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A. P. W. HARTREE-FOCK CALCULATIONS IN INSULATING CRYSTALS
F. Perrot
To cite this version:
F. Perrot. A. P. W. HARTREE-FOCK CALCULATIONS IN INSULATING CRYSTALS. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-119-C3-122. �10.1051/jphyscol:1972316�. �jpa-00215050�
A. P. W. HARTREE-F'OCK CALCULATIONS IN INSULATING CRYSTALS
F. PERROT
Commissariat h 1'Energie Atomique, 94-Villeneuve-Saint-Georges, France
RCsum6. - La methode A. P. W. est utilisee pour le calcul des etats Blectroniques des gaz rares solides et des cristaux ioniques. Le potential coulombien et l'operateur #&change Hartree-Fock sont calcules a l'aide de << sommes de Bloch >> reprksentant de manikre approchk les Btats de valence. Le traitement sensiblement exact de l'echange apparait lors du calcul des dkrivees loga- rithmiques intervenant dans les klements de matrice A. P. W. dont I'expression formelle est inchan- gee.
Le gap optique et la largeur de la dernikre bande de valence obtenus dans Ne, Ar, LiCl, NaCl et KC1 sont compares a l'experience et a de precedents calculs avec echange local et Bchange Hartree- Fock. Les conclusions essentielles tirkes de ces derniers sont confirmees qualitativement malgre d'apprkiables &carts numeriques.
La validit6 de la methode est discutee et un ordre de grandeur des incertitudes numeriques est evalue.
Abstract. - The A. P. W. method is used to compute the electronic states in solid rare gases and ionic crystals. The Coulomb part of the potential and the Hartree-Fock exchange operator are calculated using an approximate Bloch representation of the valence states. The nearly exact treatment of exchange leads to modified equations which must be solved to obtain the logarithmic derivatives included in the A. P. W. matrix elements.
The results for the optical gap and width of the highest valence band in Ne, Ar, LiCl, NaCl and KC1 are compared with experiment and calculations using a local exchange. They are also compared with previous calculations using the H. F. exchange : the main qualitative conclusions are confirmed though important quantitative differences are noted.
The validity of the approximations is then discussed and the resulting uncertainty evaluated.
I. Introduction. - Most of the band structure calculations use the Hartree-Fock-Slater approxima- tion (or some local exchange approximation) : the Hartree-Fock procedure is indeed too difficult to be carried out with some generality in solids. Weverthe- less, a {( tight binding )> calculation of KC1 with exact exchange has been given in a paper by Howland in 1958 [I]. More recently, several Hartree-Fock band structure calculations on Ar [2], K r [3], LiCl [4] and NaCl [S] have been published by Fowler, Kunz and Lipari. All these calculations apply to insulating crys- tals for the following reasons :
1) A good approximation to the valence wave functions is obtained by Bloch sums built with orbitals having the free atom (or free ion) symmetry. The calculation of the exchange integrals in then easier.
2) An analytical summation over the occupied states can be performed in the Bloch representation when the bands are filled.
3) A procedure, derived from the classical electro- statics, is able to take into account the polarization effects in insulators and to give necessary corrections to the H. F. results for comparison with experiment.
In Section 11, we propose a new nearly-exact treat- ment of the H. F. exchange for insulators in the well known A. P. W. scheme. Some results are compared with previous ones and with experiment in section 111.
The validity of the approximations is discussed in Section IV.
11. A. P. W. H. F. method. - We are not dealing with a self-consistent solution of the H. F. equation but only with an approximate one, based on two different representations of the valence states. In the H. F.
equations for the state of wave vector k appear : a) A summation over all the occupied states kt needed to calculate the Coulomb part of the potential and the exchange operator. Bloch sums are used for the representation of these valence states.
b) The wave function of the k state (valence or conduction state). This function is expanded in aug- mented plane waves :
$dr) = C: fpk(r) d i p ~ : ( r ) Y ~ ( G )
P AP
f is the A. P. W. of vector k + K,.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972316
C3-120 F. PERROT
1) APPROXIMATE REPRESENTATION OF THE VALENCE
STATES. - The %loch sum over the Nlattice vectorsx, :
is the exact wave function for a state of wave vector k in a filled band [6]. q j is a Wannier function which reduces to a free atom (or free ion) orbital q{lm in the limit of non-overlapping atoms (zero band width).
