The Molecular Orbital Theory, Chemistry tutorial


In atoms, electrons take up atomic orbitals, but in molecules they dwell in similar molecular orbitals that surround the molecule. The simplest molecule is hydrogen, which can be considered to be made up of 2 divide protons and electrons. There are 2 molecular orbitals for hydrogen; the lower energy orbital has its superior electron density between the 2 nuclei. This is the bonding molecular orbital - and is of lower energy than the two 1s atomic orbitals of hydrogen atoms making this orbital more steady than 2 separated atomic hydrogen orbitals.

The upper molecular orbital has a nodule in the electronic wave function and the electron density is low among the 2 positively charged nuclei. The energy of the upper orbital is greater than that of the 1s atomic orbital and such an orbital is termed an antibonding molecular orbital (W. Locke and the ICSTM, 1996, 1997).

Normally, the 2 electrons in hydrogen occupy the linking molecular orbital, through anti-parallel spins. If molecular hydrogen is irradiated via ultra-violet (UV) light, the molecule might absorb the energy, and endorse one electron into its antibonding orbital, and the atoms will separate. The energy levels in a hydrogen molecule can be symbolized in a diagram - explaining how the two 1s atomic orbitals join to form two molecular orbitals, one linking and one antibonding. This is exposed below - by clicking upon either the or molecular orbital in the diagram - it will illustrate graphically in a window to the right:

1891_The Energy Levels in a Hydrogen Molecule.jpg

Fig: The Energy Levels in a Hydrogen Molecule

Definitions of Molecular Orbital Theory

In chemistry, molecular orbital theory (MO theory) is a method for determining molecular structure in which electrons aren't allocated to individual bonds among atoms, but are delighted as moving under the pressure of the nuclei in the whole molecule. In this theory, each molecule has a set of molecular orbitals, in that it is supposed that the molecular orbital wave function ψf might be written as an easy weighted sum of the n component atomic orbitals χi, according to the following equation:

The cij coefficients may be computed numerically by substitution of this equation into the Schrödinger equation and application of the vibrational principle. This technique is termed the linear grouping of atomic orbitals approximation and is utilized in computational chemistry.

Another unitary alteration can be applied on the system to speed up the convergence in several computational schemes. Molecular orbital theory was seen as a opponent to valence bond theory in the year 1930s, before it was realized that the two process are intimately related and that when enlarged they become equivalent.

Some Properties of Molecular Orbital

The number of molecular orbitals is equivalent to the number the atomic orbitals comprised in the linear growth, If the  molecule  has  several  symmetry,  the  degenerate atomic orbitals  (through  the  similar  atomic  energy)  are  collection  in  linear combinations (termed  symmetry  adapted  atomic  orbitals  (SO)) that belong to the demonstration of the symmetry group, so the wave functions that explain the group are recognized as symmetry- adapted linear combinations (SALC).

The number of molecular orbitals belonging to one group representation is equal to the number of symmetry adapted atomic orbitals belonging to this representation, Within a particular representation, the symmetry adapted atomic orbitals mix more if their atomic energy levels is closer.

Examples on Molecular Orbital Theory


H2 1sσ* antibonding molecular orbital

As a simple instance consider the hydrogen molecule, H2 (see molecular orbital diagram), with the two atoms labelled H' and H". The lowest-energy atomic orbitals, 1s' and 1s", don't transform according to the symmetries of the molecule. Though, the subsequent symmetry Adapted atomic orbitals do:

1s' - 1s" Antisymmetric mixture:  negated via reflection, unchanged through other operations

1s' + 1s" Symmetric combination: unchanged via all symmetry operations the symmetric combination (termed a bonding orbital) is lower in energy than the foundation orbitals, and the antisymmetric combination (termed an antibonding orbital) is higher. Because the H2 molecule has 2 electrons, they can together go in the linking orbital, creation the system lower in energy (and therefore more stable) than 2 free hydrogen atoms. This is described a covalent bond. The bond order is equal to the number of linking electrons minus the number of antibonding electrons, separated via 2. In this instance there are 2 electrons in the bonding orbital and none in the antibonding orbital; the bond order is 1, and there is a single bond between the 2 hydrogen atoms.


