Efficient partitioning technique for computing the dynamics of intramolecular processes: Radiationless transitions in pyrazine

THE JOURNAL OF CHEMICAL PHYSICS 124, 184107 �2006�

Efficient partitioning technique for computing the dynamicsof intramolecular processes: Radiationless transitions in pyrazine

P. S. ChristopherChemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto,Ontario M5S 3H6, Canadaand Center for Quantum Information and Quantum Control, University of Toronto, Toronto,Ontario M5S 3H6, Canada

Moshe ShapiroDepartment of Chemistry, University of British Columbia, Vancouver, British, Columbia V6T 1Z1, Canadaand Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, Israel 76100

Paul Brumera�

Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto,Ontario M5S 3H6, Canadaand Center for Quantum Information and Quantum Control, University of Toronto, Toronto,Ontario M5S 3H6, Canada

�Received 28 February 2006; accepted 23 March 2006; published online 12 May 2006�

An efficient QP partitioning algorithm to compute the eigenvalues, eigenvectors, and the dynamicsof large molecular systems of a particular type is presented. Compared to straightforwarddiagonalization, the algorithm displays favorable scaling ��NT

2� as a function of NT, the size of theHamiltonian matrix. In addition, the algorithm is trivially parallelizable, necessitating no“cross-talk” between nodes, thus enjoying the full linear speedup of parallelization. Moreover, themethod requires very modest storage space, even for extremely large matrices. The method has alsobeen enhanced through the development of a coarse-grained approximation, enabling an increase ofthe basis set size to unprecedented levels �108–1010 in the current application�. The QP algorithmis applied to the dynamics of electronic internal conversion in a 24 vibrational-mode model ofpyrazine. A performance comparison with other dynamical methods is presented, along with resultsfor the decay dynamics of pyrazine and a discussion of resonance line shapes. © 2006 AmericanInstitute of Physics. �DOI: 10.1063/1.2196888�

I. INTRODUCTION

The Löwdin-Feshbach partitioning technique1–7 has along and impressive history in chemical physics. It allowsone to focus on a specific system while taking into accountits coupling to the surroundings through effective matrix el-ements. In the past the partitioning technique was mainlyused as a starting point for further approximations—throughsome perturbation strategy and/or through an estimation ofsystem-environment effects. In contrast, this paper presents anumerical method based on the partitioning technique, whichwe term the QP algorithm, which allows the accurate com-putation of bound state Hamiltonian eigenvalues and eigen-vectors for large molecular systems. Given the eigenvaluesand eigenvectors, the computation of system dynamics forlarge numbers of different initial states becomes a straight-forward task.

For the class of problems considered, the algorithm de-scribed below has a number of computationally impressivecharacteristics. Specifically, it has excellent scaling proper-ties as compared to conventional diagonalization techniques,it admits a quasicontinuum extension that allows for ex-tremely large basis expansions, it is trivially parallelizable,

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and it requires very modest storage space, even for extremelylarge matrices. As a result, the QP algorithm provides a use-ful tool to tackle the full quantum dynamics of large molecu-lar systems. As an example, we apply the method to S2 to S1

internal conversion �IC� of full 24 vibrational-mode pyra-zine, using the potential of Raab et al.8

The QP algorithm, by providing system eigenstates andeigenvalues, allows for studies of wave packet dynamics of awide range of initial preparations with little additional com-putational overhead. Moreover, by subdividing the systeminto a relevant component �“the Q space”� and a less relevantcomponent �“the P space”� the method allows for a transpar-ent physical interpretation via the direct analysis of the effectof the P space on the Q space, most often seen as resonancebroadening of the Q-space states.

In applying this method we have focused on pyrazine,whose photophysics has been a subject of long standing in-terest. Of particular interest is the ultrafast dynamics follow-ing the optical excitation of pyrazine from the ground stateS0 to the second excited singlet state S2. The process of ICfrom S2 to the first excited singlet state S1 at time scalesestimated experimentally to be 20 fs �Ref. 9� then follows. Agreat deal of theoretical work has gone into understandingthese experimental findings.8,10–12

Our interest in pyrazine also stems from past studies in

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184107-2 Christopher, Shapiro, and Brumer J. Chem. Phys. 124, 184107 �2006�

which we computationally demonstrated coherent control13

of the IC process in a four-mode model.14 In particular, wedemonstrated that the extent of phase control over the flowout of the S2 manifold depends on the participation of over-lapping resonances. At that time our studies were limited tothe four-mode model because of the enormity of the densityof states in the S1 manifold ��106 cm−1�, prohibiting thestraightforward diagonalization of the full problem by con-ventional techniques. In contrast, the QP algorithm devel-oped here allows for the treatment of the �1010 states asso-ciated with the full 24-mode problem. Our studies ofcoherent control in the 24-mode problem will be describedelsewhere.15

This paper is organized as follows. Section II describesthe theory and methodology of the QP algorithm. Specifi-cally, Sec. II A summarizes the equations of the Löwdin-Feshbach partitioning technique, Sec. II B describes the QPalgorithm and emphasizes the requirement that the P spacebe readily diagonalizable, Sec. II C outlines the coarse-grained approximation, and Sec. II D describes the pyrazinemodel used in this paper. Section III contains the results ofour investigations on pyrazine. In particular, we present theautocorrelation function and diabatic populations as a func-tion of time and compare these results with results usingwave packet propagation methods. The performance of theQP algorithm is shown to be far superior to other methods.The resonance structure of pyrazine is discussed at the end ofthis section. Finally, Sec. IV concludes the paper and in-cludes suggestions for future work.

II. THEORY, ALGORITHM, AND COMPUTATIONALDETAILS

In order to simplify the description of the QP algorithm,we present it in the framework of the IC dynamics of pyra-zine. Generalization to other molecules, and other physicalprocesses, should be obvious.

