9.4.3 Improved functional form for f - method of Methfessel and Paxton

 

Aluminium 1.0

Lithium 0.4

Tellurium 0.8

Copper, Palladium 0.4

Vanadium 0.2

Rhodium 0.2

Potassium 0.3

The method described in the last section has two shortcomings:

• The forces calculated by VASP are a derivative of the free electronic energy F (see section 9.5). Therefore the forces can not be used to obtain the equilibrium groundstate, which corresponds to an energy-minimum of . Nonetheless the error in the forces is generally small and acceptable.
• The parameter must be chosen with great care. If is too large the energy will converge to the wrong value even for an infinite k-point mesh, if is to small the convergence speed with the number of k-points will deteriorate. An optimal choice for for several cases is given in table 3. The only way to get a good is by performing several calculations with different k-point meshes and different parameters for .

These problems can be solved by adopting a slightly different functional form for . This is possible expending the step function in an complete orthonormal set of functions (method of Methfessel and Paxton [33]). The Gaussian function is only the first approximation (N=0) to the step function, further successive approximations (N=1,2,...) are easily obtained. In similarity to the Gaussian method, the energy has to be replaced by a generalized free energy functional

In contrast to the Gaussian method the entropy term will be very small for reasonable values of (for instance for the values given in table 3). The is a simple error estimation for the difference between the free energy F and the 'physical' energy . can be increase till this error estimation gets to large.