Basis set and correlation effects in configuration analysis of reactive systems

23 June 2000

Chemical Physics Letters 323 Ž2000. 416–424 www.elsevier.nlrlocatercplett

Basis set and correlation effects in configuration analysis of reactive systems Luis Rincon ´ ) , Rafael Almeida Departamento de Quımica, Facultad de Ciencias, UniÕersidad de Los Andes, Merida–5101, Venezuela ´ ´ Received 3 August 1999; in final form 25 April 2000

Abstract A methodology for the configuration analysis of multi-configurational wave-functions of a closed-shell system is presented. The molecular system is partitioned in interacting fragments, the molecular orbitals are expanded in fragment orbitals and each Slater determinant is expanded in localized reactant electronic structures. Finally, to reduce the number of relevant coefficients, a transformation of orbitals is performed. The main advantage of employing configuration analysis lays in its simplicity and compactness, specially when extended basis sets are used. As a test, the wave-function at the transition state for some nucleophilic substitution reactions is analyzed. q 2000 Elsevier Science B.V. All rights reserved.

1. On the interpretation of molecular orbital calculations Most of the classical tools for the theoretical analysis of chemical reactivity have been developed in the context of molecular orbital theory. Among them, worth mentioning, are the analysis of reactant frontier orbitals w1x, the correlation diagrams between reactants and products Žused by Woodward and Hoffmann in the formulation of the the conservation of the orbital symmetry principle w2x., the perturbational molecular orbital theory w3–7x, and the configurational analysis by using localized reactant structures w8–14x. All these theoretical approaches differ in rigor, general applicability and conceptual simplicity; nevertheless, all of them have served to sharp ) Corresponding author. Fax: q58-74-401286; e-mail: [email protected]

our intuition in dealing with the intricate factors that determine the outcome of a chemical reaction. It is a well known fact that, in general, the simplicity of these tools is lost when they are used in the interpretation of modern molecular orbital calculations in which the inclusion of the electronic correlation and the use of extended basis sets are frequent. This is a serious limitation of classical molecular orbital theory, that has inspired important discussions and ideas throughout the recent history of quantum chemistry w15–17x. In spite of this, the interpretation of molecular orbital calculations, in terms of chemical appealing ideas, still represents an open problem; probably one of the most important in theoretical chemistry. Recently, as an alternative to the traditional analysis based on orbital interactions, special attention has been devoted to the interpretation of the topology of the electron charge density w18,19x. This approach eliminates a good deal of the arbitrariness

0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 5 6 3 - 7

L. Rincon, ´ R. Almeidar Chemical Physics Letters 323 (2000) 416–424

present in the traditional ones, becoming, in this sense, appealing and rigorous. However, in the present situation, for many cases, the theoretical models obtained by orbital interaction and the topology of the charge density approaches, more than equivalent are complementary, and it is found that, in general, a deeper insight is obtained when the problem is analyzed from many points of view. Clearly, at this moment, some redefinition of the traditional tools are required in order to improve the situation regarding the interpretation of molecular orbital theory results. The question to address here is, how can one redefine the traditional tools of molecular orbital theory, in such a way, that the interpretation can be done in a simple form, no matter the theoretical level of calculation employed? In the past, other groups have addressed a similar question by employing orbital transformation techniques in the context of configuration analysis. Thus, for instance, the group of Fukui et al. w20–23x has developed an orbital transformation technique for a mono-determinantal molecular wave-function, finding that, for charge transfer processes, the interpretation is simplified due to the fact that the transformed fragment orbitals are localized in the region of interaction. On the other hand, in this letter we explore the suitability of the configuration analysis expansion when the quality of the wave-function is improved. In particular, we study the effects of a fragment orbital transformation on the wave-function expansion, when the basis set and the electron correlation increase systematically. We conclude that the advantage of employing a fragment orbital transformation lays, principally, in the compactness and simplification of the configuration analysis when extended basis sets are used. Ideally, we seek a methodology where the interpretation of the results, obtained from high quality basis set, may be done as simply as in those cases where, for example, minimal basis set are employed.

