15Adiabatic Reaction Dynamics

15.1 Reaction Kinetics and Rate Constants

Considcl an al .h i r r .ary cqui l ibr . ium systern

A + B + C +.. . . - . . .+x +Y +z ( l -5.1)

*ntt:.*, is a phcnomenological rate constant (distinguished ltom an elernentary rate constartas.defined later on)' [wr rcpresents the concentration of species w (usuaily exfresr",t inunits ol ' rnolari ty o| palt ial plessule), and each concentmtion term has associated with i t arrexponent that is sonretinres t'cl'errecl to as tlie 'molecularity' of the species. often. but rr.ralways, nrolecularities have integrar values, incruding zero. Note that since we are measurirrsa retunl [o cquilibliunr. all concentration tenns are tunctions oftime t, as are ft4 and thc r,atci tsel l .

The a prirtri prediction of ail of the variabres appea.ng on the r.h.s. of Eq. ( r.5.rt isa challcnging task' to say lhe least. This is palticularly tlue because the equiiihriurrr .,1Eq' ( l 5' | ) tnay involve the sinrulterneou. op"roiion of a large number of indiviclual c6crrrt.;rlreact ions, wi th sonre possibly involv ing very low concentrat ions of r .eact ive i r . r tcr 'c( t r ; r r r \ .the presence ol 'whish nray be di f f icurt to estabr ish exper inrentalry. In order to rr i lkc l ) r1) . ! r . ( . \ \ .a cr i t ical s inrpl i l icat ion is to break the ovemll pro. . . , dn*n into so-cal led elerrrcrrr ; r r r , . r . . ; , r .To sinrpl i fy l l l i l t ters i l t r i t . we wi l l consider only adiabat ic react ion steps, that is . r .err t . r i , , r r rtaking place on a s ingle PEs without any change in electronic state ( the [gpic , l r r . r r : . l i r r l r ; r r r tdynattr ics is c l iscussed hr icf ly in Sect ion l -5.5). For pract ical pufposes, thcrc:rre otr l r t r r , rkinds ol' elenrentary reactions: unirnolecular. and birnolecular.

where no particular stoichiorletry is inplied. when the systern is displaced from equilibriunr.by

^ddition of'rore of a particula' species, by a change in temperature and/or pressure, or.

by any other inl]uence' ernpirical observation has shown that the rate at which equilibriurnis reestablished rnay be expressecl us

I i r rc( / ) : k, , ( , ) [Al ' lBl / , lc l ' " '' ' tx l ' lYl ' lz l '

E.r:.iltiill: t'l (\\nl4t.tit,nill Cltr,ti.ttt.\.. 2il1 l)tirirrt Chrisr.Phcr J. Cronr* =--

O l{X). I iohn Wilc l , ct Sons. Lkt ISBNS: 0,-170 09i l i l -9 {crs(.d): 0-.170-09lSt-7 (rrht l

( l -5.2)

522 15 ADIABATIC REACTION DYNAMICS

small concentration of C over the course of the rneasurcment (or choosing an irreversiblereaction) and a vast excess of B. In that case, the mte expression becomes

(15.13)

which is identical to Eq. (15.7) except that ki, the so-called pseudo-filst-order rate constantwhich rnay be rneasured in exactly the fashion already described above for a normal lirst-order rate colrstallt, is the product ol'the second-order rarte constilnt ftt ilnd the eff'ectivelyconstant [Bjs when B is in excess.

Bimolecular reactions having rnore products than the single species produced from acondensation are also possible, and theil rate laws iu€ coustlucted aud tneasured in a fashionanalogous to Eqs. (15.12) and (15.13). Note that the special case of a birroleculal reactioninvolving two molecules of the sarne leactant hrs n rate law that is part icularly simple tointegrate and work with.

15.2 Reaction Paths and Transition States

Theory may play two panicular ly inrpol t :u.r t ro les in lat ional iz. ing ancl predict ing chcrnicalreact ion dynamics. As noted in t l re last sect ion, t l re l i rst stcp to understancl ing the dynam-ical behavior of a complex chernical systern is breaking down rhe ovet 'a l l systetn inro iLsconstituelrl elementary processes. From a theoretical standpoint, thc likely illtportance ol'various processes may be qualitatively assessed fronr the potential energy surllces ol'puta-tive reactions. Reactions with vely high barriers will be less likely to play an inrportant role.while low-barrier rcactions will be rnore likely to do so.

Moreover, the PES helps to define the scope of each elelnentar-y reaction. Thus, for-instance, a binroleculal conderrsation that involves the lbrrnation ol'two new bonds betweenthe reacting species rnay either proceed in a concerted tashion, with only a single predictedTS structure, or it may proceed as a stepwise process with two TS stfuctur.es; the srepwtscprocess is really two elementilry reactions - first a conclcnsation and thcn a unilroleculalrearrangement.

