molecular orbitals

Molecular Molecular OrbitalsOrbitals

An approach to bonding in An approach to bonding in which orbitals encompass the which orbitals encompass the entire molecule, rather than entire molecule, rather than

being localized between being localized between atoms.atoms.

Muhammad Asif Pakistan

Molecular OrbitalsMolecular Orbitals

Molecular orbitals result from Molecular orbitals result from the combination of atomic orbitals. the combination of atomic orbitals.

Since orbitals are wave Since orbitals are wave functions, they can combine either functions, they can combine either constructively (forming a bonding constructively (forming a bonding molecular orbital), or destructively molecular orbital), or destructively (forming an antibonding molecular (forming an antibonding molecular orbital).orbital).


Molecular OrbitalsMolecular Orbitals

Molecular orbitals form when Molecular orbitals form when atomic orbitals with similar energies atomic orbitals with similar energies and proper symmetry can overlap.and proper symmetry can overlap.

Atomic orbitals with differing Atomic orbitals with differing energies or the wrong spatial energies or the wrong spatial orientation (orthogonal) do not orientation (orthogonal) do not combine, and are called combine, and are called non-bondingnon-bonding orbitals.orbitals.


Need for MO TheoryNeed for MO Theory

Valence bond theory fails to Valence bond theory fails to explain the bonding in many simple explain the bonding in many simple molecules. molecules.

The oxygen molecule has a bond The oxygen molecule has a bond length and strength consistent with length and strength consistent with a double bond, and it contains two a double bond, and it contains two unpaired electrons.unpaired electrons.



Valence bond theory predicts Valence bond theory predicts the double bond, but not the the double bond, but not the paramagnetism of oxygen.paramagnetism of oxygen.

O=OO=O:

::

:



Resonance is another example Resonance is another example of the limitations of valence bond of the limitations of valence bond theory. Bond lengths and strengths theory. Bond lengths and strengths are intermediate between single, are intermediate between single, double or triple bonds. double or triple bonds.

Molecular orbital theory is often Molecular orbital theory is often a better approach to use with a better approach to use with molecules that have extended molecules that have extended ππ systems.systems.


Molecular Orbital TheoryMolecular Orbital Theory

In order to simplify things, we’ll In order to simplify things, we’ll consider the interaction of the consider the interaction of the orbitals containing valence electrons orbitals containing valence electrons to create molecular orbitals.to create molecular orbitals.

The wave functions of hydrogen The wave functions of hydrogen atom A and hydrogen atom B can atom A and hydrogen atom B can interact either constructively or interact either constructively or destructively.destructively.



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) ) ]]



The bonding The bonding orbital results in orbital results in increased electron increased electron density between density between the two nuclei, and the two nuclei, and is of lower energy is of lower energy than the two than the two separate atomic separate atomic orbitals.orbitals.



The The antibonding orbital antibonding orbital results in a node results in a node between the two between the two nuclei, and is of nuclei, and is of greater energy greater energy than the two than the two separate atomic separate atomic orbitals.orbitals.


Molecular Orbital TheoryMolecular Orbital TheoryThe result is an The result is an

energy level diagram energy level diagram with the bonding with the bonding orbital occupied by a orbital occupied by a pair of electrons. pair of electrons. The filling of the The filling of the lower molecular lower molecular orbital indicates that orbital indicates that the molecule is stable the molecule is stable compared to the two compared to the two individual atoms.individual atoms.



The bonding The bonding orbital is orbital is sometimes given sometimes given the notation the notation σσgg, , where the where the gg stands stands for for geradegerade, or , or symmetric with symmetric with respect to a center respect to a center of inversion. of inversion.

+

+

-

The signs on the molecular orbitals indicate the sign of the wave function, not ionic charge.



The anti-bonding The anti-bonding orbital is orbital is sometimes given sometimes given the notation the notation σσuu, , where the where the u u stands stands for for ungeradeungerade, or , or asymmetric with asymmetric with respect to a center respect to a center of inversion. of inversion.

+

+

-

The signs on the molecular orbitals indicate the sign of the wave function, not ionic charge.


Rules for Combining Rules for Combining Atomic OrbitalsAtomic Orbitals

1.1. The number of molecular orbitals = The number of molecular orbitals = the number of atomic orbitals the number of atomic orbitals combined.combined.

2.2. The strength of the bond depends The strength of the bond depends upon the degree of orbital overlap.upon the degree of orbital overlap.


