internal conversion and vibrational energy redistribution ... · internal conversion and...

This document is confidential and is proprietary to the American Chemical Society and its authors. Do not copy or disclose without written permission. If you have received this item in error, notify the sender and delete all copies.

Internal Conversion and Vibrational Energy Redistribution

in Chlorophyll A

Journal: The Journal of Physical Chemistry

Manuscript ID jp-2015-09548x.R2

Manuscript Type: Article

Date Submitted by the Author: n/a

Complete List of Authors: Shenai, Prathamesh; Nanyang Technological University, Fernandez-Alberti, Sebastian; Universidad Nacional de Quilmes, Departamento de Ciencia y Tecnologia

Bricker, William; Washington University in St. Louis, Energy, Environmental and Chemical Engineering Tretiak, Sergei; Los Alamos National Laboratory, Theory Division Zhao, Yang; Nanyang Technological University, Division of Materials Science

ACS Paragon Plus Environment

The Journal of Physical Chemistry

Internal Conversion and Vibrational EnergyRedistribution in Chlorophyll A

Prathamesh M Shenai,† Sebastian Fernandez-Alberti,∗,‡ William P Bricker,‖

Sergei Tretiak,§ and Yang Zhao∗,†

† Division of Materials Science, Nanyang Technological University, Singapore 639798,‡ Universidad Nacional de Quilmes, Roque Saenz Pea 352, B1876BXD Bernal, Argentina,

‖ Department of Energy, Environmental and Chemical Engineering, WashingtonUniversity, Saint Louis, Missouri 63130, USA, and

§ Theoretical Division, Center for Nonlinear Studies (CNLS), and Center for IntegratedNanotechnologies (CINT), Los Alamos National Laboratory, Los Alamos, New Mexico

87545, USA

E-mail: [email protected]; [email protected]

AbstractWe have computationally investigated the roleof intramolecular vibrational modes in de-termining nonradiative relaxation pathwaysof photoexcited electronic states in isolatedchlorophyll A (ChlA) molecules. To simulatethe excited state relaxation from the initiallyexcited Soret state to the lowest excited stateQy, the approach of non-adiabatic excited statemolecular dynamics has been adopted. The in-tramolecular vibrational energy relaxation andredistribution that accompany the electronicinternal conversion process is followed by an-alyzing the excited state trajectories in termsof the ground state equilibrium normal modes.The time dependence of the normal mode veloc-ities is determined by projecting instantaneousCartesian velocities onto the normal mode vec-tors. Our analysis of the time evolution of theaverage mode energies uncovers that only asmall subset of the medium-to-high frequencynormal modes actively participate in the elec-∗To whom correspondence should be addressed†Nanyang Technological University‡Universidad Nacional de Quilmes‖Washington University§Los Alamos National Laboratory

tronic relaxation processes. These active modesare characterized by the highest overlap withthe non-adiabatic coupling vectors (NACRs)during the electronic transitions. Further sta-tistical analysis of the nonadiabatic transitionsreveals that the electronic and vibrational en-ergy relaxation occurs via two distinct path-ways with significantly different timescales onwhich the hopping from Soret to Qx occursthereby dictating the overall dynamics. Fur-thermore, the NACRs corresponding to eachof the transitions have been consistently foundto be predominantly similar to a set of normalmodes that vary depending upon the transitionand the identified categories. Each pathway ex-hibits a differential timescale of energy transferand also a differential set of predominant activemodes. Our present analysis can be consideredas a general approach allowing identification ofa reduced subset of specific vibrational coordi-nates associated with non-radiative relaxationpathways. Therefore, it represents an adequateprior strategy that can particularly facilitatesmixed quantum-classical approaches.

1

Page 1 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

1 IntroductionGiven the supreme importance of natural pho-tosynthesis in sustaining life on the Earth, un-raveling its secrets has been a pivotal researcharea over many decades, especially in the lightof depleting conventional energy sources. Asophisticated biological machinery enables har-vesting of solar energy and its subsequent con-version to chemical energy inside the thylakoidmembranes in the chloroplasts of green plants.Despite the great complexity of the whole pro-cess, it is surprisingly hinged upon very few pig-ment molecules known as chlorophylls. Chloro-phylls not only act as primary photo-active pig-ments and enable energy transfer pathways tothe so-called “reaction center," but also partic-ipate in charge separation that triggers photo-chemical conversion.1,6,7 Dynamics of the pho-toexcited states in chlorophylls is thus centralto a number of aspects of photosynthesis.Previously10, we have measured the time con-

stants for high excited states to relax nonradia-tively to the lowest excited state (Qy) in Chloro-phyll A (ChlA) and B (ChlB) with ultrafasttransient absorption spectroscopy and modeledit using the non-adiabatic excited state molecu-lar dynamics (NA-ESMD) methodology64. Theslight difference in the molecular structures ofthese two chlorophylls was found to lead toslightly slower dynamics of B → Qy inter-nal conversion in ChlB in agreement with ex-perimental spectroscopic data10. Shi et al.demonstrated that the time-constant for in-ternal conversion increases with an increasein the solvent dielectric constant.11 Dong etal. further estimated the vibrational relaxationtime to be 1.5 ps for ChlA solvated in com-mon solvents such as ethyl acetate and ethylether.12,13 Even though the internal conversionin chlorophylls itself is surprisingly understud-ied, it has received significant attention in var-ious other light-harvesting pigments such asbacteriochlorophylls,14–17 carotenoids18–22 andporphyrins. Within a wider view, importanceof understanding internal conversion is evidentfrom a number of related studies in biologi-cal macromolecules such as nucleic acids23,24and many other organic/inorganic molecules or

compounds.25,30,31Nonradiative relaxation of excited electronic

states is invariably accompanied by vibrationalenergy redistribution, and a number of stud-ies point to the importance of ‘active’ vibra-tional modes in assisting internal conversionprocesses.32–36 Experimentally, such studies aregenerally based on variants of excited stateIR spectroscopy,37–39 ultrafast time-resolved IRspectroscopy,40–42 and ultrafast transient IRspectroscopy43. These techniques have beenshown to be powerful in the study of the ex-cited states decay.47–57 Much of the theoreti-cal work on intramolecular energy transfer inorganic molecules has involved classical andsemiclassical Hamiltonians using harmonic ap-proximations to identify the pathways and lifetimes of energy transfer out of a given bond ormode.44–46 Equilibrium normal modes (ENMs),typically calculated from second derivatives ofthe ground-state (GS) energy with respect tonuclear coordinates (Hessian), can be usedto elucidate vibrational motions of polyatomicmolecules.58–61 Within this theoretical frame-work, one can then address the vibrational en-ergy flow by following the time evolution of thekinetic energy of each ENM. Anharmonic ef-fects and mode couplings have, however, beenrecognized to be important towards giving riseto vibrational energy redistribution. Previousworks have been extensively discussed these as-pects in proteins26–29. In particular to ChlA,for example, density functional theory calcula-tions have been used to investigate the vibra-tional properties and to ascertain couplings be-tween important infrared-active modes of thecarbonyl groups.3,62 In this work, following upon our earlier study, we focus on characterizingthe role of vibrational energy redistribution inthe non-radiative relaxation of high-energy ex-cited states in ChlA. Our aim is to identify thenature of active normal modes related to theelectronic couplings and intramolecular energytransfer throughout the nonadiabatic excited-state molecular dynamics. We show that onlya small subset of vibrational modes of ChlA ac-tively aid the internal conversion process.The remainder of the paper is organized as

follows. The underlying theory for the NA-

2

Page 2 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

ESMD methodology is briefly described in thenext Section II. Results are presented and dis-cussed in detail in Section III before drawingconcluding remarks in the final Section IV.

