Klaas J. H. Giesbertz

Excitation energies with time-dependent density matrix functional theory: Singlet two-electron systems

Time-dependent density functional theory in its current adiabatic implementations exhibits three striking failures: (a) Totally wrong behavior of the excited state surface along a bond-breaking coordinate, (b) lack of doubly excited configurations, affecting again excited state surfaces, and (c) much too low charge transfer excitation energies. We address these problems with time-dependent density matrix functional theory (TDDMFT). 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: (a) In the fully nonadiabatic (omega-dependent, exact) formulation of linear response TDDMFT, it can be shown that linear response (LR)-TDDMFT is able to provide exact excitation energies, in particular, the first order (linear response) formulation does not prohibit the correct representation of doubly excited states; (b) 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; (c) an adiabatic approximation is formulated in which also the double excitations are fully accounted for.

Aufbau derived from a unified treatment of occupation numbers in Hartree–Fock, Kohn–Sham, and natural orbital theories with the Karush–Kuhn–Tucker conditions for the inequality constraints ni≤1 and ni≥0

In the major independent particle models of electronic structure theory-Hartree-Fock, Kohn-Sham (KS), and natural orbital (NO) theories-occupations are constrained to 0 and 1 or to the interval [0,1]. We carry out a constrained optimization of the orbitals and occupation numbers with application of the usual equality constraints summation (i) (infinity) n(i)=N and phi(i)/phi(j)=delta(ij). The occupation number optimization is carried out, allowing for fractional occupations, with the inequality constraints n(i)>or=0 and n(i)<or=1 with the Karush-Kuhn-Tucker method. This leads in all cases to an orbital energy spectrum with (only for NO and KS) possibly fractionally occupied degenerate levels at energy equal to the Lagrange multiplier varepsilon for the first equality constraint, completely occupied levels at lower energies and completely unoccupied levels at higher energies. Aufbau thus follows in all cases directly from this general derivation.

The adiabatic approximation in time-dependent density matrix functional theory: Response properties from dynamics of phase-including natural orbitals

The adiabatic approximation is problematic in time-dependent density matrix functional theory. With pure density matrix functionals (invariant under phase change of the natural orbitals) it leads to lack of response in the occupation numbers, hence wrong frequency dependent responses, in particular α(ω→0)≠α(0) (the static polarizability). We propose to relinquish the requirement that the functional must be a pure one-body reduced density matrix (1RDM) functional, and to introduce additional variables which can be interpreted as phases of the one-particle states of the independent particle reference system formed with the natural orbitals, thus obtaining so-called phase-including natural orbital (PINO) functionals. We also stress the importance of the correct choice of the complex conjugation in the two-electron integrals in the commonly used functionals (they should not be of exchange type). We demonstrate with the Löwdin-Shull energy expression for two-electron systems, which is an example of a PINO functional, that for two-electron systems exact responses (polarizabilities, excitation energies) are obtained, while writing this energy expression in the usual way as a 1RDM functional yields erroneous responses.

Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time-Dependent RDMFT (TD-RDMFT)

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.

Oscillator strengths of electronic excitations with response theory using phase including natural orbital functionals

The key characteristics of electronic excitations of many-electron systems, the excitation energies ωα and the oscillator strengths fα, can be obtained from linear response theory. In one-electron models and within the adiabatic approximation, the zeros of the inverse response matrix, which occur at the excitation energies, can be obtained from a simple diagonalization. Particular cases are the eigenvalue equations of time-dependent density functional theory (TDDFT), time-dependent density matrix functional theory, and the recently developed phase-including natural orbital (PINO) functional theory. In this paper, an expression for the oscillator strengths fα of the electronic excitations is derived within adiabatic response PINO theory. The fα are expressed through the eigenvectors of the PINO inverse response matrix and the dipole integrals. They are calculated with the phase-including natural orbital functional for two-electron systems adapted from the work of Löwdin and Shull on two-electron systems (the phase-including Löwdin-Shull functional). The PINO calculations reproduce the reference fα values for all considered excitations and bond distances R of the prototype molecules H2 and HeH(+) very well (perfectly, if the correct choice of the phases in the functional is made). Remarkably, the quality is still very good when the response matrices are severely restricted to almost TDDFT size, i.e., involving in addition to the occupied-virtual orbital pairs just (HOMO+1)-virtual pairs (R1) and possibly (HOMO+2)-virtual pairs (R2). The shape of the curves fα(R) is rationalized with a decomposition analysis of the transition dipole moments.

