Conditions for Linear Combination of Atomic Orbitals (LCAO): Theory & Sample Questions

Linear Combination of Atomic Orbitals is a quantum superposition of the atomic orbitals. It is also a technique that is used for calculating molecular orbitals in atomic chemistry. In the field of quantum mechanics, electron configurations of atoms are viewed as wave functions. According to mathematics, these wave functions are a general set of functions that describes the electron in the atom. In chemical reactions, these orbital wave functions change according to the participating atoms in the bonding. It was introduced by Sir Lennard Jones in the year 1929. 

Key Terms: Linear Combination of Atomic Orbitals, Molecular Orbital Theory, Atomic Orbitals, Molecular Orbitals, Atoms, Molecular Axis

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Molecular Orbital Theory

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The molecular orbital theory was developed by F. Hund and R.S. Mulliken in the year 1932. In the molecular orbital theory, all the atomic orbitals of combined atoms undergo mixing to form a new set of orbitals. These new orbitals are called molecular orbitals. 

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Features of Molecular Orbital Theory

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The prominent features of the molecular orbital theory are as follows-

• According to the molecular orbital theory, all the atomic orbitals of combining atoms first combine to form a new set of orbitals. They are called molecular orbitals. Once these molecular orbitals are formed, the atoms which combined to give these orbitals are lost.
• The number of atomic orbitals combined is always the same as the number of molecular orbitals formed.
• The molecules are formed by the linear combination of the atomic orbitals.
• Molecular orbitals are formed by the addition or subtraction of overlapping atomic orbitals. 
• When two atomic orbitals overlap by addition, they are called bonding molecular orbitals. The molecular orbitals formed by the negative overlapping of atomic orbitals are called anti-bonding molecular orbitals. 
• The addition of electrons in bonding molecular orbitals tends to stabilize the newly formed molecules and the subtraction of electrons in anti-bonding molecular orbital tends to destabilize the newly formed molecule. 
• The shape of the molecular orbitals depends on the atomic orbitals which were mixed.
• The atomic orbitals need to be in comparable energy and similar symmetry to form molecular orbitals. 
• Molecular orbitals have different shapes, sizes, and energies than atomic orbitals.
• The bonding molecular orbital has lower energy and is hence more stable than the anti-bonding molecular orbital.

Molecular Orbital Theory

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Linear Combination of Atomic Orbitals (LCAO)

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The linear combination of atomic orbitals can be described as a method for representing molecular orbitals. It works on the superimposition principle where constructive interference of two wave functions produces a bonding molecular orbital and destructive interference produces an anti-bonding molecular orbital. 

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Conditions Required for LCAO

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The required conditions for Linear Combination of Atomic Orbitals are- 

• Similar Energy of Combining Orbitals

The atomic orbitals of both the atoms need to have the same or similar energy. For example, a 2p orbital of an atom can combine with another 2p orbital of another atom. The 1s orbital and 2p orbital cannot combine as they have a much greater energy difference. 

• Should Have the Same Symmetry around a Molecular Axis

The atoms which are combining should have the same symmetry around the molecular axis. Else, the electron density will be sparse.

For example, all the three sub-orbitals of 2p have the same energy but still, the 2pz orbital can only combine with the 2pz orbital of another atom. 2pz of one atom cannot combine with 2px orbital of another atom due to different axis of symmetry. In general, the z-axis is called the molecular axis of symmetry.

• Proper Overlap of the Combining Orbitals

The two combining orbitals will combine to form a molecular orbital if the overlap is proper. The greater the extent of overlap of the orbitals greater will be the bond strength. 

Conditions for LCAO

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Things to Remember

• The Linear combination of atomic orbitals is a quantum superposition of the atomic orbitals. 
• The number of atomic orbitals combining is always the same as the number of molecular orbitals formed.
• Molecular orbitals are formed by the addition or subtraction of overlapping atomic orbitals. 
• The shape of the molecular orbitals depends on the atomic orbitals which were mixed.
• For LCAO, the atomic orbitals of both the atoms need to have the same or similar energy.
• The atomic orbitals should have proper overlap and the same molecular axis of symmetry in order to have the proper orientation.

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PYQs

1. The overlapping of orbitals in benzene is of the type? [KCET 2008]
2. The number of nodal planes present in σ∗s antibonding orbitals is? [KCET 2008]
3. Which of the following pairs of ions are isoelectronic and isostructural? [NEET 2015]
4. The increasing order of bond order of the following is? [KCET 2016]
5. Which of the following options represents the correct bond order? [NEET 2016]
6. The ion which is isoelectronic with CO is? [NEET 1997]
7. Among the following sets of quantum numbers. Which one is incorrect for 4d4d electron ?...[BITSAT 2005]
8. Which of the following is not permissible arrangement of electrons in an atom….[UPSEE 2019]
9. The four quantum numbers of the valence electron of potassium are :...[AP EAPCET 1998]
10. Who modified Bohr's theory by introducing elliptical orbits for electron path?...[NEET 1999]
11. The uncertainty in momentum of an electron is 1×10−5kgm/s.1×10−5kgm/s. The uncertainty in its position will be (Given, h=6.62×10−34kgm2/s)h=6.62×10−34kgm2/s)...[NEET 1999]
12. The energy of an electron in the nthnth Bohr orbit of H - atom is...[NEET 1992]
13. In Li++Li++, electron in first Bohr orbit is excited to a level by a radiation of wavelength λλ. when the ion gets deexcited to the ground state in all possible ways (including intermediate emissions), a total of six spectral lines are observed. What is the value of λλ ? (Given : h=6.63×10−34Js;c=3×108ms−1)h=6.63×10−34Js;c=3×108ms−1) ;...[JEE Main 2019]
14. In a hydrogen like atom, when an electron jumps from the MM - shell to the LL - shell, the wavelength of emitted radiation is λλ. If an electron jumps from NN-shell to the LL-shell, the wavelength of emitted radiation will be :...[JEE Main 2019]
15. In a hydrogen like atom electron makes transition from an energy level with quantum number nn to another with quantum number (n−1).Ifn>>1(n−1).Ifn>>1, the frequency of radiation emitted is proportional to….[JEE Main 2013]
16. If λ0λ0 and λλ be the threshold wavelength and wavelength of incident light, the velocity of photoelectron ejected from the metal surface is :...[JEE Main 2014]
17. Heat treatment of muscular pain involves radiation of wavelength of about 900nm900nm. Which spectral line of H-atom is suitable for this purpose ? [RH=1×105cm−1,h=6.6×10−34Js,c=3×108ms−1][RH=1×105cm−1,h=6.6×10−34Js,c=3×108ms−1]...[JEE Main 2019]
18. Given (A) n=5,mℓ=+1n=5,mℓ=+1 (B) n=2,ℓ=1,mℓ=1,ms=−1/2n=2,ℓ=1,mℓ=1,ms=−1/2 The maximum number of electron(s) in an atom that can have the quantum numbers as given in (A) and (B) are respectively:…...[JEE Main 2013]
19. For emission line of atomic hydrogen from ni=8ni=8 to nfnf = the plot of wave number (¯v)(v¯) against (1n2)(1n2) will be (The Ry dberg constant, RHRH is in wave number unit)….[JEE Main 2019]