On the other hand, consider the hypothetical molecule of He2, the atoms labelled He' and He". Again, the lowest-energy atomic orbitals, 1s' and 1s", don't change according to the symmetries of the molecule, while the subsequent symmetry adapted atomic orbitals do:

1s' - 1s" Antisymmetric combination:  negated via reflection, unchanged by other operations

1s' + 1s" Symmetric combination: unchanged through all symmetry operations

Similar  to  the  molecule  H2, the  symmetric  combination  (called  a bonding  orbital)  is  lower  in  energy than  the  basis orbitals, and  the antisymmetric  combination  (called an  antibonding  orbital) is  higher. Though, in its neutral ground state, each Helium atom encloses 2 electrons in its 1s orbital, combining for a total of 4 electrons. Two electrons fill the lower energy linking orbital, while the remaining 2 fill the higher energy antibonding orbital. Therefore, the consequential electron density around the molecule doesn't support the formation of a bond between the 2 atoms (termed a sigma bond), and the molecule does therefore not exist. Another way of looking at it is that there are two bonding electrons and 2 antibonding electrons; therefore, the bond order is 0 and no bond exists.

MO diagrams

For more complicated molecules, the wave mechanics approach loses utility in a qualitative understanding of bonding (even though is still needed for a quantitative approach).The qualitative approach of MO utilizes a molecular orbital diagram. In this kind of diagram, the molecular orbitals are symbolized via horizontal lines; the higher a line, the higher the energy of the orbital, and degenerate orbitals are situated on the similar level through a space between them. Then, the electrons to be situated in the molecular orbitals are slotted in one via one, keeping in mind the Pauli exclusion principle and Hund's rule of maximum  multiplicity (only 2 electrons, having opposite spins, per orbital; have as many unpaired electrons on one energy level as possible before starting to pair them).


The highest occupied molecular orbital and lowest unoccupied molecular orbital are often termed to as the HOMO and LUMO, correspondingly. The difference of the energies of the HOMO and LUMO, termed the band gap, can sometimes serve as a compute of the excitability of the molecule: the lesser the energy, the more simply it will be excited.

Atomic and Molecular Orbitals

Through sharing electron, molecules can shape bonds, and it is probable to regard the sharing of two electrons via 2 atoms as constituting a chemical bond. Atoms can divide one, two or three electrons (forming single, double and triple bonds). A hydrogen atom consists of a nucleus (a proton) and an electron. It isn't possible to precisely verify the position of the electron, but it is possible to compute the probability of discovering the electron at any point around the nucleus. By a hydrogen atom the probability distribution is spherical around the nucleus and it is possible to draw a spherical limit surface, within that there is a 95per cent possibility of discovering the electron. The electron has a joined energy and a fixed spatial distribution said an orbital. In the helium atom, there are 2 electrons associated by the helium nucleus. The electrons have the similar spatial allocation and energy (for example they occupy the similar orbital), but they differ in their spin (Pauli's Exclusion Principle). In common: electrons in atomic nuclei occupy orbitals of fixed energy and spatial distribution, and each orbital merely encloses a maximum of 2 electrons through anti- parallel spins.

Orbitals for Selected Molecules

This section demonstrates pictorially molecular orbitals for numerous organic and inorganic molecules. If possible - the energy level diagram is comprised and clicking upon the relevant level will produce the accompanying molecular orbital in the right-hand frame (W. Locke and the ICSTM, 1996, 1997).

1864_The Energy Levels in Different Molecules.jpg

 Fig: The Energy Levels in Different Molecules


The ethane molecule has 14 valence electrons occupying seven bonding molecular orbitals. As can be seen from the energy diagram - 4 of the molecular orbitals take place as degenerate pairs.  Like in methane - the molecular orbitals of ethane show increasing nodal structure with increasing orbital energy (W. Locke and the ICSTM, 1996, 1997).

