The conditions under which a set of atomic orbitals becomes angularly independent are investigated for both atomic and molecular systems. How these results can be applied to various molecules is considered

Joel F. Liebman

Zelek S. Herman

English

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  * Dedicated to Professor Zvonimir Maksić on the occasion of his 70th birthday. ** Author to whom correspondence should be addressed. (E-mail: jliebman@umbc.edu; zelek.herman@worldnet.att.net) CROATICA CHEMICA ACTA CCACAA, ISSN-0011-1643, ISSN-1334-417X  Croat. Chem. Acta 82 (1) (2009) 261-265. CCA-3315 Original Scientific Paper On the Angular Independence of Sets of Atomic Orbitals* Joel F. Liebmana,** and Zelek S. Hermanb,** aDepartment of Chemistry and Biochemistry, University of Maryland, Baltimore County, 1000 Hilltop Circle,  Baltimore, MD 21250 USA bHerman Scientific Consulting, 521 Del Medio Avenue, #107, Mountain View, CA 94040 USA RECEIVED JUNE 29, 2008; REVISED AUGUST 21, 2008; ACCEPTED SEPTEMBER 3, 2008  Abstract. The conditions under which a set of atomic orbitals becomes angularly independent are investi-gated for both atomic and molecular systems. How these results can be applied to various molecules is considered. Keywords: angular independence, atomic orbitals, molecular orbitals, symmetry  INTRODUCTION When presented in their real form, only the atomic or-bitals having azimuthal quantum number ℓ = 0 (the s orbitals) are spherical and have no angular dependence. The linear combination of spherical harmonics having the same azimuthal quantum number ℓ, with ℓ ≥ 1, yields atomic orbitals having angular dependence. For example, the px, py, and pz orbitals (the real p orbitals) have dumbbell shapes that change sign as value of the angular variable goes from 0 to 180o, or π radians. Simi-larly, for ℓ = 2, most of the real d orbitals have clover-leaf shapes that change sign every 90o,or π/2 radians. As ℓ increases in value, the shapes of the real orbitals be-come more complicated, e.g. the simplest drawing of an f orbital (ℓ = 3) shows a 6-petalled flower that changes sign every 60o or π/3 radians. Parenthetically, it may be noted that the shapes of f orbitals are often not drawn correctly. In 1927, Unsöld proved that if one sums the elec-tron density of all three p orbitals, where each orbital is occupied by an electron pair, then all angular depen-dence disappears; that is, the resulting total density is as spherical as found from the use of s orbitals alone.1−3 Similarly, if all five d orbitals are doubly occupied, or all seven f orbitals are doubly occupied, it was shown that the same spherical behavior results. Spherical den-sity also occurs when each orbital is singly occupied, provided that all of the electron spins are parallel. In other words, Unsöld’s theorem states that the sum of the squares of all of the angular wave functions, that is, the probability distribution, corresponding to a given value of ℓ, is a constant, independent of the orien-tation. While Unsöld’s theorem has important implica-tions for the structure of complicated atoms, most text-books on quantum mechanics do not explicitly discuss it, notable exceptions being the classic textbooks by Linus Pauling and E. Bright Wilson, Jr.,2 and by John C. Slater.3 In a previous publication by the present authors, the conditions required for the independence of sets of atomic orbitals were considered.4 In the present work, these conditions are investigated in more detail.  