Session C02: Developments of DFT from Quantum to Statistical Mechanics (II)

Finite Temperature Phase Diagrams by Nested Sampling

The partition function describes the statistical properties of a system in thermodynamic equilibrium and most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. Knowledge of the partition function, thus, gives us access to complete finite temperature phase diagram of any material. However, no one has previously been able to compute the partition function for any atomistic system with realistic models of the interatomic interactions and, indeed, it has generally been believed that such a computation is intractable. However, the introduction of the ‘Nested Sampling’ technique has changed this situation. Nested Sampling is a novel inference algorithm developed by Prof John Skilling [1] that provides a method for searching the whole of configuration space in polynomial computational time – in contrast to methods such a simulated annealing for which there is no such rigorous mathematical bound on the amount of computational time needed. Interestingly, the Nested Sampling algorithm naturally samples configuration space in a way that is closely aligned with the contribution of each region of space to the partition function – high energy regions are sparsely sampled but low energy regions are more densely sampled. In this talk I shall describe how Nested Sampling can be used to calculate partition functions and I will describe a number of applications of the methodology.
[1] AIP Conference Proceedings 735: 395–405 (2004).   

Rocksalt or cesium chloride? Investigating the relative stability of the cesium halide structures with Random Phase Approximation based methods

The Random Phase Approximation (RPA) is gaining immense popularity as the benchmark of semilocal methods such as LDAs and GGAs, as it naturally incorporates weak interaction and eliminates self-interaction error in its exchange component. By taking cesium halides as benchmarks for ionic solids, we present the comparative assessment of RPA based methods with the newly developed SCAN meta-GGA, which incorporates the intermediate-range of dispersion [1]. We also assess various long-range dispersion corrections to SCAN such as SCAN+D3 and SCAN+rVV10. We have demonstrated that RPA is quite successful in describing the ground state properties of these alkali halides, especially smaller fluorides and chlorides. Beyond RPA methods such as rALDA with an exchange-correlation kernel systematically improve upon RPA results for larger bromides and iodides. Though rALDA and rAPBE share a common formalism, rAPBE fails to predict the equilibrium properties of these halides. We have also demonstrated that the SCAN and dispersion-corrected SCAN can also accurately describe the equilibrium properties of these halides and can be good alternatives where computational power can be saved.

[1] Niraj K Nepal, Jefferson E. Bates, Adrienn Ruzsinszky, to be submitted.   

Understanding the role of image charges in mean field theories of ionic solutions.

Mean field theories for charged systems such as Poisson-Boltzmann are of great practical importance. Standard applications include the description of molecular and colloidal interactions in an aqueous environment where ions are present. Recent simulations with asymmetric ionic mixtures have shown the existence of non-trivial behavior of ions near interfaces with dielectric contrast. Near these interfaces ion concentrations are increased or depleted in ways not predicted by the Poisson-Boltzmann approach. The origin of this discrepancy can be traced to the omission, in the construction of the mean field theory, of the role of image charges. When these effects are included, a mean field theory can be formulated that addresses the observed phenomena. In this approach, the Poisson-Boltzmann equation acquires an extra term describing the presence of interfaces by means of an effective one-body short range potential.   

Deorbitalization of meta-GGA exchange correlation functionals

We explore the simplification of widely used meta-generalized-gradient approximation (meta-GGA) exchange-correlation functionals to the Laplacian level of refinement by use of approximate kinetic energy density functionals (KEDFs). Such deorbitalization is motivated by the prospect of reducing computational cost while recovering a strictly Kohn-Sham local potential framework (rather than the usual generalized Kohn-Sham treatment of meta-GGAs). Rather good results on standard molecular and condensed-phase test sets are obtained from the deorbitalized version of the SCAN functional.
The existence of deorbitalizations which yield better performance than the original meta-GGA is illustrated for molecules with the meta-GGA made very simple functional.   

The general-purpose SCAN meta-GGA for ferroelectric materials

The strongly constrained and appropriately normed (SCAN) meta-GGA was found to be accurate for geometries and energies of diversely-bonded materials. Here, we investigate its performance for the structural and electric properties of ferroelectric materials [1]. Ferroelectricity originates from the breaking of spatial inversion symmetry, and spontaneous polarization is determined by the polar structural distortion. Although DFT has been widely used for studying ferroelectric properties, an accurate prediction of the distortion is rather challenging and is limited by the adopted exchange-correlation functionals. LDA is usually more reliable than the GGAs for perovskites, but LDA usually strongly underestimate the cell volumes. The B1-WC hybrid functional was specially designed for ferroelectrics and systematically improves the calculated ferroelectric properties, however is computationally very expensive. Our results show that SCAN is comparable or more accurate than B1-WC but with much cheaper computational cost.
[1] Y. Zhang, et al., Phy. Rev. B 96, 035143 (2017)   

Liquid drops on surfaces: using density functional theory to calculate the binding potential and drop profiles and comparing with results from mesoscopic modelling

