Valence-band structure of group-iv semiconductors by means of local increments

Transcription

1 PHYSICAL REVIEW B VOLUME 55, NUMBER MAY 1997-II Valence-band structure of group-iv semiconductors by means of local increments Jürgen Gräfenstein* Max-Planck-Institut für Physik komplexer Systeme Dresden, Au enstelle Stuttgart, Postfach , D Stuttgart, Germany Hermann Stoll Universität Stuttgart, Institut für Theoretische Chemie, D Stuttgart, Germany Peter Fulde Max-Planck-Institut für Physik komplexer Systeme Dresden, Bayreuther Stra e 40, Haus 16, D Dresden, Germany Received 30 September 1996 We present a method to determine the correlated valence-band structure of covalent semiconductors that uses information from quantum-chemical ab initio calculations on appropriate molecules. In distinction from previous cluster simulations, we consider a set of comparably small molecules rather than a large cluster, which allows us to use extended one-particle basis sets. We then extract local matrix elements from these calculations and use them to set up an incremental expansion for the bulk band structure. The question how to extend our method to the determination of the conduction-band structure and the band gap is discussed. S I. INTRODUCTION The rapid progress in available computing facilities has also greatly enhanced the investigation of the electronic structure of molecules, clusters, and solids in the last years. Meanwhile, calculations based on realistic many-particle concepts are widely used for many problems. However, there is still a significant difference between the state of the art for finite and extended systems: for molecules, sophisticated quantum-chemical methods are well-established tools of investigation. These methods, which are based on selfconsistent-field SCF calculations with a subsequent inclusion of electron correlations by configuration interaction CI or related procedures, have the advantage that they are conceptually very clear and imply approximate correlated wave functions, from which every quantity of interest can in principle be derived. Many of these methods are implemented in standard quantum-chemistry ab initio program packages such as MOLPRO Ref. 1 or COLOGNE Ref. 2, which are elaborate enough to apply them routinely. In principle, quantum-chemical methods can also be applied to extended systems and, in fact, successful applications to polymers have been reported in the literature for a recent review, see Ref. 3. As regards solids, on the other hand, the method most widely used is still the Kohn-Sham formalism, 4 which is efficient and surprisingly successful for many cases but cannot be expected to be reliable for every problem of interest. Additionally, several more sophisticated approaches have been developed and used in the last years. For instance, the GW approximation of Hedin 5 has become applicable for realistic systems and can now be used, e.g., to calculate the band structure of semiconductors see, e.g., Ref. 6. Furthermore, quantum Monte Carlo methods are employed at an ever-increasing scale; see, e.g., Ref. 7. However, the application of methods analogous to the quantum-chemical ones is still in its beginning. There are indeed appropriate program packages for this purpose at the SCF level such as, e.g., CRYSTAL Ref. 8, however, their application is still a subtle business. For instance, calculations can often be made to converge only at the cost of restrictions on the basis set, which are acceptable at the SCF level but should lead to serious errors as soon as correlations are considered. Moreover, while the electron-correlation treatment of ground-state properties, in particular by the local ansatz 9 and the localincrement approach 10 methods closely related to the quantum-chemical approach, has shown considerable progress in the last few years, there are still only very few calculations of quantities related to excitations Given this situation, the question arises whether one can use the methods available for finite systems to draw information on the properties of solids. This should be realistic in particular for covalent solids, where the bonding is expected to be very similar to that in corresponding molecules. In the past, a number of cluster calculations have been done for the investigation of the properties of solids, the basic idea being the use of a large cluster in order to simulate the interior of the solid. In contrast to that work we perform here a series of calculations