The method of Murrell as modified by Golebiewski has been
applied to determine the maximum overlap hybrid orbitals for PF5
and SF4• For the former system, three cases have been considered:
cr overlap integrals of the central atom valence and F 2p orbitals are
(i) taken as angular parts of P atomic orbitals projected in the bond
directions, (ii) evaluated using SCF atomic wave functions with
equidistant F ligands, and (iii) calculated using experimental bond
lengths. The resultant overall hybridizations are compared and
discussed. For SF4, only case (iii) has been carried out. In addition,
the wave function of the lone pair orbital is obtained with symmetry
consideration and orthonormality relationships among all the hy-
. brids. This appears to be an inexpensive way to determine the
form of the lone pair and to estimate its size

King-Mo Sun

Thomas C. W. Mak

Wai-Kee Li

English

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CROATICA CHEMICA ACTA CCACAA 49 (4) 657-664 (1977) CCA-1037 YU ISSN 0011-1643 539.19 Original Scientific Paper Maximum Overlap. Hybridization in PFs and SF4 King-Mo Sun, Thomas C. W. Mak*, and Wai-Kee Li* Department of Chemistry, The Chinese University of Hong Kong, Shatin, N. T. Hong Kong Reeeived April 23, 1976 The method of Murrell as modified by Golebiewski has been applied to determine the maximum overlap hybrid orbitals for PF5 and SF4• For the former system, three cases have been considered: cr overlap integrals of the central atom valence and F 2p orbitals are (i) taken as angular parts of P atomic orbitals projected in the bond directions, (ii) evaluated using SCF atomic wave functions with equidistant F ligands, and (iii) calculated using experimental bond lengths. The resultant overall hybridizations are compared and discussed. For SF4, only case (iii) has been carried out. In addition, the wave function of the lone pair orbital is obtained with symmetry consideration and orthonormality relationships among all the hy-. brids. This appears to be an inexpensive way to determine the form of the lone pair and to estimate its size. INTRODUCTION The concept of hybridization has played an important role in the under-standing of bonding in molecules. Hybrid orbitals for molecules containing equivalent bonds can be deduced by standard applications of group theory. 1 On the other hand, for molecules possessing little or no symmetry, or for systems with non-equivalent bonds, the principle of maximum overlap is generally invoked in the construction of hybrid orbitals.2•3 An elegant method applicable to all MXk systems was first developed by Murrell4 and later simplified by Golebiewski.5 The version by Golebiewski, which is briefly described below, will be applied to PF5 and SF4 , each of which has two non-equivalent sets of hybrid orbitals. In a previous paper,6 the form and size of the lone pair orbitals in carbene-, ammonia-, and water-like systems were determined by a simple version of hybridization theory. In this note, the same is achieved for the SF4 , instead of only the s and p orbitals as in the previous cases, the approach to the deter-mination of the lone pair is entirely different. METHOD OF CALCULATION For a molecule MXk> the bonded hybrid orbitals hi on M are linear com-binations of orthonormal atomic orbitals <Pi: * Address correspondence to these authors. 658 K. -M. SUN ET AL. where n ~ k. In addition, the ligands X 1 , ..• , Xk are assumed to be a-bonded to M by means of orbitals 6 11 ••• , 6k respectively. To determine the coefficient matrix A, the overlap matrix is first constructed and then the real symmetric matrix SST (ST, being the transpose of S) diagonalized by an orthogonal matrix u: (3) where d is a diagonal matrix with real positive elements (the eigenvalues) a11 ••• , ak. The matrix A in equation (1) can then be calculated explicitly by the formula (4) where all square roots ai_,,, are taken as positive. The hybrids hi obtained in this way are the maximum overlap bonded hybrid orbitals on the central atom M.5 When the bonded hybrids are obtained, the lone pair