This is a short tutorial on how to use the CP2K-NEWTONX interface to a) generate initial conditions to compute photoabsorption spectra and b) to run non-adiabatic dynamics simulations using orbital derivative couplings. A more comprehensive tutorial on all NEWTONX features, including a documentation of the required specifications for the CP2K interface, can be found on the NEWTONX homepage, https://newtonx.org/documentation-tutorials/.

The interface enables to use electronic-structure data from CP2K and combine it with the surface hopping module of NEWTONX. Excitation energies $\Omega^M$ and excited-state eigenvectors $\mathbf{X}^M$ to describe the excited state $M$ are provided by CP2K, relying on the Tamm-Dancoff eigenvalue problem,

$$\begin{equation} \label{tda_equation} \begin{aligned} \mathbf{A} \mathbf{X}^M &= \Omega^M \mathbf{S} \mathbf{X}^M \, , \\ \sum_{\kappa k} [ F_{\mu \kappa \sigma} \delta_{ik} - F_{ik \sigma} S_{\mu \kappa} ] X^M_{\kappa k \sigma} + \sum_{\lambda} K_{\mu \lambda \sigma} [\mathbf{D}^{{\rm{\tiny{X}}}M}] C_{\lambda i \sigma} &= \sum_{\kappa} \Omega^M S_{\mu \kappa} X^M_{\kappa i \sigma} \, , \end{aligned} \end{equation}$$

with $\mathbf{S}$ representing the conventional atomic-orbital overlap matrix, $\mathbf{F}$ the Kohn-Sham matrix, $\mathbf{K}$ the kernel comprising – depending on the chosen functional – Coulomb, exchange and exchange-correlation contributions, and $\mathbf{C}$ the molecular orbital coefficients. $\mu, \nu, \dots$ denote atomic orbitals, $i, j, \dots$ occupied molecular orbitals. The corresponding excited-state gradient is obtained setting up a variational Lagrangian and taking the derivative with respect to the nuclear coordinates $\mathbf{R}$ (see also https://www.cp2k.org/howto:tddft).

By performing a TDDFPT computation, excitation energies $\Omega^M (\mathbf{R}(t))$, excited-state eigenvectors $\mathbf{X}^M (\mathbf{R}(t))$ and corresponding excited-state gradients $\nabla \Omega^M (\mathbf{R}(t))$ are provided by CP2K. On the so-defined potential energy surfaces, the nuclei are propagated classically relying on the surface hopping code of NEWTONX,

$$\begin{equation} \label{newtons_eom} \begin{aligned} \mathbf{R}(t + \Delta t) &= \mathbf{R} (t) + \mathbf{v} (t) \Delta t + \frac{1}{2} \mathbf{a}(t) \Delta t^2 \, ,\\ \mathbf{v} (t + \Delta t) &= \mathbf{v} (t) + \frac{1}{2} (\mathbf{a} (t) + \mathbf{a} (t+ \Delta t) ) \Delta t \, , \\ \mathbf{a} (t) &= - \frac{1}{m} \nabla \Omega^M (\mathbf{R}(t)) \, . \end{aligned} \end{equation}$$

The coefficients $c^M (t)$ of the total wave function $\Psi (\mathbf{R}(t))$ over all excited states $M$ are obtained implying hopping probabilities $P_{M\rightarrow N}$ of Tully's surface hopping,

$$\begin{equation}\label{surface_hopping} \begin{aligned} \Psi (\mathbf{R}(t)) &= \sum_{M} c^{M} (t) \Psi^M (\mathbf{R}(t)) \\ i \frac{{\rm{d}} c^M (t)}{{\rm{d}}t} &= \sum_N c^N (t) \left ( \delta_{MN} E_N (\mathbf{R}(t)) - i \sigma_{MN} (t) \right ) \, , \\ P_{M \rightarrow N} &= {\rm{max}} \left [ 0, \frac{-2 \Delta t}{| c^M|^2} {\rm{Re}} (c^M c^{N \ast}) \sigma_{MN} \right ] \, . \end{aligned} \end{equation}$$

