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Molecular OrbitalsMolecular 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 OrbitalsMolecular 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 OrbitalsMolecular 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 combine, and are called non-bondingnon-bonding orbitals. orbitals.
Molecular Orbital TheoryMolecular 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.
Rules for linear combinationRules 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.
Linear combination of atomic orbitalsLinear combination of atomic orbitals
Typical molecular energy levels diagram of an octahedral complex showing the frontier orbitals in the tinted box
Singly degenerate s a1g
Triply degenerate p t1u
Doubly degenerate d eg
Triply degenerate d t2g
A
Bg- identical under inversion
u- not identical
Rules for the use of MOsRules 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 TheoryMolecular Orbital Theory
Constructively:Constructively:
ΨΨ((σσ)) or or ΨΨ++ = ( = (1/√2 ) 1/√2 ) [[φφ(1s(1saa) ) + + φφ(1s(1sbb) ) ]]
Destructively:Destructively:
ΨΨ((σσ*)*) or or ΨΨ-- = ( = (1/√2 ) 1/√2 ) [[φφ(1s(1saa) ) - - φφ(1s(1sbb) ) ]]
cA = cB = 1
+. +. . .+
bondingψg
Amplitudes of wave functions added
ψg = N [ψA + ψB]
Constructive interferenceConstructive interference
The accumulation of electron density between the nuclei put the electron in a position where it interacts strongly with both nuclei.
The energy of the molecule is lower
Nuclei are shielded from each other
Amplitudes of wave functions
subtracted.
Destructive interferenceDestructive interferenceNodal 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.
+. -. ..
node
antibondingψu = N [ψA - ψB]
cA = +1, cB = -1 ψu
+ -
ΨA-ΨB
The electron is excluded from internuclear region The electron is excluded from internuclear region destabilizing destabilizing
AntibondingAntibonding
When 2 atomic When 2 atomic orbitalsorbitals combine there are 2 combine there are 2resultant resultant orbitalsorbitals..
low energy bonding orbitallow energy bonding orbital
high energy high energy antibondingantibonding orbital orbital1sb 1sa
σ1s
σ*
E1s
MolecularMolecular orbitalsorbitals
EgEg. s . s orbitalsorbitals
Molecular Orbital TheoryMolecular Orbital Theory
The bonding orbital is The bonding orbital is sometimes given the sometimes given the notation notation σσgg, where the , where the gg
stands for stands for geradegerade, or , or 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 TheoryMolecular Orbital Theory
The anti-bonding orbital The anti-bonding orbital is sometimes given the is sometimes given the notation notation σσuu, where the , where the u u
stands for stands for ungeradeungerade, or , or 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.
11.4 eV
109 nm
HH22
Location ofBonding orbital4.5 eV
LCAO of n A.O ⇒ n M.O.
Period 2 Diatomic MoleculesPeriod 2 Diatomic Molecules
For the second period, assume that, due to a For the second period, assume that, due to a better energy match, better energy match, ss orbitals combine with orbitals combine with ss orbitals, and orbitals, and pp orbitals combine with orbitals combine with pp orbitals. orbitals.
The symmetry of The symmetry of pp orbitals permits end-on- orbitals permits end-on-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.
dx2-dy2 and dxy
Cl4Re ReCl42-
A
Bg- identical under inversion
u- not identical
MOs using MOs using pp orbitals orbitals
Some texts will use the symmetry designations of Some texts will use the symmetry designations of gg (gerade) or (gerade) or uu (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 bond axis, and is designated as σσuu. Note that the designations of . Note that the designations of uu or or gg do notdo not correlate with bonding or anti-bonding. correlate with bonding or anti-bonding.
+
++ -
--
-
σg
σu
ππ Molecular Orbitals Molecular Orbitals
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.
side-by-side overlap
++
+
--
-
Molecular Orbital DiagramMolecular Orbital Diagram
This is a molecular This is a molecular orbital energy level orbital energy level diagram for the diagram for the pp orbitals. Note that the orbitals. Note that the σσ bonding orbital is bonding orbital is lowest in energy due to lowest in energy due to the greater overlap the greater overlap end-on-end.end-on-end.
2p 2p
σg
πu
πg
σu
Place labels Place labels gg or or uu in this diagram in this diagram
σg
π∗g
σ∗u
πu
Molecular Orbital DiagramsMolecular Orbital Diagrams
1.1. Electrons preferentially occupy molecular Electrons preferentially occupy molecular orbitals that are lower in energy.orbitals that are lower in energy.
2.2. Molecular orbitals may be empty, or contain Molecular orbitals may be empty, or contain one or two electrons.one or two electrons.
3.3. If two electrons occupy the same molecular If two electrons occupy the same molecular orbital, they must be spin paired.orbital, they must be spin paired.