For a film of liquid on a solid surface, the binding potential g(h) gives the free energy as a function of the film thickness h and also the closely related (structural) disjoining pressure Π = −∂g/∂h. The wetting behaviour of the liquid is encoded in the binding potential and the equilibrium film thickness corresponds to the value at the minimum of g(h). Several different methods based on density functional theory (DFT) for calculating g(h) are described. The first is the method we developed in the work of Hughes et al. [J. Chem. Phys. 142, 074702 (2015), 146, 064705 (2017)], that self-consistently calculates an effective fictitious external potential that stabilises films with non-equilibrium value of h. A second method is to calculating g(h) using a Nudged Elastic Band (NEB) approach [Buller et al., J. Chem. Phys. 147, 024701 (2017)]. A third approach is to use an overdamped nonconserved pseudo-dynamics that finds g(h) as the trajectory through the free energy landscape. We also show that all three methods generate identical results for g(h). We illustrate these methods by presenting results from a simple discrete lattice-gas model and also for a Lennard-Jones fluid and other simple liquids. The DFT used is based on fundamental measure theory and so incorporates the influence of the layered packing of molecules at the surface and the corresponding oscillatory density profile. The binding potential is frequently input in mesoscale models from which liquid drop shapes and even dynamics can be calculated. Here we show that the equilibrium droplet profiles calculated using the mesoscale theory are in good agreement with the profiles calculated directly from the microscopic DFT. For liquids composed of particles where the range of the attraction is much less than the diameter of the particles, we find that at low temperatures g(h) decays in an oscillatory fashion with increasing h, leading to highly structured terraced liquid droplets.   

Accurate Parametrization of the Non-Interacting Kinetic Energy of the Thermal Uniform Gas

There is currently tremendous interest in the XC properties of a thermal uniform gas.Here we construct an extremely accurate parametrization of the non-interacting kinetic energy for use in thermal density functional studies. Older parameterizations are either relatively crude or violate a simple relation involving the chemical potential. We use machine-learning (kernel ridge regression) to interpolate between the high- and low- temperature limits.   

Trivial Crossing Problem in FSSH: dissection of different methods on a weakly coupled spirobifluorene

Tully’s fewest switches surface hopping (FSSH) has been a popular method for simulating quantum-classical dynamics in a large variety of systems. Its popularity has also brought forward its limitations. Trivial unavoided crossings between non-interacting states that may occur in complex molecular systems present a huge numerical challenge in FSSH. At such crossings, the nonadiabatic (NA) coupling is infinite at the exact crossing point and vanishingly small elsewhere. Finite time-step numerical propagation is likely to miss such crossings and give erroneous results like unphysical long-range energy transfer. The correct treatment of trivial crossings is contingent on their proper identification. Here, we consider the performance of two different strategies developed to deal with the trivial crossing problem in FSSH. Specifically, (i) the Min-Cost adiabatic state identity tracking algorithm, and (ii) the self-consistent fewest switches surface hopping (SC-FSSH) approach have been implemented within the Non-adiabatic Excited State Molecular Dynamics (NA-ESMD) framework. The performance of the methods has been investigated for the internal conversion and energy transfer dynamics of a weakly coupled spiro-linked polyfluorene (spirobifluorene) system.   

Non-Adiabatic Dynamics in Single-Electron Tunneling Devices with Time-Dependent Density Functional Theory

The recent advance of various single-electron sources in solid-state setups has sparked interest in the investigation of electronic transport at the single-particle level. In our recent work (N. Dittmann, J. Splettstoesser, N. Helbig, arxiv:1706.04547), we put forward time-dependent density-functional theory to calculate the dynamics of interacting electrons in single-electron tunneling devices. As a physical system, we analyze a single-electron source which is built by a quantum dot tunnel-coupled to a nearby electron reservoir and driven by a time-dependent gate voltage. By using analogies with quantum-transport theory, we extract a time-nonlocal exchange-correlation potential for a Hubbard U on-site interaction on the quantum dot. The time non-locality manifests itself in a dynamical potential step, which we explicitly link to physical relaxation time scales of the electron dynamics. Finally, we discuss prospects for simulations of larger mesoscopic systems.   

Reliable prediction of the reaction barrier height with HF-DFT

Many systematic errors of semilocal DFT (GGA's and hybrids) are due to errors in the self-consistent density, and so can be reduced by using a more accurate density. For anions, radicals in solution, stretched bonds, reaction barrier heights, and even spin-crossover complexes, DFT results are substantially improved by using Hartree-Fock densities (called HF-DFT). We revisit the case of reaction barriers, showing that while HF-DFT can reduce barrier height errors by a factor of 2 or 3, even for hybrid functionals, different classes of barriers behave in different ways.   

Why Do Hybrid Density Functionals and Meta-GGAs Improve the Band Gaps of Solids in Generalized Kohn-Sham Theory?

The fundamental gap of a solid is not only an excitation energy but also a ground-state second energy difference (ionization energy minus electron affinity). For an approximate functional constructed from a non-interacting density matrix, the gap between the occupied and unoccupied one-electron energies, computed within a generalized Kohn-Sham (GKS) scheme in which the energy-minimizing exchange-correlation potential is continuous and not constrained to be a multiplication operator, has been shown [1] to equal the ground-state second energy difference within the same approximation, so long as the created electron and hole delocalize fully over one, two, or three dimensions. Thus improvements of the ground-state second energy difference from local density and generalized gradient approximations (GGAs) to meta-GGAs and then to hybrids explain the correspondingly improved band-structure gaps, which are found only in GKS and not [2] in KS theory.
[1] J.P. Perdew, W. Yang, K. Burke, Z. Yang, E.K.U. Gross, M. Scheffler, G.E. Scuseria, T.M. Henderson, I.Y. Zhang, A. Ruzsinszky, H. Peng, J. Sun, E. Trushin, and A. Goerling, Proc. Nat. Acad. Sci. (USA) 114, 2801 (2017).
[2] Z. Yang et al., Phys. Rev. B 93, 205205 (2016).