HYBRID ATOMIC ORBITALS IN ORGANIC CHEMISTRY. PART 1: CRITIQUE OF FORMAL ASPECTS

The importance of hybrid atomic orbitals, both historically and mathematically, is reviewed. Our new analysis of the original derivation of the sp³, sp², sp model reveals serious errors. Based on a critical survey of the literature, we submit six formal criteria that deprecate the use of hybrid orbitals in a pedagogical context. A sound mathematical basis of sp³ and sp² formulae does not exist; hybrid atomic orbitals have hence no legitimate role in the teaching of organic chemistry.

Keywords:
chemical education; undergraduate Organic Chemistry; misconceptions; hybrid orbitals

Authorship

Guy Lamoureux ^**e-mail: guy.lamoureux@ucr.ac.cr

Universidad de Costa Rica, Escuela de Química, San Pedro, San José, Costa RicaUniversidad de Costa RicaCosta RicaSan Pedro, San José, Costa RicaUniversidad de Costa Rica, Escuela de Química, San Pedro, San José, Costa Rica

https://orcid.org/0000-0002-2717-7506

John F. Ogilvie

Department of Mathematics, Centre for Experimental and Constructive Mathematics, Simon Fraser University, British Columbia, CanadaSimon Fraser UniversityCanadaBritish Columbia, CanadaDepartment of Mathematics, Centre for Experimental and Constructive Mathematics, Simon Fraser University, British Columbia, Canada

*e-mail: guy.lamoureux@ucr.ac.cr

SCIMAGO INSTITUTIONS RANKINGS

Figures | Formulas

Figures (1)
Formulas (15)

Thumbnail

Figure 1
Quantitatively accurate plots of hybrid atomic orbitals: a) sp³, b) sp², c) sp; each surface of constant ψ is chosen such that ψ² at that magnitude contains 0.99 of the total electronic density. The scale of each axis in expressed in unit 10⁻¹⁰ m

Thumbnail

Figure 1 Quantitatively accurate plots of hybrid atomic orbitals: a) sp³, b) sp², c) sp; each surface of constant ψ is chosen such that ψ² at that magnitude contains 0.99 of the total electronic density. The scale of each axis in expressed in unit 10⁻¹⁰ m

\begin{matrix} s = 1 \\ p_{x} = \sqrt 3 \sin (θ) \cos (ϕ) \\ p_{y} = \sqrt 3 \sin (θ) \sin (ϕ) \\ p_{z} = \sqrt 3 \cos (θ) \end{matrix}

\begin{matrix} ψ_{1, 0, 0} \\ s = \frac{2^{(\frac{1}{2})} e^{3} µ^{(\frac{3}{2})} Z^{(\frac{3}{2})} π (2 h^{2} ε_{0} - π {Ze}^{2} µ r) e^{(- \frac{π µ {Ze}^{2} r}{2 h^{2} ε_{0}})}}{8 h^{5} ε_{0} (\frac{5}{2})} \end{matrix}

\begin{matrix} ψ_{0, 1, 0} \\ p_{0} = \frac{2^{(\frac{1}{2})} e^{5} µ^{(\frac{5}{2})} Z^{(\frac{5}{2})} π^{2} {re}^{(- \frac{π µ {Ze}^{2} r}{2 h^{2} ε_{0}})} \cos (θ)}{8 h^{5} ε_{0}^{(\frac{5}{2})}} \end{matrix}

\begin{matrix} ψ_{0, 1, 1} \\ p_{1} = \frac{e^{5} µ^{(\frac{5}{2})} Z^{(\frac{5}{2})} π^{2} {re}^{(- \frac{π µ {Ze}^{2} r}{2 h^{2} ε_{0}})} \sin (θ) (\cos (ϕ) + i \sin (ϕ))}{8 h^{5} ε_{0}^{(\frac{5}{2})}} \end{matrix}

