Polyatomic Species: Hybrid & Molecular Orbitals
Polyatomic
species like methane, CH4, can be described in terms
of molecular orbital theory, however, the diagrams can be very difficult
to visualise. However, structures built up from hybrid atomic orbitals are
much easier comprehend.
Introduction: Methane,
CH4
Using the carbon and hydrogen
atomic orbitals, methane, CH4, is constructed by
overlapping the carbon's one 2s and three 2p AOs with the four hydrogen
1s AOs.
Methane's MOs have
a topology similar to the AOs of carbon, but the structure can be very
difficult to visualise, so the methane MO construction diagrams A, B
and C (below) are shown with the AOs and MOs superimposed upon line
structures of the methane. But, remember that the lines are "not
there", it is the bonding MOs that hold the molecule together.
It is possible to
devise an infinite number of MO methane constructs depending upon how
the five nuclei (which are tetrahedral with respect to each other, with
the carbon at the centre) are positioned in space with respect to the
x, y and z Cartesian axes defined px, py and pz orbitals. On other words, the four hydrogen
atoms can be regarded as sitting on a hypothetical spherical shell which
is able to freely rotate with respect to the p orbitals.
We shall examine
three methane LCAO MO constructions: A, B and C below.
MO Methane: Construction
A
In construction A,
the three MOs derived from the carbon's three p orbitals are degenerate,
as are the four hydrogen AOs, but it is not entirely obvious how
any of the hydrogen AOs are overlapping with the carbon AOs.
The hydrogens appear
to be avoiding overlap with the p orbitals.
Note, however, that these
LCAO diagrams are schematic, drawn to emphasise LCAO construction
rather than attempting to represent the orbital overlap integral.

MO Methane: Construction
B
In construction B,
the 2px and 2pz orbitals
overlap (are bonding) with respect to only two on the hydrogens,
the other two hydrogens are on a nodal plane.
The nodal plane is nonbonding
with respect to these two hydrogens.
The 2py
AO is bonding with respect to all four hydrogen AOs.

MO Methane: Construction
C
In construction C,
the 2px AO has maximum overlap with one of
the hydrogen atoms, but at the expense of overlap with the other
hydrogens and the other p orbitals.
The three p orbitals
also appear to be nonequivalent and therefore nondegenerate.
Many authors explain
the s HC bonding in methane in terms of 1s + 2px
(construction c) overlap.