Pyrazine C4N2H6 has a benzenelike structure, with thering composed of four carbons and two nitrogens, the latterbeing opposite to one another on the ring. The molecule has24 vibrational normal modes, each classified into one ofeight different irreducible representations �irrep� of the D2h

point group. Excitation from S0 preferentially populates S2

vibrational states, rather than S1 states, because the electronictransition dipole moment of the former is much larger thanthe latter. As a consequence, one can safely approximate thestate of the system after excitation as a superposition of vi-brational states belonging to the S2 manifold. Rotational lev-els are neglected in this discussion. As mentioned above, dueto the conical intersection of the S1 and S2 surfaces, popula-tion in vibrational states prepared on S2 undergo a�10–40 fs� rapid transition to the S1 states.

A. The Löwdin-Feshbach partitioning

Let H be the full Hamiltonian of the system, and ��
and �E� be its eigenstates and eigenenergies,

H�� = E�� . �1�

The full Hilbert space of the system can be divided into twosubspaces using projection operators Q and P, with P+Q=1. In the pyrazine case the Q space contains vibrationalstates of the S2 electronic manifold, and the P space com-prises the vibrational states of the S1 electronic manifold.The states �� denote the basis for the Q space with projec-tor Q given by

Q = �

�� . �2�

Similarly the �� states span the P space with projector Pgiven by

P = �

�� . �3�

Consider the nuclear Hamiltonian

H = H0 + VIC, �4�

where H0 is the sum of Hamiltonians on the two electronicsurfaces and VIC couples basis states between the two sur-faces. With this definition it is obvious that QHQ=QH0Q,PHP= PH0P, and QHP=QVICP which are described, re-spectively, as the Hamiltonian on S2, the Hamiltonian on S1,and the IC coupling. The bases �� and �� are chosen todiagonalize the noninteracting parts of the Hamiltonian:

QHQ�� = �� , �5�

PHP�� = �� . �6�

Below, we denote the dimensions of the Q space and P spaceas NQ and NP, respectively.

As summarized in Appendix A, application of the Q andP projection operators to Eq. �1� yields an eigenvalue equa-tion for the states Q��, i.e., the projection of �� onto the Qspace. Specifically,

�E� − H�E��Q�� = 0, �7�

where, for the bound state case,

H��E�� = �� + �

V��V��*

E� − ��

, �8�

V�� VIC�� . �9�

From a computational standpoint, the main advantage ofEq. �7� over Eq. �1� is that the eigenvalue problem in Eq. �7�is of dimension NQ, whereas the full eigenvalue equation inEq. �1� is of dimension of NT=NP+NQ. Since the best gen-eral diagonalization methods scale as the cube of the matrixdimension: this scaling being estimated16 as 25N3, this ap-proach provides a significant advantage.18 Hence, if NQ

�NP, as in the problems discuss below, Eq. �7� shows sig-nificant savings over straightforward diagonalization.

The price one pays in dealing with Eq. �7� is the need to

diagonalize H which is a function of the �as yet� unknown E�

eigenvalues. Indeed, this is one of the reasons why the Fes-
hbach formalism has been mainly applied to continuum

184107-3 Partitioning for intramolecular dynamics J. Chem. Phys. 124, 184107 �2006�

problems, where every energy is an eigenenergy. An algo-rithm for bound state problems based on Eq. �7� must be ableto solve these implicit eigenvalue equations efficiently.

B. The QP algorithm

One could imagine solving Eq. �7� by using self-consistent iteration. That is, one could �a� use a trial energy

E��i�, �b� diagonalize H�E�

�i��, �c� choose one of its eigenvaluesto be E�

�i+1�, and �d� iterate this procedure until the systemconverged onto a stationary point. We have already success-fully applied this approach to intramolecular vibrational re-distribution in carbonyl sulfide.19 Empirically, however, thisprocedure was found to be fairly unstable for larger prob-lems. It is difficult to obtain most of the eigenvalues in thismanner, and in addition, the number of iterations per eigen-value seemed to increase with the dimension of the P space.

Consider, instead, performing the above search on a fullvector of eigenvalues. In such a method one first selects an

energy � and then diagonalizes the entire H�� matrix toobtain a vector of eigenenergies, ��. One then computes thedifference between the input energy and the output energies.Symbolically this is represented as

� Þ H�� Þ �� Þ � �� , �10�

where

� �� = �1� − ��. �11�

Here, any � that gives a zero in one of the � componentscorresponds to an eigenvalue of the full Hamiltonian. Pro-vided that � is reasonably well behaved as a function ofenergy, finding all the � for which k��=0 amounts to find-ing all the eigenvalues of the full Hamiltonian of Eq. �1�.Further, it is relatively easy to search for solutions of k��=0: we lays down a grid of energies �i, search for pairs ofgrid points that bracket a solution, and use bisection to refinethe solution.

One problem that arises with this procedure is that theorder of energies in �� can change from one energy grid pointto the next upon iteration. Since the algorithm compares thevalues of � element by element, if the order of componentsin �� changes between grid points, this could result in miss-ing some eigenvalues.20 In order to rectify this situation, we

use the eigenvectors of H��i−1� to order the energies in thevector �� i

f. This procedure is symbolized by the expression

�i,= i−1 Þ H��i� Þ �� i�,= i Þ � �� , �12�

where = i is the eigenvector matrix of H��i�. We order �� i� by

comparing the scalar products of the various i and �i−1�eigenvectors. Specifically, the eigenvalue that is put into thekth position of �� i� is that eigenvalue whose eigenvector, � ij,has maximal overlap with the kth eigenvector of the lastiteration, � �i−1�k.

Once the solver converges on a k��=0 solution boththe eigenenergy and a vector proportional to Q�� are ob-tained. The proportionality factor is then obtained as de-
scribed in Appendix C. Further, given Q��, one can effi-

ciently compute the entire vector �� through the use of Eq.�A4� of Appendix A. Below, however, we focus solely onQ��.