2 Theoretical and Computa-tional Details

2.1 NA-ESMD background

The NA-ESMD framework63,64 allows for sim-ulation of photoinduced dynamics of large or-ganic conjugated molecules involving multiplecoupled electronic excited states, a formidabletask carried out by combining the molecu-lar dynamics with quantum transitions ap-proach65,66 with “on the fly" analytical cal-culations of excited-state energies67–69, gra-dients70,71, and non-adiabatic coupling63,72–74

terms. For this purpose, the collective elec-tron oscillator (CEO) method75–78 is applied us-ing the AM1 semiempirical Hamiltonian79,80 atthe configuration interaction singles level81,82.Previously, AM1 and similar semiempiricalmodels have been successfully applied to de-scribe excited state properties of porphyrin-based molecules83–85,80,86–88,89–91. A detaileddiscussion on the implementation, the advan-tages, and various testing parameters of NA-ESMD can be found elsewhere63,64,97.

2.2 Normal Mode Analysis

ENMs are computed at the ground state stateminimum R0 from the mass-weighted Hessianmatrix H with elements

Hij(R0) = −∂2E/∂qi∂qj, (1)

where E is ground state potential energy andqi =

√mi(Ri−R0,i) is the mass weighted carte-

sian displacement of ith atom with mass mi

and Cartesian coordinates R0,i(X0,i, Y0,i, Z0,i).Geometry optimization was performed on thechlorophyll molecule with the phytyl tail re-placed by a methyl group for computationalease, with the AM1 semi-empirical Hamilto-nian, using Gaussian 09103. Diagonalization of

H produces the set of orthonormal ENM withfrequencies νi related to the eigenvalues λi(i =1,...,3N-6) of H as νi = (

√λi/2π). ENMs are

defined in a body-fixed reference frame with theorigin at the center of mass of the molecule andaxes corresponding to its principle axes of iner-tia. Upon diagonalization of H, the set of ENMvectors {Qi}, i = 1, 3N − 6 are obtained fromthe corresponding eigenvector matrix L, whichallows to express their amplitudes throughoutthe NA-ESMD as a linear combination of theset {qi}i=1,3N as

Qi(t) =3N∑j=1

ljiqj(t) i = 1, .., 3N − 6 (2)

where lji are elements of the eigenvector ma-trix L. By differentiating this expression withrespect to time, we obtain the relationship be-tween the momenta

Qi(t) =3N∑j=1

ljiqj(t) i = 1, .., 3N − 6 (3)

In this way, the vibrational kinetic energy trans-forms in terms of the ENM momenta as

K(t) =1

2

3N−6∑i=1

(Qi(t))2 (4)

The total vibrational energy associated with agiven mode is calculated using the virial theo-rem, Ei(t) = 2Ki(t).

2.3 Molecular Dynamics Simula-tions

The ground state molecular dynamics simula-tion was then performed at 300 K with a timestep ∆t = 0.5 fs. The system was heated toallow thermal equilibration at a final temper-ature of 300 K during the first 10 ps. TheLangevin equation at constant temperature wasutilized with a friction coefficient of γ = 2.0 ps.This value has been chosen to allow an efficienttemperature coupling95 to obtain the adequateground state conformational sampling. In the 8ns long production run following equilibration,

3

Page 3 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

we sample atomic coordinates and momenta atevery 20 ps generating a set of 400 initial po-sitions and momenta for the subsequent NA-ESMD simulations. Details of the assignmentof the initial excited state for each NA-ESMDtrajectories according to a Frank-Condon win-dow centered at 2.94 eV (lowest state in theSoret band) can be found elsewhere10. As op-posed to the initial ground state simulation,NA-ESMD simuations have been carried out atconstant energy during 1.5 ps using a classicaltime-step of ∆t = 0.1 fs and Nq = 4 quan-tum time steps per classical step to simultane-ously propagate the electronic coefficients forthe 10 lowest excited states. The existence oftrivial unavoided crossings has been examinedby tracking the identities of the states104.

3 Results and DiscussionsWe have investigated the dynamics of the inter-nal conversion process of ChlA (Fig.1(a)) thattakes place after its initial photoexcitation inthe red edge (S3) of the Soret (B) band (seeFig. 1(b)). Subsequent to laser excitation, themolecular system experiences a fast B → Qx →Qy electronic energy relaxation and redistribu-tion that involves the selective participation ofspecific atomic groups and complex global mi-gration of the electronic wave function withinthe total carbon macrocycle of the porphyrinring structure10. Fig. 1(c) represents the aver-age populations of various electronic states asa function of time. Here, the population of theSoret state is taken to be a summation of thepopulations on S3 and S4 as a small percentage(< 4%) of configurations are initially excited atS4. The lowest excited state is completely pop-ulated within 1.5 ps. By fitting the simulationsdata with a sequential kinetic model we can ex-tract the time constant as τ1 = 105 fs for theSoret → Qx process and τ2 = 124 fs for theQx → Qy relaxation. The time constants ob-tained through the constant energy simulationsare close to those obtained by us in the previousstudy using constant temperature simulations.The intramolecular vibrational energy flow

in ChlA, accompanied by the electronic en-

Figure 1: (a) Molecular structure of chlorophyllA (carbon atoms in grey, oxygen in red, nitro-gen in blue, magnesium in green and hydro-gen in light grey), where R denotes the phytyltail. (b) Simulated absorption spectrum ofChlA where contributions from the 5 lowest en-ergy exited states have been included; and (c)Populations of excited states following initialexcitation centered at the lowest state in theSoret band. Results are fitted (dashed curves)with a sequential kinetic rate constant model.

4

Page 4 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

ergy transfer, is monitored by evaluating thetime evolution of the normal mode velocitiesobtained by projection of the body-fixed massweighted cartesian velocities onto the normalmodes vectors (see Section Normal Mode Anal-ysis).Fig. 2(a) and (b) shows the vibrational en-

ergy Ei along different intermediate-range andhigh-frequency ENMs (twice the kinetic energyconsidering the equipartition of energy) aver-aged over the ensemble of NA-ESMD trajecto-ries. It is observed first that the intramolecularvibrational energy redistribution is not statisti-cal, since the energy deposited in each ENMvaries significantly. It is important to stressthat, in the present work, NA-ESMD simula-tions have been performed at constant energy.Therefore, the lack of dominant vibronic relax-ation paths during the internal conversion pro-cess should lead to transient and final accumu-lations of slightly different amounts of the ini-tial excess of energy spread among all the 240vibrational normal modes of the molecule. Ex-cept for the 191th to 201th, and also 180th and182th ENMs, the energies of all other ENMsdeviate only slightly from their equilibrium val-ues during the internal conversion process. Wehave defined a normal mode as active during theinternal conversion process if, at any time dur-ing the NA-ESMD simulations, its average en-ergy exceeds twice the initial kT=0.026 eV equi-librium value, thus reflecting a strong couplingto the electronic degrees of freedom. These re-duced number of active modes experience a fastincrease in their energy reaching their maximaat about 400 fs. The maximum combined en-ergy stored in all of these active modes is 0.874eV (7049 cm−1, with the highest contributioncoming from mode 194 at 0.108 eV (871 cm−1).After that, they experience a slow energy relax-ation that is extended for more than 1 ps. Therest of the modes seems to act as a bath of cou-pled harmonic oscillators to which the excess ofenergy, transiently accumulated in these activemodes, is finally transferred.A careful analysis along each of the 240 nor-

mal modes reveals that only a small subset ofthem indexed predominantly from modes num-ber 180 (1465 cm−1), 182 (1491 cm−1) and

