# Tutorial:Triplet Excitations

In this tutorial, we will calculate triplet excitations for methane with time-propagation and Casida methods, for comparison with the spectra calculated in Tutorial:Optical Spectra from TD and Tutorial:Optical Spectra from Casida. There are fewer details here, so doing those tutorials first is advised.

## Time-propagation

We begin with a spin-polarized calculation of the ground-state, as before but with `SpinComponents`

= `spin_polarized` specified now.

`UnitsOutput`

= eV_angstrom`Radius`

= 4.*angstrom`Spacing`

= 0.18*angstrom`SpinComponents`

= spin_polarized CH = 1.097*angstrom %`Coordinates`

"C" | 0 | 0 | 0 "H" | CH/sqrt(3) | CH/sqrt(3) | CH/sqrt(3) "H" | -CH/sqrt(3) |-CH/sqrt(3) | CH/sqrt(3) "H" | CH/sqrt(3) |-CH/sqrt(3) | -CH/sqrt(3) "H" | -CH/sqrt(3) | CH/sqrt(3) | -CH/sqrt(3) %

You can verify that the results are identical in detail to the non-spin-polarized calculation since this is a non-magnetic system. Next, we perform time-propagation as before, but with the addition of `TDDeltaStrengthMode`

= `kick_spin`.

`CalculationMode`

= td`TDDeltaStrength`

= 0.01/angstrom`TDPolarizationDirection`

= 1`TDDeltaStrengthMode`

= kick_spin tmax = 10`TDPropagator`

= aetrs`TDTimeStep`

= 0.0023/eV`TDMaxSteps`

= 4350 # ~ 10.0/`TDTimeStep`

This kick mode applies a kick with opposite sign to the two spins. Whereas the ordinary kick (`kick_density`) yields the response to a homogeneous electric field, *i.e.* the electric dipole response, this kick yields the response to a homogeneous magnetic field, *i.e.* the magnetic dipole response. Note however that only the spin degree of freedom is coupling to the field; a different calculation would be required to obtain the orbital part of the response. Only singlet excited states contribute to the spectrum with `kick_density`, and only triplet excited states contribute with `kick_spin`. It is possible also to use symmetry (as described in this paper) to obtain both at once with `kick_spin_and_density`.

When the propagation completes, run ** oct-propagation_spectrum** to obtain the spectrum. The file

**now has separate columns for cross-section and strength function for each spin.**

`cross_section_vector`# nspin 2 # kick mode 1 # kick strength 0.005291772086 # pol(1) 1.000000000000 0.000000000000 0.000000000000 # pol(2) 0.000000000000 1.000000000000 0.000000000000 # pol(3) 0.000000000000 0.000000000000 1.000000000000 # direction 1 # Equiv. axes 0 # wprime 0.000000000000 0.000000000000 1.000000000000 # kick time 0.000000000000 #% # Number of time steps = 4348 # PropagationSpectrumDampMode = 2 # PropagationSpectrumDampFactor = 4.0817 # PropagationSpectrumStartTime = 0.0000 # PropagationSpectrumEndTime = 10.0004 # PropagationSpectrumMaxEnergy = 20.0000 # PropagationSpectrumEnergyStep = 0.0100 #% # Electronic sum rule = 0.000000 # Polarizability (sum rule) = 0.000009 #% # Energy sigma(1, nspin=1) sigma(2, nspin=1) sigma(3, nspin=1) sigma(1, nspin=2) sigma(2, nspin=2) sigma(3, nspin=2) StrengthFunction(1) StrengthFunction(2) # [eV] [A^2] [A^2] [A^2] [A^2] [A^2] [A^2] [1/eV] [1/eV] 0.00000000E+00 -0.00000000E+00 -0.00000000E+00 -0.00000000E+00 -0.00000000E+00 -0.00000000E+00 -0.00000000E+00 -0.00000000E+00 -0.00000000E+00 0.10000000E-01 0.28461497E-08 0.14355115E-11 -0.10581816E-12 -0.28290615E-08 -0.17883593E-10 -0.28694560E-11 0.25930434E-08 -0.25774748E-08 0.20000000E-01 0.11345196E-07 0.57352213E-11 -0.42273961E-12 -0.11276909E-07 -0.71449298E-10 -0.11464241E-10 0.10336275E-07 -0.10274060E-07 0.30000000E-01 0.25379450E-07 0.12878695E-10 -0.94916859E-12 -0.25226048E-07 -0.16044239E-09 -0.25743684E-10 0.23122471E-07 -0.22982711E-07 0.40000000E-01 0.44754476E-07 0.22832008E-10 -0.16824564E-11 -0.44482368E-07 -0.28443998E-09 -0.45640266E-10 0.40774488E-07 -0.40526579E-07 0.50000000E-01 0.69201649E-07 0.35547943E-10 -0.26189176E-11 -0.68777698E-07 -0.44285346E-09 -0.71060006E-10 0.63047589E-07 -0.62661340E-07

The physically meaningful strength function for the magnetic excitation is given by `StrengthFunction(1)` - `StrengthFunction(2)` (since the kick was opposite for the two spins). [If we had obtained ** cross_section_tensor**, then the trace in the second column would be the appropriate cross-section to consider.] We can plot and compare to the singlet results obtained in Tutorial:Optical Spectra from TD:

You can also repeat the calculation if you like for `TDDeltaStrengthMode`

= `kick_density` (the default) and confirm that the result is the same as for the non-spin-polarized calculation.

## Casida equation

While the `td` calculation required a spin-polarized ground state, the `casida` mode can take advantage of the fact that the two spins are equivalent in a non-spin-polarized system to calculate triplets without the need for a spin-polarized run. The effective kernels in these cases are:

Therefore, we will do a non-spin-polarized ground-state calculation for this run (not continuing from the spin-polarized state used for time propagation). Do ground-state and unoccupied-states runs, as in Tutorial:Optical Spectra from Casida, with the input file

`CalculationMode`

= unocc`UnitsOutput`

= eV_angstrom`Radius`

= 4.*angstrom`Spacing`

= 0.18*angstrom CH = 1.097*angstrom %`Coordinates`

"C" | 0 | 0 | 0 "H" | CH/sqrt(3) | CH/sqrt(3) | CH/sqrt(3) "H" | -CH/sqrt(3) |-CH/sqrt(3) | CH/sqrt(3) "H" | CH/sqrt(3) |-CH/sqrt(3) | -CH/sqrt(3) "H" | -CH/sqrt(3) | CH/sqrt(3) | -CH/sqrt(3) %`ExtraStates`

= 10

Then, do a Casida run by setting `CalculationMode`

= `casida`, and adding `CasidaCalcTriplet`

= `yes`. Run ** oct-casida_spectrum**, plot the results, and compare to the singlet calculation. How do the singlet and triplet energy levels compare? Can you explain a general relation between them? How does the run-time compare between singlet and triplet, and why?

Try doing a spin-polarized calculation, and compare the results to the singlet and triplet kernel Casida calculations.

## Comparison

As for the singlet spectrum, we can compare the time-propagation and Casida results. What is the main difference, and what is the reason for it?