This article was downloaded by: [University Of Pittsburgh]On: 14 November 2014, At: 13:23Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

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Transition states forhydrogen radicalreactions: LiFH as astringent test casefor density functionalmethodsOSCAR N. VENTURAPublished online: 03 Dec 2010.

To cite this article: OSCAR N. VENTURA (1996) Transition states for hydrogenradical reactions: LiFH as a stringent test case for density functionalmethods, Molecular Physics: An International Journal at the InterfaceBetween Chemistry and Physics, 89:6, 1851-1870

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M o l e c u l a r P h y s i c s , 1996, V o l . 89, N o . 6, 1851 ± 1870

Transition states for hydrogen radical reactions : LiFH as a stringent

test case for density functional methods

By OSCAR N. VENTURA

MtC-Lab, University of Uruguay, C.C. 1157, 11800 Montevideo, Uruguay

(Recei Š ed 23 February 1996 ; accepted 24 March 1996)

The Becke three-parameter Lee± Yang± Parr (B3LYP) density functionalmethod is applied to the study of the reaction Li ­ HF ! H ­ LiF ; the results

obtained are compared with experiment and previous multireference singles and

doubles con® guration interaction (MRDCI) calculations, and with singles anddoubles quadratic con® guration interaction (QCISD) and Gaussian 2 (G2)

model chemistry calculations performed also in this paper. It is found that,

using an extended 6-311 ­ ­ G(3df, 3pd) basis set, the predicted stabilizationenthalpy of the initial LiFH complex ( ® 34 ± 6 kJ mol Õ " ) and the exoergicity of

the reaction ( ® 7 ± 3 kJ mol Õ " ) are predicted in agreement with experiment

( C ® 31 kJ mol Õ " and ® 4 ± 8 ³ 8 kJ mol Õ " , respectively). These results are also inagreement with the MRDCI calculations ( C ® 31 ± 5 kJ mol Õ " and ® 9 kJ mol Õ " ,

respectively). However, the energy of the transition state with respect to the

reactants at the B3LYP level is about 25 kJ mol Õ " lower than the MRDCI result(which agrees with the QCISD and G2 values). Therefore, it is concluded that

B3LYP is not describing this reaction properly. It is shown, however, that this

defect is due mainly to the inclusion of the LYP correlation functional insteadof the Perdew± Wang originally considered by Becke for developing his

adiabatically connected functional. When B3PW91 is considered instead of

B3LYP, the height of the transition state is in better agreement with theconventional ab initio methods, although still oŒby 50 %.

1. Introduction

A lot of attention has been devoted recently to the application of density functional

theory (DFT) to molecular systems (for reviews see, e.g. [1, 2]). On the one hand, these

methods scale only as O(N # ) to O(N % ) with the size N of the system, while other

accurate methods, like coupled cluster (CC), scale even as O(N ( ). Therefore, DFT

potentially can be applied to much larger systems. On the other hand, several recent

papers have demonstrated that the accuracy of some of the more modern DFT

methods is as good as or better than that of conventional ab initio calculations.

One of the reports supporting the accuracy of modern DFT methods was

published by Hertwig and Koch [3] on 21 diatomic homonuclear molecules containing

atoms from the ® rst to the third rows of the periodic table. They showed that the

behaviour of DFT methods is from good to excellent. Specially notable is that none of

the `problematic ’ cases appearing in Hartree± Fock theory (for instance, the group II

dimers) shows up in these DFT calculations.

Martin et al. [4] have studied the basis set convergence properties and the general

performance of one of the latest DFT variants, relative to the geometries and

harmonic frequencies of some simple molecules containing only ® rst-row atoms. They

found that, using a contracted basis set of [3s2p1d] quality, the geometries are more

accurate than those obtained with coupled-cluster theory including single and double

0026± 8976 } 96 $12 ± 00 ’ 1996 Taylor & Francis Ltd

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1852 O. N. Ventura

excitations, as well as non-iterative triples (CCSD(T)). Frequencies at the DFT level

were of the same quality as those calculated at the CCSD(T) } [3s2p1d] level.

Gas phase acidities for several molecules containing atoms from the ® rst and

second rows were calculated by Smith and Radom [5] using several modern DFT

methods, and the results compared with high-level ab initio theories. They found that,

when using the 6-311 ­ G(3df, 2p) basis set, some of the DFT methods were able to

produce results of the same quality as Pople’ s advanced Gaussian 2 (G2) procedure [6].

Hobza et al. [7] studied some diŒerent types of molecular cluster (hydrogen

bonded, ionic, electrostatic and London) with two of the most recent DFT methods.

They found that at least for the ® rst two types of cluster, DFT theory using a DZP

basis set behaved reliably with respect to the prediction of geometries and stabilization

energies. They concluded, however, that DFT methods with currently available

functionals failed completely for London-type clusters (they studied Ne#, Ar

#and

benzene± noble gas complexes). Topol et al. [8] also examined speci® cally the

thermodynamics of hydrogen bonding and found that, using accurate enough basis

sets, the results for DFT calculations in ten out of twelve cases calculated agreed with

available experimental values to near chemical accuracy (i.e., C 1 kcal mol Õ " ).

Ventura et al. [9] found that DFT calculations can perform more reliably than

conventional ab initio methods in the case of compounds containing the F > O bond.

They evaluated heats of reaction for sixteen reactions involving FO, FO#, FOH and

F#O

#, showing that the average deviation from experimental values was less than

3 kJ mol Õ " . Also Baker et al. [10] studied a reaction involving FO, namely

FO ­ H2 ! FOH ­ H, among others, comparing the performance of conventional ab

initio, semiempirical and DFT methods. They demonstrated that DFT methods

provide an improvement over traditional methods, although in several cases they

noted that DFT tended to underestimate barrier heights, especially for radical

reactions.

Chemical reactions have been studied quite extensively recently. Ventura et al. [11],

for instance, demonstrated that the isomerization reaction HXCO+ ! XCOH+ (with

X ¯ H, F and Cl), which is described wrongly at the MP2 level, is described well when

DFT methods are employed. Andzelm and collaborators [12] made a study of

reactions involving small molecules, ® nding again that most DFT methods tend to

underestimate barriers for some reactions. However, they found that hybrid DFT

methods (i.e., those that include the exact Hartree ± Fock exchange as part of the

exchange functional, also called adiabatically connected functionals, ACM) behave in

a much better way. In particular, the reaction OH ­ H#! H

#O ­ H was studied

carefully by Baker et al. [13], and they concluded that only hybrid DFT methods are

able to describe the barrier appropriately. Other authors studied proton transfer

reactions (for instance, Barone et al. [14] in hydrogen bonded systems, Jursic [15]

between methane and the methyl radical) and found similar improvement in the

agreement with experimental values, over the conventional post-Hartree ± Fock

methods in common use. Somehow more complicated reactions, i.e., con-rotatory ring

opening of cyclobutene and 1,2-dihydro-1,2-diazacyclobutadienes, were studied by

Jursic and Zdravkovski [16], and in this case they found good agreement between DFT

and conventional ab initio values for the barrier of the reactions. More recently,

Nachtigall et al. [17] studied reaction and activation energies for Si > Si bond cleavage

and H#

elimination from silanes, and found that hybrid DFT methods are able to

produce results in close agreement with experiment and quadratic con® guration

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Transition states for H radical reactions 1853

interaction including singles, doubles and non-iterative triples, i.e., QCISD(T)

calculations.

