1

Ab initio Density Functional Theory Investigation of Monoclinic BiB3O6 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 BiB3O6 and the origin of the stereochemical active lone pair at Bi with local Gaussian basis sets. The calculated electronic densities in BiB3O6 picture the lone pair lobe at Bi for the first time. The formation of the lone pair lobe in BiB3O6 is studied and discussed in much detail based on the density of states of BiB3O6 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 BiB3O6 is clarified by first calculating the vibrational frequencies of BiB3O6 and assigning all modes, and then subsequently comparing BiB3O6 with the vibrations of the hypothetical isostructural compound TlB3O6 without metal lone pair electrons in it.

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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 experimentally1, 2, 3, 4, 5 and theoretically6, 7, 8, 9, 10. In

the past decades, a series of borate materials were discovered among them. Polar

monoclinic bismuth triborate BiB3O6 (BiBO) 11 is an excellent non-linear and linear

optical material due to its large effective SHG (Second Harmonic Generation)

coefficient 12.3 −⋅= Vpmd SHGeff

12, which is higher than that of most other borate-based

NLO materials currently used such as LiB3O5 (LBO) or BaB2O4 (BBO)13, 14. The NLO

susceptibility tensors of BiB3O6 were calculated with the chemical bond method15 as well

as the anion group theory16 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

[BiO4]5- group16. However, up to now, little is known about how the electronic effects in

BiB3O6, including those from 6s2 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, [BO3]3-, [BO4]5- and [BiO4]5-). A detailed understanding

of the electronic structure of BiB3O6 would contribute significantly to explain the role of

Bi (III) in BiB3O6, 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 Powell17 and of

Gillespie and Nyholm18 several decades ago, who proposed the so-called valence shell

electron pair repulsion theory (VSEPR). A classic view was invoked afterwards by

Orgel19 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 Kumar20 supported this viewpoint based on extended Hűckel

calculations by concluding that the mixing of the cationic s orbital (HOMO) and the

cationic pz orbital (LUMO) predominates for the molecular anion [BiCl6]3- and crystalline

SbI3 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

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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

BiMnO321 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 systems22,23,24,25.

In monoclinic BiB3O6, Bi (III) holds an electronic configuration [Xe] 4f145d106s2.

BiB3O6 crystallizes in space group C2 and consists of c direction-alternating layers of

[B3O6]3- rings forming sheets of corner-sharing [BO3]3- triangles and [BO4]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 [BiO4]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 BiB3O6. 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

[B3O6]3- rings and special arrangements of [BO3]3- and [BO4]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 BiB3O6 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 off-

centric electronic densities. Further studies on the 6s2 lone pair at Bi (III) in monoclinic

BiB3O6 were not performed.

First-principle calculations for periodic compounds are usually performed using plane-

wave 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 ago27,28,29,30. The advantages of the LCAO scheme based on local

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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 calculations31; 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 BiB3O6

and the origin of the stereochemically active Bi (III) lone pair. Based on the discussion of

the electronic structure of BiB3O6, 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 BiB3O6, since vibrational frequencies are relevant to

bond strengths and the hierarchy of energy distributions of atomic motions involving

different structural subunits35. We are not aware of previous computations of vibrational

frequencies although several experimental reports have already reported both Raman and

IR spectra of BiB3O632,33,34,35. 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 [BO3]3-

and [BO4]5- vibrational modes reported in other borate crystals and glasses34,35.

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

CRYSTAL0327,27,29,30,41. Calculations for both the electronic structure and vibrational

modes were performed in one primitive unit cell (denoted as pav , pbv

and pcv ) with one

bismuth, six oxygens and three borons included, which is transformed from the

crystallographic cell (denoted as av , bv

and cv ) with the double number of the

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corresponding atoms by defining the new primitive vectors baa p

vvv −= and babp

vvv+=

but maintaining the cv lattice vector as pcv .

All the basis sets were originally taken from the public EMSL database42. 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 0.870 optimized for crystalline BiB3O6; 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 0.2742 and 0.538, respectively

for crystalline BiB3O6.

We use energy-consistent scalar-relativistic ab initio effective core pseudopotentials

(ECP) of the Stuttgart-Koeln variety for Bi44 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 BiB3O6 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; 10-14 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

net45. 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

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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 weights36 were chosen.

