The generator coordinate method for a reaction coordinate coupled to a harmonic oscillator bath Frederico F. de Souza Cruz, Maurizio Ruzzi, and André C. Kersten Schmidt Citation: The Journal of Chemical Physics 109, 4028 (1998); doi: 10.1063/1.477002 View online: http://dx.doi.org/10.1063/1.477002 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/109/10?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 200.145.3.79 On: Tue, 21 Jan 2014 19:29:27 http://scitation.aip.org/content/aip/journal/jcp?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/586982248/x01/AIP-PT/JCP_CoverPg_101613/aipToCAlerts_Large.png/5532386d4f314a53757a6b4144615953?x http://scitation.aip.org/search?value1=Frederico+F.+de+Souza+Cruz&option1=author http://scitation.aip.org/search?value1=Maurizio+Ruzzi&option1=author http://scitation.aip.org/search?value1=Andr�+C.+Kersten+Schmidt&option1=author http://scitation.aip.org/content/aip/journal/jcp?ver=pdfcov http://dx.doi.org/10.1063/1.477002 http://scitation.aip.org/content/aip/journal/jcp/109/10?ver=pdfcov http://scitation.aip.org/content/aip?ver=pdfcov JOURNAL OF CHEMICAL PHYSICS VOLUME 109, NUMBER 10 8 SEPTEMBER 1998 This a The generator coordinate method for a reaction coordinate coupled to a harmonic oscillator bath Frederico F. de Souza Cruza) Departamento de Fı´sica, CFM, Universidade Federal de Santa Catarina, 88040-900 Floriano´polis, Santa Catarina, Brazil Maurizio Ruzzi Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, 01405-900 Sa˜o Paulo, S.P., Brazil André C. Kersten Schmidt Instituto Oceanogra´fico, Universidade de Sa˜o Paulo, 05508-900 Sa˜o Paulo, S.P., Brazil ~Received 9 March 1998; accepted 5 June 1998! This paper investigates the usefulness of the generator coordinate method~GCM! for treating the dynamics of a reaction coordinate coupled to a bath of harmonic degrees of freedom. Models for the unimolecular dissociation and isomerization process~proton transfer! are analyzed. The GCM results, presented in analytical form, provide a very good description and are compared to other methods like the basis set method and multiconfiguration time dependent self-consistent field. © 1998 American Institute of Physics.@S0021-9606~98!50934-8# a ul - e s th e th n s th n- o he s an an o r n to di . I ns se to ule, cs. od, e- cilla- n- ant ount ator ar ced n am- I. INTRODUCTION Tunneling on a multidimensional energy surface is challenging problem in several areas of physics. In molec physics, several studies1–12 demonstrated that purely one dimensional calculations are not able to describe the isom ization and dissociation processes, as far as the effect vibrational modes are not negligible. In nuclear physics, observed fusion cross section of heavy ions collisions at ergies below the Coulomb barrier is much larger than prediction of one-dimensional potential models, and can o be explained by introducing the coupling to other degree freedom~for a recent review see Ref. 16!. In the literature, several papers1–3,12–20 have been de- voted to the development of reliable methods to treat tunneling coupled to other modes. In this work we investigate the applicability of the ge erator coordinate method~GCM! ~Ref. 21! to treat a tunnel- ing degree of freedom coupled toN harmonic oscillators. This investigation was largely stimulated by an old paper Makri and Miller,1 where the authors have analyzed t isomerization of the Malonaldehyde molecule using a ba set methodology. We tackle the same problem using an lytical version of the generator coordinate method.22–24 We shall prove that it is possible to find an effective Hamiltoni which takes into account the role of the other degrees freedom. We will apply the method to systems which a modeled by quartic and cubic potentials coupled to harmo oscillators, although it can be applied to general Hamil nians. Section II gives a brief account of the generator coor nate method and how to obtain the effective Hamiltonian Sec. III we apply the method for two simple Hamiltonia describing the proton transfer and dissociation proces a!Electronic mail: fsc1ffs@fsc.ufsc.br 4020021-9606/98/109(10)/4028/7/$15.00 rticle is copyrighted as indicated in the article. Reuse of AIP content is sub 200.145.3.79 On: Tue, 2 ar r- of e n- e