|7. Atomic orbitals (s and
us) are centered on atomic nuclei. 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. Among them are the familiar by now
and p orbitals. Other atoms have similarly shaped 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 valence 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 effective 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
and the smaller the orbital the lower its energy. In other words, more
electronegative atoms have lower energy orbitals.
8. Molecules are constructed from atoms,
and the "new" volumes for electrons in the resulting molecules are constructed
from the atomic valence orbitals of atoms participating in bonding. The two
most commonly applied set of rules for constructing molecules out of atoms are known as the Valence Bond (VB) Theory and the Molecular Orbital (MO) Theory. The
theories use different "languages", but if carried out properly (at the
quantitative level) give essentially the same results.
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 very similar cakes at the end; we mean
similar 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
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 valence 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 (the index of
hybridization) tells us the p/s ratio in the hybrid
orbital. Remember, each new orbital is normalized, and in most
cases each orbital involved in bond making holds
one electron, indeed. 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) these hybrid or unchanged atomic orbitals on two adjacent atoms (one orbital on each) within
the molecule. In the process electrons from these orbitals are paired
in the new bond, i.e. end up with opposite spins.
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
are composed of (fractions) of many atomic orbitals on different atoms. Again, each new
orbital must be normalized,. Importantly, in this case,
the number of orbitals must be preserved, as to have "space" for the
same number of electrons before and after the mixing. After the
MOs are constructed, all available electrons are then placed in
lowest-energy orbitals, with no more than two of them (with paired
spins) per orbital.
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) can be 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). 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
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
or other markings.
13. Let's consider a molecule of H2. In the "additive" interaction
of two atomic orbitals the new orbital volume encloses both atoms. The
electrons in that volume (valence bond orbital or molecular orbital)
electrostatically interact with both
nuclei (the electrons are shared). The new orbital is of lower energy than the
atomic orbitals from which it was generated. To preserve the number of orbitals, the
"additive" mode is accompanied by the "subtractive" mode
where the new orbital volumes do not encompass both nuclei. This
orbital has a node between the nuclei.
orbital is of higher energy than the atomic orbitals from which it was
generated. The lowering of energy of the bonding
orbital (as compared to the constituent atomic orbitals) is a bit less than the corresponding
rise in energy of the antibonding orbital.
The addition of two lobes of the same sign or "color" leads to a new space that encompasses
all the lobes involved. Now it becomes clear why keeping that orbital
sign info was important. 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
visualization for our 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)
bonding (low energy)
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). These unconnected
lobes are separated by regions where the probability of finding electrons is zero, called
nodes. In such an orbital electrons
density in the space between the nuclei is diminished. The energy of that orbital is increased (in comparison to separate
atoms). Notice different color scales for the two orbitals.
The pictures we have developed here represent MO theory of H2
molecule with its two molecular orbitals (MOs). In the classical VB
theory we would limit our considerations to the bonding orbital only (we
would call it a bond) and would neglect the unoccupied antibonding one.
In a modified VB theory, we will acknowledge the "existence" of the
unoccupied (antibonding) orbitals and even put them to a good use later
14. The interaction of
any two orbitals (atomic, hybridized or molecular) is stronger (more energy lowering) 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).
Graphically, the overlap
is measured by the amount of the common spaces occupied by the
interacting (building-block) orbitals that are of the same sign minus the common
spaces occupied by the interacting (building-block) orbitals of the opposite sign. Brrrr...
That is almost incomprehensible. It may help to recall that we are
talking about interacting waves. If the waves are in-phase (+/+ or
–/–-) in a certain region of space, they will
add and the square of the new wave (i.e. the electron density) will increase
in that region. On the other hand, if the waves are out-of-phase
(+/–), they will subtract or even cancel, and the
square of the new wave will be less (or zero) in that region.
Perhaps It is best explained on a few simple
examples. Consider the interaction of 2s and 2p orbitals on the same
atom (for example, on oxygen). The s orbital has (let us say) a positive sign (marked
in red outline), and the p orbital has the top lobe positive (also
red outline) and the bottom lobe negative (marked
in blue outline). The common space
of the two orbitals is shown in tomato-red for (+/+) overlap and in aqua
for the (+/–) overlap (remember it
is all in 3D). 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
atomic orbitals discussed here are both orthogonal and normalized, or orthonormal.
Another illustration involves two p orbitals
on different atoms. In the perpendicular arrangement the (+/+)
overlap (tomato-red) is "cancelled" by the (+/–)
overlap (aqua). No bonding interaction is possible in such geometry.
On the other hand, if one of the p orbitals is shifted off the nodal
plane, the bonding interaction is possible (only +/+ overlap),
although not optimal. Such bonds are called "bent bonds". With
the same internuclear distance (r), a better overlap may be obtained
in the head-on geometry. This situation would describe a "normal"
bond. But head-on overlap does not lead to a bonding
interaction, if the signs of the lobes are mismatched as shown on the
extreme right. In this case the wavefunctions will "destruct" and a
node will form in the middle of the internuclear axis. The result
will be an antibonding combination.