Jun Yang, Michael Dolg Institute of Theoretical Chemistry, University of Cologne Greinstr. 4, D-50939, Cologne, Germany

Transcription

1 Ab initio Density Functional Theory Investigation of Monoclinic BiB 3 O 6 within a LCAO Scheme: the Bi lone pair and its role in electronic structure and vibrational modes Jun Yang, Michael Dolg Institute of Theoretical Chemistry, University of Cologne Greinstr. 4, D-50939, Cologne, Germany Abstract We present a gradient-corrected hybrid B3PW density functional theory investigation of the electronic structure of the excellent nonlinear optical crystal BiB 3 O 6 and the origin of the stereochemical active lone pair at Bi with local Gaussian basis sets. The calculated electronic densities in BiB 3 O 6 picture the lone pair lobe at Bi for the first time. The formation of the lone pair lobe in BiB 3 O 6 is studied and discussed in much detail based on the density of states of BiB 3 O 6 assuming several isostructural cells expanded and compressed isotropically with respect to the experimental geometry. The factors that lead to the stereochemically active lone pair lobe are identified as the covalent interactions between Bi 6s6p orbitals and O 2p orbitals. The influence of the lone pair electrons on lattice movements of BiB 3 O 6 is clarified by first calculating the vibrational frequencies of BiB 3 O 6 and assigning all modes, and then subsequently comparing BiB 3 O 6 with the vibrations of the hypothetical isostructural compound TlB 3 O 6 without metal lone pair electrons in it. 1

2 I. Introduction Materials exhibiting the nonlinear optical (NLO) properties and their applications are of great interest, and several scientific groups around the world have been being actively involved in research in this area both experimentally 1, 2, 3, 4, 5 and theoretically 6, 7, 8, 9, 10. In the past decades, a series of borate materials were discovered among them. Polar monoclinic bismuth triborate BiB 3 O 6 (BiBO) 11 is an excellent non-linear and linear optical material due to its large effective SHG (Second Harmonic Generation) coefficient d SHG eff = 3.2 pm V 1 12, which is higher than that of most other borate-based NLO materials currently used such as LiB 3 O 5 (LBO) or BaB 2 O 4 (BBO) 13, 14. The NLO susceptibility tensors of BiB 3 O 6 were calculated with the chemical bond method 15 as well as the anion group theory 16 in combination with an ab initio scheme. The origin of the large NLO effects was explained as the result of a dominant contribution from the [BiO 4 ] 5- group 16. However, up to now, little is known about how the electronic effects in BiB 3 O 6, including those from 6s 2 lone-pair electrons at Bi (III), actually take actions upon optical and other interesting physical properties, since most of the reports focused on computing the geometrical contributions to SHG coefficients from different individual bonds or groups (i.e., Bi-O, B-O, [BO 3 ] 3-, [BO 4 ] 5- and [BiO 4 ] 5- ). A detailed understanding of the electronic structure of BiB 3 O 6 would contribute significantly to explain the role of Bi (III) in BiB 3 O 6, and help to design and prepare NLO materials with controlled properties. The first pioneering work explaining the behaviour of electron lone pairs in terms of geometries of molecules can be traced back to the paper of Sidgwick and Powell 17 and of Gillespie and Nyholm 18 several decades ago, who proposed the so-called valence shell electron pair repulsion theory (VSEPR). A classic view was invoked afterwards by Orgel 19 who simply attributed the structural distortion to the mixing of s and p orbitals of the cations, and generally related the large dielectric response to stereochemical activities of lone pairs. Wheeler and Kumar 20 supported this viewpoint based on extended Hűckel calculations by concluding that the mixing of the cationic s orbital (HOMO) and the cationic p z orbital (LUMO) predominates for the molecular anion [BiCl 6 ] 3- and crystalline SbI 3 with trigonal distortions. However, such a simple picture has been doubted in recent years by noting that ligands also must play an important role: if the active lone pair is 2

3 formed purely by the Bi (III) cation, such a feature would be expected for all Bi (III) compounds, which is actually not the case. For example, in the monoclinic ferromagnetic BiMnO 21 3 the stereochemically active lobe-like Bi lone pair results from the admixture of Bi 6s and 6p orbitals and O 2p orbitals, which stabilizes the monoclinic structural distortion. Likewise, the structural effects of lone pairs are well studied in PbO and related systems 22,23,24,25. In monoclinic BiB 3 O 6, Bi (III) holds an electronic configuration [Xe] 4f 14 5d 10 6s 2. BiB 3 O 6 crystallizes in space group C2 and consists of c direction-alternating layers of [B 3 O 6 ] 3- rings forming sheets of corner-sharing [BO 3 ] 3- triangles and [BO 4 ] 5- tetrahedrons (Fig. 1), linked by six-fold coordinated Bi (III) 26 cations each of which has four oxygen nearest neighbours on the same side of Bi (III) (Fig. 2). The marked structural feature is that the [BiO 4 ] 5- group is a square pyramidal structure with a lone pair oriented in the opposite direction (Fig. 2), which is very favorable to produce large SHG coefficients for bulk BiB 3 O 6. It is important to point out that this distorted coordination around Bi (III) does not necessarily prove the stereochemical activity of the lone pair at Bi (III) since the [B 3 O 6 ] 3- rings and special arrangements of [BO 3 ] 3- and [BO 4 ] 5- groups impose the resulting geometry as well. A plane-wave pseudopotential calculation based on density functional theory (DFT) using generalized gradient approximation (GGA) functionals has been conducted on monoclinic BiB 3 O 6 by Lin et al. 16. Although their emphasis was not on electronic structure and lone pair formation, their calculations exhibit a spherical electronic charge distribution of lone pairs at Bi (III), which is inconsistent with the common chemical understanding that the stereochemically active lone pair should be relevant to the offcentric electronic densities. Further studies on the 6s 2 lone pair at Bi (III) in monoclinic BiB 3 O 6 were not performed. First-principle calculations for periodic compounds are usually performed using planewave basis sets in conjunction with pseudopotential and DFT techniques in solid state physics. Nowadays, some attention for solid calculations has been directed to the crystal orbital (CO) method since the general implementation of the Hartree-Fock (HF) LCAO method in the program CRYSTAL for the treatment of periodic systems was published more than two decades ago 27,28,29,30. The advantages of the LCAO scheme based on local 3

