Introduction

Molecular orbitals are generally constructed via combining atomic orbitals or hybrid orbitals from each atom of the molecule, or additional molecular orbitals from groups of atoms. A molecular orbital (MO) can be utilized to identify the electron configuration of a molecule: the spatial allocation and energy of one (or one pair of) electron(s). Most generally, a MO is symbolized as a linear combination of atomic orbitals (the LCAO-MO process); in particular in qualitative or extremely approximate usage. They are priceless in providing an easy reproduction of bonding in molecules, understood through molecular orbital theory. Most process in computational chemistry today, starts via cunning the MOs of the system. A molecular orbital explains the performance of one electron in the electric field produced through the nuclei and several average allocations of the other electrons. In the case of 2 electrons occupying the similar orbital, the Pauli Exclusion Principle demands that they have opposite spin. Essentially this is an  approximation, and extremely precise descriptions of the molecular electronic wave function don't have orbitals.

Definition of Molecular Orbital

In chemistry, a molecular orbital (MO) is a numerical function that explains the wave-like behavior of an electron in a molecule. This function can be utilized to compute chemical and physical properties these as the probability of locating an electron in any precise region. The utilize of the term 'orbital' was 1^st utilized in English by Robert S. Mulliken in the year 1925 as the English translation of Schrödinger's use of the German word, 'Eigenfunktion'. It has since been equated through the 'region' produced by the function. They can be quantitatively computed using the Hartree-Fock or Self-Consistent Field technique

Molecular Orbital (MO) Description of Homonuclear Diatomic Molecules

In this section, the molecular orbital description of homonuclear diatomic molecules will be discussed. In the 1^st approximation only the atomic orbitals having similar energy will combine to form the MO. In this approximation the simplest way to shape molecular orbital is to combine the equivalent orbitals on 2 atoms (for instance 1s + 1s, 2s + 2s, etc.). The suitable combinations are

(1s) = 1s_A + 1s_B

(1s)* = 1s_A - 1s_B

(2s) = 2s_A + 2s_B

(2s)* = 2s_A - 2s_B

(2p) = 2px_A + 2px_B

(2p)* = 2px_A - 2px_B

According to this approximation, the 1s orbitals from 2 dissimilar atoms will form σg(1s)* and σu(1s)* just like the MO's of hydrogen molecule. In the similar way 2s orbitals from 2 different atoms will form σg(2s)* and σu(1s)* MO. Such 2 molecular orbitals will appear as σg(2s) and σu(2s)*. Because an atomic 2s orbital has higher energy than atomic 1s orbital σg(2s) and σu(2s)* MO will have superior energy in comparison to σg(2s) and σu(2s)* MO correspondingly. So the ordering of the molecular orbitals discussed so far will be σg(2s) < σu(2s)* < σg(2s) < σu(2s)*. Now let us converse the combination of the 2p orbitals. Except hydrogen atom 2p orbital energy is higher than the 2s orbital energy; the bonding and antibonding molecular orbital shaped from the combination of  2p  orbital  will  have  higher  energy  than σg(2s) and σu(2s)*correspondingly. If we consider the x axis as molecular axis then such 2 orbitals will be indicated as σg(2p) and  σu(2p)*.The remaining 2p orbitals for example 2py, and 2pz will form π molecular orbitals by lateral overlapping. The bonding and antibonding molecular orbital forming by overlapping 2py orbitals are symbolized πu(2py) and πg(2py)*. Such two orbital will be directed towards Y axis. The 2pz combined in the alike way will consequence in bonding and antibonding π MO but such will be expressed along the Z axis.

The bonding π orbitals arising from the combination of two 2py atomic orbitals and two 2pz will degenerate and similarly the antibonding π* MOs. Now to find out the electron configuration of molecule via placing the electrons in such MOs in accord to the Pauli's exclusion principle and Hund's rule just like multi-electronic atoms, we require knowing the energy ordering of such molecular orbitals. The energy of the molecular orbitals based on the atomic number (atomic charge) on the nuclei. As the atomic number enhances from lithium to fluorine, the energy of the σg(2p) and energy of πu(2py) and πu(2pz) orbitals approaches to each other and interchange order in going from N₂ and O₂. But in case of N₂ molecule the MO diagram, the energy of the σg(2p) orbital will be less than that of πu(2py) and πu(2pz) orbitals .

Bond order

The net bonding in a diatomic molecule is described via a quantity termed bond order, b; b = 2(n-n*) (13.9).Where n is the number of electron in the bonding orbital and π* is the number of electrons in the antibonding orbital. This is an extremely helpful quantity for describing the characteristics of bonds, since it correlates through bond length and bond strength. If the bond order between atoms of a specified pair of elements is higher than the bond length will be shorter and as a result the bond will be stronger. The bond strength is calculated via bond dissociation energy that is the energy needed to divide the atoms to infinity. Single bonds have bond order one; double bonds have two; and so on. The bond order zero specifies that there is no bond between the given pair of atoms. For instance, bond order for He₂ is zero, since there are 2 electrons in both bonding and antibonding orbital. For this cause He₂ molecule doesn't exist.

