3.29 TDDFT - linear response

The goal is to calculate excitation energies ${\omega }_I$ and corresponding oscillator strengths $f_I$ from

$𝛀{𝐅}_I={\omega }_I^2{𝐅}_I.$

(3.105)

Linear response theory (see [52]) is the basis for this calculation. We construct

${𝛀}_{ias,jbt}={\delta }_{i,j}{\delta }_{a,b}{\delta }_{s,t}{({ϵ}_a-{ϵ}_i)}^2+2\sqrt{{ϵ}_{as}-{ϵ}_{is}}{𝐊}_{ias,jbt}\sqrt{{ϵ}_{bt}-{ϵ}_{jt}}$

(3.106)

with the coupling kernel

${𝐊}_{ias,jbt}={\displaystyle \int \int {\phi }_i^{\ast }(𝐫){\phi }_a(𝐫)\left[\frac{1}{\left|𝐫-{𝐫}^{\prime }\right|}+f_{xc}(𝐫,{𝐫}^{\prime })\right]{\phi }_j^{\ast }({𝐫}^{\prime }){\phi }_b({𝐫}^{\prime })𝑑𝐫𝑑{𝐫}^{\prime }}.$

(3.107)

In this notation I refer with the indices $i,j$ to occupied and with $a,b$ to virtual orbitals, while $s$ and $t$ denote the spin. The input energies $ϵ$ are obtained from either ground state Hartree-Fock or DFT calculations. Going by this rule we contruct the matrix $𝛀$ which then is solved for eigenvalues and eigenvectors. The excitation energies ${\omega }_I$ then follow from From the eigenvectors ${𝐅}_I$ the oscillator strengths $f_I$ can be obtained from

(3.108)

with the $\widehat{𝐗}$ being the spatial operator for the X direction and the others respectively with ${\mathrm{\Psi }}_0$ being the all electron ground state wave function and ${\mathrm{\Psi }}_I$ being the all electron excited state wave function for the state $I$ with excitation energy ${\omega }_I$.

For the TDHF calculation mode, the kernel ${𝐊}_{ias,jbt}$ is modified to become

${𝐊}_{ias,jbt}=(ias|jbt)+{\delta }_{s,t}(ij|ab)$

(3.109)

that has only the bare Coulomb part $(ia|jb)$ and the exact exchange part $(ij|ab)$ from Hartree-Fock theory. This Hartree-Fock Kernel creates a non-Hermitian matrix $𝛀$. Please not that this calculation mode is as yet only available for single processor runs due to the lack of a non-Hermitian parallel eigenvalue solver.

As of now, only the pw-lda $f_{xc}$ kernel is available for TDDFT calculation in aims, as well as other LDA lernels from libxc [194]. ²² 2 http://libxc.gitlab.io Additionally, for the full TDDFT calculation it is possible to choose functionals at will that are available from this libxc. Importantly, the initial DFT calculation that generates the input energies $ϵ$ for 3.107 is not restricted to only the same XC functional as the subsequent LR-TDDFT calculation. Different ground state functionals can be employed, though we recommend careful benchmarking for any new combination. For example, it might come in handy to be able to mix different hybrid functionals or do a DFT ground state calculation with a functional that does not provide an $f_{xc}$ kernel and still be able to perform a TDDFT calculation on top of that using the supported LR-TDDFT $f_{xc}$ kernel.

Reference [201] provides extensive benchmarks, identifying basis sets that are applicable both to LR-TDDFT and BSE@GW calculations for molecules in FHI-aims.

In particular, for light-element molecules, we recommend the “tier2+aug2” basis sets, currently found in:

species_defaults/non-standard/Tier2_aug2

for these calculations.

With the keyword neutral_excitation the user can specify the calculation mode for the linear response theory.
Also the keyword empty_states should be set to 1000, or the keyword calculate_all_eigenstates should be used, to make sure the code generates all possible empty states provided from the basis set. This number will also be reduced automatically by the code to the maximum number that can be generated from the basis set.

With the keyword excited_states the user can specify the number of excited states that will be printed in the output. The default for this is 50 if there are that many. Normal production runs will generally have many more (depending on the basis set and the number of electrons involved) that can easily reach beyond 10000. So to avoid a really huge output from this routine this defaut is set rather low. Feel free to choose any number of your liking, if it should be too large it will automatically be defaulting to all excited states. At the end of a calculation the file TDDFT_LR_Spectrum_(singlet/triplet).dat will be written into the directory the FHI-aims program was run in. It contains all computed excitation energies and the corresponding oscillator strengths.