Session B1: Recent Advances in Density Functional Theory II

Correlation Energy of the Homogeneous Electron Gas from Adiabatic Connection Fluctuation-Dissipation Theory including Exact Exchange kernel

We have developed an expression for the electronic correlation energy via the Adiabatic Connection Fluctuation-Dissipation Theorem (ACFDT) going beyond the Random-Phase Approximation (RPA) by including exact exchange contribution to the kernel (RPAx). Our derivation is valid and efficient for general systems. It is based on an eigenvalue decomposition of the time dependent response function of the Many Body system in the limit of vanishing coupling constant, evaluated by Density Functional Perturbation Theory. We tested the accuracy of this approximation on the homogeneous electron gas. Within RPAx, the correlation energy of the homogeneous electron gas improves significantly with respect to the RPA results up to densities of the order of $r_s \approx 10$. However, beyond this value, the RPAx response function becomes pathological and the approximation breaks down. We have also evaluated the dependence of the correlation energy on the spin magnetization of the system. Both RPA an RPAx are in excellent agreement with accurate Quantum Monte Carlo results.   

Molecular dissociation within the adiabatic connection fluctuation dissipation framework

The adiabatic connection fluctuation dissipation (ACFD) framework provides an exact expression for the correlation energy in terms of the dynamical density-density response function where the latter can be approximated using time-dependent density functional theory or many body perturbation theory. The first level of approximation is the so-called random phase approximation (RPA) which already incorporates many desirable features known to be difficult to capture with standard correlation functionals. For example, it contains the weak van der Waals forces, and the problem of large static correlation errors which appear in the dissociation limit of molecules are completely absent within the RPA. However, many properties are in quantitative error with experiment and certain features of strong electron correlation are missing. We will here show how the inclusion of exchange effects in the response function yields correlation energies, van der Waals coefficients and molecular dissociation energies in excellent agreement with experimental values. Some attention will be given to how an open-shell atom should be described in the dissociation limit. This further allows one to analyze a given approximate ACFD functional in terms of a so-called fractional charge analysis.   

Coupled-Cluster and Linear-Response Time-Dependent Density-Functional Theory Perspectives on Particle-Particle Random-Phase Approximation

The particle-particle random-phase approximation (pp-RPA) recently attracts extensive interests in quantum chemistry recently. Pp-RPA is a versatile model to calculate ground-state correlation energies, and double ionization potential/double electron affinity. We inspect particle-particle random-phase approximation in different perspectives to further understand its theoretical fundamentals. Viewed as summation of all ladder diagrams, the pp-RPA correlation energy is proved to be analytically equivalent to the ladder coupled-cluster doubles (ladder-CCD) theory. With this equivalence, we can make use of various well-established coupled-cluster techniques to study pp-RPA. Furthermore, we establish linear-response time-dependent density-functional theory with pairing fields (TDDFT-PF), where pp-RPA can be interpreted as the mean-field approximation to a general theory. TDDFT-PF is closely related to the density-functional theory of superconductors, but is applied to normal systems to capture exact N plus/minus 2 excitations. In the linear-response regime, both the adiabatic and non-adiabatic TDDFT-PF equations are established. This sets the fundamentals for further density-functional developments aiming for pp-RPA. These theoretical perspectives will be very helpful for future study.   

Exchange-correlation energies from pairing matrix fluctuations and the particle-particle Random Phase Approximation

Despite their unmatched success for many applications, commonly used local, semi-local and hybrid density functionals still face challenges when it comes to describing long-range interactions, static correlation and electron delocalization. Density functionals of both the occupied and virtual orbitals are able to address these problems. The particle-hole Random Phase Approximation (ph-RPA), for instance, has recently known a revival as a density functional approximation, justified by the adiabatic-connection-fluctuation-dissipation (ACFD) theorem. We formulate an adiabatic connection for the correlation energy in terms of pairing matrix fluctuations, similar in form to the ACFD theorem. With numerical examples of the particle-particle Random Phase Approximation (pp-RPA), the lowest-order approximation to the pairing matrix fluctuation, we illustrate the potential of density functional approximations based on this adiabatic connection. The pp-RPA is size-extensive, self-interaction free, fully anti-symmetric, describes the strong static correlation limit in $\mathrm{H_2}$ and eliminates delocalization errors in $\mathrm{H_2^+}$ and other single-bond systems. It gives good non-bonded interaction energies -- competitive with the ph-RPA -- with the correct $R^{-6}$ asymptotic decay as a function of the separation $R$, much better atomization energies than the ph-RPA, and reaction energies of similar quality. The adiabatic connection in terms of pairing matrix fluctuations thus paves the way for promising new density functional approximations.   

Double, Rydberg and Charge Transfer Excitations from Pairing Matrix Fluctuation and Particle-Particle Random Phase Approximation

Double, Rydberg and charge transfer (CT) excitations have been great challenges for time-dependent density functional theory (TDDFT). Starting from an $(N\pm2)$-electron single-determinant reference, we investigate excitations for the $N$-electron system through the pairing matrix fluctuation, which contains information on two-electron addition/removal processes. We adopt the particle-particle random phase approximation (pp-RPA) and the particle-particle Tamm-Dancoff approximation (pp-TDA) to approximate the pairing matrix fluctuation and then determine excitation energies by the differences of two-electron addition/removal energies. This approach captures all types of interesting excitations: single and double excitations are described accurately, Rydberg excitations are in good agreement with experimental data and CT excitations display correct 1/R dependence. Furthermore, the pp-RPA and the pp-TDA have a computational cost similar to TDDFT and consequently are promising for practical calculations.   

