Application of the Electron Localization Function to ...

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Application of the Electron Localization Function to Radical Systems JUNIA MELIN,1 P. FUENTEALBA2 1

Departamento de Quı´mica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago de Chile 2 Departamento de Fı´sica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago de Chile Received 15 August 2002; accepted 29 October 2002 Published online 00 month 0000 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/qua.10515

ABSTRACT: An application of the topological electron localization function (ELF) analysis to free radical systems is presented. A separation of the ELF function into its ␣-spin and ␤-spin contributions has been performed. Methyl and phenyl radicals, ortho-, meta-, and para-benzyne biradicals, and their corresponding radical anions have been chosen with the aim to validate the new ELF␣ and ELF␤ proposed functions. The results show that the ELF separation yields complementary information about the localization of the unpaired electron. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 92: 000 – 000, 2003

Key words: electron localization function; free radicals; unpaired electrons; ␣-spin; ␤-spin

Introduction

T

he electron localization function (ELF) has proven useful for describing the nature of the chemical bond in a wide variety of situations. The ELF performs a partition of real molecular space into chemically meaningful regions. This function was introduced by Becke and Edgecombe [1] using arguments based on the conditional pair probability function. The function was reinterpreted by Savin et al. [2] using arguments based on the excess of kinetic energy density due to the Pauli exclusion principle. Later, Savin and Silvi [3] developed a Correspondence to: P. Fuentealba; e-mail: [email protected]

International Journal of Quantum Chemistry, Vol 92, 000 – 000 (2003) © 2003 Wiley Periodicals, Inc.

topological analysis of the 3-D function. The ELF topological analysis has been applied to several systems, from atoms to solids [4 – 8]. In this work, the ELF analysis will be applied to different radical systems with the aim to describe the region of the space where the radical electron(s) is more likely to be found. This is possible using the ELF because from a formal point of view there is no difference between the excess of kinetic energy due to a localized electron pair or one localized electron. Further, the original definition of the ELF considers it in two separate contributions, one for the ␣-spin and the other for the ␤-spin. Hence, in this article the ELF associated with both densities will be calculated separately. Kohout and Savin [9] also analyzed the spin-dependent ELF data for atoms.

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MELIN AND FUENTEALBA The starting point for this study was the methyl radical because it has a simple structure and extensive reactivity. The phenyl radical is also a simple and interesting aromatic molecule, for which the total and separated ELF analysis will be discussed. Following the application to aromatic systems, a family of benzyne isomers have been studied with the topological ELF analysis to compare the singlet and triplet description of biradicals with different correlation effect. From the early 50s 1,2-didehydrobenzene or ortho-benzyne has been established as an intermediate specie in elimination reactions of halobenzenes [10]. The meta- and para-isomers can be generated by flash photolysis [11, 12]. Recently, Sanders and coworkers [13] measured their infrared spectra. Contrary to other organic biradicals that have unselective reactivity, the para-benzyne has a particularly selective bioactivity and has been widely exploited for anticancer drugs design [14, 15]. The stereoselectivity of these specie is the result of the fact that its ground state is an open-shell singlet. However, it is not easy to measure some properties of this kind of systems like the singlet– triplet energy gap ⌬ES–T. Therefore, quantum calculations are especially important for those systems. Further, from a purely theoretical point of view the benzyne isomers are a challenge to current calculation methods. For instance, the para-benzyne system has been considered for a long time an interesting benchmark in electronic structure calculations. For quantum chemical calculations, parabenzyne undergoes a phenomenon known as spatial symmetry breaking, which means that the wave function obtained from standard single-determinant methods fails to transform as an irreducible representation of the molecular point group (C2v). Crawford et al. published a thorough study on this problem [16]. One of their important conclusions points out that UDFT calculations give a reasonable description of para-benzyne. The ELF analysis will be also applied to ortho-, meta-, and para-benzyne negative ions or distonic radical anions. This radical anions are required to measure electron affinities and singlet–triplet splitting of the corresponding neutral biradicals by photoelectron spectroscopy [17].

Theoretical Framework

another electron. Later, a different and interesting interpretation of this function was given by Savin et al. [18]. They interpreted the ELF in terms of the excess of local kinetic energy density due to the Pauli repulsion T(r) and the Thomas–Fermi kinetic energy density for a homogeneous electron gas Th(r)

冋 冋 册册

ELF共r兲 ⫽ 1 ⫹

T共r兲 Th共r兲

2 ⫺1

.