Some Important Consequences from Molecular Orbital

Theory calculation whether a molecule exists or not: From MO theory we can predict whether a diatomic molecule exists or not by simply calculating the bond order. If the value is greater than zero the molecule will exist. For example the ground state electronic configuration of He₂⁺ is σ(1s) 2σ(1s)*2 , and the bond order is 0.5 . So He₂⁺ ion exists but because the bond order of the He2 molecule is zero it will not exist.

Lithium and beryllium molecules: The 6 electrons from 2 lithium atoms will fill in the molecular orbital according to Aufbau principle. Four will fill in the σg(1s) and σg(1s)* by no bonding. The last 2 electrons will enter in the σg(2s) orbital. Hence the bond order in the lithium molecule will be one and the electronic pattern will be KK σg(2s) 2 where K stands for the K (1s) shell. The electronic configuration of beryllium molecule will be KK σ (2s)2 σ (2s)*2.

Since  the  number  of  electrons  in  the  bonding  and  anti-bonding molecular  orbital  is  equal, the bond order will be zero. Hence like diberilium molecule (Be₂) molecule does not exist. Experimentally it is found that lithium is diatomic and beryllium is mono-atomic in the gaseous phage. Paramagnetic property of oxygen molecule:  One of the most impressive successes of the molecular orbital theory is the prediction of correct electronic configuration of oxygen molecule. The experimental result shows that the oxygen molecule is paramagnetic, the net spin of oxygen molecule corresponds to two unpaired electron having same spin. According to molecular theory  the ground state of oxygen molecule is KK σ(2s)2 σ(2s)*2 σ(2px)2 π(2py)2π (2pz)2 π(2py)1π(2pz)1.

According to Hund's rule two electrons having parallel spun will occupy two degenerate π (2py) and π (2pz) orbital. This explains the magnetic behavior of oxygen molecule.

Molecular Orbital Theory of Heteronuclear Diatomic Molecules

Since in this case the energies of the atomic orbital (AO) on the two atoms from which the molecular orbital are formed are different, the combination of atomic orbital to form molecular orbital will be different from that in the case of homo-nuclear diatomic molecules. In order to build up the molecular orbital explanation of hetero nuclear diatomic molecule we require considering the fact that dissimilar kinds of atoms have diverse capacities to magnetize the electrons. So essentially, the electronegativity of the bonded atoms plays very important role in the action of hetero-nuclear bond. The bonding electron will be more stable in the occurrence of the nuclei of the atom having greater electronegativity, for example the atom having lower energy.  Probability of discovering the bonded electrons will be more near that nucleus. The electron cloud will be distorted towards that nucleus and therefore the MO will resemble that AO more than the AO on the less electronegative atom. This explanation can simply explain the polarity of the hetero-nuclear molecules.

Hydrogen fluoride

Hydrogen fluoride is an easy hetero-nuclear diatomic molecule. Because the  electronegativity  of  fluorine  atom  is  much  higher  than  that  of hydrogen atom, the  energy of 1s orbital of hydrogen atom is much higher than the atomic orbital of fluorine  atom. The molecular orbital will be shaped due to the grouping of hydrogen 1s orbital through 1s, 2s and 2p orbitals of fluorine. The pictorial demonstration of the MO diagram is specified fig. There are 8 valence electrons that occupy 4 molecular orbitals. The  2 highest energy MO's are degenerate, are π-type formed only from two 2p orbitals of fluorine atom and therefore have no electron density connected by the hydrogen atom, for example they are Non-Bonding Orbitals (NBO). In Lewis theory such 2 electrons are symbolized as two "Lone Pairs". We can see from the MO diagram that the electron density isn't uniformly distributed about the molecule. There is a much superior electron density around the fluorine atom because of extremely  electronegativity  of  fluorine  atom,   and  in  each  bonding molecular  orbital,  fluorine  will  obtain  a  greater  share  of  the  electron density.

Huckel Theory: Bonding in Polyatomic Molecules

In case of polyatomic molecules the combinations amongst the atomic orbitals are complicated.  One widely used approximated theory to describe the molecular orbitals in polyatomic molecule is Huckel molecular theory (HMO). Huckel MO theory is depending on the treatment of Π electrons in conjugated molecules. Here Π orbitals are treated individually from the σ orbitals which provide the framework of the molecule. The Π electron Hamiltonian is merely written as a sum of one- electron terms. It follows that the total energy is the sum of one-electron energies. The sum is over all electrons. Because each molecular orbital is twice as occupied (for a normal sealed shell hydrocarbon that is the class we shall restrict ourselves to for now), this is twice the calculation of power terms for each molecular orbital. Each molecular orbital i is now explained as a linear combination of atomic orbitals. Now the next task is to compute the coefficient of the atomic orbitals to find out the molecular orbitals and their energies.

Let us illustrate the theory with the example of ethene:

The number of atoms (for example C atoms) (n) is 2, so the molecular orbitals are of the form: Because there are 2 atomic orbitals, 2 molecular orbitals, will shaped due to the mixture of the orbitals, one will be bonding and other will be antibonding. The electrons will in the bonding orbital of the system. As we know that each C atom always contribute one electron to the system for a neutral hydrocarbon, so the number  of  electrons  is then equivalent to the number of atoms, and the number of occupied orbitals (m) is n/2.

The best molecular orbitals are those that provide the minimum total energy. The aspects of the process are beyond the scope of this learn substance.

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