(sec:mcd.detailed)= # Simulation of (Magnetic) Circular Dichroism and Absorption Spectra (sec:mcd.general.detailed)= ## General description of the program ORCA can now simulate optical spectra that include spin-orbit coupling contributions at all levels of theory by using a common implementation. {cite}`foglia_mcd_2022` Following the energy-loss approach, the absorption cross section for a transition between states $\tilde{P}$ and $\tilde{Q}$ can be expressed as: $$\begin{split} \sigma_{\tilde{P}\tilde{Q} } = \frac{4 \pi^2}{c (E_{\tilde{Q} }-E_{\tilde{P} }) } |T_{\tilde{P}\tilde{Q} }|^2, \end{split}$$ where c is the speed of light; $E_{\tilde{P} }$ and $E_{\tilde{Q} }$ are the energy of the states ${\tilde{P} }$ and ${\tilde{Q} }$, respectively; $T_{{\tilde{P} }{\tilde{Q} }}$ is the transition moment between states ${\tilde{P} }$ and ${\tilde{Q} }$ and it can be computed with different expressions based on the applied approximation. Under a dipolar approximation to the light-matter interaction, the transition moment takes the form: $$\begin{split} T_{{\tilde{P} }{\tilde{Q} }} =& { \sum_{i=1}^N \bra{{\tilde{P} }} \mathcal{E} \cdot \hat{\textbf{p} }_i % \ket{{\tilde{Q} }} }, \\ \end{split}$$ where the sum runs over all electrons $i$; $\hat{\textbf{p} }_i$ is the linear momentum operator; and $\mathcal{E}$ is the polarization vector of the incident light. In order to take into account all the electric and magnetic mechanisms in the transition, it is necessary to use the full field-matter interaction operator (FFMIO). For transition moment, it leads to the equation {eq}`eqn:exactTM`. $$\begin{split} T_{{\tilde{P} }{\tilde{Q} }} =& \frac{e}{m_e} \sum_{i=1}^N \bra{{\tilde{P} }} \mathcal{E} \cdot \Big[ e^{i\textbf{k}\cdot \hat{\textbf{r} }_i } %\textbf{\hat{r}_i} } % \hat{\textbf{p} }_i \Big]\ket{{\tilde{Q} }} % \end{split} $$ (eqn:exactTM) where $\hat{r}_i$ is the position operator of i-th electron and $\textbf{k}$ is the wave vector that points in the direction of the light propagation whose magnitude is related to the wavelength by $\lambda = 2 \pi / |\textbf{k}|$. For free-rotating molecules, it is necessary to consider all possible orientations of the molecule with respect to the direction of the incident light. In some cases, such as the absorption of linear-polarized light under a dipolar approximation, the effect of the orientation can be averaged analytically. However, numerical integration over some selected molecular orientations, labeled as $o$ in equation {eq}`eqn:OrientAver`, is generally necessary. $$\begin{split} <\sigma_{{\tilde{P} }{\tilde{Q} }}> = \sum_o w_o \frac{4 \pi^2}{c (E_{{\tilde{Q} }}(o)-E_{\tilde{P} }(o)) } |T_{{\tilde{P} }{\tilde{Q} }}(o)|^2 \end{split} $$ (eqn:OrientAver) where $w_o$ is the weight of the orientation in the quadrature. The implementation has been designed to compute the absorption of circularly-polarized light on systems under the effect of an additional external magnetic field, B, which modifies the states $\tilde{P}$ and $\tilde{Q}$ for each orientation $o$ through a Zeeman perturbation. The computed results are presented as the difference in the absorption of the left (-) and right (+) circularly-polarized light ($\Delta$f$_{osc}$) and as the sum of the oscillator strength (f$_{osc}$), which corresponds to the linearly-polarized light absorption. The molecular orientations are constructed by using rotation matrices with three Euler angles: $\chi$, $\theta$, and $\phi$. Herein, $\chi$ (the rotation angle on a plane perpendicular to the direction of external magnetic field/incident light) is integrated analytically whereas $\theta$ and $\phi$ are taken on a grid. Finally, the states $\tilde{P}$ and $\tilde{Q}$ are obtained from QDPT by expanding the states over non-relativistic eigenstates of $\hat{H}_{0}$ ($\{I,J\}$) and the coefficients of the expansion ($d_{I\tilde{Q} }$) are obtained from the diagonalization of the complex matrix, which contains $\hat{H}_{0}$ as well as the SOC and Zeeman contributions (and SSC, if it is implemented in the selected electronic structure theory) expressed in $\{I,J\}$. $$\begin{split} \bra{\Psi_I^{SM} }\hat{H}_{0}+\hat{H}_{SOC}+\hat{H}_{Zeeman}\ket{\Psi_J^{S'M'} }= \delta_{IJ}\delta_{SS'}\delta_{MM'}E_I^S + \bra{\Psi_I^{SM} }\hat{H}_{SOC}+\hat{H}_{Zeeman}\ket{\Psi_J^{S'M'} } \end{split} $$ (eqn:mcdqdpt) (sec:mcd.example.detailed)= ## Running and analyzing MCD calculations in TDDFT module A minimum input to compute the Magnetic Circular Dichroism (MCD) requires setting the keyword DoMCD to true and to include an intensity for the external magnetic field B (in Gauss). An example input for the TD-DFT module is as follows: ```{literalinclude} ../../orca_working_input/MCD.inp :language: orca ``` In the output file of this job, the estimated oscillator strengths (f$_{osc}$) and the difference between left and right circularly polarized light absorption ($\Delta$f$_{osc}$) are provided: ``` ------------------------------------------------------------------------------ MCD Transitions via transition electric dipole moments B = 50000.00 Gauss T = 300.000 K ------------------------------------------------------------------------------ dfosc