To say that an overall plocess irrvolves two dill 'elent TS stluctules presupposcs, howevei.some sort of trajectory thrt thc reacting system rnaps out on thc PES. In gcnc|al, wherrc l lemists th ink of a system rnoving on a PES, they tend to th ink about l p i l r t icular path cal le( tthe rninimum-enelgy path (MEP) or sorlletimes the iutr-insic relcrlon coordinate (lRC). ThcMEP is the path downwards f lor l a saddle point to a nr i r r inrunr that would bc lb l lowed h1,a ball rolling on at surface if its velocity wele infinitely dautped at cvcry point; arr exarnplcol 'such a path is given in Figuie 1.4. When the potent ia l energy sur l 'uce is explessed inrnass-weighted coordinates, the MEP is also the path thlt lollows tlre steepest gladient arcvery point. The rnass-weighted Cartesian coordinates lbr an atont are sinrply the Cerltesiancoordinates scaled by the square root ol ' the atorrr ic u.r i iss. Mass-wcighted intct 'nal coordinatcsci ln be generated by diagonal izat ion ol ' the nrass-weighted Caltesian coofdiuate fb lce const l r r rr t ratr ix. This coordinate system is et very convenierr t one in which to wolk s ince t l re gladientsl i r l r t tarry electronic structurc methods are avai lable to f i rc i l i tate the fb l lowine o1' the MEP.

-dlAl : ra,IAI

I5.2 REACTION PATHS AND TRANSITION STATES 523

tt is wor.th digressing for a moment to note that following an MEP is often crucial to

understanding the natule of a TS structure. Sometimes, when a molecule has a single inrag-

inary frequency, visualization of the corresponding normal mode does not necessarily makc

it obvious what the reaction coordinate is. It can often happen that the TS structure that

has been located conesponds to some process other than the one of interest, e.g., a TS

structure tbl the internal rotation of a methyl group may be found when the desired TS

structure was for some bond-making or bond-breaking process. In such a case, following

the MEP will lead, in each dirrection, to the ultimate minimum energy structures connected

by the TS structure. On complex potential energy surfaces, such connections can be critical

to understanding the ovelall topology of the PES (see, for instance, Gustafson and Cramer'

l99s).Although the MEP and its connection to TS structurc(s) is tremendously useful as a

conceptual tool, it can also be somewhat rnisleading to the extent that it focuses analysis on

the PES itself. It should always be kept in mind that the equilibria and kinetics of reacting

systefns ale nearly always governed by the free energy of populatiorts of molecules, and not

the potential energy of single molecules. To the extent the tiee energy describes a thermaldistribution of particles cornposing the reacting system, one may think of the system as acloud hovering over the PES, with the density of the cloud thinning as it rises according toBoltzmann statistics. Within the cloud, individual molecules may be exchanging energy withone another to rise and fall relative to the PES, but the net distribution remains dictated bytelnperature. A reacting systern may be thought of as a cloud over the PES headed towardsa mountain pass whose saddle point is the TS structure. However, the passage of the cloudover the pass need by no means take place directly over the TS structure. Depending onhow wide the pass is and how tall the cloud is, many cloud particles may be able to passarbitlarily tar to the left and right of the TS structure (when the pass is very narrow onesays that the reaction has an entropic bottleneck, meaning that little variation in degrees offreedom other than the reaction coordinate is permitted).

So, while the TS stlucture, by virtue of being a stationary point on the PES, can bcinformative lbout the height of the pass, and local topology (by Taylor expansion of thesurface about the stationary point), it is only one representative of the populatiort of moleculespassing liorn reactants to products. As such, one should be rather careful not to confuse thcTS stnrcrure, which is thc stat ionary point, with the transit ion state. which may be somewhatmore rigolously delined lbr an N-atonr systenl as a surfuce having 3N - 7 degt'ecs ol'

freedorn (i.e., one less than the reactants) through which the reactive flux is maxirnizctl.'l-lr:rtis, the ratio ol'the nurnbcr of nrolecules crossing the surface in the direction reactilnls .*

products to the numbel crossing in the opposite direction in a given time interval is trtitxirttrrl.To Inake the distinction between the TS structure and the transition state morc clcar. it ir

helptul to retul'n to a somewhat older term for the latter, namely, the'activated cotttPlt'r'.The rernaindel of this chapter will hew to this distinction as closely as possihlc.

Returning to kinetics, while theory ctn be advantageously used to decotrtp()sc l (r)nl l) l ( ' \

systenr into i ts consti tuent series ofelementary reactions, we have not yct t lcstr i l t t ' r l ; rrrrrelat ionship between a theoretical quanti ty associated with the individual clcnlcntir l r ' ( i r( tr i 'n\and their fblward and reverse rate constal l ts. I t is axiomatic that leactiurts rvi l l r l r i r l r r ' i l t rr ,r

526 15 ADIABATIC REACTION DYNAMICS

out the partition function for the reaction coordinate degree of freedom (see Eq. ( 10.2g)) andwnte

k,: kt Qx u-u1, rJat t lkrT

' l_r-hqlkargo'

where Q+ is the reduced partition function over the 3N - 7 bound degrees of freedom andar1

.is the 'vibrational frequency' associated with the reaction coordinate. If we use a powerseries expansion for the exponential function of r,.r1 on the r.h.s. of Eq. (15.20), truncarrngafter the first two terms, we have

