| 7. Atomic orbitals (s and
p for
us) are centered on atoms. They serve as "building
blocks" for orbitals found in molecules.
And at the beginning was hydrogen... Yes, the shapes of
orbitals were first calculated for the hydrogen atom. These are the familiar by now
s
and p orbitals. Other atoms have similar orbitals. What changes is their size
and energy. Eventually, we also get d and f orbitals with really funny
shapes, but they are not so important in organic chemistry where we are dealing
mainly with the elements from the first two rows of the periodic table. |
For illustration purposes the relative sizes of several
atomic orbitals are presented in the table above. In general, the
orbital size increases going down in the periodic table (the increase in
the main quantum number). Within the rows, the orbital size
decreases going to the right. The positive nuclear charge increases in
this direction; the electrons are, therefore, attracted more strongly to the
nucleus and get closer to it (on average). This effect is especially pronounced in the
size of s orbitals. The electrons in these orbitals, however, screen the nuclear
charge, and the electrons in p orbitals are influenced less (smaller size change).
In a qualitative sense, the energies of these orbitals follow the same trend as
their sizes. Within the same row, s orbitals have lower energies than
p,
and the smaller the orbital the lower its energy.
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8. The molecules are constructed from atoms,
and the "new" volumes for electrons in the resulting molecules are constructed
from the atomic orbitals of atoms participating in bonding. Two basic set of rules used
are known as the Valence Bond (VB) Theory and the Molecular Orbital (MO) Theory. The
theories use different language, but are equivalent.
Molecules are collections of atoms. The positions of the
nuclei give the molecule its 3D shape, and the new spaces for electrons are constructed
from the atomic orbitals. We use two recipes to do that: these are just like baking
instructions. Regardless which one we use, we get the same cake at the end; we mean the
same final electronic structure. |
9. In VB, atomic orbitals on a given atom are
premixed (hybridized), if necessary, and then used to form bonds, pairwise between atoms, one bond at a
time.
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In this procedure (here an American chemist was
responsible) we first "prepare" all the atoms for bonding. We take all (or just
some) of the atomic orbitals of a given atom and make new combinations out of them (like
making a cocktail from several pure drinks). So in addition to pure atomic orbitals
(s or p, "pure drinks") we can also have variety of hybrids
("mixed drinks"), spx, where x tells us the p/s ratio in the hybrid
orbital. Remember, each new orbital must be normalized (i.e. the probability of
finding an electron within it should be one), and the number of orbitals must be preserved
(see below). Then we use these prepared atoms with their hybrid or atomic orbitals to make
new bonds, one at at a time. Each bond is made by superimposing (overlapping, see
below) these hybrid or unchanged atomic orbitals on two adjacent atoms (one orbital on each) within
the molecule. |
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10. In MO, atomic orbitals of all atoms are mixed to form molecular
orbitals that span many atoms (or even the whole molecule).
No premixing in here. Just take the atomic orbitals as
they are, and make "super cocktails"; i.e. directly form molecular orbitals that
are composed of (fractions) of many atomic orbitals on different atoms. Again, each new
orbital must be normalized (probability of finding an electron within it is unity), and
the number of orbitals must be preserved. |
11. The driving force for bonding is to
achieve a particularly stable electronic configuration (see noble gases) when the
atoms have two (for H), or eight electrons (for B, C, O, N, F) in their valence shells.
That configuration can be achieved by totally giving up electrons to the bonding partner
(or taking them away). In such a case we deal with ionic bonds that are rather
uncommon in organic chemistry. In most cases the octet configuration (or doublet for
H) is achieved by sharing of electrons. This kind of bonding is called covalent.
Nothing to add here. Sharing is the key term! |
12. The sharing (bonding) is accomplished
via superposition (overlap) of atomic or hybrid orbitals to form bonds (VB) or molecular
orbitals (MO). The overlap (see below) is accomplished in an "additive"
manner (bonding) or in a "destructive" (subtractive) manner (antibonding).
To share electrons new orbitals within the molecule must
allow the electrons to get close to the nuclei involved in sharing (bonding). We construct
these orbitals using atomic or hybrid orbitals as building blocks. One simple
procedure would just add all the volumes of building orbitals together. This generates
some problems, however. Let's say we start with two orbitals on two neighboring
atoms. These orbitals may accommodate up to four electrons. The new orbital we have just
made can accept at most two electrons (of different spins, see above). So in the
process, we would have "lost" space for two electrons. This example
immediately tell us that we must produce two new orbitals not just one! Or, in
general, the number of new orbitals must be equal the number of the building-block
orbitals. The water-wave analogy becomes handy again: when two waves encounter
each other, they superimpose in a "predictable" fashion. If they are in the same
phase (in-sync) they reinforce (or add to) each other, and if they are in
the opposite phase (out-of-sync) they cancel (subtract from) each other.
