# Abstracts

Short Talks

Julian Braun - Boundary Conditions for Atomistic Simulations of Crystal Defects
Andrij Vasylenko - Transport calculations in nanostructures, using the latest developments in computing scattering times from DFT at the anharmonic level
Sarah Wishart - Uncertainty Quantification for Classical Effective Potentials
Onat Berk - Representing Potential Energy Surface of Li-Si Alloys using Neural Networks
Simon Etter - Linear Scaling Approximate Factorisation and Selected Inversion for Electronic Structure Models
Petr Grigorev - Dislocation glide in tungsten studied by mixed DFT and MD approach
Roger Haydock - Non-reflective boundary conditions for cluster calculations
Sami SIRAJ-DINE - Construction of Wannier functions for topological insulators

Albert Bartok

Learning interactions from microscopic observables

Designing new materials needs a detailed understanding of the structure and processes of matter on the atomistic scale, because all macroscopic properties ultimately depend on microscopic interactions. For such studies, quantum mechanical modelling combined with atomistic simulations has been proven to be predictive in addition to being able to explain experimental phenomena.
However, larger length and timescales are not easily accessible due to the non-linear growth in computational resources required to numerically solve the quantum mechanical equations. We would like to enable fast simulations without a compromise in accuracy by using machine learning techniques to fit the quantum mechanical model. To realise this aim, we have developed the Gaussian Approximation Potentials framework, which uses microscopic data from quantum mechanical calculations on small systems to create fast, accurate and scalable models. Apart from data, the other main ingredient needed to fit Gaussian Processes are kernels. In my talk I will discuss kernels that are designed to compare atomic structures and show examples from molecular and condensed matter systems. These kernels are used to define a set of interatomic potentials or models, and a Bayesian approach determines which is the most likely, based on
the data as evidence.

George Booth

Combining 'Chemistry' and 'Physics' approaches for stronger correlation?

Stronger correlation effects represent a significant limitation in the applicability of DFT, despite numerous important manifestations of the phenomina from phase transitions to magnetic interactions. These correlations missing in traditional DFT can be found both in the solid state,
as well as isolated molecular systems, but beyond-DFT approaches in physics and chemistry communities have generally differed significantly. While in physics communities and the
solid state, techniques such as Dynamical Mean-field Theory, GW and quantum Monte
Carlo have prevailed, for molecular systems, wavefunction-based techniques including Coupled-Cluster and 'Complete Active Space' approaches have dominated. We consider why these differences have arisen, why they have remained, and discuss a number of techniques to consider them in the context of the other. This leads us to a number of new approaches, both for chemical and solid-state systems, which can significantly extend their scope, accuracy and applicability, with a more unified framework of correlated techniques.

Garnet Chan

Representations and complexity of classical and quantum simulations of electronic structure.

I will discuss and recap some slightly surprising, as well as some entirely unsurprising, results concerning the complexity of electronic structure calculations on classical and quantum computers, and the role of representations in this complexity.

Huajie Chen

Numerical Analysis of Finite Temperature DFT

We study finite dimensional approximations of the Mermin-Kohn-Sham equation, which is derived from the finite temperature DFT model. For general numerical discretization methods, we prove the convergence of the finite dimensional approximations and derive the optimal a priori error estimates. We also provide numerical simulations for several molecular systems that support our theory.

Thierry Deutsch

The Flexibility of Daubechies Wavelets for Electronic Structure Calculations

T. Deutsch (1), L. Genovese (1), S. Mohr (2), L. Ratcliff (3), S. Goedecker (4)

(1) L_Sim - Univ. Grenoble Alpes, CEA, INAC, F-38000 Grenoble, France

(2) CASE Group, BSC, E - 08034 Barcelona, Spain

(3) ALCF, ANL, Illinois 60439, USA

(4) Basel University, Switzerland

Since 2008, the BigDFT project consortium has developed an ab initio Density Functional Theory code based on Daubechies wavelets. These are a compact support multiresolution basis, optimal for expanding localised information, and form one of the few examples of systematic real space basis sets.

In recent articles, we presented the linear scaling version of BigDFT code [1], where a minimal set of localized support functions is optimized in situ. Our linear scaling approach is able to generate support functions for systems in various boundary conditions, like surfaces geometries or system with a net charge. The real space description provided in this way allows to build an efficient, clean method to treat systems in complex environments,and it is based on a algorithm which is universally applicable [2], requiring only moderate amount of computing resources.

