Transition dipole moment

, is the electric dipole moment associated with the transition between the two states.

In general the transition dipole moment is a complex vector quantity that includes the phase factors associated with the two states.

Its direction gives the polarization of the transition, which determines how the system will interact with an electromagnetic wave of a given polarization, while the square of the magnitude gives the strength of the interaction due to the distribution of charge within the system.

The SI unit of the transition dipole moment is the Coulomb-meter (Cm); a more conveniently sized unit is the Debye (D).

For a transition where a single charged particle changes state from

where q is the particle's charge, r is its position, and the integral is over all space (

The transition dipole moment is a vector; for example its x-component is

In other words, the transition dipole moment can be viewed as an off-diagonal matrix element of the position operator, multiplied by the particle's charge.

When the transition involves more than one charged particle, the transition dipole moment is defined in an analogous way to an electric dipole moment: The sum of the positions, weighted by charge.

If the ith particle has charge qi and position operator ri, then the transition dipole moment is:

For a single, nonrelativistic particle of mass m, in zero magnetic field, the transition dipole moment between two energy eigenstates ψa and ψb can alternatively be written in terms of the momentum operator, using the relationship[1]

This relationship can be proven starting from the commutation relation between position x and the Hamiltonian H:

Similar relations hold for y and z, which together give the relationship above.

While the comparison can be very useful, care must be taken to ensure that one does not fall into the trap of assuming they are the same.

In the presence of an electric field, such as that due to an electromagnetic wave, the two charges will experience a force in opposite directions, leading to a net torque on the dipole.

The magnitude of the torque is proportional to both the magnitude of the charges and the separation between them, and varies with the relative angles of the field and the dipole:

depends on the charge distribution within the atom, the strength of the electric field, and the relative polarizations of the field and the transition.

In addition, the transition dipole moment depends on the geometries and relative phases of the initial and final states.

When an atom or molecule interacts with an electromagnetic wave of frequency

, it can undergo a transition from an initial to a final state of energy difference

through the coupling of the electromagnetic field to the transition dipole moment.

When this transition is from a lower energy state to a higher energy state, this results in the absorption of a photon.

A transition from a higher energy state to a lower energy state results in the emission of a photon.

, is omitted from the electric dipole operator during this calculation, one obtains

orbital is allowed because the integral defining the transition dipole moment is nonzero.

Such a transition occurs between an even and an odd orbital; the dipole operator,

This result is reflected in the parity selection rule for electric dipole transitions.

of an electronic transition within similar atomic orbitals, such as s-s or p-p, is forbidden due to the triple integral returning an ungerade (odd) product.

Such transitions only redistribute electrons within the same orbital and will return a zero product.

If the triple integral returns a gerade (even) product, the transition is allowed.