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Theoretical study on the electronic structures of various F centers in MgO crystals1
Journal of Molecular Structure (Theochem) 451 (1998) 81±88
Theoretical study on the electronic structures of various F centers in MgO crystals 1 Eisaku Miyoshi*, Yoshiki Miyake 2, Shinichi Katsuki 3, Yoshiko Sakai Department of Applied Physics, Faculty of Engineering, Kyushu University, Ropponmatsu, Fukuoka 810, Japan Received 26 December 1997; accepted 29 January 1998
Abstract Ab initio SCF-MO calculations were performed on low-lying electronic states of various F centers in MgO crystals by using embedded cluster models. Because a simple cluster model embedded with point charges alone gave poor results, we generated environmental potentials (EPs) in a spectral representation scheme for Mg 21 and O 22 ions and used them instead of point charges around the clusters under consideration. Using embedded cluster models with the EPs and point charges, we calculated the low-lying electronic states of various F centers in MgO. The calculated order in energy was found to be: the 1 A1 state of the surface F center , the 1 E state of the surface F center , the 1 B3u state of the bulk M center (F aggregates) , the 1 T1u state of the bulk F center. The energy separation of the 1 E, 1 B3u , and 1 T1u states from the lowest state were 1.22, 2.46 and 4.05 eV, respectively, which explains the observed bands. q 1998 Elsevier Science B.V. All rights reserved. Keywords: MgO crystals; F centers; Ab initio; Environmental potentials
1. Introduction There have been many experimental studies [1±7] on the excited electronic states of oxygen vacancies in MgO crystals. Chen et al. [1] concluded that, although there is little doubt that an intense optical absorption band at 4.96 eV is due to oxygen vacancies that may be both positively charged F 1 centers (one electron trapped at the vacancies) and neutral F centers (two electrons trapped at the vacancies), there is some uncertainty whether the broad absorption bands at * Corresponding author. Tel. 1 81 92 7264788; Fax: 1 81 92 7264788.. 1 Dedicated to Professor Sigeru Huzinaga on the occasion of his 70th birthday. 2 Present address: NEC Hiroshima, Higashi Hiroshima 739-01, Japan. 3 Present address: Fukuoka Jogakuin Junior College, Fukuoka 816, Japan. 0166-1280/98/$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S0 166-1280(98)001 62-6
3.49, 2.16 and 1.27 eV are due to F-aggregate centers. The intense band was also observed by Kappers et al. [3] and was resolved at liquid-nitrogen temperatures into two peaks at 4.96 and 5.03 eV, which were attributed to F 1 and F centers, respectively. Recently, Wu et al. [7] investigated thermally generated defects in ultrathin MgO ®lms on Mo(100) using high-resolution electron-energy-loss spectroscopy (HREELS) in the 0 , 9 eV energy region. In their study, the three distinct loss features observed at 5.33, 3.58 and 1.15 eV were attributed to electronic transitions associated with F (or F 1) centers, F aggregates (M centers), and surface F centers, respectively. On the other hand, theoretical calculations have emphasized the ground-state relaxation of the oxygen vacancies in MgO [8, 9]. There have been few theoretical calculations for electronic transitions associated with oxygen vacancies. Summers et al.
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[4] calculated the absorption energy of the F centers of MgO to be 4.8 eV by the use of a two-electron wavefunction including an electronic polarization effect. Very recently, Tatewaki and Miyoshi [10] have studied the electronic structure of an LiF crystal using an ab initio MO cluster approach, where 2 Li1 n Fm clusters (m, n 13 or 14) were considered to be embedded in the ionic crystal, which is composed of point charges. The calculated band gap of LiF including a correlation correction was 13.8 eV, which is in excellent agreement with experimental values of 13.6 eV [11] and 14.2 eV [12]. The calculated bulk exciton band was 13.3 eV above the valence band, which also agrees well with the observed value of 13.5 eV [13]. They also calculated that the surface exciton state was 11.4 eV above the valence band, while the experimental values are 10.5 eV [13] or 9.1 ^ 0.6 eV [14]. Thus, they have shown that the ab initio MO cluster approach can properly describe the electronic structures of ionic crystals as far as band gaps and the exitation energies of the low-lying exciton states. By using the same method, Miyoshi and Huzinaga [15] investigated the excited states of the U center (H - ion instead of halogen ion) in NaCl. They found that the calculated transition energy for the dipole-allowed state agreed well with the observed value, within about 0.5 eV. The important conclusion they reached is that the correlation energy of two 1s electrons (H 2 ion) in the ground state is almost cancelled by the correlation-energy difference of the remaining electrons between the ground state and excited states. Adachi and Kosugi [16] also examined the excited states of F centers in NaCl using a similar method in the framework of SCF MO calculations. The calculated 1s-2p excitation energy was in good agreement with the experimental data. Thus, molecular orbital studies using embedded cluster models surrounded by point charges can properly describe the electronic structures of defect states in ionic crystals so far. In this study, we have used a similar simple embedded cluster model to investigate the electronic structures of bulk and surface F centers in the MgO crystal. The results of this treatment are discussed in Section 2. The simple embedded cluster model did not describe the difference in the electronic structures between the surface and bulk F centers in MgO. The failure of the model is attributed to the fact
that the excited electron in the model penetrates to the surrounding point charges, which will be shown in Fig. 2(a±c). Accurate modeling of the system requires that electrons only be allowed to penetrate the vicinity of neighboring ions in a manner consistent with the Pauli exclusion principle. In other words, the excited orbitals should be orthogonal to the neighboring ion core orbitals that reside at the lattice points occupied by the point charges in the crystal. In order to take into account such effects, we use a technique, which is brie¯y described in Section 3, incorporating environmental potentials (EPs) [17] in a spectral representation scheme [18±21]. The calculated results obtained by the embedded cluster model with the environmental potentials (EPs) for bulk and surface F centers and M centers are also discussed in Section 3. Concluding remarks are given in Section 4.
