Recipe for Norm-Conserving PSeudopotential database 19-95

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National Institute for Research in Inorganic Materials (Old) (New)

K. Kobayashi

Contents
Recipe
Introduction
What is a psudopotential?
Advantage
Description
Advantage of this database.
Data formats and Usage
How to use data?
Summary
Attention
Acknowlegments
Reference
Parameters
Update

Introduction

At present, I have a plan to construct the database for norm-conserving pseudopotentials(NCPS95,97). This has been already released to public as distributed by CD-R media(10/6,1998). The purpose of this database is to design new materials in the first-principles electronic structure calculation by using it. At present, although it is impossible to construct psudopotentials for all kinds of atoms in the periodic table, I have prepared pseudopotentials about more than 60 kinds of atoms. If you have a program of the Car-Parrinello (or plane wave pseudopotential) type electronic structure calculation, it is possible to perform the optimization of the electronic and ionic structures for arbitrary systems by using the NCPS95,97.

What is a psudopotential?

The basic concept of psudopotential is a frozen core approximation which assumes that the electronic states of core electrons are insensitive to the neighboring atomic configuration. This assumption is correct in many cases in solid state physics.

Advantage of using pseudopotentials:

(a) Using the pseudopotentials, it is possible to use the plane wave basis set in the electronic structure calculation because of the smoothed potential in the core region.
(b) It is relatively easy to formulate Hellmann-Feynman forces and stresses in plane wave basis.
(c) As for total energy per atom, five effective digits are sufficient in most cases in the psudopotential approach because of the absence of the core contribution, while at least eight effective digits are required in all-electron calculations.
(d) No ghost bands(*1) appear in a wide range of a calculated energy by using the correct separable pseudopotential. On the contrary, in the all-electron linearized band calculations(LMTO,LAPW), there is a limitation of the available energy range within about 1 Ry.
(e) Pseudopotential is full potential.

Description of this pseudopotential database

Traditionally, empirical pseudopotentials are constructed to reproduce experimental results. Such empirical pseudopotentials have a weak point that the charge density (wave function) does not coincide with that of the real atom even outside the core region. In 1979, Hamann, Schülter and Chiang[1] proposed a new ab-initio pseudopotential. The following conditions are imposed in this pseudopotential:
(1) Real and pseudo valence eigen-values of atom agree with each other in arbitrary atomic configurations.
(2) In an outside of cutoff radius, an amplitude of the pseudo-wave function coincide with that of a real atomic-wave function.
(3) Since the pseudo-wave function does not have a node, the plane wave basis are available in the electronic structure calculation.
(4) In an inside region of cutoff radius, the norm of the pseudo-wave function coincides with that of the real wave function.
(5) The first order energy derivative for a logarithmic derivative of the pseudo- and atomic-wave functions coincide with each other at cuttoff radius.

In the database of pseudopotentials, optimized pseudopotentials originated from Troullier and Martins[2](TM) are adopted without a few exceptions. It is realized to reduce a number of plane waves in the electronic structure calculation.

At present, pseudopotentials for H,Li,Be,B,C,N,O,F,Na,Mg,Al,Si,P,S,Cl,K,Ca,Sc,Ti,V,Cr,Mn,Fe,Co,Ni,Cu,Zn,Ga,Ge,As,Se,Br,Rb,Sr,Y,Zr ,Nb,Mo,Tc,Ru,Rh,Pd,Ag,Cd,In,Sn,Sb,Te,I,Cs,Hf,Ta,W,Re,Os,Ir,Pt,Au,Hg,Tl,Pb,Bi,Po are available.

Li,Na,Mg,Al,Si,P,S,Cl are BHS-type pseudopotentials[3], the other is TM-type. The electronic band structures (see [NPT-NCPS95]) and bulk properties [TableI][TableII][TableIII][TableIV][TableV](png files, 5-23KB) for above pseudopotentials are calculated. Calculational results are reasonable on the comparison with other theoretical and experimatal results. As for results of some pseudopotentials, please see in some references of papers[4][5].
As a result, the case of largest discrepancy of present and experimental results is Pt by 3.7% for the lattice constant. In some cases (for examples: Sc, Y, Au, Ir, Pt), the discrepancies of present and experimental(theoretical) results are not so small(about 2.5 - 3.9 %). However, in almost cases, discrepancies are within 2% as for lattice constants and 15 or 20% as for bulk modulus. As for Pd etc.,they will be improved at next version up. As for Li,Na,In and all transition metals(including Cu, Ag, Au), it is considered a partial core correction[6](PCC) in order to improve bulk properties depending on a non-linear effect of exchange-correlation terms for valence and core charges. As for K,Ca,Rb,Sr and Ga,In,Tl, they have shallow 3d, 4d or 5d core states. In this database, these pseudopotentials are constructed to consider those outer core electrons as valence states. In these cases, calculated equilibrium lattice constants are slightly overestimated within 1 or 2%. Generalized gradient correction(GGA)[7] is not considered. These pseudopotential data are simply numerical. It is not worthy without any band calculation programs. It is necessary to prepare a program of band calculation which is possible to use pseudopotentials of this database.