on comparably small molecules less than 10 atoms of the material under consideration plus a number of hydrogen atoms to saturate bonds. We determine local matrix elements appearing in an energy band calculation by employing an incremental scheme. In a sense, our approach employs the idea of how the atomic levels form bands when the atoms are combined to a solid: if two atoms approach each other, the degeneracy of their levels is lifted by the formation of bonding and antibonding orbitals. If two of these dimers are brought close together, the degeneracy between these bonding and antibonding orbitals is lifted again, etc. Continuing this procedure, one obtains more and more orbitals, the energies of which are concentrated in groups. In the limit of infinitely many atoms these are the band energies. It is not possible to keep track of this process by a series of calculations for larger and larger clusters with reasonable computational effort and accuracy, and therefore we employ /97/55 20 / /$ The American Physical Society

2 55 VALENCE-BAND STRUCTURE OF GROUP-IV a scheme that reduces energy band calculations to the determination of local matrix elements. This scheme is closely related to the local-increment approach used in Ref. 10 to calculate the correlation part of the bond energy in diamond and silicon and has been presented in brief in Ref. 19. Here we shall outline it in more detail and present applications to several materials. Our work emphasizes the usefulness of basing calculations for solids on direct-space rather than reciprocal-space quantities. There are both methodological as well as practical reasons for such an approach: on one hand, certain approximations are much easier to formulate in direct than in reciprocal space see, e.g., Ref. 20, on the other hand, for objects where the crystalline symmetry is disturbed, such as, e.g., surfaces or impurities see, e.g., Ref. 21 a direct-space approach is most appropriate. This paper is organized as follows. In Sec. II we describe how the SCF band structure can be derived from molecular calculations. Section III describes how correlations are included. In Sec. IV we discuss the problem of the gap energy and conduction-band structure. Section V contains the details of the calculations and a discussion of the results, Sec. VI gives a brief summary. II. THE SCF BAND STRUCTURE Band energies of solids can be regarded as energy differences between the N-particle ground state and a state with N 1 orn 1 particles, respectively, where one delocalized Bloch electron has been added or removed. At the SCF level, for this kind of process, Koopmans theorem holds, and the task of determining the band energies can be reduced to finding the one-particle energies of the respective Bloch states. For our purposes, however, we use a somewhat different approach which is more complicated at first glance but has the advantage that it can be extended to the correlated case. Keeping this in mind, we formulate a method in which the band energies are expressed in terms of Hamiltonian matrix elements between many-particle states. Moreover, since we want to determine these matrix elements from molecular calculations, the extra electron or hole in the (N 1)-particle states involved should be localized. We start with the SCF ground state, which we assume to be a closed-shell one, k a k a k 0, where k and label the quasimomentum and band index, respectively, and a k is the creation operator for the respective Bloch state. The energy of is E 0. The following considerations are for the valence band; the transfer to the conduction band is obvious. We introduce the (N 1)-particle states k a k, and write the valence-band energies in the form k k H k E 0. 3 As mentioned above, we want to base our considerations on states with localized extra electrons or holes. To this end, we 1 2 subject the k to a transformation analogous to the Wannier transformation. 22 This transformation yields localized states centered at a certain lattice vector R: Rn c Rn. Here n refers to different localized states within a unit cell, c Rn is the annihilation operator that generates the (N 1)-particle state from the N-particle ground state. Since Wannier transformation is not unique, the Rn are to some extent ambiguous. For our purpose we require that the holes are well localized and that each hole is assigned to, and centered at, a certain bond in the semiconductor. We now rewrite the band energies in terms of the Rn. To this end we introduce the quantities H R R with H R R R n H Rn RR nn E 0. 