 3, 5 and 7 has been B = 0cm 1. As such, the non-Condon enhancement of vibronic
structure whereby richer phonon profiles in spectra and longer quantum beats in photon echo signals may be linked to the perseverant coherent excitonic transfer in photosynthetic complexes. [21–26,28] 4
Chemical Physics 525 (2019) 110402
M. Toutounji
quantum coherence may be attributed to the non-Markovian characterization of the bath [25,26] interacting with the pigments (chlorophyll), emergence of new quantum beats in photon echo and new phonon profiles in frequency-domain signals (e.g., pump-probe [7], HB, and FLN spectra) due to non-Condon effects might also ratify and support sustaining this quantum coherence. Surely, non-Markovian effects [25] help retain long-lived quantum coherence because the slow bath (non-Markovian) will feed in coherence back into the system at an efficient rate. This is because both the non-Markovian character and non-Condon effects necessitate non-separation of timescales of both the system and the bath, thereby leading to a pronounced quantum entanglement. [25] Accounting for both non-Condon and non-Markovian effects may sustain quantum coherence through richer PSB (and PSBH) and sustained coherence memory coming from the bath an d thereby replenishing the system, respectively. As such, perhaps both effects should be consolidated for more accurate justification of the long-lived quantum coherence in light harvesting systems. [24] Future work in this direction is in progress. Finally, further future work will look at the non-Condon effects on stimulated 3-pulse photon echo (SPE) signal peak shift. [27] One may, in light of the above, clearly infer that HTVC will increase the SPE peak shift, as the non-Condon effects seem to sustain and enhance the vibronic structure. This is yet to be ratified by the author. Finally, and in light of the above, electron-vibration coupling seems to be a pivotal role in excitation energy transfer [28] with or without HTVC, Duschinsky mixing effect [6], or Fermi mixing resonance [4] in photosynthetic complexes.
Fig. 7. Non-Condon hole-burned (blue) absorption and fluorescence line narrowed (brown) spectra of model systems exhibit noticeably asymmetry using HT coupling (kj = 0.30) and thus recovering the homogeneous structure shown in Fig. 6. Note that ZPH is broader than that in Fig. 6 with richer PSBH.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
4. Concluding remarks and outlook Important nonlinear frequency-domain signals like HB and FLN are often utilized in condensed-phase systems to unmask the hidden homogeneous structure by inhomogeneous broadening. These two 4wave mixing techniques (HB and FLN) play a fundamental role in probing optical electronic dephasing through the position, width, and intensity of the ZPH and PSBH. An exponential function was employed to represent the electronic transition dipole moment in order to account for the linear absorption-emission asymmetry mainly caused by nonCondon effects which arise due to HTVC. In fact, very recently it was rigorously proved that this is indeed the case. [7] Motivated by our first paper [6] results, this paper has shown that this asymmetry can also arise in nonlinear absorption and emission, namely HB and FLN, due to HTVC as well should Condon approximation fail, rendering a ZPH whose width is twice that of its respective ZPL and broader PSBH (richer with more activated phonons due to HTVC). In closing, an intriguing observation may be drawn in relation to excitation dynamics and coherent energy transfer time scales in light harvesting molecules, e.g. photosynthetic complexes. [21,23] The photon echo [6] and pump-probe [7] calculations revealed that the inclusion of non-Condon effects yield richer vibrational structure as opposed to that hat arising in the Condon regime. Additionally, this observation is further ratified by new richer PSBH appearing in the HB and FLN spectra upon relaxing the Condon approximation. As such, non-Condon effects (emanated by Herzberg-Teller vibronic coupling) may play an appreciable factor in explaining the longevity of coherence transfer across these complexes despite the surrounding dissipative environment. While some studies have suggested that this long
