Three-dimensional quantum time-dependent study of thephotodissociation dynamics of Na � � �FH=D

Gil Katz a, Yehuda Zeiri b, Ronnie Kosloff a,*

a Department of Physical Chemistry and the Fritz-Haber Research Center, The Hebrew University, 91904 Jerusalem, Israelb Department of Chemistry, Nuclear Research Center – Negev, P.O. Box 9001, Beer-Sheva 84190, Israel

Received 9 January 2002; in final form 2 April 2002

Abstract

A time-dependent three-dimensional nonadiabatic computation study of the photodissociation of the van der Waals

Na � � �FH molecule was performed for total J ¼ 0. A very low probability of photo-reaction to produce NaF+H wasobserved from most initial conditions. Enhancement of the NaF+H product was observed for the isotopically sub-

stituted Na � � �FD. The three-dimensional calculations are in qualitative agreement with the two-dimensional previousstudy. Calculated excited state lifetimes were in the range of �100 ps. Excitation of the bend and the van-der Waalsstretch significantly shortened these lifetimes without increasing the reaction yield. � 2002 Elsevier Science B.V. All

rights reserved.

1. Introduction

Photochemical reactions are unique because ofthe ability of the light field to switch the chemicalprocess on and off. This property allows a choiceof a specific stereochemical configuration beforeinitiating the reaction. For binary chemical en-counters, the weakly bound van der Waals mole-cule of the reactant can create a launching pointfor stereospecific photochemical reactions [1]. Thisidea is behind the experimental study of thephotodissociation of the Na � � �FH [2], Li � � �FH[3] and Li � � �FCH3 [4] van der Waals molecules.Using light to initiate a chemical reaction has

additional consequences. Absorption of light in the

UV/visible region leads to excited electronic sur-faces. Chemical encounters on excited electronicsurfaces may go through a maze of multiple elec-tronic surfaces. This is in contrast to typicalchemical reactions which are well described on theground electronic surface. A theoretical frameworkof reactions on multiple electronic surfaces shouldsupply insight into the reaction mechanism, andtime scales and also be able to predict productyields and lifetimes. An understanding of thesephotochemical reactions goes beyond the Born–Oppenheimer approximation and is a challenge totheory. By building on the success of quasi-classicaltrajectory methods in describing ground state dy-namics, semiclassical approximations, whichamend the classical trajectory with the nonadia-batic electronic transitions, seemed to be a prom-ising direction. Thus a comprehensive semiclassical

27 June 2002

Chemical Physics Letters 359 (2002) 453–459

www.elsevier.com/locate/cplett

* Corresponding author. Fax: +972-2-6513742.

E-mail address: ronnie@grid.fh.huji.ac.il (R. Kosloff).

0009-2614/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0009-2614 (02 )00741-8

study was performed for the Na � � �FH!NaFþH photochemical reaction, by comparingdifferent surface hopping approximations [5].However all schemes studied gave a very high yieldof reaction products in conflict with experimentalobservations. This discrepancy became the moti-vation for a collinear nonadiabatic quantum study[6]. Both quantum and semiclassical studies em-ployed the two lowest potential energy surfacesusing the same potentials and the diabatizationscheme of the semiclassical investigation. A sub-stantial disagreement between the semiclassical andquantum mechanical methods was observed for theNa � � �FH(D) photodissociation. The branchingratio between reactive and nonreactive channelswas qualitatively different. The quantum approachsuggested that practically no reaction occurs unlessthe FH vibrational mode is excited to v ¼ 3, whilethe semiclassical calculations predicted a reactionyield of 99% even at v ¼ 0. The lifetime of thephotoexcited state was another example of dis-crepancy between the quantum and quasi-classicalcalculations. Moreover, in the quantum calcula-tion, replacement of H by D led to a marked in-crease in the reaction probability; in thesemiclassical methods, on the other hand, the re-action probability diminished [6].It is well documented that nonadiabatic dy-

namics is very sensitive to the dimensionality ofthe problem [7,8]. In addition, based on electronicstructure calculations, the ground state of theNa � � �FH molecule is bent [3]. Since the compar-ison between the fully quantum calculation andthe semiclassical methods was restricted to a col-linear configuration, it became necessary to extendthe investigation to three dimensions.The present work reports the results of three-

dimensional (3D) quantum mechanical simulationsof the Na � � �FH(D) photodissociation process.The purpose is to calculate reaction yields, excitedstate lifetimes and final state distributions.

