Molecular Orbitals Molecular Orbitals An approach to bonding in which orbitals An approach to bonding in which orbitals encompass the entire molecule, rather than encompass the entire molecule, rather than being localized between atoms. being localized between atoms.
Molecular Orbitals Molecular Orbitals Molecular orbitals result from the Molecular orbitals result from the combination of atomic orbitals. combination of atomic orbitals. Since orbitals are wave functions, they can Since orbitals are wave functions, they can combine either constructively (forming a combine either constructively (forming a bonding molecular orbital), or destructively bonding molecular orbital), or destructively (forming an antibonding molecular orbital). (forming an antibonding molecular orbital).
Molecular Orbitals Molecular Orbitals Molecular orbitals form when atomic orbitals Molecular orbitals form when atomic orbitals with similar energies and proper symmetry can with similar energies and proper symmetry can overlap. overlap. Atomic orbitals with differing energies or the Atomic orbitals with differing energies or the wrong spatial orientation (orthogonal) do not wrong spatial orientation (orthogonal) do not combine, and are called non-bonding non-bonding orbitals. orbitals. combine, and are called
Molecular Orbital Theory Molecular Orbital Theory In order to simplify things, we’ll consider the In order to simplify things, we’ll consider the interaction of the orbitals containing valence interaction of the orbitals containing valence electrons to create molecular orbitals. electrons to create molecular orbitals. The wave functions of hydrogen atom A and The wave functions of hydrogen atom A and hydrogen atom B can interact either hydrogen atom B can interact either constructively or destructively. constructively or destructively.
Linear combination of atomic orbitals Linear combination of atomic orbitals Rules for linear combination Rules for linear combination 1. Atomic orbitals must be roughly of the same energy. 2. The orbital must overlap one another as much as possible- atoms must be close enough for effective overlap. 3. In order to produce bonding and antibonding MOs, either the symmetry of two atomic orbital must remain unchanged when rotated about the internuclear line or both atomic orbitals must change symmetry in identical manner.
Typical molecular energy levels diagram of an octahedral complex showing the frontier orbitals in the tinted box Singly degenerate s a 1g Triply degenerate p t 1u Doubly degenerate d e g Triply degenerate d t 2g
B g - identical under inversion A u - not identical
Rules for the use of MOs Rules for the use of MOs * When two AOs mix, two MOs will be produced * Each orbital can have a total of two electrons (Pauli principle) * Lowest energy orbitals are filled first (Aufbau principle) * Unpaired electrons have parallel spin (Hund’s rule) Bond order = ½ (bonding electrons – antibonding electrons)
Molecular Orbital Theory Molecular Orbital Theory Constructively: Constructively: φ (1s + φ φ (1s [ φ ) + Ψ ( or Ψ Ψ + = ( 1/√2 ) ] Ψ ) or + = ( 1/√2 ) [ ) ] (1s a (1s b (σ σ) a ) b ) Destructively: Destructively: φ (1s - φ φ (1s [ φ ) - Ψ ( or Ψ Ψ - = ( 1/√2 ) ] Ψ *) or - = ( 1/√2 ) [ ) ] (1s a (1s b (σ σ*) a ) b )
Constructive interference Constructive interference + . . +. +. ψ g bonding c A = c B = 1 ψ g = N [ ψ A + ψ B ] Amplitudes of wave functions added
The accumulation of electron density between the nuclei put the electron in a position where it interacts strongly with both nuclei. Nuclei are shielded from each other The energy of the molecule is lower
node . . - +. -. + ψ u c A = +1, c B = -1 antibonding ψ u = N [ ψ A - ψ B ] Destructive interference Destructive interference Nodal plane perpendicular to the H-H bond Nodal plane perpendicular to the H-H bond axis (en density = 0) axis (en density = 0) Energy of the en in this orbital is higher. Energy of the en in this orbital is higher. Ψ A- Ψ B Amplitudes of wave functions subtracted.
The electron is excluded from internuclear region The electron is excluded from internuclear region destabilizing destabilizing Antibonding Antibonding
When 2 atomic When 2 atomic orbitals orbitals combine there are 2 combine there are 2 resultant resultant orbitals orbitals. Eg. s Eg . s orbitals orbitals σ * 1 s E high energy high energy antibonding antibonding orbital orbital 1s 1s b a σ 1 s Molecular Molecular orbitals orbitals low energy bonding orbital low energy bonding orbital
Molecular Orbital Theory Molecular Orbital Theory The bonding orbital is The bonding orbital is + - sometimes given the sometimes given the notation σ σ g , where the g g notation g , where the stands for gerade gerade , or , or stands for + symmetric with respect symmetric with respect to a center of inversion. to a center of inversion. The signs on the molecular orbitals indicate the sign of the wave function, not ionic charge.