So, to get two new orbitals we "add" or "subtract" the
wavefunctions of the building-block orbitals and look at the results. We do it pictorially here, remembering that signs of orbital wavefunctions are encoded with colors. |
13. In the "additive" interaction
of (for example) two atomic orbitals the new orbital volume encloses both atoms. The
electrons in that volume (valence bond orbital or molecular orbital) interact with both
nuclei (the electrons are shared). The new orbital is of lower energy than the
atomic orbitals used to generate it. To preserve the number of orbitals, the
"additive" mode is accompanied by the "destructive"
("subtractive") mode where the new orbital volumes do not encompass both
nuclei. The electrons in such an orbital cannot be shared, i.e. the orbital has a
node (or more precisely: a nodal plane) between the nuclei. This (antibonding)
orbital is of higher energy than the atomic orbitals from which it was
generated. To the very crude approximation, the lowering of energy of the bonding
orbital as compared to the constituent atomic orbitals is the same as the corresponding
rise in energy of the antibonding orbital.
The addition of two lobes of the same sign or "color" (now you know why
keeping that info was important) leads to a new space that encompasses
all the lobes involved. The electrons may move around the nuclei
enclosed within that new volume and interact with both of them (Coulombic
interaction). That is what we call sharing. It lowers the orbital energy, as
compared to the separate atom situation. All being said and done, now every atom can
pretend to be a noble ...gas. Here is a simple demo for H2 molecule: on the left
we have B&W 3D representations of the orbitals (note that lobe signs are marked with
continuous or dotted lines), and on the right we have the cross sections of the two orbitals
with wavefunctions mapped out in color (the black dots approximate the
position of the nuclei, although they are out of proportion in size). |
antibonding (high energy)
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bonding (low energy)
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The subtraction of two lobes of the same sign (or addition
of lobes of different sign or "color") gives two separate,
"unconnected" volumes (of different signs or colors). This unconnected
lobes are separated by regions where the probability of finding electrons is zero, called
nodal planes (or just nodes). In such an orbital electrons cannot get close to both the
nuclei involved in the bonding. They are "imprisoned" in the space around just
one atom The energy of that orbital is increased (in comparison to separate
atoms). Notice different color scales for the two orbitals. |
14. The interaction of two
orbitals (atomic, hybridized or molecular) is stronger (more energy lowering, see the
point above) if the orbitals are closer in energy, and if the overlap between them
(interpenetration of their spaces) is larger.
Now we are talking about the quality of sharing. We
measure that by how much the energy is lowered by the bonding process. It mainly depends
on the relative energies of the building-block orbitals and their overlap. The
strongest interaction is between the orbitals of equal energy; orbitals of very different
energies will not interact (or interact very weakly).
The overlap
is measured by the amount of the common space occupied by the
interacting building-block orbitals that are of the same sign minus the common
space occupied by the building-block orbitals of the opposite sign. Brrrr...
It is best explained using a couple of illustrations. Consider the interaction of
two pz orbitals of equal energy making a σ bond.
To see it better we gave the two orbitals different colors (black and red) and marked the
algebraic sign of their lobes with shading (let's say the shade corresponds to
"+").

If the distance between the atoms is too large (r1)
the overlap is small: there is only small common part in the center. At a certain
distance (r2) the overlap will be at the maximum: the positive lobes overlap
fully. But if we push the atoms too close (r3) orbitals
will interpenetrate too much: the positive lobes will overlap with
negative lobes. This "mixed-sign" interpenetration (+/−)
subtracts from the same sign interpenetration (+/+, or −/−).
Or in different words, the overlap is characterized by the amount (how
much) of the common space) and the sign (+/− is negative; +/+ or −/−
is positive). The net results is what counts.
Here, look at 2s and 2p orbitals on the
same atom (for example, on oxygen). The s orbital has (let us say) a negative sign, and the p orbital has the top lobe positive (shaded) and the bottom lobe negative.
The common space
of the two orbitals is shown in color in the second picture (green and purple, remember it
is in 3D). The green space is a positive contribution (−/−) and the purple space is
a negative contribution (+/−). The two contribution are equal in size. Thus, the net
overlap is zero! (Indeed, for those who like to keep it sophisticated we call such orbitals orthogonal, and all orbitals discussed here are both orthogonal and normalized, or orthonormal).
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