UME NUMBER - Wake Forest University

VOr. UME 43, NUMBER 20

$2 +QvEMBER 197$

Norm-Conserving Pseudopotentials

?ze D. H. Hamalm, M. Schluter, and C. Chiang

Bell Laboratories, Murray Hill,

Jersey 07974

(Received 1 August 1979)

A very simple procedure to extract pseudopotentials from ab initio atomic calculations

is presented. The pseudopotentials yield exact eigenvalues and nodeless eigenfunctions

which agree with atoxnic wave functions beyond a chosen radius x,. Moreover, logarith-

mic derivatives of real and pseudo wave functions and their first energy derivatives

' agree for r &r, guaranteeing excellent transferability of the pseudopotentials.

Pseudopotentials were originally introduced to

like approaches" in which the normalized

simplify electronic structure calculations by

pseudo wave functions have to be orthogonalized

eliminating the need to include atomic core states to core states and renormalized in order to yield

tahnedmt.h'e

strong potentials responsible Two roughly distinct lines of

for binding recent de-

gaciocnu.r'ateProcphearrtgye

densities outside the core re-

(3) guarantees, through Gauss's

velopment are discernable: In one, ion pseudopotentials of enforced smoothness were empir-

theorem, that the electrostatic potential produced

outside r, is identical for real and pseudo charge

' ically fitted to reproduce experimental energy

bands. Consequently, wave functions were only

distributions. Property (4) guarantees that the scattering properties of the real ion cores are

approximately described. In the other, the ortho- reproduced Mlitk rnininzum error as bonding or

gonalized-plane-wave (OPW) concept underlying

banding shifts eigenenergies away from the atom-

the pseudopotential method was used to derive

' "first principles" pseudopotentials from atomic

calculations. These latter potentials are gen-

erally "hard core" in character, that is, strongly

ic levels. A central point of our approach is that these two aspects of transferability are re-

lated by a simple identity. The method permits

the potentials to be intrinsically "soft core" with

repulsive at the origin. The resulting wave func-

tions generally exhibit the correct shape outside

a continuous range of compromise between po-

tential strength and the "core radius" r, . Soft-

the core region; however, they differ real wave functions by a normalization

frfoamctotrh.e'

core potentials are advantageous in band-structure calculations employing any kind of Fourier

It is the purpose of this Letter to demonstrate

analysis.

that the normalization and hard-core problems

The derivation of the identity relating proper-

can be solved simultaneously, while also max-

ties (3) and (4) is closely analogous to that of the

imizing the range of systems in which a pseudo-

Friedel sum rule and an identity discussed by

potential gives accurate results.

Shaw and Harrison' in connection with OP%-like

The new family of energy-independent pseudo-

pseudopotentials. We find (in atomic units)

potentials introduced here have the following de-

sirable properties:

(1) Real and pseudo valence eigenvalues agree

for a chosen "prototype" atomic configuration.

(2) Real and pseudo atomic wave functions agree

beyond a chosen "core radius" r, .

(3) The integrals from 0 to r of the real and pseudo charge densities agree for r &r, for each

valence state (norm conservation).

To derive a convenient family of pseudopoten-

tials with properties (1-4), we first carry out

an ab lation

initio

via a

Hseelrfm-caonn-sSisktiellnmt anf-ulilkl-ecoreproatgormamc, a'lcu-

(4) The logarithmic derivatives of the real and

pseudo wave function and their first energy deriv

atives agree for r &r,.

Properties (3) and (4) are crucial for the pseudo-

potential to have optimum transferability among

a variety of chemical environments in self-con-

sistent density

calculations

is treated

in

as a

which

real

phthyesicpasleudoobjecchta.r'ge

This approach starids in contrast to earlier OPW-

with use of a local approximation for the exchange

and correlation potential. We retain both the po-

tential V(r) and also u, (r), defined as r times the

- valence wave function. We choose an analytic

cutoff function f(x) which approaches 0 as x ~,

approaches cuts off for

1 at least

x-1. For

as fast as x' as x-0, and

each L, we choose a cutoff

r,,, radius

typically 0.5 to 1.0 times the radius

r, . of the outermost peak of u, We then form the

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VOLUME 4$, NUMBER 20

PHYSIC%I. REVIEW j.ETTERS

12 NovEMBER 1979

potential

- V,P'(r) = [1 f(r/r,,)] V(~)+ c,f(r/r,,),

(2)

which converges to V(r) for r &x,,, and adjust the

constant c, so that the nodeless solution u? of

the radial Schrodinger equation with V?~' has

. energy c? equal to the original eigenvalue &,

It is clear that property (1) is now satisfied, and that the normalized function se,,satisfies

property (2) within a multipbcative constant,

p( &~ ) (t') = Q ( (t)~'

since both satisfy the same differential equation

and homogeneous boundary condition for r &x,.

