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An introduction to the use of voronoi polyhedra in the study of organic reactions in solution
Journal of Molecular Structure (Theochem), 123 (1985) 383-389 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
AN INTRODUCTION TO THE USE OF VORONOI POLYHEDRA IN THE STUDY OF ORGANIC REACTIONS IN SOLUTION”
VERA BELLAGAMBA
and ROBERTA
ERCOLI
Zstituto di Chimica Fisica, Universitci di Sassari, V. Vienna 2, 07100 Sassari (Italy) ALDO GAMBA and MASSIMO SIMONETTA Zstituto di Chimica Fisica dell’Uniuersit~ Milano (Italy)
di Milan0 e Centro CNR, V. Go@
19-20133
(Received 21 September 1984)
ABSTRACT Voronoi polyhedra are employed in the discussion on solvent effect in simple dissociation reactions. INTRODUCTION
In recent years an extensive amount of experimental and theoretical research has been devoted to the investigation of the influence of solvent on organic chemical reactions. In the field of theoretical models for solvation we have studied, in a series of papers [l-8], the solute-solvent interaction through a limited number of solvent units (representative of the first solvation shell) in the near vicinity of the solute. The effect of solvent on the energetics of the dissociation process was investigated by calculating the dissociative path in solution. In particular the geometry of the solvent cage was optimized at several points of the reaction coordinate by minimizing the total energy of the supermolecule formed by the solute and a number of surrounding solvent molecules. The model was used many times in different situations and the results were always consistent with experiments and/or expectations. In recent years Voronoi polyhedra, first defined by mathematicians long ago [ 9, lo], have been proposed in the literature [ 111 as an aid for studying solvation structure. ln fact, the volume of the union of all polyhedra associated with the solute atoms represents the co-volume of the solvent, and the area of the external faces of the polyhedra measures the extent to which each solvent molecule coordinates each atom of the solute. Therefore it seemed interesting to apply this new measure of solvation to our structures, representative of the fiit solvation shells, previously determined by the application of quantum mechanical methods to our model. With regard to a systematic exploration of the new technique, we present aDedicated to Professor Raymond Daudel on the occasion of his 65th birthday. 0166-1280/85/$03.30
0 1985 Elsevier Science Publishers B.V.
384
here a preliminary study on haloalkyl solvolytic reactions, considering fluoromethane (CHBF) and methylene lithium fluoride (CH,LiF) as prototypes of this kind of reaction. Their dissociation mechanism in solution was previously investigated in water [ 11, hydrofluoric acid [4] and methane [6] in the case of CH3F, and in water [8] for CH2LiF. The possibility of defining simple regions in an unambigous manner so that an immediate interpretation of the solvent cage can easily be allowed is not the only goal of our investigation. Our main interest is to find well defined regions of space in which the electronic populations could be integrated. This is done with a view to improving the calculation of atomic charges, which are generally obtained by approximate techniques: e.g., Mulliken population analysis [ 121 assigns electrons to atoms by determining the number of electrons in each basis function centered on each atom, and totally neglects their spatial extent. The problem of atomic charge evaluation is of major importance in the case of solvated molecules and the usual treatments do not give reliable answers. Polyhedra construction Given a set of atoms Ai, i = l--n, the Voronoi polyhedron & around the atom Ai is the set of points closer to Ai than to any Aj, j # i. The details of the geometrical properties of these polyhedra are reported, for instance, in ref. 13 along with a scheme for their construction. Following the procedure described there, in the present work the Voronoi polyhedra were constructed in two steps, the first consisting of the building of the direct polyhedra (see ref. 13 for their definition). Each face was determined starting from the closest side to the projection of Ai on the face itself. In the second step Voronoi polyhedron was determined by cutting the direct polyhedron and defining the new faces. The Cartesian coordinates thus calculated are appropriate for use in the ORTEP program [14] for drawing the figures. RESULTS