hybrid may be determined with symmetry consideration and orthonormality relationships among all the hybrid orbitals. RESULTS AND DISCUSSION A. The PF 0 System With reference to the coordinate system for PF5 shown in Figure 1, the following P atomic orbitals (<Pi) have the proper symmetry to contribute to the formation of the five hybrids: 3s, 3p,, 3dz2, (3px, 3py), and (3dx2- y2 , 3d_xy) . Of these seven orbitals, the first three are fully utilized-, i.e., each with a weight of 1, while the last two pairs compete with each other to give a total weight of 2. To construct S, the following three situations are considered: Case (i): In the simplest type of approximation2 , the a-type overlap inte-grals may be taken as projections of the angular parts of P atomic orbitals along the P-F bonds, giving the relative value 1, (3)'1•, and (5)'1• for < 3s (P) J I 2p (F) > ,<3p (P) J 2p (F) >, and < 3d (P) J 2p (F) >, respectively. Thus the overlap matrix has the form: MAXIMUM OVERLAP HYBRIDIZATION 659 ·3s 3px 3py 3pz 3dz' 3d ' ' 3dxy x-y 61 0 0 (3)''' (Slv' 0 '5f;/) 62 0 0 -(3)'" (SJ''' 0 s = 63 '312 (3)v'/2 0 -(Sl''/2 (15)'h/4 64 -3/2 (3)"12 0 -(Sl'n/2 (lSY"/4 -3(51112 /4 65 0 -(3)v' 0 -!SY"/2 -(1st212 0 z Figure 1. Coordinate system for PFs Case (ii): The five P-F bonds are assigned the same length of l.555A the average of two experimental P-F bond distances.7 The overlap integrals are calculated using the SCF P and F wave functions of Berry et al.8 Nume-rical values < 3s (P) I 2p (F) > = 0.246590, < 3p (P) I 2p (F) > = 0.233804, and < 3d (P) I 2p (F) > = - 0.055623 are then used in constructing the overlap matrix. Case (iii): Using the experimental values7 of 1.577 A and 1.534 A for the axial and equatorial P-F bonds, respectively, and again employing the wave functions of Berry et al., a total of six overlap integrals are calculated. The numerical values are: < 3s (P) I 2p (.F) >(axial) = 0.242270, < 3s (P) I 2p (F) > (eq\.;~torialj = ·0 .250824 ~ < 3p (P) I 2p (F) > (a~ial) = 0.234372; < 3p (P) I 2p (F) >(equatorial) = 0.232903; -:: 3d (P) I 2p (F) >(axial) = -0.053345, < 3d (P) ! 2p (F) >(equatorial) = -0.057935 Once the overlap matrix is constructed, the calculation of the hybrid orbitals is straightforward; the results for all three cases are given in Table I. In case (i), since the elements of the overlap matrix involve only integers and their square roots, the coefficients of the resultant hybrids are expressible 660 K.-M. SUN ET AL. in closed algebraic forms. The overall hybridization sp17' 9 d19' 9 agrees with the result obtained by Duffey9 in a less general way. In addition, Golebiewski also suggested that this is the hybridization involved in PC15, SbC15, SF., and ClF3 •10 This scheme indicates that the 3px and 3py orbitals contribute 4/9 each to the hybridization, while the 3dx2-y2 and 3dxy orbitals contribute 5/9 each. This . over-emphasis of d-orbital participation results from the assumption that the P 3d orbitals overlap more effectively than the 3p and 3s orbitals with the F 2p orbitals. Inspection of the numerical values of the overlap integrals given above shows that this simplifying assumption is unrealistic. As expected, the results of cases (ii) and (iii) are very similar since the bond lengths usecl in the evaluation of overlap integrals are nearly equal. In both instances, the (3px, 3py), orbitals contribute more then 900/o to the overall hy-bridization, while the (3dx2-y2, 3dxy) pair contributes less than 100/o. The relatively insignificant part played by the latter is a consequence of the inef-fective a-overlap of the F 2p and the rather diffuse P 3d orbitals. If it were known beforehand that the overlap integrals between _P 3d and F 2p orbitals were of such a small magnitude, the (3dx2-y2, 3dxy) pair might then be safely ignored, leaving only 3s, 3px, 3py, 3p,_, and 3d,2 orbitals as a basis. Indeed, this then becomes a rather standard exercise in quantum che-mistry.11 However, it must be emphasized that the overlap integrals must be calculated before making such an assumption; there may be some