Experimental EvidenceExperimental Evidence

Photoelectron spectroscopy Photoelectron spectroscopy (PES) is a technique in which a beam (PES) is a technique in which a beam of ultraviolet light with an energy of of ultraviolet light with an energy of 21 eV is used to irradiate molecules. 21 eV is used to irradiate molecules.

The energy is high enough to The energy is high enough to eject electrons. The kinetic energy eject electrons. The kinetic energy of the emitted electrons is of the emitted electrons is measured, and used to determine measured, and used to determine the energy level of the electron.the energy level of the electron.



The technique The technique allows for the allows for the measurement of measurement of specific ionization specific ionization energies (I). Each energies (I). Each ionization energy ionization energy represents the represents the removal of an removal of an electron from a electron from a specific molecular specific molecular orbital.orbital.



Electrons Electrons in lower energy in lower energy levels require levels require more energy to more energy to be removed, and be removed, and are ejected with are ejected with less kinetic less kinetic energy. energy.

hhννoo = I + E = I + Ekinetickinetic


Period 2 Diatomic Period 2 Diatomic MoleculesMolecules

For the second period, assume For the second period, assume that, due to a better energy match, that, due to a 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 orbitals permits end-on-end overlap along permits end-on-end overlap along the bond axis, or side-by-side the bond axis, or side-by-side overlap around, but not along, the overlap around, but not along, the internuclear axis.internuclear axis.


MOs using MOs using pp orbitals orbitals

With the With the xx axis as the bond axis, the axis as the bond axis, the ppxx orbitals may combine constructively or orbitals may combine constructively or destructively. The result is a destructively. The result is a σσ bonding bonding orbital and a orbital and a σσ anti-bonding orbital. anti-bonding orbital.

+

++ -

--

-



The designation The designation σσ indicates indicates symmetric electron density around the symmetric electron density around the internuclear (internuclear (xx) axis. The + and – signs ) axis. The + and – signs indicate the sign of the wave function, indicate the sign of the wave function, and not electrical charges.and not electrical charges.

+

++ -

--

-



Some texts will use the Some texts will use the symmetry designations of symmetry designations of gg (gerade) (gerade) or or uu (ungerade) instead of indicating (ungerade) instead of indicating bonding or anti-bonding. bonding or anti-bonding.

+

++ -

--

-



For these orbitals, the bonding For these orbitals, the bonding orbital is orbital is geradegerade, or symmetric , or symmetric around the bond axis.around the bond axis.

+

++ -

--

-

σg



For these orbitals, the anti-bonding For these orbitals, the anti-bonding orbital is asymmetric about the bond orbital is asymmetric about the bond axis, and is designated as axis, and is designated as σσuu. Note that . Note that the designations of 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 The orbital overlap side-by-side is less than that of overlap along the bond axis than that of overlap along the bond axis (end-on-end). As a result, the bonding (end-on-end). As a result, the bonding orbital will be higher in energy than the orbital will be higher in energy than the previous example.previous example.

side-by-side overlap

+

+

+

-

-

-



ππ orbitals are asymmetric with respect orbitals are asymmetric with respect to the bond axis. There is electron density to the bond axis. There is electron density surrounding the bond axis, with a node surrounding the bond axis, with a node along the internuclear axis.along the internuclear axis.


+

+

+

-

-

-



Some texts use the subscripts Some texts use the subscripts gg and and uu instead of bonding and anti-bonding. In instead of bonding and anti-bonding. In this example, the bonding orbital is this example, the bonding orbital is ungerade, or asymmetric about a center of ungerade, or asymmetric about a center of symmetry.symmetry.


+

+

+

-

-

-πu



The anti-bonding orbital is The anti-bonding orbital is geradegerade, or symmetric about a center , or symmetric about a center of symmetry.of symmetry.


+

+

+

-

-

-πu

πg


Molecular Orbital Molecular Orbital DiagramDiagram

This is a This is a molecular orbital molecular orbital energy level energy level diagram for the diagram for the pp orbitals. Note that orbitals. Note that the the σσ bonding bonding orbital is lowest in orbital is lowest in energy due to the energy due to the greater overlap greater overlap end-on-end.end-on-end.

2p 2p

σg

πu

πg

σu


Molecular Orbital Molecular Orbital DiagramDiagram

The The alternate alternate notation is notation is provided on the provided on the right side of the right side of the energy level energy level diagram.diagram.

2p 2p

σg

πu

πg

σu


Molecular Orbital Molecular Orbital DiagramsDiagrams

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 Molecular orbitals may be empty, or contain one or two electrons.contain one or two electrons.

3.3. If two electrons occupy the same If two electrons occupy the same molecular orbital, they must be spin molecular orbital, they must be spin paired.paired.

