21.6 Modeling High Pressure with GOSTSHYP

The effects of high pressure on individual molecules can be modeled using the GOSTSHYP (Gaussians On Surface Tesserae Simulate HYdrostatic Pressure) method. This method uses the Cosmo infrastructure and can thus be used analogously. Physically, the external pressure is modeled through Gaussian potentials on a cavity. An additional, pressure-dependent contribution is self-consistently added onto the electronic energy:

\[\begin{align*} E = E^0 + E^{\text{GOST}}. \end{align*}\]

This is done using an efficient integral-direct algorithm, which optimizes the amplitudes \(p_j\) for normalized Gaussian overlaps \(\tilde{g}_j\), as modeled through equilibrium conditions for hydrostatic pressure:

\[\begin{align*} E^{\text{GOST}} = \sum_j p_j \tilde{g}_j \end{align*}\]

The amplitudes are dependent on the user-defined pressure \(p_{\text{inp}}\) and the area of the tessera \(a_j\) through: \[\begin{align*} p_j = \frac{p_{\text{inp}} \, a_j}{\tilde{f}_j} \end{align*}\]

The two auxiliary vectors \(\mathbf{\tilde{g}}\) and \(\mathbf{\tilde{f}}\) are computed in every iteration as functionals of the density matrix.

In practice, a GOSTSHYP calculation can be run similarly to a COSMO calculation. An additional keyword defining the external pressure needs to be set as:

$cosmo
   pressure unit= value
   gauss

Any conventional unit for pressure may be chosen (unit = bar, kbar, MPa, GPa, ...). The user-defined pressure in this unit is given after the equal sign as value. The use of a Lebedev cavity (gauss) is strongly recommended in order to obtain fully consistent derivatives - however, other cavities may also be used. GOSTSHYP is compatible with any choice of cavity.

The choice of reasonable vdW radii is crucial for GOSTSHYP. A majority of the electron density must be contained within the chosen cavity, otherwise erroneous negative amplitudes (\(p_j < 0\)) may arise. In this case, the calculation crashes and prints out the atoms for which negative amplitudes occured. Please increase the vdW radii for these atoms in order to proceed. If you really want to allow for negative amplitudes, you can activate ’negative_amplitudes’ in the $cosmo keyword.

It is possible to add an additional enthalpy-like (pV) contribution for vdW cavities. Consistent gradients are available. Please add ’enthalpy’ to the $cosmo keyword if you want to use it. If you only want to use the enthalpy-like contribution without GOSTSHYP, you can enter ’only_enthalpy’ instead.

GOSTSHYP is compatible with most modules for which COSMO is also available. This includes:

• Hartree-Fock and DFT using the Dscf and Ridft modules. Molecular gradients are also available, using Grad and Rdgrad, respectively. Geometries can be optimized using the jobex script.

• Electronic excitations (including the GOSTSHYP response) for Hartree-Fock and DFT using the Escf module.

• Post-Hartree-Fock methods MP2 and CC2 for ground states using the Ricc2 module. As always, a converged HF solution needs to be present.

• Geometry Hessians can be calculated using NumForce. Please note that analytical second derivatives are currently NOT available with GOSTSHYP.