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Subsections
7.2 Energy gradients
Having calculated the total energy, both the densitykernel
and the expansion coefficients for the localised orbitals
are varied. Because of the nonorthogonality
of the support functions, it is necessary to take note of the tensor
properties of the gradients [163], as noted in section 4.6.
The total energy depends upon
both explicitly and through the
electronic density . We use the result

(7.20) 
From equations 7.6, 7.16 and 7.18 we have that

(7.21) 
and therefore

(7.22) 
The sum of the Hartree and exchangecorrelation energies,
depends only on the density so that

(7.23) 
The functional derivative of the Hartreeexchangecorrelation energy with
respect to the electronic density is simply the sum of the Hartree and
exchangecorrelation potentials,
. The
electronic density is given in terms of the densitykernel by

(7.24) 
so that we obtain

(7.25) 
Finally, therefore

(7.26) 
Defining the matrix elements of the KohnSham Hamiltonian in the representation
of the support functions by

(7.27) 
the derivative of the total energy with respect to the densitykernel is
simply

(7.28) 
Again we can treat the kinetic and pseudopotential energies together,
and the Hartree and exchangecorrelation energies together. We use the
result that

(7.29) 
We define the kinetic energy operator
, whose matrix elements are

(7.30) 
Since the operator is Hermitian,

(7.31) 
Therefore
The derivation for the pseudopotential energy is identical with the
replacement of by the pseudopotential operator, and so the result
for the sum of these energies is just

(7.33) 
Again this gradient is derived by considering the change in the
electronic density.

(7.34) 
Therefore

(7.35) 
The gradient of the total energy with respect to changes in the
support functions is

(7.36) 
where is the KohnSham Hamiltonian which operates on
.
Next: 7.3 Penalty functional and
Up: 7. Computational implementation
Previous: 7.1 Total energy and
Contents
Peter Haynes