5 Note that the matrix elements and thus the H R R are uniquely defined, in spite of the ambiguity of the Wannier transformation. The band energies for a certain k vector are the eigenvalues of the matrix 1 H nn k N 0 R 4 e ikr H R. 6 This procedure is very similar to the usual linear combination of atomic orbitals LCAO approach, with the difference that it is based on many-particle states and furthermore, that valence and conduction bands are treated separately. Note that we have reduced the calculation of the band energies to the calculation of a relatively small number of local matrix elements H R, which in turn are related to (N 1)-particle states with the hole centered at a bond. These matrix elements can be determined approximately from calculations for appropriate molecules. To this end, we perform a SCF calculation for a molecule large enough to contain the bonds under consideration. We localize the occupied orbitals of the ground-state wave function mol of the molecule, e.g., by the Foster-Boys procedure, 23 so that they are centered at different bonds and denote them by the bond indices I and J in the following. The desired quantities can then be calculated approximately as H IJ I H J IJ E mol, I c I mol. In order to show that this procedure does not depend on a particular molecule used for that purpose we perform this calculation for different molecules. As we shall see later, the results are practically the same in all cases. The matrix elements in Eq. 7 can be determined directly. With the extension to correlated states in mind, we proceed, however, in a slightly different way. To determine all the matrix elements between the states 1, 2,..., n, we diagonalize the Hamiltonian in the space spanned by these functions. This yields eigenenergies 1,2,... n and corresponding eigenfunctions 7 8

3 JÜRGEN GRÄFENSTEIN, HERMANN STOLL, AND PETER FULDE 55 n i m 1 im m, with im being a unitary matrix. Because of i H j i ij it is n H IJ i 1 ii ij i, i i E mol The i are determined by the structure of the system. By this procedure, one can calculate all matrix elements occurring in one molecule in one step, and also the matrix elements between one pair or another subset of the bonds only. At the SCF level, the results are, of course, exactly the same as before. As soon as correlations are taken into account, however, this will change, and we will do this calculation for different subsets of bonds in order to take correlations into account in an incremental way. III. INCLUSION OF CORRELATIONS The formalism described so far does not take into account the correlations between the electrons. That is, neither does it reflect the fact that there are correlations in the N-particle ground state, which furthermore change as soon as an electron is added or removed, nor does it describe the polarization cloud around the extra electron or hole. In the present section we want to show how this can be remedied. As in the SCF case, the band energies k can be represented by energy differences between the N-particle state and (N 1)-particle states: k k H k E 0, where E 0 is the energy of the correlated ground state and k is the correlated counterpart of k. Due to the correlation effects, there is no such simple representation for these states as Eq. 4. Nonetheless, we can subject them to the same unitary transformation as in the SCF case, obtaining states Rn, where from the correlated ground state an electron has been annihilated in a certain bond, with the other electrons having responded to this process. From these states, the correlated band structure can be determined in the same way as the SCF band structure was from the Rn. The counterpart of the matrix elements H Rnn in Eq. 5 is denoted by H Rnn. The Rn can be represented as Rn e S R Rn, 11 where the operator S R contains excitations of any number of particles. As is usual and reasonable for weakly correlated systems, we restrict ourselves to one- and two-particle excitations, which leads to the following expression for S R : S R R1 R2 Rn ir 2 n 1 2 R 1 Ri c R1 ic R2 n R 1 Ri 1,R 2 Ri 2 R3 c Rn R1 i 1,R 2 i 1,R 4 Rn R1 2 i c 1 R2 i 2 2 R 3 n 1,R 4 n 2 c R3 n 1 c R4 n The first sum describes one-particle excitations, which generate a polarization cloud around the hole, while the second part describes the correlations in the ground state and their changes due to the creation of the hole. We assume the k, and