exploring organic chemistry with dft: radical, organo-metallic, and bio-organic applications

Exploring Organic Chemistry with DFT:Radical, Organo-metallic, and Bio-organic ApplicationsFernando Bernardi, Andrea Bottoni* and Marco Garavelli*

Dipartimento di Chimica ™G.Ciamician∫, Universita× di Bologna, via Selmi 2, 40126 Bologna, Italy , e-mail: [email protected]

Abstract

In this review we report the results of DFT investigationswhich have been carried out in different fields of organicand organometallic chemistry, including radical reactivity,structure and reactivity of organometallic compounds, andbiochemical/biophysical properties of long chain unsatu-rated systems. Many of the most popular non-localcorrected functionals (e.g. B3LYP, BHLYP, BLYP, BP86)have been benchmarked both versus experimental andhigh level ab initio (e.g. MP2, MP4, CAS-SCF/CAS-PT2)data, resulting in an impressive agreement. The DFTapproach appears to be a powerful tool, which can be usedas a valid alternative to more traditional correlatedmethods, to achieve mechanistic information of chemical/physical interest in the modelling of organic and biochem-ical systems.

In particular, in the examples selected in this review, wediscuss the results obtained for the addition reaction ofalkyl radicals to double bonds and for the hydrogen/chlorine abstraction reaction by alkyl and silyl radicalsfrom various organic substrates. Moreover, binding inter-actions (i.e. geometries and energies) in organometalliccompounds are shown to be satisfactorily reproduced viaDFT and examples of nickel-catalyzed [2� 2] cycloaddi-tion reaction and homogeneous Ziegler-Natta catalysis areinvestigated. Finally, a DFT modelling for the singlet-oxygen quenching ability and radical trapping activity ofcarotenes is presented. The simulated data provide arationale for the protective action of carotenes observed inbiological tissues and elucidates the physical and chemicalmechanisms involved in the reactivity of carotenes versusoxygen and radicals.

1 Introduction

In the last two decades, density functional theory (DFT) [1 ±11] has emerged as a practical and versatile tool to obtainaccurate information on molecular systems of chemicalinterest. The popularity of DFT-based methods stems inlarge measure from its computational expedience thatallows to acquire, even for large molecules, thermochemicaldata, force fields and frequencies, transition state structures,NMR, PES, ESCA, IR and Raman spectra. Thus thisapproach, which includes the dynamic correlation effects,represents a valid alternative to the HF theory, or to moretraditional post-HF methods such as Moller-Plesset theory,coupled-cluster or configuration interaction, which arehighly demanding in terms of CPU time.

Nowadays DFT is put into practice almost exclusivelyusing the Kohn-Sham equations [4], which are formallysimilar to the Hartree-Fock (HF) equations. The funda-

mental approximation adopted in the practical applicationsof DFT is the Local Spin Density (LSD) [5] where ahomogeneous electron gas model for the electron density isused. In spite of this crude approximation, the LSDapproach is able to provide useful results for a variety ofinorganic and organic molecular systems. In particular,results that are in better agreement with the experimentthan those obtained from HF computations, have beenobtained for transition metal complexes, transition metalclusters, polymers, metal surfaces and interfaces [6]. Otherclasses of problems, however, are not satisfactorily descri-bed at the LSD level [7]. For instance, bond energies aretypically overestimated and the geometry of moleculescontaining hydrogen bonds is not correctly reproduced. Oneway to correct the errors of the local density approximationhas been to introduce density-gradient terms (nonlocalcorrections). Since in this way the exchange and correlationcontributions are much more accurately described, thenonlocal methods afford much better estimates of bondenergies than the simpler local approaches [8 ± 9]. Becke, forinstance, has demonstrated that the nonlocal methodsprovide bond energies of the same quality as thosecomputed with the Pople×s G2 method [10]. Also, it hasbeen demonstrated that the nonlocal approach can be

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* To receive all correspondence

Key words: DFT, radical reactions, organometallic compounds,carotenes, singlet-oxygen

F. Bernardi, et al.

satisfactorily applied to the study of transition metalcomplexes where it can improve significantly the descrip-tion of metal-ligand and metal-metal bonds [11]. Theresearch carried out in this field during the last two decadeshas lead to the development of a variety of local and non-local functionals, which can be used within the Kohn-Shamformalism. These functionals have been implemented invarious quantum-mechanical packages [12 ± 16] and arenow available for molecular routine computations.

The objective of this review is to discuss a number ofrepresentative examples where the DFT results, obtainedusing various functionals and localized basis sets, arecompared to experimental data or to the results obtainedwith traditional correlated methods. This type of calibration,to establish strengths and weaknesses of the various func-tionals, is nowadays particularly important since DFTmethods are more and more replacing the HF approach inthe standard description of medium and large-size mole-cules. In the next section, to illustrate the potentiality of theDFT approach, we shall present the results obtained forsome important classes of molecular systems and chemicalreactions. So doing, we hope to clearly show that DFT maybe routinely used as an efficient tool to explore and analyzeorganic reactivity, as well as to gain useful information ofmechanistic interest in biochemical and biophysical prob-lems. As it will be demonstrated below, when properlycombined and complemented with technical skills andcomputational knowledge, DFT can frequently provideresults of impressive accuracy and reliability. In particular,we shall discuss the results of DFT applications (analyzingalso the performances of various functionals) in the case of(a) radical reactions, (b) structure of organometallic com-pounds, (c) transition metal catalyzed reactions, and (d)reactivity of bio-organic molecules polyenic systems.

2 Computational Details

The DFT computations reported in this paper have beencarried out using various hybrid and pure functionals asimplemented in the Gaussian 92/DFT [14], Gaussian 94 [15]and Gaussian 98 [16] series of programs. Following theGaussian formalism these functionals are denoted asBHLYP, B3LYP (hybrid functionals) and BLYP, BP86(pure functionals) and can be written in the general form:

a1E(S)x � a2E(HF)x� a3E(B88)x� a4E(LOCAL)c

� a5E(NON-LOCAL)c (1)

In expression 1 E(S)x is the Slater exchange [3, 4, 17],E(HF)x the Hartree-Fock exchange, E(B88)x the Becke×s1988 non-local exchange functional corrections [18], E(LO-CAL)c is a local correlation functional and E(NON-LOCAL)c is the gradient-corrected correlation contribu-tion. In the hybrid Becke×s three-parameter exchangefunctional B3LYP [18] E(LOCAL)c corresponds to theVosko, Wilk and Nusair (VWN) local correlation functional

[19] and E(NON-LOCAL)c to the correlation functional ofLee, Yang and Parr (LYP) [20] which includes both local andnon-local terms. The coefficients in expression 1 are thosedetermined by Becke: a1� 0.80, a2� 0.20, a3 � 0.72, a4� 0.19and a5 � 0.81. The other hybrid method (BHLYP) ischaracterized by the following parameters: a1� 0.50, a2 �0.50, a3� 0.50, a4 � 0.00 and a5� 1.00.

The two pure DFT functionals differ only in the corre-lation term. The functional denoted as BLYP has thefollowing expression:

E(S)x�E(B88)x�E(LYP)c (2)

while that denoted as BP86 corresponds to:

E(S)x�E(B88)x�E(P86)c (3)

where E(P86)c includes the local functional of Perdew alongwith his gradient corrections [21].

The CAS-SCF computations have been carried out eitherwith the package available in the Gaussian programs [14 ±16] or using the MOLCAS [22] software. For the CAS-SCFwave-functions the dynamical correlation energy effectshave been included using the multi-reference perturbationapproach suggested by Andersson et al. (CAS-PT2) [23].The active space used will be specified in the examplesdiscussed below.

The molecular structures and the minimum energy paths(MEP) reported in the following section have been fullyoptimized using the gradient methods available in theGaussian packages [14 ± 16]. Frequency computations havebeen carried out at the HF, MP2, DFTand CAS-SCF levelsof theory to determine the nature of the various criticalpoints and to evaluate the energy thermal corrections. In thecase of an unrestricted Moller-Plesset approach, projectedMP2 and MP4 energies have been used. The use of projectedvalues, when dealing with an unrestricted approach, isdiscussed below.

Localized basis sets have been used in all the computa-tions employing, when possible, the basis sets available inthe Gaussian pakages [14 ± 16], as documented below foreach reported example.

In many examples reported here, HF, MP2 and DFTcomputations have been carried out using unrestrictedwavefunctions. This is the elective procedure for studyingopen shell species (such as radicals, diradicals, 1O2, triplets,etc.) as the ones investigated in this paper. Unfortunately,the resulting wavefunction is not, in general, an �S2�eigenstate. This means that its energy is spin-contaminated(generally, contamination arises by the first higher multi-plicity state, i.e. the T1 state for spin-contaminated singletenergies).

Although unrestricted DFT methods in general (and theB3LYP functional in particular) have the capability ofreducing the effects of spin contamination (see, for instance,the results discussed in section 3), if such a contaminationremains significant, results can be corrected by spin-

Quant. Struct.-Act. Relat., 21 (2002) 129

Exploring Organic Chemistry with DFT: Radical, Organo-metallic, and Bio-organic Applications

projection. For this purpose, we have used the approximatespin-correction procedure proposed by Yamaguchi et al.[24a ± b], where the singlet spin-corrected (1E(SC)) energy isevaluated by computing the unrestricted DFT singlet (1E(UB))and triplet (3E(UB)) energies, and applying the formula:

�(UB)� cs1�� cT

3� (4)

1E(SC)� 1E(UB)� fSC[1E(UB)�3E(UB)] (5)

fSC�c2

T

1 � c2S

�1�S2�

3�S2� � 1�S2� (6)

As will be pointed out in the following sections, thisprocedure has been recently applied by Houk et al. [24c]in the study of the two-step diradical mechanism of theDiels-Alder reaction between butadiene and ethylene. Wehave chosen this spin-correction procedure because it seemsto provide reasonable energies for singlet diradicals andspecies, which are similar to those presented here.