ly of e f is a- f e ic - - n s. The formal developments of this section were then applied the isomerization process of the malonaldehyde molec Sec. IV, and the unimolecular dissociation, Sec. V. In Se IV and V the results are compared to the basis set meth1 semiclassical tunneling method3 and multi-configuration time dependent self-consistent field.2 In Sec. VI we present some final remarks. II. THE GENERATOR COORDINATE METHOD Let us consider a general Hamiltonian of the form H~p,x,p,q!5H1~q,p!1H2~x,p!1V~x,q!, ~1! where H1~q,p!5 p2 2m 1U~q! ~2! is related to the tunneling degree of freedom and H2~x,p!5( i 51 N h~xi ,p i ! ~3! is the Hamiltonian describing the intrinsic degrees of fre dom. They are assumed as an ensemble of harmonic os tors. The coupling term is of the form: V~x,q!5( i 51 N v i~q!xi . ~4! The main problem in the treatment of multidimensional tu neling is how to reduce the system dynamics to the relev degree of freedom and, at the same time, take into acc the effects of the other degrees of freedom. The gener coordinate method, originally derived to face the nucle many-body problem, can be useful to define this redu effective dynamics. The crucial point in this method is how to define a appropriate variational subspace where the relevant dyn 8 © 1998 American Institute of Physics ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 1 Jan 2014 19:29:27 th e o- l b y in io s w t . nd th t al o o on th e u e e c ro ia- tive liz- ef- tive ta- o- ve nt to tive be 4029J. Chem. Phys., Vol. 109, No. 10, 8 September 1998 de Souza Cruz, Ruzzi, and Schimdt This a ics can be realized. This is done by the introduction of following ansatz for the system wave function~the Griffin- Hill-Wheeler ansatz!, C~q,x!5E f ~a,b1 ,b2 , . . . ,bN!uw~q,a!& ^ P i 51 N uf~xi ,b i !&da db1 , . . . ..,dbN , ~5! whereuw(q,a)& is a generator state associated to the tunn ing degree of freedom,uf(xi ,b i)& are generator states ass ciated to the harmonic oscillators andf (a,b1 ,b2 . . . bN) is the so-called weight function. The ansatz above@Eq. ~5!# defines a variational wave function. The generator states constitute a nonorthogona sis. In some cases the variational subspace generated b ansatz corresponds to the exact Hilbert space of the orig problem. In this case the solution of the variational equat d^CuHuC& ^CuC& 50 ~6! leads to the exact solution, and the anzatz above can be as just another representation of the initial problem. Ho ever, this method is quite powerful when it is necessary reduce the problem to some relevant degree of freedom our specific case this should be done on physical grou that is, on an educated guess. In nuclear and molecular physics the usual picture of tunneling energy surface is that the reaction coordinate or fission ~fusion! degree of freedom, corresponds to two v leys connected through a saddle point on a barrier. The thogonal degrees of freedom, usually depicted as parab with pronounced curvatures, are naturally taken as harm oscillators. Within that framework we can suppose that harmonic oscillators are kept along the path close to th minimum energy states. Thus we can impose a minim condition, along the path, for the orthogonal degrees of fr dom: db i ^w~q,a!u ^ ^P i 51 N f~xi ,b i !uHuw~q,a!& ^ uP i 51 N f~xi ,b i !&50. ~7! The condition above defines a functional relation betwe the b i anda and we can rewrite the anzatz in Eq.~5! as: uC~q,x!&5E f ~a!ua&da, ~8! where ua&5uw~q,a!& ^ uP i 51 N f~xi ,b i~a!&. ~9! The introduction of Eq.~8! in the variational equation ~6! leads to the well known Griffin-Hill Wheeler equation, E 2` ` $^auHua8&2E^aua8&% f ~a8!da850. ~10! As long as it is possible to define this variational spa as a complete and closed subspace,23 the Griffin-Hill- Wheeler problem is equivalent to the solution of the Sch¨- rticle is copyrighted as indicated in the article. Reuse of AIP content is sub 200.145.3.79 On: Tue, 2 e l- a- the al n een - o In s, e he - r- las ic e ir m e- n e dinger equation for the projected Hamiltonian onto the var tional space. Further on we call this subspace as the effec space. The projection operator can be obtained by diagona ing the overlap kernel of the generator states,^aua8&, E 2` ` ^aua8&uk~a8!da852plkuk~a!, ~11! whereuk(a) is the eigenfunction andlk the eigenvalue. This allows us to define the momentum representation of the fective space with basis vectors as uk)5 1 Ap E da uk~a!ua& Alk , ~12! where we use round bras and kets for states in the effec space.