Molecular orbital diagram

A molecular orbital diagram, or MO diagram, is a qualitative descriptive tool explaining

that can take place.

History

Qualitative MO theory was introduced in 1928 by Robert S. Mulliken^[4][5] and Friedrich Hund.^[6] A mathematical description was provided by contributions from Douglas Hartree in 1928^[7] and Vladimir Fock in 1930.^[8]

Basics

Molecular orbital diagrams are diagrams of molecular orbital (MO)

(CO
₂), a MO diagram may show one of the identical bonds to the central atom. For other polyatomic molecules, an MO diagram may show one or more bonds of interest in the molecules, leaving others out for simplicity. Often even for simple molecules, AO and MO levels of inner orbitals and their electrons may be omitted from a diagram for simplicity.

In MO theory molecular orbitals form by the

ionic

. A second condition for overlapping atomic orbitals is that they have the same symmetry.

MO diagram for dihydrogen. Here electrons are shown by dots.

Two atomic orbitals can overlap in two ways depending on their phase relationship (or relative signs for real orbitals). The phase (or sign) of an orbital is a direct consequence of the wave-like properties of electrons. In graphical representations of orbitals, orbital phase is depicted either by a plus or minus sign (which has no relationship to electric charge) or by shading one lobe. The sign of the phase itself does not have physical meaning except when mixing orbitals to form molecular orbitals.

Two same-sign orbitals have a constructive overlap forming a molecular orbital with the bulk of the electron density located between the two nuclei. This MO is called the bonding orbital and its energy is lower than that of the original atomic orbitals. A bond involving molecular orbitals which are symmetric with respect to any rotation around the bond axis is called a sigma bond (σ-bond). If the phase cycles once while rotating round the axis, the bond is a pi bond (π-bond). Symmetry labels are further defined by whether the orbital maintains its original character after an inversion about its center; if it does, it is defined gerade, g. If the orbital does not maintain its original character, it is ungerade, u.

Atomic orbitals can also interact with each other out-of-phase which leads to destructive cancellation and no electron density between the two nuclei at the so-called nodal plane depicted as a perpendicular dashed line. In this

as in σ*. For a π-bond, corresponding bonding and antibonding orbitals would not have such symmetry around the bond axis and be designated π and π*, respectively.

The next step in constructing an MO diagram is filling the newly formed molecular orbitals with electrons. Three general rules apply:

• The Aufbau principle states that orbitals are filled starting with the lowest energy
• The Pauli exclusion principle states that the maximum number of electrons occupying an orbital is two, with opposite spins
• Hund's rule
  states that when there are several MO's with equal energy, the electrons occupy the MO's one at a time before two electrons occupy the same MO.

The filled MO highest in energy is called the

_g⁺ for dihydrogen. Sometimes, the letter n is used to designate a non-bonding orbital.

For a stable bond, the bond order defined as

${\displaystyle \ {\mbox{bond order}}={\frac {({\mbox{number of electrons in bonding MOs}})-({\mbox{number of electrons in antibonding MOs}})}{2}}}$

must be positive.

The relative order in MO energies and occupancy corresponds with electronic transitions found in

.

s-p mixing

The phenomenon of s-p mixing occurs when molecular orbitals of the same symmetry formed from the combination of 2s and 2p atomic orbitals are close enough in energy to further interact, which can lead to a change in the expected order of orbital energies.[9] When molecular orbitals are formed, they are mathematically obtained from linear combinations of the starting atomic orbitals. Generally, in order to predict their relative energies, it is sufficient to consider only one atomic orbital from each atom to form a pair of molecular orbitals, as the contributions from the others are negligible. For instance, in dioxygen the 3σ_g MO can be roughly considered to be formed from interaction of oxygen 2p_z AOs only. It is found to be lower in energy than the 1π_u MO, both experimentally and from more sophisticated computational models, so that the expected order of filling is the 3σ_g before the 1π_u.^[10] Hence the approximation to ignore the effects of further interactions is valid. However, experimental and computational results for homonuclear diatomics from Li₂ to N₂ and certain heteronuclear combinations such as CO and NO show that the 3σ_g MO is higher in energy than (and therefore filled after) the 1π_u MO.^[11] This can be rationalised as the first-approximation 3σ_g has a suitable symmetry to interact with the 2σ_g bonding MO formed from the 2s AOs. As a result, the 2σ_g is lowered in energy, whilst the 3σ_g is raised. For the aforementioned molecules this results in the 3σ_g being higher in energy than the 1π_u MO, which is where s-p mixing is most evident. Likewise, interaction between the 2σ_u* and 3σ_u* MOs leads to a lowering in energy of the former and a raising in energy of the latter.^[9] However this is of less significance than the interaction of the bonding MOs.

