# Time-Dependent DFT

This is a short tutorial on how to run linear-response time-dependent density functional theory (LR-TDDFT) computations for absorption and emission spectroscopy. The TDDFT module enables a description of excitation energies and excited-state computations within the Tamm-Dancoff approximation (TDA) featuring GGA and hybrid functionals as well as semi-empirical simplified TDA kernels. The details of the implementation can be found in Strand2019 and in Hehn2022 for corresponding excited-state gradients. Note that the current module is based on an earlier TDDFT implementation Iannuzzi2005. Please cite these papers if you were to use the TDDFT module for the computation of excitation energies (Strand2019, Iannuzzi2005) or excited-state gradients (Hehn2022).

## Brief theory recap

The implementation in CP2K is based on the Tamm-Dancoff approximation (TDA), which describes each excited state \(p\) with the excitation energy \(\Omega^p\) and the corresponding excited-state eigenvectors \(\mathbf{X}^p\) as an Hermitian eigenvalue problem

\[\begin{split} \mathbf{A} \mathbf{X}^p &= \Omega^p \mathbf{S} \mathbf{X}^p \, , \\ \sum_{\kappa k} [ F_{\mu \kappa \sigma} \delta_{ik} - F_{ik \sigma} S_{\mu \kappa} ] X^p_{\kappa k \sigma} + \sum_{\lambda} K_{\mu \lambda \sigma} [\mathbf{D}^{{\rm{\tiny{X}}}p}] C_{\lambda i \sigma} &= \sum_{\kappa} \Omega^p S_{\mu \kappa} X^p_{\kappa i \sigma} \, . \end{split}\]

The Hermitian matrix \(\mathbf{A}\) contains as zeroth-order contributions the difference in the Kohn-Sham (KS) orbital energies \(\mathbf{F}\), and to first order kernel contributions \(\mathbf{K}\) which comprise –depending on the chosen density functional approximation – Coulomb \(\mathbf{J}\) and exact exchange \(\mathbf{K}^{\rm{\tiny{EX}}}\) contributions as well as contributions due to the exchange-correlation (XC) potential \(\mathbf{V}^{\rm{\tiny{XC}}}\) and kernel \(\mathbf{f}^{\rm{\tiny{XC}}}\),

\[\begin{split} F_{\mu \nu \sigma} [\mathbf{D}] &= h_{\mu \nu} + J_{\mu \nu \sigma} [\mathbf{D}] - a_{\rm{\tiny{EX}}}K^{\rm{\tiny{EX}}}_{\mu \nu \sigma} [\mathbf{D}] + V_{\mu \nu \sigma}^{\rm{\tiny{XC}}} \, , \\ K_{\mu \nu \sigma} [\mathbf{D}^{{\rm{\tiny{X}}}p}] &= J_{\mu \nu \sigma} [\mathbf{D}^{{\rm{\tiny{X}}}p}] - a_{\rm{\tiny{EX}}} K^{\rm{\tiny{EX}}}_{\mu \nu \sigma}[\mathbf{D}^{{\rm{\tiny{X}}}p}] + \sum_{\kappa \lambda \sigma'} f^{\rm{\tiny{XC}}}_{\mu \nu \sigma,\kappa \lambda \sigma'} D_{\kappa \lambda \sigma'}^{{\rm{\tiny{X}}}p} \, . \end{split}\]

\(\mathbf{S}\) denotes the atomic-orbital overlap matrix, \(\mathbf{C}\) the occupied ground-state KS orbitals and \(\mathbf{D}\) and \(\mathbf{D}^{\rm{\tiny{X}}}\) ground-state and response density matrices,

\[\begin{split} D_{\mu \nu \sigma} &= \sum_k C_{\mu k \sigma} C_{\nu k \sigma}^{\rm{T}} \, , \\ D_{\mu \nu \sigma}^{{\rm{\tiny{X}}}p} &= \frac{1}{2} \sum_{k} ( X^p_{\mu k \sigma} C_{\nu k \sigma}^{\rm{T}} + C_{\mu k \sigma} (X^p_{\nu k \sigma})^{\rm{T}} ) \, . \end{split}\]