Another very important technical issue in DFT-basedinvestigations concerns wavefunction stability. For theenergies to be meaningful, we must check that theyrepresent the correct variational solution of the SCFprocedure, i.e. they must be local minima in the wave-function-coefficients space (with the specified degrees offreedom taken into consideration). Therefore, it is alwayscrucial to check wavefunction stability in order to grant thecorrectness of the corresponding energy.

3 Radical Reactions

3.1 The Addition of Alkyl and Halogenoalkyl Radicals tothe Ethylene Double Bond

Several experimental [25a ± d] and theoretical [25e ± g]mechanistic studies have been carried out over the lastthree decades on these radical reactions that are a powerfultool to form new C�C bonds and represent the key step inmany polymer processes. We report here the results that wehave obtained in a computational study on reaction (7)

.R�H2C�CH2� products (7)

where

R� .CH3,.CH2CH3,

.CH(CH3)2,.C(CH3)3

.CH2F, .CF3,.CCl3

These reactions are particularly suitable for a theoreticalinvestigation since they have been experimentally inves-tigated in the gas phase [25a ± d, g] and, consequently, theobtained activation energies are not affected by the solventeffect and can be directly compared with the theoreticalvalues. To carry out this computational study the unre-stricted Hartree-Fock, MP2 and MP4 (UHF, UMP2 andUMP4) methods and the unrestricted DFT approach withtwo pure (BLYP, BP86) and two hybrid (BHLYP, B3LYP)

functionals have been used. To verify the validity of a singlereference approach in describing the transition state region,the potential energy surfaces for the reactions involving.CH3,

.CH2CH3, and .CH2F have been re-investigated at theCAS�SCF level of theory. The active space is that requiredto describe correctly the forming of the new C�C �bond andthe simultaneous breaking of the C�C � bond. It consists ofthree electrons in three orbitals i.e. the � (doubly occupied)and the �* (empty) orbitals associated with the C�C olefinbond and the singly occupied p� orbital associated with thenon-bonding electron of the alkyl and halogenoalkylradicals. The 6-31G* basis set [14 ± 16] has been used in allcomputations.

For the various transition states the values of the mostrelevant geometrical parameters and of the correspondingenergies relative to reactants are collected in Table 1. Theseenergies include the zero-point vibrational energy correc-tions (ZPE) and can be compared to the experimentalactivation energies collected in the table. The UMP4 energyvalues are reported in parenthesis and have been obtainedfrom single-point computations on the UMP2 optimizedstructures. A schematic representation of the transitionstate structure is given in Figure 1.

At all computational levels the structure of the transitionstate is not very sensitive to the nature of the attackingradical. For the simplest system (R� .CH3), at the UHFlevel, the angle of attack of the approaching radical to theolefin (� ra angle) is 109.1�. As a consequence of theformation of the new C�C bond a considerable rehybrid-ization of the carbon atom takes place and a lengthening ofthe olefin bond, which is losing its double bond character, isobserved. The forming new C�C bond and the olefin bondare 2.232 and 1.383 ä respectively. These parameters do nochange very much with the increasing size of the attackingalkyl radical. A point of interest in these computationsconcerns the adequacy of the Hartree-Fock theory indescribing the transition states. For .CH3,

.CH2CH3 and.CH2F the geometrical parameters obtained with the CAS-SCF method are almost identical to the corresponding UHFvalues. This result is in agreement with the form of the CAS-

130 Quant. Struct.-Act. Relat., 21 (2002)

Figure 1. A schematic representation of the transition structurefor the addition of an alkyl radical to the ethylene double bond.The values of the geometrical parameters r, a and � ra arereported in Table 1.

F. Bernardi, et al.

SCF wavefunction that is dominated by the SCF config-uration. The inclusion of the dynamic correlation at theUMP2 level has the effect of making the r distance longerand the C�C double bond shorter (for .CH3, for instance, rand a become 2.261 and 1.344 ä respectively). Thus theUMP2 predicts earlier transition structures than the UHFmethod. The UMP2 geometrical parameters comparerather well with those obtained with the various DFTfunctionals. The most important changes found are a furtherlengthening of the r distance with a consequent increase ofthe reactant-like character of the transition state. Thesegeometrical modifications are small when the two hybridfunctionals are used, but become more important with thepure BLYP and BP86 functionals.

Even if the fundamental structural features of thetransition states are satisfactorily described by the UHFmethod, at this computational level the activation energiesEa are strongly overestimated and the inclusion of the

dynamic correlation effects is essential to obtain reasonableEa values. This is not surprising, since the accurate pre-diction of the energy barriers for radical reactions is adifficult problem and it is well known that high levels oftheory are needed to reproduce the experimental results[26]. A significant decrease of the activation barriers isobserved when projected UMP2 energies are considered.However, even if a general better agreement with theexperiment is found, in two cases ( .CH3 and .CH2F) theactivation barriers are still overestimated and for .C(CH3)3,.CF3 and .CCl3 they are quite underestimated. Also, thetrend of the computed activation energy that decrease alongthe series .CH3� .CH2CH3 � .CH(CH3)2 � .C(CH3)3, is indisagreement with that of the experimental values whichremain almost constant. In addition, the trend observedwhen we compare .C(CH3) (4.6 kcal mol�1) and .CH2F(6.1 kcal mol�1) is in contrast with the experiment (7.1 and4.4 kcal mol�1 respectively). Furthermore, it is worth to


Table 1. Transition state optimized structuresa and energies relative to reactants (Ea)b computed for the reaction .R�C2H4 at variouslevels of theory. The experimental activation energies (Ea,exp)are also reported. cThe symbols used for geometrical parameters are thosereported in Figure 1.

UHF CAS-SCF UMP2 (UMP4) BHLYP B3LYP BLYP BP86

R� .CH3 (Ea, exp. � 7.3 kcal mol�1)r 2.245 2.245 2.261 2.310 2.363 2.423 2.473a 1.382 1.378 1.344 1.351 1.356 1.364 1.358� ra 109.1 109.8 109.6 109.4 109.8 110.3 110.3Ea 11.6 16.3 8.9 (9.8) 8.2 6.6 5.2 4.1R� .CH2CH3 (Ea, exp.� 6.9 kcal mol�1)r 2.232 2.234 2.256 2.288 2.334 2.384 2.433a 1.383 1.379 1.344 1.353 1.358 1.367 1.361� ra 109.9 110.8 111.0 110.9 111.6 112.2 112.5Ea 11.5 16.6 7.6 (8.7) 8.5 7.2 5.9 4.7R� .CH(CH3)2 (Ea, exp.� 6.9 kcal mol�1)r 2.210 ± 2.249 2.265 2.305 2.344 2.399a 1.385 ± 1.343 1.355 1.361 1.371 1.364� ra 110.7 ± 110.3 110.8 111.4 111.9 111.4Ea 11.8 ± 5.9 (7.2) 8.5 7.5 6.6 4.9R� .C(CH3)3 (Ea, exp.� 7.1 kcal mol�1)r 2.200 ± 2.249 2.250 2.280 2.305 2.361a 1.387 ± 1.343 1.357 1.365 1.375 1.367� ra 111.6 ± 111.4 111.9 109.2 112.9 112.4Ea 12.5 ± 4.6 (6.0) 8.7 8.0 7.4 5.3R� .CH2F (Ea, exp. � 4.3 kcal mol�1)r 2.242 2.241 2.250 2.299 2.355 2.419 2.484a 1.379 1.376 1.343 1.350 1.355 1.363 1.358� ra 110.3 110.9 111.5 111.5 111.7 111.9 111.6Ea 8.6 13.5 6.1 (7.0) 6.3 4.3 3.7 2.7R� .CF3 (Ea, exp.� 2.4 kcal mol�1)r 2.299 ± 2.291 2.374 2.442 2.535 2.689a 1.372 ± 1.338 1.343 1.349 1.356 1.349� ra 106.6 ± 105.9 105.5 105.7 105.2 105.2Ea 4.5 ± 1.2 (1.9) 1.9 1.0 1.0 1.1R� .CCl3 (Ea, exp.� 6.3 kcal mol�1)r 2.199 ± 2.238 2.237 2.266 2.296 2.362a 1.383 ± 1.342 1.355 1.362 1.371 1.363� ra 108.4 ± 107.4 108.3 109.2 108.8 109.5Ea 10.1 ± 4.1 (5.1) 7.4 6.4 5.3 3.6

a Bond lengths are in ängstroms and angles in degrees. b Values in kcal mol�1. c See Ref. 25g.


point out that the UMP2 barriers are in better agreementwith the experiment than the corresponding UMP4 values.

The DFT barriers strongly depend on the functional usedin the computations. The BHLYP functional still over-estimates the barriers even if the trend along the series .CH3

� .CH2CH3 � .CH(CH3)2 � .C(CH3)3 � .CH2F, � .CF3,� .CCl3 is in much better agreement with the experiment.This trend does not vary when the two pure BLYP and BP86functionals are used, but in both cases the activationenergies become underestimated. The best agreementwith the experiment is found at the B3LYP level. With thisfunctional the difference between the experimental andtheoretical value is in all cases, except .CF3, less than1 kcal mol�1.

