Hyperfine interactions and lattice distortion of the F center in KCl, NaCl and LiCl crystals

Journal of Molecular Structure (Theochem) 580 (2002) 65±73

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Hyper®ne interactions and lattice distortion of the F center in KCl, NaCl and LiCl crystals A.A. LeitaÄo a,*, R.B. Capaz b, N.V. Vugman b, C.E. Bielschowsky a a

Instituto de QuõÂmica, Universidade Federal do Rio de Janeiro, Cidade UniversitaÂria, CT Bloco A, Rio de Janeiro, 21949-900 RJ, Brazil b Instituto de FõÂsica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, 21945-970 RJ, Brazil Received 4 January 2001; accepted 14 March 2001

Abstract Detailed embedded-cluster quantum-chemistry (ROHF, UHF and UMP2) calculations of hyper®ne interactions for the F center in KCl, NaCl and LiCl are presented. Our values for the isotropic hyper®ne interaction show a good general agreement with experiments. We show that a careful treatment of the lattice environment, local lattice distortion, spin polarization and correlation are important for this good agreement. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Ionic crystals; F center; Embedded cluster calculations; Electronic structure; Hyper®ne interactions

1. Introduction The F center in alkali halides (an electron trapped in an anionic vacancy) is the `textbook' point defect. It has been extensively studied since the early days of solid state physics [1±19], both because of its simplicity and its striking effect on the optical properties of these materials. Perfect alkali halides are transparent to the visible light due to the large energy gap between valence and conduction bands, from 6 to 13 eV depending on the crystal [20,21]. When these crystals are exposed to radiation, they get colored, because electrons or holes are trapped by defects in the crystal creating new electronic occupied and unoccupied states. The F center is the simplest and most abundant of these defects and absorbs between 2.5 and 4 eV. For having such well understood properties, the F center is the perfect benchmark for testing new ideas and techniques of cluster-like electronic structure * Corresponding author.

calculations of point defects in ionic crystals. In particular, it is the ideal system for studying the effects of lattice environment in defect properties, which are manifested both in the long-range nature of the Coulomb lattice potential of ionic crystals and also (and specially) in the strong local coupling between the trapped electron and its neighboring ions. Indeed, the F center corresponds to a limiting situation of this in¯uence since the trapped electron is not attached to any speci®c nucleus and therefore must be very sensitive even to small lattice relaxation. For these reasons, the F center has been used to test different methods for electronic structure, such as oneelectron model calculations [1±8], many-electron LCAO spin-restricted open-shell Hartree±Fock (ROHF) calculations [9,10], spin-unrestricted shell Hartree±Fock (UHF) calculations [11,12] (the former within the muf®n-tin potential (MSXa)), ROHF calculations with improved boundary conditions [12±14] and GW quasiparticle calculations in periodic supercells [15]. Most of the theoretical

0166-1280/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0166-128 0(01)00596-6

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Fig. 1. NaCl Cluster used to set up Env1. The Na atoms are the six ®rst neighbors to the vacancy and the Cl atoms are the ®rst neighbors of these Na atoms. The total charge of the cluster is 213.

calculations have been mainly concerned with the position of the ground and excited defect levels, i.e. with the optical absorption spectra. In particular, two recent works provided interesting information about the effect of lattice environment. Adachi and Kosugui [14] correlated the dependence of the F center orbital size for the ground and excited states with the size of the cluster considered in the quantum-mechanical calculations. They surrounded the cluster with point charges according to the procedure suggested by Evjen [22]. In a later work, Miyoshi et al. [16] studied the problem of the F center in MgO by considering the cluster surrounded by an environmental potential (EP). They concluded that, for the low-lying electronic states of surface and bulk F centers, a Hartree± Fock embedded cluster model with an EP satisfactorily describes the problem. In the work of Illas et al. [17], careful MRCI calculation for the electronic structure and optical properties of the F center in MgO crystals were performed, considering a quantum mechanical system composed by the vacancy, a valence [Mg 21]6[O 22]12 cluster surrounded by eight Mg 21 atoms with 3s basis and pseudopotentials, a set of 24 pseudopotentials representing the core of the Mg 21 ions that surrounds the