Within the current implementation, symmetrization and orthogonalization of the response density matrix is ensured at each step of the Davidson algorithm. The current implementation features to approximate the exact exchange contribution of hybrid functionals using the auxiliary density matrix method (ADMM). Furthermore, the standard kernel can be approximated using the semi-empirical simplified Tamm-Dancoff approximation (sTDA), neglecting in this case XC contributions and approximating both Coulomb and exchange contributions \(\mathbf{J}\) and \(\mathbf{K}\) using semi-empirical operators \(\boldsymbol{\gamma}^{\rm{\tiny{J}}}\) and \(\boldsymbol{\gamma}^{\rm{\tiny{K}}}\) depending on the interatomic distance \(R_{AB}\) of atoms \(A\) and \(B\),

\[\begin{split} \gamma^{\rm{\tiny{J}}}(A,B) &= \left ( \frac{1}{(R_{AB})^{\alpha} + \eta^{-\alpha}} \right)^{1/\alpha} \, , \\ \gamma^{\rm{\tiny{K}}}(A,B) &= \left ( \frac{1}{(R_{AB})^{\beta} +( a_{\rm{\tiny{EX}}}\eta)^{- \beta} } \right )^{1/\beta} \, , \end{split}\]

that depend on the chemical hardness \(\eta\), the Fock-exchange mixing parameter \(a_{\rm{\tiny{EX}}}\) and powers of \(\alpha\) and \(\beta\) for either Coulomb and exchange interactions.

Within the current implementation, oscillator strengths can be calculated for molecular systems in the length form and for periodic systems using the velocity form (see Strand2019).

Based on the TDA eigenvalue problem, excited-state gradients can be formulated based on a variational Lagrangian for each excited state \(p\),

\[\begin{split} L [\mathbf{X}, \mathbf{C}, \Omega, \bar{\mathbf{W}}^{\rm{\tiny{X}}}, \bar{\mathbf{Z}}, \bar{\mathbf{W}}^{\rm{\tiny{C}}} ] &= \Omega - \sum_{\kappa \lambda k l \sigma} \Omega ( X_{\kappa k \sigma }^{\rm{T}} S_{\kappa \lambda } X_{\lambda l \sigma} - \delta_{kl} ) \\ &- \sum_{kl \sigma} ( \bar{W}_{kl \sigma}^{\rm{\tiny{X}}} )^{\rm{T}} \sum_{\kappa \lambda} \frac{1}{2} ( C_{\kappa k \sigma}^{\rm{T}} S_{\kappa \lambda} X_{\lambda l \sigma} + X_{\kappa k \sigma}^{\rm{T}}S_{\kappa \lambda} C_{\lambda l \sigma}) \\ &+ \sum_{\kappa k \sigma}( \bar{Z}_{\kappa k \sigma})^{\rm{T}} \sum_{\lambda} ( F_{\kappa \lambda \sigma}C_{\lambda k \sigma} - S_{\kappa \lambda } C_{\lambda k \sigma} \varepsilon_{k \sigma}) \\ &- \sum_{kl\sigma} (\bar{W}^{\rm{\tiny{C}}}_{kl \sigma})^{\rm{T}} ( S_{kl \sigma} - \delta_{kl})\, . \end{split}\]

introducing Lagrange multipliers \(\bar{\mathbf{W}}^{\rm{\tiny{X}}}\), \(\bar{\mathbf{W}}^{\rm{\tiny{C}}}\), and \(\bar{\mathbf{Z}}\) to ensure stationarity of the corresponding ground-state (GS) equations and to account for the geometric dependence of the Gaussian orbitals and thus requiring to solve the Z vector equation iteratively.