K+Kp I ( )+- 1" '7

"-(ut t -ut t t /hrl ta+ Q^

Notice that the only two unknowns remaining are ki and ro+. In this case, rhe vibrationalfrequency tr1 should not be thought of as the imaginary frequency that derives from thestandard harmonic oscillator analysis, but rather the real inverse time constant associatedwith motion along the reaction coordinate. However, it is exactly motion along the reactioncoordinate that converts the activated complex into product B. That is, k+ = ,u.+. Eliminatingtheir ratio of unity from Eq. (r5.2t) leads to rhe canonical rST exoression

*, : b!Qr "- tu i . , ' - .u^, , t / t t l' hQe-

fre-wtu-ttn'tl*"t'

( r5.20)

( l -s.2 I )

(15.22)

/r<r.rr \" l

ksT /

k, : kBT Q+

e- (Jt . , , , (J. t .u-ut t t r ,n l' h QeQs-

A point ofoccasional confusion ar ises wi th respect to uni ts. In Eq.(15.22), a l l portronsare uni t less except for ksT/h, which has uni ts of sec-r , ent i re ly consistent wi th the uni tsexpected for a unirnolecular rate constant. In Eq. (15.23), the same is t rue wi(h resDecr rothe r .h.s. , but a bimolecular rate constant has uni ts of concentrat ion*r sec-r , which seernsparadoxical. The point is that, as with any thermodynamic quantitp orc r-nust pay croseattention to standard-state conventions. Recall that the magnitucle of the translationai partitionfunction depends on specification of a standard-state vorume (or pressure, uncler ideal gascondi t ions). Thus, a more complete way to wr i te Eq. (15.23) is

For the bimolecular reaction case involving reactants A ancl B, the derivation above gener-alizes to

( 1s.23)

kt:ksT Qtl t QxQs

(t5.24)Q|, Q' i , , - tu 1, - t l to - IJ t r t / kr l

Q+." -

where the various Q" terms have values of one and carry the standard-state volurne unrrsused for the rranslational partition function (the same generalization applies to Eq. (r5.22),

I5.3 TRANSITION-STATE THEORY 527

but it is sufficiently rare for a unimolecular reaction to have different standard-strtc voluttlcsfor the activated complex and the reactant that one rarely gives thought to this point). ('lrc

must be taken then such that if the molecular translational partition function is conrputctlfor a volume of, say, 24.5 L (the volume occupied by one mole of an ideal gas at 29tl Kand I atm pressure), and the rate constant is in, say, molecules cm-3 sec-1, the appropriltcconversion in standard states is made.

In a very general form, then, we have the canonical expression

where R refers generically to either unimolecular or bimolecular reactants, and AV+ is thedifference in zero-point-including potential energies of the reactants and TS structure. Whenworking in molar quanti t ies, Eq. (15.25) becomes

,. keT Q+ Q"a .._p+tr,,rn: h o^g* '

, kBT Qr QL . . - tvr tnrK: -- e

h Qp Q+'o

( 15.25)

( 15.26 )

in which case one often absorbs the standard-state partition functions back into the expo-nential to write

, ksT,Ac,t tprk: _!__"-aw rr , (15.27\

where AG'tx is ret'elred to as the free energy of activation. Note that using Eq. (10.6) we

may also write

( 15.28)

15.3.1.1 Relatiotr between theory and experinent

Operationally, the theoretical computation of a rate constant using TST typically enrploysEq. (15.26). One locates all necessary stationary points -one TS structure and one or twominima - and evaluates their energies and their partition functions under the rigid-rotor-harmonic-oscillator approximation. Experiment, on the other hand, measures rate constantsaccording to the methodologies outl ined in Section 15.1, typical ly with the goal of dcrivingsuch quantities as the free energy of activation. However, the experimental da(a rnly heanalyzed in a variety of ways, and it is critically important to ensure experimental/thcolcticrrlcomparisons are made under consistent conditions.

One analysis of experimental data involves carrying out rate constant measurcrncuts ;rl :lseries of temperatures, and then plott ing ln(klT) against l /Z (a so-cal led Eyring pkrt). Wcmay rearrange Eq. (15.28) to

k: W"- l , r"+IRr e^s"r IRlt

'"(;): #.#-'' ' '(f) { | 5. .1 ' ) r

530 I.5 ADIABATIC REACTION DYNAMICS

To a rough approximation, then, in the limit of a lirlly bloken bond in the TS structurc, rheprirnary KIE is

./lrigr',

- "-]r,.i"1,. --;;,'l^".

/ \hclvy( r5.34)

where c,-rR't refers to the frequency in the reactants of the bond being broken in the TS srructure.One of the laryest possible difl'erences in isotopic tiequencies involving elenrents occurs forhydrogen/deuterium substitutions, with X-H bonds typically having stretclring li.equenciesabout 357o larger than X-D bonds. Using this relarionship and a light isor.ope fi.equency ol3l00crn-r, Eq. (15.34) suggests thar the maxinrurn primary KIE lbr a hydrogen/deuterium-substituted system at 298 K is about 7. ol course, il thc bond is not lirlly br.oken in theTS structure, smaller values may beobserved/cornputed. Larger values uray also be observed,owingtoquantummechanical tunneling,ersdescribecl inSecti<ln l-5.3.3.Whenheaviererementsare used, isotope effects becotne snral ler, bu( a number of experirnental techniques have provento be sufficiently accurate to lrclsure very srnall difl'erences (see, lilr instance, Keating et n/.l 999).