4.4. When occupying degenerate molecular When occupying degenerate molecular orbitals, electrons occupy separate orbitals, electrons occupy separate orbitals with parallel spins before pairing.orbitals with parallel spins before pairing.



Although molecular orbitals Although molecular orbitals form from inner (core) electrons as form from inner (core) electrons as well as valence electrons, many well as valence electrons, many molecular orbital diagrams include molecular orbital diagrams include only the valence level.only the valence level.



For OFor O22, , there will be there will be a total of 12 a total of 12 valence valence electrons that electrons that must be must be placed in the placed in the diagram.diagram.



For OFor O22, , there will be there will be a total of 12 a total of 12 valence valence electrons that electrons that must be must be placed in the placed in the diagram.diagram.

2p 2p

2s 2s


MO Diagram for OMO Diagram for O22

2p 2p

2s 2s

σg

σ*u

σg

πu

π*g

σ*u The molecular orbital diagram for oxygen shows two unpaired electrons, consistent with experimental data.


Bond OrderBond Order

Bond order is an indicator of the Bond order is an indicator of the bond strength and length. A bond bond strength and length. A bond order of 1 is equivalent to a single order of 1 is equivalent to a single bond. Fractional bond orders are bond. Fractional bond orders are possible.possible.

The bond order of the molecule =The bond order of the molecule =(# e(# e-- in bonding orbtls) in bonding orbtls) - - (# e(# e-- in anti- in anti-bonding orbtls)bonding orbtls)

22 22Muhammad Asif Pakistan


2p 2p

2s 2s

σg

σ*u

σg

πu

π*g

σ*u

The bond order of O2 is:

8-4 = 2 2

This is consistent with a double bond.



2p 2p

2s 2s

σg

σ*u

σg

πu

π*g

σ*uThis energy level diagram works well for atoms in which the 2s and 2p levels are fairly far apart. These are the elements at the right of the table: O, F and Ne.



Oxygen is paramagnetic, consistent Oxygen is paramagnetic, consistent with having two unpaired electrons. In with having two unpaired electrons. In addition, addition, photoelectron spectroscopyphotoelectron spectroscopy (PES) (PES) can be used for determining orbital can be used for determining orbital energies in molecules. The molecule is energies in molecules. The molecule is bombarded with UV or X-rays to remove an bombarded with UV or X-rays to remove an electron from the molecule. The kinetic electron from the molecule. The kinetic energy of the emitted electron is measured energy of the emitted electron is measured and subtracted from the incident radiation and subtracted from the incident radiation to determine the binding energy of the to determine the binding energy of the electron.electron.


Photoelectron Photoelectron SpectroscopySpectroscopy

The result is a spectrum of The result is a spectrum of absorptions which are correlated to the absorptions which are correlated to the molecular orbitals of the molecule. In molecular orbitals of the molecule. In addition, electrons ejected from bonding addition, electrons ejected from bonding orbitals show more vibrational energy orbitals show more vibrational energy levels than electrons emitted from anti-levels than electrons emitted from anti-bonding or non-bonding orbitals.bonding or non-bonding orbitals.


MO diagram for Li MO diagram for Li through Nthrough N

The elements on the left side of The elements on the left side of period 2 have a fairly small energy period 2 have a fairly small energy gap between the 2s and 2p orbitals. gap between the 2s and 2p orbitals. As a result, interaction between s As a result, interaction between s and p orbitals is possible. This can and p orbitals is possible. This can be viewed in different ways. be viewed in different ways.



In some approaches, the s orbital on In some approaches, the s orbital on one atom interacts with the p orbital on one atom interacts with the p orbital on another. The interaction can be another. The interaction can be constructive or destructive.constructive or destructive.



In another approach, the s and p In another approach, the s and p orbitals on the orbitals on the samesame atom interact in atom interact in what is called what is called orbital mixingorbital mixing. .

Either approach yields the same Either approach yields the same result. The result. The σσ bonding and anti- bonding and anti-bonding orbitals are raised in energy bonding orbitals are raised in energy due to the interaction with a due to the interaction with a pp orbital.orbital.



σg

σg

σ*u

σ*u

πu

π*g


MO diagram for NMO diagram for N22

σg

σg

σ*u

σ*u

πu

π*g

N2 has 10 valence electrons.


Experimental EvidenceExperimental EvidenceThe The

photoelectronic photoelectronic spectrum of spectrum of nitrogen is nitrogen is consistent with a consistent with a molecular orbital molecular orbital approach. approach.

Electrons Electrons emitted from emitted from bonding orbitals bonding orbitals show vibrational show vibrational excitations.excitations.