3.2 Hydrogen Abstraction from Halo-substituted Methanesby Methyl Radical

We focus now our attention on the hydrogen abstractionreaction from fluoro-, chloro and bromomethanes bymethyl radicals (Eq. 8):

CHnXm� .CH3 � .CHn-1Xm�CH4 (8)

where m� 1, 2, 3, n� 3, 2, 1 for X�F, Cl and m� 1, 2, n� 2,1 for X�Br. For these reactions an irregular ordering of theactivation energies has been experimentally observed [25d].The activation energy of fluoromethanes decreases fromCH3F to CH2F2, then increases again for CHF3 [25a ± d, 27].For chloromethanes and bromomethanes the activationenergy decreases regularly along the series CH3Cl�CH2Cl2�CHCl3 and CH3Br�CH2Br2. In the former case the Ea

trend parallels that of the C ± H bond energies, while in thelatter cases this trend becomes opposite [27]. For thesereactions a computational investigation [27] on the structureand energy of reactants and transition states has beencarried out at the UHF, UMP2 and unrestricted DFT levelswith the 6-31G* basis set. The three BHLYP, B3LYP andBLYP functionals have been used.

In all cases it has been found that the hydrogenabstraction from halomethanes proceeds in one step. Thetransition state, which is schematically represented inFigure 2, is characterized by a collinear or nearly collinear


Table 2. Transition state optimized structuresa and energies relative to reactants (Ea)b computed for the hydrogen abstraction fromhalomethanes by methyl radicals at various levels of theory. The experimental activation energies (Ea, exp) are also reported.c The symbolsused for geometrical parameters are those reported in Figure 2.

UHF UMP2 BHLYP B3LYP BLYP

CH3F� .CH3 (Ea, exp� 11.8 kcal mol�1)a 1.353 1.319 1.325 1.320 1.320b 1.360 1.348 1.358 1.388 1.418Ea 28.85 21.48 15.77 10.71 7.53CH2F2� .CH3 (Ea, exp� 10.4 kcal mol�1)a 1.356 1.320 1.324 1.316 1.308b 1.349 1.344 1.354 1.390 1.430Ea 28.54 16.15 14.64 9.12 5.65CHF3� .CH3 (Ea, exp� 13.6 kcal mol�1)a 1.376 1.349 1.353 1.345 1.337b 1.322 1.312 1.320 1.351 1.387Ea 30.13 17.33 16.03 10.31 6.67CH3Cl� .CH3 (Ea, exp� 9.4 kcal mol�1)a 1.354 1.320 1.326 1.323 1.324b 1.349 1.339 1.348 1.374 1.399Ea 28.27 16.07 15.42 10.75 7.92CH2Cl2� .CH3 (Ea, exp� 7.2 kcal mol�1)a 1.348 1.308 1.314 1.303 1.295b 1.345 1.345 1.355 1.393 1.432Ea 25.94 12.49 12.48 7.64 4.72CHCl3� .CH3 (Ea, exp� 5.8 kcal mol�1)a 1.343 1.298 1.302 1.287 1.272b 1.343 1.349 1.361 1.407 1.460Ea 23.60 9.14 9.78 4.94 2.03CH3Br� .CH3 (Ea, exp� 10.1 kcal mol�1)a 1.357 1.331 1.331 1.328 1.326b 1.343 1.325 1.339 1.365 1.391Ea 28.62 16.80 15.86 11.35 8.64CH2Br2� .CH3 (Ea, exp� 8.7 kcal mol�1)a 1.352 1.320 1.317 1.303 1.292b 1.338 1.328 1.348 1.387 1.427Ea 26.26 12.58 12.74 7.95 5.02

a Values in ängstroms. b Values in kcal mol�1. c See Ref. 27.

F. Bernardi, et al.

arrangement of the three atoms involved in the process. Thevaluesof thebreakingand formingbonds (parametersaandb)are reported in Table 2. For fluoromethanes, at the UHFlevel, the transition state slightly advances toward theproduct when the number of fluorine atoms in the substrateincreases. At the UMP2 level, with the inclusion of thedynamic correlation, the transition state becomes morereactant-like. A similar effect has been observed at the DFTlevel with all three functionals. Similar results have beenobtained for chloro and bromomethanes: in both cases theinclusion of correlation determines an increase of thereactant-like character of the transition structure.

The experimental (Ea, exp) and computed (Ea) activationenergies at various levels of theory are reported in Table 2(the activation energies have been evaluated as the differ-ence between the energies of the transition states and thoseof reactants and include the ZPE corrections). The UHFactivation barriers are in all cases overestimated. This resultis similar to that discussed in the previous section for theaddition to the olefin bond. The projected UMP2 values aresignificantly lower than the corresponding UHF data, butthey are still quite overestimated (the error with respect tothe experiment is in all cases larger than 55%). Once again atthe DFT level the energy barriers depend significantly onthe type of functional. Even if with the hybrid BHLYPfunctional the barriers are still overestimated, their trendfrom CHF3 (16.04 kcal mol�1) to CH3F (15.77 kcal mol�1) isnow the same as that experimentally found. On the otherhand the pure BLYP functional provides Ea values which arequite underestimated. The best agreement with the experi-ment is obtained at the B3LYP level of theory, where the

difference between the computed and experimental value isin all cases within 1.5 kcal mol�1. As mentioned in theprevious section, a further point of interest is represented bythe capability of the DFT (B3LYP) method of reducing theeffects of spin contamination. This is evident from thecomparison of the �S2� values obtained at the UHF, UMP2and unrestricted B3LYP levels and collected in Table 3.

3.3 Hydrogen and Chlorine Abstraction fromChloromethanes by Silyl Radicals.

While alkyl radicals react with haloakanes mainly viahydrogen abstraction, many heteroatom-centered radicalspreferentially abstract a halogen atom from organic halides(see Eq. 9).

R �nM

.�X�R�R×nM-X�R .

M�B, Si, Ge, P, transition metal; X�F, Cl, Br, I (9)

In particular, the halogen abstraction carried out by silylradicals has been widely investigated and a large amount ofexperimental data on the reactivity of silicon-centeredradicals toward various organic halides is now available [28,29]. For instance Cadman et al. studied the relative rates ofchlorine abstraction from alkyl chlorides by the trimethyl-silyl radical. These authors determined a barrier of 4.06 kcalmol�1 for the chlorine abstraction from H3CCl [29]. Aloni etal. found that the activation barrier for the chlorineabstraction by trichlorosilyl radicals from chloromethanedecreases when the number of halogen atoms increases [29].

In this section we present the results of a theoretical study[29] of the hydrogen and chlorine abstraction by the silyl H3Si.

and trichlorosilyl Cl3Si . radicals from ClCH3, Cl2CH2 and Cl3CH. The computations have been carried out using the UHFmethod, the Moller-Plesset perturbation theory MP2 up to


Table 3. Values of �S2� computed at the UHF, UMP2 andunrestricted DFT (B3LYP) levels with the 6-31G* basis set.

UHF UMP2 B3LYP

CH3F� .CH3 0.7890 0.7629 0.7571CH2F2� .CH3 0.7885 0.7627 0.7570CHF3� .CH3 0.7882 0.7624 0.7569CH3Cl� .CH3 0.7888 0.7629 0.7571CH2Cl2� .CH3 0.7879 0.7627 0.7669CHCl3� .CH3 0.7861 0.7621 0.7566CH3Br� .CH3 0.7896 0.7634 0.7572CH2Br2� .CH3 0.7891 0.7636 0.7570

Figure 2. A schematic representation of the transition structurefor the hydrogen abstraction from halomethanes by methylradical. The values of the geometrical parameters a and b arereported in Table 2.

Figure 3. A schematic representation of the transition structurefor the hydrogen and chlorine abstraction from chloromethanesby silyl and trichlorosilyl radicals. The values of the geometricalparameters a and b are reported in Tables 4 and 5.


second order (UMP2) and the unrestricted DFT methodwith the two functionals B3LYP and BLYP. A schematicrepresentation of the transition structures corresponding tothe hydrogen (TS1) and chlorine (TS2) abstraction from thechloromethane molecule is given in Figure 3. The values ofthe forming (a) and breaking (b) bonds computed atdifferent levels of theory are collected in Tables 4 for H3Si .

and Table 5 for Cl3Si .. In these tables the values of theactivation energies (Ea) are also reported. These activationenergies have been computed using the following equation:

Ea��H≥� nRT (10)

where R is the gas constant, T the absolute temperature, nrepresents the molecularity of the reaction (2 for the caseinvestigated here) and �H≥ is the activation enthalpy. Themolecular enthalpy is computed as H�E�Hth, where E isthe quantomechanical energy and the thermal corrections toenthalpy (Hth) is given by:

Hth� ZPE�Evib �Erot�Etr�RT(11)

In Eq. 11 Evib, Erot and Etr are the vibrational, rotational andtranslational contributions to the energy, respectively.

For the three substrate molecules considered here, bothhydrogen and chlorine abstractions proceed in one step andthe three atoms involved in the process are characterized bya collinear arrangement. Furthermore, while in the hydro-gen abstraction the two fragments H3Si . and Cl3Si . approachthe hydrogen atom of the substrate in a staggered con-formation, in the chlorine abstraction they are arranged inan eclipsed conformation.