The reliable and accurate simulation of vibrations for crystals is still at a less

developed stage than for molecules46, 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 bsh-

script 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

diagonal elements 2,

21

idE

M αα ∂∂ and the nondiagonal elements

ji ddE

MM ,,

21

βαβα ∂∂∂ where i

must be different from j. The three-point numerical derivative formula (equation (1)) was

adopted to compute the diagonal elements,

2,

00

2,

2 )()(,,

i

dd

i d

EEEE

dE ii

αα

αα

∆

−+−=

∂∂ ∆−∆

(1)

where id ,α∆ is the displacement (0.001 a.u. in our calculations) along the Cartesian i-

coordinate of atom α and E0 is the energy at the equilibrium geometry. Totally N6 single

point calculations have to be performed for the entire N3 diagonal elements each of

which needs 2 complete single point rounds. The nondiagonal elements were computed

according to equation (2):

j

jiji

ji ddgdg

ddE

,

,,,,

,,

2

2)()(

β

βαβα

βα ∆

∆−−∆=

∂∂∂ (2)

where )( ,, ji dg βα ∆± is defined as the gradient along the i-coordinate of atom α at the

positive and negative displacement along the j-coordinate of atom β respectively

(equation (3)):

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i

jdjdji d

dEdEdg ii

,

,,,, 2

)()()( ,,

α

βββα

αα

∆

∆±−∆±=∆±

∆−∆ (3)

Totally )13(6 −NN single point calculations had to be performed for the entire

2)13(3 −NN nondiagonal elements each of which needs 4 complete single point rounds.

Therefore, the total number of single point calculations for frequencies is 118 2 +N (e.g.

1801 single point rounds for BiB3O6 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 BiB3O6

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

br

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.

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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 -16.6 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 BiB3O6. 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 6p1/2-6p3/2 splitting is 2.10 eV, i.e., 6p1/2 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 BiB3O6. Such a

conclusion is qualitatively consistent with the anion group theory16 which shows that the

contributions of the [BiO4]5- group are about ten times greater than those of the [BO3]3- or

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[BO4]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 [BO3]3- are responsible for the bottom of the conducting band, where the

states of borons (B8) in the center of tetrahedral units [BO4]5- are completely excluded

and greatly populate only at higher energy levels than 7.0 eV. This may imply that

[BO3]3- is easier to be optically polarized than [BO4]5- since the latter requires relatively

stronger electronic excitations. This supports the chemical intuition for BiB3O6 that π

electrons in the planar [BO3]3- can be more easily moved while such movements are

considerably restricted in the three-dimensionally extended [BO4]5-.

B. Bi-O interactions and stereochemically active lone pair in BiB3O6

Let us first study the charge distributions in BiB3O6 (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 6s26p3 configuration as reference with 5 electrons

for neutral Bi and 6s2 for trivalent Bi (III). However, a Bi 6p-shell population of 0.908 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 BiB3O6 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 2.629 Å,

2.390 Å and 2.223 Å as well as 2.295 Å, 2.087 Å and 1.941 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.

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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 0.100 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 BiB3O6 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 [BO3]3- and [BO4]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

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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 0.484 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

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indicated in Fig. 6. During the shrinking of the BiB3O6 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 2.037 (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 BiB3O6

In this section, the vibrational frequencies of BiB3O6 were calculated and the

vibrational modes were fully assigned due to the calculated vibrating vector analyses.

The nuclear site group method48 was applied to decompose the symmetry representation

of the Brillouin zone-center vibrational modes. In the crystal structure of BiB3O6, one Bi

and one B atoms occupy the C2 sites, and the remaining two B and six O atoms occupy

the C1 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 conditions49,48. The additional term, which essentially depends on the electronic

dielectric tensor ∞ijε and the Born dynamic charge tensor *

,ijZα associated with the atom α,

13

the induced polarization direction i and the displacement direction j, is required to

correctji dd

EMM ,,

21

βαβα ∂∂∂ .

The assignments of these vibrational modes are listed in Table 4. The low frequencies

from 136.73 cm-1 to 261.45 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 311.07 cm-1 to 477.16

cm-1, the bending and rocking vibrations of tetrahedral [BO4]5- groups dominate the

spectrum. For higher frequencies between 602.27 cm-1 and 1029.43 cm-1 the rocking and

bending vibrations are contributed not only by the [BO4]5- groups but also the by [BO3]3-

groups. The highest frequencies from 1109.15 cm-1 to 1556.75 cm-1 are attributed to the

stretching vibrations of triangle [BO3]3- groups. Moreover, the frequencies at 674.56 cm-1,

688.22 cm-1 and 1109.15 cm-1 are also partly contributed by the stretching vibrations of

tetrahedral [BO4]5- groups. It is found again in our calculation that the frequencies of in-

plane bending vibrations are higher than those of out-of -plane bending vibrations.

