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Calculation of band gaps in molecular crystals using hybrid functional theory
Chemical Physics Letters 368 (2003) 319–323 www.elsevier.com/locate/cplett
Calculation of band gaps in molecular crystals using hybrid functional theory W.F. Perger
*
Electrical Engineering and Physics Departments, Michigan Tech University, Houghton, MI 49931, USA Received 28 October 2002; in final form 20 November 2002
Abstract Hybrid functional theory is applied for calculation of band gaps in the molecular crystals anthracene, pentaerythritol (PE), pentaerythritol tetranitrate (PETN), and cyclotrimethylene trinitramine (RDX). The B3LYP hybrid functional is observed to produce band gap estimates in reasonable agreement with experiment for anthracene and RDX. This approach, which has been successfully used recently for other materials, is efficient and practical, which is especially important for these large molecular crystals. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Density-functional theory (DFT) in the localdensity approximation (LDA) [1,2] and, more recently, the generalized gradient approximation [3] have been successful for calculating the total energy in molecules and solids. However, accurate band gap calculations for semiconductors and insulators remains a problem of contemporary interest [4]. The ability of DFT to provide accurate, ab initio, ground-state properties in molecules is well known, and in cases where correlation is not strong, it may be reasonable to treat solids and molecules in the same way [5], that is, with the same theoretical methods. Molecular crystals, such as those studied here, would seem to fall into this category. Encouraged by the recent work of *
Fax: +9064872949. E-mail address: [email protected] (W.F. Perger).
Muscat et al. [6] which showed that the hybrid functional approach of Becke [7] can be used to improve the agreement with experiment for calculated band gaps for a wide variety of materials, the same approach is applied here to the molecular crystals anthracene, pentaerythritol (PE), pentaerythritol tetranitrate (PETN), and cyclotrimethylene trinitramine (RDX). Molecular crystals have not been as thoroughly investigated as other materials owing to the typical computational complexity involved. At the time of one of the first ab initio molecular crystal studies, on urea, just over 10 years ago [8], a system with 30 atoms per unit cell was at the achievable limit. During the intervening years, the advances made in utilizing symmetry to improve performance of the CR Y S T A L program [9], the progress made in basis sets and correlation theory, especially at the molecular level, and the improvements in computer hardware, have made first-principles studies of rela-
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 8 7 9 - 1
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W.F. Perger / Chemical Physics Letters 368 (2003) 319–323
tively large molecular crystals possible. The purpose of this Letter is to add to the growing body of knowledge on the use of DFT as a practical means of achieving accurate calculations for band gaps in materials, in this case the molecular crystals anthracene, PE, PETN and RDX. In addition to the relevance of the band gap as an important material property, the need for an accurate determination of the band gap assumes a special significance for the case of energetic materials such as RDX and PETN. In RDX, for example, a possible mechanism proposed by Gilman [10] for the shock-todetonation transition is a strain-induced reduction of the band gap, ultimately leading to the breaking of the N–NO2 bond. Accurate band gap calculations are therefore of central importance for determining if reasonable wave front pressures could cause such a reduction in the band gap. For comparison, different model potentials and their effect on the band gaps and the molecular highest occupied molecular orbital (HOMO)–lowest unoccupied molecular orbital (LUMO) splitting will be presented.
2. Molecular crystals studied Anthracene, C14 H10 , forms a monoclinic structure, space group P 21 =a with two molecules per unit cell. The internal (fractional) co-ordinates used in the current study for both the carbon and hydrogen atoms were taken from Cruickshank [11]. PE, CðCH2 OHÞ4 , crystallizes in the tetragonal structure with space group I4. The internal coordinates were obtained from Eilerman and Rudman [12]. PETN, ½CðCH2 ONO2 Þ4 is a molecular crystal with 29 atoms per molecule, with two molecules per unit cell, and belongs to the tetragonal P 421 c space group. The internal co-ordinates used were from Kitaigorodskii [13]. RDX, ½ðCH2 NNO2 Þ3 has 21 atoms per molecule, eight molecules per unit cell, and belongs to the orthorhombic Pbca space group, with internal co-ordinates from Choi [14]. This results in 1176 atomic orbitals for the 168 atoms per unit cell and for this size of a system, it becomes especially important to utilize efficient numerical techniques and accurate physical models in order to obtain results in a
reasonable time-frame. This fact serves to motivate using practical computational approaches, such as the hybrid functional method, which was recently shown to provide good band gap estimates for semi-conductors, semi-ionic oxides, and transition metal oxides [6].
