Equivariant analytical mapping of first principles Hamiltonians to accurate and transferable materials models

Supporting data for https://arxiv.org/abs/2111.13736.

ACEhamiltonians.jl code

This is an archived copy of the ACEhamiltonians.jl code to accompany the paper arXiv:2111.13736.

See https://github.com/ACEsuit/ACEhamiltoniansExamples for examples of how to use this code.

The code is written in Julia and requires v1.6 or later. To install the Julia depenendencies:

$ cd ACEhamiltonians.jl
$ julia
julia> import Pkg
julia> Pkg.activate(".")
julia> Pkg.instantiate()

The scripts test/plots.jl, test/fcc-to-bcc.jl and test/vacancy.jl which produce all the plots in the paper can then run as, e.g.

julia --project=. test/plots.jl

Training data

The training_data folder contains the atomic structure, Hamiltonian and overlap matrices stored in HDF5 format with the following schema:

• Data Group : aitb/
• Datasets :
  • H : Real-space Hamiltonian Matrix. Type: Float64. Shape: Tensor(# of TB Cells, # of Rows, # of Columns)
  • S : Real-space Overlap Matrix. Type: Float64. Shape: Tensor(# of TB Cells, # of Rows, # of Columns)
  • energy : Energy. Unit: eV. Type: Float64. Shape: Scalar
  • freeenergy : Free Energy. Unit: eV. Shape: Scalar
  • unitcell : Unit cell vectors. Type: Float64. Shape: Matrix(3,3)
  • positions : Atom positions. Type: Float64. Shape: Array(3)
  • forces : (Optional, if available) Forces. Type: Float64. Shape: Array(3)
  • metadata : JSON String including dictionary of information of FHIaims calculation (k-points, basis sets), TB Cells, Cutoff, Orbital definitions.,

The molecular dynamics and FHI-aims parameters are described in the manuscript.

On-site models

The onsite_models_ord2 folder contains our correlation order 2 models for the on site blocks of the Hamiltonian, in a JSON format readable by the ACE.jl and ACEhamiltonians.jl Julia packages. There are separate files for the Hamiltonian (*_H.json) and overlap (*_S.json) models. The JSON files also contain training and test sets and associated errors as plotted in Figure 3 in our manuscript.

Models have a unique identifier (UUID) which is a hash of the input parameters and training data. The mapping from (order, max_degree) to UUID is as follows:

(2,4)  - 13427527590286463256
(2,5)  - 10538156191357510769
(2,6)  - 1646489440533135164
(2,7)  - 12130775482127724115
(2,8)  - 12487060958610974041
(2,9)  - 2653067664384673997
(2,10) - 1143382251563115664
(2,11) - 4564001820340015372
(2,12) - 9474261500251782658

Off-site models

The offsite_models_ord1 and offsite_models_ord2 folders contain our order 1 and order 2 offsite models for Hamiltonian and overlap matrices. The mapping from (H_order, H_max_degree) + (S_order, S_max_degree) to UUID is as follows:

(1,6) + (1,8)    -    7014526518680934587
(1,7) + (1,9)    -    8594416159488562244
(1,8) + (1,10)   -    10204186688118368371
(1,9) + (1,11)   -    13078304848585360574
(1,10)+ (1,12)   -    14750835312950641338
(1,11)+ (1,13)   -    9883802224093245794
(1,12)+ (1,14)   -    3907899412408606585
(1,13)+ (1,15)   -    201683837542179657
(1,14)+ (1,16)   -    277744202775070779
(2,6) + (1,8)    -    4699475053563592071
(2,7) + (1,9)    -    489637409713831432
(2,8) + (1,10)   -    18034631670613263469
(2,9) + (1,11)   -    720654516759450160
(2,10)+ (1,12)   -    15214900801060024044
(2,11)+ (1,13)   -    13798832597295943078
(2,12)+ (1,14)   -    13162803789413134473

FCC only

Onsite models:

2  6 5311732756869418284
2  7 13030014632886405308
2  8 5820099621734447846
2  9 10161014511878227635
2 10 11298425190201843107
2 11 9932031839231628354
2 12 9447261873515969583

Optimised FCC model 16110190062237887798

BCC only

Onsite models:

2  6 8949023800586845770
2  7 8045797268444730200
2  8 6919809282139600809
2  9 9935027806122780319
2 10 6376963380608532713
2 11 5001375576268070883
2 12 9678585765722197901

Optimised BCC model 10293566074413000591

FCC+BCC optimised models

Onsite models

2  6 2154760103892646619
2  7 6450474921309693835
2  8 14227277988574899288
2  9 476820595195218567
2 10 5364136683220082110
2 11 14619519825012606580
2 12 14181614899005838824

Offsite FCC+BCC optimised model - 4570230078043807257

Model errors

The model_errors directory contains summarised model errors for the training and testing errors for the models listed above.

Reference data

Reference electronic structure data computed for the BCC and FCC crystals, along the Bain path and for the relaxed vacancy is stored in the reference_data folder. The Hamiltonian and overlap matrices are stored as compressed binary HDF5 files. The format and metadata can be viewed with the h5dump utility, or read in using the supplied Julia code (or indeed from other languages).

Predicted data

The predicted_data/FCC and predicted_data/BCC folders contain HDF5 files with the results of all model predictions shown in the manuscript on the FCC and BCC crystal structures. predicted_data/FCC-to-BCC contains the results of predictions along the Bain path with the optimized model described in the manuscript and predicted_data/vacancy contains the vacancy calculations.