191-201 (1665-1908 cm−1) are strongly excitedduring the excited state relaxation process asshown in Fig. 2(a). The energy plots in Fig.2(b) along any of the remaining normal modesin a similar frequency bracket suggest that theyare not involved in the excited state dynamics.We have thus identified a few vibrational modesthat are intricately associated with Soret to Qy

internal conversion in ChlA. These modes canbe associated to the 1597-1620 cm−1 band ob-served in the IR spectra of most chlorophyll sys-tems2,3. This assignment is based on previouswork performed by Wang and co-workers3. De-spite the frequency difference between the ex-perimental IR band and the calculated one, theassignment can be performed based on the com-mon relative nuclear displacements between vi-brations associated with this experimental bandand relative nuclear displacements involved inthe active normal modes. Based on the nor-mal mode assignments performed by Boldt etal5, this band is predominantly associated withCaCm, CaCb and CbCb stretching vibrations(see Fig.1(a))4,5.Fig. 2(c) and (d) shows the time-dependence

of the norm of different ENM forces averagedover the trajectories. Thereof we can point outthe role that gradients/forces on the excitedstates play during this intramolecular vibra-tional energy redistribution. Upon vertical ex-citation to the Soret band, an ultrafast transferof electronic to vibrational energy is observedas an increase of the forces in the direction ofthe modes previously identified as active normalmodes. The relative low initial values of theseforces increase in time until they reach theirmaxima at about 400 fs. This behavior indi-cates that the redistribution of the energy doesnot take place as the result of the initial ultra-fast adiabatic relaxation that occurs right afterphotoexcitation, but rather as a consequence ofnon-adiabatic dynamics.In order to further elucidate the role played

by these vibrational normal modes in the inter-nal conversion process, we investigate the corre-spondence between such active modes and thenonadiabatic coupling vectors (NACRs) definedas

dIJ = 〈ψI(r, R)|∇RψJ(r, R)〉 (5)

5

Page 5 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

Figure 2: Time evolution of the total energy oftrajectories along (a) normal modes 180, 182,191-201; and (b) all the normal modes from 171-210 excluding those plotted in (a); time evolu-tion of the norm of the normal mode forces aver-aged over the trajectories for (c) normal modes180, 182, 191-201; and (d) all the normal modesfrom 171-210 excluding those plotted in (c).

where r and R are the electronic and nuclearvector coordinates, respectively, and ψI(r, R) isthe Ith CI electronic state.The direction of NACR can be interpreted as

the nonadiabatic contribution to the directionof the main driving force on the nuclei duringelectronic transitions33. Fig. 3 depict the pro-jection of the forces on the direction of NACRthroughout the NAESMD simulations. Afterphotoexcitation, the molecular system enters inregions of the configurational phase space withstrong nonadiabatic couplings. The nonadia-batic contributions to the nuclear forces rapidlyincrease in time. At earlier times, the directionof NACR(Soret-Qx) represents a higher con-tribution than the NACR(Qx-Qy) one. Afterthe first 100 fs of the simulations, the nonadi-abatic contributions of either NACR(Soret-Qx)and NACR(Qx-Qy) persist at relatively highervalues during the next 400 fs. After that, bothcontributions decay while the system adiabat-ically relaxes on the Qy state. Therefore, thenonadiabatic contribution of the nuclear forces

Figure 3: Time evolution of the projection ofthe forces on the direction of NACR averagedover the trajectories.

seems to dictate the fast intramolecular vibra-tional energy redistribution that occurs duringthe internal conversion process. During nona-diabatic dynamics in the vicinity of level cross-ings, both the Pechukas forces and redistribu-tion of electronic energy excess upon quantumtransitions are generally raising classical mo-menta of nuclei in the direction of NACR vec-tors. This ensures an effective flow of electronicenergy into specific vibrational modes alignedwith NACR vectors.It is interesting at this point to associate each

step of the Soret to Qy internal conversion withthe participation of the different active nor-mal modes in the corresponding concomitantintramolecular vibrational energy pathways. Tothis end, from the ensemble of excited state tra-jectories, we first analyze two sets of NACRs -one corresponding to the Soret to Qx transi-tion and the other corresponding to the Qx toQy relaxation. Each NACR vector, expressedin body-fixed reference frame, is projected onthe basis of normal modes {Qi}i=1,3N−6. In Fig.4 we show the distributions of the overlap be-tween NACR vectors and various selected nor-mal modes. Fig. 4(a) indicates a considerableoverlap (> 20%) between a substantial subsetof Soret to Qx NACRs and only a few normalmodes indexed 192 and 194 while the overlapwith the other modes is relatively small. Onthe other hand, for Qx to Qy transition, theoverlaps between the NACRs and only the nor-

6

Page 6 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

mal modes 180, 191, 192 and 193 are signifi-cant as shown in Fig. 4(b). This implies thatthe nuclear motions that modulate the elec-tronic couplings, and therefore contribute tothe ultrafast electronic energy transfer, involvedirections given by only a few normal modesof intermediate-to-high frequencies. Besides,these modes that overlap the most with theNACR vectors, are in agreement with our pre-viously identified active modes due to the sig-nificant accumulation of the excess vibrationalenergy during the relaxation process.

Figure 4: The distribution of the values of over-laps between specific normal modes and the setof NACRs corresponding to (a) Soret to Qx and(b) Qx to Qy transitions.

It is worth to mention that, according tothe surface hopping prescription65,105,106, ad-justments to the nuclear velocities are requiredin order to conserve total energy following hopsbetween electronic states. The direction inwhich the velocity Ri of each of the ith atom inthe molecular system is commonly rescaled, cor-responds to the direction of the NACR vector.Via an important test of reliability, we have en-sured that changing the rescaling prescriptiondoes not alter the results of the dynamics (see,Supplementary Information).