~For details see Refs. 23 and 24!. The effective Hamiltonian in the momentum represen tion is Heff5ŜHŜ†, ~13! where the projection operatorŜ is Ŝ5E dkuk)~ku. ~14! Heff can also be written as a function of the effective m mentum and position operators~see details in Ref. 23!: Heff5ŜS ( m50 ` 1 2m : P̃mH̃ ~m!~Q̃!: D Ŝ†, ~15! where the ordering :. . . . : is ~16! and ~17! where $P̃,Q̃% are the canonical operators in the effecti subspace such that P̃uk)5kuk), ~18! Q̃uk)52 i ]/]kuk), ~19! Q̃ux)5xux). ~20! We should stress that for analytical results it’s importa to solve exactly the overlap eigenvalue problem in order obtain a complete and closed representation of the effec subspace. As it was noted elsewhere24 this is possible for any overlap of the Hilbert-Schmidt-type. Nevertheless it could ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 1 Jan 2014 19:29:27 aw a d rly to to th on fa he re r- th eter onic lete xact. ing: n set n- s of iag- we e- , by or- y- ent i- tes non- u- er- the - e the ing and set n- 4030 J. Chem. Phys., Vol. 109, No. 10, 8 September 1998 de Souza Cruz, Ruzzi, and Schimdt This a hard to find the solution in some practical cases. Such dr backs can however be circumvented by a numerical appro or a generalized Griffin-Hill-Wheeler anzatz.25 III. THE MODEL Now we use our method to study a system-bath mo which is well known in the literature. For theproton transfer case it consists of a quartic double well potential linea coupled to an harmonic oscillator and for thedissociation case it is a cubic potential barrier coupled to the oscilla That is: H~x,q!5H re~q!1Hosc~x!1cxq ~21! with H re~q!5H P2 2m 1 1 2 aoq22 1 3 boq3 dissociation P2 2m 2 1 2 aoq21 1 4 doq4 proton-transfer ~22! and Hosc~x!5 p2 2m 1 1 2 mv2x2. ~23! For the sake of simplicity we shall restrict ourselves the case of coupling to a single oscillator. As discussed Ref. 1, a term F~q!5 c2q2 2mv2 , ~24! is sometimes introduced to guarantee that the height of barrier in the original energy surface remains roughly c stant with increasing value of the coupling constant. As as we are interested in the comparison of our method others presented in the literature, we will mention whet this term is introduced or not. To construct our generator states, we start from a di product of two coherent states,26 ua& ^ ub&[ua,b&, related respectively to the$q,x% degrees of freedom. That is: ub&5exp@ba†2b* a#uf0&, whereuf0& is the harmonic oscillator ground state with cha acteristic widthA(\/2mv), and ba†2b* a5Amv \ S b2b* A2 D x2 i Am\v S b1b* A2 D p. ~25! For the reaction degree of freedom, we have ua&5exp@aA†2a* A#uw0&, where aA†2a* A5AmV \ S a2a* A2 D q2 i Am\V S a1a* A2 D P anduw0& is the harmonic oscillator ground state related to pair $A,A†%. The width of the a coherent state is rticle is copyrighted as indicated in the article. Reuse of AIP content is sub 200.145.3.79 On: Tue, 2 - ch el r. in e - r to r ct e A(\/2mV), whereV, calledgenerator frequency,is a free parameter. The results will depend strongly on this param and its choice will be discussed later. The reasons to employ coherent states for the harm oscillator is natural, because they do form an overcomp basis and the variational space generated by them is e For the reaction coordinate the reasons are the follow first, a real parameter coherent state is a localized functio~a Gaussian!, and as it was already noticed by the basis approach1 it can provide a good description for barrier pe etration problems; second, coherent states have a serie useful analytical properties and they especially have a d onalizableGaussian overlap.23 In order to obtain a one parameter generator state, shall enforce the condition~7!: ]b^a,buHua,b&50, which gives us b52xa, ~26! where x5 c \v A \ mv A \ mV . This minimization leads to the constraint relation b tween the generator coordinatea andb, where we assumed without loss of generality, thata is real. It was proved elsewhere23 that the same effective space is generated either complex or real