Finally, Sola’ et al. [18] performed a comparative analysis of density distributions

derived from conventional ab initio and DFT methods, and showed that they are

similar in general. In particular cases where the Hartree± Fock methodology fails,

however, they found that DFT is able to give more accurate results than MP2.

The present feeling is then that some of the latest DFT methods (i.e., ACM

methods) can behave as reliably or as better than conventional ab initio calculations in

a fraction of the time needed for the latter (or, generally, consuming less computational

resources for the same type of system). However, more experience is needed for the

validation of these methods, specially in cases for which the results can be confronted

with accurate conventional ab initio calculations and, particularly, in relation to

reaction barriers of radical systems involving hydrogen atoms.

In this paper we describe the application of DFT to one system which shows the

requirements of the former paragraph : the Li ­ HF ! LiF ­ H reaction. This reaction

has been extensively studied experimentally [19 ± 22] and theoretically [23 ± 27]. It

presents a deep well [22] in the entrance channel, corresponding to a nonlinear [23 ± 27]

LiFH complex, with an experimental stabilization energy of about 300 meV

(29 ± 0 kJ mol Õ " ) and enthalpy of about 317 meV (30 ± 6 kJ mol Õ " ) [22]. Theoretically,

energies for this complex were calculated at 10 ± 2 kJ mol Õ " [23], 19 ± 3 kJ mol Õ " [24],

28 ± 9 kJ mol Õ " [25], 21 ± 6 kJ mol Õ " [26], and 28 kJ mol Õ " [27]. The reaction proceeds via

a highly nonlinear transition state [23 ± 27] which has been calculated at 42 kJ mol Õ " in

the older CI calculation of Chen and Schaefer [24] or at 24 kJ mol Õ " in the more recent

multireference con® guration interaction including singles and doubles (MRDCI [28,

29]) calculation of Aguado et al. [27] (reduced to 8 ± 5 kJ mol Õ " when zero-point energies

are taken into account). No experimental value of the energy of activation exists, but

an estimation of the height of the barrier may be obtained from the experimental work

of Lee et al. [19]. In fact, they observed products at a nominal collision energy as low

as 2 ± 2 kcal mol Õ " (9 ± 2 kJ mol Õ " ) which implies either a low barrier (as the calculations

of Aguado et al. [27] suggest) or that tunnelling through a higher barrier is playing an

important role in this system. The subject is even more complicated by the fact that in

the opinion of some of the previous theoretical studies (Balint-Kurti and Yardley [23],

Chen and Schaefer [24], Sua! rez et al. [30]) the barrier located on the reaction surface

is due to an avoided crossing between the surfaces corresponding to the ionic

(Li+ ­ HF Õ ) and covalent (Li ­ HF) structures.

Eventually, the reaction proceeds past the transition state to the products,

LiF ­ H, which are enthalpically 4 ± 6 ³ 8 kJ mol Õ " [19] below the reactants, a result to

be compared with the recent theoretically calculated exoergicity [27] of about

9 kJ mol Õ " for this reaction. A weak product complex HLiF has been pointed out by

Chen and Schaefer [24] and calculated by Aguado et al. [27] to lie at an energy of

5 kJ mol Õ " below the separated H ­ LiF products.

DFT calculations on lithium compounds are not very frequent in the literature. A

recent report on a comparative study of diŒerent theoretical methods for calculating

properties of organolithium compounds was published by Pratt and Kahn [31]. They

employed a special version of DFT methods which seems to be less accurate than more

modern versions, but anyhow, they found a good agreement between the DFT results

and available conventional ab initio and experimental data. Sola’ et al. [18] included

LiF, LiH and Li#

in their comparative analysis of the conventional ab initio and DFT

electronic density distributions. Alikhani et al. [32] performed study M ± C#H

%

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1854 O. N. Ventura

complexes, with M an alkali metal (Li, Na and K). They found that the Li complex was

a weak complex presenting (in contrast to Na and K complexes) a C# v

structure. Both

CCSD and DFT calculations were performed, showing the essential agreement

between them, but also the large basis set dependence of the interaction energy.

In two related papers, Paniagua and Aguado [33] and Sua! rez et al. [34] applied

DFT methods to the LiFH problem. In the ® rst case, they applied DFT methods only

to introduce correlation energy on top of a restricted Hartree± Fock (RHF) calculation.

They found that the basic features of the RHF surface were preserved after the

addition of correlation energy in such a way. Sua! rez et al. [34] reported a more

extended study, comparing the results obtained using HF theory, M ù ller± Plesset

theory at second and third order (MP2, MP3 [35]), and con® guration interaction at the

single (CISD) and multireference (MRDCI) levels, with those obtained by using two

more modern DFT methods, but only for the introduction of correlation energy. Their

conclusions were that DFT methods show overall agreement with the CI methods,

except for the transition state being a little too high ( C 3 kcal mol Õ " ) with respect to

the best MRDCI calculation.

The DFT results reported in this paper using one of the latest methods available

show a marked discrepancy with all the previous conventional and DFT-corrected

calculations for the LiFH system. Therefore, they provide an interesting case for

discussing the relative merits and accuracies of DFT and ab initio methodologies.

2. M ethods

The main calculations reported in this paper were performed using the B3LYP

method as coded in the Gaussian 94 set of computer programs [36]. B3LYP is a hybrid

or ACM DFT method, because the exchange potential [37, 38] includes previous pure-

DFT exchange functionals plus the exact Hartree ± Fock exchange, with a few

parameters (three) adjusted according to some experimental values of a few selected

compounds (the same set used in the Gaussian 2 theory of Pople et al. [6]). The

correlation energy functional in B3LYP is that of Lee, Yang and Parr [39]. B3LYP has

been the preferred and most accurate DFT method employed up to now in the works

cited previously, because of its inclusion in the widely distributed G92 } DFT program,

However, a word of caution is necessary because in the ACM method as proposed

originally by Becke the gradient-corrected correlation energy functional employed was

that of Perdew and Wang (PW 91 [67, 68]). Becke has recently published [40] a new

modi® cation of ACM in which he includes a new correlation functional (Bc95) and

reduces the parameters needed in the exchange functional to only one. The

thermochemistry in the G2 benchmarks is then improved over the B3LYP results.

However, the improvements are small (about 10 %) and B3LYP should give results of

comparable quality.

