Module 3.5: Modern Bonding Theories

3.5. Modern Bonding Theories

3.5.1 Valence Bonding Theory

I Valence Bond Theory (VBT): A theory that considers the interactions between separate atomic orbitals as they are brought to form molecules

it is a description of covalent bonding in terms of atomic orbitals
takes into consideration the wave property of electrons and the uncertainty principle
the valence bond theory is still a localized model as the electron density is predicted to be between atoms

II There are two types of bonding classified by the VBT:

$\sigma$ -bonding: the overlap of one region on the internuclear axis between two atomic orbitals (Ref. Figure 26)
1. the greater the overlap of the atomic orbitals, the stronger the bond
2. all single bonds in the VBT are classified as $\sigma$ -bonds
3. other orbitals such as 2s or $2p_{z}$ orbitals can also form $\sigma$ -bonds.
$\pi$ -bonding: the overlap of two regions between two atomic orbitals (Ref. Figure 27)
1. contains a single nodal plane in the internuclear axis
When there are multiple bonds, $\pi$ -bonding and $\sigma$ -bonding are both used:
1. for single bonds, it is only one $\sigma$ -bond.
2. for double bonds, it is one $\sigma$ -bond and one $\pi$ -bond.
3. for triple bonds, it is one $\sigma$ -bond and two $\pi$ -bonds.

III Hybridization: a linear combination of atomic orbitals to account for observed geometry

electron promotion occurs in the first step of hybridization
1. for methane ( $CH_{4}$ ), one 2s electron is placed to an empty 2p orbital forming four unpaired electrons
following electron promotion, the atomic orbitals merge into hybrid orbitals
1. hybrid orbitals result from the constructive and destructive interference of atomic orbitals (Ref. Figure 28)
2. these hybrids are degenerate in energy but in different orientations

3.5.2 Types of Hybridization

IV There are many different types of hybridization:

$sp^{3}$ hybridization: three p orbitals and one s orbital combine to form four equivalent $sp^{3}$ hybrid orbitals:
1. each hybrid orbital has 75% p character and 25% s character
2. each $sp^{3}$ orbital has two lobes with a massive size inequality
3. as electron density is concentrated on one side of the nucleus, more efficient overlap can be achieved and stronger bonds are formed compared to absence of hybridization
4. the resulting geometry for $sp^{3}$ hybrid orbitals is tetrahedral
$sp^{2}$ hybridization: two p orbitals and one s orbital combine to form three equivalent $sp^{2}$ hybrid orbitals:
1. each hybrid orbital has 66% p character and 33% s character
2. each $sp^{2}$ has two lobes with a large size inequality (but not as much as $sp^{3}$ )
3. $sp^{2}$ hybridization leaves behind one p orbital; trigonal planar geometry
sp hybridization: one p orbital and one s orbital combine to form two equivalent sp hybrid orbitals:
1. each hybrid orbital has 50% p character and 50% s character
2. each sp orbital has two lobes with a much smaller size inequality
3. sp hybridization leaves behind two p orbitals; linear geometry
other theorical hybridizations such as $sp^{3}d^{2}$ and $sp^{3}d$ exist
1. usage of these hybrid orbitals to justify electron promotion and defiance of the octet rule can be prevented with charge-separated species (Ref. Figure 30/31)
2. ab initio data reveals that d orbital participation is very small for these molecules

V The valence bond theory has a few flaws:

it is still a localized model
hybridization of more complex orbitals such as $sp^{3}d^{2}$ and $sp^{3}d$ leads to violation of the octet rule which does not match observations
cannot describe the paramagnetic property of dioxygen.
many period 3 elements such as sulfur and phosphorus do not follow match hybridization bond angles and observations due to poor orbital overlap

VI Despite its flaws, the valence bond theory is still used in conjunction with the molecular orbital theory, due to the convenience and simplicity that hybridization provides.

3.5.3 Molecular Orbital Theory

VII Molecular Orbital Theory (MO Theory): A theory that considers all electrons to be delocalized throughout the entire molecule

Molecular orbitals: regions of space (wavefunctions) that an electron may occupy throughout a molecule
a molecular orbital is formed from the linear combination of atomic orbitals (LCAO).

VIII when two atomic orbitals (such as 1s) combine, two molecular orbitals form

a bonding molecular orbital is formed when constructive interference (Ref. Figure 34) of the atomic orbitals occur
1. this interaction reinforces each other and electron density is high between the two atomic nuclei
2. this buildup of electron density between the nuclei reduces internuclear repulsion
3. because the electron occupies a greater volume, it has lower kinetic energy
4. because the electron is attracted by two nuclei, it has lower potential energy
an antibonding molecular orbital is formed when destructive interference (Ref. Figure 35) of the atomic orbitals occur
1. this interaction results in a nodal plane with zero electron density between the two atomic nuclei
2. this lack of electron density raises internuclear repulsion and destabilizes the molecular orbital
3. the magnitude of energy raised from antibonding orbitals is larger than the magnitude of energy lowered from bonding orbitals.
Do not forget: (total # of molecular orbitals) = (total # of atomic orbitals used)

IX A molecular orbital energy-level diagram is constructed to illustrate the relative energies of the atomic orbitals and the molecular orbitals.

electrons are added according to the Aufbau principle.

X Bond order (h): the net number of bonds with consideration of antibonding interactions $b = \frac{1}{2} \times (N_{e}-N^{*}_{e})$ (Eq. 28)

an asterisk (*) is commonly used to indicate antibonding
$N_{e}$ illustrates the number of electrons in a bonding orbital.
bond order can often be correlated with bond distances and bond dissociation energies

XI. A basis set of orbitals are atomic orbitals that are available for orbital interactions

for $p_{z}$ orbitals, only direct overlap is possible and forms $\sigma$ and $\sigma^{*}$ orbitals
for $p_{x}$ and $p_{y}$ orbitals, only sideways overlap is possible; forms $\pi$ and $\pi^{*}$ orbitals
because $\pi$ -orbitals are less efficient at bonding then $\sigma$ -orbitals, they are higher in energy.