We assume that, for valence bands in insulating crystals, the true Wannier functions yj(r - X,) are well approximated by non overlapping functions qnlm(r - X,), with atomic symmetry and a radial part Pn, to be determined. The equations to be satisfied by the Pnl are obtained from the variational principle, applied to the total energy calculated with one electron wave functions of type (2). According to the non-over- lapping assumption, there is no exchange between functions centered at different sites and the average effect of the whole lattice on one ion is just a constant shift of the potential. So the equations for the Pnl are similar to the H. F. equations for the corres- ponding complete-shell free ion, but must be solved in a finite sphere Q. The radius of this sphere is chosen to be proportional to the mean value of r in the outer shell of the free ion and the sum of the ionic volumes is equal to the total volume of the solid.
The self-consistent solution of the equation gives the Pnl,i (i for each kind of ion) which are different from the radial parts of the free ion, owing to the ((pres- sure effect )).
2) COULOMB PART OF THE POTENTIAL. - The Cou- lomb potential used in the A. P. W. H. F. method is a
cc muffin-tin )) potential. The A. P. W. spheres Si (with radius proportional to the ionic sphere radius) just touch at the cell boundaries (if possible) but never overlap. Within each sphere, the electronic potential is solution of Poisson's equation with the corresponding spherical electronic density :
The constant potential outside the spheres is obtain- ed by the method of De Cicco, using the (( Ewald problem )) technique [7], [8].
3) EXCHANGE OPERATOR. - The exchange energy for a state of wave vector k is :
x $ 2 j i 1 2 (4)
As assumed above, the wave function of the calculated state k is an A. P. W. expansion of type (I), but the occupied states are Bloch sums @k,,j,i of type (2) (i refers to the kind of ion). A detailed treatment of the
integrals has been given in reference [9]. Neglecting the contribution to the integrals of the region outside the A. P. W. spheres, where @k,,j,i is small, the final expression of E; is :
L
E: = -
z z
c ~ t ( k )Sj
P n l , i ( ~ ~ ) R:(P~) xn l i I' L Si P >
x Pn,,i(~2) RIS(P~) P: dp1 P; dp2 ( 5 ) with
and
The integration in (5) are now to be performed in one A. P. W. sphere Si. The main feature of this result is that, as a consequence of the complete-shell structure of the ions, there is no matrix elements of the exchange operator between radial parts R: with different orbi- tal quantum numbers.
4) CALCULATION OF THE LOGARITHMIC DERIVA- TIVES. - Using eq. (5) for the exchange energy, the A. P. W. radial part R: within a given ion i is solution of the following equation
= C C ~ ~ ( 1 0 , Ib) P n , , , , i ( ~ > x
n'l' L
In the classical A. P. W. method proposed by Slater, the corresponding equation is homogeneous with a local exchange. Eq. (6) is solved by iterations. The logarith- mic derivative d&/dp (Si), computed on the A. P. W.
sphere as a function of energy Ek; is then introduced ina standard A. P. W. program.
It can be shown that two states of same reduced wave vector are orthogonal.
111. Results. - We shall restrict ourselves here to give the values of AE, the band gap, and of A&, the highest-band width, for solid rare gases Ne and Ar and the ionic compounds LiCI, NaCl and KCI. (The highest valence bands are 2 p in Ne, 3 p in Ar, and 3 p C1- in the ionic crystals.) The results are listed in Table I.
We cannot give here a detailed discussion of the results but only their most important features.
1) COMPARISON WITH EXPERIMENT. - All the A. P. W. H. F. gaps are much wider than the experi- mental ones (from 4 eV in Ne to 6 eV in LiC1). This
Slater Exchange Other H. F.
Calcul.
Exp. 21.42 14.16 9.4 8.97 8.5
- - - -
-
0.65Values of the band gap BE (upper line) and band width AE (lower line) in eV. (") Ref. [lo]. (b) Ref. [Ill.
(? Ref. [12]. ( d ) Ref. [2]. (? Ref. 141. (f) Ref. [5]. (g) Ref. (11.
proves the very great importance of the correlation effects and gives a qualitative confirmation to the results of Fowler, Kunz and Lipari. It has been shown in a previous paper [9] that a satisfying explanation of the difference between A. P. W. H. F. and experimen- tal gaps in solid rare gases can be obtained by a semi- classical treatment of the polarization effects and an empirical computation of the correlation energy of the highest valence level. A similar work for the ionic compounds is in progress.
Concerning the band width, the comparison is uneasy because experimental data are rare and inac- curate. Our value for KC1 compares favorably with experiment.