Numerical DATA format of this database is available only for Kleinmann-Bylander(KB) type separable form[8](Non-separable type format is not prepared).

It is a test version(beta-version) and there is no guarantee to give correct results usually for any calculational cases. It is not investigated to a variety of environment as a examples of complex compounds, surfaces, interfaces, clusters, etc., as for a problem of transferability.

I have no responsible for demerit and incorrect results using the data of pseudopotentials (see [No Warranty]).

Advantage of this database.

In this database of pseudopotentials, it is checked that there are no ghost bands in bulk calculations (see [Band Structures]) using the K-B type separable form. The bulk properties agree with other theoretical and experimental data within 2% for lattice constants in most cases. As for Li,Na,Ga(Non-3d),In(Non-4d) and transition metals(including Cu,Ag,Au), the partial core correction is considered to improve a underestimation of lattice constants, an overestimation of magnetic moments (for Fe, Co and Ni)[5][9] or other effects due to non-linear effect of exchange-correlation terms for valence and core charges. In ferro-magnetic Fe, Co and Ni cases, the formula of MJW[10] is used for the exchange-correlation potential. As for transition metals, two types of pseudopotentials are prepared, one is considered with PCC, the other is not considered with PCC. Since in the original all-electron atomic calculation, a scalar relativistic effect is considered, the relativistic effect is sufficiently considered in valence electrons for pseudopotentials. Already, pseudopotentials for over 60 kinds of atoms have been prepared and are available to calculate the band structure calculation using them for a variety of systems within a limitation of your computer resources and man power.
Maybe, it is easy to carry out the electronic structure calculation for Si or Al, even for a workstation or PC. It is possible to calculate systems including several tenths number of atoms. Using other atoms except for Si, Al, it is relatively difficult to calculate the electronic structures because of requiring a more number of plane waves. If you calculate systems which are consisted of more than one handred of atoms or including atoms for first-elements or transition or noble metals, you should submit jobs of the electronic structure calculation on high-performance computer systems(parallel, super, etc.).

Data formats and Usage

< Data formats >

(a) Implimented pseudopotentials
BHS type data
Li,Na,Mg,Al,Si,P,S,Cl
TM type optimized pseudopotentials
H,He,B,C,N,O,F,Ne,Ar,K,Ca,Sc,Ti,V,Cr,Mn,Fe,Co,
Ni,Cu,Zn,Ga,Ge,As,Se,Br,Kr,Rb,Sr,Y,Zr,
Nb,Mo,Tc,Ru,Rh,Pd,Ag,Cd,In,Sn,Sb,
Te,I,Xe,Cs,Ba,Hf,Ta,W,Re,Os,Ir,Pt,Au,Hg,
Tl,Pb,Bi,Po,At,Rn

Description

(1) As for Li,Na,transition metals(including Cu,Ag and Au) pseudopotentials of two types are prepared, one is the PCC considered, the other is without PCC.
(2) As for Ca,Ga,In,Rb,Sr pseudopotentials of two types are prepared, one is the PCC considered, the other is considering outer core electrons as valence states.
(3) As for the transition and noble metals and some other pseudopotentials, non-local parts are s, p and d. The other is only s and p are non-local and d is used as a local potential.
(4) In a case of s,p and d non-local, a local potential is a Vcore(r) as shown in BHS paper[3].
(5)As for Li,C,O, thses were constructed by Dr. Y. Morikawa.
(6) The program for constructing TM type pseudopotentials were coded by Dr. Y. Morikawa, and the all-electron atomic calculation program was coded by Prof. A. Hasegawa.
(7) The scalar relativistic effect is considered in all-electron atomic calculation (see reference Koelling and Harmon[11]). This all-electron atomic calculation program is coded by Prof. A. Hasegawa.
(8) Wigner form[12] for exchange-correlation term is used in non-magnetic (no spin polarized) calculation.
(9) No spin-orbit interaction is considered.
(10) GGA[7] is not considered.

(b) Data formats

Numerical data of one pseudopotential consists of two or three files. Units of every numerical data are atomic units. 1 a.u. is 27.2 eV(Energy, Hartree), 1 a.u. is 0.529177A(Length).