4 Gaussian basis sets are twofold with respect to plane-wave methods: firstly, some popular hybrid functionals such as B3PW and B3LYP can be utilized, whereas it is impossibly combined with plane-wave basis sets implemented in currently popular plane-wave codes although hybrid functionals provide a crucial improvement over LDA and GGA functionals in cases of molecular ab initio calculations 31 ; secondly, relatively light atoms such as oxygens and borons can be accurately described by means of all-electron treatments so that small- or large-core pseudopotential approximations are imposed only on heavy cations like bismuth where relativistic effects are large. This not only allows a more accurate Hamiltonian but is also very advantageous for cases where the semi-core shell electrons in constituent atoms of some NLO materials notably take responses under external polarizing fields. In this paper, we concentrate on investigating the electronic band structure of BiB 3 O 6 and the origin of the stereochemically active Bi (III) lone pair. Based on the discussion of the electronic structure of BiB 3 O 6, we turn to explore how lone pair electrons at Bi (III) influence the vibrational modes. A theoretical study of vibrational modes would correlate the electronic structure including the lone pair effects and NLO properties for such a frequency conversion material like BiB 3 O 6, since vibrational frequencies are relevant to bond strengths and the hierarchy of energy distributions of atomic motions involving different structural subunits 35. We are not aware of previous computations of vibrational frequencies although several experimental reports have already reported both Raman and IR spectra of BiB 3 O 32,33,34,35 6. The assignments of individual vibrational modes are first fully prescribed according to vibrating vectors analysis, while they were only incompletely available from experimental comparisons with the corresponding [BO 3 ] 3- and [BO 4 ] 5- vibrational modes reported in other borate crystals and glasses 34,35. II. Methods The first principle electronic structure calculations were performed utilizing DFT with the hybrid functional B3PW 36, 37, 38, 39, 40 implemented in the package CRYSTAL03 27,27,29,30,41. Calculations for both the electronic structure and vibrational modes were performed in one primitive unit cell (denoted as a v p, b v p and c v p ) with one bismuth, six oxygens and three borons included, which is transformed from the crystallographic cell (denoted as a v, b v and c v ) with the double number of the 4

5 corresponding atoms by defining the new primitive vectors but maintaining the c v lattice vector as c v p. v a p v v v v v = a b and = a + b b p All the basis sets were originally taken from the public EMSL database 42. For boron atoms, the 6-311G* Gaussian function basis set was applied for all cases except the frequency calculations where 6-31G * was utilized. For oxygen atoms, however, the basis sets were selected as a compromise between the accuracy and computational time: the single point electronic structure and related property calculations were done with the Dunning contraction (10s, 6p)/[5s, 3p] 43 augmented by one additional d function with the exponent of optimized for crystalline BiB 3 O 6 ; the calculations for vibrational frequencies were most expensive and were performed with 6-31G* basis set by optimizing exponents of the outmost sp and d orbitals as and 0.538, respectively for crystalline BiB 3 O 6. We use energy-consistent scalar-relativistic ab initio effective core pseudopotentials (ECP) of the Stuttgart-Koeln variety for Bi 44 to diminish the computational complexity and to incorporate the most important relativistic effects. The large-core ECP with five valence electrons was used for trivalent Bi, i.e., the 1s-5d shells were included in the ECP core, while all others were treated explicitly (6s6p shells). The reference data used to determine the spin-orbit averaged relativistic potential have been taken from relativistic all-electron (AE) calculations using the so-called Wood-Boring (WB) scalar-relativistic Hartree-Fock (HF) approach. Both AE WB and ECP calculations have been performed with an atomic finite-difference HF scheme in order to avoid basis set effects in the determination of the ECP parameters. A (4s4p1d)/[2s2p1d] valence basis set was applied with an optimized outmost p orbital exponent of 0.09 for crystalline BiB 3 O 6 in all computational cases. The following tolerances were employed in the evaluation of the infinite Coulomb and Hartree-Fock exchange series: 10-7 for the Coulomb overlap, HF exchange overlap, Coulomb penetration and the first exchange pseudo-overlap; for the second exchange pseudo-overlap. The Fock matrix has been diagonalized at 24 k-points within the irreducible Brillouin zone corresponding to a shrinking factor of 4 in the Monkhorst net 45. In order to improve the convergence, a negative energy shift of 1.0 hartree to the diagonal Fock/KS matrix elements of the occupied orbitals was added to reduce their 5

