Morse potential
Encyclopedia
The Morse potential, named after physicist Philip M. Morse
Philip M. Morse
Philip McCord Morse , was an American physicist, administrator and pioneer of operations research in World War II. He is considered to be the father of operations research in the U.S.- Biography :Morse graduated from the Case School of Applied Science in 1926 with a B.S. in physics. He earned his...

, is a convenient model for the potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

of a diatomic molecule. It is a better approximation for the vibrational
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

structure of the molecule than the quantum harmonic oscillator
Quantum harmonic oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity
Anharmonicity
In classical mechanics, anharmonicity is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in simple harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmonic oscillator and the anharmonicity can be...

of real bonds and the non-zero transition probability for overtone and combination bands. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface.

## Potential Energy Function

The Morse potential energy function is of the form

Here is the distance between the atoms, is the equilibrium bond distance, is the well depth (defined relative to the dissociated atoms), and controls the 'width' of the potential (the smaller the larger the well). The dissociation energy
Bond dissociation energy
In chemistry, bond-dissociation energy or D0, is one measure of the bond strength in a chemical bond. It is defined as the standard enthalpy change when a bond is cleaved by homolysis, with reactants and products of the homolysis reaction at 0 K...

of the bond can be calculated by subtracting the zero point energy  from the depth of the well. The force constant of the bond can be found by Taylor expansion of around to the second derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

of the potential energy function, from which it can be shown that the parameter, , is
,

where is the force constant at the minimum of the well.

Of course, the zero of potential energy is arbitrary, and 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 Morse potential is usually written in the form

where is now the coordinate perpendicular to the surface. This form approaches zero at infinite and equals at its minimum. It clearly shows that the Morse potential is the combination of a short-range repulsion and a longer-range attractive tail.

## Vibrational States and Energies

To write the stationary state
Stationary state
In quantum mechanics, a stationary state is an eigenvector of the Hamiltonian, implying the probability density associated with the wavefunction is independent of time . This corresponds to a quantum state with a single definite energy...

s on the Morse potential, i.e. solutions and of the following Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

:

it is convenient to introduce the new variables:

Then, the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

takes the simple form:

Its eigenvalues and eigenstates can be written as:
where ; and is Laguerre polynomial:

There also exists the following important analytical expression for matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

elements of the coordinate operator (here it is assumed that and )

The eigenenergies in the initial variables have form:

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

Whereas the energy spacing between vibrational levels in the quantum harmonic oscillator
Quantum harmonic oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

is constant at , the energy between adjacent levels decreases with increasing 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 where 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 that remains bound. For energies above , all the possible energy levels are allowed and the equation for is no longer valid.

Below , 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 and can be directly related to the parameters for the Morse potential.

## Solving Schrödinger's equation for the Morse oscillator

Like the quantum harmonic oscillator
Quantum harmonic oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

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