Diatomic MO diagrams

A diatomic molecular orbital diagram is used to understand the

The energies of the electrons are further understood by applying the

of the molecule. [13]

.

Dihydrogen

The smallest molecule, hydrogen gas exists as dihydrogen (H-H) with a single covalent bond between two hydrogen atoms. As each hydrogen atom has a single 1s atomic orbital for its electron, the bond forms by overlap of these two atomic orbitals. In the figure the two atomic orbitals are depicted on the left and on the right. The vertical axis always represents the orbital energies. Each atomic orbital is singly occupied with an up or down arrow representing an electron.

Application of MO theory for dihydrogen results in having both electrons in the bonding MO with electron configuration 1σ_g². The bond order for dihydrogen is (2-0)/2 = 1. The

The dihydrogen MO diagram helps explain how a bond breaks. When applying energy to dihydrogen, a molecular electronic transition takes place when one electron in the bonding MO is promoted to the antibonding MO. The result is that there is no longer a net gain in energy.

The superposition of the two 1s atomic orbitals leads to the formation of the σ and σ* molecular orbitals. Two atomic orbitals in phase create a larger electron density, which leads to the σ orbital. If the two 1s orbitals are not in phase, a node between them causes a jump in energy, the σ* orbital. From the diagram you can deduce the bond order, how many bonds are formed between the two atoms. For this molecule it is equal to one. Bond order can also give insight to how close or stretched a bond has become if a molecule is ionized.^[12]

Dihelium and diberyllium

(He-He) is a hypothetical molecule and MO theory helps to explain why dihelium does not exist in nature. The MO diagram for dihelium looks very similar to that of dihydrogen, but each helium has two electrons in its 1s atomic orbital rather than one for hydrogen, so there are now four electrons to place in the newly formed molecular orbitals.

The only way to accomplish this is by occupying both the bonding and antibonding orbitals with two electrons, which reduces the bond order ((2−2)/2) to zero and cancels the net energy stabilization. However, by removing one electron from dihelium, the stable gas-phase species He⁺
₂ ion is formed with bond order 1/2.

Another molecule that is precluded based on this principle is diberyllium.

Dilithium

MO theory correctly predicts that dilithium is a stable molecule with bond order 1 (configuration 1σ_g²1σ_u²2σ_g²). The 1s MOs are completely filled and do not participate in bonding.

Dilithium is a gas-phase molecule with a much lower

which considers the environment of each orbital due to all other electrons, both the 1σ orbitals have higher energies than the 1s AO and the occupied 2σ is also higher in energy than the 2s AO (see table 1).

Diboron

The MO diagram for diboron (B-B, electron configuration 1σ_g²1σ_u²2σ_g²2σ_u²1π_u²) requires the introduction of an atomic orbital overlap model for

orthogonally

). The p-orbitals oriented in the z-direction (p_z) can overlap end-on forming a bonding (symmetrical) σ orbital and an antibonding σ* molecular orbital. In contrast to the sigma 1s MO's, the σ 2p has some non-bonding electron density at either side of the nuclei and the σ* 2p has some electron density between the nuclei.

The other two p-orbitals, p_y and p_x, can overlap side-on. The resulting bonding orbital has its electron density in the shape of two lobes above and below the plane of the molecule. The orbital is not symmetric around the molecular axis and is therefore a

degenerate

) and can have higher or lower energies than that of the sigma orbital.

In diboron the 1s and 2s electrons do not participate in bonding but the single electrons in the 2p orbitals occupy the 2πp_y and the 2πp_x MO's resulting in bond order 1. Because the electrons have equal energy (they are degenerate) diboron is a

paramagnetic

.

In certain diborynes the boron atoms are excited and the bond order is 3.

Dicarbon

Like diboron,

Dinitrogen

With nitrogen, we see the two molecular orbitals mixing and the energy repulsion. This is the reasoning for the rearrangement from a more familiar diagram. The σ from the 2p is more non-bonding due to mixing, and same with the 2s σ. This also causes a large jump in energy in the 2p σ* orbital. The bond order of diatomic nitrogen is three, and it is a diamagnetic molecule.^[12]

The bond order for

(broad), the 2σ_g electrons at 37 eV (broad), the 2σ_u electrons at 19 eV (doublet), the 1π_u⁴ electrons at 17 eV (multiplets), and finally the 3σ_g² at 15.5 eV (sharp).