2. Configuration analysis Configuration analysis w8–14x is a powerful tool in which the molecular orbital wave-function is expanded in electronic structures, where the electrons are distributed in fragment molecular orbitals, very much in the same way as a molecular orbital wave-

417

function can be expanded in valence bond resonant structures w24x. As an interpretational tool, this technique relies on the partition of molecular systems in interacting fragments. In some sense, these expansions may be thought of as a middle ground between molecular orbital and valence bond theories. Thus, the molecular fragment orbitals are more localized than the canonical molecular orbitals Žwhich depend on the spatial symmetry group., but less localized than the hybrid atomic orbitals of classical valence bond theory. These expansions may be interpreted on the basis of the Donor–Acceptor formalism of Mulliken w25x, which is ideal for the interpretation of the electronic structure of charge-transfer molecular systems. The algorithm described in this letter is an extension of other ones reported in the literature for the one-determinantal case w8–14x. The method is implemented for multi-configurational wave-functions of the form, NCONF

Cs

Ý

Ci c i .

Ž 1.

I

In this equation, each of the NCONF configurations is represented by a string of integer numbers, indicating the occupation of the molecular orbitals, and the wave-function c i of each configuration is given by a spin-adapted combination of 2 Ns r2 Slater determinants, where Ns is the number of single occupied molecular orbitals. In order to describe the method, let us consider a composite molecular system AB, which is divided in two closed-shell molecular subsystems A and B. The configurational analysis requires the AB eigenvector molecular orbital matrix, C, the basis set overlap matrix, S and the eigenvector matrices of each fragment, CA and CB , calculated with the same geometry as in the whole molecule. Regardless of the level of calculation employed for the molecular wave-function, for each of the A and B fragments, their occupied and unoccupied orbitals are defined using Hartree–Fock reference wave-functions; therefore the CA and CB matrices correspond to the one determinantal case Ž RHF .. The outline of the computational procedure is as follow: The first step is the expansion of molecular orbitals in fragment orbitals. Thus, if CBF is the block

L. Rincon, ´ R. Almeidar Chemical Physics Letters 323 (2000) 416–424

418

matrix that contains the fragment eigenvector matrix defined by: CBF s

ž

CA 0

0 , CB

/

Ž 2.

and SBF is the corresponding fragment overlap matrix Žobtained from the full overlap matrix S deleting the interfragment elements.: SBF s

ž

SA 0

0 , SB

/

Ž 3.

the eigenvector matrix in the fragment orbital basis, C F, can be obtained by the transformation w26,27x, C F s Ž CBF .

y1

†

C s Ž CBF . SBF C ,

Ž 4.

where the second equality stems from the fact that, for each of the fragments X Ž A or B ., the following identity is satisfied: C X† S X C X s 1, and therefore Ž CBF . †SBF CBF s 1. From this last identity is clear that Ž CBF .y1 s Ž CBF . †SBF w6x.

Similarly, the overlap matrix between fragment molecular orbitals, S F, is found to be, †

S F s Ž CBF . SCBF .

Ž 5.

In a second step, each Slater determinant, of the wave-function Ž1., is expanded in the localized reactant electronic structures of Fig. 1. These structures are classified in the following manner: Ø Zeroth order structure, F 0 , which is written as an antisymmetric product of the occupied fragment orbitals,  i4 in A and  k 4 in B. Here neither electron transfer between orbitals in different molecular fragments nor electron excitation between orbitals within the same molecular fragment take place. Ø Mono- wDi-xtransferred structures, F Ž i l . wF Ž i l,i l .x, F Ž k j . wF Ž k j,k j .x, where one Žor two. electrons are transferred from the ith occupied orbital in A to the lth unoccupied one

™ ™

™

Fig. 1. Electronic structures used in the configuration analysis.

