# eXtended Tight Binding

This is a short tutorial on how to run GFN1-xTB computations. The details on the theory and the original implementation by Grimme can be found in Grimme2017. Please cite this paper if you were to use the GFN1-xTB module.

## Brief theory recap

The semi-empirical GFN1-xTB energy expression comprises contributions due to electronic (EL), atom-pairwise repulsion (REP), dispersion (DISP), and halogen-bonding (XB) terms,

\[ E_{\rm{\tiny{GFN1-xTB}}} = E_{\rm{\tiny{EL}}} + E_{\rm{\tiny{REP}}} + E_{\rm{\tiny{DISP}}} + E_{\rm{\tiny{XB}}} \, . \]

### 1. Electronic energy

The electronic energy contribution,

\[ E_{\rm{\tiny{EL}}} = \sum_i^{\rm{\tiny{occ}}} n_i \langle \Psi_i | h_0 | \Psi_i \rangle + \frac{1}{2} \sum_{A,B} \sum_{{l}^A}\sum_{{l'}^B} p_l^A p_{{l'}}^B \gamma_{AB,ll'} + \frac{1}{3}\sum_{A} \Gamma_A q_A^3 - T_{\rm{\tiny{el}}} S_{\rm{\tiny{el}}} \, , \]

contains zeroth-order contributions based on a zeroth-order Hamiltonian \(h_0\), the valence molecular orbitals \(\Psi_i\), occupation numbers \(n_i\) as well as second-order contributions which are optimized self-consistently as well as third-order diagonal contributions. The second order contributions are described using the semi-empirical electron repulsion operator \(\gamma_{AB,ll'}\) which depends on the interatomic distance of atoms \(A\) and \(B\) as well as further empirical parameters that are specific for different angular momenta \(l\) and \(l'\). The monopole charges of the second-order expression are optimized self-consistently,

\[ p_l^A = p_l^{A_0} - \sum_{\nu}^{N_{\rm{\tiny{AO}}}} \sum_{\mu \in A, \mu \in l} S_{\mu \nu } P_{\mu \nu} \, , \]

referring to the atomic orbital overlap matrix \(\mathbf{S}\) and the density matrix \(\mathbf{P}\).

The remaining diagonal terms represent a cubic charge correction based on the Mulliken charge \(q_A\) of atom \(A\) and the charge derivative \(\Gamma_A\) of the atomic Hubbard parameter \(\eta_A\). Furthermore, the electronic temperature times entropy term \(T_{\rm{\tiny{el}}}S_{\rm{\tiny{el}}}\) enables fractional orbital occupations.

### 2. Repulsion

Repulsion is described via an atom-pairwise potential,

\[ E_{\rm{\tiny{REP}}} = \sum_{AB} \frac{Z_A^{\rm{\tiny{eff}}} Z_B^{\rm{\tiny{eff}}} }{R_{AB}} \exp^{- (\alpha_A \alpha_B)^{1/2} (R_{AB})^{k_f}} \, , \]

with the effective nuclear charge \(\mathbf{Z}^{\rm{\tiny{eff}}}\) as well as the global or element-specific parameters \(k_f\) and \(\alpha\).

### 3. Dispersion

Dispersion is included by the well-established D3 method in the BJ-damping scheme Grimme2010.

### 4. Corrections

Corrections for element-specific interactions are possible using either a halogen-bonding correction term (XB) or a generic nonbonding potential correction (NONBOND). Note that the generic nonbonding potential correction is CP2K specific and thus the so-obtained energy differs from the original GFN1-xTB method,

\[ E_{\rm{\tiny{GFN1-xTB+NONBOND}}} = E_{\rm{\tiny{GFN1-xTB}}} + E_{\rm{\tiny{NONBOND}}} \, . \]

## The GFN1-xTB input section

The most important keywords and subsections of section XTB are:

- DO_EWALD: keyword to activate Ewald summation for periodic boundary conditions (PBC); has to be switched to true in case of PBC

- USE_HALOGEN_CORRECTION: keyword to switch off contribution \(E_{\rm{\tiny{XB}}}\) to correct halogen interactions, default is to include this correction

- CHECK_ATOMIC_CHARGES: the cubic charge diagonal contribution is checked to be numerically stable by switching the keyword to true.

- DO_NONBONDED: add a generic correction potential to correct bond- or atomic-specific interactions

- PARAMETER: it is possible to add this section with corresponding keywords to modify xTB parameters

The additional keywords COULOMB_INTERACTION, COULOMB_LR and TB3_INTERACTION are for debugging purposes only and it is recommended to use the default options here.

