V : . : ; . . I OFL ORNL P 2193 no : , JE W , o : . . . . . in SO } --- TEET 10 || 1.25 1.4 w MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS -1963 ODNL 1 -17 Camas 2193 COUF-610529-da old, ww se MASTER JUN 27 100 66-63 Note: This is a draft of a paper which will be presented at i correr- ence on The Calculation of the properties of Vacancies and Interstitials, May 1-5, 1966. Proceedings of the conference will be published. Contents of this draft should not be quoted or referred to without permission of the authors. Methods for Calculation of the Electronic Structure of Defects in Insulators U. Öpik and R. F. Wood LEGAL NOTICE This report no sepadu un account of Goverratat sponsored work. Neither who waited Stalos, nor the Commission, nor way person uting on behalf of the Commission: A. Makes may warranty or representation, expressed or imolled, with respect to the accu- racy, comploteness, or usefulness of the laformation contained in this report, or that the wo of way laformadoa, apparatus, mottod, or process disclosed in this roport may not Infringe privately owned rights; or B. Ascumas day Madiuues with mospect to the use of, or lor dainagu rumulting from the use of any info. nation, appurata, method, or proces disclosed la this report. de wind to the above, "person scttag on badall of the Commissioo" lacludes my on- plogue o contractor of the Commission, or employee of much coatructor, to the extent het sach employs or contractor of a Commission, or employme of much coatric or preparu, diseminatas, or provides acces. d), any Information partant to be employmert or coolnict with the Commission, or wo employ meat with such contractor. RELEASED FOR ANNOUNCEMENT IN NUCLEAR SCIENCE ABSTRACTS Solid State Division Oak Ridge Nationai Laboratory operated by Union Carbide Corporation for the U.S. Atomic Energy Commission Oak Ridge, Tennessee April 1966 ..mer om .. "Walk V AVI N ithin dis homething that he w here come to e et to the one Methods for Calculation of the Electronic Structure of Defects in Insulators U. ÖPikt and R. ). WOOD Solid State Division, Oak Ridge National Laboratory Oak Ridge, Tennessee The comments which we shall make in this paper apply most directly to calculations of the electronic structure of the r center in alkali halides. We expect them, however, to be valid for a large number of de- fects in insulators, such as the U, M, ani R centers in ionic crystals, the hydrogen interstitial in CaF, an yttriuin ion in CaF, etc. In calculating the electronic structure of the i center in the alkali halides two methods of representing the defect wave function, WA, have generally been used. In the LCAO-DO method the defect orbital (DO) is expressed as a linear combination of atomic orbitals (LCAO) on the neighboring ions. Thus, - We = L Cuisi(I - BU, (1) V.) . . 'N where $(r - Rj) is the ;-th unoccupied atomic orbital on the v-th icn. This leads to a secular equation Tcvjlnlw,k) - (vejlu,k)el = 0 (2) for the one-electron energies, €. The difficulty with this method is that the functions duri are quite diffuse, whereas vo will usually be rather Research sponsored by the U. S. Atomic Energy Commission under contract with Union Carbide Corporation. "Visiting scientist from Queens University, Belfast, North Ireland. -2- well localized and hence the sum in Eq. (1) must contain several orbitals on each ion in order to give the proper degree of localization. In addi- tion to the orthogonality problems encountered in evaluating (v, j1w,k) une has a somewhat related problem in calculating the matrix elements of the Hamiltonian, (v,jlnlu,k), in that they may have large contributions from regions of space in which ys itself is practically vanishing. The point which we wish to make here is neither new nor surprising, but perhaps neeus reemphasizing occasionally. Stated very simply, it is that it is not economi- cal to express a localized function in terms of a basis set of diffuse func- ........ ..menas ... tions, and conversely. Perhaps the most familiar case of this principle ..... occurs in band theory, where the OPW method circumvents the need to express the highly localized core states in terms of plane waves. In calculating We for defects in alkali halide crystals, it seems more useful and natural to use what we shall call the defect centered (DC) method. Ir. this method, yg is written as va(I) = N($,(?) - ļolv,j)«,j(?)] , (3) vej in which $ (r) is a fairly smooth function which may be chosen in various ways. The sum over v,j ensures that i wiil be orthogonal to the occupiedi ion orbitals, (I) = 0; (F - Rj). Equation (3) assumes, in effect, thai no 0,,, do not overlap with each other, and this simplification inust be investi.' gated in each case. If these overlaps are not negligible, it mighi prove convenient to first use the method of symmetrical orthogonalization to trans- form to a set of mutually orthogonal core functions. Expressing yg as in Eq. (3) and evaluating (Valhlus), one obtains in addition to the familiar point ion approximation a contribution from exchange www .1 " 'J I ' . . . I. .TUA VIA A - 3- with the core orbitals, a coulomb term due to penetration of , into the core region, and a contribution, arising from the overlap terms in Eq. (3), which we shall refer to as the overlap energy. Table I shows the magnitude of these contributions in a number of cases. Since these terms are not negligible and do not even approximately cancel each other, we now wish to consider methods of calculating them. In fact, at the present level of approximation, only the exchange terms present a really difficult problem. A part of the one-electron Hamiltonian is the exchange operator, hex, defined by vi (r')u(r') hexu(?) = { { ¢vi (t) sviprai dr (4) where all wave functions are assumed real. To understand the difficulty, consider that each of the , in Eq. (4) is supposed to be a Hartree-Fock free-ion orbital. The preferred method of calculating these orbitals now is to expand them in terms of Slater orbitals with non-linear variation param- e'. rs.? This expansion may contain six or seven such orbitals in the case of positive alkali ions. In order to calculate the exchange integrals be- tween , and ; accurately, it is customary to use elliptic coordinates. This method, though quite accurate, is very time consuming, involving as it does the calculation of the exchange between the defect electron and charge distributions given by products of the individual Slater vrbitals. Further- . more, since may also be expressed as a sum of orbitals * = { **OK (5) the number of two-center, two-electron integrals which must be calculated at " 1 st E T A - - - - each stage in the minimization procedure, can become very large. Even on - - -- - - - -- a fast computer the computational time involved in this method becomes a very serious iactor. Therefore, in the remainder of this paper we shall - - sketch a method which circumvents this difficulty and chen end by giving a few comparisons of accuracy and time using the two methods. Let une(t,0,0) = } P(n&\r)Yem (0,0) (6) be the orbital of an electron outside the spherically symmetrical closed shells of an ion, 1,0,being the spherical polar coordinates centered at the nucleus, I and m being the azimuthal and orbital me onetic quantuin nun- bers, respectively, and Yen being a spherical harmonic. The Hartrve-Fock equation for the radial function P(nelr), belonging to an energy value E, is .. 2Z (r lll + 12} Paner) 2&' + 1 Cecil Yk (ne,n'2'/r)] n'ka * * P(n'l"\r) - { Ene,nie P(n'elr) = - 2ĖP (neſr), n! " where Y, (ne,n'l'lr) = rok s rikp(nelr')P(n'e'lr')dr' + + pk+29 -k-4p(nel 1"}P(nie" |") de 2.(r) is Hartree's "effective nuclear charge for potential," primed quantities refer to the core orbitals, the constants Ceenk are defined by Condon and S . . .".- . .5- Shortley“ (section 9°), and the so-called non-diagonal parameters, Ene nie, are determined so as to make the function P(nelr) orthogonal to the core functions P(n'!!r). Defining 82 by 02;(7) - FCÄTET QL!20";*? & Ceex** (ne ,n'e'\) - - Ž ®eer Tene , n'e}P (n'e'|o), (8) we may call -82 (8)/r an effective exchange potential energy, and rewrite the Hartree-Fock equation as 2365702.".263, 4, 1)2E} P(ne]) = 0. (9) In agreement with what has been previously found by other alithors (see, e.g., Biermann and Lubeck,' and Stone°), we find that the func- tion 82, (r) depends strongly on the azimuthal quantum number l but only slightly on the energy, E, provided that the energy E is well above the energies of the core orbitals. We can thus replace the operator -p+22p(r) + hex (10) by an operator U, defined so that for an eigenfunction v. (r) of the square of the orbital angular momentum belonging to an eigenvalue fece + 1), STU U VO(P) = -=-+(2(r) + 62%(r)]v(7) (11) ozce) being the function defined by (8) for that value of l. Having tabulated -6- the functions only for each of the ic.'s, we can use expansions of pok and Puki in spherical harmonics about the ionic nuclei to evaluate the matrix elements, , (12) - - - - - - - * thus avoiding the need for performing the time-consuming calculations of the matrix elements of hex coming from Eq. (4). In tabulating the functions: 82(r), we have to smooth out the singularities that arise at the zeros of P(ner). The functions are so insensitive