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
5
T.-C. Dinh, T. Renger, J. Chem. Phys. 145 (2016) 034105. C.K. Chan, J. Chem. Phys. 81 (1984) 1614. V. Khidekel, S. Mukamel, Chem. Phys. 240 (1995) 304. G.J. Small, J. Chem. Phys. 54 (1971) 3300. Y. Niu, Q. Peng, C. Deng, X. Gao, Z. Shuai, J. Phys. Chem. A 114 (2010) 7817. M. Toutounji, Chem. Phys. 521 (2019) 25. M. Toutounji, Chem. Phys. 2019, Accepted. D. Matysushov, J. Chem. Phys. 150 (2019) 074504. Xunmo Yang, Theo Keane, Milan Delor, Anthony J.H.M. Meijer, Julia Weinstein, Eric R. Bittner, Nat. Commun. 8 (2017) 14554. R. Jankowiak, J. Hays, G.J. Small, Chem. Rev. 93 (1993) 1471. M. Reppert, V. Naibo, R. Jankowiak, J. Chem. Phys. 133 (2010) 014506. J. Pieper, P. Artene, M. Rätsep, M. Pajusalu, A. Freiberg, J. Phys. Chem. B 122 (2018) 9289. A. Kell, X. Feng, M. Reppert, R. Jankowiak, J. Phys. Chem. B 117 (2013) 7317. O. Rancova, R. Jankowiak, D. Abramavicius, J. Phys. Chem. B 122 (2018) 1348. R. Jankowiak, O. Rancova, J. Chen, A. Kell, R.G. Saer, J.T. Beatty, D. Abramavicius, J. Phys. Chem. B 120 (2016) 7859. A. Baiardi, J. Bloino, V. Barone, J. Chem. Theory Comput. 9 (2013) 4097. M. Bonfanti, J. Petersen, P. Eisenbrandt, I. Burghardt, E. Pollak, J. Chem. Theory Comput. 14 (2018) 5310. T. Renger, A. Klinger, F. Steinecker, M. Busch, J. Numata, F. Müh, J. Phys. Chem. B 116 (2012) 14565. J. Hays, P.A. Lyle, G.J. Small, J. Phys. Chem. 98 (1994) 7337. M. Toutounji, J. Phys. Chem. C. 114 (2010) 20764. Y.-C. Cheng, G.R. Fleming, Annu. Rev. Phys. Chem. 60 (2009) 241. T. Renger, R. Marcus, J. Chem. Phys. 116 (2002) 9997. G. Engel, T.R. Calhoun, E.L. Read, T.K. Ahn, T. Mancal, Y.C. Cheng, R.E. Blankenship, G. Fleming, Nature 446 (2007) 782. Akihito Ishizaki, Graham R. Fleming, Proc. Natl. Acad. Sci. U. S. A 106 (2009) 17257. M. Thorwart, J. Eckel, J. Reina, P. Nalbach, S. Weiss, Chem. Phys. Lett. 478 (2009) 244. P. Rebentrost, A. Aspuru-GuzikJ, Chem. Phys. 134 (2011) 101103. W. Zhang, T. Meier, V. Chernyak, S. Mukamel, Phil. Trans. R. Soc. London A 356 (1998) 405. P. Malý, V. Novoderezhkin, R. van Grondelle, T. Mančal, Chem. Phys. 522 (2019) 69.
Contents lists available at ScienceDirect
Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Linear and nonlinear Herzberg−Teller vibronic coupling effects. II: Holeburning and fluorescence line narrowing spectroscopy
T
Mohamad Toutounji Department of Chemistry, P. O. Box 15551, UAE University, Al Ain, United Arab Emirates
ABSTRACT
Part I of this series showed that Herzberg-Teller vibronic coupling manifests itself in linear spectroscopy through the breakdown of the mirror image symmetry between the absorption and emission spectra, whereby a nuclear exponential function was used to represent the molecular electronic transition dipole moment. This part tests the applicability of the framework developed in Paper I to absorption-emission asymmetry in nonlinear signals such as hole-burned and fluorescence line narrowed spectra. Model calculations of nonlinear absorption and emission spectra showing asymmetry is provided. Hole-burning and fluorescence line narrowing signals are to exemplify nonlinear absorption and emission spectra so as to reveal the anticipated asymmetry in 4-wave mixing experiments, and probe the consequent zero-phonon hole and the phonon-sideband hole. These calculations prove to be efficient and fast, exhibiting remarkable numerical stability. The observation of richer phonon-side band (PSB) in linear spectra, phono-side band hole (PSBH) in Hole-burning and fluorescence line narrowing signals, and emergence of new quantum beats over longer time scale (despite the dissipative medium) under the non-Condon regime are linked to the long quantum coherence phenomenon and stimulated photon peak shift in photosynthetic systems.