™ ™™

L. Rincon, ´ R. Almeidar Chemical Physics Letters 323 (2000) 416–424

in B, or from the k th occupied orbital in B to the jth unoccupied one in A. Ø Mono- wDi-xexcited structures, F Ž i j . wF Ž i j,i j .x, F Ž k l . wF Ž k l,k l .x. These represent the cases, where one Žor two. electrons are excited from the ith occupied orbital in A to the jth unoccupied one in the same fragment, or from the k th occupied orbital to the lth unoccupied one in B. The calculation of the coefficients of these electronic structures in each Slater determinant is performed by means of a generalization of a technique previously proposed in the literature w8–14x. Briefly, it involves expanding each molecular orbital Slater determinant of Eq. Ž1. in terms of fragment orbital Žoccupied and unoccupied. determinants, and then projecting the resulting expression on the wave-function representation of each localized electronic structure. Thus, for the order zero structure the result for the coefficient, C0 , is,

™ ™™

™

™

C0 s DA Ž 0 . DB Ž 0 . T Ž 0 <0 . .

™

Ž 6.

Here DAŽ0. and DB Ž0. are the determinants of a matrix that contains the coefficients of the occupied fragment orbitals of F 0 in the columns, and the rows represent the occupied molecular orbitals, with alpha Žmatrix DA. and beta Žmatrix DB . spin in each Slater determinant. For the one determinantal case, DAŽ0. s DB Ž0.. T Ž0 <0. is the determinant of a matrix that contains the elements of the overlap matrix between the occupied fragment orbitals of F 0. For the monotransferred and monoexcited structures, the coefficients, Ci ™ l ,Ck ™ j , and Ci ™ j , Ck ™ l , are given by

™ u. q DAŽ o ™ u. T Ž 0 <0 . T Ž o ™ u < o ™ u .

C o ™ u s 0.5 DA Ž 0 . DB Ž o =DB Ž 0 .

™ , Ž 7. where DAŽ o ™ u. is the determinant obtained from the matrix DAŽ0. by replacing the row of the occuqT Ž o

2 u <0 .

1r2

pied fragment orbital o by other of the unoccupied u. In Eq. Ž7., the indices, o and u are used, both, for excitation and charge transfer configurations. In a

419

™

similar manner, T Ž0 < o u. is found from the matrix of T Ž0 <0., by replacing the column corresponding to the occupied orbital o by other of the unoccupied u, while the determinant T Ž o u < o u. is obtained by replacing both, the row and the column corresponding to o for those of u. ™ ll, Ckk™™ jj, and Cii™™ jj, Ckk™™ll, The coefficients, Cii™ of the ditransferred and diexcited structures are found to be of the form,

™ ™

™ uu s DAŽ o C oo™

™ u. DB Ž o ™ u. T Ž o ™ u< o ™ u. .

Ž 8.

Finally, for all electronic configurations, c i , the coefficients of each of the localized electronic structures are collected, calculating in this way the total coefficient corresponding to each of these structures in the multi-configuration wave-function Ž1.. The third, and last step, involves the simultaneous transformation of the fragment orbitals, in order to greatly reduce the number of relevant excitation and charge transfer coefficients. In this study, we use the Amos and Hall transformation w28,29x, in which a new set of fragment orbitals in A, c f A , and B, c f B, are generated from the original fragment orbitals, f iA , and f lB. The method requires the generation of an interacting matrix P, whose element Pi l refers to the delocalization of the ith occupied orbital of the fragment A into the lth unoccupied orbital of the fragment B. Notice that, an analogous interacting matrix, Pk j , may be defined for the delocalization of the occupied orbitals in B into the unoccupied in A. In this work, the element Pi l Žor Pk j . is taken as the coefficient Ci ™ l Žor Ck ™ j ., obtained from the configuration analysis performed in the previous step. Once the matrix P is generated, the positive-definite Hermitian matrix P †P is calculated. If U and G are, respectively, the matrices of the eigenvectors and eigenvalues of P †P, such that, P †PU s UG ,

Ž 9.

it can be shown w28,29x that, the new transformed fragment orbitals are given by: r2 c f A s gy1 Ý f

i

Ý Pi l Ul f fiA , l

c f B s Ý Ul f f lB , l

Ž 10 .

