Ab initio molecular dynamics: Theory and Implementation

Figure 16. Four patterns of spin densities n α t , nβ t , nα m , and nβ m corresponding to the two spin–restricted determinants |t〉 and |m〉 sketched in Fig. 15, see text for further details. Taken fromRef. 214 .whereasand{ 1−2[1 ]2 ∇2 + V H (r) + V ext (r)+ Vxc[n α α m(r), n β m(r)] − 1 } n+12 V ∑xc[n α α t (r), n β t (r)] φ a (r) = Λ aj φ j (r) , (307){ 1−2[1 ]2 ∇2 + V H (r) + V ext (r)j=1+ Vxc β [nα m (r), nβ m (r)] − 1 n+12 V xc α [nα t (r), nβ t}φ (r)] ∑b (r) = Λ bj φ j (r) . (308)are two different equations for the two singly–occupied open–shell orbitals a andb, respectively, see Fig. 15. Note that these Kohn–Sham–like equations featurean orbital–dependent exchange–correlation potential where Vxc[n α α m, n β m] =δE xc [n α m, n β m]/δn α m and analogues definitions hold for the β and t cases.The set of equations Eq. (306)–(308) could be solved by diagonalization of thecorresponding “restricted open–shell Kohn–Sham Hamiltonian” or alternatively bydirect minimization of the associated total energy functional. The algorithm proposedin Ref. 240 , which allows to properly and efficiently minimize such orbital–dependent functionals including the orthonormality constraints, was implementedin the CPMD package 142 . Based on this minimization technique Born–Oppenheimermolecular dynamics simulations can be performed in the first excited singlet state.j=1107