O 22 ions and ®nally 292 point charges added to describe the long term Colombian potential. Even so, some discrepancies for the excitation energies were found. Calculations of the electronic structure of F centers 2 1 for characterization of Li14 F13 clusters [18], formation energy [19] and optical properties [17] for F centers in MgO surfaces have also been recently reported. This series of interesting works has discussed problems related to the calculation of excitation energies for which the authors claim that correlation and spin-polarization effects are not important. For other problems, such as calculation of hyper®ne interactions, correlation and polarization are essential and have not been suf®ciently discussed. In fact, previous many-electron cluster UHF calculations of hyper®ne interactions on the F center have not included correlation effects [11,12]. Also, both calculations did not include lattice relaxation. They present severe discrepancies, up to 300%, with experimental results. In the gas phase several recent works [23±28] have shown the importance of spin-polarization and correlation effects, the need of an appropriate choice of the Gaussian basis set and discussed the problem of spin-contamination in UHF calculations. Including

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Fig. 2. NaCl cluster plus TIPs used to set up Env2. The cluster showed in Fig. 1 was surrounded by 38 TIPs representing the Na 1 ions ®rst neighbors of Cl atoms. The total charge of the set (cluster plus Tips) is 125.

these effects, the calculations have been able to predict accurately the experimental results. The objective of the present work is to provide an extensive discussion on the importance of polarization, correlation and lattice relaxation effects for the calculation of hyper®ne interactions in the ®rst neighbors of the F center. For this purpose, we perform ROHF, UHF and Moller±Plesset second-order perturbation theory based on UHF reference (UMP2) [29] cluster calculations in different host crystals. Details of the calculations will be presented in Section 2. 2. Theoretical methods Adachi and Kosugui [14] and Miyoshi et al. [16] have shown that for the ground state of the F center a [Vac(cation)6(anion)12] 72 cluster …Vac ˆ vacancy†; surrounded by point charges, is large enough to ensure the convergence of the defect orbital with cluster size. This cluster consists of the vacancy, its six ®rst-neighbor cations and its twelve second-neighbor anions. Following Berrondo and Rivas-Silva [30], we add six more anions, arriving at a [Vac(cation)6(anion)18] 132 quantum mechanical cluster, shown in Fig. 1.

A 6-311 1 G p Gaussian basis set is used for the cations, which includes polarization and diffuse functions that are necessary to properly describe the delocalized orbital of the trapped electron. We have also tested using additional basis functions at the vacancy site but no signi®cant changes in the electronic structure were observed. The 18 anions were represented by the LAN1DZ basis set and pseudopotential [31±33]. The long-range Coulomb potential of the in®nite crystal is taken into account by embedding the cluster in a (11 £ 11 £ 11) cube of point charges. The charges inside the cube were 11 or 21 according to the replacement of a cation or a anion. Fractional charges on the cube faces, edges and corners were used to improve convergence of the Madelung potential at the cluster center and to keep the neutrality of the system [22,34]. This kind of environment was called Env1. Several works [17±19] have shown the convenience of using total ion potentials (TIPs) for ionic systems. For this reason, in the NaCl cluster, we surrounded the 18 Cl 2 ions by 38 LAN1DZ [31±33] pseudopotentials representing the 38 neighboring Na 1 ions as showed in Fig. 2. These pseudopotentials were originally built to represent the 10 core electrons of

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A.A. LeitaÄo et al. / Journal of Molecular Structure (Theochem) 580 (2002) 65±73

the Na atom. This cluster plus the TIPs were embedded by point charges like the environment Env1 and was called Env2. In other words, Env2 was made by cluster plus TIPs plus point charges. The point charges are absent in Figs. 1 and 2. For the LiCl clusters, instead of using cation pseudopotentials, we were able to introduce 38 Li atoms with 6-31G basis set functions that surrounded the 18 Cl 2 ions. This new LiCl cluster had the same size of cluster plus TIPs (Fig. 2) in Env2 for NaCl F center. The new LiCl cluster plus the point charges scheme was called environment Env3. Unfortunately, the calculations for the KCl cluster presented convergence problem when additional 38 pseudopotentials representing the 38 K 1 ions were added, probably due to the huge size of the calculations. As consequence, only Env1 model was used for the KCl F center. Geometry, electronic structure and hyper®ne interactions are determined by ab initio Unrestricted Hartree±Fock (UHF) calculations followed by the unrestricted Mollet±Plesset perturbation theory (UMP2) [24]. In order to analyze the different contributions to the hyper®ne interactions, Restricted Hartree±Fock Open Shell (ROHF) and Unrestricted Hartree±Fock (UHF) wavefunctions on the equilibrium geometry are also determined. The calculated non-projected kS^ 2 l values for the UHF wavefunctions are between 0.750 and 0.751 for the different crystals considered in this work and between 0.7500 and 0.7505 for the UMP2 wavefunctions. The isotropic hyper®ne coupling is determined for the cations that surrounds the F center by calculating [26]: Aiso …X† ˆ …4P=3†g e be gX bX kSZ l21 r…X† where g is the electron g-factor, b e is the Bohr magneton, gX and b X are the analogous quantities for nucleus X and r (X) is the spin density on the nucleus X. 3. Results In order to investigate the effects of lattice relaxation on the hyper®ne structure, we calculate the total