We will present how the flexibility of this approach is helpful in providing a basis set that is optimally tuned to the chemical environment surrounding each atom.

In addition than providing a basis useful to project Kohn-Sham orbitals informations like atomic charges and partial density of states, it can also be reused as-is, i.e. without reoptimization, for charge-constrained DFT calculations within a fragment approach [3]. We will demonstrate the interest of this approach to express highly precise and efficient calculations for the computational setup of systems in complex environments [4].

[1] J. Chem. Phys. 140, 204110 (2014)
[2] Phys. Chem. Chem. Phys., 2015, 17, 31360-31370
[3] J. Chem. Phys. 142, 23, 234105 (2015)
[4] J.Chem. Theory Comput. 2015, 11, 2077

Alessandro De Vita

Predicting QM-accurate forces for chemo-mechanical materials modelling

Phenomena of interest for technology such as metal embrittlement, brittle fracture or stress corrosion are beyond the reach of direct First-Principles Molecular Dynamics (FPMD) modelling, although they need its level of precision. In most situations, using classical MD is not a viable faster alternative, as suitably general and accurate (e.g., “reactive”) force fields are not available, nor is it clear how to produce fitting databases a priori guaranteed to contain the information necessary to describe all the chemical processes which might be encountered along the dynamics. “QM/MM, space-embedding” techniques have also long existed, that save computing time by combining quantum and classical descriptions, while still guaranteeing the needed accuracy [1]. These can achieve further efficiency by adopting predictor-corrector (“time embedding”) schemes while remaining free from extrapolation issues. However they are still
very computationally demanding. The situation could improve by introducing Machine Learning (ML) force fields e.g., based on Gaussian Processes (GP). These are no safe extrapolators, but offer a natural way to tackle the validation problem, since the expected error on the predicted forces on atoms can be easily calculated. This makes it in principle possible to compute new QM
information only if needed, to avoid extrapolation, using online machine learning (“on the fly”) optimally efficient schemes [2]. Eventually, it would be desirable to completely dispose of QM calculations. However, it is not clear whether highly accurate and transferable ML force fields could also avoid being impractically slow without renouncing traditionally desirable properties such as energy conservation. Another issue generally relevant in practical applications is whether ML force fields could be at the same time as fast and more accurate than traditional force fields. These issues can to some extent be explored by GP techniques using “covariant” force kernels [3] equivalent to “n-body” interatomic potentials, and/or more resolving kernels constructed from these.

[1] F. Bianchini, J.R. Kermode, and A. De Vita,
Modell. Simul. Mater. Sci. Eng. 24, 045012 (2016).
[2] Z. Li, J. R. Kermode and A. De Vita, Phys. Rev. Lett., 114, 096405 (2015).
[3] A. Glielmo, P. Sollich, and A. De Vita, Physical Review B 95, 214302 (2017).

Ralf Drautz

From density functional theory to magnetic analytic bond order potentials

Density functional theory (DFT) provides a solid basis for the simulation of materials properties. The computational expense of DFT makes the sampling of thermodynamic observables or the calculation of dynamic variables difficult. We coarse grain the interatomic interaction from DFT at two levels of approximation to allow for faster and larger simulations. First, a tight-binding model is derived from a second-order expansion of DFT in a minimal basis. The parameters in the tight-binding model are obtained directly from minimal basis DFT calculations. In a second step the tight-binding model is approximated locally and analytically, resulting in the analytic BOPs. Because of the derivation of BOPs from DFT, the contributions of magnetism and charge transfer to bond formation are directly taken into account. The BOPs are orders of magnitude faster than DFT and allow for the direct sampling of thermodynamic observables. I will discuss the application of the BOPs to simulating finite temperature properties in iron, in particular the ferromagnetic to paramagnetic phase transformation.

Geneviève Dusson

A posteriori error analysis and post-processing methods for linear and nonlinear eigenvalue problems

To determine the electronic structure of a molecular system, one often needs to solve a nonlinear eigenvalue problem. Recently, different methods have been proposed, on the one hand to obtain a guaranteed error bound on the discretization error for such problems (e.g. with a posteriori analysis), and on the other hand to propose accurate solutions at a rather low computational cost, by means of post-processing or two-grid methods. These methods are suited for different discretization methods, for example planewaves, finite elements, and for different problems, such as linear and nonlinear eigenvalue problems.

