Molecular orbital theory, bond order and magnetism

What is molecular orbital theory?

In plain words: valence bond theory keeps each electron pair between two atoms. MO theory instead mixes the atoms’ orbitals into new orbitals that belong to the whole molecule. Each mixing gives one orbital that holds the nuclei together (bonding) and one that pushes them apart (antibonding). Fill them with the molecule’s electrons and count: more electrons in bonding orbitals than in antibonding ones means a bond.

Main features

The salient features of MO theory (NCERT Section 4.7, pages 125–126):

1. Electrons in a molecule are present in molecular orbitals, as electrons of atoms are in atomic orbitals.
2. Atomic orbitals of comparable energies and proper symmetry combine to form molecular orbitals.
3. An electron in an atomic orbital is influenced by one nucleus; in a molecular orbital it is influenced by two or more nuclei. An atomic orbital is monocentric, a molecular orbital polycentric.
4. The number of molecular orbitals formed equals the number of atomic orbitals that combine.

When two atomic orbitals combine, two molecular orbitals are formed.

NCERT Chemistry Part I (Class 11), Section 4.7, page 126

5. The bonding molecular orbital has lower energy, and hence greater stability, than the corresponding antibonding molecular orbital.
6. A molecular orbital is a wave function over a group of nuclei; its squared magnitude gives the one-electron probability density.
7. Molecular orbitals are filled by the aufbau principle, obeying Pauli’s exclusion principle and Hund’s rule, exactly as atomic orbitals are filled.

Linear combination of atomic orbitals (LCAO)

The Schrödinger equation cannot be solved exactly for any system with more than one electron, so molecular orbitals are built approximately, by the linear combination of atomic orbitals (LCAO) (NCERT Section 4.7.1, page 126). For H₂, the two 1s orbitals ψ_A and ψ_B are added or subtracted. The following expressions omit normalisation factors; in general LCAO also uses unequal coefficients:

Qualitatively, this is interference of the electron waves (NCERT pages 126–127):

• Bonding orbital: the waves reinforce each other (constructive interference), so electron density builds up between the nuclei. At a fixed internuclear distance the nucleus–nucleus repulsion is unchanged, but the full electronic energy changes, and the bonding combination is lower in energy than the separated 1s reference in this elementary picture.
• Antibonding orbital: the waves cancel (destructive interference). Most of the density lies away from the space between the nuclei, and there is a nodal plane (zero electron density) between them, and the total energy is less favourable than for the bonding combination. Electrons placed here destabilise the molecule.

NCERT’s simple picture raises σ* above the atomic orbitals by as much as σ is lowered, so the total energy of the two molecular orbitals equals that of the two atomic orbitals (page 127). Beyond the NCERT text: a fuller treatment shows the antibonding orbital is raised more than the bonding one is lowered. So when bonding and antibonding electrons are equal in number, the elementary model predicts no net covalent bond, as for He₂. Weak dispersion-bound helium dimers nevertheless exist; their binding requires physics beyond this elementary count.

Conditions for the combination of atomic orbitals

Appreciable interaction between atomic orbitals requires compatible symmetry and energies; in the elementary treatment (NCERT Section 4.7.2, page 127):

1. They have the same or nearly the same energy. A large energy mismatch usually makes mixing weak, rather than mathematically forbidden; similar-energy orbitals mix most effectively (for two different atoms what matters is energy, not the label: in HF the H 1s orbital combines with an F 2p orbital of similar energy).
2. They have the same symmetry about the molecular axis (taken as z). A 2p_z orbital of one atom combines with 2p_z of the other, but not with 2p_x or 2p_y, even though their energies are the same.
3. They have nonzero, appreciable overlap. For compatible orbitals, stronger overlap generally increases bonding/antibonding splitting; the antibonding combination still has a node.

