TDDFT in Chemistry and Biochemistry III - TDDFT.org

Strategies for Solving Matrix EquationsDirect methodsDimension of matrix equation is N occ · N virt ,(A + B) iajb = (ε a − ε i )δ ij δ ab + 2(ia|jb) + 2f xciajb − c x[(ja|ib) + (ab|ij)](A − B) iajb = (ε a − ε i )δ ij δ ab + c x [(ja|ib) − (ab|ij)]Electron-repulsion integrals in Mulliken notation,∫(ia|jb) = fiajb H = d 3 r d 3 r ′ 1ϕ i (r, σ)ϕ a (r, σ)|r − r ′ | ϕ j(r, ′ σ ′ )ϕ b (r ′ , σ ′ )Computational cost of direct solution scales as N 6 ;Requires computation and simultaneous storage of all matrix elements(A + B) iajb , (A − B) iajb , scales as N 5 ;Feasible for small systems only.Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School 2010 17 / 22