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L. Rincon, ´ R. Almeidar Chemical Physics Letters 323 (2000) 416–424

where g i is the ith element of the diagonal matrix G . After the transformation is performed and the new fragment orbitals are obtained, the number of relevant coefficients in the configuration analysis should be considerably reduced. In general, this transformation should be repeated until the desirable compaction of the configuration analysis is achieved Žabout 3–5 iterations were needed for the calculations shown below.. Before continuing, we must mention that this kind of transformation was extensively used by Fujimoto and collaborators w20–23x, as an extension of the frontier orbital analysis. For some chemical relevant systems, they employed this transformation Žboth in an iterative and non-iterative form. to simplify the description of charge transfer process, and in the analysis of the local characteristics of the orbital interactions in terms of a few number of ‘paired fragments orbitals’. Here, on the other hand, we use this transformation for the compaction of the configuration analysis at different levels of theory and several basis set. For all cases shown in the next section, the wave-functions were calculated using the Gaussian-94 package w30x.

3. Inspection of the configuration analysis at the transition state in some nuclophilic substitution reactions The attack of an hydride to a hydrogen molecule is the archetypal example of a nucleophilic substitution reaction w31x, and constitutes an excellent reaction test for any interpretational theoretical-computational tool in chemical reactivity. Thus, as a first application, we have chosen to perform the configurational analysis at the transition state ŽTS. of this reaction. Since for the purpose of the present discussion, the energetic of the problem is not of main importance; we have not included it in the tables shown below. However, let us acknowledge that, for some theoretical levels Žor basis set. employed here, the energetic description is rather poor. Table 1 displays the coefficients calculated at RHF and CASSCF levels with three different basis sets. For each set of results, the second row shows the relative value of each coefficient respect to that of the zeroorder electronic structure. There, in order to facilitate the analysis, the results obtained employing a similar basis set, but a different level wave-function, are put

Table 1 Coefficients of the configuration analysis for the H2 q Hy at the transition state. Fragment: A s H2 , B s Hy.

™j

™™ jj

Cii

Ck

™l

Ckk

™™ ll

Ci

™l

™™ ll

Cii

Ck

™j

Ckk

™™ jj

MethodrBasis set

C0

Ci

RHFrSTO-3G

0.6202 1.0000

0.1151 0.1855

0.0105 0.0169

– –

– –

– –

– –

y0.4015 y0.6473

0.1119 0.1804

CASSCFŽ4,3.rSTO-3G

0.5542 1.0000 0.9937

0.1124 0.2028 –

y0.0806 y0.1454 y0.1115

– – –

– – –

– – –

– – –

y0.5241 y0.9457 –

0.0160 0.0288

RHFr6-31G

0.6274 1.0000

0.1004 0.1600

0.0080 0.0128

0.0740 0.1180

0.0042 0.0067

0.0466 0.0743

0.0016 0.0026

y0.3398 y0.5416

0.0920 0.1466

CASSCFŽ4,6.r6-31G

0.4811 1.0000 0.9825

0.1298 0.2698 –

y0.0912 y0.1896 y0.1699

0.0714 0.1484 –

0.0363 0.0755 0.0655

0.0183 0.0380 –

0.0153 0.0318 –

y0.5207 y1.0823 –

0.0433 0.0900 –

RHFr6-311q q GŽ3df,3pd.

0.7133 1.0000

0.2087 0.2926

0.0305 0.0428

0.0988 0.1385

0.0066 0.0093

0.1436 0.2013

0.0142 0.0199

y0.4701 y0.6590

0.1549 0.2172

CASSCFŽ4,9.r6-311q qGŽd,p.

