# Model Hamiltonians¶

As of the 2019 release, the DFTB engine supports two different classes of model Hamiltonians, Grimme’s extended tight-binding, and the classic Slater-Koster based DFTB. All of these model Hamiltonians are obtained by applying tight-binding approximations to the DFT total energy expression.

## Slater-Koster based DFTB¶

The efficiency of Slater-Koster based DFTB stems from its use of an optimized minimum valence orbital basis that reduces the linear algebra operations, and a two center-approximation for the Kohn-Sham potential that allows precalculation and storage of integrals using the Slater-Koster technique. This makes DFTB orders of magnitude faster than DFT, but requires parameter files (containing the integrals) for all pair-wise combinations of atoms in a molecule. Many elements can be handled with the parameter sets included in the distribution. Alternatively, sets of parameters in the SKF format can be downloaded and used from third party sources.

There are three flavors of Slater-Koster based DFTB available in our implementation:

- The “plain” DFTB Hamiltonian as introduced by Porezag and Seifert without a self-consistency cycle.
- The second order self-consistent charge extension SCC-DFTB (recently also called DFTB2), which accounts for density fluctuations and improves results on polar bonds. Note that the self-consistent calculations is about an order of magnitude slower than calculations with the “plain” DFTB Hamiltonian.
- The third order extension known as DFTB3, which improve the description of hydrogen-bonded complexes and proton affinities. Note that DFTB3 calculations are only marginally slower than SCC-DFTB based calculations.

Note that since these methods have been respectively parametrized, it is important to specify a matching parameter set when applying one of these models.

## Extended tight-binding (xTB)¶

The extended tight-binding (xTB) model Hamiltonian as recently been introduced by Grimme and coworkers. It makes similar approximations as Slater-Koster based DFTB, but instead of using precalculated integrals, xTB employs a (small) basis of Slater-type orbitals and uses an extended Hückel-like approximation for the Hamiltonian.

The DFTB Engine in AMS2019 supports the GFN1-xTB parameterization of xTB, which is optimized for geometries, frequencies and non-covalent interactions and covers all elements of the periodic table up to radon.

## Input keywords¶

The following keys allow you to select a model Hamiltonian and control different aspects of how the stationary Schroedinger equation is solved.

```
Model [DFTB | SCC-DFTB | DFTB3 | GFN1-xTB | NonSCC-GFN1-xTB]
```

`Model`

Type: Multiple Choice Default value: SCC-DFTB Options: [DFTB, SCC-DFTB, DFTB3, GFN1-xTB, NonSCC-GFN1-xTB] Description: Selects the Hamiltonian used in the DFTB calculation: - DFTB/DFTB0/DFTB1 for classic DFTB without a self-consistent charge cycle - SCC-DFTB/DFTB2 with a self-consistency loop for the Mulliken charges - DFTB3 for additional third-order contributions. - GFN1-xTB for Grimme’s extended tight-binding model in the GFN1 version. - NonSCC-GFN1-xTB for a less accurate but faster version of GFN1-xTB without a self-consistency cycle The choice has to be supported by the selected parameter set.

Different parameters may be suitable for different model Hamiltonians. It is important to choose the appropriate parameter set for the type of calculation and molecular system under study, see parameter sets.

```
ResourcesDir string
```

`ResourcesDir`

Type: String Description: The directory containing the parameter files. Absolute starting with / or relative to $ADFRESOURCES/DFTB otherwise. This key is required for the Slater-Koster based DFTB models, but optional for xTB.

**Examples:**

`ResourcesDir Dresden`

- Uses the resource directory $ADFRESOURCES/DFTB/Dresden.
`ResourcesDir /home/myusername/myparamsdir`

- Uses the specified path /home/myusername/myparamsdir as the resource directory.

**NOTE:** Each resource directory must contain a file called *metainfo.yaml*, which specifies the capabilities of the parameter set. For details see metainfo.yaml.

```
DispersionCorrection [None | Auto | UFF | ULG | D2 | D3-BJ]
```

`DispersionCorrection`

Type: Multiple Choice Default value: None Options: [None, Auto, UFF, ULG, D2, D3-BJ] Description: This key is used to specify a empirical dispersion model. Please refer to the DFTB documentation for details on the different methods. By default no dispersion correction will be applied. Setting this to auto applies the dispersion correction recommended in the DFTB parameter set’s metainfo file. Note that the D3-BJ dispersion correction is always enabled when using the GFN1-xTB model Hamiltonian.

