Functional Benchmarking

Overview

“The right answer for the right reason at the right price.” John P. Perdew

what functional to use?

  • Need to balance physical accuracy with computational cost.

  • Also want to make sure trends are correct

Need to focus on properties of interest.

  • Binding energies

  • Isolated molecules (bond distances and angles)

  • Equilibrium bond distance (between adsorbate and surface)

  • Reaction barriers

  • Reaction energetics (gibbs free energy changes)

  • Vibrational frequencies

  • Infrared Spectroscopy

  • TS geometries

  • Minimum energy geometries

  • Things to consider when benchmarking (1)

  • Is data directly comparable to ab initio (temp, pure sample, etc)

  • Is data diverse

  • What are the error bars for experimental measurements?

  • Use several sources to prevent bias

How do we address the physical phenomena that influence performance of real catalyst that are not included in DFT calculations? Do we need to worry about self-interaction errors? ( hybrid functionals can partially correct for this) Does our system have strong electron correlation?

  • Known Systems with self-interaction errors (2)

  • actinides

  • various transition-metal oxides that have partially filled d or f shells

Magnesium is a group 2 alkaline earth metal ( electron configuration [Ne] 3s2)

However in MgO, we are interested in the impact of oxygen vacancies and different substitutions of Pt. These systems should be explored for strong electron correlation

Properties - Physical Accuracy

Need to quantify how accurate DFT predictions are for specified property of interest.

Types of functionals

  • LDA - Local Density Approximation

  • exchange-correlation energy at any point in space is a function of the electron density at that point in space only (hence local) and can be given by the electron density of a homogeneous electron gas of the same density (homogeneous gases have already been calculated)

  • Tends to overestimate binding energies

  • Typically underestimates exchange and ovestimates correlation (helps errors cancel)

  • LSDA - Local spin density approximation

  • General application of LDA which includes spin dependence

  • GGA - Generalized Gradient Approximation

  • Includes local density and the gradient of the density

  • Accounts for inhomogeneous varying nature of electron density

  • two broad classes of GGAs:

  1. Emperical - Fitted to large training sets

#. Non-Emperical - Derived from first principles using constraints known from quantum mechanics *known to overestimate the adsorption energies of molecules on transition metals

  • Meta-GGA

  • Includes higher order density gradients

  • includes the orbital kinetic energy density, which is computed from orbitals that are functionals of the density

  • Due to inclusion of orbital kinetic energy density with electron density and its gradients, meta-GGAs have more flexibilty

  • Hybrids

  • Combine exact exhange from Hartree-Fock with GGA method

  • Optimizing functional fitting coefficients is usually performed on experimental data

  • Hybrid-metta GGA

  • Hybrid mixed with meta GGA

  • ONIOM - Our own n-layered Integrated molecular Orbital and Molecular mechanics

  • Computational hybrid method that enables different ab initio or semi-empirical methods to be applied to different parts of a molecule/system in combination to produce reliable geometry and energy at reduced computational cost

  • Can combine different functions like (PBE/HSE)

Functional Selection

LDA

B3LYP

  • Contains a portion of Hartree-Fock exchange and cannot be used on solids

PBE - Perdew-Burke-Ernzerhof

Constructed to satisfy 11 exact constraints.

RPBE - Revised Perdew-Burke-Ernzerhof

PBEsol

SCAN - Strongly Constrained and Appropriately Normed Semilocal Density Functional

SCAN was constructed by Perdews team and is the first meta-GGA that is fully constrained to all 17 known exact constraints that a semi-local function can satisfy

It is not fitted to any bonded system. It is fitted to norms, non-bonded systems such as atoms in which it can be accurate for the exchange and correlation energies separately, and not just their sum as in bonded systems.

SCAN performs well in: * atomization energies * lattice constants of solids * Short range weak interactions (hydrogen-bonds and vdW interactions for closed shell molecules)

  • No semilocal functional can capture long-range vdW interactions (need correction)

SCAN performs better than PBEsol and PBE for the reactions tested (ref 5 - Shows the benchmark database-Barrier Heights for Heavy Atom Transfer, Nucleophilic Substitution, Association, and Unimolecular Reactions - hydrogen and non-hydrogen transfer gas-phase reactions)

From ref. 7, SCAN is proposed to work better than PBE for defects in semiconductors, surface properties of metals, formation energies and structural phase transitions in semiconductors. Good at predicting band gap.

Typically a much lower computational cost than Hybrid functionals.

Limitations

  • no SCAN-specific pseudopotentials are available for use in VASP

Regularized SCAN functional

From ref. 8, proposed modifications to functional form to eliminate numerical instabilities.

HSE - Heyd-Scuseria-Ernzerhof

RPA - Random Phase Approximation

|Bohm and Pines’ RPA accounts for the weak screened Coulomb interaction and is commonly used for describing the dynamic linear electronic response of electron systems |Electrons are assumed to respond only to the total electric potential V(r) which is the sum of the external perturbing potential Vext(r) and a screening potential Vsc(r). |The external perturbing potential is assumed to oscillate at a single frequency ω, so that the model yields via a self-consistent field (SCF) method [4] a dynamic dielectric function denoted by εRPA(k, ω).

  • RPA tends to underestimate binding energies

BEEF-vdw

References

  1. https://youtu.be/Ey00F_vsIiY (Benchmarking DFT and beyond-DFT methods for thermodynamics and electronic properties - Geoffroy Hautier)

  2. General Performance of Density Functionals Sérgio Filipe Sousa, Pedro Alexandrino Fernandes, and Maria João Ramos The Journal of Physical Chemistry A 2007 111 (42), 10439-10452 DOI: 10.1021/jp0734474

  3. https://youtu.be/03Y0v4Ys3_A (SCAN meta-GGA: predictive power of 17 constraints)

  4. Strongly Constrained and Appropriately Normed Semilocal Density Functional. Jianwei Sun, Adrienn Ruzsinszky, and John P. Perdew. Phys. Rev. Lett. 115, 036402 – Published 14 July 2015

  5. Benchmark Database of Barrier Heights for Heavy Atom Transfer, Nucleophilic Substitution, Association, and Unimolecular Reactions and Its Use to Test Theoretical Methods. Yan Zhao, Núria González-García, and Donald G. Truhlar. The Journal of Physical Chemistry A 2005 109 (9), 2012-2018. DOI: 10.1021/jp045141s

  6. Accuracy of Density Functional Theory for Predicting Kinetics of Methanol Synthesis from CO and CO2 Hydrogenation on Copper. Maliheh Shaban Tameh, Albert K. Dearden, and Chen Huang. The Journal of Physical Chemistry C 2018 122 (31), 17942-17953. DOI: 10.1021/acs.jpcc.8b06498

  7. https://templeefrc.org/scan-overview

  8. Albert P. Bartók and Jonathan R. Yates , “Regularized SCAN functional”, J. Chem. Phys. 150, 161101 (2019) https://doi.org/10.1063/1.5094646

  9. Michal Bajdich, Jens K. Nørskov, and Aleksandra Vojvodic Phys. Rev. B 91, 155401 – Published 1 April 2015