For some of the calculations reported here, several other exchange± correlation

functionals were employed. Two other correlation functionals, besides LYP, were

employed in the ACM methods. These were, on the one hand, the PW91 correlation

functional [67, 68] to give the B3PW 91 exchange± correlation functional (the original

ACM of Becke [37]) and, on the other hand, B3P86, where the correlation functional

is the older one of Perdew and Wang [69, 70]. Finally, gradient-corrected methods also

were used where the exchange functional is not of the ACM type, but Becke’ s 1988

exchange functional [71]. This exchange functional, combined with the three

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Transition states for H radical reactions 1855

Figure 1. MRDCI [27] and B3LYP relative energies and enthalpies (in kJ mol Õ " ) with respect

to Li ­ HF of the initial complex (LiFH), transition state (TS), ® nal complex (HLiF) andproducts (H ­ LiF) in the reaction path for Li ­ HF ! H ­ LiF.

correlation functionals leads to the three methods BLYP, BP86 and BPW91, as

identi® ed in what follows.

Several basis sets were used in this research. Pople’ s basis sets, 6-31G(d, p), 6-

311G(df, pd), 9-311 ­ ­ G(2df, 2pd) and 6-311 ­ ­ G(3df, 3pd) [41] were used for the

production calculations. Other combinations of valence, diŒuse and polarization

functions were employed for assessing the basis set convergence of the transition state

energy. Grev and Schaefer [42] have argued that the 6-311G basis set is not of valence

triple-zeta quality. Regrettably, the correlation consistent basis sets of Dunning and

co-workers [43], which could be of great value in improving the quality of the

calculations, have not been developed for lithium. On the other hand, Bauschlicher

and Partridge [44] have shown recently that use of Pople-type basis sets oŒer some

advantages over correlation consistent ones for the calculation of additive corrections

(as is done in G2). Just to give a higher degree of reliability to the results of this paper,

the extended DZ ANO basis set of Roos [45, 46] was employed for the atoms and the

diatomics (allowing the calculation of the exoergicity of the reaction at a more

accurate level). This basis set is of the non-segmented type and therefore extremely

expensive to use except with specially designed computer codes : the reason they were

not adopted for the general investigation. The study was extended, however, to include

the non-contracted set and perform geometry optimizations and energy calculations

with it for the transition state and the reactants. Geometry optimizations were

performed in all cases using gradient techniques [47], with tight SCF and optimization

thresholds and using a ® ne grid for the DFT procedures. Minima and transition state

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1856 O. N. Ventura

were characterized as usual, according to the number of negative values of the second-

derivative matrices (i.e., 0 or 1), calculated analytically.

Two other ab initio procedures were employed to support the B3LYP calculations.

One was the QCISD method of Pople et al. [48], which is essentially a simpli® ed

coupled cluster type of calculation. Since geometry optimizations using the B3LYP

method are precise enough according to both former experience and the internal

evidence in this work, QCISD calculations were performed at the B3LYP } 6-

311 ­ ­ G(3df, 3pd) optimum geometries. It is not expected that the results would

change signi® cantly if QCISD } 6-311 ­ ­ G(3df, 3pd) optimum geometries were

employed instead. In fact, Bauschlicher and Partridge [49] and also M orokuma and

colaborators [50] have proposed replacing conventional ab initio optimum geometries

and frequencies with B3LYP ones in the G2 procedure [51], as a faster and more

accurate replacement, relying then on the quality of this method for geometry

optimization.

The Gaussian 2 method, one of several `model chemistries ’ developed recently

[51], was employed also. It is known that these methodologies allow accurate estimates

of molecular energies, generally to within a target accuracy of 10 kJ mol Õ " . Smith and

Radom [5] have compared the accuracy of B3LYP with respect to G2 [6] and G2(M P2)

[52] (the procedures employed here) in the calculation of gas-phase acidities, and

concluded that all of them are of similar accuracy. This is the main reason for the use

of G2 and G2(M P2) in this work. Note that both G2(M P2) and G2 are somehow

diŒerent approximations to QCISD(T) } 6-311 ­ G(3df, 2p) calculations. Replacement

of the components of the G2 procedure (i.e., correlation energy, geometry opti-

mization, zero-point energy) with the values derived using more sophisticated methods

was shown to be unnecessary for the improvement of accuracy by Curtiss et al. [53].

A recent comparison of model chemistries [60] showed that G2 predicts heats of

formation of molecules and bond dissociation energies with an average error of about

1 kcal mol Õ " ( C 4 kJ mol Õ " ).

3. Results and discussion

3.1. Initial results

The essential result of this paper is displayed graphically in ® gure 1. Collected in

this ® gure are the relative M RDCI energies and enthalpies (with respect to Li ­ HF)

provided in the papers by Aguado and coworkers ([27] and references therein)

compared with the relative B3LYP } 6-311 ­ ­ G(3df, 3pd) energies and enthalpies at

0 K calculated in this work and shown in table 1. The one discrepancy observed

immediately in ® gure 1 between the M RDCI and DFT calculations is that the latter

predicts a much lower activation energy for the reaction, due to a much larger

stabilization energy of the transition state with respect to the reactants. At this energy

level, B3LYP calculations for the initial complex (LiFH), transition state (TS) and

® nal complex (HLiF) are respectively about 7, 25 and 12 kJ mol Õ " lower than the

M RDCI results. Inclusion of zero-point energies reduces the separation to about

3 kJ mol Õ " and 24 ± 5 kJ mol Õ " for LiFH and the TS, respectively. Nonetheless, the

energy diŒerence for the TS remains very large, and the origin of this diŒerence must

be examined.

Before trying to decide whether the DFT results are de ® nitely ¯ awed (assuming, of

course, that the MRDCI calculations are correct), one can look for indirect clues to the

accuracy of the B3LYP method in this speci® c case. One of the reasons for a possible

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Transition states for H radical reactions 1857

failure in the DFT calculations (as well as in conventional ab initio theory) is that the

basis set employed may not be su� ciently extended. The basis set dependence of DFT

methods has been studied in other cases, most notably Hertwig and Koch [3], Martin

et al. [4] and Krystya! n [54], who concluded in general that convergence is faster than

with conventional ab initio procedures.