The apparent difficulties
with each of the LCAO constructions disappear when it is realised that
the net carbonhydrogen orbital overlap integral in methane is the same
for any spatial orientation of the hydrogen nuclei with respect to the
2px, 2py and 2pz
orbitals.
Molecular orbital calculations
using software such as Spartan or
Gaussian are performed by assigning
the five nuclei positions in space, and then constructing wavefunctions
for (or adding electrons into) the molecular orbitals, where the MOs are
constructed as a linear combination of the contributing atomic orbitals,
the LCAO approximation. Internuclear geometries are varied until the
energy is minimised:
Ethane
For symmetry reasons, the MOs
of ethane, CH3CH3, are rather
different to those of methane as the πbonds
and π*antibonds
are present.
Ethane possesses 8 atoms, 14
(valence) electrons and 7 MOs (four bonding and three antibonding). These
larger numbers mean that it is easier to visualise ethane with preconstructed
MOs, rather than attempting to show the full AO to MO construction.
 Carboncarbon sigma
bonding primarily arises from an (endon/endon) px
+ px > sigma MO. This is the first (and lowest
energy) ethane MO, the 1sigma MO.
 This C–C sigma
bonding is counteracted by an antibonding 1sigma* MO, although this
MO is bonding with respect to the hydrogen atoms.
 Superimposed upon
the 1sigma and 1sigma* MOs are a degenerate pair of bonding πy
and πz
MOs, but these are counteracted (again along the CC bond axis) by a
degenerate pair of antibonding πy*
and πz*
MOs.
 Sitting (in energy)
between the π
and π*
MOs is the px/px sigma MO.
 Thus – as
a first approximation – the only MO which net bonds the two
carbon atoms together is the sigmax MO formed
by the end on interaction of the two px carbon
AOs. The carboncarbon bond of ethane is a typical carboncarbon single
bond and can rotate freely.
Ethene (Ethylene)
Ethene, H2C=CH2,
has six atoms and 12 valence electrons in six MOs.
There are three sigma MOs (two
bonding and one antibonding) which are similar to the equivalent MOs in
ethane, but the
π MOs are rather different.
 The πz
bonding MO is counteracted by a πz*
antibonding MO.
 This leaves the highest
energy bonding MO occupied with electrons (the alkene's HOMO) as the
py + py –> π MO. This alkene π MO dominates the chemistry of alkenes by inhibiting bond rotation, acting
as an electron rich Lewis base centre, allowing addition reaction to
occur, etc.
 Also important is
the antibonding π*
antibonding MO, the alkene's LUMO.
Ethyne (Acetylene)
Ethyne, HCCH,
has four atoms and 10 valence electrons in four molecular orbitals:
Epichlorohydrin
Epichlorohydrin
is an organic molecule of intermediate complexity (and low symmetry).
The MOs include:
Arrrrrrrgh... Molecular Orbitals Make My Head Hurt!
Mine too.
Richard
Feynman said (here):
"I think I can safely say that nobody understands quantum mechanics."
Put another way:
 Electron interactions obey
various selection rules and as a result give rise to involved patterns
with an intrinsic beauty and sometimes extraordinary complexity.
 The electron interaction
patterns can be calculated/predicted/modelled using advanced mathematical
routines running on fast computers.
 However, that that does
not mean that electron interactions – quantum mechanics –
can be fully known and understood in an intellectual sense. Ultimately,
it is just how our world works.
 Actually matters are even
more involved than implied above. As discussed elsewhere
in the chemogenesis web book wavefunctions are mathematically complex
entities, and atomic and molecular orbitals cannot be directly observed
in principle.
An analogy from engineering: Finite Element Analysis
Finite element analysis
(FEA) is a technique for modelling mechanical structures such as: bridges,
cranes and conrods. A mathematical model of the object is created in
a CAD program, which is then subjected to virtual loads. The displacement
of the structure – the degree of bending – is determined by
treating the object as millions of tiny, interconnected triangles, the
finite elements. Overall stress is determined by considering the stresses
of each finite element. The results can be presented as stress diagrams:
Image
captured from L N Engineering.
Finite elements
analysis is conceptually simple, unlike quantum mechanics, but it plays
a similar role for the engineer as computational chemistry software
does for the chemist.
Engineers can design
a conrod and predict the stresses in the solid, physical item. Likewise,
a chemist can design a molecule in a software package like Spartan and
predict bondlengths, bond angles, van der Waals surfaces, conformers,
rotormers, etc. with great confidence.
An engineer can
zoom in and view each hypothetical finite element, if necessary. Likewise,
the chemist can look inside the calculations and visualise the contributing
molecular orbitals, should she so wish. There is more on
finite element analysis in Wikipedia
Hybridization
Atomic and molecular orbitals
are derived from the Schrödinger wave
equation, and are actually wavefunctions (waves). Waves are well understood mathematically,
and can be added together or subtracted from each other. Consider two
sine waves, the product is a superposition, here:
In molecular orbital theory,
atomic orbitals on adjacent atoms are added together to give a linear
combination of atomic orbitals (the LCAO
approximation) which are used to construct molecular
orbitals.
However, there is an alternative
approach: The s, p and d atomic orbitals can be added together (mixed
or hybridized) to produce hybrid atomic orbitals. Then, atoms in
various compatible hybridization states can be joined together into polyatomic
molecules. Hybridization, part of valance
bond (VB) theory, was presented by Linus
Pauling in 1930.
Actually, the MO and VB approached
are, in principle, exactly equivalent. It is just that the LCAO approximation
is very much easier to implement mathematically and it methodology has
been extensively developed over the past 80 years.
On the other hand, VB theory
and hybrid atomic orbitals are much easier for the human brain to conceptualise
and understand.
The s and p atomic orbitals
can be added together in various ways:
2s + 2p + 2p + 2p
sp^{3}
+ sp^{3} + sp^{3} + sp^{3} =>
sp^{3} hybridised centre
2s + 2p + 2p + 2p
sp^{2}
+ sp^{2} + sp^{2} + p => sp^{2}
hybridised centre
2s + 2p + 2p + 2p
sp
+ sp + p + p => sp hybridised centre
sp^{3}, sp^{2}
and sp hybridisation can be associated with carbon, nitrogen, oxygen atomic
centres with various charge states. (Only these atoms are discussed here,
there are many types of hybridisation.)
 The various centres can
be associated with well known functional groups, reactive intermediates
and fragments.
 They can be associated with
equivalent VSEPR centres, even though these are constructed in an entirely
different way, not involving waves.
Atomic
Centre

