- Transition dipole moment
The Transition dipole moment or Transition moment, usually denoted scriptstyle{mathbf{d}_{nm for a transition between an initial state, scriptstyle{m}, and a final state, scriptstyle{n}, 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 theDebye (D).Definition
The transition dipole moment for the scriptstyle{m, ightarrow, n} transition is given by the relevant off-diagonal element of the dipole matrix, which can be calculated from an integral taken over the product of the wavefunctions of the initial and final states of the transition, and the dipole moment operator,
:mathbf{hat{d = eleft(sum_i x_i, sum_i y_i, sum_i z_i ight),
where the
summation s are over the positions of theelectron s in the system. Giving the transition dipole moment::mathbf{d}_{nm} = int Psi^*_n ,, mathbf{hat{d ,, Psi_m , d^3r = langlePsi_n|,mathbf{hat{d,|Psi_m angle,
where the integral is, in principle over all space, but can be restricted to the region in which the initial and final state wavefunctions are non-negligible.
Analogy with a classical dipole
A basic, phenomenological understanding of the transition dipole moment can be obtained by analogy with a classical dipole. 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 case of two classical point charges, scriptstyle{+q} and scriptstyle{-q}, with a displacement vector, scriptstyle{mathbf{r, pointing from the negative charge to the positive charge, the electric dipole moment is given by
:mathbf{p} = qmathbf{r}.
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 nettorque on the dipole. The magnitude of the torque is proportional to the magnitude of the charge, the separation and varies with the relative angles of the field and the dipole,:mathbf{ au}| = |qmathbf{r}||mathbf{E}|sin heta.
Similarly, the coupling between an electromagnetic wave and an atomic transition with transition dipole moment scriptstyle{mathbf{d}_{nm, 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.
Origin
When an atom or molecule interacts with an electromagnetic wave of frequency scriptstyle{omega}, it can undergo a transition to a higher energy state by absorbing a
photon or a lower energy state by emitting a photon, provided that the energy difference between the initial and final states scriptstyle{(Delta E)} is the same as the photon energy scriptstyle{(hbaromega)}. The presence of the electromagnetic field can induce an oscillatingelectric dipole moment , referred to as the transition dipole moment. If the charge, scriptstyle{e}, is omitted one obtains scriptstyle{mathbf{R}_alpha} as used inoscillator strength .Applications
The transition dipole moment is useful for determining if transitions are allowed. For example, the transition from a bonding scriptstyle{pi} orbital to an antibonding scriptstyle{pi^*} 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 is an odd function of scriptstyle{mathbf{r, hence the integrand is an even function. The integral of an odd function over symmetric limits returns a value of zero, while for an even function this is not "necessarily" the case.References
cite web
last =
first =
authorlink =
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title = IUPAC compendium of Chemical Terminology
work =
publisher = IUPAC
date = 1997
url = http://www.iupac.org/goldbook/T06460.pdf
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doi =
accessdate = 2007-01-15
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