(1)

This expression has a convenient scaling factor to obtain values between 0 and 1. The key in the kinetic energy density ELF interpretation is the T(r) factor, which represents the difference between the kinetic energy density of the real system and the von Weizsa¨ cker one, which is exact for a boson system where the Pauli exclusion principle does not apply: T共r兲 ⫽

1 2

冘 兩ⵜ␸ 共r兲兩 ⫺ 81 兩ⵜ␳␳共r兲共r兲兩 2

i

2

(2)

i

T h ⫽ C f ␳ 共r兲 5/3 C f ⫽ 2.871,

(3)

where

␳ 共r兲 ⫽

冘 兩␸ 共r兲兩 . i

2

(4)

i

In the last expression, the ␸i(r) are the Hartree–Fock or Kohn–Sham occupied orbitals. The chemical interpretation of the ELF says that small values of ELF indicate a space region where electrons are delocalized, while high values correspond to a region of the space where the electrons are localized. The validity of this interpretation has been confirmed in many cases [2, 4, 5, 8]. The gradient field of the ELF divides the molecular space into basins of attractors (⍀i) that are nonoverlapping volumes having various associated properties. For instance, the average electron population in a specific basin is given by ៮ 共⍀ i 兲 ⫽ N

冕

␳ 共r兲dr.

(5)

⍀i

The ELF function was originally proposed by Becke and Edgecombe [1] as a measure of the probability of finding an electron in the neighborhood of

2

In this work the ELF will be calculated for orbitals ␸i,␴ and density ␳␴ separately and denoted ELF␴,

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APPLICATION OF ELF TO FREE RADICAL SYSTEMS with ␴ ⫽ ␣ or ␤. A high value of the ELF␴ is now interpreted as describing a region of the space where a radical electron can be localized. Note, however, that in this simple formulation the sum of both ELF␴ does not yield the total ELF value.

Computational Details All calculations have been done using the Gaussian 98 package [19] at the BLYP level of theory, which uses the exchange functional proposed by Becke [20] and the correlation functional proposed by Lee, Yang, and Parr [21]. The most popular hybrid functional B3LYP yields qualitatively similar results with respect to the calculation of the ELF. However, it has been shown [22] that the stability of the wave function is better for the BLYP functionals. The 6-31G(d, p) basis set has been used and all the geometries have been fully optimized without finding any imaginary frequency. The ELF topological analysis and its associated properties have been calculated using the TopMod series of programs [23]. The results can be visualized with the vis5d program [24], which gives a graphic representation of the different basins using an arbitrary color scale.

Results and Discussion

F1 F2-F7

The consequences of the unpaired electron(s) localization will be discussed and analyzed using the ELF. Three different kinds of radical systems have been chosen with the goal of validating the separation into ␣ and ␤ spin contributions. In Figure 1 the aromatic structures and corresponding atoms numbering are presented. In Figures 2–7 ELF␣ ⫽ 0.6, ELFtotal ⫽ 0.6, and ELF␤ ⫽ 0.6 isosurfaces for all molecules are displayed, in combination with the electron population of each basin calculated with Eq. (5). The radical basins V(C) are associated with only one carbon atom, whereas bonding basins represent a bond region connecting two atomic nucleus. The color convention for ELF pictures is the following: Red volumes symbolize the core electron region and each of them contains one carbon atom, green depicts the COC bonding basins V(C, C), blue depicts the space region where the radical electrons are localized, and sky-blue corresponds to the COH bond basin V(C, H), where the hydrogen atom is hidden into the basin. Core and V(C, H) basins have an average electron population of

FIGURE 1. Structures with atoms label reference. around 2.0 e. These numerical data have been excluded in all figures to simplify the analysis. The methyl radical has been included in this study as an example of a simple monoradical system. The results of applying the original ELF analysis were unexpected. We did not find any basin associated with the region of the space where it should be most probable to localize the radical electron. Figure 2 shows the isosurface and the numerical results of the total ELF and their separate ELF␣ and ELF␤ spin contributions. The ELF␣ function correctly describes the region of the space where the unpaired electron should be found. The calculations have been done with different basis sets (STO-3G, 3-21G*, 6-31G**, and 6-311⫹⫹G**) retaining the level of theory. Only small basis without polarization function can describe the radical electron region, but the electron population still does not agree well with the Lewis structure. We also improve the integration grid in the TopMod analysis; however, the radical valence basin does not appear in the total ELF. Note also that the electron population over the COH basins is of around 2.3 e (the difference between the three values is due to inaccurate numerical integration). It is also interesting to observe that the basin associated with the radical electron appears in the total ELF when the hydrogen atoms are taken out of the plane. The phenyl radical will be used as a reference system to compare with other aromatic radicals.