To satisfy (2)-(4), we now modify the interme-

~,, diate pseudo wave function

to

(4)

x'" where g, (x) cuts off to zero for x &1, and behaves

as

at small x. The chosen asymptotic be-

havior of f(x) and g(x) guarantees the potential to

be finite at the origin. ~, is the smaller solution

I ,,be of the quadratic equation resulting from the con-

dition that

normalized,

(5)

The final pseudopotential V,,I"producing the nodeless eigenfunction ze? at eigenvalue &, is now found by inverting the radial Schrodinger equation.

To form the final bare-ion pseudopotential, the

valence pseudo charge density is found with use

of the m? wave functions in the same configuration as the original atom calculation. The Coulomb and exchange-correlation potentials due to this density are then calculated and subtracted from each V?P'. Analytical expressions contain-

' ing few parameters can subsequently be fitted to

the numerical potential functions. The choice of cutoff functions

f(x) = exp(- x~),

g, (x) =x"' exp(- x'),

has provided excellent results in tests on a variety of atoms. Use of the method is illustrated in

Fig. 1, in which s, P, and d real and pseudo

wave functions are compared for Mo in the config-

uration' 4d"'5s' 5p'". It is clear that each pair of functions converges rapidly for r &~,,. The

" corresponding bare-ion pseudopotential in Fig. 1

is clearly "soft core. It is weak and nearly identical for s and p, but has a strong attractive d well, similar to earlier published pseudopo-

0.8-- 0.6--

0.2--

0.0 ; I 1

-0.2,1

0.4-- 0.2-- 0.0 ,,'

0.4

0.2

0.0, ;

~l

-0.2-- rw 0.0

I

I

I

I

I

I

I

I

I

I

I

--2.0

-4.0

CO

CL~ -6.0

-8.0

10.0

I

I

I

I

I

I

I

I

I

I

I

I

.4 1.2 2.0 2.8 3.6 4.4

R (a. u. )

l. F'iG. comparison of pseudo wave functions (solid

lines) and ab initio full-core atomic valence wave func-

tions (broken lines) for Mo. The lower panel shows the corresponding bare-ion pseudopotentials.

" tentials for the 4d transition series.

Property (4) is illustrated for the Mo atomic pseudopotentials in Fig. 2. The logarithmic derivatives of the real and pseudo wave functions,

regular at the origin, are compared at r = 3 a.u.

for a ~1-a.u. energy range straddling the atomic eigenvalues. For l =2, the agreement is so good

over the entire range that the curves are indis-

tinguishable. For l =1, the approach to a core state at --1.38 a.u. produces deviations towards

the low-energy end of the range. For positive energies the real and pseudo scattering phase

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VOLUME 4$, NUMBER 20

PHYSICAL REVIEW LETTERS

].2 NovEMBER 1979

10.0-- 8.0-- 6.0-- 40--

2.0 0.0

D

o -20

4.0 -6.0

4.0 =

(A

20 0.0 K -2.0--

-4.0-- I-- -6.0--

0

oC3 40--

2.0 0.0 -2.0 -4.0

-8.0

-1.2 -.8

-4

0

1

.8

E NERGY (a.U.)

FIG. 2. Energy dependence of logarithmic derivatives at &=3.0 a.u. for Mo ab initio full-core atomic wave

functions (broken lines) and pseudo wave functions {sol-

id lines) as shown in Fig. 1.