0.5196

0.1939

y0.1143

0.1264

0.0016

0.0764

0.0009

y0.6304

0.0071

1.0000 0.9608

0.3732 –

y0.2199 y0.1720

0.2433 –

0.0031 0.0213

0.1470 –

0.0017 –

y1.2132 –

0.0014

L. Rincon, ´ R. Almeidar Chemical Physics Letters 323 (2000) 416–424

together. Before continuing, let us point out that the CASSCF Ž4,3.rSTO y 3G and CASSCF Ž4,6.r6 y 31G calculations are equivalent to the full-CI, and that for extended basis set cases, the orbitals used in the configuration analysis are those with non-negligible eigenvalues of P †P Žwhich correspond to the paired fragments orbitals of Fukui et al. w20–23x.. For the purpose of our discussion, the most outstanding feature of the method is that it leads to about the same number of relevant structures at the TS, regardless of the level of theory employed in the calculation. To fully appreciate the importance of this result, let us recall that the number of unoccupied orbitals in the hydrogen molecule fragment Žand therefore the number of Ci ™ j and Ck ™ j coefficients in the configuration analysis. increases from 1 with the STO y 3G basis set, to 3 with the 6 y 31G basis set, to 15 with the 6 y 311 q q GŽ d, p . basis set, and to 35 with the 6 y 311 q q GŽ3df,3 pd . basis set. On the other hand, with the methodology presented here, there is only one Ci ™ j and Ck ™ j relevant for the analysis of the wave function; thus, reducing the number of coefficients involved to that of a minimal basis set calculation. We think that this simplification is linked with the transformation of the LUMO orbital, su) , of H2 . This effect is shown in Fig. 2 for the multi-configurational level and the three basis sets employed here. There we can see that for the extended basis set case, the transformation removes part of the symmetry of the canonical LUMO orbital, in such a way to increase the size of the orbital lobe oriented toward Žand thus the overlap with. the occupied orbital of the incoming hydride. This interacting orbital resembles the valence bond hybrid orbital, suggesting a link in the way: Molecular Orbital Theory-Configuration Analysis- Valence Bond Theory. Similar result was obtained for the transformed orbitals of a RHF wave-function describing a charge transfer systems, by using the transformation of Fukui et al. w23x. In Table 1, the foremost terms in the configuration analysis correspond to the zero-order structure, C0 , and the one-electron transfer or excitation structures. Among these, as it should be expected, the one electron transfer structure, Ck ™ j , representing the transfer of one electron from Hy Žfragment B . to the H2 Žfragment A. is the most important one. For multiconfigurational wave functions, its value is

421

Fig. 2. Transformed LUMO su) orbital in the H2 fragment: Ža. CASSCF Ž4,3.rSTOy3G, in this case the orbital correspond to the canonical su) orbital, Žb. CASSCF Ž4,6.r6y31G, and Žc. CASSCF Ž4,9.r6y311qqGŽ d, p ..

comparable to C0 and tends to become greater when the active space and the basis set employed are increased. This result suggests to us that, for all

422

L. Rincon, ´ R. Almeidar Chemical Physics Letters 323 (2000) 416–424

levels of calculations, an avoided curve crossing between these two configurations occurs at a geometry near the TS w12x. Also, it is clear that the coefficient of the two-electron transfer configuration, ™ jj, is much smaller than its one-electron equivaCkk™ lent and decreases appreciably when the basis set and the active space are increased. It can also be noticed that, for these kind of calculations, the dou™ jj, has a larger value ble excitation coefficient, Cii™ than the one obtained for the equivalent RHF case. At this point, one may wonder how much in the variation of the coefficients at the CASSCF level is due to intra–fragment electron correlation effect and how much to inter–fragment interaction effect. Attempting to answer this question, we have computed, at this level of calculation, the values of the configuration analysis coefficients at the limit of large inter–fragment separation, while keeping the H y H distance in fragment A frozen at the transition state geometry. The results are displayed as the third row in the CASSCF results. Let us point out that, in this limit, only the intra-fragment electron correlation effects are present. From those results, we notice that C0 is smaller than one, and only the values corresponding to inter-fragment double-excitations are different from zero. Furthermore, we notice that this decrease in the value of C0 is far smaller than that observed by comparing the RHF and CASSCF results. Thus, for the 6 y 311 q q G)) basis set, the CASSCF result is about 27% smaller than the RHF one; however, the decrease in C0 only due to intrafragment electron correlation effect is of only about 4%. In contrast to this, the values obtained for the double excitation coefficients, at the large distance limit, mostly account for the full CASSCF results at the transition state; which lead us to think that their larger values can be mainly attributed to electron correlation effects Žinstead of intermolecular interaction.. In order to clarify, even further, how the configuration analysis results depend on the molecular geometry and calculation level, we have performed the following test. At the CASSCF level, the optimum transition-state geometry is obtained, then, with this geometry, we have recalculated the molecular wave-function at the RHF level and performed with it the configuration analysis. The results show that the coefficient values are practically identical to those found for the RHF calculation at its optimized