4.4. When occupying degenerate molecular When occupying degenerate molecular orbitals, electrons occupy separate orbitals orbitals, electrons occupy separate orbitals with parallel spins before pairing.with parallel spins before pairing.
Molecular Orbital DiagramsMolecular 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 moleculesFirst period diatomic molecules
σ1s2HE
nerg
yHH2
1s 1s
σg
σu*
Bond order = ½ (bonding electrons – antibonding electrons)
Bond order: 1
σ1s2, σ*1s2 HeE
nerg
yHeHe2
1s 1s
σg
σu*
Molecular Orbital theory is powerful because it allows us to predict whether molecules should exist or not and it gives us a clear picture of the of the electronic structure of any hypothetical molecule that we can imagine.
Diatomic molecules: The bonding in He2
Bond order: 0
Second period diatomic moleculesSecond period diatomic molecules
σ1s2, σ*1s2, σ2s2
Bond order: 1
Li
Ene
rgy
LiLi2
1s 1s
1σg
1σu*
2s 2s
2σg
2σu*
σ1s2, σ*1s2, σ2s2,
σ*2s2
Bond order: 0
Be
Ene
rgy
BeBe2
1s 1s
1σg
1σu*
2s 2s
2σg
2σu*
Diatomic molecules: Homonuclear Molecules of the Second Period
SimplifiedSimplified
SimplifiedSimplified
Diamagnetic??
2σg
2σu*
3σg
1πu
1πg*
3σu*
MO diagram for BMO diagram for B22
Li : 200 kJ/molF: 2500 kJ/mol
Same symmetry, energy mix-the one with higher energy moves higher and the one with lower energy moves lower
2σg
2σu*
3σg
1πu
1πg*
3σu*
B BB2
2s 2s
2σg
2σu*
2p
2p
3σg
3σu*
1πu
1πg*
(px,py)
HOMO
LUMO
MO diagram for BMO diagram for B22
Paramagnetic
1σg
1πu
1πg
1σ g
1πu
1π g
C2
DiamagneticParamagnetic ?X
1σg
1πu
1πg
1σg
1πu
1πg
Li2 to N2O2 and F2
General MO diagrams
Distance between b-MO and AO
Heteronuclear Diatomics….
� The energy level diagram is not symmetrical. � The bonding MOs are
closer to the atomic orbitals which are lower in energy.
� The antibonding MOs are closer to those higher in energy.
c – extent to which each atomicorbitals contribute to MO
If cA>cB the MO is composed principally of φA
Rules for Combining Atomic Rules for Combining Atomic OrbitalsOrbitals
For heteronuclear molecules:For heteronuclear molecules:
1. The bonding orbital(s) will reside 1. The bonding orbital(s) will reside predominantly on the atom of lower orbital predominantly on the atom of lower orbital energy (the more electronegative atom).energy (the more electronegative atom).
2. The anti-bonding orbital(s) will reside 2. The anti-bonding orbital(s) will reside predominantly on the atom with greater orbital predominantly on the atom with greater orbital energy (the less electronegative atom).energy (the less electronegative atom).
HFHF
The 2s and 2pThe 2s and 2pxx orbitals orbitals
on fluorine interact with on fluorine interact with the 1s orbital on hydrogen. the 1s orbital on hydrogen.
The pThe pyy and p and pzz orbitals orbitals
on fluorine lack proper on fluorine lack proper symmetry to interact with symmetry to interact with hydrogen, and remain as hydrogen, and remain as non-bonding orbitals.non-bonding orbitals.
HFHF
The anti-bonding The anti-bonding orbital resides primarily on orbital resides primarily on the less electronegative the less electronegative atom (H).atom (H).
Note that the Note that the subscripts subscripts gg and and uu are not are not used, as the molecule no used, as the molecule no longer has a center of longer has a center of symmetry.symmetry.
Carbon monoxideCarbon monoxide
In carbon In carbon monoxide, the bonding monoxide, the bonding orbitals reside more on orbitals reside more on the oxygen atom, and the oxygen atom, and the anti-bonding the anti-bonding orbitals reside more on orbitals reside more on the carbon atom. the carbon atom.
Carbon monoxideCarbon monoxide
CO is a highly CO is a highly reactive molecule with reactive molecule with transition metals. transition metals. Reactivity typically Reactivity typically arises from the arises from the hhighest ighest ooccupied ccupied mmolecular olecular oorbital (HOMO), when rbital (HOMO), when donating electrons.donating electrons.
Carbon monoxideCarbon monoxide
When acting as an When acting as an electron pair acceptor, electron pair acceptor, the the llowest owest uunoccupied noccupied mmolecular olecular oorbital (LUMO), is rbital (LUMO), is significant.significant.
Carbon monoxideCarbon monoxide
When acting as an When acting as an electron pair donor, electron pair donor, the the hhighest ighest ooccupied ccupied mmolecular olecular oorbital rbital (HOMO), is (HOMO), is significant.significant.