" shifts must differ as a consequence of Levinson's

theorem, and the energy scale here is apparently also set by the core binding energies. Band-

structure methods such as augmented plane wave and Korringa-Kohn-Rostoker depend on the lattice potential solely through the logarithmic de-

rivatives at the muffin-tin radius, so that plots

such as Fig. 2 give a direct measure of the range

over which the pseudopotential mill yield accurate bands. Explicit results of band-structure

' calculations employing these new pseudopoten-

tials will be presented elsewhere. Tests for

bulk Si indicate that real" and pseudo charge

densities (with r, =0 75r. ) agree to better than

& 1o, excluding spheres of radius 1.5 a.u. about

each atom. Corresponding band-structure ener-

gies agree to better than 0.05 eV over a 20-eV

range. Another test of transferability involving both

properties (3) and (4) is given by comparing selfconsistent excited configurations of real and

pseudo atoms. Such comparisons are shown in

Table I for excited, singly and doubly ionized

configurations of Mo, Si, and O. The ion-core pseudopotential is that of the ground-state con-

figuration' for which agreement is exact by construction. The excellent agreement of the eigen-

values (i.e., deviations smaller than O.l eV over

a 30-eV energy range) clearly illustrates the ef-

fectiveness of the properties built into these pseudopotentials over a wide region of the periodic table. It should be noted that nodeless wave

functions such as the 0 2P have been constructed

by the same procedure, yielding a strongly attractive but nonsingular ion pseudopotential. The pseudopotentials introduced here permit accurate self-consistent calculations, and have

r, the flexibility of a "quality" parameter which

can be chosen appropriately for the intended application. The procedure to produce them is exceedingly simple and can be added in the form

TABLE I. Atomic eigenvalues in atomic units for pseudopotentials constructed r with ~& = 0.75& &. D denotes the deviations from the corresponding full-core ab

initio results. The signer exchange approximation is used.

Configuration

0 2s'2p' 0 2s22ps

Q 2s 2p Si 3s 3p

Si 3s'3p' Si 3s'3p' Mo 4d 5s 5p

Mo 4d55s Mo 4d 5s

--0.8818 --1.4337 --2.1181 --0.4261 --0.6981 --1.0455 --0.2108 --0.3599 --0.6421

Energy p

-0.3490

--0.8945

---01..15777687

--0.4321

---00..07856651

--0.2243 --0.4675

--0.2740

---0.3936 0.7780

--0.0011 0.0005

--0.0001 --0.0018

0.0007 0.0031

mp

mp

--0.0013

-0.0003 ---00..00000357

0.0001

0.0004 0.0025 --0.0006

-00..00002025

0.0025 --0.0016 --0.0006

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Vor. UMs 43, NUMaER 20

PHYSICAL REVIEW LETTERS

12 NovEMBER 1979

of a simple subroutine to any existing local-density atom program.

We have profited from many helpful discussions with G. &. Baraff and V. Heine. One of ns (C.C.)

is a Bell Laboratories Summer Research Associate.

See, M. L. Cohen and V. Heine, in Solid State Physics, edited by H. Ehrenreich, F. Seitz, and D. Turn-

bull (Academic, New York, 1970) Vol. 24, p. 38, and

J. V. Heine and D. Weaire, ibid. , p. 249.

A. Appelbaum and D. R. Hamann, Phys. Rev. B 8,

1777 (1978); M. Schluter, J. R. Chelikowsky, S. G. Louie and M. L. Cohen, Phys. Rev. 3 12, 4200 (1975).

C. F. Melius and %. A. Qoddard, III, Phys. Hev, A

10, 1528 (1974).

A. Zunger and M. L. Cohen, Phys. Rev. Lett. 41, 53

(1978).

R. W. Shaw and W. A. Harrison, Phys. Rev. 163, 604

(1967).

Modifications of an OPW-derived pseudopotential to satisfy a similar norm condition starting from a nonlocal (Hartree-Fock) real potential were recently pro-

posed by A. Redondo, %. A. Goddard, III, and T. C. Mc-

Qill, Phys. Rev. 8 15, 5038 (1977). These potentials

are derived by a basis-function method, and remain

singular at r =0. Application of this approach to local

potentials was later discussed by A. Zunger (to be pub-

lished) .

'F. Herman and S. Skillman, in Atomic Stmctlre Cal-

J. culation (Prentice-Hall, Englewood Cliffs, N. , . 1963)

uratDio.nnR. . Hamann and M. Schluter, to be published.

In order to define a p-like pseudopotential, the Mo

atom was excited frown its 4d Gs ground-state config-

See, (1978);

e.g., S. G. R. J'acobs,

Louie, Phys. Rev. Lett. 40,

J. Phys. C 1, 1296 (1968).

1525

See, L. I. Schiff, Qgantum Mechanics, (McGraw Hill,

New York, 1968) p. 353.

D. R. Hamann, Phys. Rev. Lett. 42, 662 (1979).

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