14 H.Eschrig 2.2.2 Partitioning of Unity The use of a local basis makes it desirable to have the density and the effec- tive potential as lattice sums of local contributions.This is not automatically provided:the density comes out form summation over the occupied KS or- bitals (2.22)as a double lattice sum,and the effective potential has anyhow a complicated connection with the KS orbitals.The decisive tool here is a partitioning of unity in r-space. There may be chosen: a locally finite cover of the real space R3 by compact cells i,that is, R3=U:i and each point of R3 lies only in finitely many i, -a set of n-fold continuously differentiable functions fi(r)with suppfiC i, that is fi(r)=0 for ri, -0≤f(r）≤1and∑if(r)=1 for allr. In the actual context,;=Rs indexed by atom positions. 0 0 0 Fig.2.3.A locally finite cover of the R2 by squares. Figure 2.3 shows a locally finite cover of the plane by a lattice of overlap- ping squares. 2.2.3 Density and Potential Representation The decomposition of the density n)=∑n(r-R-) (2.31) Rs is obtained by an even simpler one-dimensional partitioning along the line joining the two centers of a two-center contribution.14 H. Eschrig 2.2.2 Partitioning of Unity The use of a local basis makes it desirable to have the density and the effec￾tive potential as lattice sums of local contributions. This is not automatically provided: the density comes out form summation over the occupied KS or￾bitals (2.22) as a double lattice sum, and the effective potential has anyhow a complicated connection with the KS orbitals. The decisive tool here is a partitioning of unity in r-space. There may be chosen: – a locally finite cover of the real space R3 by compact cells Ωi, that is, R3 = ∪iΩi and each point of R3 lies only in finitely many Ωi, – a set of n-fold continuously differentiable functions fi(r) with suppfi ⊂ Ωi, that is fi(r) = 0 for r ∈ Ωi, – 0 ≤ fi(r) ≤ 1 and i fi(r) = 1 for all r. In the actual context, Ωi = ΩRs indexed by atom positions. ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ❝ ΩRs Fig. 2.3. A locally finite cover of the R2 by squares. Figure 2.3 shows a locally finite cover of the plane by a lattice of overlap￾ping squares. 2.2.3 Density and Potential Representation The decomposition of the density n(r) =  Rs ns(r − R − s) (2.31) is obtained by an even simpler one-dimensional partitioning along the line joining the two centers of a two-center contribution.