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energy and Aiso as a function of the ®rst-neighbors' displacements from their equilibrium positions, keeping all other atoms ®xed. Figs. 3 and 4 show, Aiso and the total energy as a function of the ®rst-neighbors' displacements, respectively. Notice that the total energy surface is very ¯at in the vicinity of the minimum. Nevertheless, Aiso is very sensitive to even small displacements. This suggests that a careful treatment of lattice relaxation is important to describe hyper®ne effects in an accurate manner. In particular, NaCl cluster presents a local in¯ection for Aiso tendency, probably due to the greater packing of the NaCl lattice when compared to the other two. Table 1 shows the ®nal UMP2 values for geometry and Aiso. The experimental values for Aiso are also included in the ®gures and in Table 1. The minimum in the total energy vs. displacement plot and the agreement between calculated and measured Aiso at the minimum can be seen as two independent determinations of the lattice distortion around the F center for these crystals. For the NaCl cluster, the calculated geometric distortion and (Aiso) for the ®rst neighbors changed when passing from Env1 to Env2, the latter Ê and the calculations presented a distortion of 20.05 A Aiso of 56.8 MHz. These results show that for calculating distortion and Aiso in NaCl clusters, total ion potentials are necessary to improve the environment short range interaction. We have performed calculation at LiCl F center using environment Env3 and UHF method and the geometry distortion and Aiso Ê and 15.8 MHz, respectively. value were 20.01 A These results indicate that Env1 is suf®cient to represent an F center in LiCl crystal and also that relaxation is signi®cant only for F center in NaCl crystal. We now focus on the importance of spin-polarization and correlation effects for describing hyper®ne interactions. Table 2 shows the calculated values for Aiso in the ®rst-neighbor cations for the different crystals, as well as the values for the isolated atoms, within three distinct methods: the ROHF method, which does not consider spin-polarization or correlation effects; the UHF method, which considers spinpolarization effects and part of the non-dynamical correlation effects; and the UMP2 method, which

Fig. 3. Isotropic hyper®ne interaction on the ®rst neighbors to the F center, for the distinct crystals as a function of lattice relaxation. The calculations were made using environment Env1 and UMP2 method. Positive values on the x axis mean outward radial displacements of the ®rst neighboring ions. Lines are drawn as a guide to eye.

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Table 1 Calculated displacement and Aiso on ®rst neighbors as calculated by UMP2 embedded method using environment Env1 Crystal

Ê) Displacement (A

Aiso without displacement (MHz)

Aiso with displacement (MHz)

Experimental Aiso (MHz) a

KCl NaCl LiCl

0.03 0.04 (20.05 b) 0.02 (20.01 c)

19.9 45.4 (53.6 b) 17.0 (15.5 c)

18.8 42.9 (56.8 b) 16.5 (15.8 c)

20.0 61.0 19.1

a b c

Ref. [7]. Results using Env2/UMP2. Results using Env3/UHF.

takes into account part of both spin-polarization, nondynamical and dynamical correlation effects. Table 2 shows that Aiso values for the isolated atoms changes signi®cantly from ROHF to UHF calculations and less signi®cantly from the UHF to the UMP2 result. A very similar picture occurs for the ®rst-neighbor ions in the F center. In abstract, the calculations indicate that relevant dynamical correlation effects occur only for K atom and KCl F center. Although no signi®cant spin contamination was observed in the present atomic UHF calculations, these results indicate that polarization of the innershell spin-orbitals is clearly present. In order to analyze this effect in more detail, we calculate, with the UHF wavefunctions, the contributions to Aiso from each atomic shell, using the following expression: X 2 Aiso …X† ˆ …4P=3†g e be gX bX kSZ l21 …uwni" …X†u ni

