Lecture on First-principles Computations (11):

The Linearized Augmented Plane-wave Method

任新国 (Xinguo Ren)

中国科学技术大学量子信息重点实验室

Hefei, 2016.10.28

Key Laboratory of Quantum Information, USTC

Recall: orthogonalized plane waves and pseudopotential methods

Problems: rapid oscillation of wave functions near the nuclei.

𝜒𝒌+𝑮OPW 𝒓 =

1

𝛺𝑒𝑖 𝒌+𝑮 ⋅𝒓 −

𝑗

𝑢𝑗 ∣ 𝒌 + 𝑮 𝑢𝑗 𝒓

● Orthogonalized plane waves

● The pseudopotential (PP) method

𝑉𝑛𝑢𝑙 𝐫 + 𝑉Hxc 𝐫 𝑉𝑖𝑜𝑛PP 𝐫 + 𝑉Hxc

𝑣,PP 𝐫

Pseudopotential for ions

Pseudopotential for valence electrons

Augmented plane waves (APW)

Muffin-tin (MT) sphere

Interstitial(I)region

Multi-atom unit cell

𝑅𝑀𝑇α

α

βI

J. Slater (1937)

𝜒𝒌+𝑮APW 𝒓, 𝑬 =

𝑒𝑖 𝐤+𝐆 ⋅𝐫, 𝐫 ∈ 𝐼

𝐿

𝑎𝐿α 𝐤 + 𝐆 𝑢𝑙α 𝑟, 𝐸 𝑌𝐿 𝐫𝛂 , 𝐫 ∈ 𝑀𝑇α

1

2m−𝑑2

𝑑𝑟2+𝑙 𝑙 + 1

𝑟2+ 𝑉𝑠 𝑟 − 𝐸 𝑟𝑢𝑙α 𝑟, 𝐸 = 0

𝐿 = 𝑙,𝑚

Spherically averaged crystal potential

The APW method

Requiring the APW to be continuousat the muffin-tin boundary

Basic formula:

𝑒𝑖 𝒌+𝑮 ⋅ 𝐑𝛂+𝐑𝑀𝑇𝛂

=

𝐿

𝑎𝐿α 𝒌 + 𝑮 𝑢𝑙α 𝑅𝑀𝑇α , 𝐸 𝑌𝐿 𝑹𝑀𝑇

α

𝑒i𝒑⋅𝒓 = 4π

𝐿

𝑖𝑙 𝑗𝑙 𝑝𝑟 𝑌𝐿∗ 𝒑 𝑌𝐿 𝒓

𝑎𝐿α 𝒌 + 𝑮 = 4π𝑖𝑙𝑒 𝒌+𝑮 ⋅𝑹𝜶𝑗𝑙 ∣ 𝒌 + 𝑮 ∣𝑅𝑀𝑇

α

𝑢𝑙α 𝑅𝑀𝑇α , 𝐸

𝑌𝐿∗ 𝒌+ 𝑮

Bessel function

Ce

Sjoestedt, Nordstroem, Singh, Solid State Comm, 2000

Spherical Bessel functions

Solution to the Heimholtz equation:

−1

𝑟2𝑑

𝑑𝑟𝑟2𝑑𝑗𝑙 𝑘𝑟

𝑑𝑟+𝑙 𝑙 + 1

𝑟2𝑗𝑙 𝑘𝑟 = 𝑘2𝑗𝑙(𝑘𝑟)

∇2 − 𝑘2 𝜓 𝒓 = 0

The regular solution (finite at 𝒓 = 0):

𝜓 𝒓 ~ 𝑗𝑙 𝑘𝑟 𝑌𝑙𝑚( 𝑟)

Spherical Bessel function, satisfying, 𝑗0 𝑥 =sin(𝑥)

𝑥

𝑗1 𝑥 =sin(𝑥)

𝑥2−cos(𝑥)

𝑥

……

APW as basis functions

χ𝒌+𝑮APW 𝒓, 𝐸 =

𝑒𝑖 𝐤+𝐆 ⋅𝐫, 𝒓 ∈ 𝐼

𝐿

𝑎𝐿α 𝐤 + 𝐆 𝑢𝑙α 𝑟, 𝐸 𝑌𝐿 𝐫𝛂 , 𝒓 ∈ 𝑀𝑇α

ψ𝑛𝐤 𝐫 =

𝑮

𝑐𝑛𝑮 𝒌 χ𝐤+𝐆APW 𝒓, 𝐸

𝐆

𝑐𝑛𝐆 𝐤 𝑒𝑖 𝐤+𝐆 ⋅𝐫, 𝐫 ∈ 𝐼

𝐿

𝐴𝐿α 𝐤 𝑢𝑙α 𝑟, 𝐸 𝑌𝐿 𝐫𝛂 , 𝐫 ∈ 𝑀𝑇α

=

𝐴𝐿α 𝐤 =

𝐆

𝑎𝐿α 𝐤 + 𝐆 𝑐𝑛𝐆 𝐤

𝐴𝐿α 𝐤 are not independent variational parameters; they are fixed by matching conditions.

