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Hybrid atomic orbitals in organic chemistry. Part 1: critique of formal aspects |

Guy Lamoureux I. Universidad de Costa Rica, Escuela de Química, San Pedro, San José, Costa Rica Recebido em 13/04/2019 Endereço para correspondência
*e-mail: guy.lamoureux@ucr.ac.cr RESUMO
The importance of hybrid atomic orbitals, both historically and mathematically, is reviewed. Our new analysis of the original derivation of the Palavras-chave: chemical education; undergraduate Organic Chemistry; misconceptions; hybrid orbitals.
In 1931, Pauling The term ‘hybrid atomic orbitals’ and the related process ‘hybridization’ were introduced by Mulliken The use and reliability of hybrid atomic orbitals (which we abbreviate as HAO) has since become challenged. We divide our critique into two parts; the present analysis arose from our endeavor to answer these two relevant scientific questions. In Part 1, does this concept of HAO from the 1930s have mathematical and logical bases? In Part 2, what are the practical problems with this concept as a pedagogical model and how can we overcome these challenges? In this article, we present modern calculations about hybrid atomic orbitals, we provide irrefutable evidence, based on Schrödinger’s time-independent equation for the hydrogen atom, that HAO lack justification, and we list six logical errors and a further critique of the hybridization model.
We here focus our attention on the purported tetrahedral and trigonal hybrid functions of carbon atoms because these are the forms most commonly invoked in organic chemistry, but our analysis and conclusions are applicable equally to other HAO. Digonal or For the purpose of his introduction of hybrid functions, Pauling According to the direct solution (in SI units with conventional symbols, including µ as the reduced mass of the atomic system) of Schrödinger’s temporally independent equation for an atom of atomic number r,θ,ϕ) follow (we retain the relation to Pauling’s nomenclature above; quantum numbers radial k, azimuthal l and equatorial m derive from the three coordinates r,θ,ϕ respectively, with energy quantum number n = k + l + 1 for this system of coordinates).The latter two formulae that contain p_{1} + p_{-1})/√2, we obtain a purely real quantity,whereas a corresponding difference p_{1} − p_{-1})/√2 yields a purely imaginary quantity,with p_{0}, also purely real.According to the latter three expressions, on removing their common factors the remaining parts depending on angular variables θ and ϕ are, including which are the same as Pauling's formulae apart from numerical factors and, particularly notably, the presence of is common to all four original functions, the remaining part of the radial dependence with associated constants is not; instead of just radial distance of which the former term is 2 We proceed to assess the disparities between Pauling's definitions in the set stated above and the expressions in our set obtained directly from the solution of Schrödinger's equation for an atom with one electron. The angular parts of p agree exactly between the two sets, but the angular part of _{z}p _{y}must contain i. Pauling claimed to distinguish correctly the radial parts as R for _{n0}(r)s and R for his three _{n1}(r)p, although he provided neither justification nor evidence of this claim. Without the radial part, the mathematical relation of angular wave functions to the overall ψ is limited._{x}, p_{y}, p_{z}^{11}According to his definitions of sp^{3} orbitals.Functions r, θ, ϕ), have infinite extent in all directions in coordinate space, except that p functions, and others with k or l > 0, have zero value on nodal planes or surfaces separating regions of positive phase from those of negative phase. Like these functions s and p, the hybrid functions have nodal surfaces between regions of positive and negative phase, and they retain formally an infinite extent. For the purpose of attributing an explicit shape to any such function, we might specify a magnitude of its amplitude that is a small fraction of the maximum amplitude at any point in space, and then form a surface of that constant function to be viewed in Cartesian coordinates x,y,z. Whereas that surface for function s becomes hence spherically symmetric, the surfaces for functions p and _{x}, p_{y}p are cylindrically symmetric about the indicated Cartesian axes. In contrast, tetrahedral hybrid functions _{z}te have surfaces cylindrically symmetric about one or other axis that is a body diagonal between opposite corners of a cube of which the origin of coordinates is at its center; functions te