Since there is no guarantee that all the solutions ofk��=0 will be found with a given grid of energy points,there is no guarantee that all the eigenstates of the systemwill be found. However, it is an empirical fact that the largerthe Q-space overlap of a particular eigenstate ��, the easierit is to find that particular eigenstate using the QP algorithm.Thus, fortunately, the most important eigenstates necessaryto describe the Q-space dynamics are relatively easy to find.In practice, in order to assess the quality of the eigenvectorcoverage, we employ the metric

��F

�� = ��F

��Q��Q�� , �13�

where F is the set of labels of eigenstates that have beenfound by the algorithm. As F approaches a perfect descrip-tion of the Q-space dynamics, the expression of Eq. �13�approaches ��.

Although we saw no degeneracy in the numerical resultspresented below, for completeness we comment on how de-generacy manifests itself in the QP algorithm. If the fullHamiltonian H has a degeneracy then this would appear as a

degeneracy in the effective Hamiltonian H. For example, if

the full Hamiltonian has a double degeneracy at Ed, H�Ed�would have a doubly degenerate eigenvalue. Given this be-havior, what happens if the full Hamiltonian has a degen-eracy greater than NQ? Clearly the formalism is exact for allHamiltonians, yet it is perfectly possible to envisage Hamil-

tonians with degeneracies larger than NQ, whereas H�E� is ofdimension NQ. The resolution to this paradox proven in Ap-pendix B is that given a set of M �NQ degenerate eigen-states, one can always reconstruct this set so that at most NQ

of them have a nonzero Q-space overlap. In this way the

degeneracy of H�E� cannot exceed NQ.The performance of the QP algorithm must be measured

against the performance of straightforward diagonalizationtechniques which scale16 as 25NT

3, with NT=NQ+NP. In orderto estimate the performance of the QP algorithm, one con-ceives of using KNT grid points,21 where K is a positiveconstant. One then searches these grid points for � zeros.

The total numerical effort required by the QP algorithmcan thus be computed as follows.

�1� The numerical effort of computing each � can be writ-ten as C1NQ

3 +C2NQ2 NP �with C1 and C2 constants typi-

cal to the problem�. This number is made up of the

effort of building H, which scales as NQ2 NP �H has NQ

2

matrix elements, each being a sum of NP terms�, and

the effort in diagonalizing H, which scales as NQ3 .

�2� The effort involved in computing KNT� vectors isKNT�C1NQ

3 +C2NQ2 NP�.

�3� The effort in refining NT solutions. If M is the averagenumber of � evaluations required to refine a solution,we obtain that the total cost of the refinement phase isMNT�C1NQ

3 +C2NQ2 NP�.

�4� The sum total of items �2� and �3� �noting that K and M



are independent of NT� is therefore NT�K+M��C1NQ3

+C2NQ2 NP�.

Thus for NP�NQ, the QP algorithm scales like NP2 NT

2, giv-ing linear speedup over straightforward diagonalization. Wesee that the performance22 of the QP algorithm is superior toconventional methods only if NQ is small compared to NP,i.e., when a smaller subspace is coupled to a far larger sub-space. In addition to possessing a good scaling behavior, theQP algorithm is highly parallelizable, requiring very modeststorage space. These parameters are especially favorable tothe clusters of personal computer nodes available nowadaysto most researchers.

In assessing the algorithm performance, we have not in-cluded the time associated with diagonalizing the P space.Here it is assumed that the P space is diagonal or can bediagonalized, as in the pyrazine case below, in times shortcompared to the other operations.

In the parallelized version of the algorithm each nodereceives a copy of the Hamiltonian and a list of energy gridpoints to search. Each node can then work in isolation until ithas completed its grid point list. Then, at the end of thecalculation, all the resultant eigenvectors/eigenvalues areharvested from the nodes. This itinerary requires very littleinternode communication, allowing for highly efficient par-allelization.

The success of the QP algorithm depends critically on

our ability to efficiently compute the �E�− PHP�−1 part of H�see Eq. �A7��. This is trivially the case if the PHP matrixcan be diagonalized analytically �e.g., the P space is com-posed of harmonic modes�. Even if PHP cannot be diago-nalized analytically, the method is very efficient if one candiagonalize individual pieces of PHP, as illustrated belowfor pyrazine. Alternatively, if this is not the case, but thevarious P-space couplings are relatively weak compared tolevel spacing, one can use perturbation theory to obtain anew basis which approximately diagonalizes PHP. In firstorder perturbation theory, for example, the coupling Hamil-tonian elements are given by

��VIC��1�� = ��VIC�� + ��W�

��VIC��H0�� − ��

,

�14�

where ��1�� is the first order correction to ��, W� is a win-dow of nearby states23 which does not include �, and wherewe assume that there is no degeneracy in the Hamiltonian�the degenerate formulas being obvious extensions�.

Given that the P space is diagonal, and thus containsmany zeros, one might assume that there are direct eigenstatemethods that scale better than NT

3, and thus that there is adirect route to diagonalizing the full Hamiltonian. This is,however, not the case. There are, of course, methods thatallow one to diagonalize large matrices efficiently for certainhighly zeroed matrix symmetries �tridiagonal, banded, etc.�.However, to the authors’ knowledge, there is no way to effi-ciently diagonalize, directly, matrices of the symmetry that ispresented in this paper. Furthermore, as is presented in the
next section, the QP algorithm allows a coarse-grained ap-

proximation that allows one to consider matrices with di-mension greater than 1010, i.e., matrices so large that thereare no known methods to compute the dynamically impor-tant eigenstates of the system.

C. Coarse graining

Although the QP algorithm allows one to handle a muchhigher number of states than in straightforward diagonaliza-tions, the actual density of states in the P space can still betoo large even for this method. Such is the case for the 24-mode pyrazine treated below. Fortunately, because the den-sity of states is so high, we are able to invoke a quasicon-tinuum approximation by which we effectively coarse grainthe Hamiltonian. The idea is to divide up the energy axis intosmall bins Am, centered at energy �m, and represent all thecoupling Hamiltonian matrix elements in each bin by asingle term. This process can radically reduce NP with, fortime scales of interest to us, negligible effect on the results.