Function

Formula

Hybridisation
State

Hybridisation
Diagram

VSEPR
Centre

VSEPR
Diagram

C

Alkane

CH4

sp^{3}


Tetrahedral
AX4


C^{–}

Carbanion

H3C^{–}

sp^{3}


Trigonal
pyramidal
AX3E


C

Alkene
(part of)

H2C=

sp^{2}


Trigonal
planar
AX3


C^{+}

Carbenium
ion

H3C^{+}

sp^{2}


Trigonal
planar
AX3


C

Alkyne
(part of)


sp


Linear
AX2


N

Amine

NH3

sp^{3}




N^{+}

Ammonium
ion

[NH4]^{+}

sp^{3}


Tetrahedral
AX4


N

Imine
(part of)

HN=

sp^{2}


Angular
AX2E


N

Nitrile
(part of)


sp


Terminal


O

Water

H2O

sp^{3}


Angular
AX2E2


O^{+}

Oxonium
ion

[H3O]+

sp^{3}


Trigonal
pyramidal
AX3E


O

Carbonyl
(part of)

O=

sp^{2}


Terminal


O^{+}

Acyl
cation
(part of)


sp


Terminal


The various hybridised atomic
centres can be "plugged together" to produce larger polyatomic
structures. The rule is that likejoinswithlike:
sp^{3} + sp^{3}
centres plug together to give structures like: CH3CH3,
CH3NH2, CH3OH,
CH3CH2^{–} and
CH3NH3^{+}.
sp^{2} + sp^{2}
centres plug together to give structures like: CH2=CH2,
CH2=NH and CH2=O. Notice that
the porbitals overlap to produce a πmolecular
orbital, often called a πbond or even just
a 'double bond'.
sp + sp centres plug together
to give structures like acetylene (ethyne) and hydrogen cyanide. Notice
that the two porbitals overlap to produce two πmolecular
orbitals, often called a 'triple bond'.
A molecule like morphine is
constructed form hydrogen, carbon, nitrogen and oxygen with the C, O &
N atoms in sp^{3}, sp^{2} and sp states of hybridization:
The VB/VSEPR atomic centres
are available as plastic plugtogetheratomiccentres in numerous molecular
model kits:
Understanding Molecular Structure: A Smörgåsbord
of Theories
In this author's opinion it
is usually too difficult to understand the molecular orbital structure
of low symmetry multiatom species like epichlorohydrin. We have computers
to run sophisticated software that can keep track of the multiple arrays
of complex numbers and calculate energies and equilibrium geometries.
Chemists mix & match theories
and make conceptual simplifications when constructing models of molecules.
Polyatomic organic molecules
are constructed by plugging sp^{3}, sp^{2} and sp hybridized
centres together. This is literally true when building plastic
models of molecules!
The functional group approach,
discussed on the next page
of this chemogenesis web book, considers large organic molecules as
consisting of functional groups (FGs): esters, aldehydes, ketones, carboxylic
acids, aromatic rings, etc. When separated by –CH2–
(methylene) groups each FG is deemed to behave as a discrete entity.
The Hückel approximation
of sigmapi separability,
or σπ
separability, assumes that (as
π electrons are at a much higher energy than the σskeleton
electrons) the σ and
π electrons have no influence upon each other. The
σ skeleton of an organic molecule is described using hybridizedVSEPR
atoms with the Hückel πsystem
functional groups superimposed 'on top'.
Frontier molecular orbital
(FMO) theory models the πsystem
functional groups. For example, the triene ester, below has a diene
HOMO and an alkene LUMO that are able to undergo DielsAlder cycloaddition
to give a bicyclic structure (more
here):
Diatomic Species by MO Theory 
Organic πSystems & Functional Groups

© Mark R. Leach 1999
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