THE REQUIREMENTS FOR ANGULAR INDEPENDENCE In 1995, Icke showed that angular dependence vanishes when an s orbital and some suitable real orbitals having ℓ ≥ 1 are added together.5 It is possible to understand this phenomenon by considering the combination of atomic orbitals as a sum of cosine functions:        010cos 1 21 cos 2cos 2 2 cos ( 1) 2nnjs N j πN ππ n π                   (1) where Θ is an arbitrary angular coordinate that applies either to an atom or a molecular system with Dnh point-262 J. F. Liebman and Z. S. Herman, Angular Independence of Orbitals Croat. Chem. Acta 82 (2009) 261. group symmetry. Here the leading terms N0 + 1 reflect the different normalization of the s orbital from those other orbitals that have some angular dependence. Moreover, Eq. (1) may be rewritten as    0n ns N S     (2) where    1cos 1 2nnjS j π        (3) One may now ask to what does this sum converge for the angle variables 0 ≤ 2πΘ ≤ 2π (i.e., 0 ≤ Θ ≤ 1) as n goes to infinity. That is, what is the value of sn(Θ) for arbitrary angles Θ and for limiting values of n? Owing to the nature of the cosine function, the situation for Θ = 0 and Θ = 1 are identical. Thus, this is an example of periodic boundary conditions. Since i is defined as and the real part of any complex number may be written as  (4) the following identity holds for all (angles) α  (5) Then the series in Eq. (3) can thus be rewritten as        1 exp 2exp 2 2exp 1 2nS i πi πi n π                    (6) Moreover, for any two arbitrary complex numbers u and v,  (7) accordingly,  (8) Now define  exp 2x i π     (9) so that Eq. (8) may be rewritten as   2 11 nnS x x x x          (10) Therefore, for all finite n and x ≠ 1,      1 / 1nnS x x x       (11) It is now possible to investigate under which con-ditions the sums Sn(Θ) and Sn(x) have values equal to zero. This situation occurs if two conditions are met: 1nx   (12) and 1x   (13) Thus,  (14) At this stage, one must consider whether Θ is a ra-tional or irrational number. If Θ is rational, then must equal m/n, where m and n are integers, and Θ is real). By periodicity or modulus n, only those cases where m does not exceed n need be considered. Since it is also required that x does not equal 1, one may exclude the cases where Θ equals either 0 or 1. Does this mean that all (or more precisely, almost all) rational numbers between 0 and 1 result in a sum of 0? In other words, does this mean that exclusive of the end points result in a sum of 0? For the values of real x chosen, it was required that Sn(Θ) and Sn(x) equal 0. The defining equation for Sn(x) is an nth order equation in x. There are but n roots, which have already been found. For integer m and n→∞, the set of numbers m/n includes all rational numbers be-tween 0 and 1. If n is set equal to p!, then the set of numbers m/n for all m not equal to n includes all quo-tients m′/n′ not equal to 0 and 1 for all n′ ≤ p. As p →∞, and setting p!= n which in turn goes to ∞, all rational numbers between 0 and 1 are included. Thus, it would seem as if an infinite number of angles sum to zero. Next, one must consider the infinite set of num-bers between 0 and 1 not yet discussed, namely, the irrational values of x. The size of the set of such num-bers is recognized to equal ℵ1, and is larger than the number of rational numbers, recognized to equal ℵ0 - indeed, ℵ1 is infinitely larger than ℵ0. What is their sum corresponding to Sn(Θ) and Sn(x)? Since each irrational number is arbitrarily close to a rational number and 1 a bi a     cos expα ia        u v u v       1 exp 2exp 2 2exp 1 2ni πS i πi n π                        exp 2 1, 1nx i πn x     J. F. Liebman and Z. S. Herman, Angular Independence of Orbitals       263    Croat. Chem. Acta 82 (2009) 261. since the sum for all rational numbers is 0, then one might surmise that the sum for irrational numbers also has the value 0 if the sum does not diverge. More pre-cisely, can the sum diverge? If the sum Sn(x) were to equal 0 at all points - at least for ”large enough” n, and not just the rational values of x, then its derivative Sn′(x) would equal 0 as well. This may be determined by re-cognizing that Sn(x) is of the form N(x)/D(x). From simple calculus,  (15) By construction, D(x) never equals 0 while N (x) always does. Recall xn =1; x ≠1. Therefore,  (16) and so N′ (x) must equal 0. This means that  (17) Since xn = 1, one obtains  (18) Remembering that n becomes arbitrarily large while x is never larger than 1 means that Eq. (18) is never realized. Sn′(x) never equals 0, and