Pulay-Kerker mixing

Obtaining a new charge density or potential

If one uses directly the potential V derived from n(r) ( which is created from the new Y)for the creation of the H the process is unstable. An early solution was to mix in only a small part of the new V_sc or n(r) into the old one. This process can be stable as long as only a very small fraction of the new V_sc or n(r) is used. The convergence in these cases can be very slow. It was recognized that the restriction of self-consistency is basically the solution of a nonlinear set of equations. One algorithm to solve a non-linear set of equations is by a Broyden method, which uses a linear combination of a certain number of differences of the input V_in(r) and the output V_out(r). This process is similar to a quasi-newton algorithm for minimizing the difference between V_in(r)- V_out(r).

            Advances of Pulay and D.D. Johnson (essentially the same method) have improved over the Broyden method especially when an incomplete partial iterative diagonalization is used in place of a full exact diagonalization. For systems with a long dimension, even these newer methods have problems converging the long wavelength components. Another improvement by kerker is a mixing in k-space. Since the small changes in the smaller k values cause larger changes in the real-space potential, the smaller components have smaller mixing coefficients.

Motivation

            The idea behind the Pulay and D.D. Johnson is to perform a least square fit of the past potential differences in order to create a Hessian to find the new direction to move. Another view is that the subspace of the past gradients is used to find the best search direction. In the Broyden method, one assumes that the potential difference is completely accurate at each SCF cycle. A Hessian is formed from the potential differences. If the electronic minimization (diagonalization) is inaccurate, the Hessian becomes significantly inaccurate. The practical reason that the electronic minimization would be inaccurate is for efficiency reasons. At the beginning of the SCF cycle the potential is far from the final result and high accuracy seems to be wasteful. With these least square methods, the algorithm can adapt and give greater importance to the directions that minimize the gradient.

Algorithm

            The specific implementation of the Pulay and D.D. Johnson methods can differ. The implementation of the method in PARATEC comes in 2 parts. The first part is the Pulay mixing. Defining

D_I=  VOUT_I – VIN_I

 and

W_I = D_I+1 - D_I,

we solve the equation (W’W) X= (WD_I). X comprises the new linear coefficients for the past potential differences. These coefficients are used to create a new VOUT and VIN.

The second part is a Kerker mixing according to the formula

V_new(k) =  ( V_in(k) (1-) + V_out(k) ),

with E_k = kinetic energy at point k.

Results

            For results see the write-up on the comparisions of total energy minimization method.