geometry. Next, the inverse test is done, namely, the optimum geometry is obtained at the RHF level, the CASSCF total molecular wave-function calculated, and, with it, also the configuration analysis coefficients. In this latter case, the values obtained are quite close to those of a full CASSCF calculation. These results indicate that the difference in the values of the configuration analysis coefficients is mainly due to the inclusion of more determinants Žand thus, of electron correlation effects. in the total molecular wave-function. Finally, the quality of the employed transformed orbitals can be estimated by computing the sum of the population in these orbitals. Ideally this sum should be 4; however, in our case we obtain 3.9948 for RHFr6y 31G, 3.9621 for RHFr6 y 311 q q GŽ3df,3 pd ., 4.0022 for C A SSC F Ž 4,6 . r 6 y 31G and 4.0152 for CASSCF Ž4,9.r6 y 311 q q GŽ d, p .. Table 2 shows the configuration analysis at the TS for some Židentity and nonidentity. nucleophilic substitution reactions Žincluding H 2 q Hy. . All of the identity reactions Ž Xyq R y X ., have been calculated with the same basis set and active space Ž CASSCF Ž4,6.r6 y 31G .. On the other hand, in order to ensure that a barrier is obtained for all the nonidentity reactions, we have used the CASSCF Ž4,4.r6 y 31 q GŽ d . level. For these reactions we have performed the configurational analysis at the TS for the two nonequivalent definition of reactant fragments. In all the studied cases, the charge transfer coefficients between transformed orbitals, Ci ™ l and Ck ™ j , showed in Table 2, are at least five order of magnitude greater than other charge transfer coefficients. Similarly, the shown transformed mono-excited coefficients are at least two order of magnitude greater. Several tendencies may be drawn from this table. First, it can be noticed that the most important contribution Žbeside the one of order zero. is the one-electron transfer coefficient, Ck ™ j , between the donor Xy and the acceptor R y Y. Since the value of this coefficient gives a measure of the amount of electron transfer at the TS, again, this is not a surprising result given the kind of reactions studied here. Additionally, for the examples where the methyl group is involved, it may be seen that, the more electronegative the nucleophyle, the less charge it will tend to transfer in the TS and the smaller the Ck ™ j value will be. To evidence this point even

L. Rincon, ´ R. Almeidar Chemical Physics Letters 323 (2000) 416–424

423

Table 2 Coefficients of the configuration analysis for nucleophilic substitution reactions, Y y R q Xy and Mulliken charges on X Ž q Ž X .., R Ž q Ž R .. and Y Ž q Ž Y .. at the transition state. Fragment: A s Y y R, B s Xy. Reaction y

C0

Ci

™j

™™ jj

Cii

Ck

™l

Ckk

™™ ll

Ci

™l

™™ ll

Cii

Ck

™j

Ckk

™™ jj

qŽ X .

qŽ R.

qŽ Y .

H2 qH

0.4811 0.1298 1.0000 0.2698

y0.0912 y0.1896

0.0714 0.0363 0.0183 0.0153 y0.5207 0.0433 y0.5254 q0.0509 y0.5254 0.1484 0.0755 0.0380 0.0318 y1.0823 0.0900

F2 q Fy

0.4999 0.1249 1.0000 0.2498

y0.1737 y0.3475

0.0041 0.0020 0.0154 0.0007 y0.7025 0.0216 y0.3912 y0.2176 y0.3912 0.0082 0.0040 0.0308 0.0014 y1.4053 0.0439

Cl 2 q Cly

0.5469 0.1348 1.0000 0.2465

y0.1454 y0.0137 0.0009 0.0309 0.0002 y0.6549 0.0331 y0.4327 y0.1347 y0.4327 y0.2659 y0.0251 0.0016 0.0565 0.0004 y1.1975 0.0605

