Molecular structure 2.1 The Born-Oppenheimer approximation
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1 Molecular structure 2.1 The Born-Oppenheimer approximation All theories of molecular structure make the simplification that the nuclei are treated as stationary while the electrons move in their field (Born-Oppenheimer approximation. Nuclei are much heavier than electrons and move much slower). The SE is then solved for a fixed distance of the nuclei. Plotting the molecular energy versus the bond length results in a molecular potential energy curve. It allows it to identify the equilibrium bond length and the bond dissociation energy Potential energy curve A molecular potential energy curve. The equilibrium bond length R e corresponds to the energy minimum. The dissociation energy is related to D e. 1
2 2.2 Valence bond (VB) theory VB theory was the first quantum mechanical theory of bonding to be developed. It has undergone much less computational development than molecular orbital theory. It includes concepts such as spin pairing, s and p bonds, and hybridization. That language is particularly widespread in the description of the properties and reactions of organic compounds Wavefunction and electron density The hydrogen molecule has two electrons and two protons. Wavefunction: =A(1)B(2) and =A(2)B(1) are equally valid descriptions (electrons are indistinguishable). Valence bond wavefunctions refer to two electrons simultaneously. The illustration represents the atomic orbital for electron 1 in black and that of electron 2 in blue. 2
3 2.2.1 Wavefunction and electron density A better description is a superposition of both states: =A(1)B(2)±A(2)B(1) The combination with lower energy is the one with a + sign, so the valence bond function is =A(1)B(2)+A(2)B(1) (enhanced electron density between the nuclei) Interference between the black and the blue contributions results in an enhanced (two-electron) density in the internuclear region Molecular potential energy curve The molecular potential energy curve for the hydrogen molecule showing the variation of the energy of the molecule as the bond length is changed. The calculated curve refers to the valence-bond model. 3
4 2.2.3 The role of electron spin The overall wavefunction, (1,2)=(A(1)B(2)+A(2)B(1))s(1,2) must be antisymmetric, i.e., (1,2) = - (2,1). The only s(1,2) fulfilling this is the one corresponding to paired electron spins s a (1,2). The electrons establish a so called s bond (cylindrical symmetry around the internuclear axis). The orbital overlap and spin pairing between electrons in two collinear p orbitals that result in the formation of a s bond. 2.3 Homonuclear diatomic molecules Essential features of VB theory: pairing of the electrons and accumulation of electron density in the internuclear region. This can be applied to homonuclear diatomic molecules. Example: N 2. Valence electron configuration: 2s 2 2p x1 2p y1 2p z 1 Convention: the internuclear axis is the z-axis. The two 2p z orbitals then point toward each other and establish a s bond. The remaining 2p orbitals do not have cylindrical symmetry around the internuclear axis. They merge to form two p bonds. 4
5 2.3.1 p-bonds A p-bond results from orbital overlap and spin pairing between electrons in p orbitals with their axes perpendicular to the internuclear axis. The bond has two lobes of electron density separated by a nodal plane Nitrogen molecule The structure of bonds in a nitrogen molecule: there is one s bond and two p bonds. The overall electron density has cylindrical symmetry around the internuclear axis. 5
6 2.4 Polyatomic molecules s bonds are formed by spin pairing of electrons in atomic orbitals with cylindrical symmetry about the internuclear axis p bonds are formed by pairing electrons occupying orbitals of the appropriate symmetry. Example: H 2 O valence electron configuration of O: 2s 2 2p x2 2p y1 2p z 1 The two unpaired 2p electrons can pair with the 1s electron of H to establish a s bond Water molecule A first approximation to the valence-bond description of bonding in an H 2 O molecule. Each s bond arises from the overlap of an H1s orbital with one of the O2p orbitals. This model suggests that the bond angle should be 90 o, which is significantly different from the experimental value. 6
7 2.4.2 Promotion Problem: Valence configuration of C is 2s 2 2p x1 2p y 1 predicting that carbon should be capable of forming only two bonds not four. So what about CH 4 for example? This problem can be solved within the model of promotion and hybridization: One 2s electron is promoted to a 2p orbital resulting in 2s 1 2p x1 2p y1 2p z1. This configuration would still predict one distinct bond established by the 2s electron Hybridization in methane The concept of hybridization solves this contradiction. The four orbitals establish four equivalent so called sp 3 orbitals by specific linear combination, which cannot be distinguished. tetrahedral symmetry. An sp 3 hybrid orbital formed from the superposition of s and p orbitals on the same atom.there are four such hybrids: each one points towards the corner of a regular tetrahedron. The overall electron density remains spherically symmetrical. Superposition of wave functions 7
8 2.4.3 sp 3 hybrid orbitals A more detailed representation of the formation of an sp 3 hybrid by interference between wavefunctions centred on the same atomic nucleus. (To simplify the representation, the radial node of the 2s orbital has been ignored) Methane molecule Each sp 3 hybrid orbital forms a s bond by overlap with an H1s orbital located at the corner of the tetrahedron. This model accounts for the equivalence of the four bonds in CH 4. 8
9 2.4.5 Hybridization in ethene To describe the structure of an ethene molecule each C atom is promoted to a 2s 1 2p 3 configuration. The superposition of an s orbital and two p orbitals then leads to sp 2 hybrid orbitals. (a) An s orbital and two p orbitals can be hybridized to form three equivalent orbitals that point towards the corners of an equilateral triangle. (b) The remaining unhybridized p orbital is perpendicular to the plane Ethene A representation of the structure of a double bond in ethene; only the p bond is shown explicitly. 9
10 2.4.7 Ethyne sp hybridization A representation of the structure of a triple bond in ethyne; only the p bond is shown explicitly. The overall electron density has cylindrical symmetry around the axis of the molecule Hybridization schemes 10
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