Types of molecular orbitals

Molecular orbitals of diatomic molecules are labelled σ (sigma), π (pi), δ (delta) and so on (NCERT Section 4.7.3, page 127):

• σ orbitals are symmetrical about the bond axis. Two 1s orbitals give σ1s and σ*1s; with z as the bond axis, two 2p_z orbitals also give σ orbitals, σ2p_z and σ*2p_z.
• π orbitals are not symmetrical about the bond axis: those from 2p_x and 2p_y have positive lobes above and negative lobes below the molecular plane. They are π2p_x, π2p_y and π*2p_x, π*2p_y. A bonding π orbital has more electron density above and below the internuclear axis; the antibonding π* orbital has a node between the nuclei.

NCERT’s text on page 127 sends the reader to Fig. 4.20(a) on page 124 for the σ1s and σ*1s contours, but Fig. 4.20 is on page 128.

The sum/difference entries below refer to the chosen same-phase 1s functions; for other orbital drawings, coefficients must account for lobe signs.

Energy level diagram

The 1s orbitals of two atoms give σ1s and σ*1s. In the same way the 2s and 2p orbitals (eight atomic orbitals on two atoms) give eight molecular orbitals (NCERT Section 4.7.4, pages 127–128):

• antibonding: σ*2s, σ*2p_z, π*2p_x, π*2p_y;
• bonding: σ2s, σ2p_z, π2p_x, π2p_y.

Their order, found from spectroscopic data for the second-period diatomic molecules, comes in two forms (pages 128–129):

• O₂ and F₂: σ1s < σ*1s < σ2s < σ*2s < σ2p_z < (π2p_x = π2p_y) < (π*2p_x = π*2p_y) < σ*2p_z
• Li₂, Be₂, B₂, C₂, N₂: σ1s < σ*1s < σ2s < σ*2s < (π2p_x = π2p_y) < σ2p_z < (π*2p_x = π*2p_y) < σ*2p_z

The key difference: up to N₂, σ2p_z lies above the π2p pair. NCERT’s Fig. 4.21 (page 131) labels the two cases “with 2s–2p mixing” and “without 2s–2p mixing”. In the lighter atoms the 2s and 2p_z orbitals (same σ symmetry) are close in energy and mix, which pushes σ2p_z up; the π orbitals cannot take part, and in O and F the 2s–2p gap is larger.

Electronic configuration and molecular behaviour

The distribution of electrons among the molecular orbitals is the electronic configuration of the molecule, and it tells us a great deal (NCERT Section 4.7.5, page 129). Write N_b for the number of electrons in bonding orbitals and N_a for the number in antibonding orbitals.

Elementary bonding prediction: N_b > N_a gives positive net covalent bond order. This count alone does not establish thermodynamic stability, resistance to electron loss or the absence of weak correlation/dispersion binding.

Bond order (b.o.) is defined as one half the difference between the number of electrons present in the bonding and the antibonding orbitals

NCERT Chemistry Part I (Class 11), Section 4.7.5, page 129

• Stability by bond order: a positive value predicts net covalent bonding in this model; zero means no net covalent bond in this model, not universal nonexistence.
• Nature of the bond: bond orders of 1, 2 and 3 correspond to single, double and triple bonds.
• Bond length and bond enthalpy: bond order is an approximate measure of bond length; the bond length decreases as the bond order increases (NCERT Section 4.7.5, page 129), and the bond enthalpy increases (NCERT Section 4.3.4, page 109).
• Magnetic nature: if every occupied molecular orbital holds two electrons, the substance is diamagnetic (repelled by a magnetic field); if one or more holds a single electron, it is paramagnetic (attracted), as O₂ is.

Full treatments of H₂, He₂, Li₂, C₂ and O₂, and of the row B₂ to Ne₂ with bond energies and lengths, are on the next topic, bonding in homonuclear diatomic molecules (NCERT Section 4.8); the examples below only count electrons.