Looking into table 1, where the optimum geometries are displayed, and comparing

with the experimental data available for the diatomics HF, HLi and LiF, one can see

that B3LYP } 6-311 ­ ­ G(3df, 3pd) calculations predict bond lengths as accurate or

more so than the MRDCI calculation. Other properties for HF and LiF show similar

behaviour. Harmonic frequencies, calculated by DFT at 4098 cm Õ " and 901 cm Õ "

respectively, are reported experimentally as 4138 cm Õ " and 910 cm Õ " [55]. The dipole

moment calculated for HF using B3LYP } 6-311 ­ ­ G(3df, 3pd) (1 ± 83 D ; D ¯ debye

E 3 ± 335 64 ¬ 10 Õ $ ! C m) is well in agreement with the experimental 1 ± 80 D measured

by M uenter and Klemperer [56], while the parallel and perpendicular components of

the dipole polarizability (calculated at 6 ± 27 au and 4 ± 45 au, respectively) compare well

with the benchmark calculations at the CCSD(T) } aug-cc-pVDZ (6 ± 28 au and 4 ± 29 au)

or CCSD(T) } aug-cc-pVTZ (6 ± 36 au and 4 ± 89 au) levels by Peterson and Dunning [57]

(although they compare less well with the limit values of 6 ± 32 au and 5 ± 13 au). (1 au

polarizability F 6 ± 486 67 ¬ 10 Õ % ! C # m # J Õ " .) M oreover, the experimental equilibrium

dissociation energy of HF is 141 ± 6 kcal mol Õ " [57] (although quoted in [60] as

135 ± 1 ³ 0 ± 2 kcal mol Õ " ), while the B3LYP calculation places it at 140 ± 0 kcal mol Õ " (the

CCSD complete basis set limit [57] is 139 ± 5 kcal mol Õ " ). In the case of LiH, there is no

experimental value for the bond dissociation energy, but the value calculated by

B3LYP ( ® 58 ± 4 kcal mol Õ " ) is in agreement with the G2 value of [60] ( ® 56 ± 6 kcal

mol Õ " ). In principle, therefore, these diatomic molecules are described reasonably well

by the B3LYP } 6-311 ­ ­ G(3df, 3pd) calculations (however, this is not a surprise,

because they are included in the G2 benchmark set used to parametrize B3LYP).

From table 1 it is also clear that the results do depend on the basis set chosen. This

is re¯ ected clearly in the evolution of the bond lengths, in particular the FH bond

length in the transition state (by the way, note that the bond lengths evolve towards the

M RDCI values when the basis set is improved). It is also evident in the energy

diŒerences among diŒerent species. Figure 2 shows, for instance, the energies of

stabilization of the LiFH initial complex with respect to HF and Li as a function of the

basis set employed. Also shown in this ® gure is the energy of reaction for LiF ­ H !Li ­ HF. It appears from the geometries in table 1 as well as from the two curves in

® gure 2 that a reasonable convergence has been reached with respect to the basis set.

M oreover, if one compares the values of the best energies with the experimental values

available, one sees that there is reasonable agreement. In the case of the exoergicity of

the reaction, after adding the zero-point corrections for the diatomics, one obtains a

value of D H ¯ ® 7 ± 3 kJ mol Õ " , to be compared with the experimental value of

® 4 ± 8 ³ 8 kJ mol Õ " . The DFT value is even better than the MRDCI ( ® 9 kJ mol Õ " ). In

the case of the stabilization energy of LiFH, the experimental energy value is about

® 29 ± 0 kJ mol Õ " (there is a non-quanti® ed experimental uncertainty in this result) and

the enthalpy value is about ® 30 ± 6 kJ mol Õ " . The B3LYP } 6-311 ­ ­ G (3df, 3pd)

energy value is ® 34 ± 8 kJ mol Õ " , transformed to an enthalpy value of ® 34 ± 6 kJ mol Õ " .

Clearly, this is also a reasonable result and proves that, at least for the minima, the

calculations are converged with respect to the basis set.

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1858 O. N. Ventura

Tab

le1.

En

ergie

so

fth

ed

iŒer

ent

spec

ies

dis

cuss

edin

this

pap

er,

usi

ng

the

B3L

YP

met

ho

ds

an

dd

iŒere

nt

basi

sse

ts.a

B3L

YP

b

MR

DC

Ic

dp

dfp

d2d

f2p

d3d

f3p

dR

oo

s[s

pd

]E

xp

d

H®

0±5

00

273

®0

±502

156

®0±5

02

257

®0

±502

257

®0

±502

355

Li

®7

±490

978

®7

±491

288

®7±4

91

325

®7

±491

325

®7

±491

951

[®7

±492

936]

F®

99

±715

531

®99

±754

713

®99±7

61

652

®99

±761

681

HF E

®100

±427

463

®100

±471

759

®100±4

86

078

®100

±486

982

®100

±498

194

(®100

±477

566)

[®100

±499

322

r0

±925

40

±920

30±9

22

50

±922

00

±921

70

±921

0±9

16

8

[0±9

92

1]

LiF E

®107

±417

567

®107

±459

165

®107±4

70

532

®107

±471

437

®107

±482

966

(®107

±469

502)

r1

±550

01

±560

51±5

70

11

±570

41

±571

91

±588

1±5

63

9H

Li

E®

8±0

82

357

®8

±086

279

®8±0

86

584

®8

±086

686

(®8

±083

453)

r1

±615

01

±591

71±5

89

81

±589

41

±603

1±5

98

LiF

H

E®

107

±944

823

®107

±984

431

®107±9

90

513

®107

±991

586

(®107

±982

170)

r(L

iF)

1±8

24

1±8

49

1±8

61

1±8

55

1±9

31

r(F

H)

0±9

55

0±9

49

0±9

52

0±9

52

0±9

31

r(H

Li)

2±2

32

2±3

02

2±3

26

2±3

18

2±3

65

h102

±2106

±2107±1

106

±9106

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Transition states for H radical reactions 1859H

LiF

E®

107

±926

501

®107

±969

391

®107979

953

®107

±980

967

(®107

±976

499)

r(L

iF)

1±6

08

1±6

14

1±6

15

1±6

15

1±6

19

r(F

H)

1±6

13

1±7

05

1±7

47

1±7

45

1±8

84

r(H

Li)

1±8

30

1±8

16

1±8

47

1±8

46

2±0

21

h69

±21

66

±366±5

66

±570

TS E

®107

±926

373

®107

±968

222

®107±9

77

627

®107

±978

631

[®107

±993

623]

(®107

±975

097)

r(L

iF)

1±6

23

1±6

61

1±6

73

1±6

71

[1±6

67]

1±6

93

r(F

H)

1±4

96

1±3

67

1±3

15

1±3

16

[1±3

01]

1±3

02

r(H

Li)

1±8

11

1±7

70

1±7

94

1±7

93

[1±7

95]

1±8

09

h70

±970

±872±7

72

±7[7

3±2

]73

aE

ner

gie

sin

au

,d

ista

nces

inA/

an

dan

gle

sin

deg.;

valu

es

inp

are

nth

ese

sco

rresp

on

dto

en

thalp

ies

at

0K

.b

Th

eco

de

for

the

basi

sse

tsis

as

foll

ow

ing

:d

p¯

6-3

1G

(d,p

);d

fpd

¯6-3

11G

(df,

pd

);2d

f2p

d¯

6-3

11

­­

G(2

df,

2p

d);

3d

f3p

d¯

6-3

11

­­

G(3

df,

3p

d);

Ro

os

¯exte

nd

edd

ou

ble

zet

aA

NO

basi

sse

to

fW

idm

ark

an

dR

oo

s[4

5,

46];

valu

es

inb

rack

ets

co

rresp

on

dto

the

un

co

ntr

act

edR

oo

sb

asi

sse

t.c

MR

DC

Ire

sult

sw

ith

an

exte

nd

edsp

dse

tfr

om

[27].

dE

xp

erim

enta

lvalu

es

tak

en

fro

m[5

5].

