The outline of B2O3 crystal structures is review ed in Section 2.1.
3.3.2.1 BASIS SET EFFECTS
The basis set plays a crucial role in the description of crystalline orbitals. Starting from the stan d ard Pople's basis sets, the exponents of o u te r shell are reo p tim ized . The reo p tim iz ed ex p o n en ts an d the M ulliken charges are com pared in Table 3.2 and 3.3.
i. Minimal basis set
As generally recognized, the M ulliken charges obtained em ploying STO-3G tend to be sm aller than w ith other basis sets (Hehre et al 1986). The bigger problem is that the STO-3G result in B2O3-II is not consistent
w ith the results of the other basis sets. The M ulliken charge on 0 (2 ) seem s to be too small and th at on 0 (1) is larger than expected. 0 (1) is tw o-fold coordinated by boron w hich is the sam e in B2O3-I, w hile 0(2)
has three-fold coordination (see Figure 2.3). The two charge distributions are expected to be different, w hich is difficult to express by using m inim al basis set. Therefore, caution is n eed ed w hen a m inim al basis set is applied to three-fold coordinated oxygen or four-fold coordinated boron.
ii. S-plit-valence and polarization function
As even B2O3 crystals represent a large system, basis set better than
6-21G, such as 8-51G, are not possible in the present version of CRYSTAL. For both polym orphs, the 6-21G basis set m ay be assum ed to be good basis set, as Dovesi et al (1987) and N ada (1990) suggested for Si02. The 3-21G basis set gives almost the sam e result as that for 6-21G.
We inv estig ated the effect of a d d in g a single G au ssian d -ty p e function to boron, because such a polarization function p ro v ed to be u sefu l in Si0 2 for describing the disto rtio n of cation orbitals in the
M ulliken charges are reduced unexpectedly. The reason for this tren d could be the BSSE (Basis Set Superposition E rror ) (see, for exam ple, C lark 1985). That is, the extra orbitals on boron are used to im prove the description of the charge distribution around oxygen.
iii. Pseudopotentials
All electron calculations restrict the feasible system size. In ord er to overcom e this problem , several types of p seu d o p o ten tials hav e been developed. In these pseudopotential techniques the role of core electrons are substituted by the effective core potentials and only the orbitals of valence electrons are calculated. H ere only one available set, PS-31G (Bouteiller 1988), is tested. The original PS-31G sets w ere optim ized for atom s and tested only on sm all molecule. Therefore, the exponents of outerm ost shell w ere reoptim ized. The M ulliken charges tu rn ed o u t to be larger than those calculated w ith the all electron cases. This m ay be d u e to electro n rich second shells, w hich are com posed of th ree gaussians, w hereas the second shell in the other all-electron calculations w ere com posed of tw o gaussians. The o rd er of the charges seem s reasonable, b u t it is difficult to evaluate these pseudopotentials, because they have not been fully tested for the crystalline state. H ow ever, these results show th at the pseudopotential technique is prom ising. If refined these pseudopotentials could become a pow erful tool for calculations on larger systems.
3.3.2.2 GEOMETRY OPTIM IZATION
In o rd er to check the accuracy of o u r calcu latio n s, u n it cell dim ensions w ere optim ized using the 3-21G basis set. The experim ental lattice p aram eters and atom ic param eters are given in Table 3.4. The interatom ic distances and interbond angles are also reported in Table 3.5,
w hile the calculated energies are sum m arized in Table 3.6. The errors in the u n it cell volum e are 1.5% for B203-I and 1.5% for B2O3-II. The errors
in the lattice param eters are 1% for either crystal structure. The atomic param eters are also varied in a point by point m anner u sing the 3-21G basis set. The calculated results are show n Table 3.7. R egarding to the B-O b ond lengths, only the error in B-O(l) b ond (the shortest B-O bond) of B2O3-II is 10% and all the other errors are w ithin 5%. O n the other hand,
all the errors in O-B-O bond angles are w ithin 5%. W hen the general accuracy of 3-21G basis set is taken into account, these calculations reproduce both crystal structures well. The optim ized geom etry is also discussed in C hapter 4.