## Simple examples

### GFN1-xTB ground-state energy for

The following input is an examplary standard input for calculating GFN1-xTB ground-state energies.

    &GLOBAL
      RUN_TYPE  ENERGY
      PROJECT_NAME xtb
      PRINT_LEVEL  MEDIUM
      PREFERRED_DIAG_LIBRARY SL
    &END GLOBAL
    &FORCE_EVAL
     METHOD QS
    &DFT
      &QS
       METHOD XTB
       &XTB
        CHECK_ATOMIC_CHARGES F    ! Keyword to check if Mulliken charges are physically reasonable
        DO_EWALD  T               ! Ewald summation is required for periodic structures
        USE_HALOGEN_CORRECTION T  ! Element-specific correction for halogen interactions (Cl, Br) with (O, N)
       &END XTB
      &END QS
      &SCF
       SCF_GUESS RESTART
       MAX_SCF 50
       EPS_SCF 1.E-6
       &OT ON
         PRECONDITIONER FULL_SINGLE_INVERSE
         MINIMIZER DIIS
       &END
       &OUTER_SCF
         MAX_SCF 200
         EPS_SCF 1.E-6
       &END OUTER_SCF
      &END SCF
     &END DFT
      &SUBSYS
        &TOPOLOGY
          COORD_FILE_FORMAT  xyz
          COORD_FILE_NAME  input.xyz
          CONNECTIVITY OFF
          &CENTER_COORDINATES
          &END CENTER_COORDINATES
         &END TOPOLOGY
        &CELL
          ABC  21.64 21.64 21.64
          ALPHA_BETA_GAMMA 90.0 90.0 90.0
          PERIODIC XYZ
        &END CELL
      &END SUBSYS
    &END FORCE_EVAL

The so-obtained output is listing information on the chosen system-specific parameters. Note that parameters can be changed manually by adding a PARAMETER section to the XTB section and specifying corresponding keywords for the specific parameters with the adjusted values or by giving the path to the modified parameter file, adding the keywords PARAM_FILE_PATH and PARAM_FILE_NAME.

                      #####   #####        #          ####### ######
                     #     # #     #      #              #    #     #
                     #     # #           #    ##   ##    #    #     #
                     #     #  #####     #      ## ##     #    ######
                     #   # #       #   #        ###      #    #     #
                     #    #  #     #  #        ## ##     #    #     #
                      #### #  #####  #        ##   ##    #    ######

     xTB| Parameter file                                              xTB_parameters
     xTB| Basis expansion STO-NG                                                   6
     xTB| Basis expansion STO-NG for Hydrogen                                      4
     xTB| Halogen interaction potential                                            F
     xTB| Halogen interaction potential cutoff radius                         20.000
     xTB| Nonbonded interactions                                                   F
     xTB| D3 Dispersion: Parameter                                         dftd3.dat
     xTB| Huckel constants ks kp kd                        1.850     2.250     2.000
     xTB| Huckel constants ksp k2sh                                  2.080     2.850
     xTB| Mataga-Nishimoto exponent                                            2.000
     xTB| Repulsion potential exponent                                         1.500
     xTB| Coordination number scaling kcn(s) kcn(p) kc     0.006    -0.003    -0.005
     xTB| Electronegativity scaling                                           -0.007
     xTB| Halogen potential scaling kxr kx2                          1.300     0.440

Analogously to any other self-consistent field optimization (SCF) method, the output also includes the energy and convergence during the SCF steps with the finally converged GFN1-xTB energy.

     SCF WAVEFUNCTION OPTIMIZATION

      ----------------------------------- OT ---------------------------------------
      Minimizer      : DIIS                : direct inversion
                                             in the iterative subspace
                                             using   7 DIIS vectors
                                             safer DIIS on
      Preconditioner : FULL_SINGLE_INVERSE : inversion of
                                             H + eS - 2*(Sc)(c^T*H*c+const)(Sc)^T
      Precond_solver : DEFAULT
      stepsize       :    0.08000000                  energy_gap     :    0.08000000
      eps_taylor     :   0.10000E-15                  max_taylor     :             4
      ----------------------------------- OT ---------------------------------------

