Charge or potential mixing¶
| Mixing type | Mixing parameters |
|---|---|
| ‘simple’ | betamix |
| ‘Anderson’ | betamix, mixdim |
| ‘Broyden1’ | betamix, mixdim, brank |
| ‘Broyden2’ | betamix, mixdim, brank |
| ‘Broyden’ | betamix, mixdim, brank |
| ‘Kerker’ | betamix |
| ‘Pulay’ | betamix, mixdim |
| ‘Pulay+Kerker’ | betamix, mixdim |
When the exchange-correlation functional is an explicit function of the charge density \(\rho\), (LDA or GGA), the ground state electron density is a fixed point of the Kohn-Sham map \(f_{\mu}\), i.e., \(\rho\) satisfies the nonlinear equation
Such a fixed point can be obtained by solving the above equation using a quasi-Newton method which produces a sequence of approximations
where \(r^{(k)} = \rho^{(k)} - f_\mu (\rho^{(k)})\) is the residual, and \(B_k\) is an approximation to the inverse of the Jacobian of \(\rho - f_{\mu}(\rho)\) at \(\rho^{(k)}\).
When \(B_k\) is chosen to be \(B_k = \beta I\), where \(I\) is the identity and \(0 < \beta < 1\), \(\rho^{(k+1)}\) is simply a convex combination of \(\rho^{(k)}\) and \(f_\mu (\rho^{(k)})\). This type of update is often known as simple mixing with \(\beta\) being the mixing parameter. It can be invoked by setting the mixtype parameter in the input arguments of the mixing function to ‘simple’, and the betamix parameter (corresponding to \(\beta\) ) to a small constant between 0 and 1.
For many problems, the use of simple mixing will make the SCF iteration converge if the mixing parameter \(\beta\) is sufficiently small. However, converge can be extremely slow. A more effective way to approximate the inverse of the Jacobian is to use a Broyden-like scheme that can be written as
where
with \(s_k = \rho^{(k)}-\rho^{(k-1)}\) and \(y_k = r^{(k)}-r^{(k-1)}\). Here \(Y_k^{\dagger}\) denotes the pseudo-inverse of \(Y_k\). Such a Jacobian inverse approximation yields an updating formula:
where \(g=Y_k^{\dagger}r^{(k)}\). This type of update is known as the Anderson mixing. It can be invoked by setting mixtype to ‘anderson’ with an appropriately chosen mixing parameter betamix, which is set to 0.7 by default. In addition to betamix, one can also specify the integer parameter mixdim, defines the maximum number of columns allowed in \(S_k\) and \(Y_k\), and corresponds to the value of $ell$ above. This parameter is set to 10 by default.
KSSOLV2.0 also provides the classical Broyden methods for approximating the Jacobian (Broyden I) and the inverse of the Jacobian (Broyden II). In the latter case, the approximation \(B_k\) is updated by
Note that the difference between the Broyden and Anderson methods is that in the Anderson method \(B_k\) is always reset to \(\beta I\) for each \(k\).
For metallic systems, a more appropriate approximation to the inverse of the Jacobian has the following representation in the reciprocal space,
for some screening costant \(\zeta\), where \(G\) is a reciprocal space grid point. We will refer users to references[a,b,c] for details of such an approximation. The updating scheme associated with this type of approximation is known as Kerker mixing.
Finally, mixing can be applied to either the electron density or the potential that depends on the electron density. In KSSOLV2.0, the choice can be made in the SCF function.