(1) Only s and p are non-local case

For examples a carbon pseudopotential, there are three files.

C001.DAT CNKB.DAT CND.DAT

C001.DAT is a input data file for coordinates of atoms and lattice parameters etc. In this case, the most important parameters are the last line of file.


4.00000000 0.15000000
4.00000000 1.52220000 -0.52220000 9.28000000 3.69000000 <-- Last line

Last four parameters 1.5222, -0.5222, 9.28, 3.69 represent a function (Vcore(r))as follows,

Vcore(r)=-Z*(C1*erf(A1*r) + C2*erf(A2*r))/r.

Here, C1=1.5222, C1+C2=1, A1=9.28, A2=3.69.

CNKB.DAT is a data for s, p non-local pseudopotentials. Head lines of this file as follows,

421 0 0.0104166666666667
0.387120133174D+00 -0.115786648758D+00
0.119684022029D-03 0.123483200711D-03 0.127402978271D-03 0.131447182927D-03
0.135619764419D-03 0.139924797866D-03 0.144366487744D-03 0.148949171993D-03
0.153677326257D-03 0.158555568247D-03 0.163588662261D-03 0.168781523830D-03
0.174139224519D-03 0.179666996885D-03 0.185370239581D-03 0.1912545.


The first data '421' is a number of log-mesh points. Next is a dummy parameter. Third data is a width of log-mesh for useful to integrate to a radial part. The maximum core radius of log-mesh is 60.0a.u. In the integration of radial part, r**2 is transformed to r**3 at log-mesh, in which r**2 means the transformation from cartesian coordinates to radial coordinates(Integral[dr3]=Integral[r2dr]Integral[cosAdA]Integral[dB]). Forth, fifth data in the second line are values of denominator of KB separable form.
Next is 421 data of log-mesh points for the core radius. In CNKB.DAT, there are three 421 data. The first is data of log-mesh points for the core radius, the second is data of non-local s pseudopotential, and the third is data of non-local p pseudopotential.

  [CNKB.DAT]

  Head data

  421 data (core radius)

  421 data (s)

  421 data (p)

KB separable form

Y_s(r) :s pseudo-wave function

Y_p(r) :p pseudo-wave function

Vnl_s(r) :non-local s pseudopotential

Vnl_p(r) :non-local p pseudopotential

|Vnl_s(r)|Y_s(r)><Y_s(r)|Vnl_s(r)|
---------------------------------- separable form of non-local s
......... <Y_s(r)|Vnl_s(r)|Y_s(r)>

|Vnl_p(r)|Y_p(r)><Y_p(r)|Vnl_p(r)|
---------------------------------- separable form of non-local p
......... <Y_p(r)|Vnl_p(r)|Y_p(r)>

In CNKB.DAT, KB means non-local separable s and p data, above parts of |Vnl_s(p)(r)|Y_s(p)(r)> are stored.

Non-local parts of s and p are made as follows,

Vnl_s(r)=V_s(r)-V_d(r),

Vnl_p(r)=V_p(r)-V_d(r).

(Attention)Pseudo-wave function has already normalized. This means the pseudo-wave function includes (4*pai)**0.5 and r as follows,
At present(11/12,1998), this is checked again that really it is necessary to be (4*pai)**0.5 for normalization, or not. (9/8,2000)It is solved, please see below. The pseudo-wave function includes (4*pai)**0.5 and r.

P_s(r):real pseudo-wave function

Y_s(r)*Y_s(r)=4*pai*P_s(r)*P_s(r)*r*r.

The case for p is the same.
[As for 4*pai]
A formula of KB separable form (l = 0: s case) is as follows,

4*pai*(2l+1)*P_l(cos(z))*|Vnl_s(r)|Y_s(r)><Y_s(r)|Vnl_s(r)|/<Y_s(r)|Vnl_s(r)|Y_s(r)>

Since |Vnl_s(r)|Y_s(r)><Y_s(r)|Vnl_s(r)|, and <Y_s(r)|Vnl_s(r)|Y_s(r)> include 4*pai, respectively, they cancel each other. At the last, 4*pai of a head of above formula only remains.
(9/29,2000)It is not necessary to consider "they cancel each other". A new 4*pai never appear as "(4*pai)**2*(2l+1)*P_l(cos(z))" because Y_s(r) is defined to be normalized as Int[Y_s(r)*Y_s(r)]dr = Int[4*pai*P_s(r)*P_s(r)*r*r]dr = 1.
In CND.DAT, d pseudopotential V_d(r) and V_d(r)-Vcore(r) are stored.