## The LR-TDDFT input section

To compute absorption spectra, parameters defining the LR-TDDFT computation have to be specified in the TDDFPT subsection. Furthermore, RUN_TYPE has to be set to `ENERGY` and the underlying KS ground-state reference has to be specified in the DFT section.

The most important keywords and subsections of TDDFPT are:

- KERNEL: option for the kernel matrix \(\mathbf{K}\) to choose between the full kernel for GGA or hybrid functionals and the simplified TDA kernel

- NSTATES: number of excitation energies to be computed

- CONVERGENCE: threshold for the convergence of the Davidson algorithm

- RKS_TRIPLETS: option to switch from the default computation of singlet excitation energies to triplet excitation energies

- RESTART: the keyword enables the restart of the TDDFPT computation if a corresponding restart file (.tdwfn) exists

- WFN_RESTART_FILE_NAME: for a restart of the TDDFPT computation, the name of the restart file has to be specified using this keyword

To compute excited-state gradients and thus corresponding fluorescence spectra, the excited state to be optimized furthermore has to be specified by adding the subsection `EXCITED_STATES` of the section `DFT`.

## Simple examples

### Excitation energies for acetone

The following input is a standard input for calculating excitation energies with the hybrid functional PBE0. It should be noted that it is possible to compute the exact exchange integrals for hybrid functionals analytically at however high computational costs. It is therefore recommended to use the Auxiliary Density Matrix Method (ADMM) Guidon2010 to approximate the exact exchange contribution. For GGAs, the corresponding ADMM sections are not required and it is sufficient to specify the chosen functional in the subsection `&XC_FUNCTIONAL` of the `&XC` section.

