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The density-based action-integral functional introduced by Runge and Gross [Phys. Rev. Lett. 52, 997(1984)] in their foundation of time-dependent density-functional theory (TDDFT) is re-examined. Based on an obvious expansion of the original definition, it becomes apparent that the action-integral functional is both trivial and non-stationary. It cannot be used to establish equations of motion for the time-evolution of quantum systems at the density-function level., Comment: 5 pages

In a recent paper [Phys. Rev. A 77, 062511 (2008)], it was advocated to modify the variational principle for the action-integral functional in the Runge-Gross foundation of time-dependent density-functional theory. This was criticised in a subsequent paper [Phys. Rev. A 62, 052510 (2010)] by the present author. In a Comment [Phys. Rev. A 83, 046501 (2011)], it is argued that the criticism is unfounded. This is a response as given to the four specific points raised in the Comment, clarifying and confirming the essence of the original criticism.

According to a recent paper by G. Vignale [Phys. Rev. A 77, 062511 (2008), arXiv:0803.2727], the problems arising in the original derivation of time-dependent density functional theory (TDDFT) based on the Runge-Gross (RG) action-integral functional (AIF) are due to an incorrect variational principle (VP). This argument and the proposed modification of the VP are critically analyzed. The more fundamental problem, though, is the indefiniteness of the RG AIF. In contrast to a widely held belief, that indefiniteness is not eliminated in the variational procedure, which unwittingly is corroborated by Vignale's initial point.

Single reference coupled-cluster (CC) methods for electronic excitation are based on a biorthogonal representation (bCC) of the (shifted) Hamiltonian in terms of excited CC states, also referred to as correlated excited (CE) states, and an associated set of states biorthogonal to the CE states, the latter being essentially configuration interaction (CI) configurations. The bCC representation generates a non-hermitian secular matrix, the eigenvalues representing excitation energies, while the corresponding spectral intensities are to be derived from both the left and right eigenvectors. Using the perspective of the bCC representation, a systematic and comprehensive analysis of the excited-state CC methods is given, extending and generalizing previous such studies. Here, the essential topics are the truncation error characteristics and the separability properties, the latter being crucial for designing size-consistent approximation schemes. Based on the general order relations for the bCC secular matrix and the (left and right) eigenvector matrices, formulas for the perturbation-theoretical (PT) order of the truncation errors (TEO) are derived for energies, transition moments, and property matrix elements of arbitrary excitation classes and truncation levels. In the analysis of the separability properties of the transition moments, the decisive role of the so-called dual ground state is revealed. Due to the use of CE states the bCC approach can be compared to so-called intermediate state representation (ISR) methods based exclusively on suitably orthonormalized CE states. As the present analysis shows, the bCC approach has decisive advantages over the conventional CI treatment, but also distinctly weaker TEO and separability properties in comparison with a full (and hermitian) ISR method.

The Comment by Holas et al. [A. Holas, M. Cinal, and N. H. March, Phys. Rev. A 78, 016501 (2008)] on our recent paper [J. Schirmer and A. Dreuw, Phys. Rev. A 75, 022513 (2007)]. is an appropriate and valuable contribution. As a small addendum we briefly comment on the relationship between the radical Kohn-Sham (rKS) form of density-functional theory and previous one-electron (particle) potential (OPP) developments., Comment: 2 pages

The general expectation that, in principle, time-dependent density functional theory (TDDFT) be an exact formulation of the time-evolution of an interacting N-electron system is critically reexamined. It is demonstrated that the previous TDDFT foundation, resting on three theorems by Runge and Gross (RG) [Phys.Rev.Lett.52, 997(1984)], is invalid because undefined phase factors corrupt the RG action integral functionals. Our finding confirms much of a previous analysis by van Leeuwen [Int. J. Mod. Phys. B15, 1969(2001)]. To analyze the RG theorems and other aspects of TDDFT, an utmost simplification of the KS theory is introduced, in which the density is obtained from a single KS equation for one spatial (spin-less) particle. This radical KS formulation allows us to analyze also the concept of KS-type equations derived without a variational principle. We argue that such an approach can reproduce, but not predict the time-development of the exact density of the interacting N-electron system. Besides the issue of the formal exactness of TDDFT, it is shown that both the static and time-dependent KS linear response equations neglect the particle-particle (p-p) and hole-hole (h-h) matrix elements of the perturbing operator. For a local (multiplicative) operator this does not lead to a loss of information due to a remarkable general property of local operators. Accordingly, no logical inconsistency arises here, because DFT requires external potentials to be local., Comment: Reason for replacement: Change of content essentially in subsection 5.C, plus minor adaptions in the Appendix, Sec. 1, and Sec. 6

It was noted recently that the ADM-diffeomorphism-constraint does not generate all observed symmetries for several Bianchi-models. We will suggest not to use the ADM-constraint restricted to homogeneous variables, but some equivalent which is derived from a restricted action principle. This will generate all homogeneity preserving diffeomorphisms, which will be shown to be automorphism generating vector fields, in class A and class B models. Following Dirac's constraint formalism one will naturally be restricted to the unimodular part of the automorphism group., Comment: 14 pages, Latex, Freiburg preprint THEP 94/21