6 coupling to the unoccupied set and maintained after diagonalization. A very accurate extra-large grid consisting of 75 radial points and 974 angular points was employed in the DFT calculations, where Becke grid point weights 36 were chosen. The reliable and accurate simulation of vibrations for crystals is still at a less developed stage than for molecules 46, and so far, only DFT methods with plane wave basis sets were consistently implemented for frequency calculations. Since the vibrational computation is not yet available in the released version of CRYSTAL03, we wrote a bshscript to obtain single point energies of atom-displaced crystalline structures for CRYSTAL03 package computations, and calculated the elements of the Hessian matrix according to the following straightforward formalism based on the harmonic approximation. A C language code was created to diagonalize the Hessian matrix. The elements of the Hessian matrix fall into two subsets with respect to displacements: the 2 1 E diagonal elements 2 M d α α, i and the nondiagonal elements 1 2 E d d M α M β α, i β, j where i must be different from j. The three-point numerical derivative formula (equation (1)) was adopted to compute the diagonal elements, d 2 E ( E d E0 ) + ( E 0 ), d E α i α, i = 2 2 α, i dα, i (1) where is the displacement (0.001 a.u. in our calculations) along the Cartesian i- d α, i coordinate of atom α and E 0 is the energy at the equilibrium geometry. Totally 6 N single point calculations have to be performed for the entire 3 N diagonal elements each of which needs 2 complete single point rounds. The nondiagonal elements were computed according to equation (2): 2 E d d α, i β, j = g α, i ( d β, j ) g 2 d α, i β, j ( d where g ± d ) is defined as the gradient along the i-coordinate of atom α at the α, i ( β, j positive and negative displacement along the j-coordinate of atom β respectively (equation (3)): β, j ) (2) 6

7 g α, i ( ± d β, j ) = E dα, i ( ± d β, j ) E 2 d α, i dα, i ( ± d β, j ) (3) Totally 6N (3N 1) single point calculations had to be performed for the entire 3N(3N 1) 2 nondiagonal elements each of which needs 4 complete single point rounds. Therefore, the total number of single point calculations for frequencies is 18N (e.g single point rounds for BiB 3 O 6 with the number of atoms N=10) with one additional calculation for the equilibrium geometry (see equation (1)). The threshold for the single point energy convergence was set to 10-8 a.u. to obtain a sufficient numerical accuracy. All the calculations were performed with the same set of orbital-indexed bielectronic integrals for the reference geometry in order to reduce the numerical noise for the Hessian elements. The vibrational frequencies were evaluated at the highest symmetry k=0 point (i.e. Γ point) in the reciprocal space because Raman and IR experimental data refer to the k=0 point only, where harmonic frequencies of periodic compounds can be calculated in the same way as for molecules. Frequencies at other k points (i.e. phonon dispersions) can be obtained by building up corresponding supercells, which is quite expensive at the current computational level and therefore will not be discussed here. III. Results and discussions A. Electronic structure of BiB 3 O 6 Total electronic densities were plotted in the plane (001) and (100) (Fig. 3 (a) and (b)) passing through bismuth atoms of two neighbor crystallographic cells. The lone pair electronic densities deviate from the symmetric sphere as expected and are oriented in the b r direction at the other side of Bi away from the four nearest coordinated oxygens (O4, O5 and O6, O7 in Fig. 2). We have calculated the total, orbital and atom electronic density of states (DOS) (Fig. 4) to explore the electronic structure in more detail. The orbital and atom DOS were obtained by projecting respectively the total DOS onto a set of specified atomic orbitals (AO) and of all AOs of specified atoms following a Mulliken analysis. All of these states span an energy range from -30 ev to 20 ev, whereas the core states for O 1s and B 1s orbitals were removed since we are only concerned about valence states. 7

8 It can be seen from Fig. 4 that the band structure can be divided into three well separated regions. The lowest region below -20 ev is mainly due to O 2s occupied orbitals with a slight mixture of B orbitals. The second region from ev to -5.6 ev known as the occupied states of the valence band is dominated by O 2p and also contributed by the Bi 6s orbitals with a small amount of Bi 6p at the higher energy range (more clearly seen from Fig. 7 (ii)). The Bi 6s orbitals stay very broad overlapping with the 2p states of coordinating oxygens spanning the entire valence band, which suggests, from the energy point of view, a covalent interaction between Bi 6s and oxygen 2p orbitals in the crystalline environment of BiB 3 O 6. The largest overlapping takes place over the energy range of 1.4 ev starting from the bottom of the valence band, which is ascribed to the bonding interaction between Bi and O followed by the filled antibonding interaction up to the Fermi level. The third region, i.e., the unoccupied states of the conducting band, stays above the Fermi level, which is mainly composed of Bi 6p orbitals but with slight B 2s2p character at the bottom, and of a considerable amount of B 2s2p orbitals with some donations from O 2p orbitals at the top. The optical gap between the top of occupied bands and the bottom of the unoccupied bands is 5.7 ev (218 nm) overestimating the experimental optical transmission upper energy threshold of 4.3 ev (286 nm) 12, which disagrees with the general trend that band gaps are always underestimated within the DFT scheme. We explain this irregular overshooting partly as the consequence of neglecting the spin-orbit (SO) splitting of unoccupied Bi 6p states due to the averaged SO potential in the applied ECP of Bi. At the all-electron state-averaged multi-configuration Dirac-Hartree-Fock level using the Dirac- Columbus Hamiltonian (GRASP) 47, the atomic 6p 1/2-6p 3/2 splitting is 2.10 ev, i.e., 6p 1/2 is lowered by 1.40 ev compared to the scalar relativistic 6p. A lowering of the conducting band by this magnitude would bring the theoretical estimate close the experimental value. Since occupied states of O 2p orbitals and unoccupied states of Bi 6p orbitals are the leading components for the top of valence band and the bottom of conducting band respectively, the small energy gap would favor the electronic transfer between Bi and O which is the main driving force for the excellent NLO effects of BiB 3 O 6. Such a conclusion is qualitatively consistent with the anion group theory 16 which shows that the contributions of the [BiO 4 ] 5- group are about ten times greater than those of the [BO 3 ] 3- or 8