Secondary KIEs are also typical ly rrruch smaller than pl iuraly KIEs. because thc isotopi-cal ly substi tuted modes are not lost in the TS st[r.rctutc (see Figur.e 15.3). tn addit ion,

Reaction coordinate

Figure l5'3 Sccondary KlEs arc associatcd with nornral rlotlcs othcr t lr i ln thc rcaction crnrtl inatc.one of which is slrown in this diagram. Thc lrcavy and liglrt vihrational lrcquencics hoth changc ongoing lronr the reactant (R) to the TS structure (i): hccausc in this cxantplc thc rnodc is 't ighter' inthc TS structutc, thc dil lbrencc hc(wccn tlrc lrcavy ernd l ighr ZPVEs incrcascs. and this cirr.rscs thepotential energy ol'activation to hc larger lor the l ight isotopelrcr t lr irn lhc ltcavy onc (tn cxiluple ol 'an invcrse secondlry KIE). In a rcil l Inrny-atenl systgnr tlrcrc i lrc potcntially r largc nunrher ul ;.ro6csthat wil l contribute 1rr the secorrdary KlE. sorle in a nornral l irshion antl sorlc in an invcrsc l irshi6n

| . . .

I5.3 TRANSITION-STATE THEORY 5.I I

secondary KIEs can be ' inverse' , which is to say that the l ight i l to l l l r i l lc t t t t ' t l l t t ' l t t ; t r r

atorrl rate can be less than one. In this case, no particular simplificatiorts ol l lt1. { | ). i I t rrr t

general, and each partition function rnay play a role in addition to th()sc ol tlrr' /.ii\ | t

This is parlicularly true because difTerent vibrational modes may cancel onc itttolltet rtt llt,'

secondary KlE. That is, one mode rrray lead to a large normal KIE but be carrcclctl h\' "',,|l ',

,

mode that leads to a large irrverse KlE, such that more subtle effects associlltc(l *'il lt. ';tr.rotational motion. lnay be made tnanifest.

Orre caveat that must be observed when conrparing computed and experinrentitl isrtlil|}r'

et'tects is that experinental tneasurements can sometimes be for multistep reactions. Wltclt rt

particular elemental'y reaction is not rate-deterrnining, that is, it is not the bottleneck irt tlte

over.all process. then it does uot matter whether or not that leaction has associated witlr it :r

large KIE; i t wi l l not inf luence the observed overal l rate. A separate caveat wi th l ight at t t t l ts

at low to ntoderate tetnperatules is that tunneling eft'ects may play a significant role.

L5.3.2 Variational Transition-state Theory

Canolical TST dctines the free ener-qy of the activated contplex based on the TS structtrrc.

This is convenient becar,rse, as i t is a stat ionary point , we cal l use the machi t re l 'y o l - (hc

r.igid-r.otor-harmonic-oscillator appt'oxirnation to colnpute the necessary paltition functiotts

to deline its 1r'educed-dirnensionality) iiee encrgy. However, it is by no means guarantcc(l

thar the free energy associatcd with the TS structure really is the hig,hest free energy ol'any

point along the MEP - it is only -quaranteed

that it is the highest point of Potential energy

along rhe MEP. As r simple example, it rnight be the case that the potential ene€y wclls

associated witlr sorne nornral modes tighten up after the TS structure is reached, even thou:lh

the bottoms of thosc wells rre at a point on the MEP slightly below the energy of the TS

str.ucturc. The increase inZPYE resulting fiom those tighter potentials may exceed thc tlt'o1t

in bottoln-ol-the-well enetgy such that the fi'ee energy of the non-stationary poillt is higltcr

than that of the TS structure.

Variational transition-state theory (VTST), irs its name implies, variationally nl()\'cs llr('

reference position along the MEP that is ernployed for the computation of the aclivrrlt'tl

cotnplex fr.ee energy, either backwards or forwlrds tiom the TS structure. until thc Irrlt'

coustant rs nr in imiz-ed. Notat ional ly

*vrsr17..r . ) : rn in *nr Qi ' ( ! 's) 9. \ \ c-av1{r) / t . r ( t5. \ i )' ' ' h ex e+-"-

wherc s is a posi t ion on the MEP at which kvTsr i ( evaluated. By convent ion. , t : 0 r t ' l i ts l r r

the sa6dle point , and negett ive and posi t ive values ate displaced to the react i l l ) l i t t t t l l t t t r l t r r ' l

s ides of the saddle point , t 'espect ively.