The therefore required non-adiabatic time derivative couplings $\sigma_{MN}$ can be obtained relying on semi-empirical models (Baeck-An; please cite Barbatti et al., Open Research Europe 1, 49 (2021).) or as numerical time derivative couplings (orbital time derivative (OD); please cite Ryabinkin et al., J. Phys. Chem. Lett. 6, 4200 (2015); Barbatti et al., Molecules 21, 1603 (2021).), with the corresponding molecular orbital overlap matrix $\mathbf{S}^{{\rm{\tiny{t-\Delta t,t}}}}$ being provided by CP2K,

$$\begin{equation}\label{ot_time_deriverative_couplings} \begin{aligned} \sigma_{MN}^{{\rm{\tiny{OD}}}} &= \sum_{ia} X_{ia}^{M} \frac{\partial }{\partial t} X_{ia}^N + \sum_{iab} X_{ia}^M X_{ib}^N S_{ab}^{{\rm{\tiny{t-\Delta t,t}}}} - \sum_{ija} P_{ij} X_{ia}^M X_{ja}^N S_{ji}^{{\rm{\tiny{t-\Delta t,t}}}} \\ S_{pq}^{{\rm{\tiny{t - \Delta t , t}}}} &= \frac{\langle \phi_i (\mathbf{R}(t- \Delta t )) | \phi_j (\mathbf{R} (t)) \rangle}{\Delta t} \, . \end{aligned} \end{equation}$$ $a,b, \dots$ denote virtual molecular orbitals.

The input sections for TDDFPT energy and gradient computations are described in the CP2K tutorial https://www.cp2k.org/howto:tddft. To furthermore provide the required CP2K output, subsequently read in by NEWTONX, the following print statements have to be added to the CP2K input files:

• FORCE_EVAL/PRINT/FORCES: prints the excited-state forces
• TDDFPT/PRINT/NAMD_PRINT with keyword option PRINT_PHASES: prints the excited-state eigenvectors in MO format as well as the corresponding phases.
• VIBRATIONAL_ANALYSIS/PRINT/NAMD_PRINT: prints normal modes to generate initial conditions

It should furthermore be noted that cartesian coordinates have to be provided in terms of the external file “coord.cp2k” and that the number of atoms has to be specified in the CP2K input file in the SUBSYS section.

The following tutorial to obtain photoabsorption spectra is based on https://vdv.dcf.mybluehost.me/nx/wp-content/uploads/2020/02/tutorial-2_2.pdf. For the electronic-structure calculation with CP2K, a cp2k.inp and cp2k.par file as well as a coordinate file named coord.cp2k has to be provided in a subdirectory called JOB_AD, with cp2k.inp including all required print sections stated above. Furthermore, to generate the initial conditions, the initqp_input file requires to specify iprog = 10 for CP2K and file_nmodes = cp2k.eig to refer to the corresponding output file comprising the normal modes provided by CP2K. All other keywords are to be chosen as outlined in the corresponding NEWTONX tutorial.

Examplary input files for computing the absorption spectrum of a water molecule are given below:

h2o_cp2k.inp

&GLOBAL
  PROJECT testsuite_ad
  RUN_TYPE ENERGY
  PREFERRED_DIAG_LIBRARY SL
  PRINT_LEVEL medium
&END GLOBAL
&FORCE_EVAL
 &PRINT                      # print statement for ground-state or excited-state forces
  &FORCES
  &END FORCES
 &END PRINT
 METHOD Quickstep
 &PROPERTIES
  &TDDFPT                    # TDDFPT input section to compute 10 excited states
   &DIPOLE_MOMENTS
    DIPOLE_FORM LENGTH
   &END DIPOLE_MOMENTS
   KERNEL FULL
   NSTATES 10
   MAX_ITER   100
   MAX_KV 20
   CONVERGENCE [eV] 1.0e-5
   RKS_TRIPLETS F
   &PRINT                     # NAMD print section to print excited-state eigenvectors
    &NAMD_PRINT
     PRINT_VIRTUALS T
     PRINT_PHASES T
    &END NAMD_PRINT
   &END PRINT
  &END TDDFPT
 &END PROPERTIES
  &DFT
    &QS
      METHOD GAPW
     EPS_DEFAULT 1.0E-17
    &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
    BASIS_SET_FILE_NAME EMSL_BASIS_SETS
    &MGRID
      CUTOFF 1000
      REL_CUTOFF 100
      NGRIDS 5
    &END MGRID
    &POISSON
       PERIODIC NONE
       PSOLVER MT
    &END
        &XC
     &XC_FUNCTIONAL PBE
     &END XC_FUNCTIONAL
    &END XC
  &END DFT
  &SUBSYS
    &CELL
      ABC 8.0 8.0 8.0
      PERIODIC NONE
    &END CELL
     &COORD
     @include coord.cp2k            # Coordinates are provided externally for the interface
     &END COORD
    &TOPOLOGY
     &CENTER_COORDINATES T
     &END
     NATOMS 3
     CONNECTIVITY OFF
    &END TOPOLOGY
    &KIND H
      BASIS_SET 6-311Gxx
      POTENTIAL ALL
    &END KIND
    &KIND O
      BASIS_SET 6-311Gxx
      POTENTIAL ALL
    &END KIND
  &END SUBSYS
&END FORCE_EVAL