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1860 O. N. Ventura

Figure 2. Convergence with the basis set of the energy for the reaction Li ­ HF ! H ­ LiF and

of the stabilization energy of LiFH with respect to Li ­ HF calculated using the B3LYPmethod. The ® ve basis sets in the ® rst case are, from left to right, 6-31G(d, p), 6-311(df,

pd), 6-311 ­ ­ G(2df, 2pd), 6-311 ­ ­ G(3df, 3pd) and Roos extended ANO DZP. In the

second case the basis sets are the same, except that there are no results with Roos basisset.

3.2. Comparison with con Š entional ab initio methods

One has to turn then to further comparison with other conventional ab initio

calculations to try to ® nd out the reason for the discrepancy in the energy of the TS.

First, one has to realize that the geometry of the TS obtained at the B3LYP } 6-

311 ­ ­ G(3df, 3pd) level is completely in agreement with the geometry obtained at the

M RDCI level (table 1). The FH bond clearly is the most sensible geometrical feature

with respect to the basis set, to the point that at the B3LYP } 6-31G(d, p) level this bond

is 0 ± 2 A/ larger than in the MRDCI calculation. This discrepancy is reduced by a factor

of ten when the better basis sets are used. Therefore, geometry should not be the reason

for the large discrepancy in the energy values. Thus, QCISD } 6-311 ­ ­ G(2df, 2pd)

calculations were performed at the B3LYP } 6-311 ­ ­ G(2df, 2pd) geometries to

compare the relative energies. From these results also relative energies at HF, MP2

and MP4 (SDQ) levels (by-products of the former calculation) were extracted. All

these energies are listed in table 2. Finally, also G2(MP2) and G2 calculations were

performed (data collected in table 3), which have their own incorporated procedure for

geometry optimization and inclusion of correlation energy, etc, but which essentially

are approximations to a full QCISD(T) } 6-311 ­ G(3df, 2p) calculation.

Figure 3 shows the relative energy information obtained at the diŒerent theoretical

levels, in comparison with the B3LYP } 6-311 ­ ­ G(3df, 3pd) results. The most

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Transition states for H radical reactions 1861

Figure 3. Relative energies in the reaction path with respect to Li ­ HF, obtained at the

B3LYP } 6-311 ­ ­ G(3df, 3pd), QCISD } 6-311 ­ ­ G(3df, 3pd), and G2 levels in this

paper compared with the MRDCI results of [27]. The ® ve plateaus correspond,respectively, to Li ­ HF, LiFH, TS, HLiF and H ­ LiF.

interesting facts observable in this ® gure and tables 2 and 3 are : (i) HF, MP2,

M P4(SDQ), QCISD, G2(MP2) and G2 results are in agreement in predicting a much

shallower entrance valley for the reaction than either MRDCI or B3LYP ; (ii) HF,

M P2, MP4(SDQ), QCISD, MRDCI, G2(M P2) and G2 predict a much larger

activation energy (with respect to reactants) than B3LYP ; (iii) MP2, M RDCI,

G2(M P2), G2 and B3LYP are in agreement with respect to the energy diŒerence

between products and reactants, while HF, MP4(SDQ) and QCISD results are too

high ; and (iv) MRDCI and G2 are essentially in agreement as to the energy of the ® nal

complex, while the B3LYP result is too low. These observations make quite clear then

that, despite minor diŒerences, conventional ab initio calculations are in agreement

with placing the TS about 25 kJ mol Õ " higher with respect to the reactants than the

B3LYP method.

3.3. BSSE and basis set completeness for the TS

There are two possible reasons that could explain the discrepancies observed

among the diŒerent methodologies. One of them is basis set superposition error (see

the discussion in [58] about basis set superposition error and DFT) and the other is

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1862 O. N. Ventura

Tab

le2.

To

tal

an

dre

lati

ve

ener

gie

so

fth

ed

iŒere

nt

spec

ies

dis

cu

ssed

inth

isp

ap

ero

bta

ined

at

the

QC

ISD

(FC

)}6-3

11

­­

G(2

df,

2p

d)}

}B3L

YP

}6-

311

­­

G(2

df,

2p

d)

level

.a

HF

MP

2M

P4(S

DQ

)Q

CIS

D

MR

CI

To

tal

Rel.

bT

ota

lR

el.

bT

ota

lR

el.b

To

tal

Rel.

bR

el.

b

H®

0±4

99

818

®0±4

99

818

®0

±499

818

®0

±400

818

Li

®7

±432

027

®7±4

32

027

®7

±432

027

®7

±432

027

®7±4

45

008

®7

±447

268

®7

±447

332

F®

99

±401

693

®99±5

98

561

®99

±609

992

®99

±610

438

®99±6

19

624

®99

±631

223

®99

±631

654

HF

®100

±056

376

®100±3

27

455

®100

±329

917

®100

±329

838

®100±3

48

805

®100

±351

812

®100

±351

732

LiF

®106

±978

060

27±6

®107±2

54

319

14

±0®

107

±253

163

23

±5®

107

±252

671

24±6

10

®107±2

90

131

10

±1®

107

±291

369

20

±7®

107

±290

958

21±7

HL

i®

7±9

86

047

264±0

®8±0

13

260

387

±3®

8±0

20

924

343

±7®

8±0

21

704

340±3

320

®8±0

27

058

385

±9®

8±0

37

309

324

±4®

8±0

36

863

343±7

LiF

H®

107

±492

501

®10±7

®107±7

67

430

®20

±8®

107

±770

108

®21

±4®

107

±770

393

®22±4

®28

®107±8

03

029

®24

±2®

107

±808

081

®23

±6®

107

±808

419

®24±5

HL

iF®

107

±469

099

50±6

®107±7

52

007

19

±6®

107

±752

258

25

±6®

107

±752

685

24±1

5

®107±7

88

053

15

±1®

107

±790

705

22

±0®

107

±791

207

20±6

TS

®107

±459

628

75±5

®107±7

48

389

29

±1®

107

±749

558

32

±5®

107

±749

695

31±9

24

®107±7

84

272

25

±0®

107

±787

815

29

±5®

107

±788

025

29±0

aT

ota

len

erg

ies

inau

,re

lati

ve

en

erg

ies

ink

Jm

olÕ

";

MR

CI

calc

ula

tio

ns

are

fro

m[2

7];

the

®rs

ten

try

for

each

row

isth

ere

sult

usi

ng

the

fro

zen

-co

reap

pro

xim

ati

on

;th

ese

con

den

try

inclu

des

all

the

ele

ctr

on

s(n

ofr

ozen

-co

reap

pro

xim

ati

on

).b

Rela

tive

ener

gie

sw

ith

resp

ect

toH

F­

Li,

for

LiF

(­H

),H

Li

(­F

),L

iFH

,H

LiF

an

dT

S.