PF5-like sy-stems where the (dx2-/, dxy) pair makes a significant contribution. B. The SF4 System With reference to the coordinate system for SF4 shown in Figure 2, it is evident that all the sulphur atomic orbitals in the M shell except the 3dxy orbital have the proper symmetry to contribute to the formation of the four bonded hybrids. On the other hand, only 3s, 3p,, and 3d,2 orbitals do so for the lone pair hybrid. To construct S, among the three situations considered for PF5, only case (iii) is carried out for SF4 • In the calculation of the two-center overlap in-tegrals, the structural parameters used are those given in Figure 2;12 the sulphur atomic wave functions employed are those reported by Craig and Thirunamachandran13 (hereafter referred to as CT); the fluorine functions are the ones used in the PF5 study. The numerical values of the six basic overlap integrals are: < 3s (S) I 2p (F) > (1.646 A) = 0.22781, < 3s (S) \ 2p (F) > (1.545 A) = 0.25864; < 3p (S) \ 2p (F) > (1.646 A) = 0.26551, < 3p (S) \ 2p (1!') > (1.545 A) = 0.27807 ; < 3d (S) \ 2p (F) > (1.646 A) = 0.03482, < 3d (S) \ 2p (F) > (1.545 A) = 0.01571. The forms of the bonded hybrids (hi; i = 1, ... , 4) are given in Table II. As menti0ned earlier, there are only three coefficients to be determined for the lone Case \ Hybrid I 3s 3px hi a 0 h2 a 0 (i) hs b (2)''2/3 h4 b -('.!.)'l•/3 hs b 0 hi 0.4365 0 h2 0.4365 0 (ii) hs 0.4542 0.6926 h4 0.4542 -0.6926 hs 0.4542 0 hi 0.4315 0 h2 0.4315 0 (iii) hs 0.4574 0.6912 h4 0.4574 -0.6912 hs 0.4574 0 TABLE I The Maximum Overlap Hybrid Orbitals in PF5 Atomic Orbital Coefficients 3py 3p, d2 z dx'-/ 0 (2)' 1'/2 -c 0 0 -(2)'12/2 -c 0 (6)'1'/9 0 d (30)'1•/18 (6)'1•/9 0 d (30)1 12/18 -2(6)11•/9 0 d -(30)'1•/9 0 0.7071 -0.5563 0 0 -0.7071 -0.5563 0 0.3998 0 0.3564 -0.0824 0.3998 0 0.3564 -0.0824 -0.7997 0 0.3564 0.1648 0 0.7071 -0.5602 0 0 -0.7071 -0.5602 0 0.3991 0 0.3523 -0.0860 0.3991 0 0.3523 -0.0860 -0.7982 0 0.3523 0.1720 a = { [25 -(145)'/'J/720 }1/2 [1 -(5/29)1 /'] + { [25 + (145)1/2]/720 }'/' [1 + (5/29)1 /2 ] = 0.399451. b = [(1/3) -(2a2/3))1 /' = 0.476403. c = [(1/2) -a']'/' = 0.583472. d = -[(1/3) -b'J'/' = -0.326150. dxy 0 0 (10)1 !2 /6 -(10)''2/6 0 0 0 -0.1427 0.1427 0 0 0 -0.1489 0.1489 0 Overall Hybridization spi1/D d1o/o sp2.91so d1·osu sp2-0113 di.oss1 -;s:: )> ~ ~ c: ;s:: 0 <: M :u t"' ~ ~ ~ tP :u H t:J H N )> >-3 H 0 z O") O") ...... 662 K.-M. SUN ET AL. °I lono pairi s - F1 - l. 646 A s - Fs • 1.545 A LFsSF , • 1 01. 6' Figure 2. Coordinate system and the relevant structural parameters of SF,. pair hybrid (h1) and they can be obtained uniquely with the orthonormality relationships among all the hybrid orbitals. The form of h1 is also. given in Table II. TABLE II The Maximum Overlap Bonded Hybrids (hi, i= 1, ... , 4) and the Lone Pair Hybrid (h1) of SF4 Coefficients of Atomic Orbitals 'O ·s:: I I I I I I I .0 s Pz Px Py dz2 dx•-y2 d xz dyz » ::r: hl 0.5871 0.3390 0.0 0.7071 -0.0944 -0.1774 0.0 -0.0088 h 2 0.5871 0.3390 0.0 -0.7071 -0.0944 -0.1774 0.0 0.0088 h 3 0.3771 -0.5879 0.7058 0.0 0.0463 0.0999 -0.0437 0.0 h4 0.3771 -0.5879 -0.7058 0.0 0.0463 0.0999 0.0437 0.0 h1 0.0841 0.1317 0.0 0.0 0.9877 0.0 0.0 0.0 The profile of the lone pair orbital may be illustrated by plotting its probability density I h1 12 against distance in the direction of its maximum. Figure 3 shows the I h1 12 curve of SF4, the most interesting feature being a maximum at Tmax = 0.635 A. As suggested previously6, this provides a plausible estimate of the size of the SF4 lone pair. Since other estimates for this value are not readily available in the literature, it is noted that the rmax values for the carbene, ammonia, and water lone pairs have been reported as 0.380, 0.289, and 0.244 A, respectively6• Therefore, the present estimate for SF4 appears to fall in a reasonable range. Attention is now turned to the forms of hybrid orbitals. Since the sulphur 3s and 3p orbitals overlap much more effectively than the 3rd orbitals do with the fluorine 2p orbitals, it is not surprising that the maximum overlap model I ,:u -f.i 4 MAXIMUM OVERLAP HYBRIDIZATION 0.5 rmu i.o 2.0 • o.6JS X Figure 3. Profile of SF, lone pair ([ h 1 [' in arbitrary units) . 