The coarse graining is performed by approximating Eq.�8� as

H��E�� = �� + �

V��V��*

E� − ��

�� + m

1

E� − �m

��Am

V��V��*

= �� + m

1

E� − �m m,�,��, �15�

where m,�,��, implicitly defined in Eq. �15�, is the coarse-grained coupling matrix. The approximation invoked in Eq.�15� is good as long as the width of the bins is small; itbecomes exact as the width of the bins approaches zero.

In order to describe the system dynamics, a connectionbetween the solutions of Eq. �15� and the set of exact eigen-states �� must be established. To do so, we define thecoarse-grained projectors �� as

�� =1

��

��A�

�� , �16�

where �� is the density of states of bin A�, � is the width ofbin A�, and �� are the true eigenstates of the system. Thedefinition of Eq. �16� has the important property that

�

�� = I , �17�

where I is the identity operator.24 The ansatz that we invokein order to connect the coarse-grained states with the QPalgorithm is that the eigenstates that are obtained from the�coarse-grained Hamiltonian� QP algorithm are equal to��. Our confidence in this protocol comes from themany model systems that we have studied for which it hasproven to be correct.

The dynamics of the system are obtained from the��U�t�� evolution operator matrix elements, where in�a.u.� U�t�=e−iHt. The analogous matrix elements within the
coarse-grained dynamics are defined as


��U�t�� = �

��A�

��e−iE�t

= �

��U�t��

�

B��,�� A�

e−iE�t

��

. �18�

The above formula amounts to replacing �� byB�� ,��, its average value in bin A�, given by

B��,�� =1

��

��A�

�� . �19�

The final sum of Eq. �18� can be approximated if oneassumes that there are many eigenvalues distributed evenlyover the bin. In this case one writes

1

�

��A�

e−iE�t

��

1

��

��−�/2

��+�/2

dEe−iEt =e−i��t sin��t/2�

�t/2

� ��t� , �20�

giving, finally,

��U�t�� = �

B��,��t� . �21�

Note that with the assumption above there is no need toknow ��. Rather, the QP algorithm, in fact, delivers��. In fact, we have maintained the �� productform so that the trace of �� is manifestly equal to 1.

It is important to realize that the approximations made inderiving Eq. �21� improve as �t decreases. For all the cal-culations presented in this paper, errors introduced by theseapproximations can be shown to be negligible since the timescale we look at in this paper is such that �t is small. Thisfact can be appreciated by considering two eigenstates, oneon each side of bin A�. In this case, the contribution to thetransition amplitude of the first line of Eq. �18� is

��U�t�� = ��1��1��exp�− i�� + �/2�t�

+ ��2��2��exp�− i�� − �/2�t�

= exp�− i��t��1��1��exp�− i�t/2�

+ ��2��2��exp�i�t/2�� . �22�

We thus have that as long as �t�1,

��U�t�� exp�− i��t��1��1�� + ��2��2�� ,

�23�

which is analogous to a single term of Eq. �21� because if�t is small then ��t� e−i��t, and because in this case��=2. The above condition holds for the calculations pre-sented below where � 10−4 eV and t�2�102 eV−1

�150 fs�. Thus for our calculations the phase differences be-
tween neighboring points in a bin are negligible.

D. The pyrazine model

We now consider IC in the 24-mode model of pyrazineof Raab et al.8 The model describes pyrazine as composed ofnormal modes, coupled by terms represented as polynomialswhose coefficients are made to fit the ab initio calculation.The Hamiltonian for the system is given by

H = i�1 0

0 1��i

2� �2

�Qi2 + Qi

2� + �− 0

0 �

+ i�G1

�ai 0

0 bi�Qi +

i,j�G2

�aij 0

0 bij�QiQj

+ � 0 �10a

�10a 0�Q10a +

i,j�G4

� 0 cij

cij 0�QiQj , �24�

where �i are the normal mode frequencies of S0 ,Qi are thedimensionless normal coordinates on S0 ,2 is the energydifference between the S2 and S1 surfaces, ai, bi, aij, and bij

are the coupling constants within each electronic manifold,and �10a and cij are the coupling terms between differentelectronic states. We denote by G1 the set of normal modesthat have Ag symmetry, by G2, the set of pairs of modes thathave the same symmetry, and by G4 all pairs of modes whoseproduct is of B1G symmetry. It is important to note that themost significant contributions to the system dynamics are theharmonic oscillator terms and the linear terms ai, bi, and�10a; the other terms, aij, bij, and cij are less relevant. None-theless, all the terms in the Hamiltonian of Raab et al. areincluded in the computations below. �The linear couplingterms within each electronic state are necessary, as shown byour lack of success in modeling pyrazine dynamics using themodel of Borrelli and Peluso25 which lacks these terms.� Thestructure of the on-surface coupling allows for a relativelyeasy diagonalization of PHP because the coupling termswithin each electronic states only couple modes of the sameirreducible representation.

The PHP part of the Hamiltonian of Eq. �24� can bewritten as

PHP = i=1

8

PHiP , �25�

where i ranges over the eight irreducible representations �ir-reps�. One can diagonalize each PHiP as

PHiP��k�i�� = ek

�i��k�i�� . �26�

With the ��i�� in hand, we form the total eigenstates �� as

��s� = ��k1

�1��k2

�2�� . . . ��k8

�8�� = �i=1

8

��k�s��i� � , �27�

with

PHP��s� = Es��s� �28�

and Es=iek�s��i� . Here k�s� denotes the collection of eight k

values �k1 ,k2 , . . . ,k8� labeling the contribution to ��s� of par-ticular eigenstates of PHiP. Analogous results can be ob-tained in diagonalizing QHQ. Constructing a basis that spans
all 24 modes is then a simple matter of taking a product basis


of the individual irrep eigenstates, a process done for both S1

and S2. This procedure was facilitated by the fact that han-dling a particular irrep required diagonalizing a matrix ofonly modest size �always less than 7000 elements�.