9 [BO 4 ] 5- groups in terms of NLO tensor coefficients. Besides, it is interesting to notice from Fig. 5 that only states of bismuth and borons (B9 and B10) in the center of triangular units [BO 3 ] 3- are responsible for the bottom of the conducting band, where the states of borons (B8) in the center of tetrahedral units [BO 4 ] 5- are completely excluded and greatly populate only at higher energy levels than 7.0 ev. This may imply that [BO 3 ] 3- is easier to be optically polarized than [BO 4 ] 5- since the latter requires relatively stronger electronic excitations. This supports the chemical intuition for BiB 3 O 6 that π electrons in the planar [BO 3 ] 3- can be more easily moved while such movements are considerably restricted in the three-dimensionally extended [BO 4 ] 5-. B. Bi-O interactions and stereochemically active lone pair in BiB 3 O 6 Let us first study the charge distributions in BiB 3 O 6 (Table 1). Since the shells with a main quantum number smaller than 6 are taken as effective core potential of Bi, the valence shell electron population has a 6s 2 6p 3 configuration as reference with 5 electrons for neutral Bi and 6s 2 for trivalent Bi (III). However, a Bi 6p-shell population of is observed, which confirms a substantial deviation from a purely ionic interaction between Bi and O. For coordinating oxygens, the electrons in both s and p shells of O4 and O5 are less populated than O6 and O7 by around 0.10 and 0.14 electrons, respectively. This can be easily understood as the consequence that O6 and O7 are closer to Bi (2.087 Å) than O4 and O5 (2.390 Å) and therefore more electrons are transferred by the covalent bonding from Bi to O6 and O7 than to O4 and O5. Since Fig.4 indicates the evident states overlapping between Bi 6s and O 2p, we further investigate the origin of the lone pair electrons at Bi by examining the electron densities and DOSS and their dependences on Bi-O distances in BiB 3 O 6 with several different hypothetical cell volumes with respect to the experimental unit cell. For the reason of simplicity, only the extreme cases corresponding to 10% expansion, experimental geometry and 7% compression are presented in Fig. 6, Fig. 7 and Fig. 8 as typical examples. The Bi-O4 and Bi-O6 distances for the three variations are Å, Å and Å as well as Å, Å and Å, respectively. Further expansions of unit cells cannot be studied due to the problems to describe correctly the transition from the close-shell system to separate open-shell fragments within DFT schemes. 9

10 These total electron densities in the planes passing through the Bi, O6, O7 and Bi, O4, O5 atoms are respectively plotted in Fig. 6 (a) and (b) from (i) to (iii). The densities corresponding to O core electrons are truncated by the contour threshold of to remove the map obscurity. It can be seen that the lone pair lobe abound Bi is directed into space at the other side of Bi away from O in both O4-Bi-O5 and O6-Bi-O7 planes. The plane O4-Bi-O5 in the 10%-expanded cell indicates the most spherical lone pair lobe around Bi (Fig. 6 (i) (b)). For the experimental and 7%-compressed cells (Fig. 6 (b) from (ii) and (iii)), more lone pair electrons are pushed away by O4 and O5 out of the lobe region and the electron density of the Bi lone pair lobe turns out to be more asymmetric. This is explained as the consequence of the enhanced covalent bonding between Bi lone pair electrons and electrons around O4 and O5 due to the decreased Bi-O4 and Bi-O5 distances when the size of the unit cell is isotropically varied from the 10% expansion to the 5% compression. In the plane O6-Bi-O7, however, the shape of the lone pair lobe remains insensitive, in our cases, to the Bi-O6 and Bi-O7 distances, since O6 and O7 are much closer to Bi with keeping stronger covalent bondings in all cases than O4 and O5. It is important to point out that although the long range Bi-O and Bi-O electrostatic interaction is excluded as the direct reason that the lone pair lobes are shaped due to its much slower decay at the relatively long Bi-O distance than the covalent bonding, however, the electrostatic potential with a low symmetry in crystals broadens the energy levels of Bi 6s and 6p spanning the reciprocal space of monoclinic BiB 3 O 6 and energetically favors covalent bondings with coordinating oxygens. In Fig. 7, sets of DOS were projected onto Bi 6s and 6p orbitals as well as O4, O5 and O6, O7 atoms corresponding to the valence band from -17 ev to the Fermi level. The PDOSS can be divided into the filled Bi-O bonding (region I, Fig. 7 from (i) to (iii)) and the filled Bi-O antibonding areas (region II, Fig. 7 from (i) to (iii)). The PDOS of Bi 6s and 6p orbitals are integrated for the two regions respectively in Table 2. The integrated PDOS of oxygen 2p orbitals is concerned in Table 2 only in region I, since the B-O bondings in [BO 3 ] 3- and [BO 4 ] 5- groups take place mainly in region II which complicate the discussion of the charge distribution on O. With shortening the Bi-O distances, the number of integrated Bi 6s and O 2p states in region I remain constantly descendent and ascendant respectively as seen from Table 2, which leads to the conclusion that the 10