For a more quantitative characterization ofthe electronic and vibrational energy pathways,representative NACR vectors associated to eachI→J transition can be defined. We constructmatrices AIJ of dimension 3N × K, with Kbeing the number of trajectories with an effec-tive I → J hop. Matrices AIJ are built withcolumns representing the NACR vector at themoment of effective I → J transition in each ofthe K NA-ESMD trajectories. As these are notsquare matrices, in order to capture the repre-sentative NACR vectors of the whole set, Sin-gular Value Decomposition (SVD) of AIJ canbe performed. Essentially it means that AIJ iswritten as the product of a 3N × K column-orthogonal matrix U, a K×K diagonal matrixW with positive or zero elements (the singularvalues), and the transpose of a K ×K orthog-onal matrix V:

[AIJ

]=[UIJ

].[WIJ

].[V TIJ

](6)

Thus, the aijIJ elements of the matrix AIJ canbe expressed as the sum of products of columnsof UIJ and rows of VT

IJ , with the “weightingfactors” being the singular values wIJ

j

aijIJ =k∑

l=1

wIJl .u

ilIJ .N

jlIJ (7)

Subsequently, in this work, the ulIJ vectors,in the decreasing order of the values of wIJ

l , arereferred to as lth SVD vectors. The eigenval-ues wIJ

l indicate the correspondence betweenthe entire dataset and the individual SVD vec-tors. Here we find that the first two SVD vec-tors, shown in Fig. 5, with the largest eigen-values can sufficiently well describe more than80% of the NACRs from the ensemble. An in-tuitive understanding of the SVD vectors canbe obtained by plotting the overlap between anNACR and the first SVD vector 1 (SVD1) ver-sus the overlap between the same NACR andSVD2. Fig. 6 shows that for the Soret to Qx

transition the complete set of NACRs can bedistinctly categorized in two subsets - one (con-taining nearly 70% of the total NA-ESMD tra-jectories) which shows close resemblance with

7

Page 7 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

Figure 5: Visualization of the first two SVDvectors for the NACRs corresponding to (a)-(b) Soret to Qx and (c)-(d) Qx to Qy internalconversion.

the SVD1 and the other with a very high over-lap with the SVD2. This indicates that theSoret to Qx transition may take place alongtwo different pathways characterized by two dif-ferent NACRs and subsequently, the excitedstate dynamics along the two paths may ex-hibit important differences. Therefore, SVD1and SVD2 vectors can be considered as an aver-age representation of two distinguished originalensembles of NACR vectors, each of them iden-tified with a particular Soret to Qx transition.On the other hand, for the Qx to Qy transitionwhile the whole dataset can still be representedby the first two SVD vectors, a distinct cate-gorization does not emerge. The NACRs forthis process are uniform combinations of SVD1and SVD2. The subspace (plane) formed byboth SVD vectors contains the overall ensem-ble of NACR vectors associated to the Soretto Qx transition. According to the structuraldiagram of Chla that can be found elsewhere4,these vectors contain substantial contributionsfrom CaCm, CaCb and CbCb stretching vibra-tions4,5 that are associated to the IR spectra

band observed at 1597-1620 cm−1 2,4.

Figure 6: The overlap of each of the NACRsfrom the ensemble with the first SVD vectorversus the overlap with the second SVD vectorfor Soret to Qx (circles) and Qx to Qy (squares)transitions.

In order to ascertain whether the two dis-tinct NACRs lead to two different relaxationpathways, we segregated the excited state dy-namics trajectories based on the following cri-terion. Category 1 contains trajectories forwhich the NACRs exhibit more than 80% over-lap with SVD1 for a given transition whereasCategory 2 contains trajectories for which theNACRs show greater than 80% overlaps withSVD2. A third category relevant only to theQx to Qy transition accounts for those trajec-tories for which the NACRs are a mixture ofthe first two SVD vectors. The shaded regionsin Fig. 6 also provide a visual aid to this cate-gorizations. Fig. 7(a)-(b) shows the probabilitydistribution of the hopping time and the dif-ference in the energies at the time of hopping(∆E), respectively, for the two categories of theSoret to Qx internal conversion. The distribu-tion of ∆E can be seen to be relatively similarfor both the categories, with the average valueequal to 0.2554 eV for category 1 and 0.2343eV for category 2. In order to clarify whetherdegeneracy (with ∆E = 0) can be reached byany of these two pathways, we adopt an arbi-trary limit of ∆E < 0.1 eV to separate casesof strict degeneracy or their immediate regionsand cases with larger ∆E. With this separationin mind, we can say that less than only 16%

8

Page 8 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

of trajectories corresponding to each relaxationpathway pass through the degeneracy. The dis-tributions of the hopping times are, however,significantly different. This distinction becomesreadily apparent in the average hopping time,which is much greater for category 2 trajecto-ries (131.87 fs) than that for category 2 (90.93fs). Both of these properties are however muchmore homogeneous for the Qx to Qy relaxationacross all 3 categories as evident in the cumula-tive probability plots in Fig. 7(c)-(d). We havealso subjected all of these distributions to theKolmogorov-Smirnov test, which allows for anobjective assessment of the similarity betweenany two distributions and have found that itshows that only the distributions in Fig. 7(a)are distinct. We thus find that the Soret toQx transition occurs along two different path-ways characterized by distinct NACRs. How-ever, once the system reaches state Qx, the sub-sequent relaxation process has uniform charac-teristics. Furthermore, we have calculated thepopulation evolution on the different trajectorysubsets based on the two categories obtainedvia SVD analysis of only the Soret to Qx transi-tion. The resulting plots shown in Fig. 8 clearlypoint to the entirely different timescales of bothSoret to Qx and the Qx to Qy internal conver-sion processes in the two categories. The timeconstant extracted via fitting simulated datawith a sequential kinetic model is found to be89.0 fs for category 1 trajectories and 133.3 fsfor category 2 trajectories for the Soret to Qx

transition. For the Qx to Qy transition, thetime constants are 94.4 and 187.3 fs for cate-gory 1 and 2, respectively.Finally, we analyze the relationship between

the categorization of the NA-ESMD trajecto-ries for the two-stepped internal conversion pro-cess and the subset of the active normal modesidentified earlier. We thus calculate the over-laps between the selected normal modes andthe NACRs corresponding to each of the twotransitions and average them within differentensembles determined by the aforementionedSVD based categorization. Table I lists the ob-tained results for all those active modes whichappear in more than 90% of the trajectorieswithin each category. For easier comparison,

Figure 7: The probability distribution of the (a)hopping times and (b) ∆E at the time of hop-ping during Soret to Qx transition for differentcategories based on SVD analysis described intext. For Qx to Qy transition, the correspond-ing plots for (c) hopping times and (d) ∆E ascumulative distributions.

Figure 8: Excited state dynamics for Soret toQx internal conversion for category 1 trajecto-ries (solid curves) and category 2 trajectories(dashed curves)

9

Page 9 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

the most important modes, which show an over-lap close to or higher than 0.25, are highlightedin bold. For the Soret to Qx transition we canobserve that the active modes are distinctly dif-ferent for the two categories. Considering thatthe categorization is based on the SVD vectors,this implies that the Soret to Qx transition isassociated with two distinct NACRs that re-semble different sets of normal modes. Further-more, as Figure 4(a) shows, the normal modeswith prominent overlaps with the Soret to Qx

NACRs are indexed 192 and 194, which is con-sistent with the fact that nearly 70% of the tra-jectories belong to category 1. Modes 191, 192and 194 are indeed also those, along which thestrongest modulations of total energies are ob-served in figure 2(a). For the Qx to Qy tran-sition as well one can note the distinctness ofparticipating normal modes between category 1and 2. However, the presence of the third cate-gory lends a significantly more mixed characterthat is consistent with the results presented inFigure 4(b).

Table 1: Average values of overlaps be-tween various normal modes and theNACRs corresponding to Soret to Qx andQx to Qy transitions. Overlap values areshown for only those active modes whichappear in > 90% of the trajectories undereach category. The values in bold em-phasize the most important modes show-ing an overlap nearly equal to or greaterthan 0.25.

