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Molecular Orbitals Study Guide

11 Jan 2017Guides

Molecular Orbitals Study Guide



Molecular Orbital Theory Summary

When Atoms Bond

Most of the time MO comes from Modus Operandi, but when talking chemistry one thing is for sure: MO stands for Molecular Orbital Theory. MO defines the molecular structure by looking at the molecular orbitals taking the molecule as a reference, and not the individual atoms that comprise it. It starts from the idea that the electrons do not belong to the atom anymore or to a bond, but are under the power of the molecule in its entirety.

Let’s take it one step at a time. All matter is made of atoms. When we talk about chemical bonding, the subatomic particle of an atom that represents the most interest is the electron. Electrons occupy orbital levels. These orbits are not to be taken literally, as they do not exist in reality, but they are mathematical functions that describe the motion of an electron or a pair of electrons by using a wave function. The wave functions reveal the probability of finding an electron at a certain distance from the nucleus. When two atoms bond, the atomic orbitals combine and MO Theory analyses how the orbital shapes combine within the molecule.

Different atom interaction results in different orbitals

When bonds between atoms are formed, the orbitals within the molecule are identified using a linear combination of atomic orbitals or LCAO (yes, another interesting string of letters). This helps us put orbitals in three main categories, depending if the atoms interfere in a constructive or destructive manner. The three categories are:

  • Bonding orbitals
  • Anti-bonding orbitals
  • Non-bonding orbitals

Whenever we have a constructive interference of the two atoms, the electron density found between the atoms will bring the two nuclei of the atoms together and make a strong connection between them. The resulting molecular orbital is lower in energy and is derived from the sum of the atomic energy levels. This type of molecular orbital, the bonding orbital, is called sigma (Σ).

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If the atoms forming the molecule have a destructive interference, we can speak of an anti-bonding orbital. This type of orbital concentrates the electron density on the outer sides of the nuclei, and this results in a weakened bond between the atoms. The two nuclei are pulled away from each other so forming an anti-orbital bond. This wave function is higher in energy than the atoms and derives from their difference. The anti-bond orbital is called sigma-starred (σ*).

A non-bonding orbital is the on in which the position the electrons occupy does not interfere with the bond order between the atoms. The energy level in the molecule is the same as in the atomic system, placing the energy level of a non-bonding orbital between the bonding and the anti-bonding orbitals. They are depicted in diagrams with the letter n.

Bond order

As bonding between the atoms inside the molecule is determinant, we can understand the bonding in the molecule by calculating the bond order. First, we need to determine the difference between the number of electrons in bonding orbitals and the number of electrons in anti-bonding orbitals. The bond order is calculated by finding the half of that difference. If we take the example of hydrogen, because it has two electrons in a bonding orbital and no electron in anti-bonding orbital we calculate (2-0)/2 and we get a bond order of one. This means there is a single bond connecting the two hydrogen atoms in the H2 molecule.

Why do we need to understand Molecular Orbital Theory in the first place?

The short answer to this question is that by calculating molecular orbital you understand the geometry of a molecule. In other words, you can determine what are the bond lengths, angles and strengths. The quantum mechanical calculation of the MO depicts a picture of the wave functions and energies of the molecule. On a scale ranging from the lowest energy orbitals to the highest ones, you can place electrons in a molecule and then their orbitals and energies are calculated for a certain position of the nucleus. This allows us to determine the energy for a molecule in a given geometry and by repeating the process, finding the right geometry that gives us the minimum energy configuration, also known as the ground state of the molecule.

Molecular Orbital Terms

Molecules are formed when atoms are joined together by chemical bonds. In each atom, there is the nucleus that is made up predominantly of protons and neutrons, and which is surrounded by an electron cloud. The proton is positively charged, while the neutron contains no charge. An electron is negatively charged, and thus the electron cloud is negatively charged. This electron cloud is composed of electron energy shells. Each energy shell contains electrons that are relatively in the same energy state. These energy shells are also called orbitals. The position of any electron in this electron cloud is determined by its energy state, with electrons in low energy states occupying the electron energy shell that is closer to the nuclei, while electrons in high energy states occupy energy shells that are further from the nuclei. During the process of chemical bonding, the electrons in the outermost energy shell in the electron cloud of each atom interact together to create a molecular bond. For atoms to bond together, their outermost energy shells must be unfilled, that is, it has one or more unpaired electrons. An unpaired electron can take part in a chemical reaction and combine with another unpaired electron to form a stable electron pair which is the foundation of a chemical bond.