Dynamical second-order Bethe-Salpeter equation kernel: a method for electronic excitation beyond the adiabatic approximation

We present a dynamical second-order kernel for the Bethe-Salpeter equation to calculate electronic excitation energies. The derivation takes explicitly the functional derivative of the exact second-order self energy with respect to the one-particle Green's function. It includes naturally a frequency dependence, going beyond the adiabatic approximation. Perturbative calculations under the Tamm-Dancoff approximation, using the configuration interaction singles (CIS) eigenvectors, reveal an appreciable improvement over CIS, time-dependent Hartree-Fock and adiabatic time-dependent density functional theory results. The perturbative results also compare well with equation-of-motion coupled-cluster and experimental results.   

On the Edge of Koopmans' Theorem

It is well known that the density of an atom falls off exponentially with increasing distance to the nucleus, with a falloff length inversely proportional to the square root of the ionization energy. It is less well known what happens when the ionization energy goes to zero, which is the case if the nuclear charge is artificially reduced to the critical value. At this critical value, there is a normalizable state at the bottom of the continuum, but the density only falls off as an exponential of the square root of the radius. We calculate this state for the two-electron atom using the pseudospectral method finding the critical value of the nuclear charge to 12 digits. This is a single-center system with strong correlation, and so is a difficult test case for DFT methods.   

DFT calculations with the exact functional

I will discuss several works in which we calculate the exact exchange-correlation functional of density functional theory, mostly using the density-matrix renormalization group method invented by Steve White, our collaborator. We demonstrate that a Mott-Hubard insulator is a band metal [1]. We also perform Kohn-Sham DFT calculations with the exact functional and prove that a simple algoritm always converges [2]. But we find convergence becomes harder as correlations get stronger. An example from transport through molecular wires may also be discussed [3].\\[4pt] [1] Lucas O. Wagner, E. M. Stoudenmire, Kieron Burke, Steven R. White, Phys. Rev. Lett. 111, 093003 (2013).\\[0pt] [2] E.M. Stoudenmire, Lucas O. Wagner, Steven R. White, Kieron Burke, Phys. Rev. Lett. 109, 056402 (2012).\\[0pt] [3] J.P. Bergfield, Z.-F. Liu, Kieron Burke, C.A. Stafford, Phys. Rev. Lett. 108, 066801 (2012)   

Uniform semiclassical approximations for many-particle systems

Semiclassical analysis is used to construct uniform asymptotic approximations to the quantum one-body and kinetic energy densities of a system of noninteracting particles in a 1D potential well in the limit of infinite particle number. The approximations encode the appropriate limiting behavior of the electron density at the bulk, edge and classically-forbidden regions. High accuracy is obtained even when far from the limits assumed in the derivations. The field of density functional theory [1] is impacted in at least two ways. First, the semiclassical kinetic energy density is orbital-free, thereby providing a rare analytical development that is not based on the gradient expansion to the kinetic energy functional. Second, several results are obtained on the global and local asymptotic behavior of the quantum density and kinetic energy density everywhere in configuration space which might be established as new guiding principles for the development of approximate functionals in Kohn-Sham DFT. \\[4pt] [1] A.~Cangi P.~Elliott, D.~Lee and K.~Burke. Semiclassical origins of density functionals. {\em Phys. Rev. Lett.}, 100(25):256406   

Benchmarking Density Functional Theory with Density Matrix Renormalization Group and Lessons For Higher Dimensions

Kohn-Sham Density Functional Theory (DFT) is a mathematically exact method that requires approximation to the exchange correlation energy which may exclude features seen in experiment or provide inadequate estimates. Meanwhile, we may use Density Matrix Renormalization Group (DMRG), a numerical method which can accurately treat strongly correlated electrons in one dimension, to find exact DFT quantities such as the Kohn-Sham potential [1]. We use DMRG in one dimension as a benchmark to test new functionals. Further, recommendations for calculations in two and three dimensional systems are discussed as well as computational proof of principles [2]. [1] E.M. Stoudenmire, et.~al., {\it Phys.~Rev.~Lett.} {\bf 109}, 056402 (2012) [2] Lucas O. Wagner, et.~al., {\it Phys.~Rev.~Lett.} {\bf 111}, 093003 (2013)   

Ensemble density-functional theory for excited states: exact results versus approximations

The ensemble density-functional theory is an exact excited-state theory, but it is not used much in practice due to unsatisfactory approximated ensemble functionals. Unlike in ground-state density-functional theory, few exact conditions are known for the ensemble; because of this, the development of approximate functionals has been slow. We present a method for inverting the ensemble density to obtain the corresponding ensemble Kohn-Sham potential, and we illustrate it for highly accurate quantum Monte-Carlo densities of the helium atom. The resulting exact ensemble Kohn-Sham potential of helium shows prominent features that do not exist in known approximate ensemble functionals. In particular, the first excitation energy calculated from the exact ensemble is demonstrated to be invariant with respect to the mixing parameter of the ensemble. No known approximation has this exact property.