2. Simple embedded models In this section, we will discuss the results of simple embedded-cluster-model calculations for bulk and surface F centers in the MgO crystal. The model clusters for the F center used in the present study include up to the fourth nearest neighboring ions: for the bulk F center, FMg101 FMg6 O142 FMg14 O21 6 , 12 , 12 , 102 FMg14 O18 , where F in the cluster notations designates the F center. The clusters are embedded in the ionic crystal composed of point charges. The point charges are generated by the parallel transfer of the unit cell that is composed of positive ( 1 2e) and negative ( 2 2e) point charges. The total systems contain 3375 ions or point charges (including F centers) for the bulk F center and 1998 ions for the surface F center. Charge neutrality is obtained by using Evjen's method [22] as described in previous studies [10, 15]. The basis set for the F center site was the same as that used for the U center in NaCl [15], which was contracted as [7s6p]. Basic sets for Mg and O were taken from the work of Huzinaga et al. [23] with the coef®cients reoptimized for the Mg 21 and O 22 ions, respectively, embedded in the ionic crystal mentioned above. For the nearest neighbors Mg 21, we contracted the basis set to (433/311 *), augmenting a p-type polarization function (1 *) [23]. For others, we used
E. Miyoshi et al. / Journal of Molecular Structure (Theochem) 451 (1998) 81±88
the re-optimized values (433/4) and (43/4) for the Mg 21 and O 22 ions, respectively. Using a simple embedded cluster model, we performed restricted Hartree-Fock calculations for the ground and low-lying excited states of bulk and surface F centers, whose con®gurations are as follows: bulk F center in (Oh symmetry) ground state
1
A1g
(1s) 2,
excited state
3
A1g
(1s) 1(2s) 1,
1;3
T1u
(1s) 1(2p) 1,
surface F center (in C4v symmetry) ground state
1
excited state
1;3
A1
(1s) (2s 1 2pz) ,
1;3
E
(1s) 1(2px,y) 1.
A1
(1s) 2, 1
1
In the bulk F center, the 1 T1u state is only a spin and dipole-allowed transition state, while in the surface F center, both 1 A1 and 1 E states are symmetry allowed. The excited state, 1 A1 , was obtained as the second state in an MCSCF calculation with three con®gurations of (1s) 2, (1s) 1(2s 1 2pz) 1, and (2s 1 2pz) 2. The calculated excitation energies of the low-lying states are summarized in Table 1. In Fig. 1 we show the 1s orbital of the F center, which is localized at the F site and is almost unchanged for various excited states. Excitation energies for both 1 A1 and 1 E states of the surface F centers are larger by 0.1-0.4 eV than those of the bulk F center, while the excitation energy of the surface F center (1.15 eV) is experimentally smaller by about 4 eV than that of the bulk F center (5.33 eV) [7]. The reason for this discrepancy can be seen in Fig. 2 where we show (a) a 2p orbital of the 1 T1u state (bulk F center); and (b) a 2px orbital of the 1 E state in the zx plane normal to the surface. The excited electrons in both the 1 T1u and 1 E states penetrate the vicinity of the surrounding points charges as if there were nothing to interfere with their overlap. Thus, both the orbitals of the bulk and surface F centers become diffuse. In reality, electrons can penetrate the vicinity of the point charges only in a manner consistent with the Pauli exclusion principle. In other words, excited orbitals should be orthogonal to the ion's orbitals that reside at the point charges.
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Table 1 Excitation energies (eV) of various F centers in the MgO crystal using a simple embedded cluster model with point charges a Bulk F center Clusters
3
FMg101 6 FMg6 O142 12 FMg14 O21 12 b FMg14 O21 12 102 FMg14 O18
2.82 0.82 2.02 2.58 1.76
3.05 2.26 2.46 2.62 2.31
2.85 2.13 2.35 2.43 2.21
3
1
E 3.41 2.65 2.64 2.90 2.40
3
E 2.89 2.34 2.40 2.50 2.22
1
3
Surface F center Clusters FMg81 5 FMg5 O82 8 FMg9O 8 b FMg9O8 FMg9 O102 13 Bulk M center Cluster F2 Mg10 O202 18 b a b
1
A1
3.01
A1g
A1 2.22 1.71 1.91 1.92 1.98
1
T1u
B3u 2.06
3
T1u
B3u 1.97
F designates the F center site. [5s4p] was used for the F center site instead of [7s6p].
We also performed calculations for FMg14 O21 12 and FMg9O8 clusters using another smaller basis set [5s4p] for the F center in which the two most diffuse functions (0.013675140 and 0.0063784397) were eliminated. Calculated excitation energies agree well
Fig. 1. The 1s orbital of the 1 T1u state obtained by using a simple cluster model embedded with point charges in the bulk F center model of FMg14 O82 18 . Dotts indicate sites of point charges. Values of contours are ^ 0.08, ^ 0.04, ^ 0.02, ^ 0.01, and ^ 0.005 (a.u.) 23/2.
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with those obtained using a larger basis set [7s6p] except for the bulk 3 A1g state, where the discrepancy is more than 0.5 eV. Thus, the smaller basis set [5s4p] almost describes these states, except for the 3 A1g state. The energy of the 3 A1g state does not converge with the present cluster model calculations. We also performed HF calculations for the ground and excited states of an M center (F aggregates) consistent with defects in the two neighboring O 21 ions, using a simple embedded cluster F2 Mg10 O202 18 in an ionic cage of 4980 point charges. The cluster consists of the nearest neighbors and next nearest neighbors from both F centers. The ground and excited states of the M center are as follows (in D2h symmetry); ground state
1
excited state
1;3
Ag B3u
(1sA 1 1sB) 2 (1sA 2 1sB) 2, (1sA 1 1sB) 2 (1sA 2 1sB) 1 (2pxA 2 2pxB) 1.
