Introduction

Chemical bonds between atoms within a molecule consist of one or more electron pairs distributed among the connected atoms. Bonding electrons are not equally distributed among the atoms that have different electronegativity. Forming a chemical bond can be formally described with the following three steps:

1. Initial state
   
   In this case we can speak about partial charge of atoms. We will give the definition of partial charge intuitively rather than exactly. A simple definition of partial charge starts from a basic two-atom model in which atoms A and B have different electron structures. In the figure below, the "boundary" of the valence shell is denoted with blue and green solid lines. This boundary is chosen to create a fixed control volume with a definite probability (near to 100%) of finding all electrons belonging to the atom on the inside.

   

   Fig. 1 Initial state of two different atoms

2. Intermediate state
   
   In the intermediate state the nuclear distance of A and B decreases. As they draw closer to each other, each of their electron clouds is polarized to a different degree. Although the electrons of both A and B have partially moved outside of the control volume, this effect is greater in atom B being the more polarizable atom. The more polarizable an atom is, the more electrons will flow out of its control volume so that the atom will become more positive relative to its initial state.

   

   Fig. 2 Intermediate state of two different atoms

3. Final state
   
   In the final state the two atoms reach the equilibrium nuclear distance and the two electron clouds overlap. Using the control volumes and electron clouds in the figure, atom B becomes more positively charged and atom A becomes more negatively charged relative to their initial states. In this way, partial charge of an atom in the molecule is measured as a charge flow out of or into the atom's control volume.

   

   Fig. 3 Final state of the two different atoms

Concept of orbital electronegativity

Partial charge distribution of the molecule is governed by the orbital electronegativity of the atoms contained in the molecule. The definition of orbital electronegativity of atoms in molecules was first given by Mulliken by the following equation:

χ_ν = 0.5*(I_ν+E_ν)

where:

• χ_ν is the orbital electronegativity of the atom.

• I_ν is the ionization potential of the atom.

• E_ν is the electron affinity of the atom.

The orbital electronegativity of the molecule depends on the hybrid state and partial charge of the atom, see references of Mcweeny and Dewar. Electronegativity is a quadratic function of partial charge given by the following equation:

χ_ν = a+bq+cq²

where:

• q is the partial charge on the atom;

• a, b, and c are coefficients determined from I_ν and E_ν.

Orbital electronegativity and subsequently partial charge distribution of any molecule is calculated iteratively.

Examples

Here are some examples for partial charge calculation.

References

1. Mulliken, R.S., J. Chem. Phys., 1934, 2, 782; doi

2. McWeeny, R., Coulson's Valence, Oxford University Press, 1979

3. Dewar, M.J.S., The Molecular Orbital Theory of Organic Chemistry, McGraw-Hill, and Inc., 1969

4. Gasteiger, J.; Marsili, M., Tetrahedron, 1980, 36, 3219; doi

5. Stewart, R., The Proton: Applications to Organic Chemistry, Academic Press, Inc., 1985, 72