    &GLOBAL
      PROJECT S20Acetone
      RUN_TYPE ENERGY
      PREFERRED_DIAG_LIBRARY SL
      PRINT_LEVEL medium
    &END GLOBAL
    &FORCE_EVAL
      METHOD Quickstep
        &PROPERTIES
          &TDDFPT                          ! input section for TDDFPT
           KERNEL FULL                       ! specification of the underlying kernel matrix K
                                             ! FULL kernel is for GGA and hybrid functional computations
                                             ! sTDA kernel is referring to a semi-empirical sTDA computation
           NSTATES 10                      ! specifies the number of excited states to be computed
           MAX_ITER   100                  ! number of iterations for the Davidson algorithm
           CONVERGENCE [eV] 1.0e-7         ! convergence threshold in eV
           RKS_TRIPLETS F                  ! Keyword to choose between singlet and triplet excitations
         !  &XC                            ! If choosing kernel FULL, the underlying functional can be
         !   &XC_FUNCTIONAL PBE0             ! specified by adding an XC section
         !   &END XC_FUNCTIONAL              ! The functional can be chosen independently from the chosen
         !  &END XC                        ! GS functional except when choosing ADMM
         !  &MGRID                         ! It is also possible to choose a separate grid for the real-space
         !    CUTOFF 800                   ! integration of the response density in the TDDFT part,
         !    REL_CUTOFF 80                ! however, in general a consistent setup for GS and ES is recommended
         !  &END MGRID
          &END TDDFPT
        &END PROPERTIES
      &DFT
        &QS
          METHOD GPW
          EPS_DEFAULT 1.0E-17
          EPS_PGF_ORB 1.0E-20
        &END QS
        &SCF
          SCF_GUESS restart
          &OT
             PRECONDITIONER FULL_ALL
             MINIMIZER DIIS
          &END OT
          &OUTER_SCF
             MAX_SCF 900
             EPS_SCF 1.0E-7
          &END OUTER_SCF
          MAX_SCF 10
          EPS_SCF 1.0E-7
        &END SCF
        POTENTIAL_FILE_NAME POTENTIAL_UZH
        BASIS_SET_FILE_NAME BASIS_MOLOPT_UZH
        BASIS_SET_FILE_NAME BASIS_ADMM_UZH
        &MGRID
          CUTOFF 800
          REL_CUTOFF 80
        &END MGRID
        &AUXILIARY_DENSITY_MATRIX_METHOD       ! For hybrid functionals, it is recommended to choose ADMM
          METHOD BASIS_PROJECTION              ! the ADMM environment for ground and excited state has to be
          EXCH_SCALING_MODEL NONE              ! identical
          EXCH_CORRECTION_FUNC NONE            ! Triple-zeta auxiliary basis sets are recommended (see below)
          ADMM_PURIFICATION_METHOD NONE        ! For periodic systems (see below), only specific ADMM options
        &END AUXILIARY_DENSITY_MATRIX_METHOD   ! are available
        &POISSON
           PERIODIC NONE
           POISSON_SOLVER WAVELET
        &END
        &XC
         &XC_FUNCTIONAL PBE0
         &END XC_FUNCTIONAL
        &END XC
      &END DFT
      &SUBSYS
        &CELL
          ABC [angstrom] 14.0 14.0 14.0
          PERIODIC NONE
        &END CELL
        &COORD
           C 0.000000 1.282877 -0.611721
           C 0.000000 -1.282877 -0.611721
           C 0.000000 0.000000 0.185210
           O 0.000000 0.000000 1.392088
           H 0.000000 2.133711 0.059851
           H -0.876575 1.319344 -1.256757
           H 0.876575 1.319344 -1.256757
           H 0.000000 -2.133711 0.059851
           H 0.876575 -1.319344 -1.256757
           H -0.876575 -1.319344 -1.256757
        &END COORD
        &TOPOLOGY
         &CENTER_COORDINATES T
         &END
        &END
        &KIND H
          BASIS_SET ORB DZVP-MOLOPT-PBE0-GTH-q1  ! in general it is recommended to use larger basis  sets
          BASIS_SET AUX_FIT admm-dzp-q1          ! for the primary and auxiliary basis (TZVP/tzp)
          POTENTIAL GTH-PBE0-q1
        &END KIND
        &KIND O
          BASIS_SET ORB DZVP-MOLOPT-PBE0-GTH-q6
          BASIS_SET AUX_FIT admm-dzp-q6
          POTENTIAL GTH-PBE0-q6
        &END KIND
        &KIND C
          BASIS_SET ORB DZVP-MOLOPT-PBE0-GTH-q4
          BASIS_SET AUX_FIT admm-dzp-q4
          POTENTIAL GTH-PBE0-q4
        &END KIND
      &END SUBSYS
    &END FORCE_EVAL

In the resulting output file, there is a `TDDFPT` section reporting the steps of the calculations. The initial guess is referring to the zeroth-order KS energy differences with the printout also listing the transition from the corresponding occupied to virtual orbital.

     -------------------------------------------------------------------------------
     -                            TDDFPT Initial Guess                             -
     -------------------------------------------------------------------------------
              State         Occupied      ->      Virtual          Excitation
              number         orbital              orbital          energy (eV)
     -------------------------------------------------------------------------------
                 1               12                   13              6.63336
                 2               11                   13              9.62185
                 3               10                   13             10.34045
                 4                9                   13             10.78846
                 5                8                   13             11.05691
                 6               12                   14             12.07194
                 7                7                   13             12.42370
                 8                6                   13             12.63557
                 9               12                   15             12.65312
                10                5                   13             12.68633

After convergence of the iterative Davidson algorithm, CP2K is printing for each of the calculated excited states the excitation energy in eV, the corresponding transition dipole as well as the oscillator strength. The form of the dipole transition integrals can be chosen by modifying the keyword DIPOLE_FORM. Possible options are `BERRY` (valid for fully periodic systems only), `LENGTH` (valid for molecular systems only) and `VELOCITY` (both). When referring to the length form, the reference point to calculate electric dipole moments can be chosen in the subsection by specifying the coordinates of the reference point REFERENCE_POINT or by choosing one of the options `COM` (center of mass), `COAC` (center of atomic charges), `USER_DEFINED` (user-defined point) and `ZERO` (origin of the coordinate system) for the REFERENCE keyword.

