REVISTA COLOMBIANA DE F´ISICA, VOL.38, No.1, 2006
NORM CONSERVING PSEUDOPOTENTIALS FOR C, N AND O ATOMS C. Espejo1,a,b ,Rafael R. Rey-Gonz´ alezc Departamento de Ciencias B´ asicas, Universidad Jorge Tadeo Lozano b Departamento de F´ısica, Universidad Nacional de Colombia c Departamento de F´ısica,Universidad Nacional de Colombia (Recibido 20 de Sep.2005; Aceptado 25 de Ene.2006; Publicado 05 de Abr.2006) a
RESUMEN
En este trabajo hacemos una breve revisi´ on de la teor´ıa de los pseudopotenciales que conservan la norma. A partir de las auto energ´ıas y funciones propias de c´ alculos autoconsistentes realizados para los ´ atomos C, N y O se obtienen los correspondientes pseudopotenciales y pseudofunciones de onda. Palabras claves: Potenciales no conservativos, SIESTA, DFT.
ABSTRACT
A brief review of norm conserving pseudopotentials theory and its application for C, N and O atoms is made. From the eigenvalues and eigefunctions of all-electron calculations performed on these atoms the pseudopotentials and pseudo wavefunctions are obtained. Keywords: Norm conserving pseudopotentials, SIESTA, DFT.
1.
Introduction
In the last decade great advances involved with first-principles calculations on systems with large number of atoms have been developed, allowing reliable simulations of complex molecules and solid systems. Of particular interest is the fully self-consistent density functional method known as SIESTA [1], this method is said to be of order-N which means that the computational cost scales linearly with the number of atoms in the system. The latter represents an important step forward since in traditional ab initio calculations the computational cost scales as N 3 [2, 3]. Order-N scaling is due to several approximations, one of them is that core electrons are replaced by norm-conserving pseudopotentials [4] in their fully nonlocal form (Kleynman-Bylander form) [5]. Hence pseudopotentials are important in any implementation of SIESTA code. Our purpose with this paper is to take a glance of pseudopotential theory and show its application to C, N and O atoms. The pseudopotentials obtained here are non relativistic and pseudo wavefunctions are not spin-polarized.
2.
Norm-conserving pseudopotentials
Pseudopotentials constitutes a possible way of avoid atomic core states in firstprinciples calculations of molecules and solids. The need for avoid core states arises in view of both the expensive work of taking into account all atomic states in such calculations, and that chemical bonds in molecules and solids involve valence states to a major extent. Ab initio pseudopotential methods consider that valence electrons move in the electric potential produced by atomic nuclei together with core electrons. So the total effect 1 email:
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(Coulomb and exchange-corrrelation interactions) of core electrons may be replaced by an adequate potential: the pseudopotential. The early calculations of first-principles pseudopotentials were made within the scheme of OPW (orthogonalized plane wave) atomic calculations. The pseudopotemtials obtained in this way are strongly repulsive near the nuclei and the corresponding wavefunctions generally present the correct shape outside the atomic core but differ from correct eigenfunctions by a normalization factor [5]. In order to overcome these problems Hamann, Schl"uter and Chiang (HSC) proposed a model pseudopotential [4] that allows to obtain pseudo wave functions which are identical to real wave functions beyond a chosen core radius, rc , whose eigenvalue agree with the real energy eigenvalue. Aditionally, integrals form 0 to r of the real and pseudo charge densities agree for rgt; rc for each valence state. This property, known as norm conservation, assures that electrostatic potential caused by pseudo and real charge densities are equal outside core region. This property is satisfied if logarithmic derivatives of the real and pseudo wave function, and their first energy derivatives agree for rgt; rc [4]. Following HSC, formerly we need to perform an ab initio self-consistent all-electron atomic calculation where exchange and correlation potential are treated within local approximation of DFT. This calculation is made for a chosen atomic configuration. Pseudopotentials are obtained from all-electron potential V (r), real eigenvalues l and eigenfunctions ul (r) = rφl (r) of valence states, in the following way [4]. A cutoff function f (x) is chosen, such that f (x) → 0 as x → ∞, f (x) → 1 as x → 0, and cuts off for x ∼ 1, for example f (x) = exp(−x4 ). Then a guess potential is generated for each angular momentum: V1lps = [1 − f (r/rcl )]V (r) + cl f (r/rcl ),
(1)
where rcl is the chosen cutoff radius for angular momentum l. Note that V1l → V (r) for rgt; rcl . The nodeless solution w1l of the Schr"odinger with V1l must be proportional to ul because both satisfies the same differential equation and boundary condition for rgt; rcl : γl w1l (r) → ul (r). The constant cl is adjusted so that the correponding eigenvalue 1l of w1l be equal to the real eigenvalue l . The wavefunction is then modified in order to be identical to ul : w2l = γl [w1l − δl g(r/rcl )], (2) where g(r) behaves as xl+1 as x → 0 and approches 0 as xgt; 1, for example g(x) = xl+1 exp(−x4 ). The constant δl is adjusted from the normalization of w2l . The pseudopotential V2l is obtained by inverting the Schr"odinger equation whose nodeless solution is w2l with eigenvalue l. To construct the bare ion pseudopotentials the coulomb and exchange-correlation potential produced by the pseudo charge density are calculated and substracted from each V2l , hence the ionic pseudopotentials tends to −2zv /r for large values of r. The use of the obtained pseudopotentials in abinitio molecular or solid calculations depends on its transferability, that is if the pseudopotentials can reproduce eigenvalues and eigenfunctions corresponding to several different atomic configurations within a small error range. Generally this error range must be of 1 mRy.
