Bulk optical absorption spectrum calculations: RPA, rt-

TDDFT, and rt-TDDFT with HSE

revised at 2021.8.12support@pwmat.com

This ppt describes how to calculate optical absorption spectrum for bulk systems using RPA, rt-TDDFT/LDA and rt-TDDFT/HSE methods.

It also uses the newest available range separated hybrid functional calculation within PWmat.

Bulk system optical calculations methods in PWmat

(1) RPA method: this is based on direct Fermi Golden rule for absorption spectrum.

Here the ∂H/∂K is the momentum operator, Ei is the eigen energy of eigen state ψi.

The real part dielectric constant at ω=0 can be found from the integral:

This is the RPA calculation without the local field effect. We will give the treatment for local field effect in another module. In LDA or GGA, usually the ε1 can be rather good, but the ε2(ω) shape might not be so good due to the ignoring of correlation effects. PWmat codes for this method: plot_ABSORP_interp.x (utility program) One could use: LDA, PBE, or HSE for this calculation.

Bulk system optical calculations methods in PWmat

(2) TDDFT method: this is based on TDDFT simulations, which incorporate the correlation effect through TDDFT.

It can be calculated by perturbation TDDFT method, or rt-TDDFT method. In PWmat, currently, we provide the rt-TDDFT solution.

In a rt-TDDFT solution, the dielectric constant is calculated based on the following definition:

Here D(ω) is the dipole moment in one particular direction, and E(ω) is the external perturbing electric field. Both are Fourier transformed from t in the actual simulation.

The above formula is used for non-periodic system (see Absorption module for non-periodic system). For periodic system, we use A(t), and E(t)=dA(t)/dt, and outputcurrent J(t), and dD(t)/dt=J(t). Spatially, A(t) is a constant, and roughly speaking J(t) is also a constant in periodic system. In the current calculation, we use a sharp Gaussianfunction in time to represent E(t) (so A(t) is almost a step like function).

Bulk system optical calculations methods in PWmat

(2) TDDFT method: this is based on TDDFT simulations, which incorporate the correlationeffect through TDDFT. We can use both LDA (or GGA), and HSE for the rt-TDDFT simulations

v The PWmat can carry out rt-TDDFT with HSE functions.

v Especially, PWmat can run the range separated hybrid (RSH, also called it HSE in Pwmat) function with HSE_alpha, HSE_omega, HSE_beta parameters, representing short range mixing (HSE_alpha) and long range mixing (HSE_beta), as well as the transition position (HSE_omega, optimally tuned functional). v The RSH can describe the long range dielectric screening with HSE_beta, it is thus believed it can describe the correct Coulomb interaction in electron-hole interaction.v Theoretical, TDDFT, especially with RSH, can describe the electron-hole interaction, thus it is a bit like Bethe-Salpeter Equation.

AK Manna, .. L. Kronik, “Quantitative prediction of optical absorption in molecular solids from an optimally tunedScreened range-separated hyrbrid functional”, J. Chem. Theory Comput., 14, 2919 (2018).

Other light absorption method in the future

v Bethe_Salpeter Equation (BSE) is an established method to calculate the exciton, and light absorption spectrum in bulk systems.

v It is a bit like the Casida equation in the perturbation treatment of TDDFT. But it has the explicit electron-hole interactions (with screening).

v The hope is that, with the range separated hybrid (RSH) method, the TDDFT can also have the explicit electron-hole interaction, with the hse_beta=1/ε as the screening. So, the hope is: TDDFT+RSH can minic the BSE+GW.

v In the future, PWmat might provide an explicit diagonalization method to solve the BSE.

The steps for RPA method

v Step1: Do a JOB=SCF calculation with reasonable number of kpoints (can be either LDA, or HSE, same below).

v Step2: Do a JOB=NONSCF calculation with more kpoints (larger MP_N123). If the kpoints on JOB=SCF is large enough, one can ignore this step.

v Step3: Do a JOB=DOS calculation, with DOS_DETAIL= 1,MP_N123. This will enable the interpolation scheme for Step4.

v Step4: Run: > plot_ABSORB_interp.x (utility), with the following DOS.input (Ef can be obtained from OUT.FERMI).

Please also check manual for more details.

RPA LDA light absorption spectrumε 2

(ω)

ω (eV)

experiment

RPA-LDA

(1) We have normalize the calculated absorption spectrum with primary cell volume.

(2) One can see, the center of mass is kind of correct, which means the corresponding ε1(0) might be good.