parameter coherent states. It is imp tant to notice that the generator statesua& constitute a non- orthogonal basis. This nonorthogonality implies that the d namics, given by the Griffin-Hill-Wheeler equation~GHW! ~10!, couples the intrinsic states corresponding to differ points along the path~here defined by the generator coord nate a). Hence, if one remembers that the coherent sta corresponds to displaced harmonic oscillators states, the diagonal terms of the GHW equation would lead to the co pling between several harmonic oscillators levels. This p mits us to consider that, even with the implementation of minimum condition in Eq.~7!, the effective dynamics de scribed by Eq.~10! take into account much more than th average effect of the intrinsic harmonic oscillators along path. Therefore it allows a better description of the coupl dynamics and makes the difference between our method others like the self-consistent field and even the basis method.1,2 With the constraint relation~26! the ua,b& state can be described by a single parameter,a, yielding ua,b& [ua,b(a)&[ua) and the overlap function is ~aua8!5exp@2 1 2 ~11x2!~a2a8!2#. ~27! This overlap can be easily diagonalized@Eq. ~11!# by a Fou- rier transform giving the following eigenfunctions and eige values: uk~a!5exp@ ika#, ~28! 2plk5A 2p 11x2 expF2 1 2 k2 ~11x2! G . ~29! ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 1 Jan 2014 19:29:27 in e ze at th c d t l a r, d n se T f ta at to ss s an ro- ell ne ers. r- ed s to set ent her e es: cy e ral e pay he - on ld der the en- ctor ling nce- ec- lly m- il- r ven 4031J. Chem. Phys., Vol. 109, No. 10, 8 September 1998 de Souza Cruz, Ruzzi, and Schimdt This a Now, using Eqs.~12!, ~14! and ~15! we obtain the effective Hamiltonians, Heff5 P2 2meff 1Veff~q!, ~30! where meff5m«m~c,V!, Veff~q! 55 S 1 2 ao2 1 2 c2 mv2D q22 1 3 boq32cd q dissociation 2S 1 2 ao1 1 2 c2 mv2 2cptD q21 1 3 doq4 proton-transfer. ~31! The effect of the coupling to the oscillator is present the enhancement factor, defined as the ratio between th fective mass and the mass itself, «m~c,V!5 meff m 5 @11 ~v/V!~c/mv2!2#2 11~v/V!2~c/mv2!2 , ~32! and by terms modifying the potential which are characteri by the coefficients cd5 3 4 \bo mv ~v/V!2~c/mv2!2/~11 ~v/V!~c/mv2!2 !, ~33! cpt5 3 2 \do mv ~v/V!2~c/mv2!2 11 ~v/V!~c/mv2!2 . ~34! One can easily see that, independently of the gener frequencyV, as the coupling strengthc goes to zero, the effective Hamiltonians revert back to the reaction part of original ones. This is a desired consistency property. One also see that as a consequence of the linear coupling an particular choice of generator states the effective mass is same for the cubic~dissociation! and for the quartic potentia ~proton transfer!. In the two cases, the factor@ 1 2(c 2/mv2)#) in the quadratic coefficient can be eliminated if the origin Hamiltonian has the renormalization termF(q). The generator frequencyV is a free parameter, howeve rather than fit, we decided to choose the parameters base an educated guess. This will be discussed with the prese tion of the results for each case. Since the effective mass is common for the two ca above, we can make some remarks valid for both cases. mass is, as expected, renormalized by the presence o other degree of freedom. Thus the effective mass has enhancement factor which depends on the coupling cons c, on the parameters of the oscillator and also on the r (v/V) between the generator frequency and the oscilla frequency. It doesnot depend on the position, neverthele the enhancement factor can be quite large in some case Now we shall deal in detail with each case to present discuss our results. rticle is copyrighted as indicated in the article. Reuse of AIP content is sub 200.145.3.79 On: Tue, 2 ef- d or e an the he l on ta- s he the an nt, io r . d IV. PROTON TRANSFER CASE: SYMMETRIC DOUBLE WELL POTENTIAL We applied the method to the proton or H-transfer p cess in the malonaldehyde molecule which is fairly w modeled by the double well potential coupled linearly to o harmonic oscillator. We use two different sets of paramet In the first case,a0 andd0 were chosen so that the ba rier height is 7.8 kcal/mol, and the local minima are localiz at 60.53 Å. We consider two bath