Analysis of ab-initio total energies obtained by dierent DFT implementations Vishnu Raghuraman and Michael Widom

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Analysis of ab-initio total energies obtained by diﬀerent DFT

implementations

Vishnu Raghuraman and Michael Widom

Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213

Yang Wang

Pittsburgh Supercomputing Center, Carnegie Mellon University, Pittsburgh, PA 15213

arXiv:2210.10069v1 [cond-mat.mtrl-sci] 18 Oct 2022

Abstract

Ab-initio crystal structure prediction depends on accurate calculation of the energies of com-

peting structures. Many DFT codes are available that utilize diﬀerent approaches to solve the

Kohn-Sham equation. We evaluate the consistency of three software packages (WIEN2k, VASP

and MuST) that utilize three diﬀerent methods (FL-APW, plane-wave pseudopotential and the

KKR-Green’s Function methods) by comparing the relative total energies obtained for a set of

BCC and FCC binary metallic alloys. We focus on the impact of choices such as muﬃn-tin vs.

full-potential, angular momentum cutoﬀ and other important KKR parameters. Diﬀerent alloy

systems prove more or less sensitive to these choices, and we explain the diﬀerences through study

of the angular variation of their potentials. Our results can provide guidance in the application of

KKR as a total energy method for structure prediction.

I. INTRODUCTION

Density functional theory-based ﬁrst principles calculations are a powerful tool to uncover

useful and interesting functional properties, guide alloy design and explain experimentally

observed behavior in a variety of condensed matter systems[1–4]. This approach is highly

popular due to (among other reasons) the increased availability of high performance com-

puting facilities and eﬃcient, easy to use DFT codes. Researchers can choose among several

diﬀerent open source and commercially available ab-initio software packages. The numerical

approach used in these packages may vary signiﬁcantly and may dictate which quantities

are more convenient to calculate. For example, the KKR method ﬁts naturally with the co-

herent potential approximation, yielding eﬃcient total energies for disordered alloy systems

[5–7]. The Kubo-Greenwood equation in combination with KKR-CPA [8] allows eﬃcient

computation of residual resistivity for systems with high chemical disorder, whereas the

semi-classical Boltzmann equation may require band structure calculation of a large unit

cell [9]. The modern KKR method solves for the Green’s functions of the Kohn-Sham equa-

tion rather than the wavefunctions, i.e., the Kohn-Sham orbitals. The optimal choice of

software package hence depends on the physical properties that the user intends to calcu-

late. It is important to ensure that the value of a given physical property for a given system is

consistent, regardless of the computational method. This consistency provides conﬁdence in

the software implementation, and allows a new ﬁrst-principles code to establish it’s validity.

Many parameters like lattice constant, bulk moduli etc could be chosen to compare

diﬀerent codes [16]. Our focus is on the consistency of total energies. The total energy (as

a function of concentration) is necessary for the computational study of phase stability and

phase transitions. For the order-disorder transition in CuZn, for example, the calculated

critical temperature can be very sensitive to the energies used. In order to use any DFT

technique to study this transition, it is important to ensure the consistency of the ﬁrst-

principles energies.

While there are many ways to solve the Kohn-Sham equation numerically, we look at the

three major approaches - Full-potential linearized augmented plane Waves (FLAPW)[10–12],

plane wave based pseudo-potential [13], and the KKR Green’s Function[14, 15] techniques as

implemented in the software packages WIEN2k [17], VASP [18], and MuST [19], respectively.

Using these three codes, we calculate the total energy for diﬀerent arrangements of BCC

binaries AB where A,B ∈ {Cr, Mo, Nb, V}and of FCC binaries CD where C,D ∈ {Ag, Cu,

Ni, Pd}.

The paper is organized as follows. First we summarize the theory behind the three DFT

techniques and highlight the similarities, diﬀerences, advantages and disadvantages of each

approach. We then present our ﬁrst-principles energy calculations for the chosen structures,

analyze their consistency and how it varies among diﬀerent binary compounds. We also look

at the impact of some standard DFT input parameters on the results obtained.

II. THEORY

Density functional theory codes express the multi-electron Schr¨odinger equation, a com-

plex function of position vectors of each electron, as a single-electron equation, referred to

as the Kohn-Sham equation [20, 21]

−∇2+Veﬀ ([ρ(r)])ψi(r) = iψi(r).(1)

where the Hamiltonian is a functional of the density ρ(r) = Pi<F|ψi(r)|2and we set ¯h= 1

and me= 1/2. This implicit equation in ψhas to be solved self-consistently. However, as

mentioned previously, there are several alternative routes to solving it.

A. Full-potential Linearized Augmented Plane Wave

The LAPW method involves breaking the system into atomic (or “muﬃn-tin”) spheres,

with each sphere containing a single atom. The wavefunction in the muﬃn-tin sphere can

be expressed as a product of radial functions and spherical harmonics[17, 22]

φkn=X

[Alm,knul(r, El) + Blm,kn˙ul(r, El)] Ylm(ˆ

r),(2)

where ulis the regular solution of the radial Schrodinger equation with energy Eland ˙ul

is its energy derivative taken at the same energy. This is a linear expansion of an energy-

dependent function about the ﬁxed energy El(linearization). The subscript kn=k+Kn,

where kis the wavevector within the Brillouin zone and Knis a reciprocal lattice vector.

The coeﬃcients Alm, Blm are obtained by matching the wavefunction and it’s derivative on

either side of the muﬃn-tin boundary. In early applications of the LAPW method, the

potential inside the muﬃn-tin sphere was taken to be spherical (muﬃn-tin approximation).

Full-potential LAPW (FLAPW) removes this restriction. The consequence of assuming

spherical symmetry is explored in detail in the next section.

This form of the wavefunction is capable of describing low energy electrons (like 1s),

which are highly localized around the nucleus and are not strongly aﬀected by the electrons

from the other atoms in the crystal. Such electrons are termed core electrons. For higher

energy electrons, the interstitial region becomes important. In the interstitial region, the

wavefunction

φkn=1

√weikn·r(3)

can be expressed as a plane wave expansion. Overall, φkncan be considered as a plane wave

augmented by a linearized spherically decomposed function inside the muﬃn-tin sphere, as

speciﬁed by Equation (2). Hence the method is called Linearized Augmented Plane Wave

(LAPW). The overall solution of the system can be written as

ψk=X

cnφkn.(4)

The coeﬃcients cnare obtained using the Rayleigh-Ritz variational principle. Since this

method explicitly deals with the core and the valence electrons, it is termed as an all-

electron method. Interested readers may refer to David Singh’s book [22] to learn more

about the FLAPW method.

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摘要：

Analysisofab-initiototalenergiesobtainedbydierentDFTimplementationsVishnuRaghuramanandMichaelWidomDepartmentofPhysics,CarnegieMellonUniversity,Pittsburgh,PA15213YangWangPittsburghSupercomputingCenter,CarnegieMellonUniversity,Pittsburgh,PA152131AbstractAb-initiocrystalstructurepredictiondependsonacc...

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Analysis of ab-initio total energies obtained by dierent DFT implementations Vishnu Raghuraman and Michael Widom

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