(3) The shape might not be correct, especially the shoulder around 2.5 eV is under estimated.

(4) The turn-on threshold is red shifter probably due to LDA band gap area

Calculated using a 2-atom Si primary cell with 8x8x8 kpoints

The steps for TDDFT calculations with LDA

vWe use a two step process for rt-TDDFT calculation for absorption spectrum calculations. The etot.input looks like this:

Inside IN.TDDFTOPT, we have:

This will output the dipole momentIn k-space (actually current) in the file:OUT.MDDIPOLE.KSPACE

The A(t) function specified by tddft_time line

t (fs)

Line: tddft_time=22,… Specifies a delta like function at 0.1 fs, with a width of 0.0141 fs, and amplitude of 1.5707. 22 means: the actual A(t) is the integral of thisfunction, which looks like left.

Due to the sudden change at 0.01 fs, we like to do two TDDFT: first for 0, to 0.02fs, with a time step of 0.001fs (MD_DETAIL line). Second, Following step1, with 0.01fs step.

The steps for TDDFT calculations with LDA (symmetry considerations)

vWe use a two step process for rt-TDDFT calculation for absorption spectrum calculations. The etot.input looks like this:

This is a 2 atom primary cell example for bulk Si:

In the TDDFT etot.input, one can simply use:MPI_N123 = 4 4 4 0 0 0 2

Note, an option 2 must be used, so there will be no symmetry applied, since during the TDDFT, due to the applied A field, the symmetry will be broken.

In the etot.input, we have applied a A field in the (1,1,1) direction ( IN.A_FIELD=T, 0.002,0.002,0.002)

So, one can still try to applied some symmetry (e.g., rotation along the (1,1,1) direction).

To do this, one can distort the system, so it can only have sym. along (1,1,1). Do a calculation with:

MPI_N123=4 4 4 0 0 0 1

Here option 1 must be used to avoid the time inversion symmetry (during TDDFT calculation, there isno time inversion symmetry, which means k and –k might not be the same.

The steps for TDDFT calculations with LDA (symmetry considerations)

Here is a system we used to generate the desired OUT.KPT andOUT.SYM, using MP_N123=4,4,4,0,0,0,1 for a JOB=SCF calculation

We have changed the identity of these two atom. We have distorted the lattice, so only maintain the sym on (111)We have also displaced the atom along 1,1,1 (0.277), so make sureThere is other symm.

Now we get OUT.KPT (copy that to IN.KPT). Only 20 kpts, otherwise, we will have 64 kpts.

We also get 6 symm operations in OUT.SYMM (copy that to IN.SYMM).

All these symmetry things might depend on the system. If you are not so sure, just use: MP_N123=4,4,4,0,0,0,2

The steps for TDDFT calculations with LDA

v After you set up with IN.KPT, and IN.SYMM (if you want to save some time), Now, just do the two steps calculations, one after another.v Step one: with: MD_DETAIL = 1, 200, 0.001, 0,0 (we are not using temperature, and also the atoms are not moving (0,0,0 in atom.config). v After finished Step one, without copying any files (it will read file from TDDOS), rerun Step two with: MD_DETAIL = 11, 1000, 0.01, 0,0.

After the two runs are finished, the following files will be used to calculate the absorption spectrum:

MDDIPOLE.KSPACE: The k-space current file J(t)OUT.TDDFT_TIME: essentially the input A(t) file specified in line: tddft_time=22,…

A plot of MMDIPOLE.KSPACE

Time (fs)

We will use the J(t), A(t), and the formula

to calculate the absorption spectrum.

Note: D(ω)=J(ω)/iω, E(ω)=iωA(ω)Time (fs)

Using absorption_spec_K2step.x to calculate absorption spectrum After we have the MDDIPOLE.KSPACE and OUT.TDDFT_TIME, we next can useabsorption_spec_K2step.x (utility) to calculate the light absorption spectrum.

Besides the above two files, absorption_spec_K2step.x also needsan absorp_K.input file (left). In it, MD_step12,dt12 is from the etot.input for the two TDDFT steps and dt used. E_field is the amplitude of A_x field from the IN.A_FIELD line in etot.input, and A(3,3) is from the atom.config

The w_cut_min, and w_cut_max, are the cut-off for the ω in eV. Due to numerical error, the calculated ε2(ω) might not be accurate or even diverges for small ω less than the band gap. Thus, it is a good idea to set a [w_cut_min, w_cut_max], outside which, ε2(ω)=0. N_electron is the number of valence electron which contributes to the oscillator strength within [w_cut_min, w_cu_max]. For example, if there are deep semicore levels, but their contributions should be outside this range (or in TDDFT, we keep their adiabatic states fully occupied, not part of the evolution states), then the N_electron should not include them. A(3,3) is from atom.config to be used to calculate the volume. The ε2(ω) will be normalized using above parameter and the equation:

Broadening factor is used to control the ω broadening, a factor around 5 is recommended.