Mode Soret to Qx Qx to Qy

Cat 1 Cat 2 Cat 1 Cat 2 Cat 3180 0.269 – 0.436 – 0.289188 – 0.317 0.118 – –189 0.175 – 0.299 – 0.210191 0.300 0.215 0.391 – 0.272192 0.502 – 0.334 0.204 0.253193 – 0.238 – 0.335 0.301194 0.403 – – 0.284 0.213195 – 0.281 0.243 – –198 – – 0.172 0.162 –199 0.147 – 0.172 0.251 –200 0.188 0.272 – 0.424 0.276201 0.142 0.428 – 0.311 0.241

4 ConclusionsWe have identified the normal modes that par-ticipate actively in the intramolecular elec-tronic energy relaxation and redistribution dur-ing the internal conversion process of the pig-ment chlorophyll A after its photoexcitation tothe Soret band. The intramolecular vibrationalenergy flow, concomitant with the electronic en-ergy transfer, is monitored by evaluating thetemporal variance of the energy in each indi-vidual normal mode. Our analysis indicatesthat this process is not statistical, since theenergy deposited on each mode varies signifi-cantly. Only a small subset of intermediate-to-high-frequency normal modes, with the highestoverlap with the non-adiabatic coupling vectorsat the instant of effective hops, experience asubstantial increase in their vibrational energyduring the process. Therefore, velocity adjust-ments, performed according to the surface hop-ping prescription, in the direction of the nonadi-abatic coupling vectors can be well validated bythe confirmation that this direction is the mainvibrationally active direction during the inter-nal conversion process. Two distinct pathwaysfor nonradiative relaxation have been discov-ered, involving two different sets of active nor-mal modes and significantly different timescalesfor the S3→S2 transition. We consider that thispresent analysis can be used as a general ap-proach allowing identification of a reduced sub-set of specific vibrational coordinates associatedwith non-radiative relaxation pathways. It isour hope that results reported here shall in-spire further developments of hybrid quantum-classical molecular dynamics simulations in ex-tended conjugated molecules that deal explic-itly with the quantum mechanical treatment ofa small subset of normal modes and the con-ventional classical treatment of the remainingdegrees of freedom.

Supporting Information Available: Thedescription of a modified method for the rescal-ing of nuclear velocities is outlined in the Sup-porting Information as well as the results from asets of simulations with the modified algorithm.This material is available free of charge via the

10

Page 10 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

Internet at http://pubs.acs.org/.

Acknowledgement P.M.S and Y.Z. ac-knowledge support from Singapore NationalResearch Foundation through the Competi-tive Research Programme (CRP) under ProjectNo. NRF-CRP5-2009-04. S.F.A is supportedby CONICET, UNQ, ANPCyT (PICT-2014-2662). S.T. acknowledge support from LosAlamos National Laboratory (LANL) DirectedResearch and Development Funds. Los AlamosNational Laboratory is operated by Los AlamosNational Security, LLC, for the National Nu-clear Security Administration of the U.S. De-partment of Energy under contract DEAC52-06NA25396. We acknowledge support of theCenter for Integrated Nanotechnology (CINT),a U.S. Department of Energy, Office of BasicEnergy Sciences user facility.

References

(1) Blankenship, R. E. Molecular Mecha-nisms of Photosynthesis; Blackwell Pub-lishing: Williston, VT, USA, 2002.

(2) Katz, J. J.; Dougherty, L.; Boucher, L.In The Chlorophylls; Vernon, L. P., Seely,G. R., Eds.; Academic Press: New York,1966; pp 185-251.

(3) Wang, R.; Parameswaran, S.; Hastings,G. Density functional theory based cal-culations of the vibrational properties ofchlorophyll-a. Vib. Spectrosc. 2007, 44,357-368.

(4) Thomas, L. L.; Kim, J. H.; Cotton, T.M. Comparative study of resonance ram-man and surface-enhanced resonance ra-man Chlorophyll a spectra using Soretand red excitation. J. Am. Chem. Soc.1990, 112, 9378-9386.

(5) Boldt, N. J.; Donohoe, R. J.; Birge,R. R.; Bocian, D. F. Chlorophyll modelcompounds: Effects of ow symmetry onthe Resonance Raman spectra and nor-mal mode descriptions of Nickel(II) dihy-droporphyrins. J. Am. Chem. Soc. 1987,109, 2284-2298.

(6) Croce, R.; van Amerongen, H. Naturalstrategies for photosynthetic light har-vesting. Nat. Chem. Bio. 2014, 10, 492-501.

(7) Scholes, G. D.; Fleming, G. R.; Olaya-Castro, A.; van Grondelle, R. Lessonsfrom nature about solar light harvesting.Nat. Chem. 2011, 3, 763-774.

(8) Niyogi, K. K. Photoprotection revis-ited: Genetic and molecular approaches.Annu. Rev. Plant Physiol. Plant Mol.Biol. 1999, 50, 333-359.

(9) Ruban, A. V.; Johnson, M. P.; Duffy,C. D. P. The photoprotective molecularswitch in the photosystem II antenna.Biochim. Biophys. Acta, Bioenerg. 2012,1817, 167-181.

(10) Bricker, W. P.; Shenai, P. M.; Ghosh,A.; Liu, Z.; Enriquez, M. G. M.; Lam-brev, P. H.; Tan, H. S.; Lo, C. S.; Tre-tiak, S.; Fernandez-Alberti, S.; et al.Non-radiative relaxation of photoexcitedchlorophylls: theoretical and experimen-tal study. Sci. Rep. 2015, 5, 13625.

(11) Shi, Y.; Liu, J. Y.; Han, K. L. Investiga-tion of the internal conversion time of thechlorophyll a from S3, S2 to S1. Chem.Phys. Lett. 2005, 410, 260-263.

(12) Dong, L. Q.; Niu, K.; Cong, S. L.Theoretical study of vibrational relax-ation and internal conversion dynamicsof chlorophyll-a in ethyl acetate solventin femtosecond laser fields. Chem. Phys.Lett. 2006, 432, 286-290.

(13) Dong, L. Q.; Niu, K.; Cong, S. L. The-oretical analysis of internal conversionpathways and vibrational relaxation pro-cess of chlorophyll-a in ethyl ether sol-vent. Chem. Phys. Lett. 2007, 440, 150-154.

(14) Kosumi, D.; Maruta, S.; Fujii, R.;Kanemoto, K.; Sugisaki, M.; Hashimoto,H. Ultrafast excited state dynamics of

11

Page 11 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

monomeric bacteriochlorophyll a. Phys.Status Solidi C 2011, 8, 92-95.

(15) Kosumi, D.; Nakagawa, N.; Sakai, S.;Nagaoka, Y.; Maruta, S.; Sugisaki, M.;Dewa, T.; Nango, M.; Hashimoto, H. Ul-trafast intramolecular relaxation dynam-ics of Mg- and Zn-bacteriochlorophyll a.J. Chem. Phys. 2013, 139, 034311.

(16) Musewald, C.; Hartwich, G.; Lossau, H.;Gilch, P.; Pöllinger-Dammer, F.; Scheer,H.; Michel-Beyerle, M. E. Ultrafast pho-tophysics and photochemistry of [Ni]-bacteriochlorophyll a. J. Phys. Chem. B1999, 103, 7055-7060.