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Transition states for H radical reactions 1863

Tab

le3.

To

tal

an

dre

lati

ve

energ

ies

an

den

thalp

ies

of

the

diŒ

ere

nt

spec

ies

dis

cu

ssed

inth

isp

ap

er

ob

tain

edat

the

G2(M

P2)

an

dG

2le

vels

.a

G2(M

P2)

G2

En

ergy

bE

nth

alp

y0

Kc

En

erg

yb

En

thalp

y0

Kc

MR

CI

To

tal

Rel.

To

tal

Rel.

To

tal

Rel

.T

ota

lR

el.

DE

DH

H®

0±5

00

000

®0±4

97

639

®0

±500

000

®0

±497

639

Li

®7

±432

217

®7±4

29

856

®7

±432

217

®7

±429

856

F®

99

±628

941

®99±6

26

580

®99

±632

814

®99

±630

453

HF

®100

±347

034

®100±3

43

729

®100

±350

007

®100

±346

702

LiF

®107

±280

923

13

±4®

107±2

77

568

®4±4

®107

±284

205

12

±6®

107

±280

850

®5

±210

®9

HL

i®

8±0

21

786

®8±0

18

468

337

®8

±022

475

®8

±019

158

333

320

LiF

H®

107

±786

269

®21

±7®

107±7

81

786

®18±4

®107

±789

532

®22

±5®

107

±785

050

®19

±2®

28

®31

±5d

HL

iF®

107

±781

302

3±2

®107±7

76

053

®5±4

®107

±784

598

2±4

®107

±779

350

®6

±25

TS

®107

±775

642

24

±1®

107±7

71

668

9±8

®107

±778

899

23

±0®

107

±774

925

8±7

24

8±7

aT

ota

len

erg

ies

inau

,re

lati

ve

en

ergie

sin

kJ

mo

lÕ";M

RC

Ica

lcu

lati

on

sare

fro

m[2

7];

rela

tive

ener

gie

scalc

ula

ted

wit

hre

spect

toH

F­

Li,

for

LiF

(­H

),

HL

i(­

F),

LiF

H,

HL

iFan

dT

S.

bW

ith

ou

tze

ro-p

oin

ten

erg

y(Z

PE

)co

rrect

ion

inclu

ded

.c

Wit

hZ

PE

incl

ud

ed(t

hes

eare

the

usu

all

ycall

edG

2en

erg

ies)

.d

Zero

-po

int

en

erg

yta

ken

fro

mth

eC

Ica

lcu

lati

on

so

f[2

4].

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1864 O. N. Ventura

Table 4. Counterpoise corrected total energies of the atoms in the fragments and relative

energies of the initial complex and the transition state at the B3LYP, HF, MP2, MP4 and

QCISD levels employing the 6-311 ­ ­ G(3df, 3pd) basis set.a

Atomb B3LYP HF MP2 MP4(SDQ) QCISD

H(HF) ® 0 ± 502 261 9 ® 0 ± 499 822 ® 0 ± 499 822 ® 0 ± 499 822 ® 0 ± 499 822

F(HF) ® 99 ± 762 582 6 ® 99 ± 402 961 9 ® 99 ± 628 862 2 ® 99 ± 640 283 7 ® 99 ± 640 742 3

H(TS) ® 0 ± 502 263 1 ® 0 ± 499 830 8 ® 0 ± 499 830 8 ® 0 ± 499 830 8 ® 0 ± 499 830 8

F(TS) ® 99 ± 762 699 4 ® 99 ± 403 460 3 ® 99 ± 628 677 3 ® 99 ± 640 181 3 ® 99 ± 640 632 3

Li(TS) ® 7 ± 491 515 1 ® 7 ± 432 049 ® 7 ± 446 213 5 ® 7 ± 448 500 6 ® 7 ± 448 571 3

H(LiFH) ® 0 ± 502 265 0 ® 0 ± 499 831 ® 0 ± 499 831 ® 0 ± 499 831 ® 0 ± 499 831

F(LiFH) ® 99 ± 762 718 4 ® 99 ± 403 584 9 ® 99 ± 630 326 8 ® 99 ± 641 771 7 ® 99 ± 642 214 9

Li(LiFH) ® 7 ± 491 457 5 ® 7 ± 432 039 7 ® 7 ± 445 854 4 ® 7 ± 448 121 9 ® 7 ± 448 189 8

D E (TS) ® 0 ± 04 ( ® 0 ± 9) ­ 76 ± 9 ( ­ 75 ± 5) ­ 28 ± 9 ( ­ 25 ± 0) ­ 32 ± 5 ( ­ 29 ± 5) ­ 31 ± 9 ( ­ 29 ± 0)

D E (LiFH) ® 34 ± 1 ( ® 34 ± 8) ® 8 ± 6 ( ® 10 ± 7) ® 16 ± 9 ( ® 24 ± 2) ® 17 ± 4 ( ® 23 ± 6) ® 18 ± 4 ( ® 24 ± 5)

a Total energies in au, relative energies (with respect to HF ­ Li) in kJ mol Õ " ; relativeenergies in parentheses are the non-corrected ones.

b In parentheses is the fragment used for the calculation of the counterpoise corrected

energy of the given atom.

that the basis set, although converged for the minima, may not be converged for the

transition state.

Basis set superposition error (BSSE) may arise because the relative energies are

calculated with respect to the separated reactants Li and HF. Although it has been

shown [58] that BSSE generally is small in DFT calculations, it is worthwhile to try to

correct for it in the present circumstances of such a diŒerent picture provided by each

diŒerent method. Two procedures were followed to suppress the BSSE. On the one

hand, the barrier with respect to the stable complex LiFH was computed. Since the

transition state is a diŒerent close arrangement of the same atoms on the same PES,

it is to be expected that BSSE in¯ uence is smaller than when we refer to the energies

for the separated reactants. The results at the MP2, MP4(SDQ), QCISD, G2, MRDCI

and B3LYP levels for this barrier are, respectively, 49 ± 9, 56 ± 0, 54 ± 3, 45 ± 5, 52 and

35 ± 7 kJ mol Õ " . One sees then that conventional ab initio methods group together

predicting an activation energy (with respect to the initial complex) of about

50 kJ mol Õ " , whereas B3LYP predicts a smaller activation energy of about 36 kJ mol Õ " .

A lthough the diŒerence is now smaller (about 14 kJ mol Õ " instead of 25 kJ mol Õ " , and

even only 10 kJ mol Õ " diŒerence with respect to the G2 calculation) it is still

considerable.