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MELIN AND FUENTEALBA

C O L O R FIGURE 2. Methyl radical system. ELF ⫽ 0.6 isosurface and electron population of the most relevant basins. T1

Table I shows its geometric parameters. In Figure 3, valence basin population and the topological representation are displayed. Because the molecule is highly symmetrical (C2v) all of the bonding basins

have similar electron population. The blue basin V(C1) identifies the unpaired electron and it has an average electron population value of 1.2 e, higher than the expected from the Lewis structure. We also

C O L O R FIGURE 3. Phenyl radical system. ELF ⫽ 0.6 isosurface and electron population of the most relevant basins.

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APPLICATION OF ELF TO FREE RADICAL SYSTEMS

C O L O R FIGURE 4. ortho-Benzyne system. ELF analysis. ELF ⫽ 0.6 isosurface and electron population of the most relevant basins.

found this radical basin in the ELF␣, with the same electron population value. In the ELF␤ does no basin exists related to a single electron. The topological analysis shows clearly that the unpaired electron corresponds to an ␣ spin configuration. Of course, this is an artifact of the density calculation because the probability of finding an electron with ␣ or ␤ spin should be the same. The interpretation should be that the blue region represents a high probability of containing just one electron, no matter of its spin. The calculations for the three benzyne isomers present a singlet ground state in accordance with other theoretical [25] and experimental [26] studies.

However, the ortho- and meta-benzynes have a closed-shell singlet ground state, whereas the parabenzyne is better described as an open-shell singlet biradical. According to our calculations, the singlet state is more stable than the triplet one by 37.5, 21.6, and 2.4 kcal/mol for the ortho-, meta-, and parabenzynes, respectively. These values compare well with the experimental ones [26], 37.5 ⫾ 0.3, 21.0 ⫾ 0.3, and 3.8 ⫾ 0.4, in the same order. Table II displays the calculated bond lengths and angles for the three isomers in the singlet and triplet states. Figures 4 – 6 show the ELF isosurfaces and average electron population on each basin of all the molecules in the singlet and triplet states, where the

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MELIN AND FUENTEALBA

C O L O R FIGURE 5. meta-Benzyne system. ELF analysis. ELF ⫽ 0.6 isosurface and electron population of the most relevant basins.

color meanings remain the same. Table II shows that in the ortho-benzyne the distance C1OC2 is shorter for the singlet than the triplet state, which is due to the triple bond character. The ELF analysis in Figure 4 shows in the V(C1, C2) basin an electron population higher in the singlet than in the triplet state, which also indicates some triple character of the C1OC2 bond. However, the presence of V(C1) and V(C2) radical basins reveals that the molecule has some biradical character. The electron population of those valence basins is almost 1.0 e. All other bond basins have similar electron populations, around 3.0 e. In the triplet state analysis, the triple

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bond disappears. The geometry and topological analysis show equivalent ring bonds, indicating that all of them have the same participation in the aromatic system. The V(C1) and V(C2) radical basins, also present in the triplet state, have an electron population bigger than the singlet, whereas the V(C1, C2) basin has a smaller electron population in the triplet state than in the singlet. From the ELF␣ and ELF␤ isosurfaces we can see that in the singlet state both contributions are the same; however, they are different in the triplet case. The blue basins corresponding to the unpaired electrons are only present in the ELF␣ contribution to the triplet. In

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APPLICATION OF ELF TO FREE RADICAL SYSTEMS

C O L O R FIGURE 6. para-Benzyne system. ELF analysis. ELF ⫽ 0.6 isosurface and electron population of the most relevant basins.

Figure 5 the meta-benzyne system is depicted. The single state presents a distorted ring with a smaller C1OC2OC3 angle than that suggested for an sp2 hybridization. The distance between the dehydrocarbons C1OC3 is 2.0 Å, which is large to be considered as a formal bond but shorter than a standard distance between two nonconsecutive aromatic carbons. For the singlet state we find two radical basins located on C1 and C3 atoms. They present an electron population bigger than 1.0 e.

