M.J. CONDENSED MATTER VOLUME 4, NUMBER 1 1 DECEMBER 2001
* qacho@ hotmail.com 4 14 2001 The Moroccan Statistical Physical and Condensed Matter Society The scalar-relativistic effects of the transferability criterion
of aPseudopotential M. Nejjar and A. Qachaou*
Laboratoire de Physique de la matière condensée (LPMC) B.P : 133-14000 Kenitra- Morocco
For the purpose to improve the precision of the transferability criterion of a norm-conserving pseudopotential, in the case of heavy atoms, we have established a new expression, that takes into account the scalar-relativistic effects.
The brought correction can be extensive directly in the case of linear methods of bands structure.
I. INTRODUCTION
The pseudopotential (PP) simplify electronic structure calculations by eliminating the need to include atomic core states and the strong potentials responsible for binding them. The widely used norm-conserving pseudopotentials have several desirable properties1,13. They are calculated from ab initio self-consistent atomic potentials based on local-density-functional (LDF) theory.
They produce nodeless valence wave functions which converge to become identical to normalized full-potential wave functions beyond a chosen “core radius” Rc, and are themselves properly normalized. This latter property is essential to produce a correct bonding description in PP calculations, and correct self-consistent electrostatic and exchange-correlation potentials. Simultaneously, norm- conserving PP reproduce the scattering power of the full atom potential correctly at energies away from the bound- valence-state energy (to first order in the energy difference). This criterion expressed in the non-relativistic case, has been applied in the scalar-relativistic case by D. R. Hamann2. In this work, we present a new expression of the transferability criterion that manages the scattering power of PP using theShrodinger scalar-relativistic equation . The relativistic effects have been neglected previously by Kleinman10.
II. TRANSFERABILITY OF PSEUDOPOTENTIAL The PP ability to reproduce a single atomic state alone does not make it useful. To be useful, the core portion of the pseudopotential must be transferable to other situations where the external potential has changed. A identity expressing the transferability criterion has been introduced
3,4, as (in atomic units):
( ) ln
2 1
0 2 2 2
∫
=
−
RR
dr dr r
d d
r d φ φ
φ ε
(1)φ is the solution of the radial Shrodinger equation at energy ε (not necessarely an eigenvalue) which is regular at the origin. The radial logarithmic derivative of φ is simply related to the scattering phase shift. The consequence of (1) is that if two potentials V1 and V2 yield solutions φ1 and φ2which have the same integrated
charge inside a sphere of radius R [for φ1(R)= φ2(R)], the linear energetic variation, around ε, of their scattering phase shift (at R) is also identical. The requirement on the atomic pseudo-wave-function that it agree identically whith the full wave function for R>Rc , and thus that the scattering properties of the PP and full potential have the same energetic variation to first order when transferred to other systems5. A good PP will well reproduce the eigenvalues and the eigenfunctions in the valence region (beyond Rc) if its enegy versus logarithmic derivative curve at Rc, of the pseudo-wave-function is sufficiently close the true energy versus logarithmic derivative curve in the range of eigenvalues we are interested in.
Generally, the transferability of PP is easily tested by doing calculations for the atom in different ionized and excited states using the PP constructed for the reference configuration (which is usually the neutral atomic ground state) and comparing the result with the self-consistent all-electron results6, thus the equation (1) control the width of the permitted energy interval in the transferability of PP.
III. TRANSFERABILITY CRITERION IN THE SCALAR-RELATIVISTIC CASE
We have used a similar step to that developed by M.T.Yin and M. L.Cohen in the non-relativistic case7. The scalar-relativistic version of the Shrodinger equation 8, used in this work, contains the mass-velocity and Darwin corrections but it is averaged over the spin-orbit term16. The equation satisfied by u(r,E), which is r times the radial wave function, is (in atomic units):
0 ) ( ) 2 1 (
2
2 2 2 2
=
+ + −
+
−
−
−
u E V r M
L L
r u dr du dr dV M dr
u
d α
(2)
where,
), E V 2 ( 1 1
M= − α2 − and α is the fine structure constant, V is the atomic potential and L is the angular momentum.
4 THE SCALAR-RELATIVISTIC EFFECTS OF THE TRANSFERABILITY CRITERION … 15
In the case of a system perturbed to the first order, the equation that manages the solution u1=u+δu at energy
E E
E1= +δ is formulated as follows:
0 ) (
) 2 1 (
2
1 1 1 2 1
1 1 1 1 2 2
1 2
=
+ + −
+
−
−
−
u E V r M
L L
r u dr du dr dV M dr
u
d α
(3)
with perturbed magnitudes expressed in the form V
V
V1= +δ and MM1 =M+δ
For R0≤r≤ (R is a variable ray of a sphere, which will be able to play the Rc role), with the boundary conditions:
u
( 0 )
=u1( 0 )
=0
(4) it can be shown (Appendix) that to first order and by neglecting the term depending on δV(r) we obtain :8 '
4 ln )
(
ln )
, 2 ( 1
0 2 4
0
2 2
0
2 2
0 2 2
∫
∫
∫
∫
− +
+
∂
− ∂
−
+
−
=
∂
∂
R
R
r R
R
R
r udr u u dr dV M
dr dr u
d u E
dr dV dr M
u E V
dr u dr u
d E E R u
α α α
(5) The equation (5) represents the desired transferability equation that takes account implicitly the masse-velocity and the Darwin relativistic corrections.