consequently the R,tobe intermediately normalized, i.e., k k R R 1. This implies that in expectation values and matrix elements containing the correlated functions we have to restrict ourselves to linked clusters. Our aim is again to determine matrix elements with respect to the Rn from molecular calculations. Note that due to the polarization cloud the correlation hole generated around the hole is rather long ranged. To make the calculation feasible on the basis of smaller molecules, we employ an incremental method. For that purpose we consider configurations generated by S R when all electrons in Rn are kept frozen except in a limited, in fact small, number of localized orbitals. Only electrons in these orbitals are correlated. We will denote an N-particle state, where the bonds (Rn), (R n ),... arecorrelated, by ;(Rn),(R n ),... and a (N 1)-particle state where the hole is in bond (Rn) and the bonds (Rn), (R n ),... are correlated, by (Rn);(R n ),.... With these wave functions we then calculate a sequence of approximations for H R R.Ina (0) first step, we find an approximate value H R R from the functions ;(Rn),(R n ) and (Rn);(R n ),, (R n );(Rn),, i.e., we take into account only correlations in the two bonds (Rn), (R n ). The difference of this matrix element to the SCF matrix element, 0 H R R 0 H R R H R R we call the intrabond increment for H R R. In a next step, we add a third bond (R 1 n 1 ) it is tacitly understood that the bonds (R 1 n 1 ),(R 2 n 2 ),... introduced here and in the following are different from both (Rn) and (R n ) and from (R each other and calculate approximation values H 1 n 1 ) R R from the functions ;(Rn),(R n ),(R 1 n 1 ), (Rn);(R n ),(R 1 n 1 ), (R n );(Rn),(R 1 n 1 ). Their difference to the lowest-order approximation, R H 1 n 1 R R R H 1 n 1 0 R R H R R, which we call a one-bond increment for H R R, can be (0) regarded as a correction to H R R due to the correlation in (R 1 n 1 ). In the next step, we consider the correlations in two bonds (R 1 n 1 ) and (R 2 n 2 ) simultaneously, obtaining an (R approximation value H 1 n 1 ),(R 2 n 2 ) R R. Again we define an increment, which we call a two-bond increment, according to R H 1 n 1, R 2 n 2 R R R HR R 1 n 1, R 2 n 2 0 R HR R H 1 n 1 R R,

4 55 VALENCE-BAND STRUCTURE OF GROUP-IV TABLE II. The basis for the CRYSTAL calculations with diamond. Type Exponent Contraction coefficient FIG. 1. The molecules used for the calculations saturating hydrogen bonds omitted, X C, Si, Ge. R H 2 n 2 R R. If we continued this procedure, we would finally end up with the exact value for H R R, i.e., H R R H R R H R R R 1 n 1,R 2 n 2 H R R R H 1 n 1 R1 n R R 1 R 1 n 1, R 2 n We expect the increments to become smaller the higher their order and the larger the distance of R 1,R 2 from R,R. A reasonable approximate value for H R R is obtained by terminating the sum in Eq. 13 after an appropriate order. The quantities entering this incrementation scheme can be extracted from molecular calculations, because they contain correlations involving a limited number of bonds only. However, they cannot be obtained straightforwardly from post- SCF correlation calculations. If, for instance, one tried to compute I;J from the reference state I, then the main effect of the single and double excitations would result in a delocalization of the hole over the bonds I and J. Therefore, we proceed similarly as in the SCF case. That is, to determine the matrix element between I and J, in the first instance we do a multireference CI calculation the notation CI stands symbolically for all related quantum-chemical methods here, we shall specify below which method is used for which purpose with I and J as reference states and allow only for correlations between these two states. The resulting wave functions (IJ) 1,2 are linear combinations of I;J and J;I with the expansion coefficients being unknown for the time being and thus lead to the matrix element H (0) IJ. Next, we do a calculation with the reference states I, J, and K, yielding the eigenstates (IJK) 1,2,3, which lead to the TABLE I. The interatomic distances in Å) used in the molecular calculations. X C Si Ge X-X X-H s s s s p p d wave functions I;JK, J;IK, and K;IJ and the corresponding matrix elements and increments. This procedure is continued up to the desired order. To determine approximate values for the expansion coefficients, we project the i () onto the reference space. Then we subject the resulting vectors to a symmetric