     -------------------------------------------------------------------------------
     -  TDDFPT run converged in 10 iteration(s)
     -------------------------------------------------------------------------------

     R-TDDFPT states of multiplicity 1
     Transition dipoles calculated using velocity formulation

             State    Excitation        Transition dipole (a.u.)        Oscillator
             number   energy (eV)       x           y           z     strength (a.u.)
             ------------------------------------------------------------------------
     TDDFPT|      1       4.67815   2.4840E-08 -1.9187E-08  5.9673E-08   5.21038E-16
     TDDFPT|      2       8.73074  -8.1610E-09  1.0502E-08  9.1823E-08   1.84132E-15
     TDDFPT|      3       9.17033   1.6661E-02  2.4911E-08 -9.2612E-08   6.23670E-05
     TDDFPT|      4       9.70094  -2.3716E-09  7.2323E-07  7.0660E-01   1.18663E-01
     TDDFPT|      5       9.88658  -3.1922E-08 -4.1476E-01  1.0768E-06   4.16669E-02
     TDDFPT|      6      10.78269   3.5483E-08  4.2695E-01 -1.3123E-07   4.81550E-02
     TDDFPT|      7      10.82893   2.3885E-01 -3.0329E-08  4.5882E-09   1.51356E-02
     TDDFPT|      8      10.92800  -2.4034E-07 -2.4299E-08  5.8160E-08   1.65293E-14
     TDDFPT|      9      11.32208   8.3808E-08 -6.8844E-01  1.9166E-07   1.31469E-01
     TDDFPT|     10      11.97282   1.2605E-08  1.7029E-08  3.3026E-01   3.19930E-02

A more detailed analysis of the excitations is given subsequently, listing orbital contributions for each transition with the corresponding excitation amplitudes. The keyword MIN_AMPLITUDE regulates the threshold for the smallest amplitude to print.

    -------------------------------------------------------------------------------
     -                            Excitation analysis                              -
     -------------------------------------------------------------------------------
            State             Occupied              Virtual             Excitation
            number             orbital              orbital             amplitude
     -------------------------------------------------------------------------------
                 1   4.67815 eV
                                    12                   13               0.998438
                 2   8.73074 eV
                                    10                   13               0.995560
                                     6                   13               0.087643
                 3   9.17033 eV
                                     9                   13               0.993356
                                     7                   13               0.075589
                 4   9.70094 eV
                                    11                   13              -0.893130
                                    12                   16               0.297568
                                     5                   13               0.269023
                                    12                   28              -0.108047
                                     9                   15               0.066145
                                     9                   30              -0.050159

### Excited-state gradient for acetone

To perform an excited-state optimization for emission spectroscopy, the run type has to be set to `GEO_OPT` and the state to be optimized has to be specified in the EXCITED_STATES section. Note that the number of excited states chosen in the TDDFPT section should be larger or at least equal to the number of the chosen excited state.

     &GLOBAL
      PROJECT S20Acetone
      RUN_TYPE ENERGY_FORCE                ! The run type has to be changed to ENERGY_FORCE of GEO_OPT
      PREFERRED_DIAG_LIBRARY SL
      PRINT_LEVEL medium
     &END GLOBAL
     &PROPERTIES
        &TDDFPT
           NSTATES 10
           MAX_ITER   100
           CONVERGENCE [eV] 1.0e-7
           RKS_TRIPLETS F
           ADMM_KERNEL_CORRECTION_SYMMETRIC T   ! required keyword when using hybrid functionals and ADMM for
        &END TDDFPT                             ! the exact exchange contribution for ES gradients
     &END PROPERTIES
     ...
     &DFT
        &QS
          METHOD GPW
          EPS_DEFAULT 1.0E-17
          EPS_PGF_ORB 1.0E-20
        &END QS
        &EXCITED_STATES T
         STATE 1                     ! the excited state to be optimized has to be specified in this section
        &END EXCITED_STATES
        &SCF
          SCF_GUESS restart
          &OT
             PRECONDITIONER FULL_ALL
             MINIMIZER DIIS
          &END OT
          &OUTER_SCF
             MAX_SCF 900
             EPS_SCF 1.0E-7
          &END OUTER_SCF
          MAX_SCF 10
          EPS_SCF 1.0E-7
        &END SCF
     ...
     &END DFT

