## (Absolute) sign of magnetic dipole transition matrix elements

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bksh
Posts: 1
Joined: 23 Jan 2018, 23:38
First name(s): D.
Last name(s): B.
Affiliation: ETH Zurich
Country: Switzerland

### (Absolute) sign of magnetic dipole transition matrix elements

Dear DALTON community,

I am currently using DALTON to calculate the magnetic transition dipole moment associated with an electronic excitation $\tilde{X}$ $\rightarrow\tilde{A}$ of a chiral organic molecule ion (C3H6O+). I have calculated the ECD excitation spectrum and have a question regarding the output, in particular the section containing the elements of the magnetic transition moment. My question may be very basic (= stupid), for which I apologize in advance.

For my problem, it is necessary to know the absolute sign of this quantity.

What I need are the matrix elements of the magnetic transition dipole moment which reads in atomic units (I omit the summation over all electron coordinates for brevity):

$\hat{\vec{m}} = - \mu_B \hat{\vec{L}} = -\mu_B \hat{\vec{r}} \times \hat{\vec{p}}$

and in terms of matrix elements:

$\langle \tilde{A} | \hat{m} | \tilde{X} \rangle$ for the ground-to-exc. state transition $\tilde{X}$ $\rightarrow\tilde{A}$.

The associated interaction energy in a homogeneous field would be

$V=-\hat{\vec{m}}\cdot \vec{B}$

In a.u., $\mu_B = 0.5$.

According to the DALTON output:

Code: Select all

@               Magnetic transition dipole moments (au)
---------------------------------------

( mu_B*<0|l_i|n>, where mu_B = 0.5 is the Bohr magneton)

Symm.  Mode   Frequency         Conventional                  London
ex.st.  No.     (eV)        x        y       z         x        y        z
==============================================================================
1      1     2.0947    0.25318 -0.18998 -0.35026   0.25303 -0.18997 -0.35018
1      2     3.1485   -0.38091 -0.15630 -0.21397  -0.38105 -0.15632 -0.21400
1      3     7.9881   -0.06048 -0.16816  0.11696  -0.06054 -0.16833  0.11707
------------------------------------------------------------------------------

I need $\langle n | \hat{m}_i | 0 \rangle = -\mu_B \times \langle n | \hat{l}_i | 0 \rangle$. Knowing that the magnetic dipole op. vector changes sign on complex conjugation and that the Bohr magneton is already included, this implies that the listed matrix element $\langle 0 | \hat{l}_i | n \rangle$ corresponds directly to the quantity I need, without the necessity of further sign flips.

I would like to ask whether my interpretation of the subject is correct.

Best regards,
bksh

hjaaj
Posts: 317
Joined: 27 Jun 2013, 18:44
First name(s): Hans Jørgen
Middle name(s): Aagaard
Last name(s): Jensen
Affiliation: Universith of Southern Denmark
Country: Denmark

### Re: (Absolute) sign of magnetic dipole transition matrix elements

You can never get a specific sign for a transition matrix element. Remember that you can multiply any solution vector to the time-independent Schrödinger equation with -1, and it is still a solution and it still corresponds to the same physical state.

-- Hans Jørgen.

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