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# General formalism for kinetic energy preconditioning

We introduce a positive-definite model Hamiltonian and write the energy of the system that it describes as

 (15)

We proceed to derive exact expressions for preconditioning the minimization of Eq. (15). For suitable choice of , these same expressions may be used to improve the condition number for minimizing the true energy Eq. (3). It is worth noting that all of the occupation numbers for the model system have been set to unity. This amounts to an additional occupancy preconditioning, first introduced by Gillan [36] in the context of metallic systems and then by Marzari et al. [29] in the general framework of ensemble density-funtional theory.

Following along the same lines as in Section 2, defining

 (16)

and substituting this, Eq. (4) and Eq. (10) into Eq. (15) we obtain

 (17)

It is at this point that a tensorially incorrect diagonal approximation'' is made in Ref. [21]. In our notation, this would be given by

 (18)

where is some constant, and the first equality follows from Eq. (6). We do not make this unnecessary approximation.

Formally, as it has been defined to be positive-definite, the matrix may be expressed in terms of its unique Cholesky factor [37]:

 (19)

Substituting this into Eq. (17) gives

 (20)

where the new variables which make the energy surface spherical are given by

 (21)

In a steepest descents procedure, although the following easily generalises to the conjugate gradients method, a line minimization is performed along the steepest descents search direction to find the new values of the coefficients :

 (22)

where is chosen to minimize the energy. We wish to minimize the energy with respect to the coefficients , yet the functional is spherical (and hence preconditioned) in the new coefficients . In order to find the new values of the coefficients that minimize the energy, we use the chain rule to write
 (23)

and from this, and Eqs. (21)-(22), it may be shown that
 (24)

where we have used the relations
 (25)

and
 (26)

obtained from Eqs. (6) and (19), respectively.

Choosing the model Hamiltonian introduced in Eq. (14), and defining

 (27) (28)

Eq. (24) becomes
 (29)

where, following the discussion in Section 3, we have replaced the model energy with the true energy . We see from Eq. (29) that preconditioning is effected by premultiplying the steepest descent gradient by the matrix and postmultiplying it by .

Next: Orthogonal basis Up: Preconditioned iterative minimization Previous: Principles
Arash Mostofi 2003-10-28