First-Order Geometrical Response Equations for State ... - Jack Simons

364 BAK, BOATZ, AND SIMONSNowconsidering a simultaneous change of all orbitals and requiring that theyremain orthonormal, we find that the elements Oijmusi define a unitary matrix;that is, we caDwrite OijasOij = [exp(X)1j,where~1 ,= 8ij +.Xij +2 tXikXk; + ...,(7)Xii = -X;i. (8)According to Eqs. (4)-(6), the SAenergy for the set of infinitesimally variedorbitals then reads .ESA= L. - ')'tJ'L hklOikOjllJ ( .l:1 'ikl- mnop .) + ? ( r;1 L (mn IOp)Oim Ojn Oko OIP ) ,(9)where we have defined the one- and iwo-particIe SAdensity matrices as '" fiSA-~R')'ij = ~ UJR')'ij,Rfir SA- ~ r Rijkl = ~ UJR ijkl,R(lOa)(lOb)and the one- and two-particIe density matrices for each of the internai states asR - ~ l/')'ij = ~. ')'ij C1 R C/,R1/(Ha)rffkl==L rlkcfcf.lJ .(Hb)Stationary points in the orbital space (energy minima and saddle points) arecharacterized by the conditiondESAdX mn = O'(12)wherem ;c n. Using Eq. (8) to redlice the mimber of variables to the(M(M - 1)/2) independent variables (remember that M denotes the number oforbitals), the differentiation of Eq. (9) with respect to an arbitrary Xmnfor m ;c ngives:d;SA = 2 L (')'~hnj - ')':ljhm;)+ 4 L (r~jkl(nj Iki) - r:ljkl(mj Iki)).dAmn ; ;kl(13)