Molecular Orbital Theory Molecular Orbital Theory The anti-bonding orbital The anti-bonding orbital + - is sometimes given the is sometimes given the notation σ σ u , where the u u notation u , where the stands for ungerade ungerade , or , or stands for + asymmetric with respect asymmetric with respect to a center of inversion. to a center of inversion. The signs on the molecular orbitals indicate the sign of the wave function, not ionic charge.
H 2 H 2 11.4 eV LCAO of n A.O ⇒ n M.O. 109 nm Location of Bonding orbital 4.5 eV
Period 2 Diatomic Molecules Period 2 Diatomic Molecules For the second period, assume that, due to a For the second period, assume that, due to a better energy match, s s orbitals combine with orbitals combine with s s better energy match, orbitals, and p p orbitals combine with orbitals combine with p p orbitals. orbitals. orbitals, and The symmetry of p p orbitals permits end-on- orbitals permits end-on- The symmetry of end overlap along the bond axis, or side-by-side end overlap along the bond axis, or side-by-side overlap around, but not along, the internuclear overlap around, but not along, the internuclear axis. axis.
dx 2 -dy 2 and d xy 2- Cl 4 Re ReCl 4
B g - identical under inversion A u - not identical
MOs using p p orbitals orbitals MOs using σ u - - + + - + - σ g Some texts will use the symmetry designations of g g (gerade) or (gerade) or u u Some texts will use the symmetry designations of (ungerade) instead of indicating bonding or anti-bonding. (ungerade) instead of indicating bonding or anti-bonding. For these orbitals, the anti-bonding orbital is asymmetric about the For these orbitals, the anti-bonding orbital is asymmetric about the bond axis, and is designated as σ σ u . Note that the designations of bond axis, and is designated as u . Note that the designations of u or or g g do not do not correlate with bonding or anti-bonding. correlate with bonding or anti-bonding. u
π Molecular Orbitals Molecular Orbitals π + - - + + side-by-side - overlap The orbital overlap side-by-side is less than The orbital overlap side-by-side is less than that of overlap along the bond axis (end-on- that of overlap along the bond axis (end-on- end). As a result, the bonding orbital will be end). As a result, the bonding orbital will be higher in energy than the previous example. higher in energy than the previous example.
Molecular Orbital Diagram Molecular Orbital Diagram This is a molecular This is a molecular σ u orbital energy level orbital energy level diagram for the p p diagram for the π g orbitals. Note that the orbitals. Note that the 2p 2p π u σ bonding orbital is bonding orbital is σ σ g lowest in energy due to lowest in energy due to the greater overlap the greater overlap end-on-end. end-on-end.
g or u in this diagram Place labels g Place labels or u in this diagram σ∗ u π∗ g π u σ g
Molecular Orbital Diagrams Molecular Orbital Diagrams Electrons preferentially occupy molecular Electrons preferentially occupy molecular 1. 1. orbitals that are lower in energy. orbitals that are lower in energy. Molecular orbitals may be empty, or contain Molecular orbitals may be empty, or contain 2. 2. one or two electrons. one or two electrons. If two electrons occupy the same molecular If two electrons occupy the same molecular 3. 3. orbital, they must be spin paired. orbital, they must be spin paired. When occupying degenerate molecular When occupying degenerate molecular 4. 4. orbitals, electrons occupy separate orbitals orbitals, electrons occupy separate orbitals with parallel spins before pairing. with parallel spins before pairing.
Molecular Orbital Diagrams Molecular Orbital Diagrams Although molecular orbitals form from inner Although molecular orbitals form from inner (core) electrons as well as valence electrons, (core) electrons as well as valence electrons, many molecular orbital diagrams include only many molecular orbital diagrams include only the valence level. the valence level.
First period diatomic molecules First period diatomic molecules σ 1s 2 H H 2 H σ u * Bond order: 1 Energy 1s 1s σ g Bond order = ½ (bonding electrons – antibonding electrons)
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