2. Details of calculations

The dynamics of the photodissociation wassimulated by solving the multi-surface time-de-pendent Schr€oodinger equation

i�ho

ot

~wwe~wwg

� �¼ ~HH

~wwe~wwg

� �: ð2:1Þ

The dynamics was generated by the Hamiltonianrepresented in a diabatic form

~HH ¼ T̂Tþ V̂Ve V̂VintV̂Vint T̂Tþ V̂Vg

� �: ð2:2Þ

The operation of the kinetic energy operator T̂T wasexecuted using the Fourier method in Jacobi co-ordinates for total angular momentum J ¼ 0

T̂TintðR; r; cÞ ¼ � �h2

2lo2

oR2

�þ o2

or2þ 1

R2

�þ 1

r2

�

ð1�

� c2Þ o2

oc2þ 2c o

oc

��; ð2:3Þ

where R is the distance from the sodium atom tothe center of mass of the HF, r is the HF distanceand c ¼ cos h, where h is the angle between thevectors associated with ~RR and ~rr. The volume ele-ment in these coordinates becomes: dV ¼dRdrdc. The evaluation of the kinetic energy op-erator requires four fast Fourier transform (FFT)operations. The potential matrix elementsViðR; r; cÞ, in the diabatic representation were ta-ken from [5].The Jacobi coordinate system of the reactant

supplied an inappropriate description of theNaF+H asymptotic photo-reaction product. Forthis reason, and to establish an internal check onconvergence, the calculations were repeated em-ploying other sets of coordinate systems such asperimetric, Eckart bond coordinates and Euclid-ean. The details of these implementations are de-scribed in [9].

3. Results and discussion

The photodissociation event was launched froma bound vibrational state on the ground electronicsurface of the van der Waals complex. The lowlying vibrational eigenstates were calculated usingthe relaxation method [11]. These calculationswere performed on the ground electronic state ofthe nonadiabatic potential to supply an initialstate. In addition the eigenstates of an improved

454 G. Katz et al. / Chemical Physics Letters 359 (2002) 453–459

ground state potential were also calculated andcompared to the results of [10]. These calculationsverified the convergence and helped in the assign-ment scheme used to classify the vibrational ei-genvalues according to low frequency Na � � �FHstretch, bend, and to high frequency FH stretch.A simulation of the photodissociation dynam-

ics, assuming a weak excitation field, can be car-ried out by promoting the wavefunction from theground PES vertically to the excited electronicsurface [12,13]. The subsequent dynamics of thiswavepacket can be filtered to obtain the energy-dependent photodissociation cross-section. An al-ternative view is that the initial state is created by ashort excitation pulse of duration s, where1=s Dx, where Dx is the width of the absorp-tion band. Employing this approach, the variationof the norm of the excited wavefunction as afunction of time shown in Fig. 1, can serve as anindication of the lifetime of the excited complex.The overlap of the excited wavefunction with theinitial wavefunction can be used to obtain theabsorption spectrum [18]. The decay times are es-timated by fitting the excited state norm to a de-caying exponential.As can be observed in Fig. 1, the decay at short

times is faster. This means that the initial wave-packet contains resonances with a lifetime lessthan �300 fs. The small oscillations are the result

of the nonadiabatic mixing. The lifetimes of thedominant contribution estimated at 1 ps havelimited accuracy.Compared to the 2D results, the decay of the

excited states in the 3D case is slightly faster. Forexample, for the Na � � �FH ground initial statewð0; 0; 0Þ a lifetime of 121 ps is obtained in the 3Dcalculation compared to 172 ps in the 2D case. Thenotation used to define the initial state wIði; j; kÞcorresponds to the three vibrational quantumnumbers: i¼HF/FD stretch, j¼ bending modeand k¼Na � � �FH van der Waals stretch. For theNa � � �FD wð0; 0; 0Þ initial state, the estimated de-cay time in the 3D case is 5.1 ps, compared to 6.6ps in the 2D result.A more precise estimation of the excited state

lifetimes is obtained by decomposing the verticalFranck–Condon initial wavefunction to the con-tributions of the direct dissociation and reso-nances. The autocorrelation function CðtÞ ¼hwð0ÞjwðtÞi includes all the dynamical informationwhich evolves out of the initial wavefunction. Byfiltering CðtÞ the contribution of the major reso-nances to the dynamics can be unraveled. The filterdiagonalization (FD) approach [14–16] allows thisdynamical information to be obtained from a finitetime history. The outcome shown in Table 1 is aseries of amplitudes, energies and lifetimes filteredout from the autocorrelation function, initiated bya specific initial wavefunction. The assignment ofthe resonances was done by noticing that the highfrequency mode is still the FH stretch and that thebend and stretch low frequency modes can becompared to the assignments of the ground statewavefunctions [9,10]. The FH stretch frequencydecreased from �4500 cm�1 on the ground elec-tronic surface, to �3780 cm�1 upon excitation.To create a basis for assigning the resonances,

the first few eigenstates of the adiabatic excitedstate potential were calculated using the relaxationmethod [11]. These excited state eigenstates sup-plied nodal patterns and approximate frequenciesof the two lowest energy modes, the bend and theNa–FH-stretch. These frequencies were comparedto frequencies obtained by FD out of the auto-correlation function of a ground surface initialwavefunction promoted to the excited surface.The initial wavefunction was a direct product