To courpute the r . l r .s. of Eq. (15.35), we need to def ine how we cottrptr tc l l tc | ) ; t t l t t r , , r r

funct ion (ancl the ZPVE) fb l the non-stat ionary point s. ln th is case, wc si t t t l t lY tot t l t t r r r r ' t , ,

take advantilge ol' our clecision to treat the activated complex as a spccics lt;tr itt| i \ /

bound degrees of freedorn. ln order to define this space fbr an arbitrlrly ltoittl ort tlr, \ll l '

i :

-5-14 I5 ADIABATIC REACTION DYNAMICS

rcilctants to products in a classical system is a Heaviside function ol'the energy, as illustratedin lrigure 15.4. rn a quantum system, however, the transmission pr.obability is sigmoidal inshape, reflecting the phenomena oftunneling and non-classical reflection. Tunneling refers totlre ability of a quantum systetn having an energy below the sacldle point to .tunnei' throughthe banier to the products side, while non-classical reffection ref'er.s to the possibility that aquantum system above the saddle-point energy will suffer from destructive intert'erence ina way that prevents it from crossing to products. This situation is compared to the classicalone in Figure 15.4.

Insofar as tunneling increases the rate constant by allowing lower-energy systems to bereactive and non-classical reflection decreases the rate constant by r.eclucing the r.eactivity ofhigher-energy systems, one rnight irragine that the two could safely be assumecl to cancel.However, a therrnally equilibrated Boltzrnann population has a rnuch larger percentage of

I5.3 TRANSITION-STATE THEORY

low-energy systems than high-energy ones, so tunnel ing ef fects tencl l ( ) l ) rc( l { rnan;r l r 'o\( lnon-classical ref lect ion, and the inclusion of quantum mechanical tunncl ing e:rr h ' . . r ' r r t r r ; r l

to predicting accurate rate constants. Within a TST (or VTST) framewo[k, ttttc s'tits

5.15

Temperalu re-dependentBoltzmann distribution

-classical

wherc r is called the transmission coefficient. The transmission coefficient is l lirr\'tr,rl

of temperature and, importantly, of the shape of the PES in the region about thc uelirrltrl

cornplex. In the classical limit r : I but, particularly at low temperatures, K cln bccorrrr'

arbitrarily large.

Qualitatively, r depends on the shape of the barrier (both height and width), tlrc tttrtss

of the particle (the lighter the particle, the greater the probability of tunneling), arttl tlrt'

telnperature, the latter because i" dictates the.Boltzmann population of the reactrnt :urtl

activated-complex energies. In the context of a many-atom system, tunneling through llrc

barlier nray occur along any one or more coordinates, and the mass in question lol cltelt

case may be considered to be the reduced mass of the normal mode. Thus, tunneling clll 'ets

can be present even when the reaction coordinate itself is dorninated only by heavy-atorrt

mot iou.

Highly accurate prcdiction of transmission coefficients including many degrees of ll 'cctlorl

is a very difficult quantum mechanical problem. A simplifying approximation is to consirlcl

tunneling only in the deglee of freedom corresponding to the reaction coordinate. Within

this one-dimensional formalism, various levels of approxirnation are available.

TI ie s implest apploximat ion is that of Wigner (1932), which takes

" - / , I 'n1r, '1

andI

n- krT

In the Skodje and Truhlar apploximation, one takes for B < a

k: * tTlktT Qr Q."n "-o"/r" 'h Qa Qr'"

I f / r lm(ut) l 're(Tt : l+- l I' 24 L ksT I

u- l^ , a

y 17 y = -J!"J"-- - - ! - " l lb-at{av1 - t ' r l

sin(f ln/a) u-p

!I

\.\.\\.

\\

q- lUJEID

AVIReactant energy

Figure 15'4 Probabilities ol'rcaclion (P) lbr systems rnoving rowards a paral2olic 6arrier lirr a reac-t ion wi th a 7-ero-poinGincluding potcnt i l l encrgy ol 'act ivat ion AVj. Clasi ie l l s-ystcrns 1-) bclgwthc barricr height havc z-ero probability ol reaction and abovc the barricr hcight havc unrt proba-bi l i ty ( i .e. ' the'curve'descr ibcs a Hcavis idc lunct ion). Quantutr systcms (------) . on the other hand,have increasingly non-zero probabilitics as the barrier encrgy is approachcd lrolr be6w bcclusc oftunneling and incrcasingly less than unit probabilities as thc barricr cnergy is tpprolchc<l lronr irbovehecause of non-classical reflcction. Note that because of the Boltzrnann distrihtition of cncrgres rn athcrnral ized populat ion ofreact ing systerni ( - . - . - . - . rc lbrcncct l to the r ight ordinatc). typical ly nranynrorc lnolecules have energies in the rcgion wlrcre tunneling can iucrcase Lhe reac(ion rate than havecrlcr'{ics in the region where non-classical rcllection can reduce thc rcacliorr ratc. As il rcsult. thelornrcr is the more quantitatively intportant of the two quantuut Dnenonlena

( 15. .17)

where vr is the irnaginary frequency associated with the reaction coordinate (the notrti()n

Im(,r) means that we take only the imaginary part of the frequency, which is to siry ll);rl

we treat it as though it is a real number rather than a complex one). The Wigner colrcrti,'rr

works wcl l provided that / r l rn(r ' l ) << ABT.

A more robust approxirnat ion to r has been provided by Skodje and Truhkl f ( l ( )s l ) .

genelal iz ing ear l ier wolk by Bel l (1959) fbr parabol ic barr iers. For notat ional con\/crur 'n( i '

we tilKe )n

( l i . l f i )

( 15. l , ) I

I l \ ' l l l t

* lg.

