# What is hund's rule of multiplicity?

Introduction

Atoms, contains protons and neutrons in the nucleus at the center and electrons revolving around nucleus. The attractive force by the nucleus on the electron, is balanced, by its rotation at fixed distances (orbital) with certain energy. In a multi electrons atom, there could be multiple orbitals. Occupation or distribution, of the electrons in the available orbitals, determines the stability, physical and chemical properties of the atom.

Distribution of the electron in the orbitals is governed by certain rules ensuring the lowest energy for the electron and hence for the atom-

ü Aufbau principle, suggest the occupation of the orbital from that of low energy to higher energy.

ü Pauli’s principle, states that only two electrons can occupy an orbital and that too only with opposite spin.

ü Some of the shells have multiple degenerate (having same energy) orbitals. Hund’s rule of multiplicity, lays conditions of distribution of electrons, in the degenerate orbitals.

Hund’s Rules

Atomic spectra of atoms exhibits fine spectral lines very close to each other, indicating the presence of closely space energy levels, more than expected of Bohr and Schrodinger equations. Hund proposed three rules for the identification of the electronic configuration of the ground state (with lowest energy) and representation of the ground state of an atom by a ‘term symbol’ to understand the extra fine spectral line of the atom. The rules are based on the total spin, orbital and total angular (spin-orbit coupling) quantum number (J) of the electrons present in the outer orbitals.

The three rules are –

1. i) Electronic configuration with maximum unpaired electron (maximum multiplicity) has low energy.

Maximum multiplicity is equal to (2S + 1), where, ‘S’ is the sum of spin quantum number (S=Σms) of all outer electrons. Multiplicity of hydrogen atom is one. Hydrogen is in a singlet state. With three valence electrons, multiplicity of nitrogen is 4 (quartet).

1. ii) If the multiplicity is same (2S + 1), configuration with largest total orbital angular momentum (L = Σml) has lower energy.

iii) For a given configuration, of same multiplicity and total angular quantum number with-

1. a) Less than half filled orbitals, smallest total angular quantum number (J = |L -S|),
2. b) More than half filled orbitals, largest total quantum number (J = |L + S|) has the lowest energy and
3. c) Half-filled degenerate orbitals, L = 0 and so, J = S