fosc dfosc/fosc 0 -> 1 -0.0000000000 0.0000000003 -0.0663180503 0 -> 2 0.0000000000 0.0000000005 0.0000209714 0 -> 3 0.0000000000 0.0000000003 0.0661609417 0 -> 4 -0.0000000001 0.0000000004 -0.2991916107 0 -> 5 -0.0000000000 0.0000000006 -0.0009270335 0 -> 6 0.0000000001 0.0000000004 0.2984154405 0 -> 7 0.0000000003 0.0000000008 0.3721777693 0 -> 8 -0.0000000000 0.0000000009 -0.0002960328 0 -> 9 -0.0000000003 0.0000000008 -0.3689867758 0 ->10 -0.0000050986 0.1741092913 -0.0000292840 ``` These results may not be accurate when the energetic order of the states changes with respect to the relative orientation between the molecule and the external magnetic field. To obtain accurate results, it is necessary to perform a post-processing step for all orientations using the `orca_mapspc` program, which saves the results in a file that has the `.cis-el.dipole-length.1.mcd` extension: ```orca orca_mapspc fur-mcd.cis-el.exact.1.mcd MCD -x050000 -x155000 -n10000 -w2000 ``` In this example, we generate the spectrum (M$^{-1}$) between 50000 and 55000 cm$^{-1}$ (-x050000 -x155000), using 10000 points (-n10000) and including a broadened normalized Gaussian function with a full width at half maximum of 2000cm$^{-1}$ (-w2000). Multiple MCD calculations can be performed in one run by setting multiple values for B. Transition moments can be also obtained through ED velocity formulation and FFMIO operator by setting the keywords `DoVelocity` and `DoQuadrupole` to true, respectively: ```{literalinclude} ../../orca_working_input/MCDfull.inp :language: orca ``` The results are printed separately in the output file for each setting: ``` ------------------------------------------------------------------------------ MCD Transitions via transition electric dipole moments B = 50000.00 Gauss T = 300.000 K ------------------------------------------------------------------------------ dfosc fosc dfosc/fosc 0 -> 1 -0.0000000000 0.0000000003 -0.0663180503 . . . ------------------------------------------------------------------------------ MCD Transitions via transition velocity dipole moments B = 50000.00 Gauss T = 300.000 K ------------------------------------------------------------------------------ dfosc fosc dfosc/fosc 0 -> 1 -0.0000000001 0.0000000013 -0.0477214281 . . . ------------------------------------------------------------------------------ MCD Transitions via full Semi-classical formulation B = 50000.00 Gauss T = 300.000 K ------------------------------------------------------------------------------ dfosc fosc dfosc/fosc 0 -> 1 -0.0000000001 0.0000000016 -0.0731029604 . . . ``` Post-processing results are saved in the files having the `.cis-el.dipole-length.1.mcd`, `.cis-el.dipole-vel.1.mcd`, and `.cis-el.exact.1.mcd` extensions. $\textbf{NOTE:}$ It is worth enphasizing that the computed values of $\Delta$f$_{osc}$ correspond to the difference in absorption between left and right circularly polarized light for the selected transition moments. In the case of both ED approximations, $\Delta$f$_{osc}$ corresponds to the MCD signal. The sum of natural circular dichroism and magnetic-induced circular dichroism is obtained when the FFMIO is requested. To obtain only the MCD spectrum in an FFMIO scheme, it is necessary to subtract the natural circular dichroism by setting B to 0.0. (sec:mcd.modules.detailed)= ## Running MCD calculations in other modules The MCD implementation can also be used in other modules such as STEOM-CCSD, CAS, ROCIS, and MRCI (see the input file examples given below) by using the same keywords as those described for the TDDFT module. In the case of CAS, ROCIS, and MR-CI modules, it is necessary to include the keyword `NewMCD True`; otherwise, the previous MCD implementation will be called instead. ```{literalinclude} ../../orca_working_input/MCDMRCI.inp :language: orca ``` ```{literalinclude} ../../orca_working_input/MCDCAS.inp :language: orca ``` ```{literalinclude} ../../orca_working_input/MCDROCIS.inp :language: orca ``` ```{literalinclude} ../../orca_working_input/MCDSTEOM.inp :language: orca ``` It is important to keep in mind that the calculation of the MCD relies on the proper description of the transition moments and angular momentum for calculating the Zeeman perturbation. Therefore, the user is responsible for selecting the proper electronic structure method. (sec:mcd.keywordlist.detailed)= ## List of related keywords ```orca %selected module DoMCD False # Enables the use of the MCD module DoDipoleVelocity True # Use the electric dipolar velocity formulation # for the light-matter interaction DoFullSemiClassical False # Use the full semiclasical ligth-matter # interaction transition moments MCDGridtype 1 # Grid for the molecular orientational average # 1 = Lebedev grid (default) # 2 = Regular grid MCDLebedev 14 # Number of points if Lebedev grid is selected # 6, 12, 14(default), 26, 50, 110, 194, 302, 434, 590, 770 NPOINTSPHI 10 # Number of points for phi angle if regular grid is selected NPOINTSTHETA 10 # Number of points for theta angle if regular grid is selected B 3000.0 # Magnetic field in Gauss Temperature 300.0 # Self-explanatory. One temperature must be defined # for each B value end ```