To discuss the structure of the transition states thequantity � is introduced. This parameter, reported in thetables, conveniently describes the nature of the varioustransition structures at different computational levels. Forthe H abstraction ��R(Si-H)/ �R(C-H) where �R(Si-H)�a/R(Si-H)eq and �R(C-H)� b/R(C-H)eq. Here a and b arethe lengths of the forming Si ± H and breaking C ± H bonds,respectively and R(Si-H)eq and R(C-H)eq are the corre-sponding equilibrium distances in the product (silane) andreactant (choromethane). In a similar way, for the Clabstraction, � is defined as �� R(Si- Cl)/�R(C-Cl) where�R(Si-Cl)� a/R(Si-Cl)eq and �R(C-Cl)� b/R(C-Cl)eq. Inthis case a and b are the lengths of the forming Si-Cl andbreaking C�Cl bonds, respectively, and R(Si-Cl)eq andR(C-Cl)eq are the corresponding equilibrium distances inthe product (chlorosilane) and reactant (choromethane). Avalue of 1 for � indicates a transition state where the twobonds are broken and formed to the same extent; a valuelower or greater than 1 corresponds to a more product-likeor to a more reactant-like transition state respectively.

Inspection of Tables 4 and 5 points out the product-likecharacter of TS1 (��1) and the reactant-like character ofTS2 (��1). The trend of the computed � values indicatesthat TS1 becomes less product-like while TS2 becomes more

reactant-like when more chlorine atoms are introduced inthe substrate. The inclusion of the dynamic correlation at theUMP2 level has the effect of making TS1 more product-like(� is smaller than 1 and further decreases) and TS2 lessreactant-like (� is larger than 1 and decreases). A similareffect is observed for TS1 at the B3LYP and BLYP levels,where again � decreases with respect to the UHFand UMP2values. An opposite trend is observed for TS2: in this case �

significantly increases when the DFT is used.Since the activation barriers for the chlorine abstraction

by the silyl radical H3Si . from chloromethanes are notexperimentally available, to roughly estimate the accuracyof the various theoretical methods, the computed activationenergies reported in Table 4 can be compared with theexperimental values determined for the chlorine abstractionby the triethylsilyl radical Et3Si . from chloromethanes.These barriers are 4.06, 2.06 and 1.14 kcal mol�1 for ClCH3,Cl2CH2 and Cl3CH, respectively [29]. From Table 4 it isevident that the corresponding UHF values (22.10, 20.28


Table 4. Optimum values of the most relevant geometrical para-metersa of the transition states TS1 and TS2 with the correspondingactivation energies (Ea)b for the reaction between silyl radical andchloromethanes at various levels of theory . The symbols used forthe geometrical parameters are those reported in Figure 3.

UHF UMP2 B3LYP BLYP

(a) H3Si .�ClCH3

Hydrogen Abstraction (TS1)a 1.711 1.644 1.642 1.639b 1.445 1.512 1.552 1.600� 0.865 0.798 0.776 0.752Ea 28.33 23.78 17.43 15.69Chlorine Abstraction (TS2)a 2.568 2.435 2.541 2.606b 2.056 2.001 2.010 2.082� 1.078 1.051 1.097 1.091Ea 22.10 15.04 8.32 5.58

(b) H3Si .�Cl2CH2


(c) H3Si .�Cl3CHHydrogen Abstraction (TS1)a 1.717 1.646 1.671 1.676b 1.416 1.480 1.476 1.500� 0.881 0.814 0.827 0.816Ea 24.08 16.30 11.04 8.84Chlorine Abstraction (TS2)a 2.654 2.514 2.741 2.909b 1.991 1.950 1.935 1.925� 1.137 1.105 1.218 1.305Ea 17.80 9.88 3.73 1.55

a Values in ängstroms. b Values in kcal mol�1.

F. Bernardi, et al.

and 17.80 kcal mol�1) are much larger. However, it is worthto point out that these barriers, even if overestimated, are inall cases smaller than the corresponding barriers required bythe H abstraction (28.33, 26.37 and 24.08 kcal mol�1). Thistrend agrees with the experimental observation that silyl andchloro silyl radicals react via halogen abstraction and nothydrogen abstraction. As previously observed for the otherradical reactions, when the projected UMP2 approach isused, these barriers significantly decrease, even if they remainquite overestimated. Once again the DFTapproach providesmuch better results. The two sets of energy barriers obtainedat this level are both quite close to the experimental valuesused as a reference (Et3Si . radical). The values obtained withthe B3LYP functional are 8.32, 6.38 and 3.73 kcal mol�1, whileat the BLYP level these values become 5.58, 3.80 and1.55 kcal mol�1 respectively. Also, as found with the HF and

MP2 methods, the chlorine abstraction is highly favored withrespect to the hydrogen abstraction. However, it is difficult inthe present case to establish what functional performs better,since in these computations the effects of the ethyl groupsbonded to silicon are neglected.

The trend of the energy barriers computed for the Cl3Si .

radical (see Table 5) is the same as that found for H3Si . : thechlorine abstraction is favored and the activation energiesdecrease with the increasing number of chlorine atoms inthe substrate. An experimental estimate of the barrier forthe chlorine abstraction, which can be used as a reference,provides 6.18, 5.60 and 4.40 kcal mol�1 for ClCH3, Cl2CH2

and Cl3CH, respectively [29]. Once again the barriers arelargely overestimated not only at the UHF level (the error ishigher than 200%), but also at the UMP2 level. These valuesgreatly improve using the DFT approach. The B3LYPcomputed activation energies are 7.33, 6.18 and 4.81 kcalmol�1 for ClCH3, Cl2CH2 and Cl3CH respectively andbecome 4.98, 3.80 and 2.72 kcal mol�1 at the BLYP level.Thus the best agreement with the experiment is found at theB3LYP level where the error ranges between 9% and 18%.

As stressed in the previous section, in the case of thehydrogen abstractions from halomethanes by methyl radi-cal, it is worth discussing again the ability of DFT-basedmethods to reduce the effect of spin contamination on thewave function. This decrease of spin contamination, whichshould provide more reliable structures and energies, isevident from the �S2� values reported in Table 6 andcomputed at the UHF, UMP2, B3LYP and BLYP levelsfor both TS1 and TS2.

4 Structure of Organometallic Compounds andModelling Homogeneous Catalysis

4.1 Nickel-Ethylene Complexes

We describe in this section the results obtained in theinvestigation of the singlet potential energy surface for thebis(ethylene)-Ni complex. These clusters provide verysimple models for understanding the nature of the metal-olefin bond and for investigating the mechanism of cata-lyzed processes [30, 31]. Complexes of this type, in fact, seemto be involved in homogeneously catalyzed [2� 2] cyclo-addition reactions. The potential surface has been describedat the CAS-SCF/CAS-PT2 and DFT levels with the B3LYP,BLYP and BP86 functionals. The CAS-SCF computationshave been carried out using the atomic natural orbital(ANO) basis suggested by Bauschlicher et al. for the nickelatom and the Dunning cc-pVDZ basis for the carbon andhydrogen atoms (more details on these basis sets can befound in Ref. 30). The active space used to build the CASwave-function includes the � and �* orbitals of the twoethylene moieties and the 4s and 3d orbitals of the nickelatom. The size of the active space has been increased in thesingle-point CAS-PT2 computations on the CAS optimizedstructures. In this case for each doubly occupied 3d orbital


Table 5. Optimum values of the most relevant geometrical para-metersa of the transition states TS1 and TS2 with the correspond-ing activation energies (Ea)b for the reaction between trichlor-osilyl radical and chloromethanes at various levels of theory. Thesymbols used for the geometrical parameters are those reported inFigure 3.

UHF UMP2 B3LYP BLYP

(a) Cl3Si .�ClCH3


(b) Cl3Si .�Cl2CH2


(c) Cl3Si .�Cl3CHHydrogen Abstraction (TS1)a 1.709 1.642 1.642 1.629b 1.402 1.464 1.510 1.581� 0.901 0.830 0.804 0.762Ea 24.97 13.83 12.10 10.76Chlorine Abstraction (TS2)a 2.522 2.419 2.536 2.616b 2.034 1.969 1.989 1.995� 1.080 1.072 1.116 1.15Ea 18.55 7.38 4.81 2.72

a Values in a ngstroms. b Values in kcal mol�1.


not taking part in the bond, a correlating 4d orbital has beenadded. For the DFT computations a local spin density(LSD)-optimized basis set of double-� quality in the valenceshell plus polarization functions (DZVP) has been used [30].Four critical points, two with C2v symmetry (bent and planar)and two with D2h and D2d symmetry, respectively, have beenlocated. The structures of these complexes are schematicallyrepresented in Figure 4, while the values of the mostimportant geometrical parameters and the correspondingenergies are reported in Table 7.

The D2d structure, where the two planes containing theC�C bonds of the two ethylenes and the metal atom form adihedral angle of 90�, has the lowest CAS�SCF energy. Thebinding of the ethylene to the nickel atom causes a

significant lengthening of the C�C bond and a non-negligible rehybridization of the carbon atoms. The C�Cbond becomes 1.399 ä (1.332 ä in the free ethylene) andthe two methylene hydrogen atoms are bent 18.8� out of theethylene molecular plane (see the out of plane angle �formed by the bisector of the HCH angle and the C�Cdirection schematically represented in Figure 4b). The C2v

(planar) structure is only 6.89 kcal mol�1 above the D2d

species, while the C2v(bent) form is 11.16 kcal mol�1 higherthan the C2v(planar) complex. The highest in energy speciesis the D2h structure, which lies 36.56 kcal mol�1 above D2d.The inclusion of the dynamic correlation energy at the CAS-PT2 level does not change the energetic order of the fourstructures, but affects the various energy gaps. While theenergy differences D2d ± C2v(planar) and D2d ± C2v(bent)only slightly decrease (they are now 5.07 and 17.31 kcalmol�1 respectively), the difference D2d-D2h strongly decreas-es (it becomes 18.96 kcal mol�1).