Long-range interactions and the sign of natural amplitudes in two-electron systems

In singlet two-electron systems, the natural occupation numbers of the one-particle reduced density matrix are given as squares of the natural amplitudes which are defined as the expansion coefficients of the two-electron wave function in a natural orbital basis. In this work, we relate the sign of the natural amplitudes to the nature of the two-body interaction. We show that long-range Coulomb-type interactions are responsible for the appearance of positive amplitudes and give both analytical and numerical examples that illustrate how the long-distance structure of the wave function affects these amplitudes. We further demonstrate that the amplitudes show an avoided crossing behavior as function of a parameter in the Hamiltonian and use this feature to show that these amplitudes never become zero, except for special interactions in which infinitely many of them can become zero simultaneously when changing the interaction strength. This mechanism of avoided crossings provides an alternative argument for the non-vanishing of the natural occupation numbers in Coulomb systems.

Density-potential mappings in quantum dynamics

In a recent paper [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 time-dependent 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 because 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.

Are natural orbitals useful for generating an efficient expansion of the wave function

Abstract We investigate whether the natural orbitals (NOs) minimize ∥ Ψ - Φ ∥ 2 , where Ψ is a wave function and Φ is a full configuration interaction (CI) approximation to Ψ in a reduced basis. We will show that the NOs rarely provide the optimal orbitals for Φ , except when (1) there are only two particles or (2) only one basis function is removed in the case of fermions. Further, we will show that the CI expansion coefficients of Ψ and Φ are identical up to a global scaling factor and demonstrate how the NOs can be used to generate the orbitals that minimize ∥ Ψ - Φ ∥ 2 .

Towards nonlocal density functionals by explicit modeling of the exchange-correlation hole in inhomogeneous systems

We put forward an approach for the development of a nonlocal density functional by a direct modeling of the shape of exchange-correlation (xc) hole in inhomogeneous systems. The functional is aimed at giving an accurate xc energy and an accurate corresponding xc potential even in difficult near-degeneracy situations such as molecular bond breaking. In particular we demand that: (1) the xc hole properly contains -1 electron, (2) the xc potential has the asymptotic -1/r behavior outside finite systems, and (3) the xc potential has the correct step structure related to the derivative discontinuities of the xc energy functional. None of the currently existing functionals satisfies all these requirements. These demands are achieved by screening the exchange hole in such a way that the pair-correlation function is symmetric and satisfies the sum rule. These two features immediately imply (1) and (2) while the explicit dependence of the exchange hole on the Kohn-Sham orbitals implies (3). Preliminary calculations show an improved physical description of the dissociating hydrogen molecule. Though the total energy is still far from perfect, the binding curve from our nonlocal density functional provides a significant improvement over the local density approximation. DOI: 10.1103/PhysRevA.87.022514 (Less)

Time-Dependent Reduced Density Matrix Functional Theory

In this chapter we will give an introduction into one-body reduced density matrix functional theory (RDMFT). This is a rather new method to deal with the quantum many-body problem. Especially the development of a time-dependent version, TDRDMFT , is very recent. Therefore, there are many open questions and the formalism has not crystalized yet into a standard form such as in (TD)DFT. Although RDMFT has similarities with DFT, there are many more differences. This chapter is too short for a full introduction into the wondrous world of RDMFT, but we hope to give an idea what (TD)RDMFT might bring.