      Step     Update method      Time    Convergence         Total energy    Change
      ------------------------------------------------------------------------------
         1 OT DIIS     0.80E-01    0.5     0.01213502      -947.7483409153 -9.48E+02
         2 OT DIIS     0.80E-01    0.3     0.00675007      -951.5762826800 -3.83E+00
         3 OT DIIS     0.80E-01    0.3     0.00092877      -953.2164544959 -1.64E+00
         4 OT DIIS     0.80E-01    0.3     0.00034159      -953.2591478247 -4.27E-02
         5 OT DIIS     0.80E-01    0.3     0.00018348      -953.2687102329 -9.56E-03
         6 OT DIIS     0.80E-01    0.3     0.00009265      -953.2707750500 -2.06E-03
         7 OT DIIS     0.80E-01    0.3     0.00005495      -953.2714236504 -6.49E-04
         8 OT DIIS     0.80E-01    0.3     0.00002612      -953.2716704946 -2.47E-04
         9 OT DIIS     0.80E-01    0.3     0.00001585      -953.2717390500 -6.86E-05
        10 OT DIIS     0.80E-01    0.3     0.00001020      -953.2717664315 -2.74E-05
        11 OT DIIS     0.80E-01    0.3     0.00000564      -953.2717774258 -1.10E-05
        12 OT DIIS     0.80E-01    0.3     0.00000354      -953.2717818198 -4.39E-06
        13 OT DIIS     0.80E-01    0.3     0.00000206      -953.2717839406 -2.12E-06
        14 OT DIIS     0.80E-01    0.3     0.00000127      -953.2717844831 -5.42E-07
        15 OT DIIS     0.80E-01    0.3     0.00000077      -953.2717846336 -1.51E-07

      *** SCF run converged in    15 steps ***

      Core Hamiltonian energy:                                   -962.45147378153547
      Repulsive potential energy:                                   8.84897617161771
      Electronic energy:                                            0.76461561909348
      DFTB3 3rd order energy:                                       0.33228335538302
      Dispersion energy:                                           -0.76618599817727

      Total energy:                                              -953.27178463361872

      outer SCF iter =    1 RMS gradient =   0.77E-06 energy =       -953.2717846336
      outer SCF loop converged in   1 iterations or   15 steps

### Adding a generic correction potential

It is possible to add a generic non bonded correction potential. The potential form can be chosen freely and needs to be specified by adding the keyword FUNCTION. Included parameters and variables have to be specified using the keywords VARIABLES and PARAMETERS. The section can be repeated as often as required and enables to include pairwise, element-specific correction potentials. The implementation also features analytic gradients for this option.

    &GLOBAL
      RUN_TYPE  ENERGY
      PROJECT_NAME xtb
      PRINT_LEVEL  MEDIUM
      PREFERRED_DIAG_LIBRARY SL
    &END GLOBAL
    &FORCE_EVAL
     METHOD QS
    &DFT
      &QS
       METHOD XTB
       &XTB
        CHECK_ATOMIC_CHARGES F
        DO_EWALD  T
        USE_HALOGEN_CORRECTION T
        DO_NONBONDED T               ! Possible option to include a generic non-bonded potential
         &NONBONDED                  ! Specification of the potential, keyword can be repeated
          &GENPOT
           ATOMS Kr Br
           FUNCTION Aparam*exp(-Bparam*r)-Cparam/r**8  ! Potential formula has to be specified
           PARAMETERS Aparam Bparam Cparam             ! Parameters included in the formula above
           VALUES 70.0 1.0 0.0                         ! Explicit values for the parameters
           VARIABLES r
           RCUT 40.5
          &END GENPOT
         &END NONBONDED
       &END XTB
      &END QS
      &SCF
       SCF_GUESS RESTART
       MAX_SCF 50
       EPS_SCF 1.E-6
       &OT ON
         PRECONDITIONER FULL_SINGLE_INVERSE
         MINIMIZER DIIS
       &END
       &OUTER_SCF
         MAX_SCF 200
         EPS_SCF 1.E-6
       &END OUTER_SCF
      &END SCF
     &END DFT
      &SUBSYS
        &TOPOLOGY
          COORD_FILE_FORMAT  xyz
          COORD_FILE_NAME  input.xyz
          CONNECTIVITY OFF
          &CENTER_COORDINATES
          &END CENTER_COORDINATES
         &END TOPOLOGY
        &CELL
          ABC  21.64 21.64 21.64
          ALPHA_BETA_GAMMA 90.0 90.0 90.0
          PERIODIC XYZ
        &END CELL
      &END SUBSYS
    &END FORCE_EVAL