309_The Molecular Orbitals of Ethane (Show  Increasing).jpg

Fig: The Molecular Orbitals of Ethane (Show Increasing Nodal Structure with Increasing Orbital Energy)

Molecules with double bonds

In molecules where the number of bonding electron pairs exceeds the number of unions between atoms, the extra electrons occupy higher energy molecular orbitals than the orbitals found in molecules where the number of bonding electron pairs equals the number of unions between atoms. Such are double bonds, and the orbitals contain a nodal plane containing the atoms sharing these π-type orbitals.


The easiest alkene is ethene. Its chemistry is dominated via two 'frontier orbitals' that is the Highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO). For the ethene orbital energy diagram such are exposed as πCC for the HOMO, and π* for the LUMO.An significant property of the ethene molecule, and alkenes in general is the existence of a high barrier to rotation about the C=C which tends to hold the molecule flat (W. Locke and the ICSTM, 1996, 1997). For the energy diagram and pictorial vision of the orbitals - please see below:

1775_The Molecular Orbitals of  Ethene (Show  Increasing).jpg

Fig: The Molecular Orbitals of Ethene (Show Increasing Nodal Structure with Increasing Orbital Energy)

Molecules with triple bond


For the energy diagram and pictorial observation of the orbitals - please see below:

1526_The Molecular Orbitals of Ethyne.jpg

Fig: The Molecular Orbitals of Ethyne (Show Increasing Nodal Structure with Increasing Orbital Energy)

Molecules with electron lone pairs

Hydrogen Fluoride

An easy diatomic molecule is Hydrogen fluoride. There are 8 valence electrons that occupy 4 molecular orbitals. The two highest energy MOs are degenerate, are π-type and have no electron density associated with the hydrogen atom, for example they are Non-Bonding Orbitals (NBO) and in Lewis Theory are represented as two 'Lone Pairs'. Another important difference between Hydrogen Fluoride and preceding molecules is that the electron density isn't uniformly distributed about the molecule. There is a much greater electron density about the fluorine atom. This is since fluorine is an extremely electronegative element, and in each linking molecular orbital, fluorine will get a greater divide of the electron density (W. Locke and the ICSTM, 1996, 1997).

2440_The Molecular Orbitals of Hydrogen Fluoride.jpg

Fig: The Molecular Orbitals of Hydrogen Fluoride (Show Increasing Nodal Structure with Increasing Orbital Energy)

Valence Bond Model versus Molecular Orbital Theory

Since arguments depend on atomic orbitals center on the bonds shaped between valence electrons on an atom, they are often said to engage a valence-bond theory. The valence-bond model can't adequately explain the fact that several molecules contain 2 equivalent bonds through a bond order between that of a single bond and a double bond. The best it can do is proposes that such molecules are mixtures, or hybrids, of the two Lewis structures that can be written for these molecules.

This problem, and many others, can be overcome via using a more sophisticated model of bonding depends on molecular orbitals. Molecular orbital theory is more powerful than valence-bond theory as the orbitals reflect the geometry of the molecule to which they are applied. But this power carries a important cost in terms of the ease by which the model can be visualized.

Using the Molecular Orbital Model to Explain Why Some Molecules Do Not Exist

This molecular orbital model can be utilized to describe why He2 molecules don't exist. Combining a pair of helium atoms by 1s2 electron configurations would create a molecule by a pair of electrons in both the bonding and the antibonding molecular orbitals. The total energy of a He2 molecule would be basically the similar as the energy of a pair of isolated helium atoms, and there would be nothing to hold the helium atoms mutually to shape a molecule.

The fact that a He2 molecule is neither more nor less stable than a pair of separated helium atoms demonstrates a significant principle: The core orbitals on an atom make no contribution to the stability of the molecules that enclose this atom. The only orbitals that are important in our discussion of molecular orbitals are those shaped whenever valence- shell orbitals are joined. The molecular orbital diagram for an O2 molecule would as a result ignore the 1s electrons on together oxygen atoms and concentrate on the interactions between the 2s and 2p valence orbitals.

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