11 covalent bonding is strengthened by involving more O 2p in Bi-O bonding orbitals. This can be pictorially confirmed, from the electron densities projected into region I for the plane O6-Bi-O7 and O4-Bi-O5 in Fig. 8 from (i) to (iii), that electrons are more densely distributed in the area between Bi and O from the 10% expansion to the 7% compression. Moreover in Fig. 8, the dominance of Bi 6s states (1.648, Table 2) in region I for the 10% expansion results in the nearly spherical electronic densities around Bi in both O6-Bi-O7 and O4-Bi-O5 planes but for the case of the 7% compression where Bi 6s states (0.595, Table 2) are comparable with O 2p states (0.388 and in Table 2), the asymmetric distortion evidently take place by distributing more O 2p electrons between Bi and O atoms. Fig. 7 clearly indicates that there is an additional interaction in region II besides the Bi 6s-O 2p antibonding since the Bi 6s and filled Bi 6p states overlap even for the largest unit cell. The further evidence for the coupling between the Bi 6p and Bi 6s-O 2p pairs can be clearly observed in the visualization of the electron density for the covalent bonding region I in Fig. 8 from (i) to (iii), where the inner portion of the Bi 6s electron density is partially destructed at the one side of Bi away from the oxygen atoms by the Bi 6s-6p hybridizing. In Table 2 the Bi 6p states are only slightly changed, which is consistent with the visualization of the Bi-O antibonding electron density, as it can be seen in Fig. 9 from (i) to (iii) that the asymmetric shape of lone pair lobes projected into region II remains nearly invariant. Therefore, the lone pair lobe is formed by hybridizing Bi 6p orbitals with the antibonding Bi 6s-O 2p combination in the way that the density is concentrated, constructively at the one side of Bi. Based on the above analysis, the covalent bonding interaction of Bi 6s-O 2p in the low occupied region I decreases the Bi electron density at one side of Bi close to oxygen atoms by populating more electrons in the area between Bi and O. In addition, the subsequent antibonding of Bi 6s-O 2p mediated by Bi 6p orbitals in the high occupied region II enhances the Bi electrons at the other side of Bi against oxygen atoms. The above two interactions form the Bi lone pair lobe with the less spherical shape or equivalently more stereochemical activity. The Bi lone pair can be stereochemically activated in the way that the covalent interaction is strengthened by approaching oxygen atoms to bismuth and involving more O 2p orbitals in bonding to Bi 6s, as the cases 11

12 indicated in Fig. 6. During the shrinking of the BiB 3 O 6 unit cell, the Bi 6s states in the Bi- O antibonding region II is densified, which just compensates the Bi 6s states in the Bi-O bonding region I for the reduction of their states at the lower energy level, and the total number of Bi 6s occupied states stays constant around the average of (Table 2), which is expected for the reason that the Bi inner 6s-6p hybridization is independent on the Bi-O distance and interaction. We further notice in Table 2, by the way, that the occupied Bi 6s DOS is equally divided into bonding states (1.013, region I) and antibonding states (1.036, region II) right at the experimental cell geometry. C. Vibrational modes of BiB 3 O 6 In this section, the vibrational frequencies of BiB 3 O 6 were calculated and the vibrational modes were fully assigned due to the calculated vibrating vector analyses. The nuclear site group method 48 was applied to decompose the symmetry representation of the Brillouin zone-center vibrational modes. In the crystal structure of BiB 3 O 6, one Bi and one B atoms occupy the C 2 sites, and the remaining two B and six O atoms occupy the C 1 sites. The lattice modes are represented by 14A+16B, among which there A+2B acoustic modes. The remaining 13A+14B modes belong to the optical modes, all of which are both IR and Raman active. Table 3 lists the comparison between calculated and available experimental frequencies. The experimental values are plotted against the calculated frequencies in Fig. 10. It can be seen that the calculated frequencies are linearly related to the experimental values, which implies that the deviation stays systematic. Most of calculated frequencies are underestimated, nevertheless they still agree well with the experimental measurements for both A and B modes. The frequencies were also calculated within the HF, LDA, GGA and B3LYP approximations, all of which lead to much larger deviations than the B3PW hybrid functional. The other important origin of the underestimation is that long-range Coulomb effect due to the coherent displacement of the crystal nuclei in polar materials are completely neglected, as a consequence of imposing periodic boundary conditions 49,48. The additional term, which essentially depends on the electronic dielectric tensorε and the Born dynamic charge tensor Z α associated with the atom α, ij *,ij 12