$i

53E 15 ADIABATIC REACTION DYNAMICS

leading to regions where KIEs will change rapidly as a funcrion of temperature, which mayalso confuse interpretation. These effects are not limited to esoterically low temperatures:enzymes catalyzing proton, hydride, and hydrogen atom transfers can exhibit large ratecontributions from tunneling at biological temperatures (Kohen and Klinman l99g).

15.4 Condensed-phase Dynamics

Solvent effects on reaction coordinates have already been discussed in a general fashionin Section I 1.1.2. In terms of estimating condensed-phase rate constants, we consider threelevels of approximation. In the simplest model, referred to as separable equilibrium solvation(SES), we assume that the effects of a surrounding condensed phase are limited simply tochanging the free energy along the MEP. In that case, the condensed-phase free energy ofactivation is simply the sum of the gas-phase free energy of activation and the free energresof solvation of the activated complex and the reactant(s) (see Figure I 1.4), and we may useEg. (15.27) to compute the rate constant directly from the condensed-phase AG,'1. The freeenergy of the activated complex in solution may either be evaluated for the gas-phase TSstructure, or may be taken as the maximum free energy along the solvated MEp, which wouldbe a variational-like treatment. operationally, we may most readily compute the solvationfree energy for each point on the MEP by assuming the solvenr to be fully equilibrated tothat point and using any convenient solvation model (a continuum model being the mosrefficient choice),

At the next level of approximation, we continue to imagine the solvent to be fully equili-brated to the reacting system at every point, but instead of working with the solvated MEpfrom the gas-phase surface, we find the equilibrium solvation path (ESp) which is the MEp onthe fully solvated surface (see Figure ll.1). While both the gas-phase and solvated surfacesare defined entirely in terms of solute coordinates, the Esp may be quite different from thegas-phase MEP because solvation effects may 'push' the path in directions orthogonal tothe gas-phase reaction coordinare (see Figure ll.5). wirh the ESp in hand, TST (or vrST)analysis may be carried out in the usual way to obtain a condensed-phase rate constant.

The beauty of the prior approximations is that by assuming a mean-field influence ofsolvation we can continue to work in a phase space having the same dimensionality as thatfor the gas phase; that being the case, analysis using the tools of TST is mechanically identicalfor the two phases. When the solvent is nol fully equilibrated with rhe complete reactionpath, however, the reacting system can no longer legitimately be described exclusively interms of solute coordinates.

Note that the region where solvent is least well equilibrated to the solute is expected to bein the vicinity of the activated complex, since it has so short a lifetime. Since non-eouilibriumsolvation is less favorable than equilibrium solvation, the non-equilibrium free energy oftheactivated complex is higher than the equilibrium free energy, and the non-equilibrium lagin solvent response thus slows the reaction. This effect is sometimes refened to as solvent'friction' and can be accounted for by inclusion in the transmission factor r.

Explicit inclusion of all solvent degrees of freedom, e.g., in an MD simulation, is not avery effective approach to modeling the non-equilibrium solvent influence, however. one

I5.5 NON.ADIABATIC DYNAMICS 539

issue that should be apparent is that one can no longer really define a TS structure under

such conditions - on the enormously high-dimensional PES constructed from all solu-te and

solvent coordinates there witl be a huge number of saddle points having similar energics

in regions between reactants and products, and the related problem of running trajectories

through this potentially high-energy volume of phase space to estimate rate constants has

already been noted in Chapter 12.To simplify marters, it is usually assumed that the influence of the solvent can be modeled

with so-called effective solvent coordinates. A typical choice is to treat the solvent coordinate

as having a harmonic potential that is linearly coupled to the solute. If one extends this approach

to use an infinite number of solvent harmonic oscillators one obtains the so-called general-

ized Langevin equation for solute dynamics (zwanzig 1973). In the other limit of reducing

consideration of lhe solltte to a single coordinate, one obtains Kramers-Grote-Hynes theory

(Kramers 1940; Grote and Hynes 1980). The development of more sophisticated treatments for

the solvent coordinates in non-equilibrium solvation models remains an active area of research

15.5 Non-adiabatic Dynamics

15.5.1 General Surface Crossings

When two (or more) potential energy surfacbs conesponding to different electronic states of

a chemical system are close to one another in energy, the electronic wave function should

really be written as a linear combination of the different adiabatic wave functions. For

simplicity, let us consider the case of only two states, in which case we would write

w(Q, q) : cr(Q)f r (Q' q) + cz(Q)fz(Q' q) (ts.42)

where ry'; and lt2 are the two adiabatic states that depend on the electronic coordinates q and

the nuclear coordinates Q and the coefficients also depend on the nuclear coordinates because

the mixing of the states will vary with different geometries. The situation is illustrated for a

single internal coordinate in Figure 15.7. Note that since the coefficients ct and c2 depend

only on nuclear coordinates, each is a nuclear wave function.