1. Introduction
2. Linear spectroscopy in Hertzberg-Teller coupled systems
Non-Condon effects play an important role in spectroscopy, kinetics, and dynamics [1–9], particularly in media where Born-Oppenheimer approximation (BOA) fails. It was shown in [6] that one ramification of BOA failure is the emergence of Hertzberg-Teller coupling through the dependence of the transition dipole moment on nuclear coordinates. The model employed in [6] to treat the transition dipole moment as a function of molecular position [2,3] offers several advantages, besides obtaining a closed-form expression for the linear dipole moment time correlation function. As pointed out earlier, the model adopted in [1,6,7] features a 4-fold advantage. i) It precludes eigenstate expansion, (ii) it eliminates perturbation theory when treating non-Condon effects, (iii) starting with one-photon absorption in the time domain, thereby allowing calculating nonlinear spectra in both time- and frequency domains, and finally (iv) it is computationally fast and stable. This paper is a continuation of Paper I, for it tests the efficacy and applicability of the model presented in [1,6] upon extending it to probe nonlinear absorption-emission asymmetry in spectra, e.g. hole burned and fluorescence line narrowing. [10–15] One may readily see that this is a first time-report showing hole-burned and fluorescence line narrowed spectra taking account Hertzberg-Teller vibronic coupling (HTVC); [6,16–18] hence unique results presented herein. These siteselective experiments contribute significantly to our understanding of optical dephasing in condensed media.
Consider a two-level system making an electronic transition from a ground state g to an excited state e whereby the electronic transition is coupled to some vibrational mode j with frequency ωj, momentum Pj, and coordinate Qj. The adiabatic electronic Hamiltonian of the twolevel system is
E-mail address: [email protected]. https://doi.org/10.1016/j.chemphys.2019.110402 Received 5 June 2019; Accepted 6 June 2019 Available online 08 June 2019 0301-0104/ © 2019 Elsevier B.V. All rights reserved.
(1)
H = Hg |g g| + He |e e| where the nuclear Hamiltonians Hg andHe are
Hg =
He =
P j2 2mj
P j2 2mj
+
1 m 2
2 2 j Qj
+
1 m 2
2 j (Qj
(2a)
Dj )
2
(2b)
Dj is the linear displacement of the upper energy curve with respect to the ground state of the vibrational mode j. Expressing the nuclear Hamiltonians in dimensionless quantities
Pj = pj mj
j
, Qj = qj
mj
j
, Dj = dj
mj
j
(3)
The linear electronic transition dipole moment correlation function J (t )
Chemical Physics 525 (2019) 110402
M. Toutounji
Trel Trn (eiH t / µ e Trel Trn (e
J (t ) =
iH t /
lineshape function for mode j
H)
µe
(4)
H)
a(
where β = 1/kT, and Trel and Trn denote the trace over the electronic and nuclear degrees of freedom, respectively. The electronic transition dipole moment of mode j accounting for Herzberg-Teller (HT) is given by [1–3,6,7]
µeg (qj ) = µge (qj ) = µge0 e kj qj
(µge0 ) 2
J (t ) =
1 z|z
z |eiHg
t/
e kj
qj e iHe t /
e kj
qj
Hg |
e
j
j
f
1 2 (kj 2
b1 = e
i jt
b2 =
ie
j
( k j + dj + ( k j
i jt
= csc (
dj2 + (kj 2 + 2kj dj + dj2 )e
2k j d j
j
j
(
dj )ei
jt
)(1 + ei
dj )ei
k j + dj + ( k j
jt
)(
ei
j t+
j t+
j
j
)/
2
(8b)
1 / 2
(8c)
)
and
=1
e
(8e)
j
The linear emission time correlation function may be cast as [6]
F (t ) =
2sin ( (1
j /2)
e
j)
d2
Exp
2
+ 2dj
+
0
2(e
j
1
+
j (1
1)
e
j)
(9)
where 2
= (1
1
= kj e
ei i jt
jt
+e
e
j
(1 + ei t)(ei j
i j t+ jt
(9a)
j)
e
j
)
(9b)
and 0
= kj (1 + ei
jt
+e
j
+e
i j t+
(9c)
j)
The linear absorption and fluorescence lineshape functions are, respectively, given by
)=
1 2
dtJ (t ) e i
t
( )=
1 2
dtF (t ) e
i t
a(
j | t|/2
(10)
and f
j | t|/2
(11)
where j is a damping constant of mode j. An interesting result may be obtained from Eqs. (7) and (8) in the low T limit, i.e. T 0, leading to a tremendous simplification of the dipole moment time correlation functions
J (t ) = exp
1 ((kj 2 2
2kj dj
dj2) + (kj + dj ) 2e
i j t)
=
( )=e
=