0.0 0.2 0.4 0.6 0.8 1.0t(ps)

1.0

|<ψ

(τ)|ψ

(τ)>

|e

e

2

0.5

ψ(0,0,0)

NaFD

ψ(4,0,0)NaFH

NaFD ψ(0,0,0)

ψ(4,0,0)

NaFH

Fig. 1. A semi-log plot of the square of the wavefunction norm

hPei ¼ jhweðtÞjweðtÞij2as a function of time for a ground state

wavefunction of NaFH/D system promoted to the excited state.

The initial wavefunction is indicated on the right-hand side.

G. Katz et al. / Chemical Physics Letters 359 (2002) 453–459 455

wavefunction composed from the HF stretchwavefunctions and the ground state bend and Na–FH-stretch on the ground electronic surface. Toassist the assignment an initial state was usedstarting from the ground vibrational state on theupper adiabatic potential. To this state an addi-tional momentum kick in the direction of the bendor to the Na–FH-stretch was applied. By com-

paring the changes in amplitudes of the frequen-cies obtained in the FD method for different initialstates the frequencies were assigned to normalmodes. Due to intermode coupling for excitedstates this assignment is only approximate. Oncefrequencies were identified, the FD procedure wasrepeated with a smaller frequency window whichled to higher resolution.

Table 1

Lifetime, energies and relative amplitude obtained by the filter diagonalization method (FD) from the 3D autocorrelation function

Initial state Possible

assignment

Energy Lifetime Amplitude Lifetime Lifetime

(cm�1) (ps) (ps) (ps)

3D-FD 3D-FD 3D-FD 3D-norm 3D-norm

wNAHFð0; 0; 0Þ wNAHFð0; 0; 0Þ 0 (1976) 107.4 0.86 121 172

wNAHFð0; 0; 1Þ 112 21.30 0.05

wNAHFð0; 0; 2Þ 204 12.3 0.02

wNAHFð0; 1; 0Þ 171 7.54 0.02

wNAHFð0; 1; 1Þ 287 6.54 0.03

wNAHFð0; 1; 2Þ 374 1.76 0.006

wNAHFð1; 0; 0Þ 3798 7.5 0.002

wNAHFð2; 0; 0Þ 7245 5.3 0.0003

wNAHFð3; 0; 0Þ 10 542 4.1 0.000007

wNAHFð4; 0; 0Þ 13 622 1.7 0.000004

wNAHFð4; 0; 0Þ wNAHFð0; 0; 0Þ (1979) 109 0.05

wNAHFð0; 0; 1Þ 109 23.90 0.09

wNAHFð0; 0; 2Þ 198 11.7 0.08

wNAHFð0; 1; 0Þ 168 7.1 0.02

wNAHFð0; 1; 1Þ 284 6.7 0.04

wNAHFð0; 1; 2Þ 371 1.81 0.009

wNAHFð1; 0; 0Þ 3721 7.65 0.27

wNAHFð2; 0; 0Þ 7257 5.9 0.09

wNAHFð3; 0; 0Þ 10 598 4.3 0.06

wNAHFð4; 0; 0Þ 13 634 1.7 0.36 30 27

wNaDFð0; 0; 0Þ wNaDFð0; 0; 0Þ 0 (1171) 3.3 0.39 5.1 6.6

wNaDFð1; 0; 0Þ 3169 1.84 0.22 – –

wNaDFð2; 0; 0Þ 6081 1.27 0.13 –

wNaDFð3; 0; 0Þ 8945 0.66 0.08 –

wNaDFð4; 0; 0Þ 10 760 0.53 0.2 2.7 –

The lifetimes are compared to the those obtained by fitting the norm to an exponential decay. The energies are relative to the first

peak whose value is indicated in bold. The assignment used is: w(HF-stretch, bend, Na–FH-stretch).

Table 2

Branching ratio between product and reactant channels

The channels Initial state 3D 2D

(NaF+H)/(Na+FH) wð0; 0; 0Þ 0.00036 0.0002

(NaF+H)/(Na+FH) wð0; 1; 0Þ 0.000394

(NaF+H)/(Na+FH) wð4; 0; 0Þ 0.2058 110 000

(NaF+D)/(Na+FD) wð0; 0; 0Þ 0.38461 0.315

(NaF+D)/(Na+FD) wð4; 0; 0Þ 0.741077 –

456 G. Katz et al. / Chemical Physics Letters 359 (2002) 453–459

The longest lifetimes of the resonances obtainedby the FD method are in good agreement with thelifetime obtained from the decay of the norm. Theresults presented in Table 1 clearly show that ex-citations of the bend and the Na � � � /FH stretchmodes shorten the excited state lifetimes. Never-theless these excitations do not lead to an increasein the reaction yield, which means that the decayof the norm is due to leaks through nonadiabatictransitions back to the reactant channels (cf. Table2) on the ground electronic state.The branching ratios between the products