The subscripts A and B indicate different F center sites, which are on the x-axis. The calculated excitation energy of the spin and dipole-allowed state of the M center was calculated to be 2.06 eV, which is similar to that of the bulk and surface F centers as calculated by a simple embedded cluster model. The excited state was found to be about 2 eV below the bulk F center and about 2 eV above that of the surface F center. The reason for these discrepancies can be attributed to the arti®cially diffuse nature of the excited orbital (2pxA 2 2pxB), which is shown in Fig. 2(c). 3. Environmental potentials Defect in crystals is one of physical and chemical phenomena that are expected to be understood in terms of localized electronic states, even though the crystal itself is not isolated, but rather is vastly Fig. 2. Excited orbitals calculated by using simple cluster models embedded with point charges. (a) 2p orbitals of the 1 T1u state in the 1 bulk F center model of FMg14 O82 18 . (b) 2px orbital of the E1u state in the surface F center model of FMg9 O102 13 . (c) (2pxA 2 2pxB) orbitals in the M center model of F2Mg10O202 18 . Dots and crosses indicate sites of point charges and ions in the cluster, respectively. Values of contours are ^ 0.08, ^ 0.04, ^ 0.02, ^ 0.01, and ^ 0.005 (a.u.) 23/2.
E. Miyoshi et al. / Journal of Molecular Structure (Theochem) 451 (1998) 81±88
dispersed. Our approach is based on a ®nite cluster model, which consists of two parts: (I) a cluster where the defect state has signi®cant amplitude, and (II) the surrounding environment, which contributes only a background to the cluster. The role of the atoms in region II is especially important in the case of ionic crystals such as MgO. Firstly, they contribute to the Madelung potential at each crystal lattice point. Secondly, the electrons occupying the atoms in region II will repel all other electrons to prevent penetration into the occupied space in region II. The point charges can then serve as a substitute for ionized atoms as far as the ®rst contribution. However, the point charges can not prevent the electron orbitals from penetrating inhibited occupied states in region II. To incorporate the second contribution of the atoms in region II to our model, we divide region II into two parts, region IIa and region IIb. We then use an environmental potential [17] at each lattice point in region IIa and replace each lattice point in region IIb by a point charge, as was also used in the case of the simple embedded model. The following paragraph brie¯y explains the environmental potential. The nuclear attraction, Coulombic forces, and the exchange operator in the Fock operator is expressed as follows: 2 3 X N A A 2Jj 2 Kj 5V 1 PA (1) VEP V42 1 rA j where NA is the number of electrons in ion A and j runs A as an environover all NA electrons. We refer to VEP mental potential (EP). The projection operator V is expressed as follows with the use of a function set {xa} which is not necessarily mutually orthogonal; X uxa l S21 ab kxb u (2) V ab
Sab kxa uxb l
(3)
The operator in the following form X uxa l S21 ab kxb uOuxc l S21 cd kxd u VOV
(4)
abcd
is called a spectral representation of an operator O [18±21]. PA is a shift operator preventing the valence orbitals in region I from collapsing into the space which is spanned by the jth occupied orbitals fjA
85
belonging to the A ion in region IIa, X PA 2 21Aj ufAj lkfAj u
(5)
j
The numbers of the ions, which were represented by the EP of Eq. (1) were 336 , 310 for models of the bulk F center and 190 , 173 for models of the surface F center. EPs for the Mg 21 and O 22 ions were developed in this study. The total number of ions considered in a cluster (region I) and the EPs (region IIa) was 363 (including the F center itself) for the bulk F center, and 197 for the surface F center. For the M center (F aggregates) we substituted 366 point charges cluster with EPs. The surrounding the F2Mg10O202 18 rest of the surrounding ions remained as point charges. The resulting excitation energies for the bulk and surface F centers and the M center in the MgO crystal, calculated by using the various embedded cluster models containing EPs (region IIa) and point charges (region IIb), are listed in Table 2. The spin and dipoleallowed transition state, 1 T1u , of the bulk F center is destabilized by about 5 eV as compared to the value obtained from the simple embedded calculation. The 1 A1 and 1 E states of the surface F center are destabilized by only 0.3 and about 2 eV, respectively. A large amount of destabilization in the bulk F center Table 2 Excitation energies (eV) of various F centers in the MgO crystal calculated using embedded cluster models containing EPs and point charges a Bulk F center Clusters
3
FMg101 6 FMg6 O142 12 FMg14 O21 12 FMg14 O102 18
19.76 12.55 11.30 10.24
9.09 7.59 7.45 7.38
5.00 4.45 4.35 4.28
3
1 E 6.52 5.04 4.54 4.55
3 E 4.18 3.52 3.29 3.31
1
3
Surface F center Clusters FMg81 5 FMg5 O82 8 FMg9O 8 FMg9 O102 13
1
A1
3.33
Bulk M center Cluster F2 Mg10 O202 18 a
F designates the F center site.
A1g
A1 2.24 1.68 1.59 1.56
1
T1u
B3u 5.79
3
T1u
B3u 3.83
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E. Miyoshi et al. / Journal of Molecular Structure (Theochem) 451 (1998) 81±88
is due to the interference caused by the penetration of EPs into the vicinity of the surrounding ionic sites. This causes the excited 2p orbital in the bulk F center to become contracted, as is shown in Fig. 3(a) which is in contrast to Fig. 2(a). The excited 2px,y and 2s 1 2pz orbitals in the surface F center also become contracted, but they can diffuse into the vacuum, as is seen in Fig. 3(b). This explains why the orbitals are less destabilized than in the case of the bulk F center. The spin and dipole-allowed transition state of the M center is raised by 3.7 eV due to the use of EPs. The resulting orbital is depicted in Fig. 3(c), where a tight (2pxA 2 2pxB) orbital is seen. The results of the embedded cluster model calculations using EPs (region IIa) and point charges (region IIb) are listed in Table 3 and compared with those of the simple embedded calculations and experimental values. The lowest spin and dipole-allowed state is the 1 A1 state in the surface F center, whose symmetry is the same as the z-axis (perpendicular to the surface). The simple embedded cluster calculations determine this state to be the highest and give a wrong order of states, where all the states are within 1 eV of each other. The embedded cluster model calculations using EPs give the correct order. Although the calculated excitation energies are constantly higher than those of the experimental values by about 2 eV, the calculated energies relative to that of the lowest excited state are well consistent with experiments. The bulk F and M centers are calculated to be 4.05 and 2.46 eV above the lowest state, respectively, while the observed values are 4.18 and 2.43 eV [7] or 3.69 and 2.22 [1], respectively. Another surface 1 E state is 1.22 eV above the 1 A1 state. Chen et al. observed a broad band at 2.16 eV, which is higher than the lowest band by 0.89 eV. Thus, the broad adsorption band observed by Chen et al. can be assigned to the 1 E state of the surface F center. The overestimation of excitation energies is due to
Fig. 3. Excited orbitals calculated by using embedded cluster models containing EPs and point charges. (a) 2p orbitals of the 1 T1u state in the bulk F center model of FMg14O82 18 . (b) 2px orbital of the 1 E1u state in the surface F center model of FMg9 O102 13 . (c) (2pxA 2 2pxB) orbitals in the M center model of F2Mg10O202 18 . Crosses indicate sites of EPs or ions in the cluster. Values of contours are ^ 0.08, ^ 0.04, ^ 0.02, ^ 0.01, and ^ 0.005 (a.u.) 23/2.