663 yields that all bonded hybrids compose almost entirely of 3s and 3p orbitals,c while the lone pair hybrid has an exceptionally high d-content. An obvious point to make at this juncture is that the quality of the lone pair orbital obtained by this method depends heavily on the quality of the atomic wave functions used. [When Clementi's14 sulphur 3s and 3p functions and the 3d orbitals of CT are used in the calculation, the resulting hybrids are quite similar to those listed in Table IL Therefore, it may be said that the Clementi and CT wave functions are comparable with each other.] Among the four bonded hybrids, those for the longer bonds (h1 and h2 ) have a d-content that is three times as large as that for h3 and h4 • Again, this is a reflection on the difference between the two basic overlap integrals < 3d (S) I 2p (F) > (1.646 A) and < 3d (S) I I 2p (F) > (1.545 A). Finally, it is mentioned that applying the method desribed here to systems such as ClF3 and BrF3 , which have two lone pairs and where d-orbital parti-cipation is expected, would not yield a unique solution for the lone pair hybrids. This is because there are not enough orthonormality relationships to determine all the coefficients. Rather, some additional assumptions have to be made in order to resolve this difficulty4•15. To conclude, the present method is an attempt to obtain inexpensively a quantitative picture of the lone pair orbital in SFclike systems. In view of the vital role the lone pairs play in models of directed valence such as the VSEPR theory16, it appears worthwhile to pursue this method further , so that the lone pair orbitals of various molecular systems may be studied and compared. Acknowledgement. The authors wish to thank Professor Z. B. Maksic and the referee for valuable comments. REFERENCES 1. F. A. Cotton, Chemical Applications of Group Theory, 2nd ed., Wiley (Inter-science), New York, 1971. 2. For a recent review in this field, see M. Randi c and Z. B. Ma ks i c, Chem. Rev. 72 (1972) 43. 664 K.-M. SUN ET AL. 3. G. De 1 Re, Theoret. chim. Acta; I (1963) 188; A. Ve i 11 a rd and G. De 1 Re, ibid. 2 (1964) 55; G. D e 1 R e, U. E s p o s it o, and M. C a r p e n t i e r i, ibid. 6 (1966) 36. 4. J. N. Murr e 11, J. Chem. Phys. 32 (1960) 767. 5. A. Go le bi e w ski, Trans. Faraday Soc. 57 (1961) 1849. 6. W.-K. Li and T. C. W. Mak, J. Mal. Structure 25 (1975) 309. 7. L. S. Barte 11 and K. W. Hansen, Inorg. Chem. 4 (1965) 1777. 8. R. S. B err y, M. Tam res, C. J. B a 11 ha use n, and H. Johan s en, Acta Chem. Scand. 22 (1968) 231. 9. G. H. Duffy, J. Chem. Phys. 17 (1949) 196. 10. A. Go le bi e w ski, Acta Phys. Polon. 23 (1963) 243. 11. A. Ju 1 g and 0 . Ju 1 g, Exercises de Chimie Quantique, Dunod, Paris 1967, p. 35. 12. W. M. To 11 es and W. G. Gwinn, J. Chem. Phys. 36 (1962) 1119; K . Kimura and S. H. Bauer, ibid. 39 (1963) 3172. 13. D. P. Craig and T. Thirunamachandran, J. Chem. Phys. 45 (1966) 3355. 14. E. C 1 em en ti, a supplement to IBM J. Res. Develop. 9 (1965) 2. 15. P . G. Ly k o s and :i;I. M. S chm e is in g, J. Chem. Phys. 35 (1961) 288. 16. R. J. G i 11 esp i e, Molecular Geometry, Van Nostrand Reinhold, London, 1972. SAZETAK Hibridizacija maksimalnog prekrivanja u molekulama PF5 i SF4 K. M. Sun, T. C. W. Mak i W. K. Li Izracunani su hibridi maksimalnog prekrivanja za molekule PF5 i SF4 primjenom matricne metode Murrell-Golebiewski. Lokalizirana orbitala koja opisuje osamljeni par elektrona u molekuli SF4 dobivena je s pomocu zahtjeva simetrije i ortonormi-ranja hibrida. Pokazano je da hibridizacija u molekuli PF5 u prvoj aproksimaciji ima oblik sp3d. Odstupanje od toga kanonskog oblika vrlo je malo. Osamljeni par elek-trona u molekuli SF4 sadrfava izrazito visok udjel d-orbitala. DEPARTMENT OF CHEMISTRY THE CHINESE UNIVERSITY OF HONG KONG SHA TIN N. T . HONG KONG Prispjelo 23. travnja 1976. 

Maximum Overlap. Hybridization in PF5 and SF4

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