Dioxygen

Oxygen has a similar setup to H₂, but now we consider 2s and 2p orbitals. When creating the molecular orbitals from the p orbitals, the three atomic orbitals split into three molecular orbitals, a singly degenerate σ and a doubly degenerate π orbital. Another property we can observe by examining molecular orbital diagrams is the magnetic property of diamagnetic or paramagnetic. If all the electrons are paired, there is a slight repulsion and it is classified as diamagnetic. If unpaired electrons are present, it is attracted to a magnetic field, and therefore paramagnetic. Oxygen is an example of a paramagnetic diatomic. The bond order of diatomic oxygen is two. ^[12]

MO treatment of

.

The bond order decreases and the

Difluorine and dineon

In difluorine two additional electrons occupy the 2pπ* with a bond order of 1. In dineon Ne
₂ (as with dihelium) the number of bonding electrons equals the number of antibonding electrons and this molecule does not exist.

Dimolybdenum and ditungsten

Dimolybdenum (Mo₂) is notable for having a sextuple bond. This involves two sigma bonds (4d_z² and 5s), two pi bonds (using 4d_xz and 4d_yz), and two delta bonds (4d_{x² − y²} and 4d_xy). Ditungsten (W₂) has a similar structure.^[21][22]

MO energies overview

Table 1 gives an overview of MO energies for first row diatomic molecules calculated by the

, together with atomic orbital energies.

Table 1. Calculated MO energies for diatomic molecules in Hartrees [17]
	H2	Li2	B2	C2	N2	O2	F2
1σg	-0.5969	-2.4523	-7.7040	- 11.3598	- 15.6820	- 20.7296	-26.4289
1σu		-2.4520	-7.7032	-11.3575	-15.6783	-20.7286	-26.4286
2σg		-0.1816	-0.7057	-1.0613	-1.4736	-1.6488	-1.7620
2σu			-0.3637	-0.5172	-0.7780	-1.0987	-1.4997
3σg					-0.6350	-0.7358	-0.7504
1πu			-0.3594	-0.4579	-0.6154	-0.7052	-0.8097
1πg						-0.5319	-0.6682
1s (AO)	-0.5	-2.4778	-7.6953	-11.3255	-15.6289	-20.6686	-26.3829
2s (AO)		-0.1963	-0.4947	-0.7056	-0.9452	-1.2443	-1.5726
2p (AO)			-0.3099	-0.4333	-0.5677	-0.6319	-0.7300

Heteronuclear diatomics

In heteronuclear diatomic molecules, mixing of atomic orbitals only occurs when the

with dinitrogen) the oxygen 2s orbital is much lower in energy than the carbon 2s orbital and therefore the degree of mixing is low. The electron configuration 1σ²1σ*²2σ²2σ*²1π⁴3σ² is identical to that of nitrogen. The g and u subscripts no longer apply because the molecule lacks a center of symmetry.

In hydrogen fluoride (HF), the hydrogen 1s orbital can mix with fluorine 2p_z orbital to form a sigma bond because experimentally the energy of 1s of hydrogen is comparable with 2p of fluorine. The HF electron configuration 1σ²2σ²3σ²1π⁴ reflects that the other electrons remain in three lone pairs and that the bond order is 1.

The more electronegative atom is the more energetically excited because it more similar in energy to its atomic orbital. This also accounts for the majority of the electron negativity residing around the more electronegative molecule. Applying the LCAO-MO method allows us to move away from a more static Lewis structure type approach and actually account for periodic trends that influence electron movement. Non-bonding orbitals refer to lone pairs seen on certain atoms in a molecule. A further understanding for the energy level refinement can be acquired by delving into quantum chemistry; the Schrödinger equation can be applied to predict movement and describe the state of the electrons in a molecule.^[13][23]

NO

Nitric oxide is a heteronuclear molecule that exhibits mixing. The construction of its MO diagram is the same as for the homonuclear molecules. It has a bond order of 2.5 and is a paramagnetic molecule. The energy differences of the 2s orbitals are different enough that each produces its own non-bonding σ orbitals. Notice this is a good example of making the ionized NO⁺ stabilize the bond and generate a triple bond, also changing the magnetic property to diamagnetic.^[12]

HF

Hydrogen fluoride is another example of a heteronuclear molecule. It is slightly different in that the π orbital is non-bonding, as well as the 2s σ. From the hydrogen, its valence 1s electron interacts with the 2p electrons of fluorine. This molecule is diamagnetic and has a bond order of one.