3. Approach The CR Y S T A L 98 [9] program was used, initially with a basis set of sð6Þspð2Þspð1Þ for carbon, nitrogen, and oxygen, and sð2Þsð1Þ for hydrogen. A Gaussian basis set was chosen, a sð6Þspð2Þspð1Þ set for the carbon atom and a sð2Þsð1Þ for the hydrogen atom (Ô6–21gÕ), as it had been successfully used previously for RDX [15]. A slightly more complicated basis set was also tried, the optimized carbon basis set of Dovesi et al. [8] and the effect this had on the band gap was observed; seeing little or no effect, the simpler basis set was chosen for the remainder of this work. The basis set superposition error (BSSE) was corrected via the counterpoise method, as previously done for urea [8]. In conjunction with this, a limited basis set optimization was performed by varying the scaling of the exponent of the outermost orbital, in a manner previously described for the molecular crystals RDX and TATB [15]. In that study, it was determined that the outermost exponent for hydrogen should be 1.05, and 1.10 for carbon, nitrogen, and oxygen. In the present study it was found for RDX that exponent values of 1.05 for hydrogen and 1.03 for C, N, and O resulted in the minimum energy; for anthracene, values of 1.0 for all atoms gave the lowest energy. However, in both cases, the differences in the total energy for these various exponent values were quite small, typically on the order of 0.1–0.01 a.u. Furthermore, a greater BSSE was found for larger exponent values. Therefore, the exponent scaling chosen for this study was 1.0 for all orbitals. Once the basis sets were determined, an efficient and practical means must be adopted for solving the Schr€ odinger equation while retaining the physics of exchange and correlation. In order to study the effect of the exchange–correlation po-
W.F. Perger / Chemical Physics Letters 368 (2003) 319–323
321
tential on the band gap, several potentials were chosen. The B3LYP functional was chosen based on its success to estimate band gaps for a variety of materials [6] and has the form LSDA Fock LSDA GGA Exc ¼ Exc þ 0:2ðExc Exc Þ þ 0:72DExc ;
ð1Þ where LSDA is the local spin-density approximation and GGA is the generalized gradient approximation and the parameters were determined by Becke [7] to optimize atomization energies, ionization potentials, and proton affinities were several molecules. The B3LYP hybrid functional combines BeckeÕs three-parameter functional [7] for the exchange with the non-local correlation potential of Lee–Yang–Parr [16]. The CR Y S T A L 98 [9] program explicitly evaluates the non-local Fock energy, then blends 20% of that value into the hybrid functional, as shown in Eq. (1). A second hybrid functional used was the B3PW which uses the Becke exchange combined with the Perdew–Wang GGA (PWGGA) [17–20] correlation. Next, a LDA exchange and von Barth–Hedin (VBH) [21] correlation potential (non-hybrid) functional was chosen. Finally, for comparison, a Hartree–Fock (HF) potential was used.
Fig. 1. Density of states for anthracene, using density-functional theory and the Hartree–Fock potential.
4. Results
Fig. 2. Density of states for anthracene, using density-functional theory and the B3LYP hybrid functional.
Fig. 1 shows the total density of states (DOS) for anthracene using the simpler 6–21g basis set and Fig. 2 shows the DOS using the B3LYP hybrid DFT functional. As can be seen by comparing those figures, the band gap narrows from about 8.2 to about 3.6 eV. The band structure for the upper five valence bands and lowest five conduction bands using the B3LYP potential is shown in Fig. 3. As with the molecular crystal urea [8], some of the bands are rather flat, likely due to weak coupling of the crystalline environment on the molecular states. A comparison of the band gaps calculated for anthracene, PE, PETN, and RDX using the exchange–correlation functionals described earlieris given in Table 1. For comparison, the highest occupied molecular orbital (HOMO)–lowest
unoccupied molecular orbital (LUMO) splittings are also given, as calculated both by using the CR Y S T A L 98 and the GA U S S I A N 98 programs, where possible, using the same basis sets and model potentials. As is apparent from Table 1, the Hartree–Fock potential over-estimates the band gap relative to experiment by roughly a factor of 2 for anthracene and a factor of 4 for RDX, in agreement with previous work [5], whereas the LDA potential predicts a value which is roughly half of the experimental one, also in general agreement with previous observations [24]. A previous second-order perturbation theory calculation gave a band gap value of 5.25 eV for RDX [25], in reasonable agreement with the B3LYP band gap of 5.54 eV.