In the present work the Voronoi polyhedra were calculated for lithium, carbon and fluorine atoms in the following systems: Li’, F-, CH$, CH,Li’, CH2F; CHBF, and CH2LiF, surrounded by proper numbers of water molecules. The location of water molecules was taken from the optimized geometries of the following systems: Li+(HzO),; F-(H?O),, n = 4, 6; CHz(H,O),; CH2Li+(HZ0)6; CHzF-(Hz0)4; CH3F(HZO)ll, and CH2LiF(H20)10. They were chosen since they are involved in the initial and final stages of the following reactions +CH:W,O),
/CH2Li+(H20)6 CH,LiF(H,O),,
K WLi”(H,O),
+ F-VW),
+ F-(H,O), + CH,F-(H,O),
(1)
(2a)
(2b)
385
In these calculations the solvent units were kept rigid. Only the oxygen atoms of the water molecules were taken into account in polyhedra construction, and an alternative approach could consider the hydrogens of water as well. The results of the latter treatment will be presented in a future investigation having, as a final goal, the determination of atomic charges in solvated molecules. In all the cases considered closed Voronoi polyhedra were obtained, with the exception of the polyhedron centered on the fluorine atom of CH,F-(H,O),. Apparently, more than four water molecules are necessary for complete solvation of this ion. The resulting polyhedra are collected in Figs. l-6, and can be conveniently compared as they are represented on the same scale. The Voronoi polyhedra shown in Figs. 1 and 2 are related to undissociated (CHBF) and dissociated (CHJ and F-) species involved in reaction (1). It appears that the quantum mechanical prediction of the geometry of the solvation shell for the initial and final stages of the reaction is rather accurate with little variation of co-volumes and superficial areas, either for carbon or fluorine atoms, with the co-volume of fluorine always predominant in comparison with carbon atom, as expected. Moreover, it is worth noting the slight effect of fluorine both on the shape and the magnitude of the polyhedron centered on the carbon atom of methyl group (Figs. l(a) and 2(a)). Figures 3(a), (b), 4(a), (b) and 5(b) refer to reaction 2a. In this case the solvation of fluorine anion represents a problem: if F- is far enough from its counterion, its solvation cage should be the one represented in Fig. 2(b) rather than that shown in Fig. 5(b). In fact the coordination number for
.OLI
86 .OlO
lo, (a)
(b)
Fig. 1. Voronoi polyhedra for the carbon (a) and fluorine (h) atoms of CH,F(H,O),,. Dots represent the positions of oxygen atoms of water molecules.
386
001
0 03
004 .O3
(a)
(b)
Fig. 2. Voronoi polyhedra for the carbon atom (a) of methyl cation in water (CH,(H,O),)
and fluorine anion (b) in water F-(H,O),.
(b) Fig. 3. Voronoi polyhedra for fluorine and lithium atoms (a) and carbon atom (b) for solvated but undissociatad CH,LiF(H,O),,. a simple ion is an unambiguously defined quantity either in the case of infinite dilution or in the case of a large cluster of water containing a simple ion. However, it was found for water solutions of LiF [15,161 in the case of ionic solutions at very high concentration, or equivalently, in the case of relatively short internuclear distances between F- and Li’, that the coordination number of a simple ion might well be ambiguous. In the case of this reaction also, the comparison between the corresponding polyhedra shown in Figs. 3(a), (b), 4(a), (b) and 5(b) confirms that
a
-:6 5
7
.O5
/’
/’
6 004
I’
04 0,’
4
;//&
1
(a)
*
HZ
g
001
63
(b)
Fig. 4. Voronoi polyhedra for lithium (a) and carbon (b) atoms for CH,Li+(H,O), .
82
6
5 2
1
0% Li
83
4 0 .% j _i______-________ 7 _A* YC FD 4
l4
3
(a)
(b)
Fig. 5. Voronoi polyhedra for the lithium cation (a) and fluorine anion (b) solvated by 6 and 4 water molecules, respectively.
both the shape and the volumes of the polyhedra centered on the different atoms of CHzLiF are roughly retained as dissociation occurs. Obviously the behaviour should be confirmed by considering a larger number of solvent units. In the case of reaction (2(b)) the Voronoi polyhedron centered on the fluorine atom of CH,F-(H,O), does not close. By inspection of Fig. 6, where the polyhedron centered around the carbon atom is reported, it is
388
8
.O3
004
4 Fig. 6. Voronoi polyhedron whole anion does not close.
for the carbon atom of CH,F-(H,O),.