The resulting output file contains for each step of the geometry optimization the already explained output of the TDDFPT computation and additionally information on the excited state that is optimized as well as the iterative solution of the Z vector equations.

     !--------------------------- Excited State Energy ----------------------------!
     Excitation Energy [Hartree]                                        0.1719188002
     Total Energy [Hartree]                                           -36.3332484083
     !-----------------------------------------------------------------------------!
     !--------------------------- Excited State Forces ----------------------------!

      Iteration    Method   Restart      Stepsize      Convergence         Time
      ------------------------------------------------------------------------------
            1        PCG       F         0.00E+00      0.0231470476        0.02
            2        PCG       F         0.30E+00      0.0003972862        1.56
            3        PCG       F         0.60E+00      0.0000361012        3.11
            4        PCG       F         0.62E+00      0.0000031865        4.65
            5        PCG       F         0.63E+00      0.0000002183        6.19
            6        PCG       F         0.73E+00      0.0000000147        7.73
            7        PCG       F         0.70E+00      0.0000000014        9.27
            8        PCG       F         0.72E+00      0.0000000001       10.82
            9        PCG       F         0.71E+00      0.0000000000       12.35
     !-----------------------------------------------------------------------------!

     ENERGY| Total FORCE_EVAL ( QS ) energy [a.u.]:              -36.333248408276823

     -------------------------------------------------------------------------------

### Choosing a semi-empirical kernel

To speed up computation times for broad-band absorption spectra, a semi-empirical simplified Tamm-Dancoff (sTDA) kernel can be chosen by setting KERNEL to `sTDA`. The semi-empirical electron repulsion operators depend on several empirical parameters, which need to be adjusted depending on the system under investigation. Most importantly, the amount of exact exchange, scaled by adjusting the parameter \(a_{\rm{\tiny{EX}}}\) needs to be chosen carefully and is really crucial to ensure a well-balanced treatment of exact exchange in the GS and ES potential energy surfaces. Too large fractions of exchange in the excited state can lead to negative excitation energies. In general, a relatively small amount of exchange \(a_{\rm{\tiny{EX}}}=0.2 / 0.1\) is therefore recommended.

For details on the STDA method see also https://github.com/grimme-lab/stda.

The most important keywords of the subsection sTDA are:

- FRACTION: fraction of exact exchange \(a_{\rm{\tiny{EX}}}\)

- MATAGA_NISHIMOTO_CEXP: keyword to modify the parameter \(\alpha\) of \(\gamma^{\rm{\tiny{J}}}\)

- MATAGA_NISHIMOTO_XEXP: keyword to modify the parameter \(\beta\) of \(\gamma^{\rm{\tiny{K}}}\)

- DO_EWALD: keyword to switch on Ewald summation for the Coulomb contributions, required when treating periodic systems. (Exact exchange is treated within the minimum image convention and does not require any adjustment.)

