In order to perform the quantum chemical calculations, we used the periodic ab-initio Hartree-Fock code CRYSTAL-92. As the program uses basis sets co n stru cted from atom ic o rb ital (AO) as w ith s ta n d a rd m o lecu lar o rb ita l p ro g ra m s (for exam ple, GAUSSIAN, GAMESS, HONDO, MOP AC, AMP AC), it is easy for chemists to use. Furtherm ore, calculated properties, such as electron charge density m aps and M ulliken charges, can be analyzed using chem ical concepts. The details of the theory are described in the m onograph of Pisani et al (1988). Here, som e com putational assum ptions and restrictions especially w h en ap p ly in g the m ethod to borate crystals are m entioned.
3.2.1 SELECTION OF A MODEL
In solid state physics, plane w aves (PW) are usually used as a basis function of crystalline orbitals ly w hich satisfy Bloch's theorem:
ij/(k;r+g) = \i/(k;r) exp(ik g) (3.1)
w here r is real-space vector, k is w avevector, and g is a direct lattice vector of the crystal.
In ste a d of PW s, CRYSTAL-92 u ses Bloch fu n ctio n s (|)|i(k), constructed from a lim ited num ber of local functions 5C^(r):
4)p(k;r) = Eg %p(r-g) exp(ik g) (3.2)
The "generating" functions X|x are centred at the atom ic nuclei and are expressed as a linear com bination of G aussian-type atom ic orbitals (GTOs), sim ilar to m olecular quantum chem istry techniques.
G en erally sp eak in g , the larg e r n u m b e r of GTOs w h ic h are em ployed, the m ore accurate becom es the calculated resu lt w ith in the H artree-F ock (HF) lim it. H ow ever, in contrast to the calculation on m olecular groups, diffuse G aussian orbitals (exponent of the order of 0.2
a.u. or less) play a crucial role in crystalline-state calculations and cause two problem s (Pisani et al 1988).
The first is th at the num ber of integrals to be explicitly calculated increases dram atically w ith a decreasing exponent. The second is th at on decreasing the exponent the risks of pseudo-linear dependence increase rap id ly , d em an d in g higher precision in o rd er to avoid "catastrophic" b eh av io u r.
H ow ever, such very diffuse AOs are m uch less im portant in three- dim ensional densely packed crystals than in atom s and molecules, w here they serve to describe the tails of the electronic d istrib u tio n to w ard vacuum . Therefore, except for the case of the m inim al basis set (STO-3G), starting from the standard Pople's basis set (Pople and Binkley 1975), the exponent of the outerm ost shell is reoptim ized.
Furtherm ore, in the present version of CRYSTAL-92, large system s cannot be sim ulated, except at the m inim al basis set level, because of restrictions related to the size of vectors and matrices. Therefore, several different basis sets are applied for sim ulation of sm all B2O3 sy ste m s,
while only a m inim al basis set is applied for other larger systems.
In o rd er to overcom e the lim itation of system size in the all electron calculations, pseudopotentials techniques have been developed (Hay and W adt 1985; D urand and Barthelat 1975; Bouteiller et al 1988). This technique has also been tested for the B2O3 systems.
The other possible problem associated w ith ou r m ethod m ay come from electron correlation w hich can only be rep resen ted u sin g p o st Hartree-Fock techniques (Hehre et al 1986). In addition, how ever, several correlation correction schem es are available for the HF energy. H ere, these precise corrections are not required in this chapter, because only the relative orders in energies or charges are m ainly discussed.
3.2.2 OPTIM IZATION OF STRUCTURE
The experim ental determ ination of crystal structures has, of course, associated errors. For example, the positions of hydrogen atom s in boric acid crystals determ ined by X-ray techniques have errors of as m uch as 0.1 Â. For this, and for other reasons, it is desirable in sim ulations to relax not only u n it cell dim ensions b u t also internal coordinates. To relax the stru ctu re m eans to search for the stru ctu re w hich has the m inim um total energy. H ow ever, the autom atic relaxation of cell dim ensions or internal coordinates is not available in the present version of CRYSTAL. A lth o u g h m an u al o p tim ization is possible p o in t b y p o in t, it is n o t efficient.
Therefore, in B2O3 crystals cell dim ensions and internal coordinates
are varied in a point by point m anner and the variations in energy are tabulated. These potential energy surface data are also used for m odelling interatom ic potentials in C hapter 5. In contrast, the calculations on the o th er b o rate crystals are carried o u t u sin g fixed (ex p erim en tally determ ined) cell dim ensions and atomic positions. We note th at a full rela x atio n tre a tm e n t u sin g ab-initio, LDA tec h n iq u e s is p o ssib le em ploying the code CASTEP, as discussed in C hapter 4.
3.2.3 MULLIKEN POPULATION ANALYSIS
It m ay be useful to define the total electronic charge on a particular atom in order th at quantitative m eaning m ay be given to such concepts as electron w ith d raw in g or donating ability. The M ulliken p o p u latio n analysis is one of such m ethods, and it is often used for discussing the relative covalency and ionicity of materials.
A lth o u g h M u llik en p o p u la tio n an aly sis is em p lo y ed in th is chapter, some caution is required.
One problem concerns the definitions of 'ionicity' and 'covalency', as discussed by C atlow and Stoneham (1983). There is a considerable arbitrariness in th eir n atu re and different charge p artitio n in g schem es exist. A nother concerns the high sensitivity of the M ulliken charges to the basis-set (H ehre et al 1986) A third difficulty is th at the charges are often not com parable w ith effective charges obtained from experim ental studies, and indeed the absolute value m ay be meaningless.
In this chapter, only the relative order in the M ulliken charges at different sites an d am ong a variety of structures are used in ord er to discuss relative degrees of "ionicity' or "covalency".
3.3 PERIODIC AB-INITIO HARTREE-FOCK SIM ULATIO N