frequencies: a slowv 5298 cm21 and a fast onev52982 cm21. Those values were the same as used in Ref. 1 and this choice allows u compare our results to the results obtained by the Basis method. In this case the renormalization termF(q) is present. In order to compare our results with a time depend self-consistent field approximation, we also consider anot set of values fora0 and d0 such that the frequency at th local minima is 1530 cm21 and the barrier height is 6.3 kcal/mol.2 Our first task is to choose the generator frequencyV. In the double well case we have three ‘‘natural’’ frequenci the oscillator frequency, the bottom of the well frequen and the barrier frequency~associated to the curvature at th barrier!. Our intention was to choose among those natu frequencies a value forV. As far as we are interested in th tunneling process in the presence of the oscillator, we special attention to the oscillator frequencyv and the barrier frequencyvbarrier. It can be seen in the expressions for t effective potential and mass that (v/V) acts as a multiplica- tive factor for the coupling. Thus the choice ofV is crucial. We tried then to find a domain ofVs, within which, changes of the value ofV would not lead to a meaningful modifica tion of the dependence of the effective potential and mass the coupling constant. This would give a region which wou be the most independent of the choice ofV and which de- pends almost exclusively on the coupling constant. In or to do so, we analyzed the curvature of the potential, height of the barrier and the mass as a function of the g erator frequency and the coupling constantc, and we chose V as one of the already mentioned natural frequencies,v or vbarrier, depending on which one was in this mostV stable region. This analysis lead us to the following choices: ~a! for the fast bath case (v.vbarrier) the best choice is V5v; ~b! for the slow bath case (v,vbarrier) is V5vbarrier. In Figs. 1 and 2 we have the mass enhancement fa for the fast and slow bath cases as a function of the coup constant. We notice that for the slow bath case the enha ment can be quite large. A typical dependence of the eff tive potential on the coupling is also depicted in Fig. 3. We now compare the GCM results with numerica generated exact results and with the basis set method.1,27 First, for the slow bath case, the GCM results are co pared to the ones obtained with the effective system Ham tonian ~ESH! of the basis set method of Makri and Mille ~see Fig. 4!. We can see that the GCM results are better e ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 1 Jan 2014 19:29:27 d h he en e o lf ot tic io he to lear the ator the nd rator tial ned ob- g all start e e the e - 4032 J. Chem. Phys., Vol. 109, No. 10, 8 September 1998 de Souza Cruz, Ruzzi, and Schimdt This a when compared to the second order corrections. In the fast bath case the GCM and the second or corrections for the basis set method cannot be distinguis from the exact result~see Fig. 5!. It is worth mentioning that if we had considered only t diagonal part of the GCM kernel we would have a treatm equivalent to the self-consistent field method. For the sak completeness we also compare our results to the ones tained by the multiconfiguration time dependent se consistent field presented by the same authors2 ~Fig. 6!. In this case we use the second set of parameters for the p tial. The comparison favors again the GCM calculations. V. DISSOCIATION PROBLEM The effective potential displays a very familiar quadra correction2 (c2/ko) x2 which would disappear ifF(q) were present. There is also a more complicated linear correct FIG. 1. Mass enhancement factor«m as a function of the square of th coupling constantc2, for an oscillator frequencyv52980 cm21 and a gen- erator frequencyV5v. FIG. 2. Mass enhancement factor«m as a function of the square of th coupling constantc2, for an oscillator frequencyv5298 cm21 and a gen- erator frequencyV5vbarrier. rticle is copyrighted as indicated in the article. Reuse of AIP content is sub 200.145.3.79 On: Tue, 2 er ed t of b- - en- n, which depends explicitly on the generator frequencyV, and is quite similar to one of the quadratic corrections in t proton transfer case. The behavior of the effective