Sum rules and properties in time-dependent density-functional theory

Time-dependent quantal density-functional theory (Q-DFT) is a description of the s-system of noninteracting fermions with electronic density equivalent to that of Schrodinger theory, in terms of fields whose sources are quantum-mechanical expectations of Hermitian operators. The theory delineates and defines the contribution of each type of electron correlation to the local electron-interaction potential nu (ee)(r,t) of the s system. These correlations are due to the Pauli exclusion principle, Coulomb repulsion, correlation-kinetic, and correlation-current-density effects, the latter two resulting. respectively, from the difference in kinetic energy and current density between the interacting Schrodinger and noninteracting systems. We employ 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 exert no net force on the system; (ii) the torque of the potential is finite and due solely to correlation-current-density effects; (iii) two sum rules involving 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 the frequency dependent correlation-current-density effect. Furthermore, via adiabatic coupling constant (lambda) perturbation theory, we prove: (iv) the exchange potential nu (x)(r,t) is the work done in a conservative field representative of Pauli correlations and lowest-order O(lambda) correlation-kinetic and correlation-current-density effects; (v) the correlation potential nu (c)(r,t) commences in O(lambda (2)), and, at each order, it is the work done in a conservative field representative of Coulomb correlations and correlation-kinetic and correlation-current-density effects, (vi) we derive the integral virial theorem relating nu (ee)(r,t) to the electron-interaction and correlation-kinetic energy for arbitrary coupling constant strength lambda, and show there are no explicit correlation-current-density contributions to the energy. From this integral virial theorem we (vii) obtain the fully interacting (lambda = 1) and exchange-only (lambda = 0) integral virial theorems as special cases, the latter showing there is no explicit correlation-kinetic contribution to the exchange energy; and (viii) write expressions for the electron-interaction and correlation-kinetic actions for arbitrary coupling constant lambda in terms of the corresponding fields.