Recent advances in reduced density matrix functional theory (RDMFT) and linear response time-dependent reduced density matrix functional theory (TD-RDMFT) are reviewed. In particular, we present various approaches to develop approximate density matrix functionals which have been employed in RDMFT. We discuss the properties and performance of most available density matrix functionals. Progress in the development of functionals has been paralleled by formulation of novel RDMFT-based methods for predicting properties of molecular systems and solids. We give an overview of these methods. The time-dependent extension, TD-RDMFT, is a relatively new theory still awaiting practical and generally useful functionals which would work within the adiabatic approximation. In this chapter we concentrate on the formulation of TD-RDMFT response equations and various adiabatic approximations. None of the adiabatic approximations is fully satisfactory, so we also discuss a phase-dependent extension to TD-RDMFT employing the concept of phase-including-natural-spinorbitals (PINOs). We focus on applications of the linear response formulations to two-electron systems, for which the (almost) exact functional is known.

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Time-dependent density functional theory (TDDFT) in its current adiabatic implementations exhibits three remarkable failures: (a) completely wrong behavior of the excited state surface along a bondbreaking coordinate; (b) lack of doubly excited configurations; (c) much too low charge transfer excitation energies. These TDDFT failure cases are all strikingly exhibited by prototype two-electron systems such as dissociating H 2 and HeH . We find for these systems with time-dependent density matrix functional theory that: (a) Within previously formulated simple adiabatic approximations, the bonding-toantibonding excited state surface as well as charge transfer excitations are described without problems, but not the double excitations; (b) An adiabatic approximation is formulated in which also the double excitations are fully accounted for. DOI: 10.1103/PhysRevLett.101.033004 PACS numbers: 31.15.Aÿ, 31.15.Eÿ, 33.20.ÿt Dissociation of molecular systems poses a serious challenge within the DFT framework. This is already true for the ground state energy curve cf., e.g., Ref.[1]. For excited states, the problems are even worse; there are various kinds of excitations which are not correctly represented in TDDFT calculations. Figure 1(a) displays the failure of TDDFT (in its adiabatic BP86 variant) for the potential energy surface (PES) of the first excited state of H 2 , 1 1 u , corresponding to the g ! u orbital excitation. As highlighted in Ref.[2], this PES goes to zero instead of going asymptotically to ca. 10 eV. B3LYP does not improve the situation at all. In Fig. 1(b), we document the TDDFT problem with double excitations for the g excitation energies of dissociating H 2 . It is evident that there is an exact state (the third at R e 1:4 Bohr) which is missing in the TDDFT calculations. This is the doubly excited g 2 ! u 2 state. The double-excitation nature is indicated with dots on the PES curve, and it is clear that, in the accurate calculations, the doubly excited state becomes lower with increasing R and has avoided crossings with the second and first state, becoming the lowest excited 1 g state (2 1 g ) from ca. 2.5 Bohr onwards. This state is completely missing in the TDDFT calculations. After some initial optimism that such doubly excited states would be accurately calculated in TDDFT, it has become clear this is not the case [3][4][5]. Figure 1(c) gives an example of the charge transfer problem. The excitation energies of HeH are shown along the dissociation coordinate for the lowest three 1 excited states. At long distance, these excitations have strong charge transfer character, from He 1s to H 1s in 2 and from He 1s to H 2s; 2p z in 3 and 4 . The TDDFT excitation energies exhibit at long distance the well-known severe underestimation of charge transfer excitations [6]. The hybrid functional B3LYP improves somewhat on the pure GGA BP86. We note that the excited state energy does not have a 1=R asymptotic behavior for this type of charge transfer excited state, the fragments being He and neutral H. The...

We study model one-dimensional chemical systems (representative of their three-dimensional counterparts) using the strictly-correlated electrons (SCE) functional, which, by construction, becomes asymptotically exact in the limit of infinite coupling strength. The SCE functional has a highly non-local dependence on the density and is able to capture strong correlation within KohnSham theory without introducing any symmetry breaking. Chemical systems, however, are not close enough to the strong-interaction limit so that, while ionization energies and the stretched H2 molecule are accurately described, total energies are in general way too low. A correction based on the exact next leading order in the expansion at infinite coupling strength of the Hohenberg-Kohn functional largely improves the results.

In a recent letter [Europhys. Lett. 95, 13001 (2011)] the question of whether the density of a time-dependent quantum system determines its external potential was reformulated as a fixed point problem. This idea was used to generalize the existence and uniqueness theorems underlying timedependent density functional theory. In this work we extend this proof to allow for more general norms and provide a numerical implementation of the fixed-point iteration scheme. We focus on the one-dimensional case as it allows for a more in-depth analysis using singular Sturm-Liouville theory and at the same time provides an easy visualization of the numerical applications in space and time. We give an explicit relation between the boundary conditions on the density and the convergence properties of the fixed-point procedure via the spectral properties of the associated Sturm-Liouville operator. We show precisely under which conditions discrete and continuous spectra arise and give explicit examples. These conditions are then used to show that in the most physically relevant cases the fixed point procedure converges. This is further demonstrated with an example.

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Time-dependent density functional theory in its current adiabatic implementations exhibits three striking failures: ͑a͒ Totally wrong behavior of the excited state surface along a bond-breaking coordinate, ͑b͒ lack of doubly excited configurations, affecting again excited state surfaces, and ͑c͒ much too low charge transfer excitation energies. We address these problems with time-dependent density matrix functional theory ͑TDDMFT͒. For two-electron systems the exact exchange-correlation functional is known in DMFT, hence exact response equations can be formulated. This affords a study of the performance of TDDMFT in the TDDFT failure cases mentioned ͑which are all strikingly exhibited by prototype two-electron systems such as dissociating H 2 and HeH + ͒. At the same time, adiabatic approximations, which will eventually be necessary, can be tested without being obscured by approximations in the functional. We find the following: ͑a͒ In the fully nonadiabatic ͑-dependent, exact͒ formulation of linear response TDDMFT, it can be shown that linear response ͑LR͒-TDDMFT is able to provide exact excitation energies, in particular, the first order ͑linear response͒ formulation does not prohibit the correct representation of doubly excited states; ͑b͒ within previously formulated simple adiabatic approximations the bonding-to-antibonding excited state surface as well as charge transfer excitations are described without problems, but not the double excitations; ͑c͒ an adiabatic approximation is formulated in which also the double excitations are fully accounted for.

In this review we provide a rigorous and self-contained presentation of one-body reduced density-matrix (1RDM) functional theory. We do so for the case of a finite basis set, where density-functional theory (DFT) implicitly becomes a 1RDM functional theory. To avoid non-uniqueness issues we consider the case of fermionic and bosonic systems at elevated temperature and variable particle number, i.e, a grand-canonical ensemble. For the fermionic case the Fock space is finite-dimensional due to the Pauli principle and we can provide a rigorous 1RDM functional theory relatively straightforwardly. For the bosonic case, where arbitrarily many particles can occupy a single state, the Fock space is infinite-dimensional and mathematical subtleties (not every hermitian Hamiltonian is self-adjoint, expectation values can become infinite, and not every self-adjoint Hamiltonian has a Gibbs state) make it necessary to impose restrictions on the allowed Hamiltonians and external non-local potentials. For simple conditions on the interaction of the bosons a rigorous 1RDM functional theory can be established, where we exploit the fact that due to the finite one-particle space all 1RDMs are finite-dimensional. We also discuss the problems arising from 1RDM functional theory as well as DFT formulated for an infinite-dimensional one-particle space.