The second procedure used for suppressing the BSSE was the counterpoise

correction of Boys and Bernardi [59], but applied to the atomization energy of the

complex and each fragment considered, and then subtracting the atomization energies

themselves. In mathematical terms, if one calls AB the complex of two fragments A

and B and calls 1 the basis set of A, 2 the basis set of B, and 12 the basis set of AB, then

the stabilization energy of AB will be calculated as

D EAB" #

¯ [EAB" #

® 3a ` AB

Ea" #

] ® 3A ` AB

[EA"® 3

a ` A

Ea"]

This way of calculating the counterpoise corrected energy of AB clearly

corresponds to a type of Born± Haber cycle which, to our knowledge, has not been

used before. It has the advantage of avoiding problems related to the diŒerent

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Transition states for H radical reactions 1865

Figure 4. Convergence of the relative energy of the transition state with respect to reactants.

The relative energies (in kJ mol Õ " ) have been ordered with respect to the energy obtainedwith each basis set for the HF molecule (conveniently scaled). The line was obtained by

® tting a second-order polynomial to all the points. The maximum of the ® tting

polynomial is obtained at an energy of about 1 kJ mol Õ " .

geometries of the free fragments and the fragments in the complex. This method was

applied to the QCISD and B3LYP calculations, and the results are given in table 4.

The ® rst observation is that the counterpoise correction does not aŒect signi® cantly

either the energy of the transition state or that of the initial complex at the B3LYP level

(the overall eŒect is less than 1 kJ mol Õ " in both cases) demonstrating again that the

basis set is extended enough. In fact, from the comparison of the numbers in the tables,

it is clear that BSSE is much more noticeable in the conventional ab initio calculations.

The values found for the counterpoise correction at the DFT level are smaller than

those found, for example, by Hobza et al. [7] or Topol et al. [8] for hydrogen bonded

complexes. One can conclude then, that BSSE is not responsible for the discrepancies

between the DFT and conventional ab initio calculations reported in this paper.

The problem with the extension of the basis set is that, even if already shown that

the basis is su� ciently extended for converging the energy of the minima, it may

happen that the transition state requires a more careful treatment. Therefore, several

more basis sets were employed to optimize the energy of HF and the TS and to

calculate the relative energy of the latter with respect to the reactants. Figure 4 plots

the relative energy of the transition state with respect to the total energy of the HF

molecule for a given basis set (as a measure of the precision this basis set can reach).

It is clear that the in¯ uence of the basis set on the energy of the transition state is very

large. Addition of polarization functions on H is essential to reach a ® rst improvement

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1866 O. N. Ventura

Table 5. Total and relative energies for the transition state using the B3LYP method.a

Basis set Li HF TS D E (TS)

6-311G ® 7 ± 491 297 ® 100 ± 448 005 ® 107 ± 953 432 ® 42 ± 36-311G(d) ® 7 ± 491 297 ® 100 ± 458 549 ® 107 ± 962 433 ® 33 ± 06-311 ­ ­ G ® 7 ± 491 297 ® 100 ± 460 507 ® 107 ± 964 153 ® 32 ± 46-311G(d, p) ® 7 ± 491 297 ® 100 ± 469 728 ® 107 ± 966 038 ® 13 ± 26-311G(df, dp) ® 7 ± 491 297 ® 100 ± 471 759 ® 107 ± 968 222 ® 13 ± 66-311G(2d, 2p) ® 7 ± 491 297 ® 100 ± 473 572 ® 107 ± 972 341 ® 19 ± 66-311G(3d, 3p) ® 7 ± 491 297 ® 100 ± 478 712 ® 107 ± 975 023 ® 13 ± 26-311G(3df, 3pd) ® 7 ± 491 297 ® 100 ± 480 453 ® 107 ± 976 213 ® 11 ± 76-311 ­ ­ G(d, p) ® 7 ± 491 297 ® 100 ± 482 384 ® 107 ± 973 965 ® 0 ± 86-311 ­ ­ G(2df, 2pd) ® 7 ± 491 297 ® 100 ± 486 078 ® 107 ± 977 627 ® 0 ± 76-311 ­ ­ G(3df, 3pd) ® 7 ± 491 297 ® 100 ± 486 982 ® 107 ± 978 705 ® 1 ± 1Roos ® 7 ± 492 936 ® 100 ± 499 322 ® 107 ± 993 623 ® 3 ± 6

a Total energies in au ; relative energies in kJ mol Õ " . Small diŒerences between the energiesin this table and in table 1 are due to the precision of the calculations.

of the results, while the second large improvement is obtained by extension of the

valence basis set with diŒusion functions. The results given by the Roos basis set can

be considered the most exact of all, both because of the number of functions involved

(it is a (14s9p4d) set) and because of the balance of the basis set. Even in this case, the

energy of the transition state is under that of the reactants by more than 3 kJ mol Õ " .

Accepting that a second order correlation polynomial can be adjusted to these points

and that somehow the maximum of this curve will give the best extrapolated energy for

the TS, one can see that it is hardly above 1 kJ mol Õ " . Therefore, one can conclude

safely that basis set is not the cause of the failure.

In conclusion, then, one is faced with a severe discrepancy between conventional

ab initio and the B3LYP methods. If one is allowed to be guided by the known

experimental data, the MRDCI results are validated by the accuracy of the

stabilization energy of LiFH (28 kJ mol Õ " versus 29 kJ mol Õ " experimentally), the

value of the activation enthalpy (8 ± 5 kJ mol Õ " ), and the exoergicity of the reaction

( ® 9 kJ mol Õ " versus ® 4 ± 8 kJ mol Õ " ). Accepting that G2 is even more accurate (which

is by no means an accurate hypothesis in itself, but seems to be validated by extensive

research [60 ± 64]), then the initial complex is in a well that is a little too deep and the

® nal complex is in a well that is too shallow, a situation that could be justi® ed by the

fact that the basis set employed in the MRDCI calculations was an spd-only set (i.e.,

no f on F or Li and no d on H) with only 47 contracted functions fo the LiFH system

(for instance, the 6-311 ­ ­ G(3df, 3pd) basis set employed here has 102 contracted

functions with f functions on Li, F and d functions on H).

On the other hand, B3LYP gives apparently sensible values for the LiFH

stabilization energy and reaction exoergicity, with a reasonable value also for the

energy of the products compared with that of the reactants. However, one can

conceive that this is mainly an eŒect of the fact that the diatomic molecules considered

here were employed among those used for parameterizing B3LYP. Then again, the

energy (and, especially, the enthalpy) of the transition state is much too low compared

with the G2 or MRDCI calculations, predicting then a reaction without a barrier.

Therefore, one must conclude that the B3LYP method does not behave properly for

this system.

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Transition states for H radical reactions 1867

Table 6. Optimum geometries and relative energies for the transition state using diŒerent

exchange± correlation functionals.a

Correlation Becke 1988 Becke 3-parameter

LYP r(LiF) 1 ± 685 1 ± 671

r(FH) 1 ± 340 1 ± 316h (LiFH) 72 ± 8 72 ± 7D E ® 14 ± 7 ® 0 ± 84

P86 r(LiF) 1 ± 676 1 ± 668

r(FH) 1 ± 423 1 ± 340h (LiFH) 70 ± 1 71 ± 2D E ® 6 ± 02 ­ 2 ± 17

PW91 r(LiF) 1 ± 682 1 ± 671

r(FH) 1 ± 397 1 ± 349

h (LiFH) 70 ± 3 71 ± 2D E ® 0 ± 84 ­ 11 ± 6

a Distances in A/ , angles in deg. and relative energies with respect to

Li ­ HF in kJ mol Õ " .