The resulting output file of the initcond.pl script of NEWTONX states that the read-in cartesian normal modes are first transfered to mass-weighted normal modes.

Cartesian normal modes (1/sqrt(amu))
 
        0.00        0.00        0.00        0.00        0.00        0.00     1523.92     3851.12
 
      0.0000     -0.0492      0.0001     -0.1268      0.5632     -0.0083      0.0000     -0.0000
     -0.0886      0.0000     -0.0000     -0.0169      0.0047      0.5777      0.0000     -0.0000
     -0.0000     -0.0000     -0.0000      0.5630      0.1269      0.0155     -0.0715      0.0487
      0.0001      0.3905     -0.0004     -0.1267      0.5632     -0.0082     -0.4184     -0.5910
      0.7043      0.0008      0.7071     -0.0162      0.0040      0.5768      0.0000      0.0000
     -0.0001     -0.5885      0.0007      0.5630      0.1270      0.0155      0.5678     -0.3867
      0.0000      0.3905     -0.0004     -0.1267      0.5632     -0.0083      0.4184      0.5910
      0.7043     -0.0009     -0.7071     -0.0170      0.0051      0.5768      0.0000      0.0000
     -0.0000      0.5885     -0.0007      0.5630      0.1269      0.0154      0.5678     -0.3867
 
     3986.44
 
      0.0712
     -0.0000
      0.0000
     -0.5650
      0.0000
     -0.4222
     -0.5650
      0.0000
      0.4222
 
Mass weighted normal modes
Frequencies will be multiplied by ANH_F =    1.00000
 
        0.00        0.00        0.00        0.00        0.00        0.00     1523.92     3851.12
 
      0.0001     -0.1967      0.0006     -0.5069      2.2526     -0.0330      0.0000     -0.0000
     -0.3543      0.0000     -0.0000     -0.0677      0.0186      2.3104      0.0000     -0.0000
     -0.0001     -0.0000     -0.0002      2.2517      0.5077      0.0619     -0.2861      0.1949
      0.0001      0.3920     -0.0004     -0.1272      0.5654     -0.0083     -0.4200     -0.5933
      0.7071      0.0008      0.7099     -0.0162      0.0040      0.5791      0.0000      0.0000
     -0.0001     -0.5908      0.0007      0.5652      0.1275      0.0155      0.5700     -0.3882
      0.0000      0.3921     -0.0004     -0.1272      0.5654     -0.0083      0.4200      0.5933
      0.7071     -0.0009     -0.7099     -0.0171      0.0051      0.5790      0.0000      0.0000
     -0.0000      0.5908     -0.0007      0.5652      0.1274      0.0155      0.5700     -0.3882
 
     3986.44
 
      0.2847
     -0.0000
      0.0000
     -0.5672
      0.0000
     -0.4238
     -0.5672
      0.0000
      0.4238

The thereon based initial conditions are summarized in the output files dubbed “final_output”, comprising geometries and velocities, as examplarily given below,

 Initial condition =     1
 Geometry in COLUMBUS and NX input format:
 o     8.0    5.00630777    5.00000001    4.46399957   15.99491464
 h     1.0    6.37684065    5.00000128    5.50815661    1.00782504
 h     1.0    3.52303474    5.00000149    5.58297278    1.00782504
 Velocity in NX input format:
   -0.000089112    0.000000000   -0.000020915
    0.000417197    0.000000002    0.000694479
    0.000997296    0.000000013   -0.000362483
 Epot of initial state (eV):    0.0865  Epot of final state (eV):     19.0799
 Vertical excitation (eV):     18.9935  Is Ev in the required range? YES
 Ekin of initial state (eV):    0.0479  Etot of initial state (eV):    0.1343
 Oscillator strength:           0.1221
 State:                         10

Moreover, the output file cross-section.dat comprises the data points of the computing photoabsorption spectrum, as shown below.