Formats of data are almost same for CNKB.DAT. But, there are no data as coresponding to forth and fifth values at CNKB.DAT. After the log-mesh data, data for V_d(r) and V_d(r)-Vcore(r) are stored in order. A number of mesh points is also 421 as log-mesh, V_d(r) and Vcore(r).

In the electronic structure calculation, it is necessary to transform the local pseudopotential from a real space to a reciprocal space with an integration of a radial coordinate. In this integration, it is very difficult to perform numerically as for V_d(r) because of -Z/r form of it. In order to solve this problem, it is separated the V_d(r) integration as two parts. One is performed as for V_d(r)-Vcore(r) where this is vanished at r > rc (rc:cutoff radius), and it is possible to numerical integration accurately. The other integration is Vcore(r). It can be performed analytically.

(2) s, p and d non-local case [Please see (16 February, 2016)]

A number of files is two, because it is not necessary to prepare a local pseudopotential data file in this case.

Vnl_s(r)=V_s(r)-Vcore(r)

Vnl_p(r)=V_p(r)-Vcore(r)

Vnl_d(r)=V_d(r)-Vcore(r)

Vcore(r) is constructed in the same way of (1) as described BHS[3]. Parameters to make Vcore(r) is the last four values in the input data file for atomic coordinates and lattice parameters.
As for numerical DATA of the non-local pseudopotential, it is only added a non-local d part to (1) case.
Forth, fifth parameters in the second line as (1) change to three parameters for s, p and d of denominator of KB separable form in this case.

How to determine whether the data are for s, p, d(non-local) or s, p(non-local).
Please see the beginning part of non-local pseudopotential data as follows. 

  421  0    1.0416666666667D-02   -2.8163871763294   -3.3810025390407
  -0.46403666724182    1.1968402202890D-04    1.2348320071129D-04    1.27402978
27076D-04
    1.3144718292661D-04    1.3561976441886D-04    1.3992479786574D-04
In above data, the number of numerical values of -2.8163871763294 -3.3810025390407 -0.46403666724182 between 1.0416666666667D-02 and 1.1968402202890D-04 (the first of radial mesh data) is three and this implies the data is s, p, d(non-local). If this number is two, the data is s, p(non-local).

How to use data?

If you do not have a program which calculates electronic strutures by using pseudopotentials, this database has no any worth. At least, you have experiences of performing the electronic structure calculations by using the KB type separable form on the basis of plane waves(maybe mixed basis O.K.). Programs to treat these pseudopotential data exists at ISSP group and NAIR, in which they are independent of each other. Of course, I have a program, but it is not public now.
(6/2, 2000)At present, there is no distribution package of the electronic strcuture calculation which is available for using NCPS95,97.

You can find a program 'ps_input.f'. This is an example to read a pseudopotential data and to integrate a radial log-mesh. This file consists of two subroutines. One is 'PSEUDO' for local pseudopotentials, the other is 'KBMAT' for non-local pseudopotentials.
Furthermore, the author shows a source program of 'kbsi.f' to construct the KB separable non-local pseudopotential parts from output data of s,p and d(f) pseudopotentials. This program outputs the final data for NCPS95,97 (read in the ps_input.f).

Summary

The purpose of this database is to prepare the ghostless separable pseudopotentials. It will give a basic data to create and design new materials.
At present, separable pseudopotentials more than 60 kinds of atoms are available. It is possible to calculate the systems to be arbitrary combinations of atoms under a variety of environments(complex compounds, surfaces, interfaces, inpurities, vacancies, clusters, etc.). This will lead to prediction of new structures and/or new materials, chemical reactions and solid state properties.
The author hopes that as possible as many researcher will use this database in order to design and predict new materials.

Now, preparing the public version distributed by CD-R, if you have a interesting for this database, please send me an E-mail and see [information](E-mail address is shown at the last of this document.).

Attention to use NCPS95(97)!

NCPS95,97 is now beta-version(no warranty). It is forbidden to redistribute this database to any other users. The author will not be liable to 'NCPS95(97) users' for any loss or damages (for examples hard disk crush, mistakes of electronic band structure calculation, etc.) arising out of using this database (NCPS95,97) or from any defect or inaccuracy of NCPS95(97), inability to use NCPS95(97) or any other problems.