A similar problem, i.e., too low a transition state, was reported by Baker et al. [13]

and Andzelm et al. [12] with respect to the OH ­ H#! H

#O ­ H reaction. They

found that only the ACM method could give a relatively reasonable value for the

barrier for hydrogen transfer. This barrier was calculated at 3 ± 7 kcal mol Õ " using

the 6-31G(d) basis set and at 1 ± 8 kcal mol Õ " using a TZ2P basis set. Experimentally, the

best value for this barrier is 3 ± 0 kcal mol Õ " [65], while Francisco [66] calculated it as

5 ± 8 kcal mol Õ " at the QCISD(T) } 6-311 ­ ­ G(3df, 3pd) level. To complete the analysis

of the LiFH problem, also the transition state, as well as the reactants, for the OH ­ H#! H

#O ­ H reaction were calculated in this paper. The result obtained for the barrier

at the B3LYP } 6-311 ­ ­ G(3df, 3pd) level was 1 ± 2 kcal mol Õ " , slightly smaller than

that of Andzelm et al. [12] at the TZ2P level. Interestingly, the diŒerence between the

value in this paper and that of Francisco is about 20 kJ mol Õ " , comparable with the

diŒerence between the QCISD and B3LYP results obtained in this paper for the TS of

the LiFH system. In the case of the HOHH system, the B3LYP value is nearer to the

experimental one (or, better, to the most recent experimental one), than the QCISD(T),

although if one compares the two barriers the B3LYP value is about 20 kJ mol Õ " lower

than the QCISD(T). Therefore, for the reaction of Li with HF, B3LYP could be

thought of as nearer to the true barrier than G2 (analogous in this paper to the

Francisco QCISD(T) } 6-311 ­ ­ G(3df, 3pd) calculation).

3.4. Other functionals

The fact that B3LYP de® nitely seems to be ¯ awed for the transition state of the

LiFH radical system (as seems to be the case for other radicals involving the H atom)

does not imply that other functionals are also. Thus, a geometry optimization of the

transition state using combinations of the Becke 1988 and B3 exchange functionals

and the LYP, P86 and PW91 correlation functionals was performed. Since the B3LYP

results appeared converged with respect to the basis set when 6-311 ­ ­ G(3df, 3pd)

was used, only this basis set was employed for the calculations. B3LYP } 6-

311 ­ ­ G(3df, 3pd) optimum geometries were employed as a starting point. Local

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1868 O. N. Ventura

spin-density calculations were also attempted, but no convergence to a TS was found

and therefore the results are not reported here.

In table 6 are collected the results obtained for the relative energy of the TS with

respect to Li ­ HF using the diŒerent functionals. What is immediately obvious from

this table is that changing the correlation functional in the ACM improves agreement

with the conventional ab initio calculations. The best result is obtained using B3PW 91,

although the height of the TS ( ­ 11 ± 6 kJ mol Õ " ) is still half of that given by MRCI

( ­ 24 kJ mol Õ " ). Gradient corrected functionals cannot compare with ACM methods

(as already noted by Andzelm et al. [12]), but in this case also PW91 seems to be the

best correlation functional, moving up the BLYP result by almost 14 kJ mol Õ " .

Perdew and Burke, from an analysis of the gradient-correct functionals, also concluded

[71] that PW91 seems to be the best correlation functional available. In fact, B3PW91

also gives, sensible values for the stabilization energy of the initial complex

( ® 26 kJ mol Õ " versus ® 28 kJ mol Õ " for the M RCI calculation) and the energy of the

HLiF complex ( ­ 9 ± 0 kJ mol Õ " versus ­ 5 kJ mol Õ " at the MRCI level).

4. Conclusion

Basis-set converged, BSSE-free B3LYP calculations were performed on the

reaction path for Li ­ HF ! H ­ LiF. QCISD } 6-311 ­ ­ G(3df, 3pd) at the optimum

B3LYP geometries, and G2 calculations were performed for comparison with the

B3LYP calculations and M RDCI results of the literature. Also, other types of ACM

and gradient-corrected functionals were employed for comparison. The following

conclusions were reached.

(1) The B3LYP } 6-311 ­ ­ G(3df, 3pd) optimum geometries are well in agreement

with MRDCI calculations, except in the case of the ® nal complex LiHF where

there is a more noticeable discrepancy.

(2) The B3LYP } 6-311 ­ ­ G(3df, 3pd) for the initial LiFH complex as well as for

the products with respect to the reactants also are reasonably in agreement

with MRDCI results.

(3) The B3LYP } 6-311 ­ ­ G(3df, 3pd) relative energy of the transition state and,

to a lesser extent, that of the ® nal HLiF complex, are much lower than the

M RDCI or G2 energies.

(4) QCISD(Full) } 6-311 ­ ­ G(3df, 3pd) } } B3LYP } 6-311 ­ ­ G(3df, 3pd) relative

energies are in reasonable agreement (i.e., a 5 kJ mol Õ " interval) with the

M RDCI results, except for the energies of HLiF (21 kJ mol Õ " versus

5 kJ mol Õ " ) and the stabilization energy of the products (22 kJ mol Õ " versus

10 kJ mol Õ " ).

(5) Gaussian 2 theory predicts energies that are in general completely in agreement

with MRDCI (except in the case of LiFH for which it is a little higher).

(6) ACM functionals behave better than non-ACM ones in the prediction of the

height of the transition state. Of these functionals, N3PW91 (the original

ACM by Becke [37]) gives the best result for the TS, although the relative

energy is not more than half of that obtained with the conventional ab initio

calculations.

As a general conclusion then, one observes that conventional ab initio methods

(QCISD, MRDCI and G2) are in agreement among themselves in predicting a TS

about 25 kJ mol Õ " higher in energy than the reactants. B3LYP predicts a much

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Transition states for H radical reactions 1869

lower energy for this transition state (somewhat improved if one uses B3P86 or

B3PW 91 instead) and, consequently, a completely diŒerent qualitative picture of the

reaction, i.e., a reaction with no barrier is predicted. This is the ® rst case in which such

a severe discrepancy is observed between B3LYP and conventional methods and, thus,

the LiFH system can be useful as a benchmark for the study of improved DFT

methods. In fact, it was shown also here that B3PW91 is better than B3LYP, and

seems that it would be preferred over the latter when reactions of radical involving H

atoms are studied.

The author acknowledges the reception of a fellowship from the Alexander von

Humboldt-StiŒtung for a stay in Germany, during which this work was performed.

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