E. Miyoshi et al. / Journal of Molecular Structure (Theochem) 451 (1998) 81±88
87
Table 3 Comparison of calculated and experimental values for excitation energies (eV) of bulk and surface F centers, and bulk M centers in the MgO crystal calculated using embedded cluster models containing EPs and point charges F centers
Bulk F 1 T1u
Surface F 1 A1
Simple embedded cluster
2.31 (20.70) 7.38 (4.05) 5.33 (4.18) 4.96 (3.69)
3.01 (0.00) 3.33 (0.00) 1.15 (0.00) 1.27 (0.00)
Embedded cluster with EPs Exptl. a Exptl. b a b
1
E
2.40 (20.61) 4.55 (1.22) 2.16 (0.89)
Bulk M 1 B3u 2.06 (20.95) 5.79 (2.46) 3.58 (2.43) 3.49 (2.22)
See ref. [7]. See ref. [1].
the intense effect of the shift operator in Eq. (5), which prevents the valence orbitals of the clusters in region I from collapsing into the space that is spanned by the neighboring ion core orbitals in region IIa. However, it is dif®cult at the present stage for the operator so as properly to work, because it requires that all the core orbitals of the neighboring ions should be considered explicitly. Finally, we shall comment on the electron correlation of the F centers of the MgO crystal. The electronic state of the F center is essentially the same as the U center (H 2 ion instead of halogen ion) in the NaCl crystal, as studied by Miyoshi and Huzinaga [15]. They concluded that the correlation energy of the two 1s electrons in the ground state is almost cancelled by the correlation-energy difference of the remaining electrons between the ground state and excited states. Thus, since the F center in MgO is expected to be similar to the U center in NaCl, we do not take the correlation energy into consideration in the present study.
4. Conclusions We performed ab initio MO calculations utilizing embedded cluster models to investigate the low-lying electronic states of the F centers of the MgO crystal. A simple cluster model with surrounding point charges gave poor results. We utilized the environmental potentials (EPs) instead of point charges around the clusters con-
sidered; a concept originally proposed by Huzinaga et al. [17]. We generated EPs for Mg 21 and O 22 ions in a spectral representation scheme [18±21]. In the embedded cluster model calculations using the EPs, we have described the 1 A1 state of the surface F center as the lowest spin- and dipole-allowed state. The 1 E state of the surface F center was calculated to be 1.22 eV above the lowest state, and the 1 B3u state of the bulk M center and the 1 T1u state of the bulk F center were found to be 2.46 and 4.05 eV above the lowest state, while the corresponding experimental values are 0.89, 2.22 (2.43) and 3.68 (4.18) eV, respectively [1, 7]. Thus, ab initio MO calculations using the embedded cluster model containing EPs yield satisfactory results in describing the low-lying electronic states of various F centers in the MgO crystal. Acknowledgements This paper is dedicated to Professor Sigeru Huzinaga on the occasion of his 70th birthday. The authors would like to thank him for his hospitality and generous guidance in their research activities over many years. The authors also thank Dr M. Yoshimine for allowing us to use the ALCHEMY2 software. Some of the calculations in this study were performed on the IBM RS/6000 PS2 cluster in the Computer Center of the Institute for Molecular Science. The present research is partly supported by a Grant-inAid for Scienti®c Research from the Ministry of Education, Science, and Culture. Almost all of
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calculations in this study were performed by using the JAMOL4 program [24]. Calculations for the excited state, 1 A1 , was done by using the ALCHEMY2 program [25±27]. References [1] Y. Chen, R.T. Williams, W.A. Sibley, Phys. Rev. 182 (1969) 960. [2] Y. Chen, J.L. Kolopus, W.A. Wibley, Phys. Rev. 186 (1969) 865. [3] L.A. Kappers, R.L. Kroes, E.B. Hensley, Phys. Rev. B1 (1970) 4151. [4] G.P. Summers, T.M. Wilson, B.T. Jeffries, H.T. Tohver, Y. Chen, M.M. Abraham, Phys. Rev. B27 (1983) 1283. [5] G.H. Rosenblatt, M.W. Rowe, G.P. Williams Jr, R.T. Williams, Phys. Rev. B39 (1989) 10309. [6] Y. Chen, V.M. Orea, R. Gonzalez, R.T. Williams, G.P. Williams, G.H. Rosenblatt, G.J. Pogatshnik, Phys. Rev. B42 (1990) 1420. [7] M.-C. Wu, C.M. Truong, D.W. Goodman, Phys. Rev. B46 (1992) 12688. [8] Q.S. Wang, N.A.W. Holzwarth, Phys. Rev. B41 (1990) 3211. [9] K. Jackson, M.P. Pederson, B.M. Klein, Phys. Rev. B43 (1991) 2364.