In this talk, we shall compare some recent a posteriori error estimations together with some post-processing methods in a unified framework. We will in particular show how these methods can be related to a Taylor expansion of the equation.

Gero Friesecke

Pair densities in density functional theory

The exact interaction energy of a many-electron system is determined by the electron pair density, and is not approximated very well in standard density functional theory. In the talk, I
(1) explain the observation by M.Levy that an exact density-to-pair-density map exists,
(2) show how many common functionals including Dirac exchange arise from certain explicit approximations of this map, and
(3) present numerical and asymptotic computations of the exact map for one-parameter families of 1D homogeneous and inhomogeneous model densities varying from 'concentrated' to 'dilute'.

The pair densities are seen to develop remarkable multi-scale patterns which cross over from mean-field to strongly correlated behaviour and show strong dependence on the particle number. The simulation results are confirmed by rigorous asymptotic results in the concentrated respectively dilute limit. The former limit leads to the well-known concept of 'exact exchange', and the latter leads into optimal transport theory. Our numerical results support Becke's celebrated semi-empirical idea underlying the functional B3LYP to mix in a fractional amount of exchange, albeit not to assuming the mixing to be additive and taking the fraction to be a system-independent constant. Joint work with Huajie Chen (Beijing Normal), Codina Cotar (UCL), Claudia
Klueppelberg (TUM).

References:
H. Chen, G. F., Multiscale Model. Simul., 13(4), 1259–1289, 2015;
C. Cotar, G. F., C. Klueppelberg, arXiv 1706.05676 2017

David Gontier

Supercell method for the simulation of insulators without and with defects.

A crystal is mathematically modeled by a periodic Hamiltonian acting on the whole space. In order to compute numerically the properties of such a system, an approximation must be done. One of the most natural approximation consists into studying the Hamiltonian on a supercell, i.e. a box containing L times the periodicity of the crystal, with periodic boundary conditions. The purpose of
this talk is to prove that, in the insulating case, the error made with this approximation is exponentially small with respect to L. We will then consider the case when a local defect is present in the crystal, hence breaking the periodicity of the model.

Roger Haydock

An exact independent electron method for interacting electrons

This talk explains how to treat interacting electrons independently, but without a non-linear exchange and correlation functional. The idea is to replace independent electron states by independent operators which add electrons at definite energies to interacting states of the system. Calculations are similar to those for independent electron states, but lead to systematic approximations for correlation in cohesive energies and other properties. Results for correlation
contributions to the relative stability of transition metal structures and magnetic states illustrate the method.

James Kermode

Multiscale modelling of rare events in materials chemomechanics

‘Chemomechanical’ processes involving complex and interrelated chemical and mechanical processes that originate at the atomic scale often determine the ultimate behaviour of materials. Fracture and dislocation creep are prominent examples, and remain some of the most challenging ‘multi-scale’ modelling problems, typically requiring both an accurate description of chemical processes and the inclusion of very large model systems. I will explain how these requirements can be met simultaneously by combining a quantum mechanical description of crack tips and/or dislocation cores with a classical atomistic model that captures the long-range elastic behaviour of the surrounding crystal matrix, using a QM/MM (quantum mechanics/molecular mechanics) approach such as the 'Learn on the Fly’ (LOTF) scheme. Methodological aspects will be
illustrated with applications, focussing in particular on slow processes such as thermally or chemically activated fracture.

Salma Lahbabi

A mean-field model for disordered crystals

In this talk, we consider disordered quantum crystals in the reduced Hartree-Fock (rHF) framework. The nuclei are supposed to be classical particles arranged around a reference periodic conguration. We first study crystals with

 Perfect Crystal Disordered Crystal

extended defects, such as dislocations or doping in semi-conductors, with nuclear density

$\mu = \mu_{per} + \nu$

where ${\mu_{per}}$ is a periodic nuclear distribution corresponding to the background perfect crystal and ${\nu}$ represents the defect. We next consider a family of nuclear distributions $\mu (\omega, \cdot)$, where ${\omega}$ spans a probability space ${\Omega}$. Under some assumptions on the nuclear distribution ${\mu}$, we prove the existence of an electronic ground state ${\gamma}$, solution of the rHF equations with short-range Yukawa interaction for both types of systems [1, 2]. We obtain partial results for Coulomb interacting systems. We also consider the case of crystals with a low concentration of random defects.