    &PROPERTIES
      &TDDFPT
       KERNEL sTDA     ! switches on the semi-empirical kernel sTDA
       &sTDA
         FRACTION 0.2  ! it is crucial to adjust the fraction of exact exchange
       &END sTDA
       NSTATES 10
       MAX_ITER   100
       CONVERGENCE [eV] 1.0e-7
       RKS_TRIPLETS F
      &END TDDFPT
    &END PROPERTIES

In the output, it can be checked that the sTDA kernel was switched on and the correct parameters for \(\alpha\), \(\beta\) and \(a_{\rm{\tiny{EX}}}\) were chosen:

     sTDA| HFX Fraction                                                       0.2000
     sTDA| Mataga-Nishimoto exponent (C)                                      1.5160
     sTDA| Mataga-Nishimoto exponent (X)                                      0.5660
     sTDA| TD matrix filter                                           0.10000000E-09

     -------------------------------------------------------------------------------
     -                      sTDA Kernel: Create Matrix SQRT(S)                     -
     -------------------------------------------------------------------------------

The subsequent printout is then identical to the printout for GGA and hybrid functional kernels:

     R-TDDFPT states of multiplicity 1
     Transition dipoles calculated using velocity formulation

             State    Excitation        Transition dipole (a.u.)        Oscillator
             number   energy (eV)       x           y           z     strength (a.u.)
             ------------------------------------------------------------------------
     TDDFPT|      1       4.88982   8.2927E-09  8.6085E-09 -2.2495E-08   7.77366E-17
     TDDFPT|      2       9.05888   4.5649E-09 -4.9483E-09 -8.2739E-08   1.52940E-15
     TDDFPT|      3       9.23700   1.4100E-02 -1.2802E-08  3.2962E-08   4.49914E-05
     TDDFPT|      4       9.52129   1.1297E-08 -4.7865E-08  8.3174E-01   1.61372E-01
     TDDFPT|      5      10.01790  -9.4987E-09 -4.8324E-01 -5.0387E-08   5.73140E-02
     TDDFPT|      6      10.86728  -1.6175E-08 -3.6799E-01 -7.5353E-09   3.60529E-02
     TDDFPT|      7      10.97656  -3.7898E-01 -1.0287E-08 -3.3941E-09   3.86231E-02
     TDDFPT|      8      11.21831   4.5569E-08 -4.2529E-09  1.9115E-08   6.76117E-16
     TDDFPT|      9      11.40817   2.7777E-08 -6.8033E-01 -3.7411E-08   1.29362E-01
     TDDFPT|     10      11.82525  -1.2576E-09  1.3562E-08  3.8592E-01   4.31486E-02

It should be noted that it is possible to combine sTDA with the semi-empirical ground-state reference GFN1-xTB. However, it is then recommended to adjust all parameters Grimme2016 and to apply corrections to shift the virtual KS orbital eigenvalues. A shift can be applied by adding the keyword EV_SHIFT (for open-shell systems EOS_SHIFT).

### Periodic systems

Adjustments are required when switching to periodic boundary conditions. As already mentioned above, oscillator strengths should be calculated using the `BERRY` or `VELOCITY` formulation. When choosing the `sTDA` kernel, DO_EWALD has to be switched to true to activate Ewald summation for Coulomb contributions. When computing ES gradients using KERNEL `FULL` in combination with hybrid functionals and ADMM, only the ADMM2 method relying on basis projection is implemented in combination with the default for the exchange functional for the first-order GGA correction term.

    &AUXILIARY_DENSITY_MATRIX_METHOD
      METHOD BASIS_PROJECTION
      ADMM_PURIFICATION_METHOD NONE
      EXCH_SCALING_MODEL NONE
      EXCH_CORRECTION_FUNC PBEX
    &END

### Natural transition orbitals

Natural transition orbitals (NTOs) are printed when choosing `PRINT_LEVEL medium` in the `GLOBAL` section or when enabling the NTO_ANALYSIS section. For this purpose it is required to generate unoccupied orbitals and the keyword LUMO enables to adjust the number of unoccupied orbitals. It is possible to print the NTOs as `CUBE_FILES` or in Molden format, with the latter being activated with the keyword MOS_MOLDEN.