13 the induced polarization direction i and the displacement direction j, is required to correct 1 2 E d d M α M β α, i β, j. The assignments of these vibrational modes are listed in Table 4. The low frequencies from cm -1 to cm -1 belong to the external vibrations involving the translational motions of Bi and the polyanion groups and combined with some internal modes in terms of the rocking motion of polyanion groups. From cm -1 to cm -1, the bending and rocking vibrations of tetrahedral [BO 4 ] 5- groups dominate the spectrum. For higher frequencies between cm -1 and cm -1 the rocking and bending vibrations are contributed not only by the [BO 4 ] 5- groups but also the by [BO 3 ] 3- groups. The highest frequencies from cm -1 to cm -1 are attributed to the stretching vibrations of triangle [BO 3 ] 3- groups. Moreover, the frequencies at cm -1, cm -1 and cm -1 are also partly contributed by the stretching vibrations of tetrahedral [BO 4 ] 5- groups. It is found again in our calculation that the frequencies of inplane bending vibrations are higher than those of out-of -plane bending vibrations. We believe that the special electronic structure of BiB 3 O 6 particularly with the lone pair lobe oriented along the direction of the b v vector would pose an evident impact on lattice movements. In order to study the lone pair effect on vibrations of BiB 3 O 6, we calculated the vibrational modes of a hypothetical model compound TlB 3 O 6 with the same geometric structure as monoclinic BiB 3 O 6. The advantage of using TlB 3 O 6 as the comparison with BiB 3 O 6 is threefold. First of all, Tl (III) and Bi (III) have the same [Xe]4f 14 5d 10 core and exhibit similar relativistic effects for the outer electrons; secondly, the ionic radii for Tl 3+ and Bi 3+ are similar, i.e., 1.03 Å and 1.17 Å, respectively. The last fundamental point is based on the different stabilities of 6s orbitals at Tl and Bi. The 6s 2 6p 1 electrons at Tl (orbital energies 6s 1/2 = , 6p 1/2 = , 6p 3/2 = hartree [GRASP] 47 ) are less strongly bound than those in Bi 6s 2 6p 3 (orbital energies 6s 1/2 = , 6p 1/2 = , 6p 3/2 = hartree) and the lone pair lobe is thus removed from TlB 3 O 6. Fig. 11 indicates the complete spherical density around Tl in TlB 3 O 6 where apparently the lone pair effect does not apply. According to the vibrational modes, the comparison of calculated frequencies between BiB 3 O 6 and TlB 3 O 6 is shown in Table 5, where all the vibrational modes with lower- 13

14 frequency shifts of more than 10 cm -1 from BiB 3 O 6 to TlB 3 O 6 are marked as *. The maximum and minimum Bi-O6 (O7) distances achieved during the vibrations are indicated in Table 5 as well. It can be easily observed that all the star-marked modes correspond to the large variations of the Bi-O6 (O7) distances with respect to the equilibrium Bi-O6 (O7) value of Moreover, we have found that the sequences of modes with frequencies lower than 700 cm -1 are quite different in terms of Bi and Tl motions; however, no sequence alteration takes place at higher frequencies more than 700 cm -1. The above interesting difference associated with Bi-O6 bonds between BiB 3 O 6 and TlB 3 O 6 are suggested to be explained as the lone pair effect: since the Bi-O6 covalent interactions take place in BiB 3 O 6, more electrons are distributed in the area between Bi and O6 in BiB 3 O 6. The electrostatic repulsions between the lone pair electrons and the Bi-O6 covalent bonds increase the restoring forces in BiB 3 O 6, whereas this is not the case in TlB 3 O 6 because of its absence of lone pair electrons. Therefore, frequencies associated with large variations of the Bi-O6 distances are blue-shifted in BiB 3 O 6 with respect to TlB3O6, and the sequence of modes is altered as well according to how large such a frequency shift actually becomes. IV. Conclusions The detailed electronic structure of BiB 3 O 6 has been investigated in detail for the first time using the DFT-B3PW method in conjunction with local Gaussian basis sets. The optical gap of polar BiB 3 O 6 was calculated to be 218 nm which agrees well with the experimental optical transmission upper energy threshold of 286 nm. The overestimation of the optical gap was partly due to the neglecting of the spin-orbit splitting of Bi 6p orbitals. It is also concluded that [BO 3 ] 3- groups are easier to be optically polarized than [BO 4 ] 5- groups. The total and projected density of states as well as the total and projected electronic densities has been computed based on several DFT calculations for BiB 3 O 6 with different slight isotropic expansions and compressions of the cells. The stereochemical activity of the Bi lone pairs in BiB 3 O 6 is found to originate from the covalent interactions between occupied Bi 6s and O 2p orbitals hybridized by occupied Bi 6p orbitals in the antibonding way at the higher energy states below the Fermi level. The shape of the lone pair lobes strongly depends on the strength of the cation-anion covalent couplings. Stronger covalent Bi-O couplings lead to less spherical but more 14

15 stereochemically active lone pair lobes. The vibrational frequencies of BiB 3 O 6 have been calculated and the 27 optical vibrational modes have been theoretically assigned for the first time. The influence of lone pairs on lattice vibrations of BiB 3 O 6 has been studied by comparing with vibrations of the hypothetical isostructural compound TlB 3 O 6. It is found that the electrostatic repulsions between lone pair electrons and the covalent bonds in BiB 3 O 6 increase the restoring forces and consequently cause the blue-shift of frequencies associated with Bi-O6 bonds in BiB 3 O 6 as compared to TlB 3 O 6. Acknowledgements This work was financially sponsored by the project Graduiertenkolleg 549, "Azentrische Kristalle" at the University of Köln. The authors are grateful to Prof. Dr. L. Bohatý for valuable discussions on BiB 3 O 6. 15

16 Figure 1 The crystal structure of monoclinic BiB 3 O 6 (C2 space group). 16

17 Figure 2 The six-fold coordination of Bi (III) in monoclinic BiB 3 O 6. Bismuth is indicated by the sequence number 1. Oxygens are numbered from 2 to 7. Borons in the center of tetrahedral units [BO 4 ] 5- are labeled by 8 and those in triangular units [BO 3 ] 3- labeled by 9 and 10. O2 and O3 are translationaly identical to O6 and O7 in primitive cells. The bond distances of Bi-O2 (Bi-O3), Bi-O4 (Bi-O5) and Bi-O6 (Bi-O7) are Å, Å and Å. 17

18 b v a v (a) Figure 3 Total electron densities of plane (001) (a) and of plane (100) (b).the contour lines are plotted between the minimum of 0.0 and the maximum of 0.12 for (001) and for (100) with the step size of and , respectively. 18