For the triplet case the situation is similar to that of ortho-benzyne. ELF␣ has two monosynaptic basins associated with the radical electrons and well localized on C1 and C3 atoms, whereas ELF␤ does not present these basins. The ELF analysis and electron population of the basins for para-benzyne are shown in Figure 6. For the singlet state ELF presents two basins associated with the radical electron, which are located on atoms C1 and C4, respectively. The ELF picture clearly shows that ELF␣ and

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C O L O R FIGURE 7. Radical anions systems. ELF ⫽ 0.6 isosurface and electron population of the most relevant basins.

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APPLICATION OF ELF TO FREE RADICAL SYSTEMS TABLE I ______________________________________ Geometric parameters of phenyl radical. Phenyl C1OC2a C2OC3 C3OC4 C4OC5 C5OC6 C6OC1 C1OC2OC3b C2OC3OC4 C3OC4OC5 C4OC5OC6 C5OC6OC1 C6OC1OC2

1.39 1.41 1.40 1.40 1.41 1.39 125.75 116.66 120.65 120.13 120.13 116.66

Atom numeration as in Fig. 1. a Bond distances in Å. b Bond angles in degrees.

ELF␤ are different. One electron with ␣ spin is localized on the C1 atom, whereas the other electron with ␤ spin is localized on the C4 atom. Of course, it is again an artifact of the calculation because the probability of having an electron with ␣ or ␤ spin is the same. Recently, Kraka et al. [27] showed that the on-top density discussed by Perdew et al. [28] is able to reproduce the correct symmetry of the Hamiltonian. The ELF analysis for the triplet state is similar to the singlet one.

The term “distonic ions” was introduced some time ago [29] to characterize an ion with separated charge and radical sites. The benzyne anions belong to this class of ions. It is known that these systems present multiple low-lying excited states and usually a multiconfigurational wave function is necessary. However, a thorough study of the benzyne anions [17] demonstrated that the density functional-based methods have excellent performance in predicting the thermochemistry of the benzyne anions. In Figure 7 the ELF, ELF␣, and ELF␤ isosurfaces and electronic populations are presented. Because of the unrestricted character of the orbitals the ELF␣ and ELF␤ spin contributions are clearly different. However, both of them present two valence basins representing the charge and radical sites. These radical basins present a total population of around 4.0 e in all cases, one unity greater than that expected from the Lewis structures. However, the electron population value associated with the radical basins of ELF␣ contribution is consistent with the populations observed by the phenyl radical and the benzyne biradicals. For the ortho-radical anion there is no evidence of a triple bond between the C1 and C2 atoms, which agrees with the calculations of Nash and Squires [17]. It is interesting to note that ELF␤ contributions to ortho- and metaradical anions are similar to that of ELF␤ pictures and basin electron populations of biradical systems in its corresponding singlet states. This fact is asso-

TABLE II ______________________________________________________________________________________________ Geometric parameters for benzyne isomers. ortho-

C1OC2a C2OC3 C3OC4 C4OC5 C5OC6 C6OC1 C1OC2OC3b C2OC3OC4 C3OC4OC5 C4OC5OC6 C5OC6OC1 C6OC1OC2

meta-

para-

S

T

S

T

S

T

1.26 1.39 1.42 1.41 1.42 1.39 127.0 110.5 122.4 122.4 110.5 127.0

1.41 1.39 1.42 1.40 1.42 1.39 120.9 118.9 120.3 120.3 118.9 120.9

1.37 1.37 1.38 1.41 1.41 1.38 93.6 140.6 116.0 113.0 116.0 140.6

1.39 1.39 1.39 1.42 1.42 1.39 115.3 124.7 116.8 121.5 116.8 124.7

1.36 1.48 1.36 1.35 1.49 1.35 117.8 117.8 124.3 117.8 117.8 124.3

1.39 1.42 1.39 1.38 1.42 1.38 116.6 116.4 127.0 116.5 116.4 127.1

Atom numeration as in Fig. 1. a Bond distances in Å. b Angles in degrees.