[10] H. Tatewaki, E. Miyoshi, Surf. Sci. 327 (1995) 129. [11] R.T. Poole, J. Liesegang, R.C.G. Leckey, Phys. Rev. B11 (1975) 5179. [12] M. Piacentini, Solid State Commun. 17 (1975) 697. [13] T.E. Gallon, Surf. Sci. 206 (1988) 365. [14] P. Wurz, J. Sarnthein, W. Husinsky, G. Betz, P. Nordlander, Y. Wang, Phys. Rev. B43 (1991) 6729. [15] E. Miyoshi, S. Huzinaga, Phys. Rev. B48 (1993) 8583. [16] J.-I. Adachi, N. Kosugi, Bull. Chem. Soc. Jpn 66 (1993) 3314. [17] S. Huzinaga, S. Katsuki, O. Matsuoka, J. Math. Chem. 10 (1992) 25. [18] S. Katsuki, S. Huzinaga, Chem. Phys. Lett. 152 (1988) 203. [19] S. Katsuki, Can. J. Chem. 70 (1992) 285. [20] S. Katsuki, J. Chem. Phys. 98 (1993) 496. [21] H. Tatewaki, S. Katsuki, Y. Sakai, E. Miyoshi, J. Comput. Chem. 17 (1996) 1056. [22] H.M. Evjen, Phys. Rev. 39 (1932) 675. [23] S. Huzinaga, J. Andzelm, M. Klobukowski, E. RadzioAndzelm, Y. Sakai, H. Tatewaki, Gaussian Basis Sets for Molecular Calculations, Elsevier, Amsterdam, 1984. [24] H. Kashiwagi, T. Takada, E. Miyoshi, S. Obara, F. Sasaki, a program for SCF-MO calculations, a library program of the Computer Center of Institute for Molecular Science, Okazaki, Japan, 1987. [25] B.H. Lengs®eld III, J. Chem. Phys. 73 (1980) 382. [26] B. Liu, M. Yoshimine, J. Chem. Phys. 74 (1981) 612. [27] B.H. Lengs®eld, B. Liu, J. Chem. Phys. 75 (1981) 478.
Theoretical study on the electronic structures of various F centers in MgO crystals 1 Eisaku Miyoshi*, Yoshiki Miyake 2, Shinichi Katsuki 3, Yoshiko Sakai Department of Applied Physics, Faculty of Engineering, Kyushu University, Ropponmatsu, Fukuoka 810, Japan Received 26 December 1997; accepted 29 January 1998
Abstract Ab initio SCF-MO calculations were performed on low-lying electronic states of various F centers in MgO crystals by using embedded cluster models. Because a simple cluster model embedded with point charges alone gave poor results, we generated environmental potentials (EPs) in a spectral representation scheme for Mg 21 and O 22 ions and used them instead of point charges around the clusters under consideration. Using embedded cluster models with the EPs and point charges, we calculated the low-lying electronic states of various F centers in MgO. The calculated order in energy was found to be: the 1 A1 state of the surface F center , the 1 E state of the surface F center , the 1 B3u state of the bulk M center (F aggregates) , the 1 T1u state of the bulk F center. The energy separation of the 1 E, 1 B3u , and 1 T1u states from the lowest state were 1.22, 2.46 and 4.05 eV, respectively, which explains the observed bands. q 1998 Elsevier Science B.V. All rights reserved. Keywords: MgO crystals; F centers; Ab initio; Environmental potentials
1. Introduction There have been many experimental studies [1±7] on the excited electronic states of oxygen vacancies in MgO crystals. Chen et al. [1] concluded that, although there is little doubt that an intense optical absorption band at 4.96 eV is due to oxygen vacancies that may be both positively charged F 1 centers (one electron trapped at the vacancies) and neutral F centers (two electrons trapped at the vacancies), there is some uncertainty whether the broad absorption bands at * Corresponding author. Tel. 1 81 92 7264788; Fax: 1 81 92 7264788.. 1 Dedicated to Professor Sigeru Huzinaga on the occasion of his 70th birthday. 2 Present address: NEC Hiroshima, Higashi Hiroshima 739-01, Japan. 3 Present address: Fukuoka Jogakuin Junior College, Fukuoka 816, Japan. 0166-1280/98/$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S0 166-1280(98)001 62-6
3.49, 2.16 and 1.27 eV are due to F-aggregate centers. The intense band was also observed by Kappers et al. [3] and was resolved at liquid-nitrogen temperatures into two peaks at 4.96 and 5.03 eV, which were attributed to F 1 and F centers, respectively. Recently, Wu et al. [7] investigated thermally generated defects in ultrathin MgO ®lms on Mo(100) using high-resolution electron-energy-loss spectroscopy (HREELS) in the 0 , 9 eV energy region. In their study, the three distinct loss features observed at 5.33, 3.58 and 1.15 eV were attributed to electronic transitions associated with F (or F 1) centers, F aggregates (M centers), and surface F centers, respectively. On the other hand, theoretical calculations have emphasized the ground-state relaxation of the oxygen vacancies in MgO [8, 9]. There have been few theoretical calculations for electronic transitions associated with oxygen vacancies. Summers et al.