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W.F. Perger / Chemical Physics Letters 368 (2003) 319–323
Fig. 3. Band structure for anthracene, using density-functional theory and the B3LYP hybrid functional.
Table 1 Comparison of band gaps and HOMO–LUMO splittings in anthracene, PE, PETN and RDX for a variety of exchange and correlation potentials Potential
HF B3LYP B3PW LDA-VWN
Anthracene
PE
PETN
RDX
Egap
HO–LU
Egap
HO–LU
Egap
HO–LU
Egap
HO–LU
8.18 3.45 3.56 2.36
8.42 3.64 3.65 2.50
15.6 7.54 7.87 5.21
17.5 8.80 9.19 6.55
14.3 5.96 5.99 3.89
14.8 6.47 6.48 4.22
13.8 5.54 5.53 3.59
14.7 5.92 5.90 3.69
All values are in eV and are at the k ¼ ð000Þ-point. Experimental values for the optical band gaps are 4.4 eV for anthracene [22] and 3.4 eV for RDX [23].
In summary, the results of this study for anthracene, PE, PETN, and RDX indicate the utility of the B3LYP hybrid functional as an efficient means for calculating band gaps for these molecular crystals, as had been recently observed in other materials [6].
Acknowledgements The author would like to acknowledge the financial support of the US Office of Naval Research (MURI program) for this work. Also acknowledged are the extended visits at and collaboration with the Institute for Shock Physics, Washington State University, where Dr. Y. Gupta et al. provided the motivation for this work. Dis-
cussions with Dr. Ravi Pandey and Dr. Miguel Blanco are also gratefully acknowledged.
References [1] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [2] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [3] J.P. Perdew, K. Burke, Y. Wang, Phys. Rev. B. 54 (23) (1996) 16533. [4] I.N. Remediakis, E. Kaxiras, Phys. Rev. B. 59 (8) (1999) 5536. [5] P. Fulde, Electron Correlations in Molecules and Solids, Springer, Berlin, 1991. [6] J. Muscat, A. Wander, N.M. Harrison, Chem. Phys. Lett. 342 (2001) 397. [7] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [8] R. Dovesi, M. Causa, R. Orlando, C. Roetti, V. Saunders, J. Chem. Phys. 92 (12) (1990) 7402.
W.F. Perger / Chemical Physics Letters 368 (2003) 319–323 [9] V.R. Saunders, R. Dovesi, C. Roetti, M. Causa, N.M. Harrison, R. Orlando, C.M. Zicovich-Wilson, CR Y S T A L 98 UserÕs Manual, University of Torino, Torino, 1998. [10] J.J. Gilman, Philos. Mag. B 67 (1993) 207. [11] D.W.J. Cruickshank, Acta Cryst. 9 (1956) 915. [12] D. Eilerman, R. Rudman, Acta Cryst. B 35 (1979) 2458. [13] A.I. Kitaigorodskii, Organic Chemical Crystallography, Consultants Bureau Enterprises, New York, 1961. [14] C.S. Choi, Acta Cryst. B 28 (1972) 2857. [15] A.B. Kunz, Phys. Rev. B. 53 (15) (1997) 9733. [16] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B. 37 (1988) 785. [17] J.P. Perdew, Electronic Structure of Solids, Akademie Verlag, Berlin, 1991.
323
[18] J.P. Perdew, Y. Wang, Phys. Rev. B. 23 (1981) 5048. [19] J.P. Perdew, Y. Wang, Phys. Rev. B. 40 (1989) 3399. [20] J.P. Perdew, K. Burke, Y. Wang, Phys. Rev. B. 45 (1992) 13244. [21] U. von Barth, L. Hedin, J. Phys. C: Solid State Phys. 5 (1972) 1629. [22] P.J. Bounds, W. Siebrand, Chem. Phys. Lett. 75 (1980) 414. [23] P.L. Marinkas, J. Lumin. 15 (1977) 57. [24] J.P. Perdew, M. Levy, Phys. Rev. Lett. 51 (1983) 1884. [25] M.M. Kuklja, A.B. Kunz, J. Appl. Phys. 89 (9) (2001) 4962.