Molecular Orbitals

In the resulting molecule, a new nucleus is formed through the bonding of the individual atomic nuclei. This bonding results from the bond created when the electrons of the different atoms are shared by the different atoms. The new nuclei are the molecule is referred to as the molecular nuclei, and the electrons from the different atoms form the molecular electron cloud. However, some of the electrons in the different atoms that have formed the molecules are not assigned to any specific chemical bond between the atoms, but they are still under the influence of the molecular nuclei. These electrons orbit around the molecular nuclei and their position within the molecular electron cloud can be determined using the principles of quantum mechanics. Also, because these electrons are under the influence of the molecular nuclei, they are regarded as bonded to the molecular nuclei, and hence participate in the chemical bonding of the molecule, and are thus also described as bonded electrons.

A theory called Molecular Orbital Theory has been postulated to explain the bonding patterns of bonded electrons in the molecule, as well as calculate their position within the molecular electron cloud, as well as describe their relations to the respective atomic orbitals of the bonded atoms that make up the molecule. According to this theory, there are three types of molecular orbitals; bonding orbitals, non-bonding orbitals, and anti-bonding orbitals. Each of these types of the molecular orbital is defined hereafter, alongside other terms used to describe the properties and essence of molecular orbitals.

Molecular Orbital Terms

Chemical Bond: Force that joins two or more atoms together in a chemical compound. The main types of chemical bonds are ionic bonds, covalent bonds, and the much weaker electrostatic hydrogen bonds. Covalent bonding allows different atoms to form a molecule.

Covalent Bond: This is a chemical bond formed when unpaired electrons in the outermost energy shells of an atoms combine with another unpaired electron from another atom, so that the outermost energy shells of each atom are filled, and the paired electrons are shared equally between the bonded atoms.

Valence electrons: These are the electrons in the outermost energy shell that participate in the formation of a covalent bond.

Molecular Orbital: It is the electron cloud formed around the molecular nuclei, and contains both valence electrons and the electrons that have not been assigned to any of the chemical bonds in the molecule.

Atomic Orbital: The energy shells that form the electron cloud around the positively-charged nuclei of an atom. These energy shell are assigned hydrogenic wavefunctions using the sp3 notation system. In this system, the innermost energy shell that surrounds the atomic nucleus can only be occupied by a pair of electrons and is assigned the s notation to denote the s shell. The other outer energy shells can be occupied by a maximum of four pairs of electrons, that is electrons; and the energy shell that surrounds the s shell is assigned the p1 value, while the energy that surrounds the p1 shell is assigned the p2 value, and that which surrounds the p2 shell is assigned the p3 value.

Bonding Molecular Orbital: This is the orbital that forms when two different atomic orbitals that are in-phase to each other overlap, and consequently, the electrons in the overlapped region acquire a lower energy state than they had in their parent atomic orbitals. These low energy electrons stabilise the molecular bond. 

Overlap: A Measure of constructive interference between two atomic orbitals.

Bonding Orbital: Orbital formed when electrons in the electron cloud concentrate between the two atoms so that the resulting electron density attracts the nuclei of each atom towards one another.

Anti-bonding orbital: Orbital formed when out-of-phase atomic orbitals overlap, thus causing the resultant electron density around each atom to pull the nuclei of the two atoms away from each other. Bonding Order: This is the difference in number between bonding electron pairs (that form a bonding orbital) and anti-bonding electron pairs (that form an anti-bonding orbital). Non-bonding Orbital: It contains electrons still attached to their atomic orbitals and do not cause the individual atoms to be attracted to each other, or to move away from each other.