2kj dj
dj2 ) + (kj
d j ) 2e
i j t)
(14b)
( j)
j
1 (kj 2
j /2 2 j) +
! ( +
( j /2) 2
(15)
dj )2
(15a)
d N(
m) a (
) exp [
a( B
)]
(16)
where B is the burning frequency. The inhomogeneous broadening due to the environmental heterogeneity may be assigned a Gaussian distribution, encompassing all zero-phonon line (ZPL) frequencies,
(12)
N( ) = 1 ((kj 2 2
(14a)
1 (kj + dj )2 2
HB ( ) =
and
F (t ) = exp
dj2)
k j dj
Nonlinear spectra are indispensable tool that may be used to unmask the hidden structural and dynamical information underneath the inhomogeneous broadening. Hole-burning is one of the selective spectroscopy techniques to probe electronic dephasing of a guest molecule embedded in a solid or amorphus host. The idea of hole-burning is basically exciting a group of molecules within an inhomogeneously broadened band, leaving a hole in the ground state. The hole structure is very important experimentally for measuring pure electronic dephasing time [10]. This Section provides calculations of hole-burned (HB) absorption and fluorescence line narrowed (FLN) spectra of model systems using the lineshape functions in Eqs. (10) and (11), respectively, to test the applicability of the above developed theory in revealing the associated non-Condon effects (HT coupling) in nonlinear spectral signals, i.e., nonlinear absorption-emission asymmetry. Two key components of HB and FLN spectra, namely zero-phonon hole (ZPH), with a Franck-Condon factor (FCF) equal e 2 j (as opposed to corresponding ZPL FCF which is equal e j ) and twice the respective ZPL width, and the phonon-side band hole (PSBH) are important to probing electronic dephasing and the strength of linear elctron-phonon coupling in condensed systems. The effect of HT coupling manifests itself through HB-FLN asymmetry and the enhancement of the PSBH richer (richer vibronic structure) through enclosing more phonon profiles as opposed to that of Condon PSBH, as has clearly been pointed out in [6] and rigorously proved in [7]. (In photon echo signals, this would show longer lasting quantum beats (vibronic structure) as shown in [6].) The theory of Hays-Small [19] and Jankowiak [11,13–15] will be utilized for calculating HB and FLN spectra with the aid of Eqs. (7)–(11) or by directly using Eqs. (14) and (15). The hole-burning signal after burning time is given by [19]
(8d)
j /2)/2
1 2 (kj 2
3. Nonlinear Absorption-Emission asymmetry in Hertzberg-Teller coupled systems
(8a)
2
(14)
where
j
i j t)
=
=0
where
A = exp
+ ( j/2)2
Similarly, the linear fluorescence lineshape function in the low T limit is
(6)
(7)
)
m
and
= Tr (e Hg ) and where the canonical partition function is z|z = exp( |z|2 ) . Without loss of generality, µge0 in Eq. (5) has been set equal to unity for simplicity and tractability. Evaluating the integral in Eq. (6) yields J (t ) = Aexp [(b12 + b22 )/4 ]/(
m! (
j/2 2 j)
where
mj j / .
z d 2z
( j )m
j
m=0
(5)
where kj is an HT coupling coefficient measured in units of Using harmonic coherent states {|z } in Eq. (4) yields [6]
)=e
1 2
2
exp
m
2
2
(17)
2
Equation (17) represents a Gaussian profile with variance of ZPL frequencies ( ) centered at frequency m . While signifies the burning time, ρ is the product of the absorption cross-section, the laser burn 2
(13)
Inserting Eq. (12) in Eq. (10) produces the linear absorption 2
Chemical Physics 525 (2019) 110402
M. Toutounji
Fig. 2. Condon homogeneous absorption (green) and fluorescence (red) spectra of the insets in Fig. 1 superimposed to ratify their mirror image symmetry when there is no HT coupling.
Fig. 1. An inhomogeneously broadened absorption spectrum of a linearly coupled system with one vibrational mode whose ωj = 100 cm−1, dj = 1.1, T = 0.10 K, Δ = 120 cm−1, and γj = 20 cm−1. The green and the red spectra in the insets of the broad band are, respectively, the corresponding the Condon single-site linear absorption and linear fluorescence spectra. The inhomogeneous structureless band is calculated by convolving Eq. (17) and Eq. (10).