References

[1] Éric Cancès, Salma Lahbabi, and Mathieu Lewin. Mean-field models for disordered crystals. J. math. pures appl., 2012. In press.
[2] Salma Lahbabi. The reduced Hartree-Fock model for short-range quantum crystals with nonlocal defects. Ann. Henri Poincaré, 2013. In press.

Antoine Levitt

Numerical analysis of Brillouin zone integration methods

Spectral properties of Schrödinger operators with periodic potentials can be computed via the Bloch-Floquet theory. This expresses quantities of interest as integrals over the Brillouin zone, a d-dimensional torus, of spectral properties of a parametrized operator acting on a finite domain, yielding efficient computational schemes. I will present a numerical analysis of the methods used to compute these integrals in condensed matter physics, with particular emphasis on the case of metals. This is joint work with E. Cancès, V. Ehrlacher, D. Gontier and D. Lombardi.

Lin Lin

Green's function embedding

As Kohn-Sham density functional theory (KSDFT) being applied to increasingly more complex materials, the periodic boundary condition associated with supercell approaches also becomes unsuitable for a number of important scenarios. Green's function embedding methods allow a more versatile treatment of complex boundary conditions, and hence provide an attractive alternative to describe complex systems that cannot be easily treated in supercell approaches. In this talk, we first revisit the literature of Green's function embedding methods from a numerical linear algebra perspective. We then propose a new Green's function embedding method called PEXSI-$\Sigma$. As a proof of concept, we demonstrate the accuracy of the PEXSI-$\Sigma$ method for graphene with divacancy and dislocation dipole type of defects using the
DFTB+ software package. (Joint with Xiantao Li and Jianfeng Lu)

Jianfeng Lu

Stochastic algorithms for high dimensional quantum systems

In this talk, we will discuss our recent work on a surface hopping Gaussian beam method for high dimensional transport systems, in particular, the quantum-classical Liouville equations arising from semiclassical Schrödinger dynamics in the non-adiabatic regime. If time permits, we will also discuss a quantum kinetic Monte Carlo algorithm for spin dynamics. These methods are
based on a common design principle of stochastic approximation of a series expansion from time dependent perturbation theory. Based on joint works with Zhenning Cai (NUS) and Gero Friesecke (TUM).

TBA

Fred Manby

Connecting wavefunctions and density functional theory

Here I will describe work on converting the very rough Unsöld approximation for electron correlation into an accurate correlation functional, allowing inclusion of wavefunction-type correlation effects in approximate density functional theory in a natural way. If time allows I will also talk about recent efforts in the field to combine density functional theory with accurate
wavefunction theories in multiscale embedding methods.

Faizan Nazar

Locality of Electronic Structure Models

I will give an overview of locality results for electronic structure models and discuss their applications in materials simulations. These include the construction of QM/MM coupling methods and linear scaling algorithms. I will also introduce a lattice relaxation problem, which considers the
rearrangement of a crystal lattice after the introduction of a defect as a variational problem. This treats both point defects and dislocations. Our main result establishes the far-field decay of minimising lattice displacements, which applies to several electronic structure models including: Tight-Binding and DFT models, such as the restricted Hartree-Fock Yukawa model and a
Thomas-Fermi type model with full Coulomb interaction.

Eric Polizzi

FEAST-based scalable algorithms for real-space and real-time first-principle calculations

Realistic first-principle quantum simulations applied to large-scale atomistic systems pose unique challenges in the design of numerical algorithms that are both capable of processing a considerable amount of generated data, and achieving significant parallel scalability on modern
high-end computing architectures. From atoms and molecules to nanostructures, we discuss how the FEAST eigensolver can considerably broaden the perspectives for enabling reliable and high-performance large-scale first-principle all-electrons DFT and real-time TDDFT calculations. We
will present various simulation results with applications ranging from computational electronic spectroscopy of molecules, to plasmonic excitations in carbon-based nanostructures.

Alexey Sokol

Point defects in wide-gap semiconductors from hybrid QM/MM embedded cluster calculations

We review recent developments in our approach to modelling defect structure and properties in ionic materials with a particular focus on wide-gap semiconducting materials. We will discuss shallow and deep defect levels in the context of compact and diffuse electronic states, problems of quantum confinement in the embedded quantum mechanical clusters with implications for
nanoparticles and quantum dots, polarisation and the use of dielectric continuum. Examples will be given based on the ChemShell QM/MM implementation.