2) COMPARISON WITH CALCULATIONS USING SLATER
EXCHANGE. -The band gaps computed with Slater exchange by Mattheiss [lo] in Ar (A. P. W.), A. B.
Kunz [ I l l in the three ionic compounds (0. P. W.) and P. D. De Cicco 1121 in KC1 (A. P. W.) are closer to the experimental values than ours. This is a general result for calculations using a local exchange. Except in KCl, the H. F. S. band width As is smaller than ours.
3) COMPARISON WITH OTHER H. F. CALCULATIONS. -
The gaps given by Fowler, Kunz and Lipari, though wider than the H. F. S. ones, are smaller than ours in all cases. Their valence band widths are always greater.
There is a significant difference between the two series of H. F. results and an explanation would be useful.
IV. Discussion of the A. P. W. H. F. approximations.
- We shall now discuss the influence on our results of the three approximations made in Section 11.
1) The cr double representation )> of the valence states is not self-consistent since the functions q,,, are not exact. An accurate calculation of the correc- tions due to self-consistency is not possible, but their importance can be evaluated in the Hartree-Fock- Slater approximation, using De Cicco's results on KC1 [12]. The first iteration density of De Cicco is a superposition p:FS of free-ion densities ; his self-
consistent density is p!ES. The corrections resulting
HFS H F S
from the density variation ps, -ps are - 0.40 eV on the gap and + 0.07 eV on the C1-3 p band width.
We have performed a calculation in KC1 using the density p F S (Hartree-Fock-Slater density for single
(( compressed )) ions) : the corrections were found
+ 0.41 eV on the gap and - 0.07 eV when going from pYS to pEFS. Hence, the total correction in the pro- cess pEFS + ptFS -+ ':!p is + 0.01 eV for the gap and 0. eV for the band width. The reason why cancella- tions occur is that p, is greater than p, in the region outside the sphere while p, is smaller than p,, in this region. Of course, we do not assume that these correc- tions are adequate for the H. F. density variation
p?:-pEF. But we may think that pEF is a very rea- sonable density for a non self-consistent calculation.
2) The non-spherical part of the Coulomb potential gives corrections which have been evaluated, from De Cicco's H. F. S. results in KCI, to + 0.06 eV on the gap and to less than 0.01 eV on the band width.
The corrections in the H. F. case would certainly have the same magnitude.
3) The contribution of the region outside the A. P. W. spheres to the exchange integral (5) is neglect- ed in our method, This contribution gas been esti- mated to + 0.04 eV in Ar + 0.5 eV in KC1 for the band gap ; it is negligible for the band width.
Looking at the order of magnitude of the various numbers involved in the corrections, it seems rzason- able to think that the total error made on the band gap is probably lower than 1 eV and certainly much smaller in the rare gases than in the ionic compounds.
The uncertainty on the band width is likely not greater than some tenths of eV.
V. Conclusion. - The A. P. W. H. F. method can be applied in a rather simple way to the solid rare gases and ionic crystals. It proves the great importance of correlation effects in the insulating solids. The results, though they lead to similar qualitative conclusions, are not in numerical agreement with previous H. F.
calculations.
C3-122 F. PERROT
References
El] HOWLAND (L. P.), Phys. Rev., 1958,109, 1927. [7] SLATER (J. C.) and DE CICCO (P.), Quarterly Progress 121 LIPARI (N. 0 . ) and FOWLER (W. B.), Phys. Rev., 1970, Report MIT, 1963, 50, 46.
B 2, 3354. [8] DE CICCO (P.), Quarterly Progress Report MIT, 1964, 133 LIPARI (N. O.), Phys. Stat. Sol., 1970, 40, 691. 53, 73.
[4] KUNZ (A. B.), J. Phys. C : Sol. St. Phys., 1970,3,1542. [9] D A G E N ~ (L.) and PERROT (F.), Phys. Rev. (to be [5] LIPARI (N. 0 . ) and KUNZ (A. B.), Phys. Rev., 1971, published).
B 3, 491. [lo] MATTHEISS (L. F.), Phys. Rev., 1964, 133, 1399.
[6] ZIMAN (J. M.), Principles of the Theory of Solids [ l l ] KUNZ (A. B.), Phys. Rev., 1968,175, 1147.
(Cambridge UP, Cambridge, 1965). [I23 DE CICCO (P.), Phys. Rev., 1967,153,931.