### SCC details and spin-polarization¶

```
SCC
AdaptiveMixing [True | False]
Converge
Charge float
End
HXDamping [True | False]
Iterations integer
Mixing float
OrbitalDependent [True | False]
Unrestricted [True | False]
nDIIS integer
End
```

`SCC`

Type: Block Description: This optional section configures various details of the self-consistent charge cycle. If the model Hamiltonian does not need a self-consistent solution (e.g. plain DFTB0), none of this information is used and the entire section will be ignored. `AdaptiveMixing`

Type: Bool Default value: True Description: Change the mixing parameter based on the monitored energy. A significant increase of energy will strongly reduce the mixing. Then it will slowly grow back to the SCC%Mixing value. `Converge`

Type: Block Description: Controls the convergence criteria of the SCC cycle. `Charge`

Type: Float Default value: 1e-08 Description: The maximum change in atomic charges between subsequent SCC iterations. If the charges change less, the SCC cycle is considered converged.

`HXDamping`

Type: Bool Description: This option activates the DFTB3 style damping for H-X bonds. Note that this is always enabled if the DFTB%Model key is set to DFTB3. Not used with xTB. `Iterations`

Type: Integer Default value: 500 Description: Allows to specify the maximum number of SCC iterations. The default should suffice for most standard calculations. Convergence issues may arise due to the use of the Aufbau occupations for systems with small HOMO-LUMO gaps. In this case the use of a Fermi broadening strategy may improve convergence. Choosing a smaller mixing parameter (see DFTB%SCC%Mixing) may also help with convergence issues: it often provides a more stable but slower way to converge the SCC cycle. `Mixing`

Type: Float Default value: 0.15 Description: The parameter used to mix the DIIS linear combination of previously sampled atomic charge vectors with an analogous linear combination of charge vectors resulting from population analysis combination. It can assume real values between 0 and 1. `OrbitalDependent`

Type: Bool Description: Activates or disables orbital resolved calculations. If this key is absent the recommended settings from the parameter file’s metainfo. `Unrestricted`

Type: Bool Default value: False Description: Enables spin unrestricted calculations. Only collinear spin polarization is supported, see Theor Chem Acc (2016) 135: 232, for details. Must be supported by the chosen parameter set. Not yet compatible with DFTB3, k-space sampling periodic calculations or the xTB models. `nDIIS`

Type: Integer Default value: 20 Description: Specifies the maximum number of samples considered during the direct inversion of iteration of subspace (DIIS) extrapolation of the atomic charges during the SCC iterations. A smaller number of samples potentially leads to a more aggressive convergence acceleration, while a larger number often guarantees a more stable iteration. Due to often occurring linear dependencies within the set of sample vectors, the maximum number of samples is reached only in very rare cases.

```
Occupation
Strategy [Auto | Aufbau | Fermi]
Temperature float
End
```

`Occupation`

Type: Block Description: Configures the details of how the molecular orbitals are occupied with electrons. `Strategy`

Type: Multiple Choice Default value: Auto Options: [Auto, Aufbau, Fermi] Description: This optional key allows to specify the fill strategy to use for the molecular orbitals. Can either be ‘Aufbau’ for simply filling the energetically lowest orbitals, or ‘Fermi’ for a smeared out Fermi-Dirac occupation. By default the occupation strategy is determined automatically, based on the other settings (such as the number of unpaired electrons). `Temperature`

Type: Float Default value: 300.0 Unit: Kelvin Description: The Fermi temperature used for the Fermi-Dirac distribution. Ignored in case of aufbau occupations.

```
UnpairedElectrons integer
```

`UnpairedElectrons`

Type: Integer Default value: 0 Description: This specifies the number of unpaired electrons (not the multiplicity!). This number will then be used in the orbital-filling strategy. Has to be compatible with the total number of electrons, meaning it must be an even number if the total number of electrons is even and odd if the total number is odd. Must be an integer value. Note that this does not activate spin polarization, it only affects the filling of the orbitals.

### k-space integration¶

As of the 2019 release, the k-space integration is unified between BAND and DFTB and uses the same keys as input. See the page on k-space integration in the BAND manual for details and recommendations. Note however, that the default for DFTB is to **only sample the Gamma-point**, i.e. k-space sampling must be explicitly enabled if desired.