so Sn(x) does not equal 0 at all points. Indeed, since ℵ0 is infinitely smaller than ℵ1, Sn(x) almost never equals 0. Thus, the initial conclusion is, in fact, false. Lest the reader be unsold by the preceding discus-sion, there is an alternative way to investigate the sum given in Eq. (6) and rewritten as Eq. (7). Indeed, one may recast the sum [1 + x + x2 +…+ x n−1] as  (19) For all x chosen such that Θ = m/n, where m and n are integers,  (20) Then, either x = 0, in which case n =1, or Tn(x) = 0. Therefore, Sn(x) = 0 for all n ≠ l. Similarly, for all x chosen such that Θ = m/n, where m and n are integers,  (21) For this situation, n = 1 is an allowed solution. Other solutions are [Tn(x)] = n and [Tn(x)] = 0. Consider the individual terms, e.g., Tn,k(x) Then it might appear that  nT x n     (22) However,  [Tn,k(x)] < 1 for all k ≠ 0. Therefore, the sum of the terms  [Tn,k(x)] ≤ n, where equality is found only for n = 1. This implies that the only general solution for arbitrary integer n is Tn(x) = 0.  CHEMICAL SIGNIFICANCE The foregoing investigation of the angular independence of certain sets of atomic orbitals has implications for chemistry. First, consider homoleptic ions and mole-cules such as BBr3, [ICl4]−, and [XeF5]−, with 3-, 4-, and 5-fold symmetry and bond angles of 2π/3, π/2, and 2π/5, respectively. These species can be compared to NF3, [IF4]+, and IF5, which lack the symmetries of the first set of species. Second, there are also regular polygonal rings such as [C5H5]−, C6H6, [C7H7]+, and [C8H8]2−, with 5-, 6-, 7-, and 8-fold symmetry and bond angles (formed by two adjacent carbons and the n-fold symmetry axes) such as π/5, π/3, 2π/7, and π/4. These species contrast with [C5H5]˙, [C6H6]+, [C7H7]−, and C8H8. In other words, the geometry is not merely due to the fact that these species are of the type AB3, AB4, and AB5 for the first set and CnHn, (n = 5, 6, 7, 8) for the second. A close examination of the coefficients of the real molecular orbitals, as opposed to the complex formulation, indi-cates unequal electron densities for all atoms in systems having Dnh point-group symmetry. Interestingly, the first set of carbon-containing rings gives us an even simpler proof of our angular independence result. According to the textbook litera-ture,6 within the Hückel molecular orbital approximation for an N-carbon ring, a simple expression for the coefficient )(kmb on of the mth orbital, ψm on carbon atom k is given by  (23) Ck is the normalization factor and is the same for all k ≠ N . It is seen that k = N is a constant function because cos(0) equals 1 and so has a different normali-zation factor from the other solutions. However, the N-dimensional vector that corres-ponds to the mth molecular orbital, ψm, is orthogonal to          2' ''nD n N x D x N xS xD x         1' ' / / 1nnS x N x D x nx x   1 0nnx  / 0n x         ,0 ,1,2 , 1n n nn n nT x T x T xT x T x       n nxT x T x   2n nT x nT x   ( ) cos 2 /km kb C πmk N264 J. F. Liebman and Z. S. Herman, Angular Independence of Orbitals Croat. Chem. Acta 82 (2009) 261. other all eigenfunction solutions including ψN. Since the coefficient at all atoms in the lowest energy solution for the n-atom ring equals the constant (1/ N ), this means that the sum over all k (including N ) vanishes:  (24) However, the common factors (1/ N ) and Ck cannot equal zero. Therefore, one concludes that  (25) Thus, for specified angles, one apiece for each size of the carbon ring, and within a multiplicative factor, angu-lar independence has been demonstrated, at least for the class of ions and molecules considered. Therefore, rational numbers appear to be adequate for the understanding and, at least heuristic, validity of the conditions required for angular independence. In-deed, for a planar homocyclic system, most often carbo-cyclic CnHn with some charge or another, consider the molecular orbitals. If each member of a degenerate set of molecular orbitals is equally occupied (either both singly or both doubly occupied), then