19 b v c v (b) Figure 3 Total electron densities of plane (001) (a) and of plane (100) (b).the contour lines are plotted between the minimum of 0.0 and the maximum of 0.12 for (001) and for (100) with the step size of and , respectively. 19

20 PDOS Fermi level Bi s p PDOS PDOS O B s p s p Total DOS Energy (ev) Figure 4 Total (DOS), orbital and atom-projected electronic density of states (PDOS) for s and p orbitals on oxygens, borons and bismuths in BiB 3 O 6. 20

21 B9 B10 B8 PDOS Energy (ev) Figure 5 The contributions of the states from boron atoms at the conduction band. 21

22 (a) Plane O6-Bi-O7 (b) (i) Plane O4-Bi-O5 10% expansion Figure 6 Total electron densities for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi-O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of

23 (a) Plane O6-Bi-O7 (b) (ii) Plane O4-Bi-O5 experimental cell Figure 6 Total electron densities for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi-O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of

24 (a) Plane O6-Bi-O7 (b) (iii) Plane O4-Bi-O5 7% compression Figure 6 Total electron densities for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi-O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of

25 Cell expanded by 10% Bi_s Bi_p O4 O5 O6 O7 I II Energy (ev) (i) 10% expansion Figure 7 The DOSS was projected respectively onto Bi 6s and Bi 6p orbitals as well as O4, O5, O6 and O7 atoms in the valence band for (i) 10% expansion, (ii) experimental cells and (iii) 7% compression in BiB 3 O 6. 25

26 Experimental Cell Bi_s Bi_p O4 O5 O6 O7 I II Energy (ev) (ii) experimental cell Figure 7 The DOSS was projected respectively onto Bi 6s and Bi 6p orbitals as well as O4, O5, O6 and O7 atoms in the valence band for (i) 10% expansion, (ii) experimental cell and (iii) 7% compression in BiB 3 O 6. 26

27 Cell compressed by 7% Bi_s Bi_p O4 O5 O6 O7 I II Energy (ev) (iii) 7% compression Figure 7 The DOSS was projected respectively onto Bi 6s and Bi 6p orbitals as well as O4, O5, O6 and O7 atoms in the valence band for (i) 10% expansion, (ii) experimental cells and (iii) 7% compression in BiB 3 O 6. 27

28 (a) Plane O6-Bi-O7 (b) (i) Plane O4-Bi-O5 10% expansion Figure 8 The projected electron densities for region I of the covalent bonding interaction for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi- O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of

29 (a) Plane O6-Bi-O7 (b) (ii) Plane O4-Bi-O5 experimental cell Figure 8 The projected electron densities for region I of the covalent bonding interaction for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi- O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of

30 (a) Plane O6-Bi-O7 (b) (iii) Plane O4-Bi-O5 7% compression Figure 8 The projected electron densities for region I of the covalent bonding interaction for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi- O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of

31 (a) Plane O6-Bi-O7 (b) (i) Plane O4-Bi-O5 10% expansion Figure 9 The projected electron densities for region II of the covalent antibonding interaction for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi-O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of

32 (a) Plane O6-Bi-O7 (b) (ii) Plane O4-Bi-O5 experimental cell Figure 9 The projected electron densities for region II of the covalent antibonding interaction for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi-O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of

33 (a) Plane O6-Bi-O7 (b) (iii) Plane O4-Bi-O5 7% compression Figure 9 The projected electron densities for region II of the covalent antibonding interaction for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi-O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of

34 Slope: Slope: Experimental Frequencies (cm -1 ) Calculated Frequencies (cm -1 ) Figure 10 The correlation between the calculated frequencies and the experimental values. The solid line is the linear fitting between experimental data and calculated values. The dash line with the slope of is illustrated to show that the calculated frequencies are slightly underestimated. 34

35 (a) Figure 11 The electron density around Tl in the plane O4-Tl-O5 (a) and O6-Tl-O7 (b). The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of

36 (b) Figure 11 The electron density around Tl in the plane O4-Tl-O5 (a) and O6-Tl-O7 (b). The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of

37 Table 1. Mulliken shell populations (s and p) and atomic charge (Q) for the constituting atoms in BiB 3 O 6 BiB 3 O 6 s p Q Bi (6s) (6p) O4, O (1s2s) (2p) O6, O (1s2s) (2p) B (1s2s) (2p) B9, B (1s2s) (2p)

38 Table 2. Integrated Projected Density of States (PDOS) for region I Bi-O bonding interaction and region II Bi-O antibonding interaction in BiB 3 O 6 Region I Region II Total BiB 3 O 6 Bi 6s O4 2p O6 2p Bi 6s Bi 6p Bi 6s Bi 6p 10% expansion experimental cell % compression

39 Table 3. Calculated and experimental frequencies (cm -1 ) of monoclinic BiB 3 O 6. 13A modes in (a) and 14B modes in (b) (a) 13A modes ref. 23 ref. 25 ref. 24 calculated