2

2 uwni# …X†u † where g is the electron g-factor, b e is the Bohr magneton, gX and b X are the analogous values for nucleus, w ni"(X) and w ni#(X) are the spin-orbitals associated with the n atomic shell on the nucleus X. Table 3 shows these contributions to Aiso for the isolated atoms and also for the ®rst-neighbor atoms to the vacancy on the crystal environment. One observes that only 68% of the total calculated value for Aiso on the isolated Li atom comes from the contribution of the unpaired 2s electron, the other 32% coming from polarization of the 1s electrons. The percentage

contribution to total Aiso values related with the innershell electrons is reduced for the Na and K atoms, being 19 and 14%, respectively. Similarly, for the LiCl, NaCl and KCl crystals, only 69, 72 and 76% of the Aiso, respectively, come from the unpaired electron, the remaining contribution coming basically from inner-valence and inner-shell polarization. Therefore, for these systems, it is crucial to include polarization of the inner-shell electrons in the calculation of Aiso. The differences between the UHF and UMP2 values, presented in Table 2, indicates that dynamical correlation also contributes to the calculated Aiso values, specially for isolated potassium atom and for the K ion near the F center in the KCl host lattice. We have quoted in Table 2 the percentage differences between the UMP2 and UHF values for Aiso on the isolated atoms and also on the ®rst neighbors on the F center. These differences do not change signi®cantly from the isolated atoms to the ions in the crystal environment, indicating that little contribution for the correlation energy comes from the interaction between the unpaired and the host lattice electrons. Table 4 compares the present UMP2 values for Aiso with previous one-electron results by Harker [7], many-electron UHF-LCAO results by Kung et al. [12] and UHF-MSXa results of Yu et al. [11], as well as the experimental results. This comparison shows that previous many-electron calculations [11,12] present serious discrepancies with the experimental results. Part of the discrepancy may be related to correlation effects not considered in those calculations, but

Fig. 4. Total energy as a function of lattice relaxation for the distinct crystals. The calculations were made using environment Env1 and the UMP2 method. Positive values on the x axis mean outward radial displacements of the ®rst neighboring ions.

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Table 2 Calculated ROHF, UHF and UMP2 and experimental values for Aiso (MHz) on ®rst neighbors and for the isolated atoms Isolated atoms

ROHF

UHF

UMP2

K Na Li

148.8 611.1 271.8

183.7 752.7 391.0

214.0 783.7 395.9

12.4 30 12.6

16.5 42.1 16.3

18.8 42.9 16.5

Crystal environment K 1 ion in KCl Na 1 ion in NaCl Li 1 ion in LiCl a

Experimental a

(UMP2-UHF) (%) 16.5 4.1 1.2

20 61 19.1

13.9 1.8 1.2

Ref. [7].

the disagreement between their UHF results and ours (Table 3) indicates further problems. These may be related, in the work of Kung et al. [12], to the small basis set used, leading to an insuf®cient description of polarization effects, and also to the small quantum mechanical cluster used (composed by the vacancy and the ®rst neighbors). In the work of Yu et al. [11], the problems in the UHF calculations may be related to the muf®n-tin potentials (MSXa) used. Table 4 also shows that the one electron model results of Harker [7], where a variational calculation is performed employing model potentials to represent the host lattice, agrees with the experimental results for the Aiso in NaCl crystal but disagrees for LiCl. 4. Conclusions We have calculated the electronic structure for the F center with detailed embedded-cluster ROHF, UHF and UMP2 calculations. The ®rst neighbors relaxation and the isotropic hyper®ne interactions (Aiso) for the F

center in the LiCl, NaCl and KCl host lattices were determined and the importance of spin-polarization and correlation effects on the calculated Aiso were carefully addressed. In order to clearly distinguish between intrinsic atomic contributions and lattice effects, we have also performed analogous calculations for isolated Li, Na and K atoms. Polarization effects were shown to be important both for the isolated atoms and for the ions in the crystals. Indeed, within the UHF calculations, the unpaired electron contributed with approximately 70% of the total Aiso values, the rest coming from polarization of the inner-shell electrons. Dynamical correlation also contributed to the calculated Aiso, but to a lower extent, being more important for the KCl crystal then for the others. The calculations also showed that the Aiso values are very sensitive to even small lattice relaxation, but only in NaCl the relaxation showed to be relevant. The Aiso values proved to be a sensitive test to infer the quality of cluster models. Our ®nal values for the isotropic hyper®ne interaction using environment Env1 showed an excellent

Table 3 Calculated contributions from each shell to Aiso (MHz) on the isolated atoms and on ®rst neighbors, by UHF calculations

Li atom Na atom K atom Li ion in LiCl Na ion in NaCl K ion in KCl

Total Aiso (UHF)