IV. DISCUSSION
In a more precise calculation of electronic structure in solids constituted by heavy atoms, it is necessary to account for relativistic effects 9. Through the analysis of relativistic terms that appear in the equation (2) we notice that all relativistic effects are very more important near the nucleus . Given that electrons of valence participating to electronic properties of studied materials have very weak probabilities to be near the nucleus, it ensues that the relativistic effects would have to be more important for atomic states that for states of bands. This is not entirely exact because for an energy E the wave function
) E , r (
u has a similar shape of the corresponding atomic wave function. However, the formation of a solid produces a compression of atomic charge densities in atomic sphere.
This phenomenon can be described mainly by the different conditions of the normalization used in both cases. This normalization means that the amplitude of u(r,E) is very more important in the case of a solid than for the atomic wave function and consequently displacement of energies due to relativistic effects are very large in amplitude for solids that for atoms, for the same states.
The kinetic relativistic effect has been used in the calculation of atomic electronic structure. It has been introduced in the construction of norm-conserved pseudopotential from the Dirac equation14. To use the scalar-relativistic version of the Schrodinger equation preferably than the Dirac equation is more practical, because in the first one the relativistic effects appear only under the form of additive terms to non-relativistic term, whereas in the second the relativistic effect appear implicitly in the Dirac equation. The even remark could have be made concerning the expression (5) above in comparison with the equation (8) of the reference14. A terms appearing in the second member of the equation (5), define the relativistic effects on the transferability criterion. Thus, the fourth term remains the weakest looking the effect of the α4term. The second term represents the relativistic effects on the normalization of the charge density inside the sphere of ray R, while the third term puts in obviousness the dependence in energy of the logarithmic derivative of the pseudo-wave function in the same sphere.
We notice that the rule of transferability obtained in this work constitutes a generalisation of the expression obtained previously in the non-relativistic case. Indeed the expression (1) deduces easily from the equation (5) by a simple expulsion of all terms containing the fine structure
α.
The relativistic terms that appearing in the new rule of transferability depend all on the potential V (r). These terms modify strongly the non-relativistic contribution in the atomic potentials, when one takes into account the Coulomb strong attraction at the vicinity of the atomic nucleus.
For the PP, following the standard steps of an atomic PP construction 5, we obtain the truncated PP at the origin.
The intensity of this truncation is the main property that classifies between the hard and the soft PP. Generally, we use a PP in a solid under a non-local form (depending on the angular momentum character of the wave function), this affects directly characteristics of relativistic terms in the equation (5). Specially, systems which have 4f valence wave functions, requiring deep PP because these wave functions are nodeless in the radial direction and the PP cannot soften the electron-nuclear interaction by eliminating oscillations in the wave function and thereby reducing the corresponding kinetic energy.
The methods of band structure based on muffin-tin potential such that LAPW 11,12 and LMTO 11 depend essentially on the logarithmic derivative of the wave function at the muffin-tin ray. These methods are not habitually used for PP, because the space of energy over which the logarithmic derivative of PP will have to follow meticulously the trace of the full potential and to obtain a good transferability of the PP, represents the interval of energy where bands of the PP must be precise. The construction of norm-conserving PP does not guarantee directly that the logarithmic derivatives of the pseudo and
16 M. NEJJAR AND A QACHAOU 4
the true wave functions meet on all the previous energetic interval. Thus, all attempt to reconcile the LAPW method with the PP approach 12,15 for systems containing heavy atoms, would have to take account relativistic terms added in the transferability criterion presented here.
V. CONCLUSION
We have deduced a PP transferability rule expression including the scalar-relativistic effects. This criterion could be applied to the solids containing heavy atoms.
APPENDIX Let us consider the integral
(A1) dr u ) E V ( M 2 r
) 1 L ( L
r - 1 dr
d dr dV M dr 2 u d I
2 1 2 2 R 2 0
1
+ + −
− +α
=∫
From equation (1), we have to first order in uδ ,
(A2) udr ) E V ( M r 2
) 1 L ( L
r 1 dr
d dr dV M dr 2 u d I
2 R 0
2 2 2
δ
+ + −
∫ −
−
+ α
=
After an integration by parties, we have the expression:
[ ] [ u u ' u ' u ] dr (A3)
dr dV M 2 u ' u ' u u I
R 0 R 2
0 + ∫ α δ − δ
δ
− δ
=
Furthermore, by using the equation (2), the equation (A1) becomes
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[ 1 ( V E ) ] u dr (A4)
) E V ( 2
r dr u dr du dr
) V ( d M ) 2 V E dr ( dV M 4 u I
R 2 0
2 R
0
2 2
2
∫ δ −δ −α −
∫ +
−
δ −δ −α δ
= α
Taking account of boundary conditions (Equation (4)), and combining the two expressions (A3) and (A4), we can write
[ ]
(A5) ) V ( T r udr ' u dr u dV M 8 E
dr u ' u ' u dr u dV M 4
dr u ) E V ( E dr u u E
' u u
2 1
R
0 2
4 R
0 2
R 0
2 R 2
0 2 R R
2
∫ + δ
− δ α
∫ α δ − δ +
∫α − − δ
∫ + δ
−
=
δ
with )T(δV is the function defined by
[
1 (V E)]
u dr (A6)V
r dr ' u dr u
) V ( d M V 4 dr dV M 8 u ) V ( T
R 0
2 2
R 0
2 2
4
∫δ −α −
∫ +
−
α δ +α δ
−
= δ
The final result, in which the multiplicative constant can be factorized out, is independent of the auxiliary boundary conditions.
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