CH 4 q Hy

0.5924 0.1149 1.0000 0.1940

y0.1019 y0.0308 0.0023 0.0344 0.0008 y0.5246 0.0073 y0.5422 q0.0843 y0.5422 y0.1720 y0.0520 0.0039 0.0581 0.0014 y0.8856 0.0123

CH 3 F q Fy

0.7651 0.1230 1.0000 0.1608

y0.0180 y0.0044 0.0087 0.0329 0.0011 y0.3691 0.0166 y0.7085 q0.4171 y0.7085 y0.0236 y0.0058 0.0114 0.0430 0.0014 y0.4824 0.0217

CH 3 Cl q Cly 0.7860 0.1052 1.0000 0.1338

y0.0642 y0.0266 0.0004 0.0355 0.0007 y0.3535 0.0290 y0.7155 q0.4309 y0.7155 y0.0817 y0.0338 0.0005 0.0452 0.0009 y0.5139 0.0369

CH 3 Br q Bry 0.5131 0.0932 1.0000 0.1815

y0.0445 y0.0228 0.0033 0.0044 0.0006 y0.3572 0.0188 y0.7021 q0.4047 y0.7021 y0.0866 y0.0444 0.0064 0.0086 0.0011 y0.6954 0.0366

CH 3 F q Cly

0.5115 y0.0093 y0.0016 y0.1651 0.0137 0.0329 0.0007 y0.5504 0.0217 y0.4324 q0.3066 y0.8742 1.0000 y0.0182 y0.0031 y0.3228 0.0267 0.0643 0.0014 y1.0761 0.0424

CH 3 Cl q Fy

0.9621 y0.1255 y0.0613 y0.0164 0.0277 0.0819 0.0041 y0.1928 0.0186 y0.8742 q0.3066 y0.4324 1.0000 y0.1304 y0.0637 y0.0170 0.0288 0.0851 0.0043 y0.2004 0.0193

CH 3 Br q Cly 0.5293 y0.0674 1.0000 y0.1273

0.0226 y0.0289 0.0055 0.0031 0.0004 y0.3747 0.0127 y0.7282 q0.4432 y0.7149 0.0427 y0.0540 0.0104 0.0059 0.0008 y0.7079 0.0240

CH 3 Cl q Bry 0.5146 y0.0771 1.0000 y0.1498

0.0309 y0.0521 0.0187 0.0045 0.0004 y0.3584 0.0191 y0.7149 q0.4432 y0.7282 0.0600 y0.1012 0.0363 0.0087 0.0008 y0.6965 0.0371

further, we have displayed in the last columns of Table 2 the Mulliken charge of the species at the TS. There we can indeed see that the higher the charge on the nucleophile Žthe smaller the extension of charge transfer., less important is the relative contribution of Ck ™ j . Moreover, as it should be, the charge on X and Y are the same for the identity cases. It can also be noticed that the Ci ™ j coefficient, which corresponds to the excitation of the R y Y fragment, has a sizable contribution. For the non-identity reactions, the two possible choices of reactive fragments lead to different configuration analysis results. This is particularly clear for the CH 3 F q Cly and CH 3 Cl q Fy examples. For this

last case, as is evident by the clearly predominant value of C0 and by the small extension of the charge transferred, the TS is reached very early in the reaction coordinate. From the previous discussion, one can see that the method employed here, beside of providing an important compaction of the wavefunction that greatly facilitates the analysis of the results, also allows, from the qualitative trends of the coefficients, to obtain some leads on the electronic reorganization at the TS. Finally, let us just mention that, in the course of this study, the H 2 q Hq and some aromatic electrophilic substitution reaction have also been analyzed. For all these cases, a similar compaction of

424

L. Rincon, ´ R. Almeidar Chemical Physics Letters 323 (2000) 416–424

the wave-function is found for RHF and CASSCF levels. Mainly for reasons of space, these results are not shown in this preliminary letter.

Acknowledgements This research was supported by CONICIT through the Computational Catalysis Project ŽGrant G97000667. and the Consejo de Desarrollo Cientıfico ´ Humanıstico y Tecnologico of the Universidad de ´ ´ Los Andes ŽCHCHT-ULA.. We thank to CeCalCULA for the computational support of this research.

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