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### Can Dalton report real and imagenary components of the dipole matrix elements?

Posted: **08 Sep 2020, 02:06**

by **rlt92**

Hi, as part of my project I'm interested in computing dipole transition matrix elements between a range of states including the ground state and a set of excited states in nitrobenzene. I've been using Dalton to perform an MCSCF wavefunction calculation to compute the ground state configuration then compute the transition dipole matrix elements on the ground state, between excited states, and from ground to excited state. When I perform these calculations however I only get one real valued number representing the matrix element.

Are these values are the real component of the dipole transition matrix elements? If so how can I get the imaginary components? If not, what do these values represent and how can I retrieve the real and imaginary values? Do I need to worry about phase stability of these values between calculations when using dalton?

Attached are copies of my earlier dipole moment calculations for nitrobenzene.

### Re: Can Dalton report real and imagenary components of the dipole matrix elements?

Posted: **09 Sep 2020, 13:53**

by **hjaaj**

As all wave functions are real, also for excited states, and the dipole operator is a real operator, I do not understand why you would expect imaginary parts? -- It is a different thing if you do a damped response calculation an imaginary lifetime factor.

### Re: Can Dalton report real and imagenary components of the dipole matrix elements?

Posted: **10 Sep 2020, 09:39**

by **rlt92**

Thank you for the response. While the dipole operator is real, I guess I was expecting that the wave function would have real and imaginary components which would result in real and imaginary values of the dipole operator (spherical harmonics of a hydrogen atom come to mind as an example). Maybe this isn't the place for this but why are the wave functions considered by Dalton purely real? My first couple of thoughts is that maybe it's because we're only considering the electronic component of the wave function or maybe it's due to other approximations baked into how these wave functions are calculated?

That being said, I am interested in damped response calculations, I was looking at the *ABSORP module and it appears that it has the ability to compute the polarizability and the hyperpolarizability. I guess at some point during the calculation dipole moments might be computed but I didn't see many options related to dipole moment calculations listed in the manual related to that module.

### Re: Can Dalton report real and imagenary components of the dipole matrix elements?

Posted: **10 Sep 2020, 12:19**

by **taylor**

In the absence of magnetic fields (and ignoring relativistic effect, or approximating them in certain ways), without loss of generality the wave function can be taken as real. This has enormous computational advantages compared to ignoring this possibility, and thus most codes looking at energies and properties of molecules consider only real wave functions and real property operators.

If we take the hydrogen atom case, it is true that to obtain eigenfunctions of both l^{2} and l_{z} we need to use complex spherical harmonics. This is usually ignored in favour of eigenfunctions of l^{2} combined so as to give a real, rather than complex associated Legendre function. In other words, instead of p_{+1}, p_{0}, and p_{-1}, we use the more familiar p_{x}, p_{y}, and p_{z}. The spherical harmonics for m_{l}=0 are all real, but in general for other values they are complex. Again, the use of real p, d, etc., functions is of no consequence in calculations (assuming no magnetic fields).

If we do, for whatever reason, want a complex solution, it is straightforward to combine real functions to obtain angular momentum eigenfunctions, and similarly operators. One can thus go from a calculation of real wave functions for (say) a diatomic molecule to the appropriate eigenfunctions of l_{z}, which is the only good quantum number for electronic orbital motion in this case, and a spherical tensor form for an operator such as the dipole or quadrupole moment. Be aware that if you start to explore this it is crucial to pay attention to the phase convention used for the spherical harmonics, which is almost universally that of Condon and Shortley (functions with m_{l}=0 are all real, and the angular momentum shift operators all give a positive result). This does not always give the result naively expected: for example

p_{-1} = p_{x} - ip_{y}

but

p_{+1} = -p_{x} - ip_{y}, not p_{x} + ip_{y}.

The phase issue is irrelevant if we use only real functions and operators, but it requires attention when trying to construct the corresponding complex forms.

Best regards

Pete

### Re: Can Dalton report real and imagenary components of the dipole matrix elements?

Posted: **11 Sep 2020, 06:27**

by **taylor**

I was a little quick sending yesterday: I had made a mental note to add a remark about normalization and then forgot about it and eventually hit "submit"... Of course, the linear combinations I showed for the complex p functions in terms of real functions are not normalized and the RHSs should be multiplied by 1/sqrt(2) to accomplish this.

Best regards

Pete

### Re: Can Dalton report real and imagenary components of the dipole matrix elements?

Posted: **14 Sep 2020, 19:50**

by **hjaaj**

See reference [316] in the Dalton 2018 manual for the original derivation and explanation of "damped response theory":

[316] Patrick Norman, David M. Bishop, Hans Jrgen Aa. Jensen, and Jens Oddershede.

Near-resonant absorption in the time-dependent self-consistent field and multicongurational self-consistent field approximations.

The Journal of Chemical Physics, 115(22):10323{10334, 2001.