XII. Compounds can be classified according to their behavior in a magnetic field:

diamagnetic substances tend to move out of a magnetic field
paramagnetic substances tend to move into a magnetic field

XIII. A list of the second period homonuclear diatomics:

Diatomic Species	Electronic Configuration	Bond Order	Diamagnetic, paramagnetic	Desription
$H_{2}$	$\sigma_{g}^{2}(1s)$	1	Diamagnetic	A single $\sigma$ orbital that contains an electron pair.
$He_{2}$	$\sigma_{g}^{2}\sigma_{u}^{*2}(1s)$	0	Diamagnetic	There is no net bonding interaction so this species is extremely unstable.
$Li_{2}$	$\sigma_{g}^{2}(2s)$	1	Diamagnetic	this is in agreement with observation of gas-phase $Li_{2}$ molecules.
$Be_{2}$	$\sigma_{g}^{2}\sigma_{u}^{*2}(2s)$	0	Diamagnetic	As expected, this species is also extremely unstable.
$B_{2}$	$\pi_{u}^{1}\pi_{u}^{1}(2p)$	1	Paramagnetic	paramagnetic due to s-p orbital mixing (See below)
$C_{2}$	$\pi_{u}^{2}\pi_{u}^{2}(2p)$	2	Diamagnetic	oddly enough does not possess any $\sigma$ bonds
$N_{2}$	$\pi_{u}^{2}\pi_{u}^{2}\sigma_{g}^{2}(2p)$	3	Diamagnetic	very stable triple bond
$O_{2}$	$\sigma_{g}^{2}\pi_{u}^{2}\pi_{u}^{2}\pi_{g}^{1}\pi_{g}^{1}(2p)$	2	Paramagnetic	the energy levels changed due to $\sigma-\pi$ crossover
$F_{2}$	$\sigma_{g}^{2}\pi_{u}^{2}\pi_{u}^{2}\pi_{g}^{2}\pi_{g}^{2}(2p)$	1	Diamagnetic	weak bond so hard to find as a diatomic

XIV. As $Z_{eff}$ increases across the second period, the energy of atomic orbitals decreases

as 2s orbitals have much higher penetration than 2p orbitals, the separation of energy levels between 2s and 2p increases across the period (Ref. Figure_38)
1. for molecules like $O_{2}$ and $F_{2}$ , the energy separation between 2s and 2p were great enough that their interactions could be ignored
2. for molecules like $B_{2}$ to $N_{2}$ , the energy separation is small enough for orbital mixing to occur
orbital mixing: the interaction of molecular orbitals of similar symmetry and energy
$\sigma-\pi$ crossover: the change in energy diagram positions of $\sigma_ {g}(2p)$ and $\pi_{u}(2p)$
1. both $B_{2}$ and $O_{2}$ are paramagnetic due to this $\sigma-\pi$ crossover

XV. The following three conditions will determine the formation of molecular orbitals for heteronuclear molecular orbitals:

formation is allowed if symmetries of atomic orbitals are compatible
1. if symmetries do not match, this is known as symmetry-disallowed interaction.
2. if symmetry-disallowed, the overlap integral is zero and therefore no energy gain.
formation is efficient if region of overlap between two atomic orbitals is significant.
formation is efficient if atomic orbitals are relatively close in energy.
1. the ionic bonding model represents the scenario where the separation of energy between two atomic orbitals is so big that there is barely any orbital interaction.
2. the covalent bonding model represents the scenario where the separation of energy between two atomic orbitals is so little that there is maximum orbital interaction.

XVI. The linear combination of atomic orbitals to form diatomic molecules can be described mathematically:

$\psi = c_{A}\psi_{A} + c_{B}\psi_{B}$ (Eq. 29)

for homonuclear diatomic molecules, the coefficients $c_{A}$ and $c_{B}$ are equal.
1. In other words, equal contributions are made by both atomic orbitals.
2. this resembles a pure nonpolar bond.
for heteronuclear diatomic molecules, the coefficients $c_{A}$ and $c_{B}$ are not equal.
1. In other words, nonequal contributions are made by the two atomic orbitals.
2. this resembles a polar bond. If extreme, it can resemble an ionic bond.
3. if coefficients $c_{A}^{2}$ is large for a certain molecular orbital, this molecular orbital should resemble atomic orbital A with electron density closer to A.
4. if coefficients $c_{B}^{2}$ is large for a certain molecular orbital, this molecular orbital should resemble atomic orbital B with electron density closer to B.

XVII. As shown in Figure 39, the bonding orbital is significantly closer in energy to the atomic orbital that is lower in energy than the one that is higher in energy.

Because more electronegative atoms (higher $Z_{eff}$ ) have lower energy levels, the following can be justified:
1. the bonding orbital will have significantly more character of the electronegative element's atomic orbital that is lower in energy
2. Conversely, the antibonding orbital will have significantly more character of the electropositive element's atomic orbital that is higher in energy

XVIII. For molecular orbitals with more than two atoms, it is much more difficult to predict energy levels of molecular orbitals; a character table or a computer software is required.

This is also true for heteronuclear diatomic orbitals with elements that have atomic numbers greater than 8.

3.5.1 Valence Bonding Theory
3.5.2 Types of Hybridization
3.5.3 Molecular Orbital Theory

Module 3.5: Modern Bonding Theories

3.5. Modern Bonding Theories

3.5.1 Valence Bonding Theory

3.5.2 Types of Hybridization

3.5.3 Molecular Orbital Theory

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