ln this case, the Schrddinger equation becomes

Hw(Q, q) : Er"ulcr(Q)fr(q;Q) + cz(Q)tz(q;Q)l ( l 's '43)

where the Hamiltonian operator now includes nuclear kinetic energy as well as the nuclcar

repulsion, i.e.,

( l 5.44)

where nrr is the mass of nucleus k in atomic units, V2 is defined as in Eq. (4.4). and /t"r

and V7y are defined in Eq. (a.16) and the following discussion. To determine a givcn tr tts

a function of nuclear coordinates Q, we can multiply both sides of Eq. (15.43) on lhc lcli

nuclci

H:t-^ v i - t -H"r*v1y7 zmk

542 15 ADIABATIC REACTION DYNAMICS

them is Marcus theory (Marcus 1964). The fuil scope of Marcus theory is very broad, and weconsider here only the simplest apprication of the moder. we wiil taie the generic eiectrontransfer reaction

I5.5 NON.ADIABATIC DYNAMICS

"M^"lv^"'^

da

o, (c)

Figure l5.E Electfon-transfer reaction coordinate diagrams used in Mrcus theory. Diagram (a) rel'crs

to-" "u."

with no ner free energy of reaction, in which case the intersection of the two curves occurs

at tr/4 above the minima and is taken as the barrier to the electron transfer (a barrier associated with

solvenr reorganization in the simplest limit). When the overatl driving force is equal in magnitude to

I (b), the tio "u.u",

cross at the equilibrium solvent configuration of the first state' and reaction is

barrierless. However, when the driving force becomes still greater (c), the crossing of the two curves

proceeds to the lefr on the reaction coordinate, and occurs at higher energy than the minimum of

ih" ,"actant curve. This situation creates the inverted region where rate decreases with increasing

exergonicity

above the two equal minima. This situation in illustrated in the first leaction coordinate

diagram of Figure 15.8, and rationalizes the denominator of the exponential in Eq. (15'50):

if AGIB is zero, then the argument of the exponential is )'l4RT which is indeed the 'barrier'

for reaiiion in the system with no thermochemical driving force in either direction.

Note that Marcus theory in the form of Eq. (15.50) makes a rather surprising prediction.

If AGIB is equal to ), in rragnitude but of opposite sign, which is to say the exergonicity

of the electron transfer exactly cancels the reorganization energy, than the argument of the

exponential is zero and the rate is predicted to be diffusion-controlled. However, il 'thc

driving force becomes greater still, then the argument of the exponential returns to positivc'

and the rate is predicted to decrease (Figure 15.8). This conesponds to the so-called invcrtcd

region of Marcus theory. That is, as one of a pair of reactants in an electron-transfer reitctiott

is varied so that the reaction becomes more and more favorable in a free-energy scnsc, ihc

rate is predicted to reach a maximum and then decrease. Expeiimental verification ol'this

prediction did not occur until many years after the initial publication of the theory' in part

because the required driving force is so high and in part because of the technical challcttgcs

associated with rneasuring very large rate constants. Nevertheless, an invertetl rcgion hrr

543

A-+B+A*B- ( r5.49)

For this simple case, Marcus theory predicts the rate constant for electron transfer to be

kgr - I ^tt-

( tGiB+'1')2/4^Rr ( 15.s0) (b)

where Zas is the collision frequency for the reactants (typicaily in the range of lOe tol0l0 sec-l for reactions in non-viscous liquids at ambient temperatures), Aci, is the freeenergy change for the electron transfer, ), is the so-cailed reorganization ene?!y, n is theuniversal gas constant, and I is the temDerature

The reorganization energy term derives from the solvent being unable to reorient on thesame timescale as the elecrron transfer takes place. Thus, at the instant of transfer, the bulkdielectric portion of the sorvent reaction field is oriented to sorvate charge on species A,and not B, and over the course of the electron transfer only the optical part of the solventreaction field can relax to the change in the position of the charge isee Section 14.6). If theBorn formula (Eq. ( l l . l2)) is used to compute the sorvat ion f ree energies o[ rhe var iousequilibrium and non-equilibrium species involved, one finds that

where Aq is the amount of charge transferred (r for the reacrion of Eq. (15.49)), e- is rhefast dielectric constant (sometimes called the optical dielectri..onrtunt, equal to the squareof the index of refraction - around 2 for typical solvents;, r0 is rhe slow, or burk, dierectricconstant, rA and rB are the radii of species A and B, respectively, and r4s is the drstancebetween them at reaction. The quantity in Eq. (15.51) is sometimes caLled ),o because rtconsiders only 'outer-sphere', which is to say sorvent, reorganization. More sophisticatedapproaches can be used when inner-sphere reorganization is arso important, e.g., ior ligatedmetal systems where the metal-ligand bond lengths might vary significantly as a function ofcharge. In such instances, inner-sphere reorganization energies can often be estimated fromcalculations of relaxation energies when the geometry of the species for. the initial chargestate is allowed to relax to the final charge state.

The exact form of Eq. (15.50) is made more intuitive by considerihg the simple reactioncoordinate diagrams of Figure 15.8. In these cases, we consider two parabolic potentialenergy surfaces corresponding to the two sides ofEq. (15.a9). The reaction coordinate mav-in particularly simple instances, be thought of as a generarized sorvent coordinate. Thus, whenthe solvent is optimally configured for A- * B, the energy of the curve for state A + B-is quite high. If the free energies of the left and right sides of Eq. (15.a9) are rhe same(which would happen if A and B were different isotopes of the sarne metar, for instance),the separation of the two curves at either minimum is exactly ,1.. From the mathematics ofparabolae, this requires the intersection of the two curves to take place at l/4 energy units

) . : (Lq)2 (* - *) (* .

