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Proc. Nat. Acad. Sci. USA Vol. 73, No. 2, pp. 274-275, February 1976 Chemistry Correlation of nonorthogonality of best hybrid bond orbitals with bond strength of orthogonal orbitals (valence bonds/transition metals/spd orbitals/spdf bond orbitals/nucleonic magic numbers) LINUS PAULING Linus Pauling Institute of Science and Medicine, 2700 Sand Hill Road, Menlo Park, California 94025 Contributed by Linus Pauling, November 25,1975 ABSTRACT An expression is derived for the bond length and that with axis at 6 = a is of two spd orbitals with maximum values in two directions forming a given bond angle by consideration of the nonor- thogonality integral of two best orbitals in these directions. 'I'b(a) =1- (s + 3/2 cos ap, + (5/4)1/2 (3 cos2 a - I)d2+2 This equation is equivalent to the expression derived by for- mulating the pair of orthogonal orbitals. Similar expressions + (15/4)1/2 sin2 adxz + 31/2sin apx + 151/2 sin a cos adx, [3] are derived for spdf orbitals. Applications are made to icosa- hedral and cuboctahedral bonds and to the packing of nu- From these equations we see that the nonorthogonality inte- cleons in atomic nuclei. gral In many compounds of transition metals the atoms of these A = (+b(0)\I'b(a)) has the value metals form nine covalent bonds with surrounding atoms. These bonds can be discussed with use of sets of spd hybrid A = (5 cos2a + 2 cos a -1)/6 [4] bond orbitals directed toward the ligated atoms. I formulat- ed an equation relating the strength (bond-forming power) It is interesting that the value of A, which is obtained by of a hybrid bond orbital to the angles it makes with other integrating the product of the two functions over the surface bond orbitals and applied it in the discussion of the struc- of the sphere, is just equal, except for a numerical factor re- tures of metal carbonyls and other substances (1). This equa- lated to the normalization, to the product of the value of one tion was derived by constructing two orthogonal orbitals function along its axis and that of the other in the same di- with the maximum values (bond strengths) in the directions rection. This coincidence has the same quantum mechanical related by the bond angle. I have now found that equivalent basis as the vector model of the electronic structure of atoms. results can be obtained by consideration of the nonorthogon- Let us assume that the functions 'b(O) and 'Ib(a) can be ality integral of the two best hybrid orbitals with their axes expressed as linear combinations of the mutually orthogonal in these directions. functions I1 and I2: The set of two orthogonal orbitals *b(O) = (1 - 4 2)1/2 d + (I2 [5] For bond angle a we may consider the bond directions to * b(a) = 3I1 - (1 - 02)1/2 12 [6] lie at 6 = a/2 and sp = 00 and 1800. The bond orbitals then have the form ais + a2pz + aad,2 + a4da2+92 4 (apx + a6d,,), with al2 + a22 + a32 + a42 = a52 + a62 = 'A, to From these equations we obtain give normalization and orthogonality (2). The six pertinent angular wave functions are 1, 31/2 cos 0,(5/4)1/2 (3 cos2 6 - *1 = $(i - fl2)/(112(o) - O4h(a)l/(l - 232) [7] 1), (15/4)1/2 sin2 6 cos 2p, 31/2 sin 6 cos (p, and (15)1/2 sin 6 cos 6 cos p, respectively. Application of the Lagrange meth- with a similar equation for 2. We see that the value of A is od of undetermined multipliers (2) shows that the value of the functions in the bond direction is a maximum when the coefficients are proportional to the values of their corre- A = 23(1 - 2)1/2 [8] sponding functions in these directions. This value is and from this equation, together with Eq. 4, we obtain a S(a) = (3 - 6x + 7,5x2)1/2 + (1.5 + 6x - 7.5x2)"2 [1] quadratic equation for a2. From the equation for I(0) the value of S is seen to be 3(1 - a 2)1/2. In this way we obtain with x = cos2 (a/2). A curve showing S(a) is given in ref. 1. another equation for S(a): Nonorthogonality of best bond orbitals The maximum values of S(a) are at a = 73.148° and S(a) = 3[10.25 -(5 Cos2 C + 2cosa - 1)2/14411/2 133.6220. At these angles the orbitals are the best spd bond + 0.5]1/2 [9] orbitals, with strength 3. The best orbital with axis at 6 = 00 is This equation, although much different in form from Eq. 1, is equivalent to it. This equivalence shows that the deficit ',(°) = 1 (s + 31"2p + 5"2dz2) in bond strength associated with bond angle a can be dis- (5cos2' + 2cos6 -1) [2] cussed by use of the nonorthogonality of best bond orbitals = 2 at this angle. 