\begin{matrix} ψ_{0, 1, - 1} \\ p_{- 1} = \frac{e^{5} µ^{(\frac{5}{2})} Z^{(\frac{5}{2})} π^{2} {re}^{(- \frac{π µ {Ze}^{2} r}{2 h^{2} ε_{0}})} \sin (θ) (\cos (ϕ) - i \sin (ϕ))}{8 h^{5} ε_{0}^{(\frac{5}{2})}} \end{matrix}

p_{x} = \frac{2^{(\frac{1}{2})} e^{5} µ^{(\frac{5}{2})} Z^{(\frac{5}{2})} π^{2} {re}^{(- \frac{π µ {Ze}^{2} r}{2 h^{2} ε_{0}})} \sin (θ) \cos (ϕ)}{8 ε_{0}^{(\frac{5}{2})} h^{5}}

p_{y} = \frac{i 2^{(\frac{1}{2})} e^{5} µ^{(\frac{5}{2})} Z^{(\frac{5}{2})} π^{2} {re}^{(- \frac{π µ {Ze}^{2} r}{2 h^{2} ε_{0}})} \sin (θ) \sin (ϕ)}{8 ε_{0}^{(\frac{5}{2})} h^{5}}

p_{z} = \frac{2^{(\frac{1}{2})} e^{5} µ^{(\frac{5}{2})} Z^{(\frac{5}{2})} π^{2} {re}^{(- \frac{π µ {Ze}^{2} r}{2 h^{2} ε_{0}})} \cos (θ)}{8 ε_{0}^{(\frac{5}{2})} h^{5}}

\begin{matrix} s = 1 \\ p_{x} (θ, ϕ) = \sin (θ) \cos (ϕ) \\ p_{y} (θ, ϕ) = i \sin (θ) \sin (ϕ) \\ p_{z} (θ, ϕ) = \cos (θ) \end{matrix}

e^{(- \frac{π µ {Ze}^{2} r}{2 h^{2} ε_{0}})}

\frac{2 h^{2} ε_{0}}{e^{2} µ Z π} - r

\begin{matrix} {te}_{111} = \frac{1}{2} (s + p_{x} + p_{y} + p_{z}) \\ {te}_{1 - 1 - 1} = \frac{1}{2} (s + p_{x} - p_{y} - p_{z}) \\ {te}_{- 11 - 1} = \frac{1}{2} (s - p_{x} + p_{y} - p_{z}) \\ {te}_{- 1 - 11} = \frac{1}{2} (s - p_{x} - p_{y} + p_{z}) \end{matrix}

\begin{matrix} {te}_{1} = \frac{1}{2} (s + \sqrt 2 p_{y} + p_{z}) \\ {te}_{2} = \frac{1}{2} (s - \sqrt 2 p_{y} + p_{z}) \\ {te}_{3} = \frac{1}{2} (s + \sqrt 2 p_{x} - p_{z}) \\ {te}_{4} = \frac{1}{2} (s - \sqrt 2 p_{x} - p_{z}) \end{matrix}

\begin{matrix} {tr}_{1} = s / \sqrt 3 + \sqrt (2 / 3) p_{x} \\ {tr}_{2} = s / \sqrt 3 - p_{x} / \sqrt 6 + p_{y} / \sqrt 2 \\ {tr}_{3} = s / \sqrt 3 - p_{x} / \sqrt 6 - p_{y} / \sqrt 2 \end{matrix}

\begin{matrix} {te}_{111}, {te}_{1 - 1 - 1}, {te}_{- 11 - 1}, {te}_{- 1 - 11}, or ({sp}^{3} or {sp}^{2} as specified above), \\ {tr}_{111}, {tr}_{11 - 1}, {tr}_{1 - 11}, p_{z}, or ({sp}^{2}) \\ {di}_{11}, {di}_{1 - 1}, p_{x}, p_{y}, or (sp) \\ s, p_{x}, p_{y}, p {}_{z}, \end{matrix}

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HYBRID ATOMIC ORBITALS IN ORGANIC CHEMISTRY. PART 1: CRITIQUE OF FORMAL ASPECTS

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[1] *e-mail: guy.lamoureux@ucr.ac.cr