Reinhold Schneider

Combination of Tensor Networks (QC-DMRG) with a Coupled Cluster Method

Hierarchcial tensor approximation (HT-Hackbusch) and tensor trains (TT-Oseledets) introduced for high-dimensional approximation are closely related to tensor network (TNS) and matrix product states (MPS) in quantum physics.

In this framework the Schrödinger equation is formulated in second quantization in a discrete Fock space, and usually solved by alternating search algorithms like DMRG (density matrix
renormalization group). This provides high quality approximation of the full CI solution even in presence of strong correlation phenomena. However this tensor low rank approximation is not invariant under unitary transformations of orbital basis functions. Optimization of basis
function together with low rank tensor approximation improves the DMRG calculations further. Nevertheless this approach (DMRG) became too expensive for computing dynamical correlations. We analyse an approach which combines this approach (e.g. DMRG) and MR-SCF-CAS (multi-reference complete active space) with the tailored multi-reference coupled cluster (MRCC) ansatz, which accounts for the dynamical correlation not captured by DMRG.

The cost of the present MR-CC calculation is not larger that for standard single double coupled cluster calculations, but strong correlation effects, degenerate and nearly degenerate
states can be computed as well. The convergence analysis is based on earlier analysis of the Coupled Cluster approximation. This is joint work with F. Faulstich (TUB), O. Legeza (Budapest) A. Laestadius and S. Kvaal (Dept. Chemistry U Oslo)

Dallas Trinkle

Dislocation cores and defect interactions from first principles: Current state of the art and new challenges

Mechanical behavior, specifically plastic deformation at low and high temperatures in metal alloys is governed by the motion of dislocations. Dislocations in crystalline materials were hypothesized nearly eighty years ago, and their experimental and theoretical study has provided powerful tools
for modern materials engineering. While the long-range elastic field of a dislocation is known and straight-forward to compute, many of the strongest effects of dislocations occur in the "core"--the center of the dislocation--where elasticity breaks down, and new chemical bonding environments
can often make even empirical potential descriptions suspect. Hence, there is much effort to use the accuracy of modern density-functional theory to study dislocation cores accurately, as well as their interaction with other defects, such as solutes and boundaries. While there are a variety of possible coupling or "multiscale" techniques available, I will focus on flexible boundary conditions, which use the lattice Green's function to couple electronic structure to an infinite harmonic bulk; this approach greatly simplifies many "hand-shaking" problems, and generally provides a
computationally efficient approach. This methodology has explained solid-solution softening in molybdenum (explaining a 50-year-old mystery of metallurgy), dislocation cores in aluminum and titanium, and provided a wide range of mechanical behavior predictions for magnesium alloys. For the case of nickel-based superalloys, we use density functional theory with flexible boundary conditions to compute the dislocation core structures for a 1/2<110> Ni screw dislocation and a <110> Ni3Al screw superdislocation. This also showcases our new method for computing the lattice Green function that accounts for the topology change due to a dislocation and larger partial splitting. These dislocation core structures are a first step to studying anomalous yield
stress or creep mechanisms in Ni3Al from first-principles.

Daniel Urban

DFT inspired materials design: Towards Z phase-strengthened 12% Cr ferritic-martensitic steels

The challenge of raising the steam inlet temperature of fossil-fired power plants calls for ferritic-martensitic creep-resistant steels with a Cr content of 11-12% in order to achieve sufficient corrosion and oxidation resistance. However, it has been found that in steels strengthened by fine (V,Nb)N particles, these precipitates transform during long-term service into coarse, thermodynamically more stable Z-phase particles, CrMN (M=V,Nb,Ta), that deteriorate the steels' mechanical behavior.

We present extensive atomistic simulations, using density function theory, that help to understand the essential mechanisms underlying the formation of the detrimental Z-phases. Our results reveal that the Z-phase transformation proceeds via diffusion of Cr atoms into the MN particles and their subsequent clustering in a layered arrangement of the Z-phase. We systematically scan the configuration space of various predecessor structures by varying their morphology, stoichiometry and point defect concentration. Our results support experimental efforts to control the precipitation of the Z-phase through appropriate microstructural engineering.