The geometrical structures obtained at the DFT level withthe three functionals are almost identical to those found atthe CAS-SCF level, suggesting that, for these systems, the


Table 6. Values of �S2� computed at the UHF, UMP2 andunrestricted DFT (BLYP and B3LYP) levels with the 6-31G*basis set.

UHF UMP2 B3LYP BLYP

Hydrogen Abstraction (TS1)H3Si .�ClCH3 0.7885 0.7624 0.7562 0.7535H3Si .�Cl2CH2 0.7872 0.7622 0.7559 0.7532H3Si .�Cl3CH 0.7852 0.7821 0.7554 0.7529Cl3Si .�ClCH3 0.7883 0.7624 0.7561 0.7533Cl3Si .�Cl2CH2 0.7880 0.7626 0.7558 0.7530Cl3Si .�Cl3CH 0.7869 0.7623 0.7555 0.7527

Chlorine Abstraction (TS2)H3Si .�ClCH3 0.8543 0.7900 0.7606 0.7549H3Si .�Cl2CH2 0.8504 0.7895 0.7592 0.7538H3Si .�Cl3CH 0.8442 0.7885 0.7572 0.7529Cl3Si .�ClCH3 0.8507 0.7861 0.7605 0.7547Cl3Si .�Cl2CH2 0.8525 0.7880 0.7558 0.7530Cl3Si .�Cl3CH 0.8523 0.7883 0.7576 0.7529

Figure 4. A schematic representation of the various structureslocated for the Ni(C2H4)2 complex. The values of the geometricalparameters a, b, c, � and �� are reported in Tables 7.

Table 7. Optimized geometrical parametersa and relative energies(E)b for the D2d, C2v(planar), C2v(bent) and D2h bis(ethylene)-nickel complexes computed at various levels of theory. The energyvalues reported in parenthesis have been obtained at the CAS-PT2 level. The symbols used for the geometrical parameters arethose reported in Figure 4.

CAS(CAS-PT2) B3LYP BLYP BP86

D2d

a 1.399 1.396 1.408 1.406b 2.021 1.990 2.013 1.986c 2.021 1.990 2.013 1.986� 18.8 15.5 15.2 15.2�� 18.8 15.5 15.2 15.2E 0.00 (0.00) 0.00 0.00 0.00

C2v(planar)a 1.397 1.396 1.409 1.408b 2.022 2.009 2.029 1.996c 2.015 1.976 1.998 1.972� 18.5 15.5 15.4 16.2�� 20.2 17.8 17.9 18.3E 6.89 (5.07) 5.04 5.64 5.03

C2v(bent)a 1.397 1.381 1.398 1.397b 2.055 2.047 2.053 2.022c 2.055 2.047 2.053 2.022� 18.7 11.7 13.1 13.4�� 18.7 11.7 13.1 13.4E 18.05 (17.31) 12.80 13.24 14.09

D2h

a 1.373 1.379 1.392 1.390b 2.121 2.050 2.068 2.038c 2.121 2.050 2.068 2.038� 11.6 11.6 11.3 11.5�� 11.6 11.6 11.3 11.5E 36.56 (18.96) 12.84 14.53 15.94

a Bond lengths are in ängstroms and angles in degrees. b Values in kcalmol�1.

F. Bernardi, et al.

inclusion of the dynamic correlation does not affectcritically the geometry. A significant variation is observedonly in the C2v(bent) structure where the � angle (seeFigure 4c) significantly increases (159.2�, 158.0� and 170.1�at the BLYP, BP86 and B3LYP levels respectively). Themost interesting result is that at the DFT level the energeticorder of the four critical points is the same as that found atthe CAS-PT2 level and the energy differences do not changesignificantly. The DFT results also indicate that the BP86functional provides the best agreement with the CAS-PT2data.

Full Hessian matrix computations were performed at theDFT (BP86) level of theory to characterize the nature of thevarious critical points. These computations pointed out thatthe D2d structure is the only real minimum of the surface (allreal frequencies) and that the other critical points are saddlepoints of index 1 (C2v(planar), one imaginary frequency),index 2 (C2v(bent), two imaginary frequencies) and index 3(D2h, three imaginary frequencies).

4.2 Ethylene Dimerization Catalyzed by Ni(0) Complexes

The [2� 2] cycloadditions represent an effective tool for thesynthesis of four-membered rings. Since these reactions arecharacterized by large activation energies, they must becarried out at high temperature (400 ± 700 �C). However, inthe presence of transition metal complexes used as catalysts,these reactions proceed much faster and under milderconditions [31, 32]. A vast amount of experimental work isnow available in the literature [32], where it is demonstratedthat Rh(I), Pd(II), Pt(II), Ni(0) complexes are effectivecatalysts for [2� 2] cycloaddition reactions. These reactionshave also much theoretical interest since transition metalsapparently remove the symmetry constraints that makethem thermally forbidden in a concerted approach, accord-ing to the Woodward-Hoffman symmetry rules. To explainthis evidence two different mechanistic schemes are usuallyproposed. The first hypothesis is a concerted mechanismwhere the two new C�C bonds are formed simultaneously.The role played by the metal is that of providing suitable dorbitals, that, after combination with the olefin � orbitals,make the reaction symmetry-allowed. The second hypoth-esis corresponds to a non-concerted mechanism, whichinvolves the formation of 1 :1, and 1 :2 metal-olefin com-plexes followed by the formation of a metal-carbon �

bonded intermediate (metallacyclopentane). The metal-lacyclopentane intermediate can lead to the cyclopropaneproduct by reductive elimination.

We report the results of a DFT computational study [31]on a model-system which emulates a [2� 2] cycloadditioncatalyzed by Ni(0) complexes. The model discussed here isformed by a Ni(PH3)2 fragment that can bond either one ortwo ethylene molecules leading to the Ni(PH3)2C2H4 orNi(PH3)2(C2H4)2 complexes (1 :1 and 1 :2 metal-olefincomplexes) that are assumed to represent two possibleactive forms of the catalysts. For both complexes thereaction with an additional ethylene molecule has been

investigated using the B3LYP functional in the unrestrictedform. It has been proven, in fact, that this approach canprovide a reliable description of both the concerted anddiradical pathways in the case of [4� 2] and [2� 2] cyclo-additions [31, 24c]. All the computations have been carriedout with the pseudopotential LANL2DZ [14 ± 16] basiswhich has been demonstrated to be capable of satisfactorilydescribing the ethylene and bis(ethylene)-nickel complexes[30]. The following steps of the catalyzed reaction arediscussed: (i) The formation of the ethylene and bis(ethy-lene)-nickel complexes Ni(PH3)2C2H4 or Ni(PH3)2(C2H4)2

(active forms of the catalyst); (ii) the attack of a freeethylene on the active catalysts (formation of biradicalintermediates); (iii) the intramolecular coupling of thediradical leading to the nickelacyclopentane.

The two complexes arising from the interaction of theNi(PH3)2 fragment with one (M1) or two (M2) ethylenemolecules are shown in Figure 5, with the values of the mostimportant geometrical parameters and those of the relativeenergies. M1 is a planar tricoordinated complex, while M2 is atetracoordinated complex, 2.79 kcal mol�1 lower than M1.


Figure 5. A schematic representation of the ethylene-nickel (M1)and bis(ethylene)-nickel (M2) complexes and of the anti biradicaltransition state TS1. The energies (kcal mol�1) are relative to M2�a non-interacting ethylene molecule (asymptotic limit). Bondlenghts are in ängstroms and angles in degrees.


The most remarkable feature found in the investigation ofthe reaction surface, is that an additional ethylene moleculeattacks the complex M2 not at the metal center, but at onecarbon of the two ethylene ligands. This attack, whichinvolves the transition state TS1 (energy barrier of 35.80 kcalmol�1), leads to the formation of the intermediate M3,24.26 kcal mol�1 higher in energy than the asymptotic limit(M2� a non-interacting ethylene molecule). M3 is repre-sented in Figure 6. These structures (TS1 and M3) are bothcharacterized by an anti orientation of the attacking ethyl-ene with respect to the ethylene ligand and are similar tothose determined for the non-catalyzed reaction [31]. Thenew forming C�C bond is 1.900 ä in TS1 and becomesalmost completed (1.584 ä) in the intermediate M3. Themost interesting aspect, which characterizes the electronicstructure of TS1 and M3, is that one of the two unpairedelectrons is mainly localized on the terminal methylene andthe other on the nickel atom.

A further investigation of the reaction surface has shownthat a rotation around the new C�C bond leads from the anti

intermediate to a syn intermediate M4, which is 6.91 kcalmol�1 higher in energy than M3. In the new structuralarrangement the two unpaired electrons easily couple toform, with a negligible barrier, the nickelacyclopentane M5.This complex is 34.8 kcal mol�1 lower in energy than M4 and3.01 kcal mol�1 under the asymptotic limit. The computa-tions have demonstrated that the ring closure to form thenew Ni-C bond in the metallacyclopentane leads to theelimination of the ethylene ligand not involved in thereaction. M4 and M5 are represented in Figure 6.

An anti attack of one ethylene molecule has also beenconsidered on the M1 complex, the other possible activeform of the catalyst and it has been found that an antitransition state and an anti intermediate (TS2 and M6

represented in Figure 7) exist. However in this case theactivation energy required to form M6 from M1 is higher(39.79 kcal mol�1) than that found for M2 and, moreinteresting, almost identical to the value of 40.34 kcalmol�1 found, at the same computational level, for the non-catalyzed reaction [31]. These results suggest that a catalytic


Figure 6. A schematic representation of the anti biradicalintermediate M3, of the syn biradical intermediate M4, and ofthe nickelacyclopentane M5. The energies (kcal mol�1) are relativeto M2 � a non-interacting ethylene molecule (asymptotic limit).Bond lenghts are in ängstroms and angles in degrees.