Calculation of band gaps in molecular crystals using hybrid functional theory W.F. Perger
*
Electrical Engineering and Physics Departments, Michigan Tech University, Houghton, MI 49931, USA Received 28 October 2002; in final form 20 November 2002
Abstract Hybrid functional theory is applied for calculation of band gaps in the molecular crystals anthracene, pentaerythritol (PE), pentaerythritol tetranitrate (PETN), and cyclotrimethylene trinitramine (RDX). The B3LYP hybrid functional is observed to produce band gap estimates in reasonable agreement with experiment for anthracene and RDX. This approach, which has been successfully used recently for other materials, is efficient and practical, which is especially important for these large molecular crystals. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Density-functional theory (DFT) in the localdensity approximation (LDA) [1,2] and, more recently, the generalized gradient approximation [3] have been successful for calculating the total energy in molecules and solids. However, accurate band gap calculations for semiconductors and insulators remains a problem of contemporary interest [4]. The ability of DFT to provide accurate, ab initio, ground-state properties in molecules is well known, and in cases where correlation is not strong, it may be reasonable to treat solids and molecules in the same way [5], that is, with the same theoretical methods. Molecular crystals, such as those studied here, would seem to fall into this category. Encouraged by the recent work of *
Fax: +9064872949. E-mail address: [email protected] (W.F. Perger).
Muscat et al. [6] which showed that the hybrid functional approach of Becke [7] can be used to improve the agreement with experiment for calculated band gaps for a wide variety of materials, the same approach is applied here to the molecular crystals anthracene, pentaerythritol (PE), pentaerythritol tetranitrate (PETN), and cyclotrimethylene trinitramine (RDX). Molecular crystals have not been as thoroughly investigated as other materials owing to the typical computational complexity involved. At the time of one of the first ab initio molecular crystal studies, on urea, just over 10 years ago [8], a system with 30 atoms per unit cell was at the achievable limit. During the intervening years, the advances made in utilizing symmetry to improve performance of the CR Y S T A L program [9], the progress made in basis sets and correlation theory, especially at the molecular level, and the improvements in computer hardware, have made first-principles studies of rela-
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 8 7 9 - 1
320
W.F. Perger / Chemical Physics Letters 368 (2003) 319–323
tively large molecular crystals possible. The purpose of this Letter is to add to the growing body of knowledge on the use of DFT as a practical means of achieving accurate calculations for band gaps in materials, in this case the molecular crystals anthracene, PE, PETN and RDX. In addition to the relevance of the band gap as an important material property, the need for an accurate determination of the band gap assumes a special significance for the case of energetic materials such as RDX and PETN. In RDX, for example, a possible mechanism proposed by Gilman [10] for the shock-todetonation transition is a strain-induced reduction of the band gap, ultimately leading to the breaking of the N–NO2 bond. Accurate band gap calculations are therefore of central importance for determining if reasonable wave front pressures could cause such a reduction in the band gap. For comparison, different model potentials and their effect on the band gaps and the molecular highest occupied molecular orbital (HOMO)–lowest unoccupied molecular orbital (LUMO) splitting will be presented.
2. Molecular crystals studied Anthracene, C14 H10 , forms a monoclinic structure, space group P 21 =a with two molecules per unit cell. The internal (fractional) co-ordinates used in the current study for both the carbon and hydrogen atoms were taken from Cruickshank [11]. PE, CðCH2 OHÞ4 , crystallizes in the tetragonal structure with space group I4. The internal coordinates were obtained from Eilerman and Rudman [12]. PETN, ½CðCH2 ONO2 Þ4 is a molecular crystal with 29 atoms per molecule, with two molecules per unit cell, and belongs to the tetragonal P 421 c space group. The internal co-ordinates used were from Kitaigorodskii [13]. RDX, ½ðCH2 NNO2 Þ3 has 21 atoms per molecule, eight molecules per unit cell, and belongs to the orthorhombic Pbca space group, with internal co-ordinates from Choi [14]. This results in 1176 atomic orbitals for the 168 atoms per unit cell and for this size of a system, it becomes especially important to utilize efficient numerical techniques and accurate physical models in order to obtain results in a
reasonable time-frame. This fact serves to motivate using practical computational approaches, such as the hybrid functional method, which was recently shown to provide good band gap estimates for semi-conductors, semi-ionic oxides, and transition metal oxides [6].
