Ab initio molecular dynamics: Theory and Implementation

The best approximation to the final solution within the subspace spanned by then stored vectors is obtained in a least square sense by writingn∑c (n+1)i(G) = d k c (k)i(G) , (216)k=1where the d k are subject to the restrictionn∑d k = 1 (217)and the estimated error becomesk=1e (n+1)i(G) =n∑k=1d k e (k)i(G) . (218)The expansion coefficients d k are calculated from a system of linear equations⎛⎞ ⎛ ⎞ ⎛ ⎞b 11 b 12 · · · b 1n 1 d 1 0b 21 b 22 · · · b 2n 1d 20⎜ ... .. . . ⎟ ⎜ .=⎟ ⎜ .(219)⎟⎝ b n1 b n2 · · · b nn 1⎠⎝ d n⎠ ⎝0⎠1 1 · · · 1 0 −λ 1where the b kl are given byb kl = ∑ i〈e k i (G)|e l i(G)〉 . (220)The error vectors are not known, but can be approximated within a quadratic modele (k)i (G) = −K −1G,G ψ(k) i (G) . (221)In the same approximation, assuming K to be a constant, the new trial vectors areestimated to bec i (G) = c (n+1)i (G) + K −1G,G ψ(n+1) i (G) , (222)where the first derivative of the energy density functional is estimated to ben∑ψ (n+1)i(G) = d k ψ (k)i(G) . (223)k=1The methods described in this section produce new trail vectors that are not orthogonal.Therefore an orthogonalization step has to be added before the newcharge density is calculatedc i (G) ← ∑ kc j (G)X ji . (224)There are different choices for the rotation matrix X that lead to orthogonal orbitals.Two of the computationally convenient choices are the Löwdin orthogonalizationX ji = S −1/2ji(225)68