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Tkatchenko-Scheffler method with iterative Hirshfeld partitioning

The Tkatchenko−Scheffler (TS) dispersion correction method which uses fixed neutral atoms as a reference to estimate the effective volumes of atoms-in-molecule (AIM) and to calibrate their polarizabilities and dispersion coefficients (see Sec. 6.77.3) fails to describe the structure and the energetics of ionic solids. As shown in Ref. [133,134], this problem can be solved by replacing the conventional Hirshfeld partitioning used to compute properties of interacting atoms by the iterative scheme proposed by Bultinck [135]. In this iterative Hirshfeld algorithm (HI), the neutral reference atoms are replaced with ions with fractional charges determined together with the AIM charge densities in a single iterative procedure. The algorithm is initialized with a promolecular density defined by non-interacting neutral atoms. The iterative procedure then runs in the following steps:

  1. the Hirshfeld weight function for the step $ i$ is computed as

    $\displaystyle w_A^{i}({\mathbf{r}}) = {n^{i}_A({\mathbf{r}})}/\left({\sum_B n^{i}_B({\mathbf{r}})}\right),$ (6.101)

    where the sum extends over all atoms in the system

  2. the number of electrons per atom is determined using

    $\displaystyle N_A^{i+1} = N_A^{i} + \int \left[ n_A^{i}({\mathbf{r}}) - w_A^i({\mathbf{r}})\,n({\mathbf{r}}) \right]\,d^3{\mathbf{r}},$ (6.102)

  3. new reference charge densities are computed using

    $\displaystyle n^{i+1}_A({\mathbf{r}}) = n^{\text{lint}(N^i_A)}({\mathbf{r}})\le...
...n^{\text{uint}(N_A^i)}({\mathbf{r}})\left [ N^i_A - \text{lint}(N^i_A)\right ],$ (6.103)

    where lint$ (x)$ expresses the integer part of $ x$ and uint$ (x)=$lint$ (x)+1$.

Steps (1) to (3) are iterated until the difference in the electronic populations between two subsequent steps ( $ \Delta_A^i = \vert N_A^i-N_A^{i+1}\vert$) is less than a predefined threshold for all atoms. The converged interative Hirshfeld weights ($ w_A^{i}$) are then used to define the AIM properties needed to evaluate dispersion energy, see Sec. 6.77.3.

The DFT-TS calculation with iterative Hirshfeld partitioning (DFT-TS/HI) is invoked by setting IVDW=21. The convergence criterion for iterative Hirshfeld partitioning (in e) can optionally be defined via parameter HITOLER (the default value is 5e-5). Other optional parameters controlling the input for the calculation are as in the conventional TS method (Sec. 6.77.3). The default value of the adjustable parameter VDW_SR is 0.95 and corresponds to the PBE functional.

The PBE-TS/HI method is described in detail in J. Chem. Theory Comput. 9, 4293 (2013) and its performance in optimization of various crystalline systems is examined in J. Chem. Phys. 141, 034114 (2014).

IMPORTANT NOTES:


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