have this directional quality whether or not they contain s, but with systematic s content the amplitude is more concentrated along the body diagonal axis on one side of the origin than on the other. The objective of Pauling's construction of these tetrahedral hybrid functions was to obtain a resemblance to the structure of methane: with the atomic nucleus of carbon at the center of a cube, the hydrogen nuclei lie at alternate corners of that cube; the hybrid functions that clearly originate as hydrogen functions might then serve as bond orbitals - more accurately, bond basis functions.With the same definitions of of which the surfaces of the orbitals, according to the same criteria as before, have For trigonally directed hybrid atomic orbitals ( These three hybrid functions, of which p_{0} as above as a fourth function, we then have the above three functions that can make three bonds that are symmetric with respect to that plane.In Figure 1 appear accurate plots of
Figure 1. Quantitatively accurate plots of hybrid atomic orbitals: a) sp, b) ^{3}sp, c) sp; each surface of constant ψ is chosen such that ψ^{2}^{2} at that magnitude contains 0.99 of the total electronic density. The scale of each axis in expressed in unit 10^{−10} m
Many chemists and material scientists use these HAO for a qualitative description of geometric structure and bonding characteristics of molecules of various chemical compounds. The fact that HAO are irrelevant in these cases but that the usage continues indicates the difficulty in eliminating obsolete concepts. Tetrahedral and trigonal hybrid orbitals are open to severe criticism, of which six instances follow. 1. The formation of four real tetrahedral HAO using linear combinations of real and imaginary parts in spherical polar coordinates is mathematically impossible and logically unsound. In discarding √−1 from p can never occur together; whenever _{x}, p_{y}, p_{z}p is rotated to become _{z}p or _{x}p, the remaining two transform into an orthogonal complex couple. The premise that couple _{y}p and _{x}p as real functions is equivalent to a complex couple containing e_{y}^{±iϕ} is false.^{14}2. The combinations of HAO in various sets such as the set described as 3. Although Pauling emphasized the value of 4. Trigonal hybrid functions (to which reference is sometimes made as sp or digonal constitutes a legitimate linear combination of real functions in spherical polar coordinates, but functions having exactly the same geometric properties arise directly in Schrödinger's own solution of his equation for the hydrogen atom in paraboloidal coordinates;^{19} there is no necessity for such an arbitrary linear combination to generate the desired shape.5. If one undertakes a molecular-orbital calculation for CH one obtains exactly the same structure of CH 6. Those solutions of Schrödinger's equation in spherical polar coordinates as presented above, and which were obviously Pauling's inspiration for
During 1955-1956, a thesis criticizing the hybridization model "hybridisation...is consequently shown to be of no physical meaning" was censored and papers based on this work rejected (Pritchard We recall some pertinent quotations from the literature. In
We seek to distinguish clearly between the HAO used in the teaching of general and organic chemistry and the other uses of 'hybrid orbitals' in modern chemistry, in which these orbitals might be implemented within basis sets for these calculations. Whether such basis sets for the calculations comprise atomic orbitals or their combinations in hybrid orbitals as presented above, such functions are artifacts of those particular calculations, and have no meaning outside those contexts. Orbitals are the exact algebraic solutions of the Schrödinger's equation for an atom with one electron, It would be unwise to remove all use of 'hybrids' or 'hybridization' from chemistry; 'hybridization' used in modern calculations is different and much more rigorously defined. For instance, in their work Foster and Weinhold used rigorous algorithms of the natural-bond-orbital (NBO) method to derive natural hybrid orbitals (NHO) that describe the electronic density based on calculated wave functions,
In this article, we seek to convince readers, based on mathematical and logical arguments, that, for the teaching of organic chemistry, hybridization is an obsolete concept; the pioneers (
El Centro de Investigaciones en Productos Naturales (CIPRONA) and la Escuela de Química, Universidad de Costa Rica provided support. We thank many professors and students at UCR for helpful discussion.
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