Triatomic molecules

Carbon dioxide

. Carbon is the central atom of the molecule and a principal axis, the z-axis, is visualized as a single axis that goes through the center of carbon and the two oxygens atoms. For convention, blue atomic orbital lobes are positive phases, red atomic orbitals are negative phases, with respect to the wave function from the solution of the Schrödinger equation.^[24] In carbon dioxide the carbon 2s (−19.4 eV), carbon 2p (−10.7 eV), and oxygen 2p (−15.9 eV)) energies associated with the atomic orbitals are in proximity whereas the oxygen 2s energy (−32.4 eV) is different.^[25]

Carbon and each oxygen atom will have a 2s atomic orbital and a 2p atomic orbital, where the p orbital is divided into p_x, p_y, and p_z. With these derived atomic orbitals, symmetry labels are deduced with respect to rotation about the principal axis which generates a phase change, pi bond (π)^[26] or generates no phase change, known as a sigma bond (σ).^[27] Symmetry labels are further defined by whether the atomic orbital maintains its original character after an inversion about its center atom; if the atomic orbital does retain its original character it is defined gerade, g, or if the atomic orbital does not maintain its original character, ungerade, u. The final symmetry-labeled atomic orbital is now known as an irreducible representation.

Carbon dioxide’s molecular orbitals are made by the

(higher energy than either parent atomic orbital energy) molecular orbitals.

• MO model carbon dioxide
• 
  Atomic orbitals of carbon dioxide
• 
  Molecular orbitals of carbon dioxide
• 
  MO diagram of carbon dioxide

Water

For nonlinear molecules, the orbital symmetries are not σ or π but depend on the symmetry of each molecule. Water (H
₂O) is a bent molecule (105°) with C_2v molecular symmetry. The possible orbital symmetries are listed in the table below. For example, an orbital of B₁ symmetry (called a b₁ orbital with a small b since it is a one-electron function) is multiplied by -1 under the symmetry operations C₂ (rotation about the 2-fold rotation axis) and σ_v'(yz) (reflection in the molecular plane). It is multiplied by +1(unchanged) by the identity operation E and by σ_v(xz) (reflection in the plane bisecting the H-O-H angle).

C2v	E	C2	σv(xz)	σv'(yz)
A1	1	1	1	1	z	x2, y2, z2
A2	1	1	−1	−1	Rz	xy
B1	1	−1	1	−1	x, Ry	xz
B2	1	−1	−1	1	y, Rx	yz

The oxygen atomic orbitals are labeled according to their symmetry as a₁ for the 2s orbital and b₁ (2p_x), b₂ (2p_y) and a₁ (2p_z) for the three 2p orbitals. The two hydrogen 1s orbitals are premixed to form a₁ (σ) and b₂ (σ*) MO.

Mixing takes place between same-symmetry orbitals of comparable energy resulting a new set of MO's for water:

• 2a₁ MO from mixing of the oxygen 2s AO and the hydrogen σ MO.
• 1b₂ MO from mixing of the oxygen 2p_y AO and the hydrogen σ* MO.
• 3a₁ MO from mixing of the a₁ AOs.
• 1b₁ nonbonding MO from the oxygen 2p_x AO (the p-orbital perpendicular to the molecular plane).

In agreement with this description the photoelectron spectrum for water shows a sharp peak for the nonbonding 1b₁ MO (12.6 eV) and three broad peaks for the 3a₁ MO (14.7 eV), 1b₂ MO (18.5 eV) and the 2a₁ MO (32.2 eV).^[29] The 1b₁ MO is a lone pair, while the 3a₁, 1b₂ and 2a₁ MO's can be localized to give two O−H bonds and an in-plane lone pair.^[30] This MO treatment of water does not have two equivalent rabbit ear lone pairs.^[31]

Hydrogen sulfide (H₂S) too has a C_2v symmetry with 8 valence electrons but the bending angle is only 92°. As reflected in its photoelectron spectrum as compared to water the 5a₁ MO (corresponding to the 3a₁ MO in water) is stabilised (improved overlap) and the 2b₂ MO (corresponding to the 1b₂ MO in water) is destabilized (poorer overlap).

References

External links

• MO diagrams at meta-synthesis.com Link
• MO diagrams at chem1.com Link
• Molecular orbitals at winter.group.shef.ac.uk Link