Figure 7. A schematic representation of the anti biradicaltransition state TS2 and of the anti biradical intermediate M6,associated with the attack of one ethylene molecule on the M1

complex. The energies (kcal mol�1) are relative to M1� a non-interacting ethylene molecule. Bond lenghts are in ängstroms andangles in degrees.

F. Bernardi, et al.

effect, even if not very large for the simple model-systemdiscussed here, exists only for the 1 :2 and not for the 1 :1nickel-ethylene complexes. The 1 :1 complexes more easilycoordinate an additional ethylene at the metal center, aspreviously seen. These DFT studies enforce the hypothesis,based on experimental observation, that olefin dimerizationproceeds through a non-concerted mechanism involving anequilibrium between bis(olefin)-metal complexes and met-allacyclopentanes.

4.3 Homogeneous Ziegler-Natta catalysis

The Ziegler-Natta polymerization of olefins is an importantprocess used in the industry to obtain long polymeric chains.The process is extremely fast and proceeds with highstereoselectivity. According to the most commonly acceptedmechanism (see Scheme 1) [33], the active form of thecatalyst is characterized by a vacant site in the metalcoordination sphere which binds an olefin to form a metal-alkyl-olefin complex. In a subsequent step the olefin

molecule inserts into the metal-alkyl bond leading to anew alkyl complex characterized by a longer chain (growingchain). The resulting complex has a new vacant site on themetal that can bind another olefin molecule. Even if a greatdeal of experimental [34] and theoretical [33] work has beencarried out on this reaction, many mechanistic details arestill obscure. In particular the nature of the active form of thecatalyst and of the metal-alkyl-olefin complexes needs to beelucidated. If we consider, for instance, the commonly usedtwo-component Ziegler-Natta catalyst TiCl4-AlR3, the realactive form originated by the catalyst-cocatalyst interactioncould correspond either to a bimetallic complex or to asolvent-separated ion-pair as shown in Scheme 2.

In a recent paper [33] a DFT(B3LYP) investigation hasbeen carried out on both mechanistic hypothesis i.e. thebimetallic complex and the solvent separated ion-pair. Wediscuss here in details only the results obtained for the latterhypothesis, since in this case it is possible to compare theDFT data with those obtained at the MP2 and CAS-PT2levels of theory. A model-system formed by the Cl2TiCH�

3

cationic species interacting with one ethylene molecule hasbeen used to emulate the positive fragment of the solvent-separated ion-pair. The computations have been carried outwith two different basis sets. The simpler one is the MIDI4basis of Huzinaga augmented by two sets of p functions onthe titanium atom (exponents 0.083 and 0.028). The moreaccurate basis is formed by the 6-31G* basis for carbon,aluminium, chlorine and hydrogen and by the Watchers-Hay basis for titanium (a (14s, 11p, 6d) primitive setcontracted to [8s, 6p, 4d]). More details on these basis setsare reported in Ref. 34.

The structures of the critical points located on thepotential energy surface are represented in Figure 8. Thevalues of the most relevant geometrical parameters and theenergy values relative to reactants (Cl2TiCH�

3 � a noninteracting ethylene molecule) are reported in the figure. A� complex m1 between the Cl2TiCH�

3 moiety and theethylene molecule forms without any barrier. This complexis much more stable than reactants: 34.80 and 37.88 kcalmol�1 at the MIDI4 and 6-31G* levels respectively. TS1 is afour-centered structure corresponding to the transition statefor the ethylene insertion: this requires the overcoming of abarrier of 7.67 kcal mol�1 at the MIDI4 level (5.65 kcalmol�1 with the 6-31G* basis). The transition state leads tothe insertion product m2, which is a propyl complexcharacterized by an approximately planar four-centeredstructure. The insertion product m2 is significantly morestable than reactants: 40.35 and 45.55 kcal mol�1 are theexothermicity values obtained with the MIDI4 and the6-31G* basis, respectively

A comparison of the DFT results obtained at the twolevels of accuracy (MIDI4 and 6-31G*) indicates that boththe geometrical parameters and the energy values are notvery sensitive to the basis set. For this reason, to validate theDFT approach, the reaction has been re-investigated at theMP2 and CAS-SCF/CAS-PT2 levels with the MIDI4 basisset. The structures have been fully re-optimized at the MP2


Scheme 2.

Scheme 1.


and CAS-SCF levels and CAS-PT2 single-point computa-tions have been carried out on the CAS-SCF optimizedstructures. The CAS-SCFactive space includes the ethylene� and �* orbitals, the � and �* orbitals associated with theTi-C bond and the empty 3dz2, 3dxy and 3dyz orbitals on themetal atom.

The results obtained with the MP2 approach are verysimilar to those provided by DFTwith the same basis set. Atthis level of theory the reaction is exothermic by 39.44 kcalmol�1 and the insertion barrier is 8.73 kcal mol�1. At theCAS-SCF level the topology of the surface is identical tothat already described at the DFTand MP2 levels. This is inagreement with the fact that CAS-SCF wavefunction ismostly dominated by the SCF configuration (0.95 is thecorresponding weight). However, even if the number andthe nature of the critical points do not change, the differ-ences observed for several geometrical parameters between

the CAS-SCFand DFT values are larger than those found inthe comparison between MP2 and DFT. This is probably dueto the fact that at the CAS-SCF level the largest part of thedynamic correlation is neglected. For this reason the CAS-SCF energy values are not expected to be reliable: theexothermicity of the reaction decreases notably(�27.30 kcal mol�1), while the insertion barrier becomesmuch larger (14.54 kcal mol�1). It is interesting to note that,when the dynamic correlation is included by means ofsingle-point CAS-PT2 computations on the CAS-SCFstructures, the energy values become very similar to thosedetermined at the MP2 and DFT levels. The CAS-PT2provides, in fact, an exothermicity of 37.25 kcal mol�1 and abarrier of 6.43 kcal mol�1. These results indicate that theDFT(B3LYP) is adequate to describe this class of reactionsand confirm the importance of the dynamical correlation toobtain accurate structures and energies.

5 Polyenes, Carotenoids and Singlet-OxygenQuenching

Long-chain polyenes and the study of their chemicalreactivity and properties provide an illustrative example ofa DFT-based investigation in organic chemistry. As it will beshown below, this case-study displays quite well both theextended applicability of DFT methodologies, as well astheir limitations. Moreover, in particular conditions, it turnsout that situations which are in general not described bystandard DFT methods (e.g. excited states, etc.) may also beinvestigated, thus gaining significant information for topicalbio-chemical and bio-physical problems (which, otherwise,could be hardly inferred). This is the result of a deepknowledge and proper control of the computational toolsand algorithms, which may be suitably tuned to get non-standard DFT results. Furthermore, these examples clearlyillustrate how limited and dangerous could be the use of aDFT-based approach if not supported by a proper know-ledge of quantum-chemistry and computational methods.Although nowadays DFT is gaining more and morepopularity due to its geometric/energetic accuracy andapplicability on large molecular systems, nevertheless its useas a −black ± box× deserves attention and care. Still, whencomplemented by proper technical skills, we can frequentlytreat tricky problems, getting results with impressivereliability.

5.1 Biological Activity of Carotenes

Radical scavenging (i.e. antioxidant ability) and singlet-oxygen (1O2) quenching activity are one of the mostimportant biochemical properties of carotenoid systems.In fact, carotenoids protect vital biological structuresagainst free-radicals and 1O2 (a highly reactive and toxicform of oxygen) degradation, both in bound (e.g. in photo-synthetic centers) and unbound (e.g. in biological tissues)conditions [35 ± 44]. Though these properties may act in


Figure 8. A schematic representation of the structures ofreactants, intermediate m1, transition state TS1 and product m2

for the insertion of one ethylene molecule into the Ti-C bond inthe (Cl2TiCH3)� species. The values of the reported geometricalparameters have been obtained at the B3LYP/MIDI4, (B3LYP/6-31G*), [MP2/MIDI4] and CAS-SCF/MIDI4 computational levels.The corresponding energies (kcal mol�1) are relative to reactantsi.e. the (Cl2TiCH3)� species� a non interacting ethylene molecule(see Ref. 34 for further details). Bond lenghts are in ängstromsand angles in degrees.

F. Bernardi, et al.

concert leading to a common protective effect, still theydepend on very different processes. Thus, for example,carotenoids may quench 1O2 catalytically (i.e. carotenoidsare regenerated) via a very efficient (almost diffusion-controlled) physical pathway, i.e. an energy transfer(ET) process, followed by an intersystem crossing(ISC):

1O2 � 1carotenoid� 3O2� 3carotenoid� 3O2� 1carotenoid (�heat) (12)

In contrast to physical quenching, carotenoids may alsochemically react with 1O2 (and radicals in general). Radicalscavenging involves, in fact, real chemical reactions (chem-ical pathway) [45] leading to carotenoid oxidation. Thesepaths result in the generation of stable radical and diradicalspecies, in the destruction of carotenoids, and thus in the lossof antioxidant protection (i.e. these are not catalyticprocesses) [46]:

1O2/radicals� carotenoid� chemical pathway (13)

The specific reactions involved in the chemical pathway(Eq. 13) are still not completely understood. It is not clearwhether the observed oxidation products (which includeapocarotenal chain cleavage fragments) [46] are formed bydirect addition of 1O2 to the carotenoid system or by reactionbetween the triplet-oxygen (3O2) and the triplet carotenoid(3carotenoid) produced by the main energy transfer process(Eq. 12). Moreover, the radical trapping ability of carote-

noids could depend by some specific and reactive form ofthese systems. Finally, other competing chemical reactionscould be responsible for alternative catalytic 1O2 quenchingpathways. These processes, even if less efficient than energytransfer (Eq. 12), could be competitive with oxidationreactions.