With this in mind, we can describe the full procedure forconstructing the Hamiltonian. First, a harmonic oscillator ba-sis is selected by taking all basis states whose energy wasless than some cutoff value, here taken to be 1.9 eV, wherethe threshold of S1 is at E=0.0 eV and that of S2 is at E=0.846 eV. Next the Hamiltonian within each electronicstate for each irrep is constructed and diagonalized. This di-agonalizes both QHQ and PHP. Given ��1�� and ��2��, theeigenstates which diagonalize the decoupled S1 and S2 mani-folds, we compute QHP for an � basis defined as

��s�1�� =

�

Cs��1�� , �29�

��s�2�� =

�

Cs��2�� , �30�

�QHP�mn = ��m�2��H��n

�1�� = ��

�Cm��2� �*Cn�

�1�V��, �31�

where the C’s are the expansion coefficients of the �� statesin the bare states �� or ��. This process results in diagonalQHQ and PHP matrices and off-diagonal block QHP—allexpressed in the �� basis.

For purposes of comparing with previous results wechose the Q space to be composed of the 176 vibrationalstates of S2 that have the largest Franck-Condon overlap with�S0 ;0�, where �S0 ;0� is the ground vibrational state of S0. Wedefine the autocorrelation function as

J�t� � ��0��U�t��0�� 32�

for some initial state ��0��. The 176 basis states chosenyielded a J�t� for ��0��= �S0 ;0�, in good agreement to thatobtained in Ref. 26. In comparison, when we used a basis of28 basis states, the behavior of J after about 15 fs �the initialdecay� was noticeably different from that of Ref. 26. Thisresult highlights an important issue in selecting a Q-spacebasis for a decay problem: the resonances in the Q-spacebasis interfere with one another. Hence, the resonance lineshape of a particular Q-space state can change depending onwhich other resonances are included in the calculation.

III. RESULTS

We first studied the QP algorithm for a variety of modelsystems, choosing assorted values of NQ and NP, bare ener-gies ��, and coupling matrix elements V��. The energies�� were either chosen to be evenly spaced or randomlydistributed, and V�� was chosen to have constant, Gaussian,sinusoidal, or random functional forms, or products of these.In all these cases the QP algorithm successfully generated theeigenstates of the system, while displaying the scaling be-havior discussed above. Further, for each of the model sys-tems, we tested the coarse-grained approximation by com-paring it to an exact calculation with some NP for whichnbins�NP was satisfied. The agreement of the coarse-grained
and exact results was very good for all the calculations we

have performed. In particular, similar resonance line shapesand similar dynamical evolution, which got better with in-creasing nbins /NP, were obtained. These results support thearguments of Sec. II C.

A. Numerical performance tests

Below we present the autocorrelation function J�t� aswell as the diabatic population P2�t�,

P2�t� = ��t��Q��t�� , �33�

i.e., the probability of finding the system in any state in S2 attime t.

Figure 1 displays our computed J�t�, �Eq. �32��, andP2�t�, �Eq. �33��, as functions of time for ��0��= �S0 ;0�. OurJ�t� plot is in excellent agreement with the equivalent multi-configuration time dependent Hartree �MCTDH� results inFig. 6 of Ref. 26, with the small differences arising from ouruse of only 176 �� basis states in the Q space, yielding a�� S0 ;0��2 of about 0.8. Achieving �� S0 ;0��2=1would require an enormous number of Q states, as the �S0 ;0�state expansion converges very slowly after the first 170states. Such convergence studies were not undertaken since,in any event, a ��0��= �S0 ;0� initial condition is unphysicalinsofar as it would arise only from a ��t� laser pulse.

Unless otherwise specified, in all the calculations re-ported below the Hamiltonian was coarse grained over arange of �2 eV into 200 bins, giving rise to eigenvalues thatwere typically spaced by 10−4 eV. A comparison of theresults for 200 vs 1000 bins is shown in Fig. 1, supportingthe reliability of the 200 bin results.

We have compared the QP algorithm with two othermethods used in the past to solve the pyrazine dynamics withthe Hamiltonian of Raab et al.: the semiclassical method ofThoss et al.,26 and the quantum MCTDH method of Raabet al.8 When distributed over forty 1.8 GHz AMD Opteronprocessors, in a vanilla Beowulf configuration, the parallel-ized QP calculation of the eigenstates used to generate Fig. 1takes only 40 min. This is equivalent to 27 CPU hours werethe method to be run with a single processor. �We note that

FIG. 1. �Main� J�t� �dark line� and P2�t� �light line� for the �S0 ;0� initialstate. �Inset� Comparison of J�t� for 200 and 1000 bins.

the added serial time for diagonalizing the P-space Hamil-



tonian and coarse graining the Hamiltonian is approximately2.3 CPU hours.� For comparison, the MCTDH calculationstook 485 CPU hours on a Cray T90 shared-memorymachine.27 We conservatively estimate that modern hardwarewould give a factor of 2–3 performance increase of theMCTDH calculations. This still results in a six- to ninefoldefficiency improvement of the QP method relative to that ofMCTDH. The semiclassical method of Thoss et al., which isapparently more computationally costly than the MCTDHmethod, is outperformed by the QP algorithm by an evenlarger margin.

Even more important than the serial performance is theparallel performance of the QP algorithm. From the opera-tional perspective, the wall time, not the serial time, is therelevant quantity. We do not expect MCTDH to benefitnearly as significantly from parallelization as does the QPalgorithm. This is because the MCTDH method, in whichone solves a set of first order nonlinear coupled differentialequations, requires a large degree of internode communica-tion, thereby destroying the linear speedup with the numberof nodes that is enjoyed by the QP algorithm.

An additional important point is that the QP algorithmprovides system eigenvalues and eigenstates, which, oncedetermined, can be used to obtain assorted ��t�� at very littleadditional cost. That is, once the eigenvalues and eigenstatesare known, calculations of P2�t� of additional initial ��S2 states take less than 4 min to complete, and J�t� com-putations are on the order of seconds.

