Sum rules and properties in time-dependent density-functional theory - art. no. 042508

Time-dependent quantal density-functional theory (Q-DFT) is a description o f the s-system of noninteracting fermions with electronic density equivalen t to that of Schrodinger theory, in terms of fields whose sources are quant um-mechanical expectations of Hermitian operators. The theory delineates an d defines the contribution of each type of electron correlation to the loca l electron-interaction potential nu (ee)(r,t) of the s system. These correl ations are due to the Pauli exclusion principle, Coulomb repulsion, correla tion-kinetic, and correlation-current-density effects, the latter two resul ting. respectively, from the difference in kinetic energy and current densi ty between the interacting Schrodinger and noninteracting systems. We emplo y Q-DFT to prove the following sum rules and properties of the s system: (i ) the components of the potential due to these correlations separately exer t no net force on the system; (ii) the torque of the potential is finite an d due solely to correlation-current-density effects; (iii) two sum rules in volving the curl of the dynamic electron-interaction kernel defined as the functional derivative of nu (ee)(r,t) are derived and shown to depend on th e frequency dependent correlation-current-density effect. Furthermore, via adiabatic coupling constant (lambda) perturbation theory, we prove: (iv) th e exchange potential nu (x)(r,t) is the work done in a conservative field r epresentative of Pauli correlations and lowest-order O(lambda) correlation- kinetic and correlation-current-density effects; (v) the correlation potent ial nu (c)(r,t) commences in O(lambda (2)), and, at each order, it is the w ork done in a conservative field representative of Coulomb correlations and correlation-kinetic and correlation-current-density effects, (vi) we deriv e the integral virial theorem relating nu (ee)(r,t) to the electron-interac tion and correlation-kinetic energy for arbitrary coupling constant strengt h lambda, and show there are no explicit correlation-current-density contri butions to the energy. From this integral virial theorem we (vii) obtain th e fully interacting (lambda = 1) and exchange-only (lambda = 0) integral vi rial theorems as special cases, the latter showing there is no explicit cor relation-kinetic contribution to the exchange energy; and (viii) write expr essions for the electron-interaction and correlation-kinetic actions for ar bitrary coupling constant lambda in terms of the corresponding fields.