Morse potential

Potential energy function

The Morse potential energy function is of the form

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

where 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

which is usually written as

where r is now the coordinate perpendicular to the surface. This form approaches zero at infinite r and equals − D e at its minimum, i.e. 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.

Vibrational states and energies

Like the quantum harmonic oscillator, the energies and eigenstates of the Morse potential can be found using operator methods. One approach involves applying the factorization method to the Hamiltonian.

To write the stationary states on the Morse potential, i.e. solutions Ψ ( v ) and E ( v ) of the following Schrödinger equation:

it is convenient to introduce the new variables:

Then, the Schrödinger equation takes the simple form:

Its eigenvalues and eigenstates can be written as:

where z = 2 λ e − ( x − x e ) ;  N n = [ n ! ( 2 λ − 2 n − 1 ) Γ ( 2 λ − n ) ] 1 2 and L n ( α ) ( z ) is a generalized Laguerre polynomial:

There also exists the following important analytical expression for matrix elements of the coordinate operator (here it is assumed that m > n and N = λ − 1 2 )

The eigenenergies in the initial variables have form:

where v is the vibrational quantum number, and ν 0 has units of frequency, and is mathematically related to the particle mass, m , and the Morse constants via

Whereas the energy spacing between vibrational levels in the quantum harmonic oscillator is constant at h ν 0 , the energy between adjacent levels decreases with increasing v in the Morse oscillator. Mathematically, the spacing of Morse levels is

This trend matches the anharmonicity found in real molecules. However, this equation fails above some value of v m where E ( v m + 1 ) − E ( v m ) is calculated to be zero or negative. Specifically,

This failure is due to the finite number of bound levels in the Morse potential, and some maximum v m that remains bound. For energies above v m , all the possible energy levels are allowed and the equation for E ( v ) is no longer valid.

Below v m , E ( v ) is a good approximation for the true vibrational structure in non-rotating diatomic molecules. In fact, the real molecular spectra are generally fit to the form¹

in which the constants ω e and ω e χ e can be directly related to the parameters for the Morse potential.

As is clear from dimensional analysis, for historical reasons the last equation uses spectroscopic notation in which ω e represents a wavenumber obeying E = h c ω , and not an angular frequency given by E = ℏ ω .

Morse/Long-range potential

An important extension of the Morse potential that made the Morse form very useful for modern spectroscopy is the MLR (Morse/Long-range) potential. The MLR potential is used as a standard for representing spectroscopic and/or virial data of diatomic molecules by a potential energy curve. It has been used on N₂, Ca₂, KLi, MgH, several electronic states of Li₂, Cs₂, Sr₂, ArXe, LiCa, LiNa, Br₂, Mg₂, HF, HCl, HBr, HI, MgD, Be₂, BeH, and NaH. More sophisticated versions are used for polyatomic molecules.