We believe that the special electronic structure of BiB3O6 particularly with the lone

pair lobe oriented along the direction of the bv

vector would pose an evident impact on

lattice movements. In order to study the lone pair effect on vibrations of BiB3O6, we

calculated the vibrational modes of a hypothetical model compound TlB3O6 with the

same geometric structure as monoclinic BiB3O6. The advantage of using TlB3O6 as the

comparison with BiB3O6 is threefold. First of all, Tl (III) and Bi (III) have the same

[Xe]4f145d10 core and exhibit similar relativistic effects for the outer electrons; secondly,

the ionic radii for Tl3+ and Bi3+ 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

6s26p1 electrons at Tl (orbital energies 6s1/2=-0.4496, 6p1/2=-0.2113, 6p3/2=-0.1765 hartree

[GRASP]47) are less strongly bound than those in Bi 6s26p3 (orbital energies 6s1/2=-

0.6868, 6p1/2=-0.3384, 6p3/2=-0.261 hartree) and the lone pair lobe is thus removed from

TlB3O6. Fig. 11 indicates the complete spherical density around Tl in TlB3O6 where

apparently the lone pair effect does not apply.

According to the vibrational modes, the comparison of calculated frequencies between

BiB3O6 and TlB3O6 is shown in Table 5, where all the vibrational modes with lower-

14

frequency shifts of more than 10 cm-1 from BiB3O6 to TlB3O6 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 2.087. 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 BiB3O6 and

TlB3O6 are suggested to be explained as the lone pair effect: since the Bi-O6 covalent

interactions take place in BiB3O6, more electrons are distributed in the area between Bi

and O6 in BiB3O6. The electrostatic repulsions between the lone pair electrons and the

Bi-O6 covalent bonds increase the restoring forces in BiB3O6, whereas this is not the case

in TlB3O6 because of its absence of lone pair electrons. Therefore, frequencies associated

with large variations of the Bi-O6 distances are blue-shifted in BiB3O6 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 BiB3O6 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 BiB3O6 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 [BO3]3- groups are easier to be optically polarized than

[BO4]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 BiB3O6

with different slight isotropic expansions and compressions of the cells. The

stereochemical activity of the Bi lone pairs in BiB3O6 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

15

stereochemically active lone pair lobes. The vibrational frequencies of BiB3O6 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 BiB3O6 has been studied by

comparing with vibrations of the hypothetical isostructural compound TlB3O6. It is found

that the electrostatic repulsions between lone pair electrons and the covalent bonds in

BiB3O6 increase the restoring forces and consequently cause the blue-shift of frequencies

associated with Bi-O6 bonds in BiB3O6 as compared to TlB3O6.

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 BiB3O6.

16

Figure 1 The crystal structure of monoclinic BiB3O6 (C2 space group).

17

Figure 2 The six-fold coordination of Bi (III) in monoclinic BiB3O6. Bismuth is indicated by the sequence number 1. Oxygens are numbered from 2 to 7. Borons in the center of tetrahedral units [BO4]5- are labeled by 8 and those in triangular units [BO3]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 2.087 Å, 2.390 Å and 2.632 Å.

67 4

5

2

31

8

9

10

18

av

bv

(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 0.1185 for (100) with the step size of 0.004 and 0.00395, respectively.

19

cv

bv

(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 0.1185 for (100) with the step size of 0.004 and 0.00395, respectively.

20

-30 -25 -20 -15 -10 -5 0 5 10 15 20

PD

OS

PDO

SD

OS

Total

PD

OS

Energy (eV)

B s p

O s p

Bi s p

Fermi level

Figure 4 Total (DOS), orbital and atom-projected electronic density of states (PDOS) for s and p orbitals on oxygens, borons and bismuths in BiB3O6.

21

-1 3 7 11 15 19

Energy (eV)

B9 B10 B8

PD

OS

Figure 5 The contributions of the states from boron atoms at the conduction band.

22

(a) Plane O6-Bi-O7

(b) Plane O4-Bi-O5

(i) 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 0.100 with the step size of 0.002.

23

(a) Plane O6-Bi-O7

(b) Plane O4-Bi-O5

(ii) 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 0.100 with the step size of 0.002.

24

(a) Plane O6-Bi-O7

(b) Plane O4-Bi-O5

(iii) 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 0.100 with the step size of 0.002.