σg

πu

σ*u



σg

πu

σ*uσg

σg

σ*u

σ*u

πu

π*g


Heteronuclear Diatomic Heteronuclear Diatomic MoleculesMolecules

The more electronegative atom The more electronegative atom will have orbitals of lower energy, will have orbitals of lower energy, and therefore contribute more to the and therefore contribute more to the bonding orbitals.bonding orbitals.

The less electronegative atom The less electronegative atom has orbitals of higher energy, and has orbitals of higher energy, and contributes more to the anti-bonding contributes more to the anti-bonding orbitals.orbitals.


Rules for Combining Rules for Combining Atomic OrbitalsAtomic Orbitals

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 predominantly on the atom of lower orbital energy (the more orbital energy (the more electronegative atom).electronegative atom).

2. The anti-bonding orbital(s) will reside 2. The anti-bonding orbital(s) will reside predominantly on the atom with predominantly on the atom with greater orbital energy (the less greater orbital energy (the less electronegative atom).electronegative atom).


HFHF

The 2s and 2pThe 2s and 2pxx orbitals on fluorine orbitals on fluorine interact with the 1s interact with the 1s orbital on hydrogen. orbital on hydrogen.

The pThe pyy and p and pzz orbitals on fluorine orbitals on fluorine lack proper symmetry lack proper symmetry to interact with to interact with hydrogen, and remain hydrogen, and remain as non-bonding as non-bonding orbitals.orbitals.


HFHF

The anti-bonding The anti-bonding orbital resides orbital resides primarily on the less primarily on the less electronegative atom electronegative atom (H).(H).

Note that the Note that the subscripts subscripts gg and and uu are are not used, as the not used, as the molecule no longer molecule no longer has a center of has a center of symmetry.symmetry.


Carbon monoxideCarbon monoxide

In carbon In carbon monoxide, the monoxide, the bonding orbitals bonding orbitals reside more on reside more on the oxygen atom, the oxygen atom, and the anti-and the anti-bonding orbitals bonding orbitals reside more on reside more on the carbon atom. the carbon atom.



CO is a highly CO is a highly reactive molecule reactive molecule with transition with transition metals. Reactivity metals. Reactivity typically arises typically arises from the from the hhighest ighest ooccupied ccupied mmolecular olecular oorbital (HOMO), rbital (HOMO), when donating when donating electrons.electrons.



When When acting as an acting as an electron pair electron pair acceptor, the acceptor, the llowest owest uunoccupied noccupied mmolecular olecular oorbital (LUMO), rbital (LUMO), is significant.is significant.



When When acting as an acting as an electron pair electron pair donor, the donor, the hhighest ighest ooccupied ccupied mmolecular olecular oorbital (HOMO), rbital (HOMO), is significant.is 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 a donor and a ππ acceptor ligand. It acceptor ligand. It donates electrons from its HOMO to donates electrons from its HOMO to form a sigma bond with the metal.form a sigma bond with the metal.



Carbon monoxide accepts Carbon monoxide accepts electrons from filled electrons from filled dd orbitals on orbitals on the metal into its antibonding the metal into its antibonding (LUMO) orbital.(LUMO) orbital.



This phenomenon is called This phenomenon is called back back bondingbonding. The increased electron density . The increased electron density in the antibonding orbitals of CO causes in the antibonding orbitals of CO causes an increase in the C-O bond length and a an increase in the C-O bond length and a decrease in its stretching frequency.decrease in its stretching frequency.


MOs for Larger MOs for Larger MoleculesMolecules

Group theory is usually used to Group theory is usually used to develop molecular orbital diagrams and develop molecular orbital diagrams and drawings of more complicated molecules. drawings of more complicated molecules. When a central atom is bonded to When a central atom is bonded to several atoms of the same element (Hseveral atoms of the same element (H22O, O, BFBF33, or PtCl, or PtCl44

2-2-], group theory can be used ], group theory can be used to analyze the symmetry of the orbitals of to analyze the symmetry of the orbitals of the non-central atoms, and then combine the non-central atoms, and then combine them with the appropriate orbitals of the them with the appropriate orbitals of the central atom.central atom.



The orbitals of the non-central The orbitals of the non-central atoms are called atoms are called group orbitalsgroup orbitals. In . In considering a simple example, Hconsidering a simple example, H22O, O, we obtain group orbitals using the we obtain group orbitals using the two two 1s1s orbitals on the hydrogen orbitals on the hydrogen atoms.atoms.