Figure 2�a� displays P2�t� for several initial states, la-beled by their energy on the decoupled S2 manifold. Panel�A� shows the wide diversity of P2�t� falloff for differentinitial states. In comparing P2�t� to J�t�2 we note that thedecay of J�t�2 with time can be attributed entirely to Q-P IC

14
coupling. This means, as discussed in our earlier paper, that

the difference �P2�t�−J�t�2� indicates the amount of IC re-crossing, i.e., the amplitude that reappears in the Q spaceafter decaying to the P space. In our earlier paper on four-mode pyrazine we saw a pronounced degree of IC recross-ing. We conjectured that this was due to the limited size ofthe pyrazine model used. Figures 2�b� and 2�c� show P2�t��solid lines� and J�t�2 �dashed lines� for the five basis statesof Fig. 2�a�. It is obvious that J�t�2 follows P2�t� fairlyclosely, indicating only a moderate amount of IC recrossing,consistent with expectation.

This behavior seems in stark contrast to that of Fig. 2�d�,where P2�t� and J�t�2 for the �S0 ,0� state of Fig. 1 are shown.One might surmise that the rapid falloff of J�t�2 compared toP2�t� would be an indication of strong IC recrossing, but thisis not the case. Rather, the �S0 ,0� state is initially a superpo-sition of vibrational states in S2. Thus, the autocorrelationdecay out of the initial �S0 ,0� state reflects not just the ICbetween surfaces but also the nonstationary dynamics of theinitial vibrational wave packet on the S2 surface. The lattersource of decay is, in fact, responsible for the fast decay ofJ�t�2 shown in Fig. 2�d�.

B. Resonance structure

The resonance line shape of a basis state �� is defined as�� 2, i.e., the square of the projection of the exact eigen-states on state �� as a function of the energy E�. This lineshape, which is directly connected to observations, oftengives a good indication of the dynamics of the system. In ourearlier studies of pyrazine,14 we showed that the system reso-nance line shapes play a significant role in predicting thecoherent controllability of a system. Specifically, we showedthat a pair of system states is controllable only if either �a�the resonances of the pair overlap significantly or �b� they

FIG. 2. �a� P2�t� for several S2 states.The states are labeled according totheir energy. �b� P2�t� �solid line� andJ�t�2 �dashed line� for the states at 0.53and 0.97 eV. These are the two stateswhose resonances are pictured in Fig.3. Arrows point at J�t� and P2�t� of the0.53 eV state. �c� P2�t� �solid line� andJ�t�2 �dashed line� for states at 0.76,0.86, and 0.92 eV. The 0.76 eV stateis pictured in Fig. 6. �d� P2�t� and J�t�2

for the �S0 ;0� state in Fig. 1.

both simultaneously overlap a third resonance. In this earlier



work14 on four-mode pyrazine, we observed diffuse, struc-tureless resonances, displaying a picket-fence-like structurewith ill-defined overall shape. This behavior was contrary towhat would be expected if the S2 states decayed into a qua-sicontinuum of S1 states—the case, we believed, for pyra-zine. This contradictory situation motivated us to examinethe resonance structure of pyrazine more closely using thefull 24-mode model, as is done here.

Figure 3 shows two typical resonance line shapes ob-tained in our calculations: the left most resonance is centeredon a bare state �� of energy 0.53 eV and the resonance onthe right is centered at 0.97 eV. Note that this graph reflectsa general trend that we observed in the resonance lineshapes: the lower energy resonances tend to be more spreadout in energy, peaking at smaller energies.

FIG. 3. Resonance line shapes for 0.53 and 0.97 eV states of pyrazine.

FIG. 4. �a� Model Gaussian-shaped QHP IC couplings as a function of the bthe resonance is centered at Ec=0. �c� The same when the resonance is ceindependent couplings, centered at Ec=0.5. In panels �A�, �B�, and �C� the
has the P-state energies evenly spaced at 0.05, with QHP characterized by a con

In order to understand these observations we considercoupling a single Q state located at energy Ec to a set of Pstates. As depicted in Fig. 4�a�, we model the couplingstrengths to be a Gaussian function centered about E=0. A Qstate placed at Ec=0 yields the symmetric bimodal resonanceline shape displayed in Fig. 4�b�. As shown in Fig. 4�c�,when the resonance center is shifted to small Ec�0, thebimodal structure loses its symmetry, displaying a typicalFano-type shape �normally discussed for continuum cases�:The peak closer to Ec gets larger and the resonance tails offmore into the negative energy domain. As the width of theGaussian in Fig. 4�a� increases, the center of the right peakof Fig. 4�c� approaches Ec while the left peak vanishes. Inthe limit of infinite width, Fig. 4�c� transforms into theLorentzian shape of Fig. 4�d�, typical of constant IC cou-pling.

The above model provides a good explanation of thetrend shown in Fig. 3. When we coarse grain the QHP ICcoupling of the two bare states of Fig. 3 we obtain the curvesshown in Fig. 5. Notice how the Hamiltonian for the Ec

=0.53 eV ket �marked with the left downward arrow� hasmuch more curvature around Ec compared to the couplingfor the Ec=0.97 eV resonance �marked with the right upwardarrow�. Thus the resonance at Ec=0.53 eV seen in Fig. 3 isanalogous to the resonance in Fig. 4�c�, with the strong leftshift in the resonance due to the strong coupling elements atmore negative energies. In contrast, the resonance at Ec

=0.97 eV is more likely the resonance of Fig. 4�d�, with thevariation in the coupling much less pronounced, conse-quently yielding a single peaked Bixon-Jortner shape.

ate energies. �b� Resonance structure for the QHP given by panel �A� whend at Ec=0.5 �arrow of panel �A��. �d� Resonance characterized by energyce basis states are located at evenly spaced 0.1 energy intervals. Panel �D�

are stntereP-spa

stant coupling of 0.055.



Resonance line shapes provide an energy space perspec-tive that is complementary to the time dependent picture. For

example, it can be shown that for �� 2�e−a�E − E0�2Gauss-

ian line shapes, J�t�2=e−�t / ��2with �=�2�2a. For Gaussian

line shapes the autocorrelation lifetime � is related to thefull width at half maximum �FWHM� of the resonance lineshape as �=1.55 �eV fs� /. Similar results hold for Lorent-zian line shapes. However, when the resonance line shapehas bimodal or higher complexity, this inverse relationshipno longer holds.