Absorption spectrum results using TDDFT + LDA

TDDFT-LDA

RPA-LDA

Exp

v Compared to RPA, the onset of the absorption spectrum is earlier. v It does developed the double peak In another word, the oscillator strength has shifted downward. v But the center of mass has shifted away from the experimental result. So if ε1(0) is calculated using the integration, it will be too big.

Note, we have using more eigen statesIn the TDDFT expansion:TDDFT_DETAIL = 1, 24, 6NUM_BAND = 30It gives essentially the same results(so the TDDFT itself is fully converge.

Absorption spectrum results using TDDFT + LDA: more kpoints

Using MP_N123=4,4,4, the whole TDDFT calculations only takes a few minutes. To test the convergence, we have used: MP_N123=8, 8, 8. It takes about a hour with 120 reducedk-points Repeat the above process, we get the following absorption spectrum:

TDDFT-LDA4x4x4 kpt

TDDFT-LDA8x8x8 kpt

Exp

v Unfortunately, the 8x8x8 kpt case does not really improve the situation. v The onset is even slightly smaller. v But the first peak is indeed smaller, the two peak shape does more like the experiments.v Nevertheless, please note, for RPA calculation, one needs a huge kpt grid to get a fully converged results.

Absorption spectrum results using TDDFT + HSE (range separated HSE)

In the literature, there are claims that the optimally tune range separated hybrid (RSH) can give good optical spectrums:

Comparison betweenTDDFT-RSH and BSEFor organic crystal Light absorption.

AK Manna, .. L. Kronik, J. Chem. Theory Comput., 14, 2919 (2018).

Optimally tuned range separated hybrid functional (OT-RSH)

Now, the Pwmat can run RSH !

For a JOB=SCF calculation, the etot.input will look like this:

v Note, HSE_ALPHA is for short range mixing, there are some theoretical argument, showing that it is a good idea to set HSE_ALPHA=1/4.

v HSE_BETA is for the long range mixing, it is a good idea to set it to 1/ε(0). for silicon, we have used ε(0)=11.7 for bulk Silicon.

v The HSE_OMEGA controls the cut-off between the short range and long range. Larger HSE_OMEGA, the short range region is shorter.

v The idea of optimally tuned RSH is to change HSE_OMEGA, so we get the best electronic structure, usually to have the correct band gap for a bulk system.

v For Si, we find HSE_OMEGA=0.625 gives the correct band gap of 1.17 eV.

Rt-TDDFT using OT-RSH

Now, the Pwmat can run TDDFT under HSE (Or OT-SRH)! The etot.input for the 2 step absorption spectrum run, looks like this:

v Everything is the same as in rt-TDDFT simulation for LDA, except te XCFUNCTIONAL = HSE and the corresponding HSE_ALPHA,BETA, OMEGA

v We have used a 5 times larger IN.A_FIELD, expecting the HSE converges might not be as good as the LDA converge.

v Same as before, follow the two step procedure, after finished the first run, continue the second run without any changes for 1000 steps. v Note, for HSE, we have used: 4, 1 (first line), instead of 1, 4 (not allowed for HSE).

This the much more expensive than the LDA one. The whole calculation takes about 5 hours, while the Corresponding LDA run only takes about 20 minutes.

Rt-TDDFT OT-RSH absorption results

After the two runs, use the absorption_spec_K2step.x to get the absorption spectrum(output in absorp_spectrum.data), the same as the LDA case. The TDDFT-OT-RSH (HSE) results are shown as below.

TDDFT-LDA

TDDFT-HSE

EXP

v The TDDFT-HSE is indeed closer to the experiment, but not completely close. The center of mass are still shifted. v The amplitude of first peak is even stronger (Note that, more k-point calculates will likely reduce the first peak strength). v Interesting, for smaller ω, the TDDFT-HSE is more unstable than the TDDFT-LDA, this is probably due to the long range explicit exchange integral in RSH. v Compared to BSE, the TDDFT-HSE is still worse (at least for this Si case). v But it is possible, for other cases, the situation could be much better.