Chapter 11 - Molecular Structure

• Covalent bond - An electron pair shared between two neighboring atoms.
• Ionic bond - Here the cohesion arises from the Coulombic attraction between ions of opposite charge.
• Major approaches to the calculation of the molecular structure   * Valence bond theory   * Molecular orbital theory

The Born-Oppenheimer approximation

• Born-Oppenheimer approximation - It is supposed that the nuclei, being so much heavier than an electron, move relatively slowly and may be treated as stationary while the electrons move in their field.
• Molecular potential energy curve - Obtained when more than one molecular parameter is changed in a polyatomic molecule.
• Equilibrium bond length (Re) - The internuclear separation at the minimum of the curve.
• The bond dissociation energy (Do) - It’s closely related to the depth of the minimum below the energy of the infinitely widely separated and stationary atoms.

Valence-bond theory

• Valence bond theory - The first quantum mechanical theory of bonding to be developed.

11.1 Homonuclear diatomic molecules

• Bond in valence bond theory - A bond is regarded as forming when an electron in an atomic orbital on one atom pairs its spin with that of an electron in an atomic orbital on another atom.
• σ bond (sigma bond) - The electron distribution described by the wavefunction of this equation. It has a cylindrical symmetry around the internuclear axis. It’s called this because, when viewed along the internuclear axis, it resembles a pair of electrons in an s orbital.

• π bond - It arises from the spin pairing of electrons in two p orbitals that approach side-by-side. It is so called because viewed along the internuclear axis, a π bond resembles a pair of electrons in a p orbital.

11.2 Polyatomic molecules

• Promotion - The excitation of an electron to an orbital of higher energy.

Molecular orbital theory

• Homonuclear diatomic molecules - Formed from two atoms of the same element.
• Heteronuclear diatomic molecules - Diatomic molecules formed from atoms of two different elements.

11.3 The hydrogen molecule-ion

Linear combinations of atomic orbitals

• Linear combination of atomic orbitals (LCAO) - An approximate molecular orbital formed from a linear combination of atomic orbitals.

Bonding orbitals

The total probability density is proportional to the sum of →

• A^2 - The probability density if the electron were confined to the atomic orbital A.
• B^2 - The probability density if the electron were confined to the atomic orbital B.
• 2AB - Extra contribution to the density.   * Overlap density - It represents an enhancement of the probability of finding an electron in the internuclear region.
• Bonding orbital - An orbital which, if occupied, helps to bind two atoms together.
• σ electron - An electron that occupies a σ orbital.

Antibonding orbitals

• Antibonding orbital - An orbital that, if occupied, contributes to a reduction in the cohesion between two atoms and helps to raise the energy of the molecule relative to the separated atoms.
• Inversion symmetry - The behavior of the wavefunction when it is inverted through the center of the molecule.
• Gerade symmetry - An identical value of the wavefunction.
• Ungerade symmetry (Odd symmetry) - Same size but opposite sign of the wavefunction.

Photoelectron spectroscopy

• Photoelectron spectroscopy (PES) - It measures the ionization energies of molecules when electrons are ejected from different orbitals by absorption of a photon of the proper energy, and uses the information to infer the energies of molecular orbitals.
• Koopmans' theorem - States that the ionization energy is equal to the orbital energy of the ejected electron.

11.4 Homonuclear diatomic molecules

• Polar bond - A covalent bond in which the electron pair is shared unequally by the two atoms.
• Variation principle - If an arbitrary wavefunction is used to calculate the energy, the value calculated is never less than the true energy.
• Trial wavefunction - The arbitrary wavefunction.

11.5 Heteronuclear diatomic molecules

Molecular orbitals for polyatomic systems

11.6 The Huckel approximation

Huckel approximations

• All overlap integrals are set equal to zero.
• All resonance integrals between non-neighbors are set equal to zero.
• All remaining resonance integrals are set equal (to β).

11.7 Computational chemistry

• Hartree-Fock equations

      * The Fock operator (f1)

        * The Coulomb operator (J)

        * The exchange operator (K)

• Roothaan equations

      * F → The matrix formed from the Fock operator

        * S → The matrix of overlap integrals

Semi-empirical and ab initio methods

• Semi-empirical methods - Where many of the integrals are estimated by appealing to spectroscopic data or physical properties such as ionization energies, and using a series of rules to set certain integrals equal to zero.
• Ab initio methods - Here an attempt is made to calculate all the integrals that appear in the Fock and overlap matrices.
• The Fock matrix

• Complete neglect of differential overlap (CNDO) - In which all integrals are set to zero unless A and B are the same orbitals centred on the same nucleus, and likewise for C and D.
• Kohn-Sham equations - They are found by applying the variation principle to the electron energy.

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11.8 The prediction of molecular properties

Electron density and the electrostatic potential surfaces

• Isodensity surface - A surface of constant total electron density.
• Solvent-accessible surface - Here the shape represents the shape of the molecule by imagining a sphere representing a solvent molecule rolling across the surface and plotting the locations of the center of that sphere.
• Electrostatic potential surface (An 'elpot surface') - In which net positive charge is shown in one color and net negative charge is shown in another, with intermediate gradations of color.

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