Partial Differential Equations - Modelling and ... - ResearchGate

152 G. Bencteux et al.
m
1
n
C 1
0
C =
m
p
N b= (p+1) n/2
0
C p
N = m + ... + m
1 p
Fig. 1. Block structure of the matrices C.
n
n
0
0
H 1
0
H =
0
H p
D =
N = (p+1) n/2
b
N = (p+1) n/2
b
Fig. 2. Block structure of the matrices H and D.
A detailed justification of the choice of this structure is given in [BCHL07].
Let us only mention here that the decomposition is suggested from the localization
of electrons and the use of a localized basis set. Note that each block
overlaps only with its first neighbors. Again for simplicity, we expose the
method in the case where overlapping is exactly n/2, but it could be any
integer smaller than n/2.
The resulting minimization problem can be recast as
{ p∑
inf Tr ( H i C i C t )
i , Ci ∈M n,mi (R), m i ∈ N, Ci t C i = I mi ∀ 1 ≤ i ≤ p,
i=1
}
p∑
Ci t TC i+1 =0 ∀ 1 ≤ i ≤ p − 1, m i = N . (6)
i=1
In the above formula, T ∈M n,n (R) is the matrix defined by
{
1 if k − l = n
T kl =
2 ,
0 otherwise,
(7)
H i ∈M n,n (R) is a symmetric submatrix of H (see Figure 2), and