Acknowlegments

The author wishes to thank Dr. Y. Morikawa(JRCAT) for helpful assistance in the calculation partially for pseudopotential(Li, C, O) construction. A computer program for TM type pseudopotentials was coded by him. The original relativistic all-electron atom calculation program were coded by Prof. A. Hasegawa(University of Niigata). The author also wish to thank Prof. S. Tsuneyuki group(T. Ogitsu and K. Kusakabe[At present, Niigata univ.], ISSP) for valuable discussions. Some additional pseudopotentilas(Li,Ca) are constructed by their groups. Helpful and useful discussions and information for all-electron calculations by Dr. M. Arai(NIRIM) are gratefully acknowleged. The numerical calculations were mainly performed at NIRIM(DEC AlphaServer2100 4/200[Before 1/31, 1999], COMPAQ GS140[After 2/1, 1999]), and the author thanks the Supercomputer Center, Institute for Solid State Physics(ISSP), University of Tokyo for the facilites and use of FACOM VPP500, and the Center for Promotion of Computational Science and Engineering(CCSE, JAERI) for the facilites and use of FACOM VPP300, NEC SX4, Cray T94, etc.

Reference

(*1)In a pseudopotential calculation, there are other type ghost bands by using the Kleinman-Bylander separable form[8].

[1] D. R. Hamann, M. Schluter and C. Chiang, Phys. Rev. Lett.,43(20), 1494(1979)
[2] N. Troullier and J. L. Martins, Solid State Commun. 74, 613(1990)
;and Phys. Rev. B43, 1993(1991)
[3] G. B. Bachelet, D. R. Hamann and M. Schluter, Phys. Rev. B26(8), 4199(1982)[BHS]
[4] K. Kobayashi, Y. Morikawa, K. Terakura and S. Bluegel, Phys. Rev. B45, 3469(1992)
[5] K. Kobayashi, in proceedings of International Workshop on Computer Modeling and Simulation for Materials Design, 96(1996)
; K. Kobayashi, "Norm-conserving pseudopotential database(NCPS97)", Computational Material Science 14, 72(1999)
[6] S. G. Louie, S. Froyen and M. L. Cohen, Phys. Rev. B26, 1738(1982)
[7] A. D. Becke, Phys. Rev. A38, 3098(1988): J. P. Perdew, Phys. Rev. B33, 8822(1986)
[8] L. Kleinman and D. M. Bylander, Phys. Rev. Lett.,48(20), 1425(1982)
[9] T. Sasaki, A. M. Rappe and S. G. Louie, Phys. Rev. B52, 12760(1995)
[10] J. F. Janak, L. Morruzi and A. R. Williams, Phys. Rev. B12, 1257(1975)
[11] D. D. Koelling and B. N. Harmon, J. Phys. C: Solid State Physics 10, 3107
[12] E. P. Wigner, Phys. Rev., 46, 1002(1934)

Parameters for Vcore(r) are as follows. [Add parameters]
Formats of a1,a2,c1 correspond to Vcore(r) parameters of Table in the paper of BHS. c1 + c2 = 1. "A" and "B" indicate two types of pseudopotentials are prepared for the same atom. Except for above tabulated pseudopotentials, only s and p are non-local parts of pseudopotentials and the d potential is treated as local.
(*) means a revised Vcore parameters (7/29, 1998) in order to avoid ghost bands. A case of preparing two type pseudopotentials (A, B) is unified to one type of pseudopotential for Mn, V and Y.
Update
(August 7, 1996)
(Minor update 11/6, 1996)
(Slightly update 2/5, 1997)
(Minor update 2/28, 1997)
(Minor update 8/4, 1997)
(Very Minor update 12/16, 1997)
(Minor update 7/29, 1998)
(update 10/6, 1998)
(Minor update 11/20, 1998)
(Very Minor update 6/2, 2000)
(Minor update 8/3, 2000)
(Minor update 9/8, 2000)
(Very Minor update 9/29, 2000)
(Very Minor update 10/2, 2000)
(Very Minor update 3/25, 2001)
(Very Minor update 6/20, 2001)
(Very Minor update 4/4, 2002)
(Minor update 5/15, 2008)
(Very Minor update 9/21, 2010)
(Very Minor update 4/1, 2011)
(Very Minor update 7/9, 2012)
(Quite Minor update 11/9, 2012)
(Very Minor update 8/29, 2016)
[Contact Information]
305-0044 1-1 Namiki, Tsukuba-shi, Ibaraki, Japan

MPTG, MANA, National Institute for Materials Science(MANA/NIMS)

Kazuaki Kobayashi

Fax: (81)29-852-7449

(E-mail address)
E-mail: "kobayashi.kazuaki-@-nims.go.jp" [Please replace "-@-" with "@" for against spam-mail at 13. June, 2003.]

Web site http://www.nirim.go.jp/staff/kobayak/ (Closed!)

The Postal Code is 305-0044.
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