Answer to Question #134090 in Physical Chemistry for Nihal

Question

Explain morse potential energy curve

Answer 1

The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. ... The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface.

The Morse potential energy function is of the form

{\displaystyle V'(r)=D_{e}(1-e^{-a(r-r_{e})})^{2}}

Here {\displaystyle r} is the distance between the atoms, {\displaystyle r_{e}} is the equilibrium bond distance, {\displaystyle D_{e}} is the well depth (defined relative to the dissociated atoms), and {\displaystyle a} controls the 'width' of the potential (the smaller {\displaystyle a} is, the larger the well). The dissociation energy of the bond can be calculated by subtracting the zero point energy {\displaystyle E_{0}} from the depth of the well. The force constant (stiffness) of the bond can be found by Taylor expansion of {\displaystyle V'(r)} around {\displaystyle r=r_{e}} to the second derivative of the potential energy function, from which it can be shown that the parameter, {\displaystyle a}, is

{\displaystyle a={\sqrt {k_{e}/2D_{e}}},}

where {\displaystyle k_{e}} is the force constant at the minimum of the well.

Since the zero of potential energy is arbitrary, the equation for the Morse potential can be rewritten any number of ways by adding or subtracting a constant value. When it is used to model the atom-surface interaction, the energy zero can be redefined so that the Morse potential becomes

{\displaystyle V(r)=V'(r)-D_{e}=D_{e}(1-e^{-a(r-r_{e})})^{2}-D_{e}}

which is usually written as

{\displaystyle V(r)=D_{e}(e^{-2a(r-r_{e})}-2e^{-a(r-r_{e})})}

where {\displaystyle r} is now the coordinate perpendicular to the surface. This form approaches zero at infinite {\displaystyle r} and equals {\displaystyle -D_{e}} at its minimum, i.e. {\displaystyle r=r_{e}}. It clearly shows that the Morse potential is the combination of a short-range repulsion term (the former) and a long-range attractive term (the latter), analogous to the Lennard-Jones potential.