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In the ORCA output for a TD-DFT calculation, there are two types of spectra printed. One is called "transition electric dipole moment" and "transition velocty dipole moment".

-----------------------------------------------------------------------------
         ABSORPTION SPECTRUM VIA TRANSITION ELECTRIC DIPOLE MOMENTS
-----------------------------------------------------------------------------
State   Energy    Wavelength  fosc         T2        TX        TY        TZ  
        (cm-1)      (nm)                 (au**2)    (au)      (au)      (au) 
-----------------------------------------------------------------------------
   1   27648.0    361.7   0.719341268   8.56537  -2.87788   0.45802  -0.27090
   2   30253.3    330.5   0.184731134   2.01022  -1.38229   0.30293  -0.08792
   3   33113.8    302.0   0.000596722   0.00593  -0.06749  -0.02846   0.02383
   4   33958.2    294.5   0.016666067   0.16157  -0.23836  -0.26972   0.17890
   5   35481.0    281.8   0.001963234   0.01822   0.07587  -0.09746   0.05442
   6   36251.0    275.9   0.082087211   0.74547   0.30134   0.67418  -0.44737
   7   36892.9    271.1   0.007036359   0.06279  -0.18262  -0.04896   0.16444
   8   40296.4    248.2   0.163705579   1.33744  -1.15180   0.10363   0.00712
   9   40712.6    245.6   0.085961535   0.69511  -0.81462  -0.01219   0.17707
  10   41670.1    240.0   0.002721111   0.02150   0.08394  -0.09869  -0.06864

-----------------------------------------------------------------------------
         ABSORPTION SPECTRUM VIA TRANSITION VELOCITY DIPOLE MOMENTS
-----------------------------------------------------------------------------
State   Energy    Wavelength   fosc        P2         PX        PY        PZ  
        (cm-1)      (nm)                 (au**2)     (au)      (au)      (au) 
-----------------------------------------------------------------------------
   1   27648.0    361.7   0.179172895   0.03386   0.18109  -0.00384  -0.03240
   2   30253.3    330.5   0.036126643   0.00747   0.08535   0.01325  -0.00302
   3   33113.8    302.0   0.007496146   0.00170   0.03852   0.00888  -0.01156
   4   33958.2    294.5   0.001365959   0.00032   0.01150  -0.01299  -0.00401
   5   35481.0    281.8   0.002616796   0.00063  -0.00588   0.01407   0.02005
   6   36251.0    275.9   0.011772984   0.00292  -0.04163   0.03367  -0.00706
   7   36892.9    271.1   0.000265060   0.00007  -0.00315   0.00212   0.00724
   8   40296.4    248.2   0.075499769   0.02079   0.14370  -0.00400  -0.01129
   9   40712.6    245.6   0.028944201   0.00805   0.08950   0.00244  -0.00615
  10   41670.1    240.0   0.000916239   0.00026  -0.00984   0.01137   0.00590

The oscillator strenghts are different in each, but the wavelengths are the same.

I am working on predicting UV/visible spectra. I do not need the exact intensities (extinction coefficients), however, I need to be able to get the relative order of the intensities of the peaks right, so that I can match the TD-DFT result to the experimental result with a script. Notice that the order of the intensities between root 3 and 7 are not the same between the two spectra.

Which of the oscillator strengths are better correlated with the experimental extinction coefficient (in the experimental absorption spectrum)? (And more generally, why are the two different?)

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2 Answers 2

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You have stumbled across a problem that was described in 1983 in the following way (note that the "40 years" refers to 1943-1983):

"Over the past 40 years, many authors have tried to answer the following question: For a given pair of approximate wavefunctions $|\phi_a\rangle$ and $|\phi_b\rangle$, which form of the $f$ value, $f_L$ calculated with [the dipole length transition moment] or $f_V$ calculated with [the dipole velocity transition moment] may be expected to be more accurate?"

The phrase "transition velocity dipole moment" is sometimes used to refer to the "velocity form" of the transition dipole moment.

The above quote comes from the paper "Electric dipole oscillator strengths: length and velocity!" by Roginsky, Klapisch and Cohen published in "Chemical Physics Letters Volume 95, Issue 6, 18 March 1983, Pages 568-572" proposed that when $f_L$ and $f_V$ (the "length" and "velocity" forms of the electric dipole oscillator strength) values are different, then an appropriate linear combination of them will often be more accurate than the better of $f_L$ and $f_V$. They also define an "acceleration" version in Eq. 6 of that paper!

None of this would be necessary if exact wavefunctions were used, but usually in quantum chemistry calculations, we only have approximate wavefunctions.

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To give a little background on where these different forms come from, it is related to gauge theory in electromagnetism. We can represent an electromagnetic field through Maxwell's equations using a scalar potential $V$ and a vector potential $\mathbf{A}$. It turns out these values are not unique: for any twice differentiable function $f(x,t)$ $$ \begin{align} V' = V-\frac{\partial f}{\partial t}\\ \mathbf{A}' = \mathbf{A} + \nabla f \end{align} $$ are also valid solutions that describe the same field. This is referred to as a gauge symmetry. While different gauges (i.e. choices of $f$) are equivalent for exact calculations (more on this below), they lead to different forms of various matrix elements, most notably the electric dipole.

The common form of dipole matrix elements $-\langle a|\boldsymbol{r}|b\rangle$ (in atomic units) corresponds to using the so called length gauge (due to the dipole depending on the length operator). In the velocity gauge, these same matrix elements would be written $i\omega_{ab}^{-1}\langle a|\boldsymbol{p}|b\rangle$ (or $-\omega_{ab}^{-1}\langle a|\boldsymbol{\nabla}|b\rangle$ showing the connection to velocity).

For the exact eigenstates of the molecular Hamiltonian, these expressions are exactly equal due to the identity $$\langle a|\boldsymbol{X}|b\rangle=\omega_{ab}^{-1}\langle a|[\boldsymbol{X},H]|b\rangle$$ where $\boldsymbol{X}$ can be any operator and $[\boldsymbol{r},H]=-\boldsymbol{p}$. Since this identity/commutator only holds exactly for an exact calculation, these different forms of the dipole are not equivalent in practical calculations.

As to which form to use for your calculations, I don't think there is a great physical reason to choose one gauge over another in general; afterall in principle they are both equally valid. I'm most familiar with this gauge issue in the context of computing optical rotation (OR) and for that property the choice is usually made based on computational convenience.

The length gauge leads to an origin dependent OR without corrections, such as using gauge-including/London atomic orbitals, so it is somewhat trickier to implement properly. The velocity gauge is origin independent, but gives a nonzero OR for a static perturbation, which is nonphysical, so this static limit is usually subtracted from the results for other frequencies, essentially making any given calculation twice as expensive. So for OR, it depends whether you want a cheaper calculation or a simpler implementation.

For absorption, I expect any choice between the two will largely be an empirical and system dependent, though Nike's answer suggests some efforts have been made to mathematically motivate one choice or another.

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