40 (b) 14B modes ref. 23 ref. 25 ref. 24 calculated

41 Table 4. Assignments of vibrational modes in monoclinic BiB 3 O 6. 13A modes in (a) and 14B modes in (b) (a) 13A modes Cal. Freq. (cm -1 ) Assignments O2-Bi-O3, O4-Bi-O5 O6-Bi-O7 in-plane bending Bi translating along b O2-Bi-O3, O4-Bi-O5 O6-Bi-O7 in-plane bending Bi slightly translating along b rocking for [BO 3 ] 3- and [BO 4 ] 5- subunits [BO 4 ] 5- symmetric bending [BO 4 ] 5- antisymmetric bending [BO 4 ] 5- symmetric tetrahedral distortion [BO 3 ] 3- rocking [BO 3 ] 3- out-of-plane bending [BO 4 ] 5- rocking [BO 3 ] 3- rocking [BO 4 ] 5- symmetric stretching, bending [BO 4 ] 5- antisymmetric bending [BO 3 ] 3- in-plane bending [BO 4 ] 5- tetrahedral distortion [BO 3 ] 3- in-plane bending [BO 3 ] 3- in-plane symmetric stretching [BO 4 ] 5- stretching [BO 3 ] 3- in-plane antisymmetric stretching [BO 4 ] 5- symmetric bending and rocking [BO 3 ] 3- in-plane antisymmetric stretching [BO 4 ] 5- symmetric bending 41

42 (b) 14B modes Cal. Freq. (cm -1 ) Assignments Bi translating Polyanion antitranslating Bi translating Polyanion rocking Bi translating Polyanion rocking Bi translating Polyanion rocking [BO 4 ] 5- rocking [BO 4 ] 5- rocking [BO 3 ] 3- out-of-plane bending [BO 4 ] 5- rocking [BO 3 ] 3- in-plane bending, [BO 4 ] 5- bending [BO 4 ] 5- antisymmetric stretching [BO 4 ] 5- antisymmetric bending [BO 3 ] 3- in-plane bending [BO 4 ] 5- tetrahedral distortion [BO 3 ] 3- in-plane bending [BO 4 ] 5- tetrahedral distortion [BO 3 ] 3- in-plane bending [BO 4 ] 5- tetrahedral distortion [BO 3 ] 3- in-plane antisymmetric stretching [BO 3 ] 3- in-plane antisymmetric stretching [BO 4 ] 5- tetrahedral distortion [BO 3 ] 3- in-plane antisymmetric stretching 42

43 Table 5. The comparison between the calculated frequencies (cm -1 ) of monoclinic BiB 3 O 6 and TlB 3 O 6. The star (*) sign indicates the large frequency shifts of more than 10 cm -1 from BiB 3 O 6 to TlB 3 O 6. The numbers in the bracket are the maximum (Å) and minimum (Å) Bi-O6 (O7) distances that can be achieved in BiB 3 O 6 during the vibration. The equilibrium Bi-O6 distance is Å. A modes B modes BiB 3 O 6 TlB 3 O 6 BiB 3 O 6 TlB 3 O * (2.512, 1.670) * (2.505, 1.949) (2.366, 2.046) * (2.437, 1.884) (2.166, 2.075) (2.230, 2.020) (2.374, 1.840) (2.301, 2.068) * (2.312, 1.884) (2.122, 2.087) * (2.464, 1.740) * (2.626, 1.594) (2.210, 1.977) (2.200, 1.988) * (2.452, 1.756) (2.164, 2.054) * (2.345, 1.832) * (2.457, 1.720) * (2.309, 1.865) (2.233, 1.942) * (2.326, 1.860) (2.360, 1.818) * (2.337, 1.851) (2.180, 1.996) * (2.235, 1.949) * (2.386, 1.815) (2.201, 1.979)

44 Figure captions Figure 1 The crystal structure of monoclinic BiB 3 O 6 with C2 space group. Figure 2 The six-coordinated sphere of Bi (III) in monoclinic BiB 3 O 6. Bismuth is indicated by the sequence number 1. Oxygens are numbered from 2 to 7. Borons in the center of tetrahedral units [BO 4 ] 5- are cited by 8 and those in triangular units [BO 3 ] 3- are cited by 9 and 10. O2 and O3 are translationaly identical to O6 and O7 in primitive cells. The bond distances of Bi-O2 (Bi-O3), Bi-O4 (Bi-O5) and Bi-O6 (Bi-O7) are Å, Å and Å. Figure 3 Total electron densities of plane (001) (a) and of plane (100) (b).the contour lines are plotted between the minimum of 0.0 and the maximum of 0.12 for (001) and for (100) with the step size of and , respectively. Figure 4 Total (DOS), orbital and atom-projected electronic density of states (PDOS) for s and p orbitals on oxygens, borons and bismuths in BiB 3 O 6. Figure 5 The contributions of the states from boron atoms at the conduction band. Figure 6 Total electron densities for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi-O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of Figure 7 The DOSS was projected respectively onto Bi 6s and Bi 6p orbitals as well as O4, O5, O6 and O7 atoms in the valence band for (i) 10% expansion, (ii) experimental cells and (iii) 7% compression in BiB 3 O 6. Figure 8 The projected electron densities for region I of the covalent bonding interaction for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi- O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of Figure 9 The projected electron densities for region II of the covalent antibonding interaction for (i) 10% expansion, (ii) experimental geometry and (iii) 7% compression with respect to the experimental crystallographic unit cell in the plane O6-Bi-O7 (a) and plane O4-Bi-O5 (b) with labels in the oxygen centers. The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of Figure 10 The correlation between the calculated frequencies and the experimental values. The solid line is the linear fitting between experimental data and calculated values. The dash line with the slope of is illustrated to show that the calculated frequencies are slightly underestimated. Figure 11 The electron density around Tl in the plane O4-Tl-O5 (a) and O6-Tl-O7 (b). The contour lines are plotted between the minimum of 0.0 and the maximum of with the step size of