Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
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Figure 15. Four possible determinants |t 1 〉, |t 2 〉, |m 1 〉 <strong>and</strong> |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 <strong>and</strong> 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 <strong>and</strong> t determinants whereas their spin densitiesclearly differ, see Fig. 16. Thus, the decisive difference between the m <strong>and</strong>t functionals Eq. (302) <strong>and</strong> Eq. (303), respectively, comes exclusively from theexchange–correlation functional E xc , whereas kinetic, external <strong>and</strong> Hartree energyare identical by construction. Note that this basic philosophy can be generalizedto other spin–states by adapting suitably the microstates <strong>and</strong> the correspondingcoefficients in Eq. (301) <strong>and</strong> 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) <strong>and</strong> 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