25

-17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6

Energy (eV)

Bi_s Bi_p O4 O5 O6 O7

Cell expanded by 10%

I II

(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 BiB3O6.

26

-17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6

Energy (eV)

Bi_s Bi_p O4 O5 O6 O7

Experimental Cell

I II

(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 BiB3O6.

27

-17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3

Energy (eV)

Bi_s Bi_p O4 O5 O6 O7

Cell compressed by 7%

I II

(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 BiB3O6.

28

(a) Plane O6-Bi-O7

(b) Plane O4-Bi-O5

(i) 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 0.100 with the step size of 0.002.

29

(a) Plane O6-Bi-O7

(b) Plane O4-Bi-O5

(ii) 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 0.100 with the step size of 0.002.

30

(a) Plane O6-Bi-O7

(b) Plane O4-Bi-O5

(iii) 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 0.100 with the step size of 0.002.

31

(a) Plane O6-Bi-O7

(b) Plane O4-Bi-O5

(i) 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 0.100 with the step size of 0.002.

32

(a) Plane O6-Bi-O7

(b) Plane O4-Bi-O5

(ii) 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 0.100 with the step size of 0.002.

33

(a) Plane O6-Bi-O7

(b) Plane O4-Bi-O5

(iii) 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 0.100 with the step size of 0.002.

34

0 200 400 600 800 1000 1200 1400 16000

200

400

600

800

1000

1200

1400

1600

Slope: 1.030 Slope: 1.000

Exp

erim

enta

l Fre

quen

cies

(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 1.000 is illustrated to show that the calculated frequencies are slightly underestimated.

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 0.100 with the step size of 0.001.

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 0.100 with the step size of 0.001.

37

Table 1. Mulliken shell populations (s and p) and atomic charge (Q) for the constituting

atoms in BiB3O6

BiB3O6 s p Q

Bi 2.166 (6s) 0.908 (6p) +2.022

O4, O5 3.762 (1s2s) 4.952 (2p) -0.726

O6, O7 3.860 (1s2s) 5.093 (2p) -0.963

B8 2.304 (1s2s) 1.557 (2p) +0.809

B9, B10 2.307 (1s2s) 1.709 (2p) +0.695

38

Table 2. Integrated Projected Density of States (PDOS) for region I Bi-O bonding

interaction and region II Bi-O antibonding interaction in BiB3O6

Region I Region II Total

BiB3O6 Bi 6s O4 2p O6 2p Bi 6s Bi 6p Bi 6s Bi 6p

10% expansion 1.648 0.069 0.187 0.401 0.963 2.049 0.963

experimental cell 1.013 0.266 0.419 1.036 0.891 2.049 0.891

7% compression 0.595 0.388 0.484 1.419 0.906 2.014 0.906

39

Table 3. Calculated and experimental frequencies (cm-1) of monoclinic BiB3O6. 13A modes in (a) and 14B modes in (b)

(a) 13A modes ref. 23 ref. 25 ref. 24 calculated

203 209 202.64 189.90

271 269 269.89 235.84

316 315 316.40 261.45

394 394 395.18 371.24

440 - 439.98 402.65

575 577 580.60 602.27

649 650 652.46 640.53

- 718 715.43 688.22

837 860 857.46 748.47

1105 1097 1104.00 1020.43

1194 1194 1187.19 1109.15

- 1401 1415.03 1274.13

1485 1484 1484.69 1556.75

40

(b) 14B modes

ref. 23 ref. 25 ref. 24 calculated

135 133 136.44 136.74

168 167 167.38 171.58

- 192 187.37 177.03

- - 219.49 216.16

369 369 371.38 311.07

465 453 451.14 477.16

667 667 669.26 640.91

- 685 684.00 674.56

751 742 746.77 741.45

942 942 940.76 872.00

- 1065 - 1002.34

- 1293 1294.26 1156.83

1369 1380 1379.38 1267.33

1451 1452 1454.33 1470.18

41

Table 4. Assignments of vibrational modes in monoclinic BiB3O6. 13A modes in (a) and 14B modes in (b)

