# Workshops

## Efficient localised orbitals for large systems, strong correlations and excitations

### Organisers

- Peter Haynes
*(Imperial College London, United Kingdom)* - David D. O'Regan
*(Trinity College Dublin, Ireland)* - Simon M.-M. Dubois
*(IMCN - NAPS - UCLouvain, Belgium)* - Paolo Umari
*(University of Padova, Italy)*

### Supports

CECAM

Psi-k

Supported by CECAM JC Maxwell Node

### Description

**In parallel with continuing progress in linear-scaling methods relying on optimised, localised orbitals, recent developments have established the potential for efficient use of such functions in diverse areas including quantum transport, correlated systems and electronic excitations. This workshop aims to bring together expertise in these topics to clarify the state of the art in optimisation and localisation procedures and to focus efforts in the development of optimised local orbitals for advanced electronic structure methods.**

**Motivation**

This event continues the occasional series of CECAM workshops previously held in Lyons with the co-sponsorship of Psi-k:

- Local orbital methods for large scale atomistic simulations, July 1998.
- Local orbitals and linear-scaling ab initio calculations, September 2001.
- Linear-scaling ab initio calculations: applications and future directions, September 2007.

The emphasis of the previous workshops was the use of local orbitals within linear-scaling methods for large-scale density-functional theory (DFT) calculations. The interest in developing such methods from the early 1990s led to new efforts in the optimisation of basis sets consisting of (pseudo) atomic orbitals within the condensed matter community that had adopted the pseudopotential plane-wave methodology as its standard. A particularly fruitful outcome of the previous workshops was the interaction between the condensed matter physics and quantum chemistry communities. These basis sets are already being heavily exploited in transport calculations and there is currently interest in using them to develop more efficient methods for many-body perturbation theory within the GW approximation.

Alongside the development of more accurate atomic-type basis sets, work has also been carried out on the development of linear-scaling methods that employ a set of local orbitals optimised in situ to the unique chemical environment of each atom. These optimised orbitals have recently been shown to provide consistency in the definition of the projectors used in DFT+U calculations, and methods have also been developed to refine optimised orbitals to describe bound but unoccupied (conduction band or virtual) states in addition to the occupied (valence band or real) states required for the self-consistent determination of the ground state.

This workshop aims to widen the scope of local orbitals to explore the use of optimised local orbitals in methods for treating strongly-correlated systems (e.g. DFT+U and dynamical mean field theory) and excitations (e.g. time-dependent DFT and many-body perturbation theory). The workshop will bring together scientists with experience of optimising local orbitals, primarily from the community developing linear-scaling DFT methods, with those seeking to exploit local orbitals to expand the scope and scale of electronic structure methods that go "beyond DFT". By promoting much greater interaction between these mostly disconnected groups, progress in the development of these new methods will be accelerated to the benefit of both groups of participants: those with experience of optimising local orbitals will be introduced to new areas of application for their work; those seeking to develop new methods will benefit from that experience. This workshop is particularly timely given the recent resurgence of interest in local orbitals.

**State of the art**

The generation of localised orbitals is a matter which has concerned many branches of electronic structure theory over the past two decades and the last few years in particular have been ones of intense progress. An efficient orbital representation is typically one in which the operators of interest can be expressed with adequate accuracy, small matrix rank and, where possible, predictable matrix sparsity. Systematic improvability is a further desirable attribute. Differing criteria have been employed to optimise these orbitals, used to represent non-interacting quasiparticles, e.g., Kohn-Sham states, many-body quasiparticles or their products depending on the context.

In linear-scaling implementations of Kohn-Sham density functional theory [1-8], together with its extensions to excited state phenomena [9-11], one is often concerned with locating orbitals which are strictly localised, so that the Hamiltonian matrix is sparse, and which afford a sparse representation of the single particle density-matrix for insulators and finite-temperature metals. These orbitals may be refined in situ on a fixed underlying basis, for example to minimise the total energy, or they may be initially optimised in a pre-processing step and fixed thereafter.

In methods for strongly-correlated systems, such as DFT+DMFT, DFT+U and DFT+SIC, in their numerous incarnations, one often must define spaces to which many-body corrections or exact conditions on the exchange-correlation functional, beyond LDA-based approximations, are applied. These spaces may or may not encapsulate the effects of orbital hybridisation or the competing tendencies of localisation and delocalisation near a metal-insulator transition. Numerous orbital optimisation criteria are in use to this field [12-20], such as maximisation of measures of orbital localisation, maximisation of the Coulomb repulsion or minimisation of its anisotropy, minimisation of the total energy, recovery of many-body expectation values or minimisation of energy dependence.

In many-body perturbation theory GW calculations for the evaluation of quasiparticle properties, local orbitals allow for a reduction of the computational load. This in turn permits to implement self-consistency schemes which proved to be important for molecular systems and molecular transport problems [21,22]. The possibility of representing orbitals in terms of localised Wannier like functions has also been used to reduce the computational load of GW calculations performed with schemes and codes based on plane-waves basis sets [23]. In contrast with plane-waves basis sets local orbitals allow for all-electrons GW calculations.

Localised orbitals may furthermore form a highly efficient bases for extracting tight-binding models from the ab initio calculations. In particular, Wannier functions have been shown to provide excellent representations for Fermi surface properties [24], orbital magnetoelectric coupling [25], electron-phonon interactions [26], Van der Waals effects [27], magnetically induced lattice distortions [28], spin-wave excitation spectra [29] and many-body quasiparticles [30].

Optimisation algorithms are a matter for technical investigation and optimisation in and of themselves [31-33], often carrying over from one criterion to the next, and sessions of this workshop will place focus on these. As an example, the complications resulting from orbital non-orthogonality, advantageous as it may admit increased localisation and matrix sparsity, or constraints on the orbitals they represent, have attracted careful attention over the years [34-38].

**This CECAM workshop is a timely opportunity for users and developers of localised orbital methods, as well as workers on the orbital optimisation procedures themselves, to present the latest methods developed in their diverse areas and to learn from those in others.**

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