orthonormalization. The expansion coefficients of the resulting matrix with respect to the reference states are used as approximations to the sought expansion coefficients. In the spirit of our incrementation scheme this approach is reasonable, because in the limit that all correlations are included it would yield the exact result: in this case, the i () could be chosen so as to coincide with the Bloch functions k, the projection of which onto the reference space yields the k. Now, the relation between the k and the Rn is the same as that between the k and the Rn. In the above procedure we treat singles and doubles separately from each other. The reason for this is the following: the two-particle excitations describe the correlations of the electrons in an increasing number of bonds if we continue with our procedure. Therefore, a size-consistent method is needed to treat these correlations properly. On the other hand, the singles describe the polarization cloud around a single hole. Here size consistency does not play a role. Therefore, for the treatment of the singles we use a multireference configuration-interaction singles-only MR-CI S method, whereas we used quasidegenerate variational TABLE III. The basis sets for the CRYSTAL calculations for silicon and germanium. Silicon Germanium Type Expt. Contr. coeff. Exp. Contr. coeff. s s s p p p d

5 JÜRGEN GRÄFENSTEIN, HERMANN STOLL, AND PETER FULDE 55 FIG. 3. The SCF dashed lines and correlated solid lines band structures for silicon. FIG. 2. The SCF dashed lines and correlated solid lines band structures for diamond. perturbation theory 24 QDVPT for the doubles thereat keeping single-particle excitations suppressed. The polarization cloud around an extra particle or hole is long ranged. On the other hand, the effects arising from the farther surroundings of the hole can be expected to be mainly of electrostatic nature, and it appears reasonable to describe them by considering the respective part of the solid as a polarizable continuum. Therefore, the idea of this approach is the following: the energy of a state with an additional electron or hole more generally, of any extra multipole moment is lowered by the dielectric relaxation of the surroundings. If we neglect this relaxation outside a sphere of radius R, then the resulting energy (R) is related to the true energy ( ) by (R) ( ) CR (2L 1), where L is the order of the multipole. Thus, if we determine (R) for two values R 1,2 then the correct value ( ) results by extrapolation: R 2 R 2 /R 1 2L 1 R 1 R 2 /R 1 2L The hopping matrix element between the states Rn and R n is one-half of the energy difference between the states 1/2( Rn R n ), i.e., between two states with different positions and shapes of the hole. To leading order their energy difference is due to a dipole field implying that an equation analogous to Eq. 14 holds with L 1 for the single-excitation corrections to the hopping matrix elements. As a reasonable approximation we take into account singleparticle excitations only in a group of bonds that form approximately a sphere around the bond pair under consideration. The ratio R 2 /R 1 in Eq. 14 is then replaced by the ratio (N 2 /N 1 ) 1/3 where N 1 and N 2 are the respective number of bonds contained in the group. Then Eq. 14 leads to H s H s N 2 N 2 /N 1 2L 1 /3 H s N 1 N 2 /N 1 2L 1 / Here H s (N) is the contribution of single-particle excitations to the respective Hamilton matrix element for the case that N bonds are correlated. The final correction to the Hamilton matrix element is then H s ( ) H s (N 2 ), if N 2 is the number of all bonds that have been treated explicitly. A similar approximation has been used in Ref. 17. IV. ENERGY GAP PROBLEM A quantity of particular interest in a band structure calculation of a semiconductor is the band gap between the conduction and valence bands. Therefore the question arises whether or not the method presented here can be extended to the conduction band as well. Unfortunately, this is presently not the case for technical reasons. When a Foster-Boys localization of the virtual states is performed, the antibonding orbitals have too high an energy, which is because they contain not only contributions from what corresponds to the conduction band but also from higher bands. Therefore, one would have to perform the localization only for states in an appropriate energy range. It is, however, not possible to separate these states from the virtual spectrum because the latter is continuous. This is in marked difference to the spectrum of the occupied states, which is discrete. If one considers larger and larger molecules, then the energies of these states concentrate in regions that are approximately the band energies in the corresponding solid. If one localizes those TABLE IV. The SCF and correlated values for the Hamilton matrix elements. For the numbering of the bonds, see Fig Diamond SCF correlated Silicon SCF correlated Germanium SCF correlated

