Ab initio molecular dynamics: Theory and Implementation

free energy functional discussed in the previous section – are difficult to adapt forcases where the symmetry and / or spin of the electronic state should be fixed 168 .An early approach in order to select a particular excited state was based on introducinga “quadratic restoring potential” which vanishes only at the eigenvalue ofthe particular state 417,111 .A method that combines Roothaan’s symmetry–adapted wavefunctions withKohn–Sham density functional theory was proposed in Ref. 214 and used to simulatea photoisomerization via molecular dynamics. Viewed from Kohn–Sham theory thisapproach consists in building up the spin density of an open–shell system based ona symmetry–adapted wavefunction that is constructed from spin–restricted determinants(the “microstates”). Viewed from the restricted open–shell Hartree–Focktheory à la Roothaan it amounts essentially to replacing Hartree–Fock exchange byan approximate exchange–correlation density functional. This procedure leads toan orbital–dependent density functional which was formulated explicitely for thefirst–excited singlet state S 1 in Ref. 214 . The relation of this approach to previoustheories is discussed in some detail in Ref. 214 . In particular, the success of theclosely–related Ziegler–Rauk–Baerends “sum methods” 704,150,600 was an importantstimulus. More recently several papers 252,439,193,195,196 appeared that are similarin spirit to the method of Ref. 214 . The approach of Refs. 193,195,196 can be viewedas a generalization of the special case (S 1 state) worked out in Ref. 214 to arbitraryspin states. In addition, the generalized method 193,195,196 was derived within theframework of density functional theory, whereas the wavefunction perspective wasthe starting point in Ref. 214 .In the following, the method is outlined with the focus to perform moleculardynamics in the S 1 state. Promoting one electron from the homo to the lumo in aclosed–shell system with 2n electrons assigned to n doubly occupied orbitals (that isspin–restricted orbitals that have the same spatial part for both spin up α and spindown β electrons) leads to four different excited wavefunctions or determinants, seeFig. 15 for a sketch. Two states |t 1 〉 and |t 2 〉 are energetically degenerate tripletst whereas the two states |m 1 〉 and |m 2 〉 are not eigenfunctions of the total spinoperator and thus degenerate mixed states m (“spin contamination”). Note inparticular that the m states do not correspond – as is well known – to singlet statesdespite the suggestive occupation pattern in Fig. 15.However, suitable Clebsch–Gordon projections of the mixed states |m 1 〉 and|m 2 〉 yield another triplet state |t 3 〉 and the desired first excited singlet or S 1 state|s 1 〉. Here, the ansatz 214 for the total energy of the S 1 state is given byE S1 [{φ i }] = 2E KSm [{φ i }] − E KSt [{φ i }] (301)where the energies of the mixed and triplet determinants∫Em KS [{φ i }] = T s [n] + dr V ext (r)n(r) + 1 ∫dr V H (r)n(r) + E xc [n α2m, n β m](302)∫Et KS [{φ i }] = T s [n] + dr V ext (r)n(r) + 1 ∫dr V H (r)n(r) + E xc [n α t , n β t ] (303)2are expressed in terms of (restricted) Kohn–Sham spin–density functionals constructedfrom the set {φ i }, cf. Eq. (75). The associated S 1 wavefunction is given105