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I am trying to model isolated structures such as quantum dots or molecules in Quantum ESPRESSO (QE).

QE uses 3D periodic boundary conditions (PBC) by default. That's why in order to reduce any interaction between the molecule and its images/replicas, we use very large supercells. Now, according to the QE pw.x documentation, we can use the assume_isolated flag in the SYSTEM namelist when we want to perform calculation assuming the system to be isolated (a molecule or a cluster in a 3D supercell).

Now, the default setting is assume_isolated='none' i.e. regular periodic calculation without any correction. But if we set assume_isolated='martyna-tuckerman' or assume_isolated='makov-payne', then do we also need to set the supercell large enough?

According to this suggestion from Max-centre for Electronic properties of isolated molecules calculation, one should set the assume_isolated='martyna-tuckerman' and also set the supercell large. My question is why do we need to set the supercell large if we specify the molecule to be isolated? Isolated structure doesn't have any replicas.

Does this mean when assume_isolated is not set as 'none', then we still have the 3D PBC applied? I am asking this because if setting assume_isolated='mt' can specify the molecule as isolated, then we don't need to specify the supercell very large and the calculation should be much faster for a small supercell.

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Quantum Espresso, like all plane-wave DFT software (e.g. ABINIT, CASTEP, VASP), has to have periodic boundary conditions, because the plane-waves themselves are periodic.

When you want to study clusters (or any lower-dimensional system) the 3D periodicity is unwanted, but if you have sufficient vacuum surrounding your cluster then you would expect the results to be accurate, because the inter-cluster interaction would decay with distance. Unfortunately, this is not true for interaction forces which decay as $\frac{1}{r^2}$, such as the electrostatic interaction, because although the individual cluster-cluster interactions decay as $\frac{1}{r^2}$, there are $O(r^2)$ clusters at this distance, because they are repeated periodically in 3D space. In fact the energy per cluster does converge for neutral clusters, although it is only conditionally convergent, and it is usually calculated efficiently with an Ewald summation. (For charged systems the energy diverges, which is why plane-wave DFT codes always add a homogeneous neutralising background charge for such systems.)

When the 3D periodicity is not desirable, the electrostatic terms are no longer what you want to compute. A different way is needed to calculate these long-range interactions, which does not assume 3D periodicity, and this is where methods such as Martyna-Tuckerman[1] are useful. These methods essentially compute a correction to the usual plane-wave DFT total energy, which removes the spurious effect of the enforced 3D periodicity from the long-range electrostatic interaction.

This does not remove the periodicity of the density, potential etc. itself, and so it is still necessary to have a large enough vacuum gap that these quantities do not interact with their periodic images in neighbouring cells.

[1] G.J. Martyna and M.E. Tuckerman, "A reciprocal space based method for treating long range interactions in ab initio and forcefield-based calculations in clusters", J. Chem. Phys. 110, 2810 (1999); https://doi.org/10.1063/1.477923

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