The polygon for the
easy to understand that this is due to the unusual location of the four water molecules, all lying on the same side of the anion. This may be due to an unappropriate energy optimization procedure or to the use of an inadequate number of solvent molecules. Anyhow the Voronoi polyhedron gives a prompt view of the level of solvation and allows to correct possible lacks of the solvation model either for the whole molecule or for parts of it. The present preliminary analysis evidences two promising points: Voronoi polyhedra can be usefully used to define the minimum number of solvent units necessary to describe properly the first solvation shell of molecules and ions and, moreover, the properties of different solvents could be studied by examining the effect of each solvent in terms of the different shapes and magnitudes of the corresponding polyhedra. REFERENCES 1 P. Cremaschi, A. Gamba and M. Simonetta, J. Chem. Sot. Perkin Trans. 2 (1977) 162. 2 A. Gamba, M. Simonetta, G. B. Suffritti, I. Szele and H. Zollinger, J. Chem. Sot. Perkin Trans. 2 (1980) 493. 3 P. Demontis, R. Ercoii, A. Gamba, G. B. Suffritti and M. Simonetta, J. Chem. Sot. Perkin Trans. 2 (1981) 488. 4 P. Demontis, E. S. Fois, A. Gamba, B. Manunza, G. B. Suffritti and M. Simonetta, J. Chem. Sot. Perkin Trans. 2 (1982) 783. 5 E. S. Fois, A. Gamba, G. B. Suffritti, M. Simonetta, I. Szele and H. Zoiiinger, J. Phys. Chem., 86 (1982) 3722. 6 P. Demontis, A. Gamba, G. B. Suffritti and M. Simonetta, J. Chem. Sot. Perkin Trans. 2 (1983) 997. 7 A. Gamba, B. Manunza, C. Gatti and M. Simonetta, Theor. Chim. Acta (Berlin), 63 (1983) 245. 8 V. Bellagamba, R. Ercoli, A. Gamba and M. Simonetta, J. Chem. Sot., Perkin Trans. 2 (1985) 185. 9 G. L. Dirichlet, Z. Reine, Angew. Math., 40 (1850) 216. 10 G. F. Voronoi, Z. Reine Angew. Math., 134 (1908) 198.
389 11 E. E. David, C. W. David, J. Chem. Phys., 76 (1982) 4611; 77 (1982) 3288; 77 (1982) 6251; 78 (1983) 1459. 12 R. S. Mulliken, J. Chem. Phys., 23 (1955) 1833. 13 W. Brostow, J. P. Dussault and B. L. Fox, J. Comput. Phys., 29 (1978) 81. 14 C. K. Johnson, Oak Ridge Thermal-Ellipsoid Plot Program ORNL-3794 Second Rev. UC.4 Chemistry. 15 R. 0. Watts, E. Clementi, J. Fromm, J. Chem. Phys., 61(1974) 2550. 16 J. Fromm, E. Clementi, R. 0. Watts, J. Chem. Phys., 62 (1975) 1388.
AN INTRODUCTION TO THE USE OF VORONOI POLYHEDRA IN THE STUDY OF ORGANIC REACTIONS IN SOLUTION”
VERA BELLAGAMBA
and ROBERTA
ERCOLI
Zstituto di Chimica Fisica, Universitci di Sassari, V. Vienna 2, 07100 Sassari (Italy) ALDO GAMBA and MASSIMO SIMONETTA Zstituto di Chimica Fisica dell’Uniuersit~ Milano (Italy)
di Milan0 e Centro CNR, V. Go@
19-20133
(Received 21 September 1984)
ABSTRACT Voronoi polyhedra are employed in the discussion on solvent effect in simple dissociation reactions. INTRODUCTION
In recent years an extensive amount of experimental and theoretical research has been devoted to the investigation of the influence of solvent on organic chemical reactions. In the field of theoretical models for solvation we have studied, in a series of papers [l-8], the solute-solvent interaction through a limited number of solvent units (representative of the first solvation shell) in the near vicinity of the solute. The effect of solvent on the energetics of the dissociation process was investigated by calculating the dissociative path in solution. In particular the geometry of the solvent cage was optimized at several points of the reaction coordinate by minimizing the total energy of the supermolecule formed by the solute and a number of surrounding solvent molecules. The model was used many times in different situations and the results were always consistent with experiments and/or expectations. In recent years Voronoi polyhedra, first defined by mathematicians long ago [ 9, lo], have been proposed in the literature [ 111 as an aid for studying solvation structure. ln fact, the volume of the union of all polyhedra associated with the solute atoms represents the co-volume of the solvent, and the area of the external faces of the polyhedra measures the extent to which each solvent molecule coordinates each atom of the solute. Therefore it seemed interesting to apply this new measure of solvation to our structures, representative of the fiit solvation shells, previously determined by the application of quantum mechanical methods to our model. With regard to a systematic exploration of the new technique, we present aDedicated to Professor Raymond Daudel on the occasion of his 65th birthday. 0166-1280/85/$03.30
0 1985 Elsevier Science Publishers B.V.