* -

*) o5 s,)

.r.ti

I

546

small a rate constant, the prc-exponential is predicted to be too larye, which returns the rate'' constant to a reasonable value, and vice versa. In spite of such conrpensating err.ors, in the

case of acetone the final enot' in tlre rate is alrrosi a factor of 20. At lower'ter,rp",:u,u,.r,. this en'or would increase drarnatically.Nevertheless, the agreenrent that is obtained - which is probabry the hest one shourd

expect given the srnal l s ize of the basis set usecl in the cCSD(T) calculatronsand the possible problenrs ilssociated with bir.adical character in the merhyr vinylether pathway - suggests that the theoretically pleciicted TS s(r.uctures nr. n..rro,.

' representations of the actual tlansition states. This establishes thc concerted tratule ol threeol'the rearrangements and the stepwise nature of the firurrn.

Bibliography and Suggested Additional ReadingChuang. Y -Y.' Cramcr. C. J.. altcl Trulrlar. D. G. 199{t. "fhe Intcrl:rcc ol' Elcctronic Stnrcrure dnd

Dynanrics lbr Rcact ions in Solut ion' . I r t t . J. et tut t t t r r t t Cl tctn. .70.g&7.Chuarrg. Y.-Y.. Radhakr ishnan. M. L. , Fasr, p. L. , Cr lnrcr . c. J. , antr r ruhrar, D. G. r999. .Direct

Dytratr t ics l i r r Frcc Radical Kinct ics in Solut ior : Solvcnt El l 'cct oI thc Ri l tc Cor]stant l0r rhc Rcact ionof Mcthanol wirh Atonric Hydro_uen'. J. plns. Cltent. A. l0-1. 41J93.

Espcnsttn. J. H. 1995. Cherni tu l Kiner i ts utr t l Rcutt ior t lVI t , t l tut t i . rnt . t .2nd Edn.. McGraw-Hi l l : NcwYork.

Gi l r rct t . B' C. and Tfuhlar, D. G. 1979. 'scnr ic lassical Tunncl in-u Ci l lculur ions' . .1. pht . t . Chcnt. . g3,292t.

Hynes' J' T' 1996' 'Crossing the Transition Siate in Solution'. in So/r'cirr Elftct.s uttil Clrctrtitut Reuc-lilin', Tapia. O. and Bertriin, J. Eds., Kluwer: Dordrcclrt,2jl.

Jenscn. F. 1999. rntrctructknt to cunpututiotru! Chenistt-r,, wirey: chichestcr.Jensen' F' and Norrby, P.-o.2003. 'Transition States from Ernpirical Force Ficlds'. Thertr. cltcrn. Acc-.

109. r .Johnston, H. S. 1966. Gus pha,se Reuctiou R(ile Theo^,, Ronald press: New york.Lowry, T ' H. and Richardson. K. S. 198] | . Mat 'hunisnt und Theorr . i r t orgurt ic.Chtnt istr t . .2nd Edn..

Harper & Row: New York.Stcinfefd. J. I., Francisco, J. S.. and Hase, w. L. 1999. clrcntiutl Kirtcti<.t untl Dyttutttit,s.2nd Edn..

Prenticc Hall: Upper Saddlc Rivcr. NJ.Truhlar ' D G.. Carret t . B. C.. and Kl ippcrrstc in. S. J. 1996. 'Currcnt .status ol Transi t io l -stare ' fhcory, .

J. Phts. Chetn., 11:,11, 1277 l.Tuckcr 'S C andTruhlar.D.G. l9t t9. 'Dynatr ical Forurulat ionol Trarrs i t ionStatcTheory: Var iat ional

Transition States and Scmicrassicirr rtrnrrcring'. in Nav Trtcttrcti<.trr Ct21r.r,,,,,.lttr tJrukr.ttundinlgO4iuni<.Rctt t l i rurr . Bcrtr i in, J. and Czismadia. I . G.. Eds.. Kluwcr: Bcr l in.29l .

Worth ' G' A' and Rohb. M' A.2002. 'Apply ing Dircct Molccular Dynanr ics ru Non-adiahar ic sysrems,.Adr,. Chtnt. Phr.s.. 124. 355.

References

Af f isorr' T' C' and Truhlar. D' G. 1998. ln: Modern Mettuxls,.fitr Multitlintan.siotrul Drnutrtit..r Contlttt-_

t.tuions in Chenisttt', Thompson. D. L.. Ed.. World Scicntilic: Singaporc. (r18.Belf. R. P. 1959. Truns. Funulu.t' &r:.. 55. l.Dubnikova F. and Li f .shirz. A.2000. J. phts. Chcm. A, 104.44S9.Eckart, C. t930. Phrs. Rzr'., 35, 1303.

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I.5 ADIABATIC REACTION DYNAMICS 5J7