The biological importance of these processes represent thebackground and the motivation for a full mechanistic inves-tigation. Obviously, the first task in this quest is to define areliable model and to select a proper computational method.

5.1.1 The Diradical Intermediate Model: trans� cisThermal Isomerization Barriers in long linearPolyenes. Towards a Carotenoid Model.

It has been suggested that the radical trapping action of -carotene (and carotenes in general) might derive from areactive (diradical-type) twisted intermediate, half the wayalong the path for thermal trans� cis isomerization aboutthe central double bond [47 ± 50]. The existence of such adiradical intermediate might depend on a special resonance-type stabilization effect at the twisted region, which makesthe system stable (i.e. long living) enough to trap otherradicals. For this to happen, the diradical twisted structuremust be an energy minimum (i.e. an intermediate) on thepotential energy surface (PES), and the energy barrier fortrans� cis thermal isomerization must be small enough(�30 kcal mol�1) to be active at physiological conditions(37 �C), see Scheme 3. This could grant a small but constantamount of reactive (i.e. diradical-type) twisted species in the


Scheme 3.


environment, which immediately scavenges incoming rad-icals. Otherwise, if no twisted minimum exists on the PESand/or there is a too high-energy barrier, such a trappingeffect would be much smaller, if not completely absent.

To elucidate this hypothesis, trans� cis thermal isomer-ization about the most central double bond for a series ofeight all-trans conjugated polyenes (C2nH2n�2, where n is thenumber of double bonds) of increasing chain length (n� 1,2, 3, 4, 5, 7, 9, 11) has been investigated by means of DFT/UB3LYP [18] computations using the 6-31G* basis set [15],and the nature of the twisted structures (i.e. minima ortransition states) has been inspected. Note that the last twolongest terms of the series can be considered as violaxantin(n� 9) and -carotene (n� 11) models, see Figure 9.

This investigation has also allowed us to benchmark theaccuracy of the DFTapproach versus accurate high-level abinitio methods such as multireference MP2 computations(obtained using the CAS-PT2 methodology [22 ± 23]). Theseresults are summarized in Table 8 where the CAS-PT2 andDFT isomerization barriers are listed along with the zero-point energy (ZPE) corrections (computed at the ab initiocomplete active space self consistent field (CAS-SCF) andDFT level of theory), and along with the available exper-imental barriers. The DFT barriers agree with the CAS-PT2ones within 1 kcal mol�1 and the same happens for the ZPEcorrections, providing a very strong validation for theaccuracy of the DFT approach. Moreover, the impressiveagreement between experimental barriers and DFT ones(within 0.6 kcal mol�1) indicates that this method provides arealistic description for the energetics of the isomerization


Figure 9. Typical carotenoid structures (n is the number of conjugated C�C double bonds). The available experimental second-orderrate constants (1010 Kq M�1s�1) for the quenching of singlet-oxygen are given in parentheses [44a]. The common 9 double bonds moiety ishighlighted in bold.

Table 8. CAS-PT2 and DFT computed rotational barriersa

(�ECAS-PT2,�EDFT) for TRANS�CIS isomerization of all-transpolyenes, zero-point vibrational energy correction at CAS-SCFb/6-31G* and DFT/UB3LYP/6-31G* levels (�Ezpe), DFT zero-point energy corrected rotational barriers (�Ecor) and availableexperimental enthalpies of activation (�H≥)c.

nd �ECAS-PT2 �EDFT �Ezpe

CAS-SCF�Ezpe

e

DFT�Ecor �H≥

1 62.8 62.9 �5.0 �4.5 58.4 58.12 54.2 53.2 �3.7 �3.9 49.33 43.9 44.2 �3.2 �3.2 41.0 40.9[50c]4 38.9 39.8 �3.1 36.75 35.6 �2.9 32.7 32.17 30.8 �2.8e 28.0 27.59 27.7 �2.7e 25.0 24.5

11 25.5 �2.6e 22.9 22.4f

a All values in kcal/mol. b All CAS-SCF computations have been performedusing the full active space of �-electrons and �-orbitals. c Experimental datarefer to trans�cis rearrangements of semirigid polyenes as reported byDoering et al. [50b]. d n�number of double bonds. e Zero-point DFTvibrational energy corrections (�Ezpe) are computed (via analyticallyfrequencies on twisted critical points and all-trans equilibrium geometries)only for n� 1, 2, 3, 4, 5. For longer terms we assume �Ezpe �A�B/(n�C),where A� 2.42, B� 2.50, C� 0.20 are obtained by interpolations of thecomputed DFT zero-point energy corrections for the first three terms withan odd number of double-bonds (n� 1, 3, 5); the asympthotic behaviour ofthe selected function is suggested by the first 5 computed terms (n� 1, 2, 3,4, 5); the choice of the first three odd terms (n� 1, 3, 5) to interpolate issuggested by the necessity to obtain �Ezpe values for the longer odd terms(n� 7, 9, 11). f Extrapolated non-experimental value [50b].

F. Bernardi, et al.

in polyenes and carotenoid systems. The computed barrierfor the two longer terms (n� 9, 11) shows that this process isalready active at physiological temperature. The 22.9 kcalmol�1 barrier computed for the -carotene model agreesvery well with the 22.4 kcal mol�1 value reported by Doeringet al. [50b] and with a recent kinetic estimate of 20 kcalmol�1 [49].

Without entering too much into the details (the interestedreader should refer to the full paper [51]), here we just saythat, while the closed shell planar minima show the typicalsingle/double C�C bond alternation, the optimized centralbond twisted structures (i.e. the transition state) are formedby two orthogonal non-interacting polyenil radicals whichare characterized (for the longer terms of the series) by acentral allyl-radical type system. Anyway, these structuresdo not display any special stabilization effect, except the onedepending on radical delocalization, which is responsible forthe lowering in the barriers as increasing the length of thechain. Indeed, all the points correspond to real transitionstates (see Figure 10), and no twisted intermediates areinvolved in the process.

The involvement of a diradical-type radical scavenger wasfurther investigated exploring the triplet (T1) PES. Ananalogous isomerization process was located for the -carotene model (n� 11), see Figure 10. In this case, both theoptimized planar minima (experimental [52] and theoretical[53] evidences support a T1 PES with planar minima at thecis and trans positions) and transition structure (TS) have asimilar diradical-type character, with the TS which isdegenerate and identical to the one previously optimizedon S0. If we suppose that T1 �S0 ISC easily occurs near thispoint (where the singlet-triplet energy gap becomes verysmall) and we calculate the all-trans T1 lifetime from thecomputed energy barrier (11 kcal mol�1), we get a value of

6 s [54], which gives a surprising agreement with thatobserved for the all-trans -carotene T1 state (5 g) [55].Therefore, if a very easy ISC between the S0 and T1 twistedstructures occurs, thermal equilibration among the fourminima (two closed-shell singlet and two diradical-typetriplet minima), may generate diradical triplets with poten-tial radical scavenging activity. Anyway, also in the case ofefficient thermal equilibrium, the population of the morestable all-trans triplet minimum would be very low. In fact,the concentration ratio between the two all-trans singlet andtriplet minima is about 1011, as we can calculate in firstapproximation by their related energy gap, suggesting thatT1 is not responsible for the radical trapping action. All theseconsiderations lead to the mechanistic hypothesis that it isthe S0 closed shell minimum of carotenes to be responsiblefor the trapping action via formation of resonance-stabilizedcarbon-centered radicals or diradicals [38].

Thus, this simple investigation provides two very impor-tant pieces of information: (i) the DFT/B3LYP/6-31G*approach gives impressively good results both in the singletand triplet states, and will be the elective method for theseinvestigations; (ii) the −diradical intermediate model× forradical trapping activity must be ineffective.

5.1.2 A Further Insight: Bio-physical vs. Bio-chemical Paths

The next step in the quest for understanding caroteniodsbiochemical properties involves the computational study ofthe reactions with 1O2 (using the previously validated DFT/UB3LYP/6-31G* approach). The all-trans decaottanonaene(P9), a polyene with 9 conjugated double bonds, was selectedas a carotenoid model. In fact, the 9 conjugated double bondmoiety is a common feature in many carotenoids [36, 44a],while the substituents at the two ends of the chain may be


Figure 10. Simplified DFT singlet (S0) and triplet (T1) energy profiles for the isomerization about the central C�C bond of the -carotene model P11; � is the dihedral angle of rotation about the central double bond. MIN S0, MIN T1 represent the DFT optimizedclosed-shell S0 and diradical-type T1 (all-trans or cis) minima respectively, connected by the twisted diradical-type transition state TS (seeRef. 51 for further details).


different (see Figure 9). Moreover, several carotenoidsystems with only 9 conjugated double bonds exist, such asviolaxantin and neurosphorene, which show a singlet-oxy-gen quenching efficiency and a chemical reactivity compa-rable to that of -carotene and longer carotenoids [44a](Figure 9).