(17) Lambrev, P. H.; Miloslavina, Y.; vanStokkum, I. H. M.; Susz, A.; Tworzydlo,J.; Fiedor, J.; Huhn, G.; van Grondelle,R.; Garab, G.; Fiedor, L. J. Phys. Chem.B 2013, 117, 11260-11271.

(18) Macpherson, A. N.; Gillbro, T. Solventdependence of the ultrafast S2-S1 inter-nal conversion rate of beta-carotene. J.Phys. Chem. A 1998, 102, 5049-5058.

(19) Cerullo, G.; Polli, D.; Lanzani, G.; DeSilvestri, S.; Hashimoto, H.; Cogdell,R. J. Photosynthetic light harvesting bycarotenoids: Detection of an interme-diate excited state. Science 2002, 298,2395-2398.

(20) Polli, D.; Cerullo, G.; Lanzani, G.; DeSilvestri, S.; Yanagi, K.; Hashimoto,H.; Cogdell, R. J. Conjugation lengthdependence of internal conversion incarotenoids: Role of the intermediatestate. Phys. Rev. Lett. 2004, 93, 163202.

(21) Kosumi, D.; Komukai, M.; Hashimoto,H.; Yoshizawa, M. Ultrafast dynamics ofall-trans-beta-carotene explored by reso-nant and nonresonant photoexcitations.Phys. Rev. Lett. 2005, 95, 213601.

(22) Lustres, J. L. P.; Dobryakov, A. L.;Holzwarth, A.; Veiga, M. S2 to S1 inter-nal conversion in beta-carotene: Strong

vibronic coupling from amplitude oscil-lations of transient absorption bands.Angew. Chem., Int. Ed. 2007, 46, 3758-3761.

(23) Pecourt, J. M. L.; Peon, J.; Kohler, B.;DNA excited-state dynamics: Ultrafastinternal conversion and vibrational cool-ing in a series of nucleosides. J. Am.Chem. Soc. 2001, 123, 10370-10378.

(24) Pecourt, J. M. L.; Peon, J.; Kohler, B.Ultrafast internal conversion of electron-ically excited RNA and DNA nucleosidesin water. J. Am. Chem. Soc. 2000, 122,9348-9349.

(25) Fidder, H.; Rini, M.; Nibbering, E. T. J.The role of large conformational changesin efficient ultrafast internal conversion:Deviations from the energy gap law. J.Am. Chem. Soc. 2004, 126, 3789-3794.

(26) Yu, X.; Leitner, D. M. Vibrational energytransfer and heat conduction in a protein.J. Phys. Chem. B 2003, 107, 1698-1707.

(27) Fujusaki, H.; Straub, J. E. Vibrationalenergy relaxation in proteins. Proc. Natl.Acad. Sci. USA 2005, 102, 6726-6731.

(28) Stock, G. Classical simulation of quan-tum energy flow in biomolecules. Phys.Rev. Lett. 2009, 102, 118301.

(29) Leitner, D. M. Energy flow in proteins.Ann. Rev. Phys. Chem. 2008, 59, 233-259.

(30) Radloff, W.; Stert, V.; Freudenberg, Th.;Hertel, I. V.; Jouvet, C.; Dedonder-Lardeux, C.; Solgadi, D. Internal conver-sion in highly excited benzene and ben-zene dimer: femtosecond time-resolvedphotoelectron spectroscopy. Chem. Phys.Lett. 1997, 281, 20-26.

(31) Mataga, N.; Shibata, Y.; Chosrowjan,H.; Yoshida, N.; Osuka, A. Internalconversion and vibronic relaxation from

12

Page 12 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

higher excited electronic state of por-phyrins: Femtosecond fluorescence dy-namics studies. J. Phys. Chem. B 2000,104, 4001-4004.

(32) Lasorne, B.; Sicilia, F.; Bearpark, M.J.; Robb, M. A.; Worth, G. A.; Blan-cafort, L. Automatic generation of activecoordinates for quantum dynamics calcu-lations: Application to the dynamics ofbenzene photochemistry. J. Chem. Phys.2008, 128, 124307.

(33) Soler, M. A.; Roitberg, A. E.; Nelson, T.;Tretiak, S.; Fernandez-Alberti, S. Anal-ysis of State-Specific Vibrations Coupledto the Unidirectional Energy Transfer inConjugated Dendrimers. J. Phys. Chem.A, 2012, 116, 9802-9810.

(34) Galindo, J.; Fernandez-Alberti, S.; Roit-berg, A. Electronic Excited State Spe-cific IR Spectra for Phenylene Ethyny-lene Dendrimer Building Blocks. J. Phys.Chem. C 2013, 117, 26517-26528.

(35) Saab, M.; Doriol, L. J.; Lasorne, B.;Guarin, S.; Gatti, F. A quantum dynam-ics study of the benzopyran ring open-ing guided by laser pulses, Chem. Phys.2014, 442, 93-102.

(36) Lasorne, B.; Worth, G. A.; Robb, M.A. Excited-state dynamics.WIREs Com-put. Mol. Sci., 2011, 1, 460-475.

(37) Chen, Y.; Palmer, P. M.; Topp, M. R.Infrared Spectroscopy of Jetcooled, Elec-tronically Excited Clusters of Coumarin151: Excited-state Interactions and Con-formational Relaxation. Int. J. MassSpectrom. 2002, 220, 231-251.

(38) Weiler, M.; Bartl, K.; Gerhards, M. In-frared/ultraviolet Quadruple ResonanceSpectroscopy to Investigate Structures ofElectronically Excited States. J. Chem.Phys. 2012, 136, 114202.

(39) Seurre, N.; Le Barbu-Debus, K.; Lah-mani, F.; Zehnacker-Rentien, A.; Se-pioa, J. Electronic and Vibrational Spec-

troscopy of Jet-cooled Mcyanophenoland Its Dimer: Laser-induced Fluores-cence and Fluorescence-dip IR Spectra inthe S0 and S1 States. Chem. Phys. 2003,295, 21-33.

(40) Zhang, Y.; Kubicki, J.; Platz, M. S. Ul-trafast UV-Visible and Infrared Spectro-scopic Observation of a Singlet Vinylcar-bene and the Intramolecular Cycloprope-nation Reaction. J. Am. Chem. Soc.2009, 131, 13602-13603.

(41) Zhang, Y.; Burdzinski, G.; Kubicki, J.;Vyas, S.; Hadad, C. M.; Sliwa, M.;Poizat, O.; Buntinx, G.; Platz, M. S.Study of the S1 Excited State of para-Methoxy-3-phenyl-3-methyl Diazirine byUltrafast Time Resolved UV-Vis andIR Spectroscopies and Theory. J. Am.Chem. Soc. 2009, 131, 13784-13790.

(42) Vyas, S.; Kubicki, J.; Luk, H. L.; Zhang,Y.; Gritsan, N. P.; Hadad, C. M.;Platz, M. S. An Ultrafast Time-resolvedInfrared and UV-vis Spectroscopic andComputational Study of the Photochem-istry of Acyl Azides. J. Phys. Org. Chem.2012, 25, 693-703.

(43) Dougherty, T. P.; Heilweil, E. J. Ultra-fast Transient Infrared Absorption Stud-ies of M(CO)6 (M = Cr, Mo or W) Pho-toproducts in N-hexane Solution. Chem.Phys. Lett. 1994, 227, 19-25.