Figure 6 shows resonances for the states at energies of0.76 and 0.97 eV, whose J�t�2 are shown in Figs. 2�b� and2�c�. The FWHM, estimated from the resonances in Fig. 6,are about 0.02 eV for the 0.76 eV state and about 0.1 eV forthe 0.97 eV state. These numbers give � of approximately 78and 16 fs, respectively, which are in qualitative agreementwith results of Fig. 2. The agreement is less favorable whenone looks at the resonance line shapes of Fig. 3. Here, the0.53 eV resonance has a width of 0.2 eV, twice as wide asthe 0.97 eV resonance. Yet, the J�t�2 functions of Fig. 2�b�are approximately the same. In fact, the � from Fig. 2 for the

FIG. 5. Coarse-grained IC coupling PHQ in pyrazine as a function of en-ergy. Each point has been averaged over ten bins to smooth out very fastfluctuations.

FIG. 6. The resonance line shapes of the 0.76 and 0.97 eV S2 states in
pyrazine.

0.53 eV resonance is slightly less than that for the 0.97 eVresonance. Clearly, the decay times derived from Fano-typeline shapes are not simply proportional to −1.

IV. DISCUSSION

We have presented an efficient method based on parti-tioning theory to compute bound state dynamics of complexmolecular systems for which the P space is readily diagonal-izeable. The method has very favorable NT

2 scaling propertiesand very modest core requirements even for very large ma-trices. We have used this method to study the very substantialproblem of IC dynamics and resonance line shapes of 24-mode pyrazine. Future work will include the coherent controlof IC in 24-mode pyrazine and extending the work of Frish-man and Shapiro28 to the full quantum calculation of spon-taneous emission of complex molecules.

Two improvements of the QP algorithm can be consid-ered. As shown above, the QP algorithm scales as NQ

4 . Al-though NQ is assumed to be much smaller than NP, neverthe-less its size may represent a non-negligible obstacle. It is

possible to incorporate a procedure by which one breaks Hup into several smaller matrices and diagonalizes thesesmaller matrices as needed. A second improvement wouldresult from incorporating matrix deflation into the procedure.This would result in two advantages. First, at present, if onedoes a calculation which gives unacceptable values for themetric of Eq. �13�, one has to repeat the entire calculationwith a denser set of grid points. It would be beneficial to usethe eigenstates that have already been found in the first cal-culation in this second calculation. Second, the QP algorithmtends to find eigenstates with larger Q-space overlap morereadily than states with small Q-space overlap. That is, thelarge Q-space vectors tend to mask the smaller Q-space vec-tors. Both of these issues suggest that it would be useful ifone could project out of the original Hamiltonian the sub-spaces corresponding to eigenstates that have already beenfound. The well known technique of deflation29 does justthis.

ACKNOWLEDGMENTS

One of the authors �P.S.C.� thanks H.-D. Meyer, L. S.Cederbaum, and G. A. Worth for pyrazine data and for theiradvice on their pyrazine model. This work was supported bythe Natural Sciences and Engineering Research Consul ofCanada.

APPENDIX A: DERIVATION OF EQUATION „7…

Here we derive Eq. �7� for the bound state case, follow-ing in close analogy to the continuous case of Ref. 13. Start-ing with the time independent Schrödinger equation and us-ing Q+ P=1 we have

E�� = H�� = H�P + Q�� . �A1�

Multiplying this equation on the right by Q or P, and remem-bering that Q=Q2 and P= P2, gives two equations

�E� − PHP�P�� = PHQ�� , �A2�



�E� − QHQ�Q�� = QHP�� . �A3�

Equation �A2� can be inverted to give

P�� = �E� − PHP�−1PHQ�� , �A4�

where we assume �E�− PHP� is uniquely invertible. UsingEq. �A4� in the right hand side of Eq. �A3� gives the equation

�E� − QHQ�Q�� = QHP�E� − PHP�−1PHQQ�� . �A5�

Rearranging this equation gives Eq. �7� with the H operatorgiven by

H�E�� = QHQ + QHP�E� − PHP�−1PHQ . �A6�

Taking matrix elements of this operator, identifying QHP=QVICP, and using the spectral resolution

�E� − PHP�−1 = �

��E� − ��

�A7�

yield Eq. �8�.

APPENDIX B: DEGENERACY IN THE QP ALGORITHM

Let there be a Hamiltonian such that there is a set ofM �NQ degenerate eigenstates, ��M. We wish to show thatone can transform this set of eigenstates �by taking appropri-ate linear combinations� into two new sets

��M → ��NQ + ��M−NQ, �B1�

where all the states �� and �� are orthonormal eigenstatesof the Hamiltonian, and all the elements of the second sethave the property Q��=0.

We label ��1� , . . . , ��M��M. Assume that all eigen-states have nonzero Q-space overlap. If this were not thecase, then one would reduce the size of the initial set untilthis was true. Create a set of vectors, ��zi�, all of which,except one, are orthogonal to a vector �� in the Q space.Then we have �assuming that �� 1��0 which must be truefor at least one ��

�z1� = ��1� ,

�z2� = C2��2� − �21��1�� ,

�B2�]

�zM� = CM��M� − �M1��1�� ,

where Ci normalizes the states and

�ij =��i�� j�

. �B3�

Clearly �� zi�=0 unless i=1. Next, form a new set of vec-tors, ��yi�, that are orthonormalized via Gram-Schmidt
orthogonalization

�yM� = �zM� ,

�yM−1� = DM−1��zM−1� − �yM��yM�zM−1�� ,

]

�B4�

�yn� = Dn��zn� − k=n+1

M

�yk��yk�zn�� ,

]

�y1� = D1��z1� − k=2

M

�yk��yk�z1�� .