3. Approach The CR Y S T A L 98 [9] program was used, initially with a basis set of sð6Þspð2Þspð1Þ for carbon, nitrogen, and oxygen, and sð2Þsð1Þ for hydrogen. A Gaussian basis set was chosen, a sð6Þspð2Þspð1Þ set for the carbon atom and a sð2Þsð1Þ for the hydrogen atom (Ô6–21gÕ), as it had been successfully used previously for RDX [15]. A slightly more complicated basis set was also tried, the optimized carbon basis set of Dovesi et al. [8] and the effect this had on the band gap was observed; seeing little or no effect, the simpler basis set was chosen for the remainder of this work. The basis set superposition error (BSSE) was corrected via the counterpoise method, as previously done for urea [8]. In conjunction with this, a limited basis set optimization was performed by varying the scaling of the exponent of the outermost orbital, in a manner previously described for the molecular crystals RDX and TATB [15]. In that study, it was determined that the outermost exponent for hydrogen should be 1.05, and 1.10 for carbon, nitrogen, and oxygen. In the present study it was found for RDX that exponent values of 1.05 for hydrogen and 1.03 for C, N, and O resulted in the minimum energy; for anthracene, values of 1.0 for all atoms gave the lowest energy. However, in both cases, the differences in the total energy for these various exponent values were quite small, typically on the order of 0.1–0.01 a.u. Furthermore, a greater BSSE was found for larger exponent values. Therefore, the exponent scaling chosen for this study was 1.0 for all orbitals. Once the basis sets were determined, an efficient and practical means must be adopted for solving the Schr€ odinger equation while retaining the physics of exchange and correlation. In order to study the effect of the exchange–correlation po-
W.F. Perger / Chemical Physics Letters 368 (2003) 319–323
321
tential on the band gap, several potentials were chosen. The B3LYP functional was chosen based on its success to estimate band gaps for a variety of materials [6] and has the form LSDA Fock LSDA GGA Exc ¼ Exc þ 0:2ðExc Exc Þ þ 0:72DExc ;
ð1Þ where LSDA is the local spin-density approximation and GGA is the generalized gradient approximation and the parameters were determined by Becke [7] to optimize atomization energies, ionization potentials, and proton affinities were several molecules. The B3LYP hybrid functional combines BeckeÕs three-parameter functional [7] for the exchange with the non-local correlation potential of Lee–Yang–Parr [16]. The CR Y S T A L 98 [9] program explicitly evaluates the non-local Fock energy, then blends 20% of that value into the hybrid functional, as shown in Eq. (1). A second hybrid functional used was the B3PW which uses the Becke exchange combined with the Perdew–Wang GGA (PWGGA) [17–20] correlation. Next, a LDA exchange and von Barth–Hedin (VBH) [21] correlation potential (non-hybrid) functional was chosen. Finally, for comparison, a Hartree–Fock (HF) potential was used.
Fig. 1. Density of states for anthracene, using density-functional theory and the Hartree–Fock potential.
4. Results
Fig. 2. Density of states for anthracene, using density-functional theory and the B3LYP hybrid functional.
Fig. 1 shows the total density of states (DOS) for anthracene using the simpler 6–21g basis set and Fig. 2 shows the DOS using the B3LYP hybrid DFT functional. As can be seen by comparing those figures, the band gap narrows from about 8.2 to about 3.6 eV. The band structure for the upper five valence bands and lowest five conduction bands using the B3LYP potential is shown in Fig. 3. As with the molecular crystal urea [8], some of the bands are rather flat, likely due to weak coupling of the crystalline environment on the molecular states. A comparison of the band gaps calculated for anthracene, PE, PETN, and RDX using the exchange–correlation functionals described earlieris given in Table 1. For comparison, the highest occupied molecular orbital (HOMO)–lowest
unoccupied molecular orbital (LUMO) splittings are also given, as calculated both by using the CR Y S T A L 98 and the GA U S S I A N 98 programs, where possible, using the same basis sets and model potentials. As is apparent from Table 1, the Hartree–Fock potential over-estimates the band gap relative to experiment by roughly a factor of 2 for anthracene and a factor of 4 for RDX, in agreement with previous work [5], whereas the LDA potential predicts a value which is roughly half of the experimental one, also in general agreement with previous observations [24]. A previous second-order perturbation theory calculation gave a band gap value of 5.25 eV for RDX [25], in reasonable agreement with the B3LYP band gap of 5.54 eV.