equal electron density occurs at each atom. Accordingly, C6H6, with point-group symmetry D6h, has uniform electron density at each atom. So does [C5H5]−, with point-group sym-metry D5h, but neutral C5H5 does not, even with with point-group symmetry D5h. That the last system also will undergo Jahn-Teller distortion only exacerbates or exaggerates the nonuniformity of charges on carbon. There are some other species that are worthy of mention at this point. For example, consider the cyclo-nonatetraenide ion, [C9H9]−, which has D9h point-group symmetry in both solution, and in numerous diverse salts and complexes.7 However, the nonacyano deriva-tive, [C9(CN)9]−, is expected to have only 2-fold sym-metry because of repulsion between the substituents according to calculational theory.8 Nonetheless, this anion is still expected to remain the conjugate base of an exceptionally strong acid as well as a desirable synthetic goal for the organic chemist. Furthermore, higher species with the formula CnHn (neutral and charged, n > 9) lack Dnh symmetry. For example, calculational theory shows the still-unknown [C11H11]+ to have 6 minima, none of which have D11h symmetry.9 Indeed, the well-known and highly stable [18]-annulene C18H18, not only lacks D18h symmetry, but seemingly lacks even the customarily believed D6h point-group symmetry.10 Even for these species with n ≤ 9, when unequal occupancy of degenerate molecular orbitals occurs, there exists the possibility of blocking Dnh point-group symmetry. After all, as drawn, some of the degenerate molecular orbitals have nodes at some of the atoms and, therefore, zero electron density at those atoms. Since the other atoms have nonzero coefficients, the electron density is positive, and the orbital electron density lacks Dnh point-group symmetry.  CONCLUSION Within the spirit of Unsöld’s theorem, certain conditions have been found under which a collection of atomic orbitals or molecular orbitals are angularly independent. Acknowledgements. The authors express their sincere congra-tulations to their dear friend Professor Zvonimir B. Maksić on the occasion of his 70th birthday. We wish him and his family all the best in the years to come. We are profoundly apprecia-tive of the hospitality shown us by Zvonko and his colleague and beloved wife, Dr. Mirjana Eckert-Maksić.  REFERENCES 1. A. Unsöld, Ann. d. Phys. 82 (1927) 355−393. 2. L. Pauling and E. B. Wilson, Jr., Introduction to Quantum Me-chanics With Applications to Chemistry, McGraw-Hill, New York, 1935, p. 150. 3. J. C. Slater, Quantum Theory of Atomic Structure, Vol. 1, McGraw-Hill, New York, 1960, p. 182. 4. J. F. Liebman and Z. S. Herman, Int. J. Quantum Chem. 101 (2005) 283−286. 5. V. Icke, The Force of Symmetry, Cambridge Univ. Press, New York, 1995, pp. 95, 162. 6. A. Streitwieser, Jr., Molecular Orbital Theory for Organic Chemists, John Wiley, New York, 1961, p. 47. 7. See M. D. Walter, G. Wolmershäuser, and H. Sitzmann, J. Am. Chem. Soc. 127 (2005) 17494−17503 and references cited there-in. 8. R. Vianello and Z. B. Maksić, Eur. J. Org. Chem. (2004) 5003−5010. 9. P. M. Warner, J. Org. Chem. 71 (2006) 9271−9282. 10. C. S. Wannere, K. W. Sattelmeyer, H. F. Schaefer III, and P. v. R. Schleyer, Angew. Chem., Int. Ed. 43 (2004) 4200−4206.       11 / cos 2 / 0NkkN C πmk N   1cos 2 / 0Nkπmk NJ. F. Liebman and Z. S. Herman, Angular Independence of Orbitals       265    Croat. Chem. Acta 82 (2009) 261. SAŽETAK O kutnoj neovisnosti setova atomskih orbitala Joel F. Liebmana i Zelek S. Hermanb aDepartment of Chemistry and Biochemistry, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 USA bHerman Scientific Consulting, 521 Del Medio Avenue, #107, Mountain View, CA 94040 USA Istraživani su uvjeti pod kojima set atomskih orbitala postaje kutno neovisan, za atomske i molekulske sisteme. Razmatrano je kako se ovi rezultati mogu primjeniti na različite molekule. 

On the Angular Independence of Sets of Atomic Orbitals

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