274 Chemistry: Pauling Proc. Nat. Acad. Sci. USA 73 (1976) 275 The best twelve orthogonal bond orbitals directed toward the corners of a cuboctahedron were reported by Canon and 3.99 Duffey (4) to be of type sp2_62d5f3 38, with strength 3.921. t I 1 The bond angles are 600(4), 90°(2), 1200(4), and 180°(1), S 54.880 100.430 145.370 leading with Eq. 12 to S = 3.91881, again in good agree- 3.98 ment. It is interesting that the best sp3d5f3 bond orbital has 3.97 strength 19 + (21)1/21/(12)1/2 = 3.92095. Its nodal angles, 59.3', 107.10, and 150.50, are closer to those of the icosahe- dron than are those of the sp3d5f' functions. By applying Eq. 4 to these angles, assuming 5, 5, and 1 neighbors, respec- 400 600 800 1000 1200 1400 1600 1800 tively, we obtain for S the value 3.91907, very close to the FIG. 1. The bond strength for the best orthogonal spdf orbit- correct value. als in directions making the angle a with one another. The Friauf-polyhedron functions A rather good approximation to the deficit 3 -S(a) is The best set of sixteen orthogonal spdf bond orbitals has not given by the simple expression (5 cos2 a + 2 cos a - 1)2/96. been formulated. It is likely, however, that these orbitals are directed toward the corners of the Friauf polyhedron, which Hybrid spdf orbitals has four corners (B) at the corners of a regular tetrahedron The best spdf bond orbital in the direction 0 00 is and twelve (A) at the corners of the truncated negative tet- rahedron (5). The bond angles for AA are 50.480 (18), 95.220 (24), 117.04° (12), and 144.900 (12), for AB 58.520 'b(spdf, 0) = 8 (35 c063 + 15C2O -15 cosO-3) [10] (24), 121.480 (12), and 150.500 (12), and for BB 109.470 (6). These are rather close to the best bond angles, and they lead By forming two equivalent orbitals for bond angle a the to values of S, 3.9609 for A and 3.9558, that are close to the value of their bond strength in these directions is found to be maximum. I have not found any arrangement of sixteen spdf orbitals that gives larger values of S. S(a) = (3 + 15x - 45X2 + 35x3Y'2 + (5-15x The nucleonic magic number 126 + 45X2 - 35X3)1/2 [10] I have interpreted the magic number 126 as corresponding to a completed sp3 core of four spherons (dinucleons), a with x = cos2 (a/2). completed sp3d5p7 outer core of sixteen spherons, and a The nonorthogonality integral for two best spdf orbitals at mantle of 43 spherons (6). The best set of sp3d5f' localized angle a is orbitals is indicated by the foregoing calculation to corre- spond to the Friauf polyhedron, as I had surmised before (6). A = (35 cos3 a + 15 cos2 a-15 cos a -3)/32 [11] This polyhedron has 28 triangular faces. An outer layer with 28 spherons out from the center of these faces and 16 in the With the assumptions used in deriving Eq. 9 we obtain centers of the twelve 5-rings and four 6-rings leads to a total the expression of 64 spherons, 128 nucleons. This geometrical argument fails by one spheron to give the magic number 126, which S (a) = 410225-(35 COS3 Of + 15 cos2a-15 cosa corresponds to the completion of a quantized subshell, but the close approximation indicates that the geometry of clos- 3)2/409681/2 + 0.5 1/2 [12] - est packing has some significance in nuclear structure. The function S(a) calculated with Eq. 10 or Eq. 12 is This study was supported in part by a grant from The Education- shown in Fig. 1. The nodal angles of the best spdf bond or- al Foundation of America. bital are at 54.878030, 100.431870, and 145.368500. The bond-strength defect is zero at these angles. As before (1), I assume that the defect in S for an orbital 1. Pauling, L. (1975) "Valence-bond theory of compounds of tran- orthogonal to several others can be calculated by taking the sition metals," Proc. Nat. Acad. Sd. USA 72, 4200-4202. sum of the defects corresponding to the several bond angles. 2. Pauling, L. (1931) "The nature of the chemical bond. Applica- tion of results obtained from the quantum mechanics and from Icosahedral and cuboctahedral bonds a theory of paramagnetic susceptibility to the structure of mol- Macek and Duffey (3) formulated the best set of twelve or- ecules, " J. Am. Chem. Soc. 58,1367-1400. thogonal spdf bond orbitals directed toward the corners of a 3. Macek, J. H. & Duffey, C. H. (1961) "Bonding in icosahedral regular icosahedron. They found the orbitals to be of type complexes," J. Chem. Phys. 34, 288-290. 4. Canon, J. R. & Duffey, G. H. (1961) "Cuboctahedral bonding," sp3d5f3 and to have the strength 3.921. Each bond lies at J. Chem. Phys. 35,1657-1660. 63.43490 with five others, 116.56510 with another five, and 5. Friauf, J. B. (1927) "The crystal structure of the intermetallic 1800 with one. These angles lead, with Eq. 12, to S = compound MgCu2," J. Am. Chem. Soc. 49,3107-3114. 3.90537, quite close to the value given by Macek and Duf- 6. Pauling, L. (1965) "The close-packed-spheron theory and nu- fey. clear fission," Science 150, 297-305.