(44) Noid, D. W.; Koszykowski, M. L.; Mar-cus, R. A. Quasiperiodic and StochasticBehavior in Molecules. Ann. Rev. Phys.Chem. 1981, 32, 267-309.

(45) Schulz, I. P. A.; Sudbo, S.; Krajnovich,D. J.; Kwok, H. S.; Shen, Y. R.; Lee,Y. T. Multiphoton dissociation of poly-atomic molecules. Ann. Rev. Phys. Chem1979, 30, 379-409.

(46) McDonald, J. D. Creation and dis-posal of vibrational energy in polyatomicmolecules. Ann. Rev. Phys. Chem 1979,30, 29-50.

13

Page 13 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

(47) Hill, J. R.; Ziegler, C. J.; Suslick, K. S.;Dlott, D. D.; Rella, C. W.; Fayer, M.D. Tuning the vibrational relaxation ofco bound to heme and metalloporphyrincomplexes. J. Phys. Chem. 1996, 100,18023-18032.

(48) Hamm, P.; Lim, M. H.; Hochstrasser, R.M. Structure of the amide i band of pep-tides measured by femtosecond nonlin-earinfrared spectroscopy. J. Phys. Chem.B 1998, 102, 6123-6138.

(49) Peterson, K. A.; Rella, C. W.; Engholm,J. R.; Schwettman, H. A. Ultrafast vibra-tional dynamics of the myoglobin amide iband. J. Phys. Chem. B 1999, 103, 557-561.

(50) Dlott, D. D. Vibrational energy redistri-bution in polyatomic liquids: 3d infrared-raman spectroscopy. Chem. Phys. 2001,266, 149-166.

(51) Iwaki, L.; Dlott, D. D. Vibrational en-ergy transfer in condensed phases. In En-cyclopedia of chemical physics and phys-ical chemistry; Moore, J. H., Spencer, N.D., Eds.; IOP Publishing Ltd: Bristol,Vol. 3, pp 2717-2736, 2001.

(52) Fayer, M. D. Ultrafast infrared and Ra-man spectroscopy; CRC Press: MarcelDekker, Inc., New York, USA, 2001.

(53) Shigeto, S.; Dlott, D. D. Vibrational re-laxation of an amino acid in aqueoussolution. Chem. Phys. Lett. 2007, 447,134-139.

(54) Wang, Z. H.; Carter, J. A.; Lagutchev,A.; Koh, Y. K.; Seong, N. H.; Cahill, D.G.; Dlott, D. D. Ultrafast flash thermalconductance of molecular chains. Science2007, 317, 787-790.

(55) Shigeto, S.; Pang, Y.; Fang, Y.; Dlott, D.D. Vibrational relaxation of normal anddeuterated liquid nitromethane. J. Phys.Chem. B 2008, 112, 232-241.

(56) Fang, Y.; Shigeto, S.; Seong, N.; Dlott,D. D. Vibrational energy dynamics ofglycine, n-methylacetamide, and ben-zoate anion in aqueous (D2O) solution.J. Phys. Chem. A 2009, 113, 75-84.

(57) Nguyen, P. H.; Stock, G. Nonequilibriummolecular-dynamics study of the vibra-tional energy relaxation of peptides inwater. J. Chem. Phys. 2003, 119, 11350-11358.

(58) Brooks, C. L.; Karplus, M.; Pettitt, B.M. Proteins: A Theoretical Perspectiveof Dynamics, Structure, and Thermody-namics. Adv. Chem. Phys. 1988, 71, 1-259.

(59) Nishikawa, T.; Go, N. Normal modesof vibration in bovine pancreatic trypsininhibitor and its mechanical property.Struct. Funct. Genet. 1987, 2, 308-329.

(60) Brooks, B.; Karplus, M. Harmonic dy-namics of proteins: normal modes andfluctuations in bovine pancreatic trypsininhibitor. Proc. Natl. Acad. Sci. U. S. A1983, 80, 6571-6575.

(61) McCammon, J. A.; Harvey, S. C. Dy-namics of Proteins and Nucleic Acids;Cambridge University Press: Cambridge,U.K., 1987.

(62) Parameswaran, S.; Wang, R.; Hastings,G. Calculation of the Vibrational Proper-ties of Chlorophyll a in Solution. J. Phys.Chem. B 2008, 112, 14056-14062.

(63) Nelson, T.; Fernandez-Alberti, S.;Chernyak, V.; Roitberg, A. E.; Tretiak,S. Nonadiabatic excited-state moleculardynamics modeling of photoinduceddynamics in conjugated molecules. J.Phys. Chem. B 2011, 115, 5402-5414.

(64) Nelson, T.; Fernandez-Alberti, S.;Roitberg, A.; Tretiak, S. NonadiabaticExcited-State Molecular Dynamics:Modeling Photophysics in OrganicConjugated Materials. Acc. Chem. Res.2014, 47, 1155-1164.

14

Page 14 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

(65) Tully, J. C. Molecular-dynamics withelectronic-transitions. J. Chem. Phys.1990, 93, 1061-1071.

(66) Hammes-Schiffer, S.; Tully, J. C. Proton-transfer in solution - molecular-dynamicswith quantum transitions. J. Chem.Phys. 1994, 101, 4657-4667.

(67) Tretiak, S.; Mukamel, S. Density matrixanalysis and simulation of electronic ex-citations in conjugated and aggregatedmolecules. Chem. Rev. 2002, 102, 3171-3212.

(68) Chernyak, V.; Schulz, M. F.; Mukamel,S.; Tretiak, S.; Tsiper, E. V. Krylov-space algorithms for time-dependenthartree-fock and density functional com-putations. J. Chem. Phys. 2000, 113, 36-43.

(69) Tretiak, S.; Isborn, C. M.; Niklasson, A.M. N.; Challacombe, M. Representationindependent algorithms for molecular re-sponse calculations in time-dependentself-consistent field theories. J. Chem.Phys. 2009, 130, 054111.

(70) Furche, F.; Ahlrichs, R. Adiabatic time-dependent density functional methodsfor excited state properties. J. Chem.Phys. 2002, 117, 7433-7447.

(71) Tretiak, S.; Chernyak, V. Resonantnonlinear polarizabilities in the time-dependent density functional theory. J.Chem. Phys. 2003, 119, 8809-8823.

(72) Tommasini, M.; Chernyak, V.; Mukamel,S. Electronic density-matrix algorithmfor nonadiabatic couplings in moleculardynamics simulations. Int. J. QuantumChem. 2001, 85, 225-238.

(73) Chernyak, V.; Mukamel, S. Density-matrix representation of nonadiabaticcouplings in time-dependent densityfunctional (TDDFT) theories. J. Chem.Phys. 2000, 112, 3572-3579.

(74) Send, R.; Furche, F. First-order nona-diabatic couplings from time-dependenthybrid density functional response the-ory: Consistent formalism, implementa-tion, and performance. J. Chem. Phys.2010, 132, 044107.

(75) Mukamel, S. Photochemistry - trees totrap photons. Nature 1997, 388, 425-427.

(76) Mukamel, S.; Tretiak, S.; Wagersreiter,T.; Chernyak, V. Electronic coherenceand collective optical excitations of con-jugated molecules. Science 1997, 277,781-787.