3.
Results on C, N and O
Norm conserving pseudo potentials were calculated for carbon, nitrogen and oxygen atoms using the method proposed by HSC. All-electron self-consistent calculations and pseudopotentials generation were carried out by means of the ATOM program delivered within SIESTA package [6]. Calculated pseudopotentials are shown in figure 1, and 408
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Nitrogen
Carbon
Oxygen
15 l=0
5
l=0 l=1
-5
ps
Vl (Ry)
l=0 l=1
l=1 -Zv/r
-15 l=2
-25 l=2
-35
-55
-Zv/r
-Zv/r
-45 0
0,5
1
1,5
2
l=2
1
0,5
1,5
2
0,5
1
1,5
2
r (a.u.) Figura 1: Bare ion pseudopotentias for C, N and O atoms. In all cases Vlps → −Zv /r for rgt; rcl , where Zv is the valence charge of the atom.
transferability test results are shown in table 1. From the graphics it is clear that all bare ion pseudopotentials are soft-core, this means that there is not strongly repulsive behavior near de origin. As can be seen from the table, obtained pseudopotentials reproduce both ground state energies of the atoms and single ionized state energies within an error range ∼ 0,0005 Ry. The atomic configurations used to generate pseudopotentials were C 2s2 2p1 3d1 , N 2s2 2p2 3d1 , and O 2s2 2p3 3d1 . Energy (Ry) Configuration s C 2s2 2p2 −1,00187 C 2s2 2p1 3d1 −1,64545 −1,88102 C 2s2 2p1 N 2s2 2p3 −1,35212 N 2s2 2p2 3d1 −2,13732 −2,36822 N 2s2 2p2 O 2s2 2p4 −1,74213 −2,66470 O 2s2 2p3 3d1 O 2s2 2p3 −2,89159
p −0,39910 −1,02399 −1,25868 −0,53287 −1,29994 −1,53032 −0,67660 −1,58086 −1,80743
d ... −0,05258 ... ... −0,05278 ... ... −0,05294 ...
∆E(Ry) s 0,00007 −0,00002 0,00004 0,00011 −0,00002 0,00005 0,00024 ≈0 0,00023
p −0,00051 −000002 0,00003 −0,00025 −0,00003 0,00005 −0,00002 ≈0 0,00022
d ... ≈0 ... ... ≈0 ... ... ≈0 ...
Tabla 1: Transferability test. Atomic eigenvalues for obtained pseudopotentials and their difference ∆E from real atomic eigenvalues.
4.
Conclusions
The method proposed by Hamann, Schl¨ uter and Chiang can be used to generate norm conserving pseudopotentials with high degree of transferability. Optimus cut off radius 409
REFERENCIAS
REVISTA COLOMBIANA DE F´ISICA, VOL.38, No.1, 2006
were found near 0,50rml for l = 0, 1 and near 0,3rml for l = 2, where rml is the radius of the outermost peak of ul . ATOM program was employed to generate norm conserving pseudopotentials which will be used further in small bio-molecules calculations with SIESTA. Authors acknowledge Universidad Nacional de Colombia and Colciencias for their support.
Referencias [1] J. M. Soler, E. Artacho, J. D. Gale, A. Garc´ıa, J. Junquera, P. Ordej´ on, and D. S´ anchez-Portal, J. Phys.: Condens. Matt. 14, 2745-2779 (2002). [2] D S´ anchez-Portal, P. Ordej´ on, E. Artacho, and J. Soler, Int. J. Quantum Chem. 65, 453 (1997). [3] P. Ordej´ on, E. Artacho and J. M. Soler, Phys. Rev. B (Rapid Comm.) 53, R1044110443 (1996). [4] D. R. Hamann, M. Schl"uter, and C. Chiang, Phys. Rev. Lett. 43, 1494-1497 (1979). [5] L. Kleinman and D. M. Bylander, Phys. Rev. Lett. 48, 1425-1428 (1982). [6] http://www.uam.es/departamentos/ciencias/fismateriac/siesta/
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