potential with respect the coupling can be seen in Fig. 7, where it becomes c that the effective potential can be extremely sensitive to coupling constant, which makes the choice of the gener frequency much more delicate.28 Following the same basic arguments of the previous section we again examined behavior of the barrier height, curvature of the potential a mass as a function of the coupling constant and the gene frequency. In this case the strong sensitivity of the poten led us to chose as a first hintV5v, which seems to be a good zero order choice. In Fig. 8 we have results obtai with this choice. The results are compared to the one tained by the Makri and Miller semiclassical tunnelin model3 and the exact one. The results are good for sm values of the coupling constant, but we can see that they FIG. 3. Typical behavior of the effective potential for some values of coupling constant. FIG. 4. The tunneling splitting as function of the square coupling forv 5298 cm21, V5vbarrier. The barrier height is 7.8 kcal/mol and th minima position are60.53 Å. The GCM results~* ! are compared to the exact results~solid line!, the zero order results~dashed line! and the second order perturbation theory~dot-dashed line! obtained from the effective sys tem Hamiltonian~ESH! of the basis set method in Ref. 1. ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 1 Jan 2014 19:29:27 f t o fo s a lin on - e n i an d in hat nd re- al the il- ink th it - ffi- al or. tter a t e t ne th the d 4033J. Chem. Phys., Vol. 109, No. 10, 8 September 1998 de Souza Cruz, Ruzzi, and Schimdt This a to drift apart for values ofc2 around 0.0005~mdyn/!Å 2. This feature seems to come from the strong dependence o effective potential with the coupling constant. An analysis the main features of the effective potential showed that high coupling, the effective Hamiltonian no longer display region stable against variations ofV. We try then to intro- duce a generator frequency depending itself on the coup constant. Keeping in mind that in the small coupling regi the initial choiceV5v shows good results, we tried to iden tify V as a renormalizedv. Observing the presence of th correction on the quadratic term of the effective potential a also the mass enhancement factor, we propose the follow correction: FIG. 5. The tunneling splitting as function of the square coupling forv 52980 cm21, V5v. The barrier height is 7.8 kcal/mol and the minim position are60.53 Å. The GCM results~* ! are compared to the exac results~solid line!, the zero order results~dashed line! obtained from the effective system Hamiltonian~ESH! of the basis set method in Ref. 1. Th second order perturbation theory of the basis set method gives results cannot be distinguished from the GCM and exact ones. FIG. 6. The GCM tunneling splitting is compared to the results obtai with the multiconfiguration time dependent self-consistent field~Ref. 2! and the exact results. In this case, the barrier height is 6.3 kcal/mol and frequency at the local minima is 1530 cm21. The frequency of the bath mode is 2980 cm21. rticle is copyrighted as indicated in the article. Reuse of AIP content is sub 200.145.3.79 On: Tue, 2 he f r g d ng V5Ak02 ~c2/a0! meff 5vA12 ~c2/a0mv2! «m~c,v! . ~35! There is a good improvement with this prescription as it c be seen in Fig. 9. We stress the excellent results obtaine the small and intermediary regions. We also point out t this ‘‘intermediary regions’’ present a tunneling rate arou 10 000 times larger than the ones in the small coupling gion. So, the word ‘‘intermediary’’ is, as a matter of physic effects, a bit misleading, and the agreement achieved by method is remarkable. One of the interesting features of the effective Ham tonian is the resemblance with the one obtained by Br et al.,29 with a completely different approach. In fact, bo are identical in the small coupling region and in the lim v@vbarrier. In this very region, Brinket al. analyze the spontaneous fission of239U, and obtain very reasonable re sults. Due to the completely different approaches it is di cult to compare the higher order corrections from the form point of view, however, it seems to have a similar behavi The relationship between the two methods might be be discussed in a subsequent paper. hat d e FIG. 7. Typical behavior of the cubic effective potential as a function of square of the coupling constant for the dissociation case. FIG. 8. The GCM results, withV5v, for the log of decay rateT normal- ized by the decoupled (c50) decay rateT0 are compared to the exact an semiclassical method~SCM! results~Ref. 3!. The barrier height is 7.4 kcal/ mol and the local maximum occurs at 0.71 Å. ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 1 Jan 2014 19:29:27 o fe re is ith b ia d lly an e th th to s th th o rs an s o a w e re za th ator hat e gen- . 25. ing on ost. th- ive g R. up- on, m. and anta 96. ra 4034 J. Chem. Phys., Vol. 109, No. 10, 8 September 1998 de Souza Cruz, Ruzzi, and Schimdt This a VI. CONCLUDING REMARKS As seen in all results shown, the GCM provided a go description for the system-bath model. In the proton-trans process the results are virtually exact, and no further cor tions on the generator frequencies were needed. In the d ciation like case the effective Hamiltonian is consistent w the one obtained with a semiclassical approximation Brink et al.29 for a nuclear case, and the molecular dissoc tion rates predicted are quite as good as the ones provide the semiclassical tunneling model. In this work, the main results were obtained analytica which allowed us to analyze in a more transparent way with no numerical cost the role of the coupling to other d grees of freedom. As an example, we should mention role of the mass enhancement factor. The presence of factor which comes naturally from the method turns out be, in some cases, more important than the modification the one-dimensional effective potential. This suggests sometimes the intrinsic degrees of freedom can affect system more on its inertia than on its potential. The role the mass renormalization is also stressed in some pape20 Furthermore the analytical form of the effective Hamiltoni gave us means to choose the generator frequency on phy grounds rather than fitting. The GCM is a variational method and the accuracy the results depends on a good choice of the generator st The fact that our effective subspace~variational space! is virtually the correct one for the isomerization problem sho that coherent states do provide a good ansatz for this typ problem. In the dissociation problem we have a good ag ment for the small coupling region, however, a renormali tion of the frequency was required for large values of FIG. 9. The same as the previous figure using the normalized gene frequency@see Eq.~35!#. rticle is copyrighted as indicated in the article. Reuse of AIP content is sub 200.145.3.79 On: Tue, 2 d r c- so- y - by d - e is of at e f . ical f tes. s of e- - e coupling constant. In this case, a new choice of gener states could improve the predictions, but, it is quite sure t we will not have the simplicity of the analytical results w have with coherent states. Nevertheless we can use the eralized generator coordinate method as proposed in Ref In this case, to solve the quantum mechanical problem us the nonorthogonal basis, we will need only low dimensi matrices. So we do not expect a large computational c Such improvements are being presently investigated. So, the GCM, although based on a very heavy ma ematical foundation~one of the reasons of its accuracy!, is fairly simple to apply and it generates an analytical effect Hamiltonian which allows a rich problem analysis givin extremely good results. ACKNOWLEDGMENTS This work is partially based on the Master thesis of M. and A.C.K.S. and was made possible by partial financial s port of CAPES, CNPq and FINEP. 1N. Makri and W. H. Miller, J. Chem. Phys.86, 1451~1987!. 2N. Makri and W. H. Miller, J. Chem. Phys.87, 5781~1987!. 3N. Makri and W. H. Miller, J. Chem. Phys.91, 4026~1989!. 4W. H. Miller, J. Phys. Chem.87, 3811~1983!. 5S. Baughcum, R. W. Duerst, W. F. Rowe, Z. Smith, and E. Bright Wils J. Am. Chem. Soc.103, 6296~1981!. 6S. Baughcum, Z. Smith, E. B. Wilson, and R. W. Duerst, J. Am. Che Soc.106, 2260~1984!. 7E. M. Fluder and J. R. de La Vega, J. Am. Chem. Soc.100, 5265~1978!. 8E. Bosch, M. Moreno, J. M. Lluch, and J. Bertra´n, J. Chem. Phys.93, 5685 ~1990!. 9N. Shida, P. F. 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