never accounted for non-Condon effects and their had always been in the Condon regime. Therefore, the reported results herein should be noteworthy.) Fig. 1 shows an inhomogeneously broadened absorption spectrum of a linearly coupled system with one vibrational mode whose dj = 1.1, T = 0.10 K, Δ = 120 cm−1, and ωj = 100 cm−1, −1 γj = 20 cm . HT coupling is nil by setting kj = 0 in Eq. (12) or (13). The green and the red spectra in the upper corners of the inhomogeneous broad band are, respectively, the corresponding homogeneous (single-site) linear absorption and linear fluorescence spectra. These are Condon spectra whereby BOA is followed. The inhomogeneous structureless band is calculated by convolving Eq. (17) and Eq. (10) (or 14) using Δ = 120 cm−1. The absorption and emission spectra in the insets are obtained by inserting Eqs. (7) and (9) in Eqs. (10) and (11), respectively, after setting kj = 0 . Fig. 2 ratifies that the Condon homogeneous absorption and fluorescence spectra of Fig. 1 are mirror image of each other in case of negligible HTVC where kj = 0. Carrying out Condon (kj = 0 ) HB (blue) and FLN (orange) calculations using Eqs. (16) and (21) produces Fig. 3 in which HB (blue) and FLN (brown) spectra exhibit perfect mirror image symmetry, thereby successfully eliminating the inhomogeneous broadening, due to the surrounding environment heterogeneity, and thus recovering the homogeneous structure shown in Fig. 2. Figs. 4 and 6 are the same as in Fig. 2 but with finite HT coupling (kj = 0.1and0.30 ), thereby causing a breakdown of the mirror image symmetry that appears in both Figs. 2 and 3. Figs. 5 and 7, however, use Eqs. (14) and (15) in Eqs. (16) and (21) to calculate the non-Condon HB and FLN spectra with kj = 0.1and0.30 . Note that the HB-FLN asymmetry is caused by the HTVC coupling in Figs. 5 and 7. The non-Condon HB-FLN spectra in Figs. 5 and 7 remarkably recover the corresponding homogenous absorptionemission in Figs. 4 and 6 by eliminating the inhomogeneous broadening. Because the HTVC magnitude in Figs. 6 and 7 is greater than that in Figs. 4 and 5, the asymmetry is more pronounced. The ZPH in both Figs. 5 and 7 has lost intensity and broadened in the hole-burned spectra as opposed to their respective ZPL appearing in Figs. 4 and 6 absorption spectra, whereas the PSBH has gotten richer [1,2] due to
flux, and the quantum yield. One can get an informative hole expression under short burning time and infinite inhomogeneous broadening that the hole signal in Eq. (16) may be recast as [20]
HB ( )
d
a(
)
a( B
)
(18)
The integral in Eq. (16) is not trivial. One can however resort to Fourier analysis theory which will reveal that the integral in Eq. (16) is just a convolution and may be recast using Parseval’s identity as [20]
HB ( )
d t |J (t )|2 e
2
i t ei B t
(19)
Carrying out the evaluation of Eq. (19), and guided by the work in Ref. [20], one can arrive at the hole-burned spectrum, from which the ZPH profile ( B = 0 ) may be drawn as
ZPH ( )
e2
j
j 2
+
j
2
(20)
The FCF and fwhm of ZPH are and 2 j , respectively, as expected. An additional important spectral feature of a hole-burned spectrum besides ZPH besides the real PSBH, should be pointed out is the pseudo phonon-sideband hole (PSBH) profiles which appear on left of the ZPH, respectively. More information on experimental hole-burning spectroscopy may be found in [10]. The FLN spectrum equation which will be utilized in this work is [11]
e2 j
FLN ( ) =
d N(
m) f
(
) exp [
f
(
B
)]
(21)
The same parameters used in [6] for linear spectroscopy will be used in the forthcoming model calculations. (Note that Refs. [11,19] 3
Chemical Physics 525 (2019) 110402
M. Toutounji
Fig. 3. Condon hole-burned (blue) absorption and fluorescence line narrowed (brown) spectra of model systems exhibit perfect mirror image symmetry, thereby successfully eliminating the inhomogeneous broadening in Fig. 1 and thus recovering the homogeneous structure shown in Fig. 2.
Fig. 5. Non-Condon hole-burned (blue) absorption and fluorescence line narrowed (brown) spectra of model systems fail to exhibit mirror image symmetry, thereby successfully eliminating the inhomogeneous broadening in Fig. 1 but with very small HT coupling (kj = 0.10) and thus recovering the homogeneous structure shown in Fig. 4. The slight HB-FLN asymmetry is caused by HTVC. Note that ZPH is broader than that in Fig. 4.
