Transition (dipole) moment

Hi QChem Forum,

can anyone define the difference between the calculated Transition Moment and Transition Dipole Moment (in the Excited State Analysis) when one does a TD-DFT calculation? It is not a simple unit conversion and when those quantities are discussed in the literature they seem to be used quite synonymously. Every input or link to proper literature would be greatly appreciated. The output in question looks as follows:

Excited state 2: excitation energy (eV) = 2.4820
Total energy for state 2: -5641.73302437 au
Multiplicity: Singlet
Trans. Mom.: 0.1811 X 0.0234 Y 0.0383 Z
Strength : 0.0021162303
X: D( 437) → V( 2) amplitude = 0.3114
X: D( 441) → V( 1) amplitude = 0.9308

Singlet 2 :

CT numbers (Mulliken)
  omega =     1.0222
  <Phe> =    -0.1050

Exciton analysis of the transition density matrix
  Trans. dipole moment [D]:             0.237087

Cartesian components [D]: [ -0.230112, -0.029764, -0.048712]

Thank you

Hi Sara,

The transition dipole moment you see calculated in the normal TDDFT output is computed as the dot product of the normalized excitation vectors for that state ($X_{ia}$ for CIS/TDA, additionally $Y_{ia}$ for TDHF/RPA/full TDDFT) with the integrals for each of the three dipole operator components. The excitation vectors themselves are unitless, and the dipole integrals are in units of e*bohr. The excitation vectors are in the occupied-virtual subspace of the MO basis, while integrals are typically in the AO basis, so either the excitation vectors need to be transformed into (square) transition densities in the AO basis or the dipole integrals transformed into the occ-virt MO basis for this dot product. (As a side note, you can confirm the literature definition of the oscillator strength yourself by dotting the TDM with itself, multiplying it by the excitation energy in hartree, then multiplying again by two thirds.)

The Cartesian components (printed in units of Debye) in the Excited State Analysis section can be obtained roughly from the TDM components in the prior section by converting from e*bohr to Debye via the conversion factor from Debye - Wikipedia, and then multiplying again by 2. The sign difference is a matter of convention; changes in sign between components should be consistent. The final “Trans. dipole moment [D]” is then the vector (L2) norm of the Cartesian components. When you apply the unit conversion, it won’t match exactly, because the precision of the first output (4 digits) is so low.

Thank you, this is really helpful. I’ll check the conversion to make sure I understood your points correctly.

Those two transition dipole moments are in fact off by just a factor, which appears to be precisely half the conversion factor from Debye to atomic units.

One caveat to the calculation that Eric describes (if you were to attempt to do it yourself) is that for full TDDFT you need to recall that it’s defined with the slightly funny metric xx - yy = 1. (Note the minus sign.)