Exact Exchange in Density Functional Calculations

38 6. Projector Augmented Wave Methodwhere n is the quantum state label, containing a k index, a band index, and a spin index.This transformation yields the transformed KS equationsˆT † Ĥ ˆT | ˜ψ n 〉 = ɛ n ˆT† ˆT | ˜ψn 〉 (6.2)which needs to be solved instead of the usual KS equation. Now we need to define ˆT in a suitableway, such that the auxiliary wave functions obtained from solving (6.2) becomes smooth.Since the true wave functions are already smooth at a certain minimum distance from the core,ˆT should only modify the wave function close to the nuclei. We thus defineˆT = 1 + ∑ aˆT a (6.3)where a is an atom index, and the atom-centered transformation, ˆT a , has no effect outside acertain atom-specific augmentation region |r − R a | < rc a . The cut-off radii, rca should be chosensuch that there is no overlap of the augmentation spheres.Inside the augmentation spheres, we expand the true wave function in the partial waves φ a i ,and for each of these partial waves, we define a corresponding auxiliary smooth partial wave ˜φ a i ,and require that|φ a i 〉 = (1 + ˆT a )| ˜φ a i 〉 ⇔ ˆT a | ˜φ a i 〉 = |φ a i 〉 − | ˜φ a i 〉 (6.4)for all i, a. This completely defines ˆT , given φ and ˜φ.Since ˆT a should do nothing outside the augmentation sphere, we see from (6.4) that we mustrequire the partial wave and its smooth counterpart to be identical outside the augmentationsphere∀a, φ a i (r) = ˜φ a i (r), for r > rcawhere φ a i (r) = 〈r|φa i 〉 and likewise for ˜φ a i .If the smooth partial waves form a complete set inside the augmentation sphere, we can formallyexpand the smooth all electron wave functions as| ˜ψ n 〉 = ∑ iP a ni| ˜φ a i 〉, for |r − R a | < r a c (6.5)where Pni a are some, for now, undetermined expansion coefficients.Since |φ a i 〉 = ˆT | ˜φ a i 〉 we see that the expansion|ψ n 〉 = ˆT | ˜ψ n 〉 = ∑ iP a ni|φ a i 〉, for |r − R a | < r a c (6.6)has identical expansion coefficients, Pni a .As we require ˆT to be linear, the coefficients Pni a must be linear functionals of | ˜ψ n 〉, i.e.∫Pni a = 〈˜p a i | ˜ψ n 〉 = dr˜p a∗i (r − R a ) ˜ψ n (r) (6.7)where |˜p a i 〉 are some fixed functions termed smooth projector functions.As there is no overlap between the augmentation spheres, we expect the one center expansionof the smooth all electron wave function, | ˜ψ n〉 a = ∑ i | ˜φ a i 〉〈˜pa i | ˜ψ n 〉 to reduce to | ˜ψ n 〉 itself inside theaugmentation sphere defined by a. Thus, the smooth projector functions must satisfy∑| ˜φ a i 〉〈˜p a i | = 1 (6.8)inside each augmentation sphere. This also implies thati〈˜p a i 1| ˜φ a i 2〉 = δ i1,i 2, for |r − R a | < r a c (6.9)