6 55 VALENCE-BAND STRUCTURE OF GROUP-IV FIG. 4. The SCF dashed lines and correlated solid lines band structures for germanium. states corresponding to one band, one obtains good approximations to the Wannier states of the solid. For virtual states no such grouping of state energies can take place. One might think of remedying this by putting the molecule into a potential well. By doing so, one obtains indeed a discrete spectrum, which, however, depends sensitively on the boundary conditions. This has been confirmed by both some simple model investigations 25 and a comparison of the spectrum of virtual states calculated with different basis sets the choice of a spatially restricted basis set has similar effects as the introduction of a potential wall. The conclusion is that, in distinction to the case of the valence band, a correct description of the periodic potential is necessary to describe the behavior of the conduction electrons properly. This can be done, e.g., by exploiting an embeddedcluster approach, or by introducing periodic boundary conditions into the quantum-chemistry codes used. FIG. 5. The SCF band structure for silicon from the molecular calculations dashed lines and from CRYSTAL solid lines. V. CALCULATIONS AND RESULTS A. Details of the calculations Calculations of the SCF and correlated band structures have been done with the described method for diamond, silicon, and germanium. In these calculations, all appropriate molecules with up to five X (X C, Si, Ge atoms have been included, which yields contributions from up to thirdnearest-neighbor bonds. Figure 1 shows the X skeletons of the molecules, all free bonds being saturated with hydrogen in every case. For diamond all-electron calculations were done with the correlation-consistent valence double- basis set by Dunning, 26 whereas for silicon and germanium the relativistic energy-consistent pseudopotentials and the corresponding optimized valence basis sets of the Stuttgart group, 27 without contractions, have been used. The basis sets have been augmented by a set of d functions with an exponent of 0.40 in both cases. To get some idea how the basis TABLE V. Values of the SCF matrix elements for silicon extracted from different molecules. For the numbering of the bonds, see Fig. 8. Molecule Si 3 H Si 4 H iso Si 4 H Si 5 H iso1 Si 5 H iso2 Si 5 H iso3 Si 5 H Si 5 H 12 2 Si 5 H 12 1,1 Si 5 H

7 JÜRGEN GRÄFENSTEIN, HERMANN STOLL, AND PETER FULDE 55 TABLE VI. The correlation contributions to the values of the Hamilton matrix elements for silicon basis set with d functions. Intra refers to the intrabond increments, 1 bond to the sum of all one-bond increments, 2 bonds, to the sum of all two-bond increments, continuum means the continuum correction to the single-excitation corrections. Total means the sum of all single or double corrections, respectively; final stands for the value of the matrix element with all corrections included SCF Singles Intra bond bonds Continuum Total Doubles Intra bond bonds Total Final TABLE VII. The correlation contributions to the values of the Hamilton matrix elements for silicon basis set without d functions. For the notation, see Table VI SCF Singles Intra bond bonds Continuum Total Doubles Intra bond bonds Total Final TABLE VIII. Continuum corrections for the single-excitation contributions to the nearest-neighbor Hamilton matrix element. Diamond Silicon Germanium With d functions Without d functions H s (10) H s (34) H s ( ) H s ( ) H s (34)