3.3.2.3 D ISC U SSIO N
The average experim ental B-O bond distance of 1.372
A
in B2O3-Iagrees well w ith the length assum ed by Zachariasen (1963) for a bond of strength' 1 .0 (1.365
A),
and also w ith a value of 1.37+0.02A
w hich w asquoted as the m ean B-O distance for three-coordinated b oron b y W augh (1968).
W hen the charges of three different oxygen in B2O3-I are com pared, the charge of 0(3) is larger than those of the other tw o oxygens. G urr et al (1970) distinguished 0(3) for the other oxygen atom s by using the term ' higher coordination' of 0(3) through w hich adjacent ribbons are linked, alth o u g h 0 (3) is tw o-fold coordinated as w ell as 0 (1 ) or 0(3 ). H ere, M ad elu n g p o ten tials (i.e. the total C oulom b c o n trib u tio n s from all atoms) are calculated as -66.42, -65.49 and -67.18 for 0(1), 0 (2 ) and 0(3). The M adelung potential of 0(3) is larger than those of the other oxygens. In agreem ent w ith G urr's interpretation, it is the obvious explanation for the large M ulliken charge of 0(3).
R egarding B2O3-I, the near planar shape of the BO3 triangle and the
sm all M ulliken charges em phasize the partially covalent character, w hile the d isto rtio n of the triangle and the larger M ulliken charge of 0 (3) indicate the presence of some ionic character.
On the other hand, the average B-O distance for B2O3-II is 1.475 Â,
exactly that proposed by Zachariasen (1963) as an average tetrahedral B-O distance. H ow ever, the tetrahedron is very distorted, w ith one short B- 0(1) distance of 1.373 Â and three long B-0(2) distances of 1.507, 1.506 and 1.512 A. 0 (1) is two-fold coordinated, while 0 (2) is three-fold coordinated.
Prew itt and Shannon (1968) calculated electrostatic bon d strengths in B2O3-II using Zachariasen's table of bond strength versus B-O bond
len g th (Z achariasen 1963). The calculated n et b o n d stre n g th is 3.01 aro u n d B, 1.96 around 0(1), and 2.03 around 0(2). They concluded that the distortions are necessary to balance the electrostatic charge in the crystal.
W hen the M ulliken charges of the four different oxygens in B2O3-II
(except in the case of m inim al basis set) are com pared, the charge of 0 (1)
is found to be m uch sm aller than the others. It is also interesting to note th at the 0 - 0 distances of the four oxygens are alm ost same, although the distance of B an d 0(1) is shorter than the others. This m eans th at the tetrah ed ral arrangem ent of the four oxygens is n o t very distorted; b u t rather w ithin the tetrahedron, B approaches to 0(1). Therefore, it seems reasonable to assum e th at the difference in coordination num ber around oxygen chiefly changes B-O bond strength rather th an the 0 - 0 repulsion in the interatom ic potential m odel for the system. To check w hether the d istortion can be explained in term s of charge transfer betw een boron and oxygen atom s, w e perform ed lattice energy m inim izations using a sim ple rigid ion m odel. Such a m odel, w hich assigns th e M ulliken
charge to each atom , cannot, how ever, explain the short B-O(l) b ond length, because the B-O(l) distance becomes longer, if the charge of 0(1) is set to be sm aller since the attraction betw een B and 0 (1 ) is reduced. H ow ever, the other interpretation nam ely th at B-B rep u lsio n shortens the B-O bond length is also possible. But the results of a perfect lattice relaxation sim ulation, discussed in C hapter 5, show th at the change in the B-O bond strength can reproduce the detailed structure b etter than can be achieved by changing the B-B repulsive term.
Next, the bond lengths in B2O3-I and B2O3-II are com pared. As
Johnson et al (1982) pointed out, one short bond B-O(l) distance (1.373