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E. Miyoshi et al. / Journal of Molecular Structure (Theochem) 451 (1998) 81±88
[4] calculated the absorption energy of the F centers of MgO to be 4.8 eV by the use of a two-electron wavefunction including an electronic polarization effect. Very recently, Tatewaki and Miyoshi [10] have studied the electronic structure of an LiF crystal using an ab initio MO cluster approach, where 2 Li1 n Fm clusters (m, n 13 or 14) were considered to be embedded in the ionic crystal, which is composed of point charges. The calculated band gap of LiF including a correlation correction was 13.8 eV, which is in excellent agreement with experimental values of 13.6 eV [11] and 14.2 eV [12]. The calculated bulk exciton band was 13.3 eV above the valence band, which also agrees well with the observed value of 13.5 eV [13]. They also calculated that the surface exciton state was 11.4 eV above the valence band, while the experimental values are 10.5 eV [13] or 9.1 ^ 0.6 eV [14]. Thus, they have shown that the ab initio MO cluster approach can properly describe the electronic structures of ionic crystals as far as band gaps and the exitation energies of the low-lying exciton states. By using the same method, Miyoshi and Huzinaga [15] investigated the excited states of the U center (H - ion instead of halogen ion) in NaCl. They found that the calculated transition energy for the dipole-allowed state agreed well with the observed value, within about 0.5 eV. The important conclusion they reached is that the correlation energy of two 1s electrons (H 2 ion) in the ground state is almost cancelled by the correlation-energy difference of the remaining electrons between the ground state and excited states. Adachi and Kosugi [16] also examined the excited states of F centers in NaCl using a similar method in the framework of SCF MO calculations. The calculated 1s-2p excitation energy was in good agreement with the experimental data. Thus, molecular orbital studies using embedded cluster models surrounded by point charges can properly describe the electronic structures of defect states in ionic crystals so far. In this study, we have used a similar simple embedded cluster model to investigate the electronic structures of bulk and surface F centers in the MgO crystal. The results of this treatment are discussed in Section 2. The simple embedded cluster model did not describe the difference in the electronic structures between the surface and bulk F centers in MgO. The failure of the model is attributed to the fact
that the excited electron in the model penetrates to the surrounding point charges, which will be shown in Fig. 2(a±c). Accurate modeling of the system requires that electrons only be allowed to penetrate the vicinity of neighboring ions in a manner consistent with the Pauli exclusion principle. In other words, the excited orbitals should be orthogonal to the neighboring ion core orbitals that reside at the lattice points occupied by the point charges in the crystal. In order to take into account such effects, we use a technique, which is brie¯y described in Section 3, incorporating environmental potentials (EPs) [17] in a spectral representation scheme [18±21]. The calculated results obtained by the embedded cluster model with the environmental potentials (EPs) for bulk and surface F centers and M centers are also discussed in Section 3. Concluding remarks are given in Section 4.
2. Simple embedded models In this section, we will discuss the results of simple embedded-cluster-model calculations for bulk and surface F centers in the MgO crystal. The model clusters for the F center used in the present study include up to the fourth nearest neighboring ions: for the bulk F center, FMg101 FMg6 O142 FMg14 O21 6 , 12 , 12 , 102 FMg14 O18 , where F in the cluster notations designates the F center. The clusters are embedded in the ionic crystal composed of point charges. The point charges are generated by the parallel transfer of the unit cell that is composed of positive ( 1 2e) and negative ( 2 2e) point charges. The total systems contain 3375 ions or point charges (including F centers) for the bulk F center and 1998 ions for the surface F center. Charge neutrality is obtained by using Evjen's method [22] as described in previous studies [10, 15]. The basis set for the F center site was the same as that used for the U center in NaCl [15], which was contracted as [7s6p]. Basic sets for Mg and O were taken from the work of Huzinaga et al. [23] with the coef®cients reoptimized for the Mg 21 and O 22 ions, respectively, embedded in the ionic crystal mentioned above. For the nearest neighbors Mg 21, we contracted the basis set to (433/311 *), augmenting a p-type polarization function (1 *) [23]. For others, we used
E. Miyoshi et al. / Journal of Molecular Structure (Theochem) 451 (1998) 81±88
the re-optimized values (433/4) and (43/4) for the Mg 21 and O 22 ions, respectively. Using a simple embedded cluster model, we performed restricted Hartree-Fock calculations for the ground and low-lying excited states of bulk and surface F centers, whose con®gurations are as follows: bulk F center in (Oh symmetry) ground state
1
A1g
(1s) 2,
excited state
3
A1g
(1s) 1(2s) 1,
1;3
T1u
(1s) 1(2p) 1,
surface F center (in C4v symmetry) ground state
1
excited state
1;3
A1
(1s) (2s 1 2pz) ,
1;3
E
(1s) 1(2px,y) 1.
A1
(1s) 2, 1
1
In the bulk F center, the 1 T1u state is only a spin and dipole-allowed transition state, while in the surface F center, both 1 A1 and 1 E states are symmetry allowed. The excited state, 1 A1 , was obtained as the second state in an MCSCF calculation with three con®gurations of (1s) 2, (1s) 1(2s 1 2pz) 1, and (2s 1 2pz) 2. The calculated excitation energies of the low-lying states are summarized in Table 1. In Fig. 1 we show the 1s orbital of the F center, which is localized at the F site and is almost unchanged for various excited states. Excitation energies for both 1 A1 and 1 E states of the surface F centers are larger by 0.1-0.4 eV than those of the bulk F center, while the excitation energy of the surface F center (1.15 eV) is experimentally smaller by about 4 eV than that of the bulk F center (5.33 eV) [7]. The reason for this discrepancy can be seen in Fig. 2 where we show (a) a 2p orbital of the 1 T1u state (bulk F center); and (b) a 2px orbital of the 1 E state in the zx plane normal to the surface. The excited electrons in both the 1 T1u and 1 E states penetrate the vicinity of the surrounding points charges as if there were nothing to interfere with their overlap. Thus, both the orbitals of the bulk and surface F centers become diffuse. In reality, electrons can penetrate the vicinity of the point charges only in a manner consistent with the Pauli exclusion principle. In other words, excited orbitals should be orthogonal to the ion's orbitals that reside at the point charges.
83
Table 1 Excitation energies (eV) of various F centers in the MgO crystal using a simple embedded cluster model with point charges a Bulk F center Clusters
3
FMg101 6 FMg6 O142 12 FMg14 O21 12 b FMg14 O21 12 102 FMg14 O18
2.82 0.82 2.02 2.58 1.76
3.05 2.26 2.46 2.62 2.31
2.85 2.13 2.35 2.43 2.21
3
1
E 3.41 2.65 2.64 2.90 2.40
3
E 2.89 2.34 2.40 2.50 2.22
1
3
Surface F center Clusters FMg81 5 FMg5 O82 8 FMg9O 8 b FMg9O8 FMg9 O102 13 Bulk M center Cluster F2 Mg10 O202 18 b a b
1
A1
3.01
A1g
A1 2.22 1.71 1.91 1.92 1.98
1
T1u
B3u 2.06
3
T1u
B3u 1.97
F designates the F center site. [5s4p] was used for the F center site instead of [7s6p].