8 55 VALENCE-BAND STRUCTURE OF GROUP-IV FIG. 6. The SCF band structure for silicon from the molecule calculations with solid lines and without dashed lines d functions. set size influences the results, calculations for silicon have also been done without this function for comparison. As regards the geometry of the molecules, the X-X and X-H distances given in Ref. 28 have been used see Table I. All bond angles have been chosen as in the solid, i.e., as arccos( 1/3) For the calculations, the program package MOLPRO Ref. 1 has been used. The localized orbitals have been generated with the Foster-Boys procedure 23 contained in this program. For the case of the molecule X 5 H 1,1 12 for the notation see Fig. 1, the correlated calculations did not converge for the case with four reference states, which proved to be due to the occurrence of a threefold-degenerate state in these cases. This could, however, be easily circumvented by doing these calculations with a slightly distorted geometry, i.e., two of the X-X bonds were lengthened or shortened, respectively, by 0.01 and 0.02 Å and the energy levels were extrapolated to zero distortion. The error due to this procedure could be estimated to be less than 10 7 a.u. and is thus negligible. For comparison, the SCF band structures were calculated as well with the SCF package CRYSTAL. 8 For diamond, the all-electron basis by Roos and Siegbahn 29 has been used as a starting point. To make the SCF procedure converge, however, the most diffuse basis functions had to be modified as described in Ref. 11. For silicon and germanium, the calculations have been carried out using the same pseudopotentials and basis sets as the molecular calculations. Again, the most diffuse functions had to be compressed, which was done following Paulus. 30 The basis sets for the crystal calculations are listed in Tables II and III. FIG. 7. The correlated band structure for silicon with solid lines and without dashed lines d basis functions. FIG. 8. The numbering of the bonds used in the tables. B. Results An overview of the resulting Hamilton matrix elements is given in Table IV. It is obvious that the correlations have a remarkable effect on the matrix elements and thus on the shape of the bands. The resulting SCF and correlated band structures are shown in Figs. 2, 3, and 4. For silicon, the results are given in some more detail. First we show the values of the Hamilton matrix elements as obtained from different molecules. Table V gives an overview over these values. One sees that even for the comparably small molecules we have used the agreement of corresponding matrix elements is fairly good. So, it is justified to assume that these values are also in good agreement with the values for the solid. For the correlated matrix elements, such a comparison is, of course, not possible. However, what one can do here is to compare single energy increments. One finds that the agreement is similarly good as for the SCF matrix elements, thus our approach is justified as well for the correlated case. Next let us compare the SCF band structure resulting from our calculation with a solid-state calculation done with CRYSTAL. One sees in Fig. 5 that, while the general agreement is good, the CRYSTAL band structure exhibits some more structure, e.g., smaller effective masses at the 25 point. This indicates that to obtain a precise SCF band structure one has to take into account more coordination spheres than we have done with our molecular calculations. This is, however, difficult, because both the number of isomeric molecules and their size would increase rapidly when adding more coordination spheres. Nonetheless, our calculations prove to be reasonable and are an appropriate starting point for the inclusion of correlations, which are shorter ranged than those effects that determine the band structure at the SCF level. The same has, by the way, been observed for the binding energy: while the incrementation scheme converges well for the correlation part of the binding energy, 10 attempts to exploit such a scheme for the SCF binding energy as well 31 did not lead to satisfactory results. To combine the advantages of the solid-state and the local approaches, in the following the SCF band structures have been calculated

9 JÜRGEN GRÄFENSTEIN, HERMANN STOLL, AND PETER FULDE 55 TABLE IX. Comparison of band energies at special points in ev for our calculations and the GW calculations in Ref. 6. The energy at the point 25, i.e., the top of the valence band, has been put to zero. X 1 X 4 1 L 2 L 1 L 3 Diamond Our calc Ref Silicon Our calc Ref Germanium Our calc Ref with CRYSTAL and the correlation contributions found by our method have been added to them. Furthermore, we have studied the influence of the basis set on correlation effects. To this end, we have done the calculations for silicon both with and without d functions at the Si atoms. The resulting matrix elements for both cases are to be found in Tables VI and