The characters for The characters for the group orbitals is the group orbitals is obtained by considering obtained by considering each hydrogen as a each hydrogen as a spherical spherical 1s1s orbital. They orbital. They remain in position for remain in position for identity, are exchanged identity, are exchanged during rotation, remain in during rotation, remain in place for σplace for σxzxz (the (the molecular plane), and are molecular plane), and are exchanged for σexchanged for σyzyz..


Group Orbitals of WaterGroup Orbitals of Water

ΓΓredred and its irreducible and its irreducible representations are:representations are:



The AThe A11 representation has both representation has both 1s1s orbitals with positive wave orbitals with positive wave functions: Hfunctions: Haa+H+Hbb..

The BThe B11 representations is H representations is Haa+H+Hbb..



These group orbitals are combined These group orbitals are combined with orbitals on oxygen that have with orbitals on oxygen that have the same symmetry.the same symmetry.



The 2s and 2pz orbital on oxygen have A1 symmetry, the 2px orbital has B1 symmetry, and the 2py has B2 symmetry.Muhammad Asif Pakistan

Molecular Orbitals of Molecular Orbitals of WaterWater

Since the Since the 2p2pyy orbital on oxygen orbital on oxygen doesn’t match the symmetry of the doesn’t match the symmetry of the group orbitals of hydrogen, it will group orbitals of hydrogen, it will remain non-bonding. The other remain non-bonding. The other orbitals on oxygen will combine with orbitals on oxygen will combine with the appropriate group orbitals to the appropriate group orbitals to form bonding and antibonding form bonding and antibonding molecular orbitals.molecular orbitals.



Group theory is usually used to Group theory is usually used to develop molecular orbital diagrams develop molecular orbital diagrams and drawings of more complicated and drawings of more complicated molecules. A simplified example will molecules. A simplified example will be shown for the be shown for the ππ bonding of bonding of benzene.benzene.


ππ Bonding of Benzene Bonding of Benzene

Benzene belongs to point group Benzene belongs to point group DD6h6h. In determining the orbital . In determining the orbital combinations for combinations for ππ bonding, we need bonding, we need to obtain to obtain ГГππ by looking only at the p by looking only at the pzz orbitals on each carbon atom.orbitals on each carbon atom.We need only

consider those orbitals on carbon atoms that remain in place for a given symmetry operation.



DD6h6h EE 2C2C

66

2C2C

33

CC

22

3C3C′′22

3C3C″″22

ii 22SS33

22SS66

σσ

hh

3 3 σσdd

3 3 σσvv

ГГππ

{z axis

C′2C″2



DD6h6h EE 2C2C

66

2C2C

33

CC

22

3C3C′′22

3C3C″″22

ii 22SS33

22SS66

σσ

hh

3 3 σσdd

3 3 σσvv

ГГππ 66 00 00 00 -2-2 00 00 00 00 -6-6 00 22

{z axis

C′2C″2



DD6h6h EE 2C2C

66

2C2C

33

CC

22

3C3C′′22

3C3C″″22

ii 22SS33

22SS66

σσ

hh

3 3 σσdd

3 3 σσvv

ГГππ 66 00 00 00 -2-2 00 00 00 00 -6-6 00 22

{z axis

C′2C″2

This reduces to: B2g + E1g + A2u + E2u



Гπ: B2g + E1g + A2u + E2u

Group theory can be used to Group theory can be used to draw each of the draw each of the ππ molecular orbitals. molecular orbitals. Molecular orbitals with fewer nodes Molecular orbitals with fewer nodes are lower in energy (more bonding), are lower in energy (more bonding), and those with more nodes are higher and those with more nodes are higher in energy (more antibonding).in energy (more antibonding).




A2u fully bonding and lowest in energy

E1g degenerate bonding orbitals with one nodeMuhammad Asif Pakistan



E2u degenerate largely anti-bonding orbitals with two nodes

B2g fully anti-bonding orbital with three nodesMuhammad Asif Pakistan


A2u

E1g

E2u

B2g


Molecular Orbitals of Molecular Orbitals of ComplexesComplexes

Group theory is also used to Group theory is also used to construct molecular orbital diagrams construct molecular orbital diagrams for the complexes of metal atoms or for the complexes of metal atoms or ions. The symmetry combinations of ions. The symmetry combinations of the atomic orbitals on the ligands are the atomic orbitals on the ligands are determined, and then “matched” with determined, and then “matched” with appropriate atomic orbitals on the appropriate atomic orbitals on the central metal. Both central metal. Both σσ and and ππ bonding bonding between the metal and ligands can be between the metal and ligands can be considered.considered.


molecular orbitals

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