NaF+H and the reactants Na+FH was estimatedfrom the ratio of the accumulated flux calculatedat a dividing surface positioned in the asymptoticvalleys. As can be seen in Fig. 2 the ratio stabilizedafter approximately 1 ps. The bursts at 400 fs are

0 0.5 1 1.5time(ps)

0

0.2

0.4

0.6

0.8

(NaF

+ H

) /

(Na

+ F

H)

ψ(0,0,0)

ψ(4,0,0)

ψ(0,0,0)

ψ(4,0,0)

NaFHx10

NaFH

NaFD

NaFD

Fig. 2. The accumulated flux ratio between reactants and

products as a function of time for different initial wavefunc-

tions.

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10V(I)

0

0.1

0.2

0.3

0.4

<P

>

NaFH

ψ(0,0,0)

V(i)

0

0.1

0.2

0.3

0.4

<P

>

NaFH

ψ(4,0,0)

0 1 2 3 4 5 6 7 8 9 10

V(i)

0

0.1

0.2

0.3

0.4

<P

>

NaFD

ψ(0,0,0)

0 1 2 3 4 5 6 7 8 9 10

V(i)

0

0.1

0.2

0.3

0.4

<P

>

NaFD

ψ(4,0,0)

Fig. 3. The final vibrational distribution of the NaF products starting from Na � � �FH and Na � � �FD.

G. Katz et al. / Chemical Physics Letters 359 (2002) 453–459 457

the result of flux from fast direct reaction. Theoverall contribution of this component in the ac-cumulated flux is very small.From a qualitative point of view, the 3D results

are similar to the 2D ones. The reaction proba-bility is also found to be strongly influenced byisotopic substitution. The branching ratio betweenthe reactant and product channels is shown inTable 2 where the 3D data are compared to the 2Dresults.The qualitative trends in the 3D calculations are

similar to the 2D results. However the enhance-ment of the reaction product formation upon HFstretch excitation and the isotope substitution isless pronounced.

Starting with specific initial states a final stateanalysis was performed at the asymptotes. Theanalysis extracted the translational energy distri-bution of the products by a Fourier transform ofthe wavefunction in the asymptotic channel. Thefinal vibrational distribution of the NaF wasevaluated by accumulating the flux of the pro-jected wavefunction on the asymptotic vibrationalstates.The final state vibrational distribution of NaF

is shown in Fig. 3.These results indicate that exciting the initial

HF/D vibration results in a much hotter productvibrational distribution for both isotopes. Fig. 4shows the final translational energy distribution.

0 1 2 3Energy(eV)

0

0.1

0.2

0.3

0.4

<p>

NaFH ψ(0,0,0)

0 1 2 3Energy(eV)

0

0.05

0.1

0.15

0.2

<p>

NaFH ψ(4,0,0)

0 1 2 3Energy(eV)

0

0.1

0.2

0.3

0.4

<p>

NaFD ψ(0,0,0)

0 1 2 3Energy(eV)

0

0.05

0.1

0.15

0.2

<p>

NaFD ψ(4,0,0)

Fig. 4. The translational distribution of the Na � � �F–H/D relative motion.

458 G. Katz et al. / Chemical Physics Letters 359 (2002) 453–459

The final translational distribution seems toreflect a mapping of the FH initial vibrationwavefunction. This observation can be rational-ized by the light hydrogen atom which is respon-sible for most of the vibrational motion in the HF/D stretch. This is clearly manifested in the ob-served asymptotic relative translation.

4. Conclusions

To conclude, the 3D quantum calculationsshow the same general trends as the 2D calcula-tions. This means that the bending motion due toits low frequency is a spectator to the other degreesof freedom. These finding still leave the large dis-crepancy in the product yield between the fullyquantum and semiclassical calculations. In par-ticular, the new semiclassical calculations [17] onan improved potential energy surface still show avery high yield of NaF products and very shortexcited state lifetimes. The central role played bythe nonadiabatic effects in the dynamics suggeststhat a complete picture of the dynamics shouldinclude also the two nonreactive electronic excitedstates which correlate with the perpendicular Pexcitations on the sodium atom.

Acknowledgements

We thank J. C. Polanyi, D. Neuhauser and D.G. Truhlar for their help and assistance. This re-search was supported by the Israel Canada fund.

The Fritz Haber Research Center is supported byby the Minerva Gesellschaft f€uur die Forschung,GmbH M€uunchen, Germany.

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