We also performed calculations for FMg14 O21 12 and FMg9O8 clusters using another smaller basis set [5s4p] for the F center in which the two most diffuse functions (0.013675140 and 0.0063784397) were eliminated. Calculated excitation energies agree well
Fig. 1. The 1s orbital of the 1 T1u state obtained by using a simple cluster model embedded with point charges in the bulk F center model of FMg14 O82 18 . Dotts indicate sites of point charges. Values of contours are ^ 0.08, ^ 0.04, ^ 0.02, ^ 0.01, and ^ 0.005 (a.u.) 23/2.
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E. Miyoshi et al. / Journal of Molecular Structure (Theochem) 451 (1998) 81±88
with those obtained using a larger basis set [7s6p] except for the bulk 3 A1g state, where the discrepancy is more than 0.5 eV. Thus, the smaller basis set [5s4p] almost describes these states, except for the 3 A1g state. The energy of the 3 A1g state does not converge with the present cluster model calculations. We also performed HF calculations for the ground and excited states of an M center (F aggregates) consistent with defects in the two neighboring O 21 ions, using a simple embedded cluster F2 Mg10 O202 18 in an ionic cage of 4980 point charges. The cluster consists of the nearest neighbors and next nearest neighbors from both F centers. The ground and excited states of the M center are as follows (in D2h symmetry); ground state
1
excited state
1;3
Ag B3u
(1sA 1 1sB) 2 (1sA 2 1sB) 2, (1sA 1 1sB) 2 (1sA 2 1sB) 1 (2pxA 2 2pxB) 1.
The subscripts A and B indicate different F center sites, which are on the x-axis. The calculated excitation energy of the spin and dipole-allowed state of the M center was calculated to be 2.06 eV, which is similar to that of the bulk and surface F centers as calculated by a simple embedded cluster model. The excited state was found to be about 2 eV below the bulk F center and about 2 eV above that of the surface F center. The reason for these discrepancies can be attributed to the arti®cially diffuse nature of the excited orbital (2pxA 2 2pxB), which is shown in Fig. 2(c). 3. Environmental potentials Defect in crystals is one of physical and chemical phenomena that are expected to be understood in terms of localized electronic states, even though the crystal itself is not isolated, but rather is vastly Fig. 2. Excited orbitals calculated by using simple cluster models embedded with point charges. (a) 2p orbitals of the 1 T1u state in the 1 bulk F center model of FMg14 O82 18 . (b) 2px orbital of the E1u state in the surface F center model of FMg9 O102 13 . (c) (2pxA 2 2pxB) orbitals in the M center model of F2Mg10O202 18 . Dots and crosses indicate sites of point charges and ions in the cluster, respectively. Values of contours are ^ 0.08, ^ 0.04, ^ 0.02, ^ 0.01, and ^ 0.005 (a.u.) 23/2.
E. Miyoshi et al. / Journal of Molecular Structure (Theochem) 451 (1998) 81±88
dispersed. Our approach is based on a ®nite cluster model, which consists of two parts: (I) a cluster where the defect state has signi®cant amplitude, and (II) the surrounding environment, which contributes only a background to the cluster. The role of the atoms in region II is especially important in the case of ionic crystals such as MgO. Firstly, they contribute to the Madelung potential at each crystal lattice point. Secondly, the electrons occupying the atoms in region II will repel all other electrons to prevent penetration into the occupied space in region II. The point charges can then serve as a substitute for ionized atoms as far as the ®rst contribution. However, the point charges can not prevent the electron orbitals from penetrating inhibited occupied states in region II. To incorporate the second contribution of the atoms in region II to our model, we divide region II into two parts, region IIa and region IIb. We then use an environmental potential [17] at each lattice point in region IIa and replace each lattice point in region IIb by a point charge, as was also used in the case of the simple embedded model. The following paragraph brie¯y explains the environmental potential. The nuclear attraction, Coulombic forces, and the exchange operator in the Fock operator is expressed as follows: 2 3 X N A A 2Jj 2 Kj 5V 1 PA (1) VEP V42 1 rA j where NA is the number of electrons in ion A and j runs A as an environover all NA electrons. We refer to VEP mental potential (EP). The projection operator V is expressed as follows with the use of a function set {xa} which is not necessarily mutually orthogonal; X uxa l S21 ab kxb u (2) V ab
Sab kxa uxb l
(3)
The operator in the following form X uxa l S21 ab kxb uOuxc l S21 cd kxd u VOV
(4)
abcd
is called a spectral representation of an operator O [18±21]. PA is a shift operator preventing the valence orbitals in region I from collapsing into the space which is spanned by the jth occupied orbitals fjA
85
belonging to the A ion in region IIa, X PA 2 21Aj ufAj lkfAj u
(5)
j
The numbers of the ions, which were represented by the EP of Eq. (1) were 336 , 310 for models of the bulk F center and 190 , 173 for models of the surface F center. EPs for the Mg 21 and O 22 ions were developed in this study. The total number of ions considered in a cluster (region I) and the EPs (region IIa) was 363 (including the F center itself) for the bulk F center, and 197 for the surface F center. For the M center (F aggregates) we substituted 366 point charges cluster with EPs. The surrounding the F2Mg10O202 18 rest of the surrounding ions remained as point charges. The resulting excitation energies for the bulk and surface F centers and the M center in the MgO crystal, calculated by using the various embedded cluster models containing EPs (region IIa) and point charges (region IIb), are listed in Table 2. The spin and dipoleallowed transition state, 1 T1u , of the bulk F center is destabilized by about 5 eV as compared to the value obtained from the simple embedded calculation. The 1 A1 and 1 E states of the surface F center are destabilized by only 0.3 and about 2 eV, respectively. A large amount of destabilization in the bulk F center Table 2 Excitation energies (eV) of various F centers in the MgO crystal calculated using embedded cluster models containing EPs and point charges a Bulk F center Clusters
3
FMg101 6 FMg6 O142 12 FMg14 O21 12 FMg14 O102 18
19.76 12.55 11.30 10.24
9.09 7.59 7.45 7.38
5.00 4.45 4.35 4.28
3
1 E 6.52 5.04 4.54 4.55
3 E 4.18 3.52 3.29 3.31
1
3
Surface F center Clusters FMg81 5 FMg5 O82 8 FMg9O 8 FMg9 O102 13
1
A1
3.33
Bulk M center Cluster F2 Mg10 O202 18 a
F designates the F center site.