Contribution of the unpaired electron

Contribution of the 1s shell electrons

391 753 184 16.3 42.1 16.5

266 609 158 11.3 30.5 12.5

125 62 1 4.4 9.5 0.5

Contribution of the 2s shell electrons

Contribution of the 3s shell electrons

82 6

19

10.17 1.5

2.5

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Table 4 Comparison with previous theoretical results for Aiso (MHz) on ®rst neighbors Crystal

Kung et al. [12] UHF

Harker et al. [7] one electron model

Yu el al. [11] UHF (MSXa)

Present UMP2

Experimental a

KCl NaCl LiCl

8.23 ± 32.76

28.4 59.0 38.2

Much higher Much higher Much higher

18.8 42.9 16.5

20.7 61 19.1

a

As quoted in Ref. [7].

agreement with experiments, except for the NaCl lattice where only a reasonable agreement was achieved. For F center in NaCl lattice the improved environment Env2 produced excellent results. Our calculations represent a clear improvement over previous cluster calculations [11,12], showing that a careful treatment of different effects (spin-polarization, correlation and lattice relaxation) is crucial when using cluster models to determine Aiso. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

T. Inui, Y. Uemura, Prog. Theor. Phys. 5 (1950) 252. B.S. Gourary, F.J. Adrian, Phys. Rev. 105 (1957) 1180. T. Kojima, J. Phys. Soc. Jpn 12 (1957) 908 see also p. 918. R.F. Wood, J. Korringa, Phys. Rev. 123 (1961) 1138. F. Martino, Int. J. Quantum Chem. 2 (1968) 217 see also p. 233. È pik, R.F. Wood, Phys. Rev. 179 (1969) 772 see also p. 783. U. O A.H. Harker, J. Phys. C: Solid State Phys. 9 (1976) 2273. C.H. Leung, K.S. Song, Can. J. Phys. 58 (1980) 412. R.C. Chaney, C.C. Linn, Phys. Rev. B 13 (1976) 843. J.N. Murrel, J. Tennyson, Chem. Phys. Lett. 69 (1980) 212. H.L. Yu, M.L. de Siqueira, J.W.D. Connolly, Phys. Rev. B 14 (1976) 772. A.Y.S. Kung, A.B. Kunz, J.M. Vail, Phys. Rev. B 26 (1982) 3352.

[13] R. Pandey, M. Seel, A.B. Kunz, Phys. Rev. B 41 (1990) 7955. [14] J.I. Adachi, N. Kosugui, Bull. Chem. Soc. Jpn 66 (1993) 3314. [15] M.P. Surh, H. Chacham, S.G. Louie, Phys. Rev. B 51 (1995) 7464. [16] E. Miyoshi, Y. Miyake, S. Katsuki, Y. Sakai, J. Mol. Struct. (THEOCHEM) 451 (1998) 81. [17] F. Illas, G. Pacchioni, J. Chem. Phys. 108 (1998) 7835. [18] R.F.W. Bader, J.A. Platts, J. Chem. Phys. 107 (1997) 8545. [19] A.M. Ferrari, G. Pacchioni, J. Chem. Phys. 99 (1995) 17010. [20] V.G. Grachev, M.F. Deigen, H.I. Neymark, S.I. Peter, Phys. Status Solidi 43 (1971) 93. [21] R.T. Poole, J.G. Jenkin, J. Liesegang, R.C.G. Leckey, Phys. Rev. B 11 (1975) 5179. [22] H.M. Evjen, Phys. Rev. 39 (1932) 675. [23] D.M. Chipman, Theor. Chim. Acta 82 (1992) 93. [24] D.M. Chipman, I. Carmichael, J. Phys. Chem. 95 (1991) 4702. [25] S.A. Perera, J.D. Watts, R.J. Bartlett, J. Chem. Phys. 100 (1994) 1425. [26] T.-K. Ha, H.U. Suter, M.T. Nguyen, J. Chem. Phys. 105 (1996) 6385. [27] I. Carmichael, J. Phys. Chem. A 101 (1997) 4633. [28] J.W. Gauld, L.A. Eriksson, L. Radom, J. Phys. Chem. A101 (1997). [29] W. Chen, H.B. Schlegel, J. Chem. Phys. 101 (1994) 5957. [30] M. Berrondo, J.F. Rivas-Silva, Int. J. Quantum Chem. 57 (1995) 1115. [31] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 270. [32] W.R. Wadt, P.J. Hay, J. Chem. Phys. 82 (1985) 284. [33] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 299. [34] C. Sousa, J. Casanovas, J. Rubio, F. Illas, J. Comput. Chem. 14 (1992) 680.