Eigenvalue problem within the APW basis set

𝑮 = 𝑮𝒋 = 𝑗1𝒃𝟏 + 𝑗2𝒃𝟐 + 𝑗3𝒃𝟑,

𝑗 = χ𝒌+𝑮𝒋APW𝜓𝒏𝒌 𝒓 =

𝒋

𝑐𝑗𝑛 𝒌 𝝌𝒌+𝑮𝒋APW 𝒓

Nonlinear eigenvalue problem:

𝑗

𝑖, 𝐸 ∣ 𝐻 − 𝐸 ∣ 𝑗, 𝐸 + 𝑖, 𝐸 ∣ 𝐻𝑆 ∣ 𝑗, 𝐸 𝑐𝑗𝑛 𝒌 = 0

𝑖 ∣ 𝐻𝑆 ∣ 𝑗 = 𝑆

𝑑𝑆 χ𝐤+𝐆𝐢APW∗ 𝐫

∂

∂𝑟χ𝐤+𝐆𝐣APW 𝐫− −

∂

∂𝑟χ𝐤+𝐆𝐣APW 𝐫+

The surface term:

The secular equation: Det 𝐻 𝐸 − 𝐸𝑂 𝐸 = 0

A real example: band structure of copper

Calculated by the APW method, empirical potential, 2048 k points.G. A. Burdick, Phys. Rev. 129, 138 (1963).

APW is an all-electron method

The 𝑢𝑙 𝑟 𝑌𝑙𝑚 𝑟 are orthogonal to core states. This basis can be used to obtain true valence states in the real potential.

(1) Calculate core states separately in each SCF cycle.

(2) Use the same potential for core and valence and calculate the charge density from the sum of these.

Pros and Cons of the APW method

Exact for muffin-tin potentials, extendible to full potential in principle, but very involved in practice.

Truly all-electron method, applicable equally well for both light and heavy elements.

Energy-dependent basis set => nonlinear eigenvalue problem => solution cannot be found by one diagonalization, costly!

Linearization: the LAPW method

O.K. Andersen, Phys. Rev. B 12, 3060 (1975)

𝑒𝑖 𝐤+𝐆 ⋅𝐫, 𝐫 ∈ 𝐼

𝐿

𝐴𝐿α 𝐤 + 𝐆 𝑢𝑙α 𝑟, 𝐸𝑙 + 𝐵𝐿α 𝐤 + 𝐆 𝑢𝑙α 𝑟, 𝐸𝑙 𝑌𝐿 𝐫𝛂 , 𝐫 ∈ 𝑀𝑇α𝜒𝒌+𝑮LAPW 𝒓 =

χ𝒌+𝑮LAPW 𝒓 is required to be

continuously differentiable atthe MT boundary.

Fixed parameter

This can be achieved by choosingproper 𝐴𝐿α 𝒌 + 𝑮 , 𝐵𝐿α 𝒌 + 𝑮

𝑢𝑙α 𝑟, 𝐸𝑙 =𝑑𝑢𝑙α 𝑟, 𝐸

𝑑𝐸𝐸=𝐸𝑙

Pros and Cons of the LAPW method

● Energy-independent basis functions =>linear eigenvalue problem !

● The slope is continuous => no surface term

● More plane waves are needed to achieve the same accuracy as APW

● Not flexible enough to describe the semicore states

The semicore states

delocalized

Localized, but not confinedin the MT sphere

Confined in the MT sphere

The standard LAPW method is not flexible enough todescribe the semi-core states

The local orbital (LO) correction to LAPW

D. J. Singh, Phys. Rev. B 43, 6388 (1991)

In addition to the LAPWs, one adds LOs

Φ𝐿αLO 𝐫 =

0, 𝐫 ∈ 𝐼

𝐿

𝐴𝐿α𝐿𝑂𝑢𝑙α 𝑟, 𝐸𝑙 + 𝐵𝐿α

𝐿𝑂 𝑢𝑙α 𝑟, 𝐸𝑙 + 𝐶𝐿α𝐿𝑂𝑢𝑙α 𝑟, 𝐸𝐿𝑂 𝑌𝐿 𝐫𝛂 , 𝐫 ∈ 𝑀𝑇α

● Both the value and slope of Φ𝐿αLO 𝐫 are required to be zero at

boundary, and Φ𝐿αLO 𝐫 is normalized.

● The semicore states can now be adequately described with minimum extra cost!

What determines the basis size?

● Large MT sphere =>few plane waves

● Small MT sphere =>many plane waves

● RMTGmax is a good measure ofa converged basis

RMT

Can one do better?

Can one combined the advantages of the APW and LAPW method, i.e., to find an energy-independent basis that does not demanda noticeable increase of the planewave cutoff than the original APW method?

YES: APW+lo

The APW+lo method

Fixed energy parameter

𝜒𝒌+𝑮APW 𝒓 =

𝑒𝑖 𝐤+𝐆 ⋅𝐫, 𝐫 ∈ 𝐼

𝐿

𝑎𝐿α 𝐤 + 𝐆 𝑢𝑙α 𝑟, 𝐸𝑙 𝑌𝐿 𝐫𝛂 , 𝐫 ∈ 𝑀𝑇α

+

Φ𝑙α𝑙𝑜 𝐫 =

0, 𝐫 ∈ 𝐼

𝐿

𝐴𝐿α𝑙𝑜 𝑢𝑙α 𝑟, 𝐸𝑙 + 𝐵𝐿α

𝑙𝑜 𝑢𝑙α 𝑟, 𝐸𝑙 𝑌𝐿 𝐫𝛂 , 𝒓 ∈ 𝑀𝑇α

Sjöstedt, Nordström, Singh, Solid State Commun. 114, 15 (2000)

Determined by zero value at the boundary andthe normalization condition.

Much less plane waves are needed, but the surface term is back!

Test example

Cu

One order of magnitude speed up!

The fast approach: (L)APW+lo

Mixed approach: APW+lo for “physically important”𝑙 channels, and LAPW for the higher ones

Madsen et al. (2001)

Force acting on oneO atom in sodium electrosodalite(40 atoms/unit cell)

K. Schwartz, P. Blaha, G. Madsen, Comput. Phys. Commun. 147, 71 (2002)

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