Notice that �� yi�=0 unless i=1, and if i=1 then �� y1�� 1��0. We define ��1��y1�, and we are left with theset ��1� , �y2� , . . . , �yM�. An advantage of this set is that onecan take any linear combination of ��y2� , . . . , �yM� and thatvector is also orthogonal to ��. This begs one to repeat theabove algorithm using the set ��y2� , . . . , �yM� and consideringa different member of the Q space, ��. This would result ina new set of vectors ��1� , ��2� , �y3� , . . . , �yM�, such that��y3� , . . . , �yM� were orthogonal to both �� and ��. Natu-rally, this process can be repeated until all the Q-space barestates had been used, resulting in a set��1� , . . . , ��NQ

� , �yNQ+1� , . . . , �yM�. This last set has exactlythe properties of Eq. �B1�, where ��yNQ+1� , . . . , �yM� are allorthogonal to the whole of the Q space.

APPENDIX C: NORMALIZATION OF �D�‹

Recall that any constant multiple of an eigenstate is alsoan eigenstate. Typically, when an eigensolver completes thediagonalization, it assures that all the eigenstates are normal-ized to unity. This means that our calculation ends up with astate �D�� such that

�D��D�� = 1, �C1�

C��D�� = Q�� , �C2�

for some constant C�. That is, the solution we obtain fromthe QP algorithm is proportional to Q��. We must still com-pute the constant of proportionality, which is the subject ofthis appendix. Remembering that 1=Q+ P, Q=Q2, and P= P2, we write

�� = ��Q�� + ��P�� = ��Q2�� + ��P2�� , �C3�

��Q2�� = �C��2�D��D�� = �C��2, �C4�

��P2�� = ��QHP�E� − PHP�−1�E� − PHP�−1PHQ�� ,

�C5�

where the last equality uses Eq. �A4�. Then using the spectral
resolution, Eq. �A7� and �C5� becomes


��P2�� = �

��QH��HQ��E� − ��2 , �C6�

which using Eq. �C2� gives

��P2�� = �C��2�

�D��H��H�D��E� − ��2 . �C7�

Then with Eqs. �C7� and �C3� we can write

1 = �C��2�1 + �

�D��H��H�D��E� − ��2 � , �C8�

which allows one to compute the normalization factor C�.Note that it is only the magnitude of C� that is relevant, thephase is irrelevant. This can be seen as follows. C� sets thephase for Q��, which in turn sets the phase for ��. Considerthe propagator in terms of the eigenstates

U�t� = �

e−iE�t�� . �C9�

It can readily be seen that any eigenstate �� can be replacedby ei�� for any real � without changing the propagator ofthe above equation. Thus the phase of �� is irrelevant, andby reversing the above logic, the phase of C� is irrelevant.

1 U. Fano, Nuovo Cimento 12, 156 �1935�.2 U. Fano, Phys. Rev. 124, 1866 �1961�.3 H. Feshbach, Ann. Phys. �N.Y.� 5, 357 �1958�.4 P. O. Löwdin, J. Math. Phys. 3, 969 �1962�.5 P. O. Löwdin, J. Mol. Spectrosc. 10, 12 �1963�.6 M. Shapiro, J. Chem. Phys. 56, 2582 �1972�.7 M. Shapiro, J. Phys. Chem. 102, 9570 �1998�.8 A. Raab, G. A. Worth, H.-D. Meyer and L. S. Cederbaum, J. Chem. Phys.

110, 936 �1999�.9 V. Stert, P. Farmanara, and W. Radloff, J. Chem. Phys. 112, 4460 �2000�.

10 M. Seel and W. Domcke, J. Chem. Phys. 95, 7806 �1991�.


11 G. Stock and W. Domcke, J. Chem. Phys. 97, 12466 �1993�.12 A. L. Sobolweski, C. Woywod, and W. Domcke, J. Chem. Phys. 98,

5627 �1993�.13 M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecu-

lar Processes �Wiley, New Jersey, 2003�.14 P. S. Christopher, M. Shapiro, and P. Brumer, J. Chem. Phys. 123,

064313 �2005�.15 P. S. Christopher, M. Shapiro, and P. Brumer �in preparation�.16 A. R. Gourlay and G. A. Watson, Computational Methods for Matrix

Eigenproblems �Wiley, London, 1973�.17 Y. Saad, Numerical Methods for Large Eigenvalue Problems �Manchester

University Press, Manchester, UK, 1992�.18 We quote here the scaling of the Householder tridiagonalization followed

by the QR algorithm, which is regarded as the best general purposediagonalization method �Refs. 16 and 17�. There are several methods tocompute extreme eigenvectors/eigenvalues �e.g., Lanczos method�, butbecause we are interested in the dynamics of our system, we require all�or most� of the eigenvectors/eigenvalues.

19 A. A. Sanz, P. S. Christopher, M. Shapiro, and P. Brumer �in preparation�.20 Eigensolvers tend to order the eigenvalues in increasing order, so we

know what order the eigenvalues are in. But that does not help us pre-serve the order of the eigenvalues from one grid point to the next.

21 The grid points are put down so that their density is proportional to thebare state density of states.

22 Note, however, that the coarse graining method described in the nextsection allows one to get very good QP-algorithm performance even ifNQ NP.

23 The technique of using a window of nearby states seems to work well,based on the fact that the denominator of the second term of Eq. �14�makes nearby states the important part of the expansion.

24 Also note that this definition for �� means that Tr��=1. That is,Tr��=�� =1/��A�

1�=1.25 R. Borrelli and A. Peluso, J. Chem. Phys. 119, 8437 �2003�.26 M. Thoss, W. H. Miller, and G. Stock, J. Chem. Phys. 112, 10282 �2000�.27 A less detailed calculation was also presented by Raab et al. in the same

paper. This calculation took only 100 CPU hours and was designed tocheck the convergence of their calculations. We assume that the largercalculation is the more accurate and thus compare our computations tothese results.

28 E. Frishman and M. Shapiro, Phys. Rev. Lett. 87, 253001 �2001�.29 B. N. Parlett, The Symmetric Eigenvalue Problem �Prentice-Hall, Engle-

wood Cliffs, NJ, 1980�.


Efficient partitioning technique for computing the dynamics of intramolecular processes: Radiationless transitions in pyrazine

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