(77) Tretiak, S.; Chernyak, V.; Mukamel,S. Recursive density-matrix-spectral-moment algorithm for molecular non-linear polarizabilities. J. Chem. Phys.1996, 105, 8914-8928.

(78) Tretiak, S.; Zhang, W. M.; Chernyak,V.; Mukamel, S. Excitonic couplings andelectronic coherence in bridged naphtha-lene dimers. Proc. Natl. Acad. Sci. U. S.A. 1999, 96, 13003-13008.

(79) Dewar, M. J. S.; Zoebisch, E. G.; Healy,E. F.; Stewart, J. J. P. AM1: A new gen-eral purpose quantum mechanical model.J. Am. Chem. Soc. 1985, 107, 3902-3909.

(80) Stewart, J. J. P. Optimization of pa-rameters for semiempirical methods IV:extension of MNDO, AM1, and PM3to more main group elements. J. Mol.Model. 2004, 10, 155-164.

(81) Shavitt, I. Modern Theoretical Chem-istry; Plenum Press: New York, NY,USA, 1976.

(82) Szalay, P. G.; Müller, T.; Gidofalvi, G.;Lischka, H.; Shepard, R. Multiconfigu-ration self-consistent field and multiref-erence configuration interaction methodsand applications. Chem. Rev. 2012, 112,108-181.

15

Page 15 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

(83) Reynolds, C. H. An AM1 theoreticalstudy of the structure and electronicproperties of porphyrin. J. Org. Chem.1988, 53, 6061-6064.

(84) Zandler, M. E.; D’Souza, F. Electronicand structural properties of the metal-loporphyrin structural isomers: Semiem-pirical AM1 and PM3 calculations. J.Mol. Struct.: THEOCHEM 1997, 401,301-314.

(85) Katagi, T. Semi-empirical am1 and pm3calculations of five- and six-coordinateoxo iron (iv) porphyrin complexes. J.Mol. Struct.: THEOCHEM 2005, 728,49-56.

(86) Stewart, J. J. P. Optimization of pa-rameters for semiempirical methods I.method. J. Comp. Chem. 1989, 10, 209-220.

(87) Stewart, J. J. P. Optimization of param-eters for semiempirical methods II. appli-cations. J. Comp. Chem. 1989, 10, 221-264.

(88) Stewart, J. J. P. Optimization of param-eters for semiempirical methods III. ex-tension of PM3 to Be, Mg, Zn, Ga, Ge,As, Se, Cd, In, Sn, Sb, Te, Hg, Tl, Pb,and Bi. J. Comp. Chem. 1991, 12, 320-341.

(89) Linnanto, J.; Korppi-Tommola, J. Spec-troscopic properties of Mg-chlorin, Mg-porphyrin and chlorophylls a, b, c1, c2,c3 and d studied by semi-empirical andab initio MO/CI methods. Phys. Chem.Chem. Phys. 2000, 2, 4962-4970.

(90) Linnanto, J.; Korppi-Tommola, J. Spec-troscopic properties of mg-chlorin, mg-bacterochlorin and bacteriochlorophyllsa, b, c, d, e, f, g, and h studied by semiem-pirical and ab initio MO/CI methods. J.Phys. Chem. A 2001, 105, 3855-3866.

(91) Linnanto, J.; Korppi-Tommola, J. Quan-tum chemical simulation of excited statesof chlorophylls, bacteriochlorophylls and

their complexes. Phys. Chem. Chem.Phys. 2006, 8, 663-687.

(92) Hutter, M. C.; Reimers, J. R.; Hush,N. S. Modeling the bacterial photosyn-thetic reaction center. 1. magnesium pa-rameters for semiempirical AM1 methoddeveloped using a genetic algorithm. J.Phys. Chem. B 1998, 102, 8080-8090.

(93) Verlet, L. Computer "experiments" onclassical fluids. I. thermodynamical prop-erties of Lennard-Jones molecules. Phys.Rev. 1967, 159, 98-103.

(94) Allen, M. P.; Tildesley, D. J. ComputerSimulation of Liquids; Clarendon Press:Walton St. Oxford, United Kingdom,1987.

(95) Paterlini, M. G.; Ferguson, D. M. Con-stant temperature simulations using theLangevin equation with velocity Verletintegration. Chem. Phys. 1998, 236, 243-252.

(96) Hull, E.; Enright, W. H.; Jackson, K.R. User’s guide for DVERK - A subrou-tine for solving non-stiff ODEs. Techni-cal Report 100, Department of ComputerScience, University of Toronto, Canada,1976.

(97) Nelson, T.; Fernandez-Alberti, S.;Chernyak, V.; Roitberg, A. E.; Tretiak,S. Nonadiabatic excited-state moleculardynamics: Numerical tests of conver-gence and parameters. J. Chem. Phys.2012, 136, 054108.

(98) Wu, C.; Malinin, S. V.; Tretiak, S.;Chernyak, V. Y. Multiscale Modelingof Electronic Excitations in BranchedConjugated Molecules Using an ExcitonScattering Approach. Phys. Rev. Lett.2008, 100, 057405.

(99) Ditchfield, R.; Hehre, W. J.; Pople, J. A.Self-consistent molecular-orbital meth-ods. IX. An extended Gaussian-type ba-sis for molecular-orbital studies of or-

16

Page 16 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

ganic molecules. J. Chem. Phys. 1971,54, 724-728.

(100) Hehre, W. J.; Ditchfield, R.; Pople, J. A.;Self-consistent molecular orbital meth-ods. XII. Further extensions of Gaussian-type basis sets for use in molecular or-bital studies of organic molecules. J.Chem. Phys. 1972, 56, 2257-2261.

(101) Hariharan, P. C.; Pople, J. A. Theinfluence of polarization functions onmolecular orbital hydrogenation ener-gies. Theor. Chem. Acc. 1973, 28, 213-222.

(102) Becke, A. D. Density-functional thermo-chemistry. III. the role of exact exchange.J. Chem. Phys. 1993, 98, 5648-5652.

(103) Frisch, M. J.; Trucks, G. W.; Schlegel, H.B.; Scuseria, G. E.; Robb, M. A.; Cheese-man, J. R.; Scalmani, G.; Barone, V.;Mennucci, B.; Petersson, G. A., et al.Gaussian 09, Revision A.02; Gaussian,Inc.: Wallingford, CT, 2009.

(104) Fernandez-Alberti, S.; Roitberg, A. E.;Kleiman, V. D.; Nelson, T.; Tretiak, S.Shishiodoshi unidirectional energy trans-fer mechanism in phenylene ethynylenedendrimers. J. Chem. Phys. 2012, 137,014512.

(105) Tully, J. C. Nonadiabatic molecular dy-namics. Int. J. Quantum Chem. Symp.1991, 25, 299-309.

(106) Hack, M. D.; Jasper, A. W.; Volobuev,Y. L.; Schwenke, D. W.; Truhlar, D. G.Quantum Mechanical and QuasiclassicalTrajectory Surface Hopping Studies ofthe Electronically Nonadiabatic Predis-sociation of the Ã State of NaH2. J. Phys.Chem. 1999, 103, 6309-6326.

17

Page 17 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

Graphical TOC Entry

18

Page 18 of 18



123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

internal conversion and vibrational energy redistribution ... · internal conversion and...

Documents