Molecular Orbital Theory Details

Molecular orbital theory is a scientific deduction used to determine the structure of a molecule. This theory was obtained from the works of Frederich Hund, Robbert S. Mulliken, John Leonard Jones and John C.Slater. It is based on electrons moving under influence of nuclei in the molecule as opposed to assigning individual bonds between atoms. Two pairs of electrons occupy a molecular orbital after two atoms are bonded together. A number of orbitals in a molecule is equal to a number of orbitals in the combining atoms. Combining two atoms forms a molecule and two molecular orbitals. These are bonding and anti-bonding orbitals called sigma and sigma-starred respectively. Sigma is lower in energy than atomic orbitals, a derivative of their sum. On the other hand, sigma-starred is lower in energy than the bonding orbitals, resulting from their difference.

Quantitative Applications

The molecular orbital theory builds on electron wavefunctions of Quantum Mechanics to describe chemical bonding. In MO theory each molecule has a given set of orbitals. The quantitative application is based on the assumption that the wave function of the molecular orbital can be written as a sum of all constituent atomic orbitals. Substituting the equation into Schrodinger`s equation helps in obtaining the linear coefficients numerically. This is followed by the application of the variational principle, a mathematical technique used to construct coefficients of individual atomic orbital basis. A macro coefficient means orbitals contains more of that particular orbital. The molecular orbital is then characterised as that type. This method of sing linear combinations to quantify orbital contributions is utilised in computational chemistry. Some computational schemes can be accelerated by applying an additional unitary transformation to the system.


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Nature of Orbitals

Molecules can form bonds by sharing electrons and sharing of two atoms is characterised as chemical bonding. Atoms share one, two or three electrons to form single, double or triple bonds. The position of an electron cannot be accurately determined but the probability of finding it at any point in the nucleus can be calculated. Hydrogen atom which consists of a nucleus(proton) and an electron has a 95 percent chance of locating its electron. This electron has a fixed energy and fixed spatial distribution called an orbital. This is the same to electrons found in helium except their spin differs according to Pauli exclusion principle. Generally, nuclei electrons occupy orbitals of fixed energy and spatial distribution. Each orbital contains a maximum of two electrons with anti-parallel spins.

Type of Orbitals

As noted earlier, atoms share electrons to form single or multiple bonds. Single bonds are always sigma bonds while in multiple bonds, the first bond is sigma and the second and third bonds are pi bonds. p orbitals overlap to yield either sigma or pi molecular orbitals. The latter are formed by side by side overlap whereas sigma molecular orbitals are formed when overlapping occurs end to end. Applying Schrodinger's equation in three dimensions helps in determining the shape of each orbital. In a hydrogen atom, the first 1s orbital has the lowest energy. The remaining 2s, 2px, 2py and 2pz have the same amount of energy. In all other atoms, 2s orbital is lower energy than 2px, 2py and 2pz orbitals described as degenerate. Electrons occupy atomic orbitals in atoms but in molecules, they occupy similar molecular orbitals surrounding the molecule.

Hydrogen, a simple molecule, is considered to have two separate protons and electrons. Therefore, hydrogen has two molecular orbitals where the lower energy orbital contains a greater electron density between the two nuclei. This is the bonding molecular orbital. It's of lower energy compared two atomic orbitals of hydrogen atoms. This makes it more stable than two separated hydrogen orbitals. The upper molecular orbital with a node in the sine wave function has more energy making it an antibonding molecular orbital. Helium molecule has four orbitals where two are lower energy bonding orbitals and the other two are higher energy sigma-starred orbitals.

Bond Order

The bonding and anti-bonding electrons contained in orbitals are used to determine the bond order. This order is obtained by dividing the difference of bonding and antibonding electrons by two. For instance, hydrogen molecule has a bond order of 1 which means there is a single bond connecting two hydrogen atoms in the hydrogen molecule. Helium has a bond order of (2-2)/2 = 0 which means it's not a stable molecule.

The molecular theory is applicable in both small and large molecules. It can be employed in molecules like carbon(iv)oxide and ions like nitrate where bonding molecules are delocalized. MO theory is also useful in analysing aromatic molecules like benzene. Here all six C atoms are involved in the delocalized pi cloud enveloping the entire molecule. This theory can also be stretched to complex ions and solids including materials like semiconductors and superconductors. The occurrence of molecular orbitals varies with selective molecules. These are saturated molecules, molecules with lone electron pairs, conjugated and aromatic molecules, molecules with double or triple bonds and other intersecting molecules.