(a) 13A modes

Cal. Freq. (cm-1) Assignments

189.90 O2-Bi-O3, O4-Bi-O5 O6-Bi-O7 in-plane bending

Bi translating along b

235.84 O2-Bi-O3, O4-Bi-O5 O6-Bi-O7 in-plane bending

Bi slightly translating along b

261.45 rocking for [BO3]3- and [BO4]5- subunits

371.24 [BO4]5- symmetric bending

402.65 [BO4]5- antisymmetric bending

602.27 [BO4]5- symmetric tetrahedral distortion

[BO3]3- rocking

640.53 [BO3]3- out-of-plane bending

[BO4]5- rocking

688.22 [BO3]3- rocking

[BO4]5- symmetric stretching, bending

748.47 [BO4]5- antisymmetric bending

[BO3]3- in-plane bending

1020.43 [BO4]5- tetrahedral distortion

[BO3]3- in-plane bending

1109.15 [BO3]3- in-plane symmetric stretching

[BO4]5- stretching

1274.13 [BO3]3- in-plane antisymmetric stretching

[BO4]5- symmetric bending and rocking

1556.75 [BO3]3- in-plane antisymmetric stretching

[BO4]5- symmetric bending

42

(b) 14B modes

Cal. Freq. (cm-1) Assignments

136.74 Bi translating

Polyanion antitranslating

171.58 Bi translating

Polyanion rocking

177.03 Bi translating

Polyanion rocking

216.16 Bi translating

Polyanion rocking

311.07 [BO4]5- rocking

477.16 [BO4]5- rocking

640.91 [BO3]3- out-of-plane bending

[BO4]5- rocking

674.56 [BO3]3- in-plane bending, [BO4]5- bending

[BO4]5- antisymmetric stretching

741.45 [BO4]5- antisymmetric bending

[BO3]3- in-plane bending

872.00 [BO4]5- tetrahedral distortion

[BO3]3- in-plane bending

1002.34 [BO4]5- tetrahedral distortion

[BO3]3- in-plane bending

1156.83 [BO4]5- tetrahedral distortion

[BO3]3- in-plane antisymmetric stretching

1267.33 [BO3]3- in-plane antisymmetric stretching

[BO4]5- tetrahedral distortion

1470.18 [BO3]3- in-plane antisymmetric stretching

43

Table 5. The comparison between the calculated frequencies (cm-1) of monoclinic BiB3O6 and TlB3O6. The star (*) sign indicates the large frequency shifts of more than 10 cm-1 from BiB3O6 to TlB3O6. The numbers in the bracket are the maximum (Å) and minimum (Å) Bi-O6 (O7) distances that can be achieved in BiB3O6 during the vibration. The equilibrium Bi-O6 distance is 2.087 Å.

A modes B modes

BiB3O6 TlB3O6 BiB3O6 TlB3O6

189.90* (2.512, 1.670) 135.50 136.74* (2.505, 1.949) 84.96

235.84 (2.366, 2.046) 289.25 171.58* (2.437, 1.884) 135.37

261.45 (2.166, 2.075) 273.34 177.03 (2.230, 2.020) 194.54

371.24 (2.374, 1.840) 415.54 216.16 (2.301, 2.068) 332.85

402.65* (2.312, 1.884) 391.58 311.07 (2.122, 2.087) 306.20

602.27* (2.464, 1.740) 555.33 477.16* (2.626, 1.594) 445.47

640.53 (2.210, 1.977) 666.90 640.91 (2.200, 1.988) 642.58

688.22* (2.452, 1.756) 640.88 674.56 (2.164, 2.054) 680.34

748.47* (2.345, 1.832) 722.70 741.45* (2.457, 1.720) 704.33

1020.43* (2.309, 1.865) 1009.97 872.00 (2.233, 1.942) 871.38

1109.15* (2.326, 1.860) 1099.16 1002.34 (2.360, 1.818) 1001.64

1274.13* (2.337, 1.851) 1252.48 1156.83 (2.180, 1.996) 1148.79

1556.75* (2.235, 1.949) 1540.32 1267.33* (2.386, 1.815) 1250.43

1470.18 (2.201, 1.979) 1469.16

44

Figure captions

Figure 1 The crystal structure of monoclinic BiB3O6 with C2 space group. Figure 2 The six-coordinated sphere of Bi (III) in monoclinic BiB3O6. Bismuth is indicated by the sequence number 1. Oxygens are numbered from 2 to 7. Borons in the center of tetrahedral units [BO4]5- are cited by 8 and those in triangular units [BO3]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 2.087 Å, 2.390 Å and 2.632 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 0.1185 for (100) with the step size of 0.004 and 0.00395, 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 BiB3O6. 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 0.100 with the step size of 0.002. 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 BiB3O6. 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 0.100 with the step size of 0.002. 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 0.100 with the step size of 0.002. 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 1.000 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 0.100 with the step size of 0.001.

45

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