Fig. 4. Non-Condon homogeneous absorption (green) and fluorescence (red) spectra with very small HT coupling (kj = 0.10) where homogeneous absorption- fluorescence slight asymmetry is starting to emerge. Notably, more vibrational structure is also starting to appear.
Fig. 6. Non-Condon homogeneous absorption (green) and fluorescence (red) spectra with kj = 0.30 where homogeneous absorption- fluorescence mild asymmetry is more pronounced that that in Fig. 4. Notably, richer vibrational structure in the PSB than both Figs. 2 and 4.
HTVC than the corresponding PSB in Figs. 4 and 6. The transactions of the ZPH and PSBH in HB and FLN spectra reveal a wealth of information of electronic dephasing and linear electron-phonon coupling due to the phonon wing profile. The burning frequency in Figs. 3, 5 and 7 has been B = 0cm 1. As such, the non-Condon enhancement of vibronic
structure whereby richer phonon profiles in spectra and longer quantum beats in photon echo signals may be linked to the perseverant coherent excitonic transfer in photosynthetic complexes. [21–26,28] 4
Chemical Physics 525 (2019) 110402
M. Toutounji
quantum coherence may be attributed to the non-Markovian characterization of the bath [25,26] interacting with the pigments (chlorophyll), emergence of new quantum beats in photon echo and new phonon profiles in frequency-domain signals (e.g., pump-probe [7], HB, and FLN spectra) due to non-Condon effects might also ratify and support sustaining this quantum coherence. Surely, non-Markovian effects [25] help retain long-lived quantum coherence because the slow bath (non-Markovian) will feed in coherence back into the system at an efficient rate. This is because both the non-Markovian character and non-Condon effects necessitate non-separation of timescales of both the system and the bath, thereby leading to a pronounced quantum entanglement. [25] Accounting for both non-Condon and non-Markovian effects may sustain quantum coherence through richer PSB (and PSBH) and sustained coherence memory coming from the bath an d thereby replenishing the system, respectively. As such, perhaps both effects should be consolidated for more accurate justification of the long-lived quantum coherence in light harvesting systems. [24] Future work in this direction is in progress. Finally, further future work will look at the non-Condon effects on stimulated 3-pulse photon echo (SPE) signal peak shift. [27] One may, in light of the above, clearly infer that HTVC will increase the SPE peak shift, as the non-Condon effects seem to sustain and enhance the vibronic structure. This is yet to be ratified by the author. Finally, and in light of the above, electron-vibration coupling seems to be a pivotal role in excitation energy transfer [28] with or without HTVC, Duschinsky mixing effect [6], or Fermi mixing resonance [4] in photosynthetic complexes.
Fig. 7. Non-Condon hole-burned (blue) absorption and fluorescence line narrowed (brown) spectra of model systems exhibit noticeably asymmetry using HT coupling (kj = 0.30) and thus recovering the homogeneous structure shown in Fig. 6. Note that ZPH is broader than that in Fig. 6 with richer PSBH.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
4. Concluding remarks and outlook Important nonlinear frequency-domain signals like HB and FLN are often utilized in condensed-phase systems to unmask the hidden homogeneous structure by inhomogeneous broadening. These two 4wave mixing techniques (HB and FLN) play a fundamental role in probing optical electronic dephasing through the position, width, and intensity of the ZPH and PSBH. An exponential function was employed to represent the electronic transition dipole moment in order to account for the linear absorption-emission asymmetry mainly caused by nonCondon effects which arise due to HTVC. In fact, very recently it was rigorously proved that this is indeed the case. [7] Motivated by our first paper [6] results, this paper has shown that this asymmetry can also arise in nonlinear absorption and emission, namely HB and FLN, due to HTVC as well should Condon approximation fail, rendering a ZPH whose width is twice that of its respective ZPL and broader PSBH (richer with more activated phonons due to HTVC). In closing, an intriguing observation may be drawn in relation to excitation dynamics and coherent energy transfer time scales in light harvesting molecules, e.g. photosynthetic complexes. [21,23] The photon echo [6] and pump-probe [7] calculations revealed that the inclusion of non-Condon effects yield richer vibrational structure as opposed to that hat arising in the Condon regime. Additionally, this observation is further ratified by new richer PSBH appearing in the HB and FLN spectra upon relaxing the Condon approximation. As such, non-Condon effects (emanated by Herzberg-Teller vibronic coupling) may play an appreciable factor in explaining the longevity of coherence transfer across these complexes despite the surrounding dissipative environment. While some studies have suggested that this long
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
5
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