A1g
A1 2.24 1.68 1.59 1.56
1
T1u
B3u 5.79
3
T1u
B3u 3.83
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E. Miyoshi et al. / Journal of Molecular Structure (Theochem) 451 (1998) 81±88
is due to the interference caused by the penetration of EPs into the vicinity of the surrounding ionic sites. This causes the excited 2p orbital in the bulk F center to become contracted, as is shown in Fig. 3(a) which is in contrast to Fig. 2(a). The excited 2px,y and 2s 1 2pz orbitals in the surface F center also become contracted, but they can diffuse into the vacuum, as is seen in Fig. 3(b). This explains why the orbitals are less destabilized than in the case of the bulk F center. The spin and dipole-allowed transition state of the M center is raised by 3.7 eV due to the use of EPs. The resulting orbital is depicted in Fig. 3(c), where a tight (2pxA 2 2pxB) orbital is seen. The results of the embedded cluster model calculations using EPs (region IIa) and point charges (region IIb) are listed in Table 3 and compared with those of the simple embedded calculations and experimental values. The lowest spin and dipole-allowed state is the 1 A1 state in the surface F center, whose symmetry is the same as the z-axis (perpendicular to the surface). The simple embedded cluster calculations determine this state to be the highest and give a wrong order of states, where all the states are within 1 eV of each other. The embedded cluster model calculations using EPs give the correct order. Although the calculated excitation energies are constantly higher than those of the experimental values by about 2 eV, the calculated energies relative to that of the lowest excited state are well consistent with experiments. The bulk F and M centers are calculated to be 4.05 and 2.46 eV above the lowest state, respectively, while the observed values are 4.18 and 2.43 eV [7] or 3.69 and 2.22 [1], respectively. Another surface 1 E state is 1.22 eV above the 1 A1 state. Chen et al. observed a broad band at 2.16 eV, which is higher than the lowest band by 0.89 eV. Thus, the broad adsorption band observed by Chen et al. can be assigned to the 1 E state of the surface F center. The overestimation of excitation energies is due to
Fig. 3. Excited orbitals calculated by using embedded cluster models containing EPs and point charges. (a) 2p orbitals of the 1 T1u state in the bulk F center model of FMg14O82 18 . (b) 2px orbital of the 1 E1u state in the surface F center model of FMg9 O102 13 . (c) (2pxA 2 2pxB) orbitals in the M center model of F2Mg10O202 18 . Crosses indicate sites of EPs or ions in the cluster. Values of contours are ^ 0.08, ^ 0.04, ^ 0.02, ^ 0.01, and ^ 0.005 (a.u.) 23/2.
E. Miyoshi et al. / Journal of Molecular Structure (Theochem) 451 (1998) 81±88
87
Table 3 Comparison of calculated and experimental values for excitation energies (eV) of bulk and surface F centers, and bulk M centers in the MgO crystal calculated using embedded cluster models containing EPs and point charges F centers
Bulk F 1 T1u
Surface F 1 A1
Simple embedded cluster
2.31 (20.70) 7.38 (4.05) 5.33 (4.18) 4.96 (3.69)
3.01 (0.00) 3.33 (0.00) 1.15 (0.00) 1.27 (0.00)
Embedded cluster with EPs Exptl. a Exptl. b a b
1
E
2.40 (20.61) 4.55 (1.22) 2.16 (0.89)
Bulk M 1 B3u 2.06 (20.95) 5.79 (2.46) 3.58 (2.43) 3.49 (2.22)
See ref. [7]. See ref. [1].
the intense effect of the shift operator in Eq. (5), which prevents the valence orbitals of the clusters in region I from collapsing into the space that is spanned by the neighboring ion core orbitals in region IIa. However, it is dif®cult at the present stage for the operator so as properly to work, because it requires that all the core orbitals of the neighboring ions should be considered explicitly. Finally, we shall comment on the electron correlation of the F centers of the MgO crystal. The electronic state of the F center is essentially the same as the U center (H 2 ion instead of halogen ion) in the NaCl crystal, as studied by Miyoshi and Huzinaga [15]. They concluded that the correlation energy of the two 1s electrons in the ground state is almost cancelled by the correlation-energy difference of the remaining electrons between the ground state and excited states. Thus, since the F center in MgO is expected to be similar to the U center in NaCl, we do not take the correlation energy into consideration in the present study.
4. Conclusions We performed ab initio MO calculations utilizing embedded cluster models to investigate the low-lying electronic states of the F centers of the MgO crystal. A simple cluster model with surrounding point charges gave poor results. We utilized the environmental potentials (EPs) instead of point charges around the clusters con-
sidered; a concept originally proposed by Huzinaga et al. [17]. We generated EPs for Mg 21 and O 22 ions in a spectral representation scheme [18±21]. In the embedded cluster model calculations using the EPs, we have described the 1 A1 state of the surface F center as the lowest spin- and dipole-allowed state. The 1 E state of the surface F center was calculated to be 1.22 eV above the lowest state, and the 1 B3u state of the bulk M center and the 1 T1u state of the bulk F center were found to be 2.46 and 4.05 eV above the lowest state, while the corresponding experimental values are 0.89, 2.22 (2.43) and 3.68 (4.18) eV, respectively [1, 7]. Thus, ab initio MO calculations using the embedded cluster model containing EPs yield satisfactory results in describing the low-lying electronic states of various F centers in the MgO crystal. Acknowledgements This paper is dedicated to Professor Sigeru Huzinaga on the occasion of his 70th birthday. The authors would like to thank him for his hospitality and generous guidance in their research activities over many years. The authors also thank Dr M. Yoshimine for allowing us to use the ALCHEMY2 software. Some of the calculations in this study were performed on the IBM RS/6000 PS2 cluster in the Computer Center of the Institute for Molecular Science. The present research is partly supported by a Grant-inAid for Scienti®c Research from the Ministry of Education, Science, and Culture. Almost all of
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E. Miyoshi et al. / Journal of Molecular Structure (Theochem) 451 (1998) 81±88
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