to the energy E that a rough knowledge of E is sufficient to enable us to calculate them with the required accuracy. To summarize, the advantage of the present method is that, in deter- mining the functions 82 (r), we in effect calculate the exchange integrals once and for all, instead of having to recalculate them every time they are needed. It should be realized that our procedure replaces the original one- 1. electron Hamiltonian h by a new one, say hoff, such that only those eigen- values of h which are above the core eigenvalues, and the eigenfunctions be- longing to these, are practically identical to the corresponding eigenvalues and eigenfunctions of hoff: This is not true of the core eigenstares of the two Hamiltonians, and it can be shown that in carrying out the orthogonaliza- tion procedures, we now have to use the core eigenvalues and eigenfunctions of hoff and not those of h. This method requires the expansion of each of the "smooth" trial functions or in spherical harmonics about the nucleus of each of those ions that are treated as extended ions, and this can in itself be very time . . -7- consuming. It was therefore decided to gain speed by sacrificing a certain amount of mathematical accuracy. A computer program was written which ex- pands an arbitrary function in spherical harmonics about an arbitrary center, on the assumption that terms of degree l > L, where L < 12, are negligible. Instead of numerical integration, the program uses summations over a set of judiciously chosen points within a sphere (whose radius is input data) out- side which the expansion will be inaccurate. No spherical harmonics are. evaluated during the expansion process; instead, the spherical harmonics at the required points, multiplied by suitable weight factors, were stored as constants of the program when the program was written. The method turned out to be more accurate than had been anticipated. Depending on circun- stances, the expansion of a typical Slater-type function of not too high an azimuthal quantum number takes between 0.5 and 4.0 seconds on the C2C 16042 computer. These new methods have been tested on the F center in Kci, by com- paring the results with those obtained by calculating the exchange and over- lap integrals by more conventional methods (reference 1). The energies agree to within 0.03 ev, but the energy difference between the ground anc: the first excited state vas found to differ from that obtained by the conventional methods by only 0.005 ev. Most of the 0.03 eV inaccuracy in the energy pro- ably arose from our failure to achieve complete self-consistency in solving . the Hartree-Fock equation (7) for the free k* iun when we determined 82(r). According to some indications, the errors arising from our method of expan- sion in spherical harmonics are probably not greater than 0.003 eV. These are small sacrifices if one considers the gain in speed: a calculation which previously took over 2 hours can now be done in less than 5 minutes, provided . . T : I " . ,' IN -8. that the functions have been previously detemnined; but this probably overestimates the gain, because in the early calculations by the conven- tional methods we had used suine quantum chemistry routines which had beei written for problems in which much greater accuracy was required, and, more- over, we probably did not use these routines in the most efficient way. The determination of the functions oz", which we do separately for six values of l, takes about 7 or 10 minutes for each ion; once determined, these functions can be used any number of times. * -9- Table 1. Defects in KCl: contributions to the total energy, in ev. These are in addition to the point-ion energy. Defect and State Exchange Penetration Overlap Total F center ground -3.016 -1.715 +6.993 -4,721 F center excited -4.720 -3.178 +10.934 -1.978 U center ground -1.422 -0.828 +4.391 -28.734 . .- . - -.- i'. - ... 2 -10- References E Es 17 1. Taken from unpublished calculations of R. F. Wood and H. W. Joy on the F center and R. L. Gilbert and R. F. Wood (to be published) on the U center. See also R. F. Wood and H, W. Joy, Phys. Rev. 136, A451 (1964). The method used is that of C.C.J. Roothan, Rev. Mod. Phys. 23, 69 (1951) and 32, 179 (1960), with Slater orbitals as a basis set. 3. D. R. Hartree, The Calculation of Atomic Structures (Wiley, New York, and Chapman and Hall, London, 1957). E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Univer- sity Press, Cambridge, 1935). 5. L. Biermann and K. Lübeck, Z. Astrophys. 25, 325 (1948). 6. P. M. Stone, Phys. Rev. 127, 1151 (1962). .. - . . .. . . . . ... . - : END DATE FILMED 7 / 28 / 66 H