384
here a preliminary study on haloalkyl solvolytic reactions, considering fluoromethane (CHBF) and methylene lithium fluoride (CH,LiF) as prototypes of this kind of reaction. Their dissociation mechanism in solution was previously investigated in water [ 11, hydrofluoric acid [4] and methane [6] in the case of CH3F, and in water [8] for CH2LiF. The possibility of defining simple regions in an unambigous manner so that an immediate interpretation of the solvent cage can easily be allowed is not the only goal of our investigation. Our main interest is to find well defined regions of space in which the electronic populations could be integrated. This is done with a view to improving the calculation of atomic charges, which are generally obtained by approximate techniques: e.g., Mulliken population analysis [ 121 assigns electrons to atoms by determining the number of electrons in each basis function centered on each atom, and totally neglects their spatial extent. The problem of atomic charge evaluation is of major importance in the case of solvated molecules and the usual treatments do not give reliable answers. Polyhedra construction Given a set of atoms Ai, i = l--n, the Voronoi polyhedron & around the atom Ai is the set of points closer to Ai than to any Aj, j # i. The details of the geometrical properties of these polyhedra are reported, for instance, in ref. 13 along with a scheme for their construction. Following the procedure described there, in the present work the Voronoi polyhedra were constructed in two steps, the first consisting of the building of the direct polyhedra (see ref. 13 for their definition). Each face was determined starting from the closest side to the projection of Ai on the face itself. In the second step Voronoi polyhedron was determined by cutting the direct polyhedron and defining the new faces. The Cartesian coordinates thus calculated are appropriate for use in the ORTEP program [14] for drawing the figures. RESULTS
In the present work the Voronoi polyhedra were calculated for lithium, carbon and fluorine atoms in the following systems: Li’, F-, CH$, CH,Li’, CH2F; CHBF, and CH2LiF, surrounded by proper numbers of water molecules. The location of water molecules was taken from the optimized geometries of the following systems: Li+(HzO),; F-(H?O),, n = 4, 6; CHz(H,O),; CH2Li+(HZ0)6; CHzF-(Hz0)4; CH3F(HZO)ll, and CH2LiF(H20)10. They were chosen since they are involved in the initial and final stages of the following reactions +CH:W,O),
/CH2Li+(H20)6 CH,LiF(H,O),,
K WLi”(H,O),
+ F-VW),
+ F-(H,O), + CH,F-(H,O),
(1)
(2a)
(2b)
385
In these calculations the solvent units were kept rigid. Only the oxygen atoms of the water molecules were taken into account in polyhedra construction, and an alternative approach could consider the hydrogens of water as well. The results of the latter treatment will be presented in a future investigation having, as a final goal, the determination of atomic charges in solvated molecules. In all the cases considered closed Voronoi polyhedra were obtained, with the exception of the polyhedron centered on the fluorine atom of CH,F-(H,O),. Apparently, more than four water molecules are necessary for complete solvation of this ion. The resulting polyhedra are collected in Figs. l-6, and can be conveniently compared as they are represented on the same scale. The Voronoi polyhedra shown in Figs. 1 and 2 are related to undissociated (CHBF) and dissociated (CHJ and F-) species involved in reaction (1). It appears that the quantum mechanical prediction of the geometry of the solvation shell for the initial and final stages of the reaction is rather accurate with little variation of co-volumes and superficial areas, either for carbon or fluorine atoms, with the co-volume of fluorine always predominant in comparison with carbon atom, as expected. Moreover, it is worth noting the slight effect of fluorine both on the shape and the magnitude of the polyhedron centered on the carbon atom of methyl group (Figs. l(a) and 2(a)). Figures 3(a), (b), 4(a), (b) and 5(b) refer to reaction 2a. In this case the solvation of fluorine anion represents a problem: if F- is far enough from its counterion, its solvation cage should be the one represented in Fig. 2(b) rather than that shown in Fig. 5(b). In fact the coordination number for
.OLI
86 .OlO
lo, (a)
(b)
Fig. 1. Voronoi polyhedra for the carbon (a) and fluorine (h) atoms of CH,F(H,O),,. Dots represent the positions of oxygen atoms of water molecules.
386
001
0 03
004 .O3
(a)
(b)
Fig. 2. Voronoi polyhedra for the carbon atom (a) of methyl cation in water (CH,(H,O),)
and fluorine anion (b) in water F-(H,O),.