Note

k-space integration is currently not supported together with the extended tight-binding (xTB) model Hamiltonians, i.e. periodic calculations with xTB are restricted to sample only the gamma-point.

```
KSpace
Quality [GammaOnly | Basic | Normal | Good | VeryGood | Excellent]
Regular
NumberOfPoints integer_list
End
Symmetric
KInteg integer
End
Type [Regular | Symmetric]
End
```

`KSpace`

Type: Block Description: Options for the k-space integration (i.e. the grid used to sample the Brillouin zone). Note that k-space sampling calculations are currently not supported with the xTB model Hamiltonians. `Quality`

Type: Multiple Choice Default value: GammaOnly Options: [GammaOnly, Basic, Normal, Good, VeryGood, Excellent] Description: Select the quality of the K-space grid used to sample the Brillouin Zone. If ‘GammaOnly’, only one point (the gamma point) will be used. For the other options, the actual number of K points generated depends on the size of the unit cell. The larger the real space cell, the fewer K points will be generated. The CPU-time and accuracy strongly depend on this option. `Regular`

Type: Block Description: Options for the regular k-space integration grid. `NumberOfPoints`

Type: Integer List Description: Use a regular grid with the specified number of k-points along each reciprocal lattice vector. For 1D periodic systems you should specify only one number, for 2D systems two numbers, and for 3D systems three numbers.

`Symmetric`

Type: Block Description: Options for the symmetric k-space integration grid. `KInteg`

Type: Integer Description: Specify the accuracy for the Symmetric method. 1: absolutely minimal (only the G-point is used) 2: linear tetrahedron method, coarsest spacing 3: quadratic tetrahedron method, coarsest spacing 4,6,... (even): linear tetrahedron method 5,7.... (odd): quadratic method The tetrahedron method is usually by far inferior.

`Type`

Type: Multiple Choice Default value: Regular Options: [Regular, Symmetric] Description: The type of k-space integration grid used to sample the Brillouin zone (BZ) used. ‘Regular’: simple regular grid. ‘Symmetric’: symmetric grid for the irreducible wedge of the first BZ (useful when high-symmetry points in the BZ are needed to capture the correct physics of the system, graphene being a notable example).

### xTB specific keywords¶

A few keywords only apply to the xTB model Hamiltonian.

```
XTBConfig
SlaterRadialThreshold float
useXBTerm [True | False]
End
```

`XTBConfig`

Type: Block Description: This block allows for minor tweaking. `SlaterRadialThreshold`

Type: Float Default value: 1e-05 Description: Threshold determining the range of the basis functions. Using a larger threshold will speed up the calculation, but will also make the results less accurate. `useXBTerm`

Type: Bool Default value: False Description: Whether to use the Halogen bonding (XB) term. This is not advised as it has a non-continuous PES.

Note

The GFN1-xTB implementation in AMS currently does not implement the electronic entropy term from the article by Grimme et al. It therefore gives slightly different energies (but not gradients!) for systems with partially occupied molecular orbitals.

### Technical options¶

```
Purify
Enabled [True | False]
Iterations integer
Tolerance float
End
```

`Purify`

Type: Block Description: By default (when the DFTB%Purify section is not present), the next step’s density matrix is calculated from molecular orbitals obtained as eigenvectors of the charge-dependent Hamiltonian. An alternative way to obtain the density matrix is using an iterative purification procedure enabled by this keyword. Note also that molecular orbitals are not calculated when using the density matrix purification method, meaning that any subsequent calculations using molecular orbitals (e.g. TD-DFTB or Mayer bond order analysis) are incompatible with this keyword. Density matrix purification can be considerably faster than diagonalization for molecular of gamma-only periodic systems with large HOMO-LUMO gaps. Note that density matrix purification is faster for systems where the density matrix is sparse. The fill-in of the density matrix is printed to the standard output during the calculation. If it is too large (e.g. >50%), it is probably faster to use normal diagonalization. `Enabled`

Type: Bool Default value: False Description: By default (when purification is not used), the next step’s density matrix is calculated from molecular orbitals obtained as eigenvectors of the charge-dependent Hamiltonian. An alternative way to obtain the density matrix is using an iterative purification procedure. Note that molecular orbitals are not calculated when using the density matrix purification method, meaning that any subsequent calculations using molecular orbitals (e.g. TD-DFTB or Mayer bond order analysis) are incompatible with this keyword. Density matrix purification can be considerably faster than diagonalization for molecular of gamma-only periodic systems with large HOMO-LUMO gaps. Density matrix purification is faster for systems where the density matrix is sparse. The fill-in of the density matrix is printed to the standard output during the calculation. If it is too large (e.g. >50%), it is probably faster to use normal diagonalization. `Iterations`

Type: Integer Default value: 50 Description: Set the maximum number of steps in the purification cycle. If the desired tolerance is not reached by then, the calculation is aborted. `Tolerance`

Type: Float Default value: 1e-08 Description: Set the purification convergence threshold. Purification is considered converged when the trace of the density matrix becomes equal to the total number of electrons within the specified tolerance.

```
StoreMatrices [True | False]
```

`StoreMatrices`

Type: Bool Default value: False Description: Determines whether the Hamiltonian and overlap matrices are stored in the binary result file.