350
W. Piskorz and F. Zasada
Due to the non-completeness of the basis set, the following errors occur: for
the localised basis sets, it is the basis set superposition error (BSSE) while for the
planewave basis sets, the error named the Pulay stress. In both cases, the basis set
changes when the set of atom changes (former case) or when the unit cell size or
shape changes (latter case). Generally, both errors are lowered when the size of the
basis set increases. For the avoidance of the Pulay stress during the optimisation
of the cell volume, the series of calculations for the varied cell size are performed
(with the same cut-off energy hence with different number of plane waves) and the
resultant energy versus size of the cell function is fitted to some equation of state,
typically to the Birch–Murnaghan [60] EOS.
2.3.3 Computed Observables and Other Quantities
Having chosen the computational method, a plethora of observables and other properties can be calculated, e.g. SCF-derived properties like the band structure and density
of electronic states (DOS), or the optimisation of the cell parameters and atom positions, miscellaneous spectroscopic properties—interpretation of XAFS, population
(in the PBC world, the dominating are Bader [61, 62] and DDEC [63–65]) and bond
order (DDEC) analyses, etc. Besides static parameters, the dynamic properties can be
calculated including the phase stability [66, 67], thermal properties, and the Raman
or dipole infrared spectroscopies.
In the studies of the solid interfaces, the scanning tunnelling microscopy (STM) is
one of the most important techniques and the most common theory used to simulate
the STM images is the Tersoff–Hamann theory [68], (TH). Its conceptual simplicity is based on the s-wave approximation, namely the replacement of the electronic
structure of the tip by single orbital of spherical symmetry (i.e. orbital s). Thanks
to this approximation, the tunnelling current is proportional to the substrate local
density of states integrated over the range of voltage given by the bias voltage. The
drawback of TH method is the neglect of the real electronic structure of the tip (and
its modification induced by sticking of miscellaneous species to the tip during experiment) and the limited resolution. Its main advantage over the more accurate Bardeen
approximation is speed: the change of simulated current (“microscope settings”)
changes the image at an instant, and the integration itself is also relatively fast. On
the other hand, the Bardeen formalism [69], originally derived from Oppenheimer’s
scattering theory, is used for the cases when the failure of TH is expected, like in
[70], where the Hofer’s bSKAN [71] implementation of Bardeen theory was used
together with VASP code.
2.3.4 Atomistic Thermodynamics
The computational modelling of the nanocrystal morphology is most commonly
realised by the Wulff construction based on the (free) energy calculations at the
(most commonly) DFT+U level of theory. The Wulff [72] construction is based on
the theorem that the minimum surface energy of the convex polyhedron is achieved
W. Piskorz and F. Zasada
Due to the non-completeness of the basis set, the following errors occur: for
the localised basis sets, it is the basis set superposition error (BSSE) while for the
planewave basis sets, the error named the Pulay stress. In both cases, the basis set
changes when the set of atom changes (former case) or when the unit cell size or
shape changes (latter case). Generally, both errors are lowered when the size of the
basis set increases. For the avoidance of the Pulay stress during the optimisation
of the cell volume, the series of calculations for the varied cell size are performed
(with the same cut-off energy hence with different number of plane waves) and the
resultant energy versus size of the cell function is fitted to some equation of state,
typically to the Birch–Murnaghan [60] EOS.
2.3.3 Computed Observables and Other Quantities
Having chosen the computational method, a plethora of observables and other properties can be calculated, e.g. SCF-derived properties like the band structure and density
of electronic states (DOS), or the optimisation of the cell parameters and atom positions, miscellaneous spectroscopic properties—interpretation of XAFS, population
(in the PBC world, the dominating are Bader [61, 62] and DDEC [63–65]) and bond
order (DDEC) analyses, etc. Besides static parameters, the dynamic properties can be
calculated including the phase stability [66, 67], thermal properties, and the Raman
or dipole infrared spectroscopies.
In the studies of the solid interfaces, the scanning tunnelling microscopy (STM) is
one of the most important techniques and the most common theory used to simulate
the STM images is the Tersoff–Hamann theory [68], (TH). Its conceptual simplicity is based on the s-wave approximation, namely the replacement of the electronic
structure of the tip by single orbital of spherical symmetry (i.e. orbital s). Thanks
to this approximation, the tunnelling current is proportional to the substrate local
density of states integrated over the range of voltage given by the bias voltage. The
drawback of TH method is the neglect of the real electronic structure of the tip (and
its modification induced by sticking of miscellaneous species to the tip during experiment) and the limited resolution. Its main advantage over the more accurate Bardeen
approximation is speed: the change of simulated current (“microscope settings”)
changes the image at an instant, and the integration itself is also relatively fast. On
the other hand, the Bardeen formalism [69], originally derived from Oppenheimer’s
scattering theory, is used for the cases when the failure of TH is expected, like in
[70], where the Hofer’s bSKAN [71] implementation of Bardeen theory was used
together with VASP code.
2.3.4 Atomistic Thermodynamics
The computational modelling of the nanocrystal morphology is most commonly
realised by the Wulff construction based on the (free) energy calculations at the
(most commonly) DFT+U level of theory. The Wulff [72] construction is based on
the theorem that the minimum surface energy of the convex polyhedron is achieved