Force Pulay

This, the well-known Hellmann-Feynman theorem [128,129], can then be used for the calculation of the first derivatives. In nonnal situations, however, the use of an incomplete atom-centered (e.g., atomic orbital) basis set means that further terms, known as Pulay forces, must also be considered [130]. 

Unfortunately, this only holds for the exact wavefunction and certain other types ol leavefuiiction (such as at the Hartree-Fock limit). Moreover, even though the Hellmarm-Feynman forces are much easier to calculate they are very unreliable, even for accurate wavefunctions, giving rise to spurious forces (often referred to as Pulay forces [Pulay l )S7]). 

When performing variable-cell AIMD simulations with plane-wave basis sets, problems originate from the fact that the basis set is not complete with respect to the cell vectors.71 This incompleteness can introduce fictitious forces (Pulay forces) into asys and lead to artificial dynamics. To overcome this problem, one must ensure that asys is well converged with respect to the basis set size. In general, it is found that one needs to employ a plane-wave kinetic... 

In order to solve the Kohn Sham equations, an expansion of the one-electron wave functions on a basis set is performed. Both localized basis sets and plane wave ones are currently used. Localized basis sets have the advantage of their small size. However they are attached to the atomic positions, which yields non-zero Pulay forces in geometry optimization and molecular dynamics. Plane waves, on the other hand, provide a uniform sampling of space, whatever the specific conformation of the system they are independent of the atomic positions, but they require the use of pseudopotentials to mimick core electrons and a very large number of vectors is necessary in standard surface calculations. 

A. Finite-basis sets The molecular wave functions are represented as a finite sum of atom-centred four-component basis functions, which causes a spurious force often called orbital basis correction (OBC) (also known as Pulay force (Pulay 1983)). For an atom centred at Ra it reads... 

However, almost exclusively the solutions to the Kohn-Sham or Hartree-Fock equations are expanded in a finite basis set as in Eq. (13). This means that neither the Kohn-Sham nor the Hartree-Fock equations are solved exactly and, therefore, the Hellmann-Feynman theorem is only approximately valid. In practical calculations the deviations are so large that the forces calculated from Eq. (43) are essentially of no use. Pulay21 has shown that one has to include an extra term, the so-called Pulay force,... 

A significant advantage of plane wave basis sets is that they are independent of the nuclear positions. This means that the problem of basis set superposition error (Section 5.10) does not occur, and the calculation of the gradient of the energy is easy, as it is given directly by the Hellman-Feynman force, i.e. there are no components associated with the change of basis function position ( Pulay forces ). 

Although it is by no means a restriction or a requirement of the method [34], PWs are often used as a convenient basis set for the coding of the CPMD EOM, since they have several good properties (i) accuracy can be systematically improved in a fully variational way, (ii) PWs are independent from atomic positions (i.e. no Pulay forces [22] involved), (iii) they can be easily distributed in parallel processing. However, we must observe that the fact that PWs they are not localized can lead to... 

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