Surface Hopping with NEWTON-X

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/.

Brief theory recap

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,

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

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 Time-Dependent DFT).

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,

\[\begin{split} \mathbf{R}(t + \Delta t) &= \mathbf{R} (t) + \mathbf{v} (t) \Delta t + \frac{1}{2} \mathbf{a}(t) \Delta t^2 \, ,\\ \mathbf{v} (t + \Delta t) &= \mathbf{v} (t) + \frac{1}{2} (\mathbf{a} (t) + \mathbf{a} (t+ \Delta t) ) \Delta t \, , \\ \mathbf{a} (t) &= - \frac{1}{m} \nabla \Omega^M (\mathbf{R}(t)) \, . \end{split}\]

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,

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

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 (2016).), with the corresponding molecular orbital overlap matrix \(\mathbf{S}^{{\rm{\tiny{t-\Delta t,t}}}}\) being provided by CP2K,

\[\begin{split} \sigma_{MN}^{{\rm{\tiny{OD}}}} &= \sum_{ia} X_{ia}^{M} \frac{\partial }{\partial t} X_{ia}^N + \sum_{iab} X_{ia}^M X_{ib}^N S_{ab}^{{\rm{\tiny{t-\Delta t,t}}}} - \sum_{ija} P_{ij} X_{ia}^M X_{ja}^N S_{ji}^{{\rm{\tiny{t-\Delta t,t}}}} \\ S_{pq}^{{\rm{\tiny{t - \Delta t , t}}}} &= \frac{\langle \phi_i (\mathbf{R}(t- \Delta t )) | \phi_j (\mathbf{R} (t)) \rangle}{\Delta t} \, . \end{split}\]

\(a,b, \dots\) denote virtual molecular orbitals.

General input setup

The input sections for TDDFPT energy and gradient computations are described in Time-Dependent DFT. 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.

A) Initial conditions and photoabsorption spectra

The following tutorial to obtain photoabsorption spectra is based on section 2 of 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. Furthermore, a vibrational analysis computation has to be performed to provide cartesian normal modes, with the input file including the corresponding NAMD print section.

Examplary input files for computing the absorption spectrum as well as for performing a vibrational analysis for a single water molecule with CP2K are given below:

&GLOBAL
  PROJECT excited_states_for_h2o 
  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
                                    # Coordinates are provided externally for the interface
    &COORD
      @include coord.cp2k
    &END COORD
    &TOPOLOGY
      &CENTER_COORDINATES T
      &END
      NATOMS 3                       # specifying number of atoms for NEWTONX
      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

&GLOBAL
  PROJECT normal_modes_for_h2o
  RUN_TYPE VIBRATIONAL_ANALYSIS      #computing normal modes to generate initial conditions
  PREFERRED_DIAG_LIBRARY SL
  PRINT_LEVEL medium
&END GLOBAL
&FORCE_EVAL
  &PRINT
    &FORCES
    &END FORCES
  &END PRINT
  METHOD Quickstep
  &DFT
    &QS
      METHOD GAPW                   # GAPW enables comparison with all-electron molecular program codes like Turbomole
      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
                                    # coordinates must be provided as external file for NEWTONX
    &COORD
      @include coord.cp2k
    &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
&VIBRATIONAL_ANALYSIS
  &PRINT
    &NAMD_PRINT                      # keyword to enable printing of cartesian normal modes
    &END NAMD_PRINT
  &END PRINT
  DX 0.001
&END VIBRATIONAL_ANALYSIS

The input file cp2k.par includes all specifications regarding the executable and parallelization setup.

 parallel = 16
 exec = cp2k.psmp

Furthermore, a initqp_input file has to be generated for NEWTONX following the instructions given in the NEWTONX tutorial. Specifications for CP2K in the initqp_input file are the following:

  • The file comprising the normal modes of the CP2K frequency computation – for the above input provided as normal_modes_for_h2o-VIBRATIONS-1.eig– has to be specified as file_nmodes = normal_modes_for_h2o-VIBRATIONS-1.eig.

  • The electronic structure program has to be specified as CP2K by defining iprog = 10.

&dat
 nact = 2
 iprog = 10
 numat = 3
 npoints = 500
 file_geom = geom
 file_nmodes = normal_modes_for_h2o-VIBRATIONS-1.eig
 anh_f = 1
 rescale = n
 temp = 0
 ics_flg = n
 chk_e = 1
 nis = 1
 nfs = 11
 kvert = 1
 de = 100
 prog = 14
 iseed = 0
 lvprt = 1
/

After providing the excited-state CP2K computation based on input file h2o_cp2k.inp in the subdirectory JOB_AD, the normal modes normal_modes_for_h2o-VIBRATIONS-1.eig of the frequency computation and the initqp_input file for NEWTONX, the script initcond.pl of NEWTONX can be executed to generate initial conditions. The resulting initcond-output file of NEWTONX, it is first stated that the read-in cartesian normal modes are transferred 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 external output files for each state, dubbed “final_output_XXX”, comprising information on the various 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

Based on the initial conditions, the broadened photoabsorption spectrum can be computed with the nxinp script. As outlined in section 2.7 of the cited NEWTONX tutorial, the so-obtained output file cross-section.dat comprises the data points of the computed photoabsorption spectrum as visualized below:

B) Non-adiabatic dynamics using orbital determinant derivatives