(b) Fig. 3. Voronoi polyhedra for fluorine and lithium atoms (a) and carbon atom (b) for solvated but undissociatad CH,LiF(H,O),,. a simple ion is an unambiguously defined quantity either in the case of infinite dilution or in the case of a large cluster of water containing a simple ion. However, it was found for water solutions of LiF [15,161 in the case of ionic solutions at very high concentration, or equivalently, in the case of relatively short internuclear distances between F- and Li’, that the coordination number of a simple ion might well be ambiguous. In the case of this reaction also, the comparison between the corresponding polyhedra shown in Figs. 3(a), (b), 4(a), (b) and 5(b) confirms that
a
-:6 5
7
.O5
/’
/’
6 004
I’
04 0,’
4
;//&
1
(a)
*
HZ
g
001
63
(b)
Fig. 4. Voronoi polyhedra for lithium (a) and carbon (b) atoms for CH,Li+(H,O), .
82
6
5 2
1
0% Li
83
4 0 .% j _i______-________ 7 _A* YC FD 4
l4
3
(a)
(b)
Fig. 5. Voronoi polyhedra for the lithium cation (a) and fluorine anion (b) solvated by 6 and 4 water molecules, respectively.
both the shape and the volumes of the polyhedra centered on the different atoms of CHzLiF are roughly retained as dissociation occurs. Obviously the behaviour should be confirmed by considering a larger number of solvent units. In the case of reaction (2(b)) the Voronoi polyhedron centered on the fluorine atom of CH,F-(H,O), does not close. By inspection of Fig. 6, where the polyhedron centered around the carbon atom is reported, it is
388
8
.O3
004
4 Fig. 6. Voronoi polyhedron whole anion does not close.
for the carbon atom of CH,F-(H,O),.
The polygon for the
easy to understand that this is due to the unusual location of the four water molecules, all lying on the same side of the anion. This may be due to an unappropriate energy optimization procedure or to the use of an inadequate number of solvent molecules. Anyhow the Voronoi polyhedron gives a prompt view of the level of solvation and allows to correct possible lacks of the solvation model either for the whole molecule or for parts of it. The present preliminary analysis evidences two promising points: Voronoi polyhedra can be usefully used to define the minimum number of solvent units necessary to describe properly the first solvation shell of molecules and ions and, moreover, the properties of different solvents could be studied by examining the effect of each solvent in terms of the different shapes and magnitudes of the corresponding polyhedra. REFERENCES 1 P. Cremaschi, A. Gamba and M. Simonetta, J. Chem. Sot. Perkin Trans. 2 (1977) 162. 2 A. Gamba, M. Simonetta, G. B. Suffritti, I. Szele and H. Zollinger, J. Chem. Sot. Perkin Trans. 2 (1980) 493. 3 P. Demontis, R. Ercoii, A. Gamba, G. B. Suffritti and M. Simonetta, J. Chem. Sot. Perkin Trans. 2 (1981) 488. 4 P. Demontis, E. S. Fois, A. Gamba, B. Manunza, G. B. Suffritti and M. Simonetta, J. Chem. Sot. Perkin Trans. 2 (1982) 783. 5 E. S. Fois, A. Gamba, G. B. Suffritti, M. Simonetta, I. Szele and H. Zoiiinger, J. Phys. Chem., 86 (1982) 3722. 6 P. Demontis, A. Gamba, G. B. Suffritti and M. Simonetta, J. Chem. Sot. Perkin Trans. 2 (1983) 997. 7 A. Gamba, B. Manunza, C. Gatti and M. Simonetta, Theor. Chim. Acta (Berlin), 63 (1983) 245. 8 V. Bellagamba, R. Ercoli, A. Gamba and M. Simonetta, J. Chem. Sot., Perkin Trans. 2 (1985) 185. 9 G. L. Dirichlet, Z. Reine, Angew. Math., 40 (1850) 216. 10 G. F. Voronoi, Z. Reine Angew. Math., 134 (1908) 198.
389 11 E. E. David, C. W. David, J. Chem. Phys., 76 (1982) 4611; 77 (1982) 3288; 77 (1982) 6251; 78 (1983) 1459. 12 R. S. Mulliken, J. Chem. Phys., 23 (1955) 1833. 13 W. Brostow, J. P. Dussault and B. L. Fox, J. Comput. Phys., 29 (1978) 81. 14 C. K. Johnson, Oak Ridge Thermal-Ellipsoid Plot Program ORNL-3794 Second Rev. UC.4 Chemistry. 15 R. 0. Watts, E. Clementi, J. Fromm, J. Chem. Phys., 61(1974) 2550. 16 J. Fromm, E. Clementi, R. 0. Watts, J. Chem. Phys., 62 (1975) 1388.