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MELIN AND FUENTEALBA ciated with the qualitative validity of the frozen orbitals approximation. Concluding, this work has shown that it is possible to evaluate the ELF separately for ␳␣ and ␳␤ electron densities. The proposed ELF␣ and ELF␤ functions complement and, in some cases, improve the chemical information contained in the ELF. For open-shell systems, the ELF␣ yields a qualitative useful description of the space region where is most probable to find an unpaired electron. ACKNOWLEDGMENT The authors thank Professor Andreas Savin for helpful discussions and suggestions. This work was supported by FONDECYT (Fondo Nacional de Desarrollo Cientifico y Tecnologico, Chile) Grant 1010649. J.M. extends thanks to CONICYT (Consejo Nacional de Ciencia y Tecnologia, Chile) for a graduate fellowship.

References 1. Becke, A. D.; Edgecombe, K. E. J Chem Phys 1990, 92, 5397. 2. Savin, A.; Becke, A. D.; Flad, D.; Nesper, R.; Preuss, H.; von Schnering, H. Angew Chem Int Ed Engl 1991, 30, 409. 3. Silvi, B.; Savin, A. Nature 1997, 371, 683. 4. Berski, S.; Silvi, A.; Latajka, Z.; Leszczynski, J. J Chem Phys 1999, 111, 2542. 5. Beltran, A.; Andres, J.; Noury, S.; Silvi, B. J Phys Chem A 1999, 103, 3078. 6. Bonini, N.; Trioni, M. I.; Brivio, G. P. J Chem Phys 2000, 113, 5624. 7. Oliva, M.; Safont, V. S.; Andres, J.; Tapia, O. Chem Phys Lett 2001, 340, 391.

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8. Chamorro, E.; Santos, J. C.; Gomez, B.; Contreras, R.; Fuentealba, P. J Chem Phys 2001, 114, 23. 9. Kohout, M.; Savin, A. Int J Quantum Chem 1996, 60, 815. 10. Roberts, J. D.; Simmons, H. E.; Carlsmith, L. A.; Vaughan, C. W. J Am Chem Soc 1953, 75, 3290. 11. Berry, R. S.; Clardy, J.; Schafer, M. E. Tetrahedron Lett 1965, 6, 1003. 12. Berry, R. S.; Clardy, J.; Schafer, M. E. Tetrahedron Lett 1965, 6, 1011. 13. Marquardt, R.; Balster, A.; Sanders, W.; Kraka, E.; Cremer, D.; Radziszewiski, D. J. G. Angew Chem 1998, 110, 1001. 14. Nicolaou, K. C.; Smith, A. L.; Wendeborn, S. V.; Hwang, C.-K. J Am Chem Soc 1991, 113, 3106. 15. Nicolaou, K. C.; Dai, W. M. Angew Chem Int Ed Engl 1991, 30, 1387. 16. Crawford, T. D.; Kraka, E.; Staton, J. F.; Cremer, D. J Chem Phys 2001, 114, 10638. 17. Nash, J. J.; Squires, R. R. J Am Chem Soc 1996, 118, 11872. 18. Savin, A.; Jepsen, O.; Flad, J.; Andresen, O. K.; Preuss, H.; von Schnering, H. G. Angew Chem 1992, 31, 187. 19. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B. et al. Gaussian 98, Revision A.9; Gaussian, Inc.: Pittsburgh, PA, 1998. 20. Becke, A. D. Phys Rev A 1988, 38, 3098. 21. Lee, C.; Yang, W.; Parr, R. G. Phys Rev B 1988, 45, 13244. 22. Schreiner, P. R. J Am Chem Soc 1998, 120, 4184. 23. Noury, S.; Krokisdis, X.; Fuster, F.; Silvi, B. Comput Chem Oxford 1999, 23, 597. 24. Hibbard, B.; Kellum, J.; Paul, B. vis 5d, version 5.1; Visualization Project, University of Wisconsin-Madison Space Science and Engineering Center, 1990. 25. Cramer, C. J.; Nash, J. J.; Squires, R. R. Chem Phys Lett 1997, 227, 311. 26. Wenthol, P. G.; Squires, R. R.; Lineberger, W. C. J Am Chem Soc 1998, 120, 5279. 27. Grafenstein, J.; Hjerpe, A. M.; Kraka, E.; Cremer, D. J Phys Chem A 2000, 104, 1748. 28. Perdew, J. P.; Savin, A.; Burke, K. Phys Rev 1995, 51, 4531. 29. Yates, B. F.; Bouma, W. J.; Radom, L. J. J Am Chem Soc 1984, 106, 5805.

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APPLICATION OF ELF TO FREE RADICAL SYSTEMS AQ1

City?

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