Our computational results [56] indicate that carotenoidscan be involved in different types of reactions, which includeenergy transfer, 1,2-addition, T1 dissociation and triplet-triplet recombination. A summary of the correspondingreaction pathways is illustrated in Figure 11. It can be seenthat the energy transfer process involves an almost barrier-less path; therefore this catalytic physical quenching(Eq. 12) is the preferred route. Nevertheless, secondarybut concomitant low energy barrier reactions (Eq. 13) occurvia direct attack of singlet-oxygen upon the double bonds of

the carotenoid model. These processes lead to diradicalsystems where singlet and triplet states are degenerate.While ring closure reactions on S0 produce 1,2-additiondioxetane intermediates, (which may then decompose to thefinal observed carbonyl chain cleavage oxidation products[46]) an efficient and competitive S0� T1 ISC may alsooccur at the diradical minima configuration [56], anddeactivated triplet oxygen, together with the starting singletground state carotene, is produced through a dissociationprocess on T1. Singlet diradical formation followed by ISCand triplet dissociation represent an alternative chemicallymediated catalytic quenching of singlet-oxygen which seemsto be more favored than oxidation reactions. This alter-native process may act together with the more efficientphysical pathway, reducing competitive oxidation whichresults in the loss of carotenoids and thus of antioxidant


Scheme 4.

Figure 11. Summary of the DFT energy profiles (spin-projected values in kcal mol�1) of the main reaction paths computed for thelonger model-system (O2�P9). Available experimental energies are given in frames. The relative positions of the triplet (T1) and singlet(S0) states of reactants, intermediates, products and transition states (TS1, TS2, TS3 and TS4) are shown. P9/Sing-Min and P9/Trip-Min representthe optimized all-trans planar minima in the S0 and T1 states respectively (see Ref. 56).

F. Bernardi, et al.

protection. Carotenoid regeneration can in fact occurs alsothrough a competitive chemical singlet-oxygen quenchingpath. Our results suggest the general reaction pattern shownin Scheme 4.

Note the good agreement between the DFT-computedand the available experimental data (see Figure 11). More-over, to assess the accuracy of the computations, we havealso investigated the same process using a shorter polyenesystem, the all-trans hexatriene (P3). This model reaction(1P3 � 1O2) allowed both DFT [15] and multi-referenceMoller-Plesset perturbation theory (CAS-SCF/CAS-PT2/6-31G*) computations [22 ± 23], resulting in a reasonableagreement (see Ref. 56 for details).

6 Singlet Ground and Excited States via DFT in theEnergy Transfer Process

Though, in general, standard DFT methods cannot describeexcited states, still special situations exist that allow excitedstate computations, and the energy transfer problempresented above is one of these lucky cases. While the S0

wavefunction for the starting reactant refers to a singlet-

oxygen plus singlet-carotene coupling (1P9� 1O2), the S0

wavefunction for the relaxed energy transfer product refersto a triplet-oxygen plus triplet-polyene coupling (3P9� 3O2).Due to the abrupt change in the S0 wavefunction spindensities on going from a singlet-singlet to a triplet-tripletcoupling description, it was possible to follow (all along thepath) each coupling situation as a DFT-stable wavefunction.In this way, the two diabatic components of the process werecomputed, and consequently the singlet ground (S0) stateand the singlet excited (S*) state surfaces along the energytransfer path [57] were estimated. Thus, both the singletground and the singlet excited states could be described, seeFigure 12a.

It is worth to say that energy transfer efficiency is not onlya matter of energetics, but it also depends on vibronicinteractions, short-distance interactions (such as overlap ofthe electron clouds), long-range antenna-type dipole-dipoleinteractions, according to the type (short range or longrange) of energy transfer involved [58]. However, the factthat we have obtained an almost zero energy barrier is ademonstration of the efficiency of such a process (at leastfrom an energetic point of view). This result is consistentwith the observation of a very fast process (approaching the


Figure 12. Linear interpolated DFTenergy profiles (values in kcal mol-1) for a) the energy transfer process in the long model-system (O2

�P9), and b) the same hypothetical process in the small model-system (O2�P3). Dotted lines in a) represent the diabatic (singlet-singletand triplet-triplet coupling) components of the energy transfer path. The relative positions for the singlet ground (S0) and singlet excited(S*) states of the hypothetical reactants and products are shown. P3/Sing-Min and P3/Trip-Min represent the optimized all-trans planar minima inthe S0 and T1 states of the P3 system, respectively (see Ref. 56).


diffusion control limit [36, 44a], see Figure 9). Moreover, ifwe consider longer polyenes, which have a smaller verticalS0� T1 excitation energy, we should expect an even easierprocess because the crossing between the two diabaticsurfaces should occur closer to the singlet-singlet reactant,due to the smaller energy gap separation between the groundand excited singlet states. In fact, long chain carotenoids suchas dodecapreno -carotene (19 double bonds) or decapreno-carotene (15 double bonds) have higher singlet-oxygenquenching rate constants than short chain carotenoids such asviolaxantin (9 double bonds) [44a] (see Figure 9).

The short model system (1P3 � 1O2) allowed us to accessthe differences in chemical reactivity between long-chainpolyene (such as carotenes) and shorter conjugated chainstoward singlet-oxygen. Though the computed reaction pathsand their trends are almost the same (see Ref. 56 for details),a striking difference appears: 1P3 does not allow an energytransfer path. The triplet-triplet product (3P3 � 3O2) of thehypothetical physical quenching is associated with thesinglet excited state S*, and the two diabatic curvesdescribing the singlet-singlet and triplet-triplet couplingnever cross (see Figure 12b), in contrast to the behaviorfound in the longer system, which gives rise to curve crossingand to the energy transfer channel on S0, (see Figure 12a).This depends on the larger singlet-triplet energy gapseparation in short conjugated polyenes such as P3, whichprevents singlet-singlet/triplet-triplet curve crossing. As forthe longer system, we have carried out computations for thetriplet-triplet coupling (S* state) due to the intrinsic stabilityof the corresponding wavefunction (DFT wavefunctionstability has been checked both in the singlet-singlet (S0) andtriplet-triplet (S*) states).

An interesting question may naturally arise from theseresults, i.e. how many double bonds are needed for theenergy transfer process to exist and how many for it tobecome efficient with respect to the other chemical (oxygenaddition) paths. An answer just based on the results for theinvestigated short (1P3) and long (1P9) systems is certainlynot conclusive. Still, from the energy profiles shown inFigure 12b, we know that shifting the two diabatic curves(describing the singlet-singlet and triplet-triplet couplings)one to the other of only 9.5 kcal mol�1 will result in curvecrossing and possibly in the existence of an energy transfer(even if not energetically efficient) process. We guess thismay happen from 5 or 6 conjugated double bonds polyenes.However, for a competitive physical quenching, the singlet-singlet S0 and triplet-triplet S* states have to be very closedeach other already at the reactant FC region. In fact, this isthe condition to have a sudden crossing and to prevent asignificant energy barrier (this simply means that the singlet-triplet energy gap for the polyene has to be as close aspossible to that of O2, which is exactly what happens incarotenes). Since a barrier (even if very small indeed) is stillobservable along the interpolated path of our P9 modelsystem (see Figure 12a), we guess that 9 (or at most 8) is theminimum number of conjugated double bonds in a polyeneto have energetically favored energy transfer.

7 Perspectives of DFT in Organic Chemistry

In this review we have reported the results of DFT inves-tigations carried out in different fields of organic and organo-metallic chemistry. In particular, we have discussed examplesof radical reactivity, structure and reactivity of organometalliccompounds, and biochemical/biophysical properties of unsa-turated (e.g. carotenoid) systems. We have demonstrated thatthe DFT approach represents a powerful tool, which can beused as a valid alternative to more traditional correlatedmethods such as Moller-Plesset perturbation theory, config-uration interaction methods, coupled-cluster methods whichrequire, when applied to molecules of chemical interest,strong and often untenable computational effort. The com-putational expedience, which characterizes DFT-based meth-ods, makes this approach particularly interesting since it ispossible to obtain an accurate description, which includescorrelation energy, of medium and large size molecules asthose involved in a reliable simulation of organic reactions orin the modelling of bio-organic systems. The examplesselected in this review, which are representative of importantorganic, organometallic and bio-organic reactions, have beeninvestigated using different DFT functionals. These func-tionals (B3LYP, BHLYP, BLYP, BP86) are within the mostpopular non-local corrected functionals, which can be easilyfound in many commercially available quantum chemistrypackages. In all cases we have proved that they are capable ofproviding data which reproduce satisfactorily the experimen-tal results or the data obtained at higher levels of theory.

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[54] The rate constant is estimated using the simple Arreniusequation (k�A e-Ea/RT, where A is the pre-exponential factor)at the physiological temperature of 310 K. For unimolecularreactions the lifetime (�) is calculated as the inverse of the first-order rate constant (��k�1), and a standard unimolecularpreexponential factor of 1013 s�1 (see Benson, S. W.; Thermo-chemical Kinetics; Wiley, New York, 1976) has been used.

[55] Hashimoto, H., Koyama, Y., Ichimura, K., and Kobayashi, T.,Time-Resolved Absorption-Spectroscopy of the Triplet-StateProduced from the All-Trans, 7-Cis, 9-Cis, 13-Cis, And 15-CisIsomers of -Carotene, Chem. Phys. Lett. 162, 517 (1989).

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[57] The diabatic components of the adiabatic electronic eigen-functions describe the energy of a particular spin-coupling(see Olivucci, M. Ph. D Thesis, University of Bologna, 1988),while the adiabatic functions represent the surfaces of the realstates (the singlet ground and excited states in this case).

[58] Gilbert, A. and Bargott, S., Essentials of Molecular Photo-chemistry; Blackwell Scientific Pubblications, Oxford, 1991;pp. 167 ± 181.

Received on December 4, 2001; accepted on February 27, 2002


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