Molecular Orbital Problems and Solutions

Molecular orbital

Molecular orbital refers to chemistry discipline of the mathematical function which describes the wave-like characteristic of an electron within a molecule. A molecular orbital is used in a calculation of physical and chemical properties of finding the probability of the location of the desired electron in a particular region.

The scientific term orbital was established in the year 1992 by Robert S. Mulliken. It was an abbreviation for a single orbital wave function. The orbital wave function is used in explaining the region of space through which the function has a meaningful amplitude at an elementary level. Molecular orbital modelling is done through a combination of atomic orbitals from every atom of a molecule or different molecular orbitals resulting from a conglomeration of atoms. They can easily be quantitatively calculated via Hartree-Fock techniques.

Brief overview

Regions within a molecule where the electron occupies the orbital can be represented by the use of a molecular orbital. Molecular orbitals are found through a combination of different atomic orbitals. This provides the probability of locating the electron within an atom.

Molecular orbital furthermore can be used to show electron configuration within an atom. The majority of today’s technique in computational chemistry starts by computing the system of a molecular orbital. It explains the character of a single electron within the electronic field produced by the nuclei and other standard distribution of some electrons. Pauli Principle requires that in a situation where double electrons occupy the same orbital, they should have opposite rotation. This is just a mere approximation but there is a more accurate explanation of molecular electronic wave functions that do not have orbitals.

Orbitals in a molecule come about as a result of permitted interaction among atomic orbitals which occur when symmetries belonging to the atomic orbital correctly fit each other. An overlap between double atomic orbitals figures out the usefulness of atomic orbital interaction. The efficiency of the atomic orbital is very significant if their energy is closely related. Also, molecular orbital should be equal in number just like those of the atoms being combined to make up the molecule.

Linear combination of atomic orbitals

This refers to the quantum super positioning of the atomic orbitals and also a method for computing molecular orbitals in a discipline called quantum chemistry. Wave functions refer to the description of electronic configuration in quantum mechanics. Mathematically, the mentioned Wave functions form a base set functions that describe the electrons of a particular atom. Modification occurs to the wavefunctions when they are subjected to various chemical reactions. Here, the electron cloud outlook is modified as per the category of the atom taking part in the chemical bond.

A Linear combination of atomic orbitals was introduced in the year 1929 by Sir John Lennard-Jones. He described the bonding present in diatomic molecules belonging to the first major row of the chemistry periodic table. This had been described by a scholar by the name Linus Paulin for H2+.

Mathematical description of linear combination of atomic orbitals

To describe this mathematically, we assume that the number of atomic orbitals and the number of atomic orbitals which are in the linear expansion are equal. A given number of atomic orbitals combines and as a result leads to the formation a particular number of molecular orbitals. Hartree-Fock method is utilised to determine the expansion coefficients. Therefore, the orbitals are then expressed in terms of a linear combination of the basis. Basis functions are single electron function that may either be centred or not on the nuclei of component atom belonging to the molecule. The basis functions are called atomic orbitals. Atomic orbitals used in the process are those of the hydrogen atoms. This is because they are known analytically.

The Linear combination of coefficients is then obtained through reduction of total energy of the system. This approach used quantitatively is referred to us Hartree-Fock technique. Since the establishment of computational chemistry, LCAO technique usually does not consider the actual optimisation of wave function but a qualitative discussion that is very important in rationalising and predicting the outcome results determined by modern methods.

The outlook of the molecular orbitals with their energies is approximately determined through comparison of the atomic orbitals of the atoms and also imposing recipes called level repulsion. The graphs drawn to bring a clear description are known as correlation diagrams. Koopmans theorem is crucial in determining the atomic orbital energies. The theorem is important in the energy calculation or conducting the experiment. It is carried out through the use of orbitals and molecules that participate in the bonding. This is sometimes known as Symmetry Adaptation Linear Combination (SALC).The starting step in carrying out the process is allocating the molecule a point group.

Water is the most common example that shows C2v symmetry. The bonding reducible representation for water is then determined. Every operation within the point group is done on the molecule. The characteristic of the stated operation is the number of the bonds that are not moved. There is then decomposition of reducible representations into various irreducible representations. The obtained irreducible representations are equivalent to the symmetry of orbitals that takes part in the process.


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