Ab initio molecular dynamics: Theory and Implementation

Figure 15. Four possible determinants |t 1 〉, |t 2 〉, |m 1 〉 and |m 2 〉 as a result of the promotion of asingle electron from the homo to the lumo of a closed shell system, see text for further details.Taken from Ref. 214 .by|s 1 [{φ i }]〉 = √ 2 |m [{φ i }]〉 − |t [{φ i }]〉 (304)where the “microstates” m and t are both constructed from the same set {φ i } ofn + 1 spin–restricted orbitals. Using this particular set of orbitals the total densityn(r) = n α m(r) + n β m(r) = n α t (r) + n β t (r) (305)is of course identical for both the m and t determinants whereas their spin densitiesclearly differ, see Fig. 16. Thus, the decisive difference between the m andt functionals Eq. (302) and Eq. (303), respectively, comes exclusively from theexchange–correlation functional E xc , whereas kinetic, external and Hartree energyare identical by construction. Note that this basic philosophy can be generalizedto other spin–states by adapting suitably the microstates and the correspondingcoefficients in Eq. (301) and Eq. (304).Having defined a density functional for the first excited singlet state thecorresponding Kohn–Sham equations are obtained by varying Eq. (301) usingEq. (302) and Eq. (303) subject to the orthonormality constraint ∑ n+1i,j=1 Λ ij(〈φ i |φ j 〉 − δ ij ). Following this procedure the equation for the doubly occupied orbitalsi = 1, . . ., n − 1 reads{− 1 2 ∇2 + V H (r) + V ext (r)+ V αxc [nα m (r), nβ m (r)] + V βxc [nα m (r), nβ m (r)]− 1 2 V αxc[n α t (r), n β t (r)] − 1 2 V βxc[n α t (r), n β t (r)]} n+1∑φ i (r) = Λ ij φ j (r) (306)j=1106