VII, respectively. One notices that the differences in the SCF matrix elements are comparably small but there are large differences as soon as correlations are considered, which holds particularly true for single excitations. The latter is due to the fact that singles, above all, describe the polarization cloud, for the description of which one needs functions with an angular momentum higher by at least one than for the original functions; i.e., for p functions one needs to include at least d functions here. Figure 6 shows the SCF band structures based on the molecular calculations, with the picture for the CRYSTAL band structures being similar. Figure 7 shows the correlated band structures for the two cases. Table VIII shows the values of the continuum corrections according to Eq. 15. One sees that the continuum corrections, on the one hand, do make a contribution to the matrix elements; on the other hand, this contribution is small enough to consider the approximation justifiable. For the other hopping matrix elements, the continuum corrections cannot be calculated reasonably with the number of bonds available. In Table IX, we have compared the results of our calculation with those of a GW calculation by Hott. 6 The agreement between the results proves to be good. VI. CONCLUSIONS We have presented a method to determine the valenceband structure for covalent solids based on quantumchemical calculations on relatively small molecules. Local matrix elements have been extracted from these calculations both at the SCF and correlated levels using an incremental scheme, transformation to delocalized states subsequently yields the bulk band structure. The most important assumption behind this method, viz., that the electrons behave very much the same in these molecules as in a solid to be described, could be confirmed both by comparing the values for certain intermediate results obtained from different molecules and by a comparison of the final results with other calculations. Unfortunately, a straightforward extension of this approach to the conduction band has proven not possible at present. There is no simple way to account for the different behavior of an additional electron in the solid and in a molecule, in particular as regards the SCF level. Nonetheless, the presented method appears promising. On the one hand, it seems promising to make some modifications with existing quantum-chemical methods and the corresponding codes such that the behavior of a conduction electron in the periodic potential of a solid can be modeled properly, as is done by Shukla. 32 On the other hand, the incremental scheme based on Wannier states can be used in connection with a solid-state calculation as well, if only the program used allows the generation of localized Wannier orbitals. Also, we would like to point out that the theory presented here can be reformulated in terms of cumulants and projection techniques. 33 *Present address: Göteborgs Universitet, Institutionen för Teoretisk Kemi, Kemigården 3, S Göteborg, Sweden. 1 H.-J. Werner and P.J. Knowles, computer code MOLPRO92, Institut für Theoretische Chemie, Universität Bielefeld and School of Chemistry, Physics, and Environmental Science, University of Sussex, 1992, with contributions by J. Almlöf, R.D. Amos, A. Berning, C. Hampel, R. Lindh, W. Meyer, A. Nickla, P. Palmieri, K.A. Peterson, R.M. Pitzer, H. Stoll, A.J. Stone, P.R. Taylor; H.-J. Werner, and P.J. Knowles, J. Chem. Phys. 89, ; Theor. Chim. Acta 78, ; P. Knowles and H.-J. Werner, Chem. Phys. Lett. 145, E. Kraka, J. Gauss, F. Reichel, L. Olsson, Z. Konkoli, Z. He, and D. Cremer, computer code COLOGNE 94, Göteborgs Universitet, Göteborg, J.-Q. Sun and R.J. Bartlett, J. Chem. Phys. 104, P. Hohenberg and W. Kohn, Phys. Rev. 136, B ; W. Kohn and L.J. Sham, Phys. Rev. 140, A L. Hedin, Phys. Rev. 139, A R. Hott, Ph.D. thesis, Universität Stuttgart, 1990; Phys. Rev. B 44, X.P. Li, D.M. Ceperley, and R.M. Martin, Phys. Rev. B 44, R. Dovesi, C. Pisani, V.R. Saunders, and C. Roetti, computer code PISA, Gruppo di Chimica Teorica, Università degli Studi di Torino and United Kingdom Science and Engineering Research Council Laboratory, Daresbury, 1992; R. Dovesi, C. Pisani, and C. Roetti, Hartree-Fock Ab-initio Treatment of Crystalline Systems, Lecture Notes in Chemistry Vol. 48 Springer-Verlag, Berlin, We note that for polymers SCF programs are available, which are listed, e.g., in the review 3.

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