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W.F. Perger / Chemical Physics Letters 368 (2003) 319–323
Fig. 3. Band structure for anthracene, using density-functional theory and the B3LYP hybrid functional.
Table 1 Comparison of band gaps and HOMO–LUMO splittings in anthracene, PE, PETN and RDX for a variety of exchange and correlation potentials Potential
HF B3LYP B3PW LDA-VWN
Anthracene
PE
PETN
RDX
Egap
HO–LU
Egap
HO–LU
Egap
HO–LU
Egap
HO–LU
8.18 3.45 3.56 2.36
8.42 3.64 3.65 2.50
15.6 7.54 7.87 5.21
17.5 8.80 9.19 6.55
14.3 5.96 5.99 3.89
14.8 6.47 6.48 4.22
13.8 5.54 5.53 3.59
14.7 5.92 5.90 3.69
All values are in eV and are at the k ¼ ð000Þ-point. Experimental values for the optical band gaps are 4.4 eV for anthracene [22] and 3.4 eV for RDX [23].
In summary, the results of this study for anthracene, PE, PETN, and RDX indicate the utility of the B3LYP hybrid functional as an efficient means for calculating band gaps for these molecular crystals, as had been recently observed in other materials [6].
Acknowledgements The author would like to acknowledge the financial support of the US Office of Naval Research (MURI program) for this work. Also acknowledged are the extended visits at and collaboration with the Institute for Shock Physics, Washington State University, where Dr. Y. Gupta et al. provided the motivation for this work. Dis-
cussions with Dr. Ravi Pandey and Dr. Miguel Blanco are also gratefully acknowledged.
References [1] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [2] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [3] J.P. Perdew, K. Burke, Y. Wang, Phys. Rev. B. 54 (23) (1996) 16533. [4] I.N. Remediakis, E. Kaxiras, Phys. Rev. B. 59 (8) (1999) 5536. [5] P. Fulde, Electron Correlations in Molecules and Solids, Springer, Berlin, 1991. [6] J. Muscat, A. Wander, N.M. Harrison, Chem. Phys. Lett. 342 (2001) 397. [7] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [8] R. Dovesi, M. Causa, R. Orlando, C. Roetti, V. Saunders, J. Chem. Phys. 92 (12) (1990) 7402.
W.F. Perger / Chemical Physics Letters 368 (2003) 319–323 [9] V.R. Saunders, R. Dovesi, C. Roetti, M. Causa, N.M. Harrison, R. Orlando, C.M. Zicovich-Wilson, CR Y S T A L 98 UserÕs Manual, University of Torino, Torino, 1998. [10] J.J. Gilman, Philos. Mag. B 67 (1993) 207. [11] D.W.J. Cruickshank, Acta Cryst. 9 (1956) 915. [12] D. Eilerman, R. Rudman, Acta Cryst. B 35 (1979) 2458. [13] A.I. Kitaigorodskii, Organic Chemical Crystallography, Consultants Bureau Enterprises, New York, 1961. [14] C.S. Choi, Acta Cryst. B 28 (1972) 2857. [15] A.B. Kunz, Phys. Rev. B. 53 (15) (1997) 9733. [16] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B. 37 (1988) 785. [17] J.P. Perdew, Electronic Structure of Solids, Akademie Verlag, Berlin, 1991.
323
[18] J.P. Perdew, Y. Wang, Phys. Rev. B. 23 (1981) 5048. [19] J.P. Perdew, Y. Wang, Phys. Rev. B. 40 (1989) 3399. [20] J.P. Perdew, K. Burke, Y. Wang, Phys. Rev. B. 45 (1992) 13244. [21] U. von Barth, L. Hedin, J. Phys. C: Solid State Phys. 5 (1972) 1629. [22] P.J. Bounds, W. Siebrand, Chem. Phys. Lett. 75 (1980) 414. [23] P.L. Marinkas, J. Lumin. 15 (1977) 57. [24] J.P. Perdew, M. Levy, Phys. Rev. Lett. 51 (1983) 1884. [25] M.M. Kuklja, A.B. Kunz, J. Appl. Phys. 89 (9) (2001) 4962.