The highest occupied molecular orbital of CO is a molecular orbital which puts significant electron density on the carbon atom.
The lowest unoccupied molecular orbital of CO is the π* orbitals. The lobes of the LUMO are larger on the carbon atom than on the oxygen atom.
CO as a LigandCO as a Ligand
Carbon monoxide is known as a Carbon monoxide is known as a σσ donor and donor and a a ππ acceptor ligand. It donates electrons from its acceptor ligand. It donates electrons from its HOMO to form a sigma bond with the metal.HOMO to form a sigma bond with the metal.
CO as a LigandCO as a Ligand
Carbon monoxide accepts electrons from Carbon monoxide accepts electrons from filled filled dd orbitals on the metal into its antibonding orbitals on the metal into its antibonding (LUMO) orbital.(LUMO) orbital.
CO as a LigandCO as a Ligand
This phenomenon is called This phenomenon is called back bondingback bonding. The . The increased electron density in the antibonding orbitals of increased electron density in the antibonding orbitals of CO causes an increase in the C-O bond length and a CO causes an increase in the C-O bond length and a decrease in its stretching frequency.decrease in its stretching frequency.
MOs for Larger MoleculesMOs for Larger Molecules
Group theory is usually used to develop Group theory is usually used to develop molecular orbital diagrams and drawings of molecular orbital diagrams and drawings of more complicated molecules. When a central more complicated molecules. When a central atom is bonded to several atoms of the same atom is bonded to several atoms of the same element (Helement (H22O, BFO, BF33, or PtCl, or PtCl44
2-2-], group theory can ], group theory can
be used to analyze the symmetry of the orbitals be used to analyze the symmetry of the orbitals of the non-central atoms, and then combine of the non-central atoms, and then combine them with the appropriate orbitals of the central them with the appropriate orbitals of the central atom.atom.
MOs for Larger MoleculesMOs for Larger Molecules
The orbitals of the non-central atoms are The orbitals of the non-central atoms are called called group orbitalsgroup orbitals. In considering a simple . In considering a simple example, Hexample, H22O, we obtain group orbitals using O, we obtain group orbitals using
the two the two 1s1s orbitals on the hydrogen atoms. orbitals on the hydrogen atoms.
Group Orbitals of WaterGroup Orbitals of Water
The AThe A11 representation has both representation has both 1s1s orbitals with orbitals with
positive wave functions: Hpositive wave functions: Haa+H+Hbb..
The BThe B11 representations is H representations is Haa+H+Hbb..
Molecular Orbitals of WaterMolecular Orbitals of Water
Since the Since the 2p2pyy orbital on oxygen doesn’t match orbital on oxygen doesn’t match
the symmetry of the group orbitals of hydrogen, the symmetry of the group orbitals of hydrogen, it will remain non-bonding. The other orbitals it will remain non-bonding. The other orbitals on oxygen will combine with the appropriate on oxygen will combine with the appropriate group orbitals to form bonding and antibonding group orbitals to form bonding and antibonding molecular orbitals.molecular orbitals.
4p
4s
3d
x2-y2 z2 xy xz yzNB MOs
Bonding MOs
Six donor orbitals
6NH3 each donating 2 e-s
Antibonding MOs
Atomic orbitals in metal ion Atomic orbitals in ligand ionMolecular orbitals
Molecular Orbital diagram for [CoIII(NH3)6]3+
∆ο
Oh σ bonding
4p
4s
3d
x2-y2 z2 xy xz yz NB MOs
Bonding MOs
Six donor orbitals
6 F- each donating 2 e-s
Antibonding MOs
Atomic orbitals in metal ion Atomic orbitals in ligand ionMolecular orbitals
Molecular Orbital diagram for CoF63-
∆ο
Oh σ bonding
Clearly good σ donor ligandResult in good M-L overlap Strongly antibonding eg set
t2g
eg
t2g
ML6
σ-onlyML6
σ + π
Stabilization
(empty π-orbitals on ligands)
∆o∆’o ∆o has increased
Case 1 (CN-, CO, C2H4)empty π-orbitals on the ligandsM→L π-bonding (π-back bonding)
t2g (π)
t2g (π*)
eg
t2g
eg
t2g
ML6
σ-onlyML6
σ + π
Case 2 (Cl-, F-) filled π-orbitals on the ligandsL→ M π-bonding
(filled π-orbitals)
Stabilization
Destabilization
t2g (π)
t2g (π*)
eg
∆’o
∆o∆ o has decreased
Strong field Weak field
Putting it all on one diagram.
Summary:strong σ- or π-donor weak field ligands.
π-acceptors strong field ligands.
π donor ligands lower in E than t2g.
π acceptor ligands higher in E than t2g.
Or