Solved examples

Common mistakes to avoid

Quick revision summary

• MO theory (Hund and Mulliken): electrons occupy molecular orbitals over the whole molecule; n atomic orbitals give n molecular orbitals.
• LCAO: σ = ψ_A + ψ_B (bonding, density between nuclei, lower energy); σ* = ψ_A − ψ_B (antibonding, nodal plane, higher energy).
• Combining orbitals need similar energy, the same symmetry about the axis and good overlap.
• σ orbitals are symmetrical about the bond axis, π orbitals are not.
• Order up to N₂: … σ*2s < π2p < σ2p_z < π*2p < σ*2p_z; for O₂ and F₂, σ2p_z comes below π2p.
• Bond order ½(N_b − N_a): positive predicts net covalent bonding in the elementary model; comparable higher-order bonds tend to be shorter and stronger; unpaired electrons mean paramagnetic.

Questions students ask

What is the difference between valence bond theory and molecular orbital theory?

Valence bond theory describes a bond as the overlap of two atomic orbitals holding an electron pair localised between two atoms, while molecular orbital theory combines the atomic orbitals into new orbitals that spread over the whole molecule and fills them with all of its electrons. Valence bond theory, with hybridisation and resonance, gives a simple picture of shapes and of single, double and triple bonds; elementary molecular orbital theory conveniently explains what the simple localised pair model does not, such as the paramagnetism of O₂ and the bond order of odd-electron F₂⁺ (1.5). Both are quantum mechanical approximations and usually agree on the number of bonds.

Is molecular orbital theory correct?

Molecular orbital theory is an approximation, not an exact description: the LCAO method that NCERT uses builds molecular orbitals approximately because the Schrödinger equation cannot be solved exactly for any molecule with more than one electron. Within that limit it is very successful, correctly predicting, for example, that O₂ is paramagnetic and that He₂ has no net covalent bond in the elementary diagram (although a very weak dispersion-bound dimer is known), and refined versions of it are the basis of most modern computer calculations on molecules. Valence bond theory, refined in the same way, gives equally good results, so the two are best seen as complementary descriptions of the same bonds.

What is a non-bonding molecular orbital?

A non-bonding molecular orbital is one whose energy stays about the same as that of the atomic orbital it comes from, because that orbital finds no partner of matching symmetry and similar energy on the other atom; electrons in it neither strengthen nor weaken the bond and are left out of the bond order. In HF, with z as the bond axis, the H 1s orbital combines with fluorine’s 2p_z, but fluorine’s 2p_x and 2p_y have the wrong symmetry to combine with 1s, so they stay non-bonding and hold two of fluorine’s three lone pairs. The term is beyond the NCERT text, whose examples are homonuclear diatomic molecules.

What is the difference between molecular orbital theory and VSEPR theory?

VSEPR theory predicts only the shape of a molecule, from the number of electron pairs round its central atom, while molecular orbital theory describes the energies of all the electrons in the molecule and so predicts its bond order, bond length, stability and magnetism. VSEPR says nothing about bond strength, stability or magnetism, such as why the N≡N bond is so much stronger than the F–F bond; at NCERT level, molecular orbital theory is applied only to diatomic molecules, which are linear anyway, so it is not used to find shapes. For the shapes of polyatomic molecules use VSEPR theory or hybridisation.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Write the important conditions required for the linear combination of atomic orbitals to form molecular orbitals (NCERT Exercise 4.34, page 135).

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   (1) The combining atomic orbitals must have the same or nearly the same energy. (2) They must have the same symmetry about the molecular axis. (3) They must have nonzero overlap compatible with their symmetry.

2. Q2. Give three differences between a bonding and an antibonding molecular orbital.

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   A bonding MO forms by addition of the atomic wave functions, has lower energy than the atomic orbitals and has high electron density between the nuclei, so its electrons stabilise the molecule. An antibonding MO forms by subtraction, has higher energy and a nodal plane between the nuclei, so its electrons destabilise the molecule.

3. Q3. Find the bond order of He₂⁺. Can it exist? Is it paramagnetic?

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   He₂⁺ has 3 electrons: σ1s² σ*1s¹. Bond order ½(2 − 1) = 0.5, which is positive, so it can exist. It has one unpaired electron, so it is paramagnetic.