DECISIÓN FINAL

The choice of the basis set used to describe the atomic species in the crystal is a very important preliminary step in a calculation. The experience gained in molecular quantum chemistry can be largely exploited: in particular the atomic core can usually be described by the standard atomic solutions (Clementi and Roetti, 1974). As regards valence shells molecular basis sets can be used as a convenient starting point. They usually perform well in the case o f covalent crystals, while in the case of metallic or ionic compounds the valence must be completely redefined. Semi-ionic compounds, like most o f the silicates, require particular care, and the problem will be discussed in details in Chapter 5.

In all cases diffuse Gaussian orbitals (with exponent coefficients o f less than 0.15 a.u.) have a critical effect on periodic HF calculations: they cause a dramatic increase in the number of integrals that are evaluated and they increase the risk o f pseudo-linear dependence. In periodic calculations, however, very diffuse AO’s do not play the same important role as in molecular calculations. In the latter case they are used to describe the tails o f the electronic distribution in the vacuum, which o f course does not apply for infinite systems.

Many calculations have been performed using relatively poor basis sets, such as the STO-3G sets proposed by Pople and co-workers (Hehre et al., 1969, 1970; Pietro et al. 1980,1981). Split valence basis sets, such as the 6-21 G set (Binkley et al, 1981; Gordon et al, 1982) often provide accurate results, especially when the variational freedom is increased by adding polarization function in the form o f d-orbitals (Hariharan and Pople, 1973; Pietro et al.,

1982).

3 .3 .4 C a lc u la tio n o f o b se r v a b le q u a n titie s in th e H F approxim ation

In this section we consider the ground state energy and the electronic properties that can be evaluated by CRYSTAL.

3.3.4.1 Ground State Energy

The ground state energy EQ o f a system is one o f the most interesting and important parameters that can be extracted from an electronic structure calculation. Equilibrium geometries, relative stabilities o f different phases, reaction paths, formation energies can be derived in terms o f the variation o f EQ as a function of the internal coordinates o f the system.

The total energy per cell can be expressed as a sum o f kinetic (E^), exchange (E ) and Coulomb interaction (E J terms.

A C

They can be synthetically expressed as:

(where the superscripts ee, en, ne and nn refers to electron-electron, electron- nuclear, nuclear-electron and nuclear-nuclear interactions, respectively), and where T and X have been defined in equation (3.3.18, 3.3.21) and the terms in the expression of E correspond to the interaction between the charge distributions

V

(nuclear and electronic) in the zero cell and the overall charge distribution throughout the crystal.

Ek = S i2g P1 2 8 T1 2 8 •

(3.3.27) (3.3.26)

(3.3.28)

Comparison of EQ with experimental energies requires some care. First o f all the HF total energies are affected by the absence o f correlation terms, as will be discussed in the next section. Furthermore we must consider the effects o f incomplete basis sets, numerical approximations and, for heavy atoms,the absence o f relativistic corrections. In any case, the total HF energies must be extrapolated to room temperature (or the temperature at which the experiment was carried out), the nuclear zero-point energy subtracted and the isolated atoms energies added, in order to obtain a quantity that may be compared with experimental measurements. We note that this procedure implicitly assumes the validity o f the Born-Oppenheimer approximation, i.e. the separation o f nuclear and electronic motion.

Problems relating to the basis set may arise when evaluating the HF binding energy. The isolated atomic energies should be evaluated to the same degree o f accuracy as the total HF crystal energy. However, using the same AOs for the isolated atoms as for the crystalline species, leads to an overestimation o f the binding energy, since the variational freedom is larger in the crystal, where valence orbitals are shared by a large number o f neighbouring atoms. This is a well known problem in quantum chemistry, often referred to as “basis set superposition error” (BSSE). In order to correct it, the counterpoise method (Boys and Bemardi, 1970) may be used: the reference atomic energy is obtained using all AOs o f that atom supplemented by the valence AOs o f the surrounding atoms.

Notwithstanding these difficulties, it is often possible to compare energies o f ‘similar’ systems, or o f different geometries o f the same crystal: within these limits, the errors tend, to a large extent, to cancel. It is however clear that particular care and caution are necessary when properties directly related to the energy are studied and discussed.

3.3.4.2 Physical observables: electronic properties

It is useful to recall some general and interesting results produced by the density matrix formalism applied to the problem o f determining the mean value of an observable. This formalism is discussed at length in many reference books, for example in Landau and Lifshitz (1965).

The mean value of any observable that corresponds to a one-electron operator <^l>, can be expressed in terms o f the density matrix o f first order

< f !> = J [ f , ( l ) P^XpXj') ] x , ^ dXj (3.3.29)

where PjCxj.Xj') =

J

The mean value o f any observable that corresponds to a two-electron operator

^ 2 can always be expressed in terms o f the density matrix o f second order:

< f 2>

= V jJ

1 1 2 2

(3.3.30)

Since the study o f the observables o f many-electron systems always refers to one- and two-electron operators, and considering that the first order density matrix can be expressed in terms of the second-order density matrix, we can conclude that the mean value o f any observable can be expressed in terms o f the second order density matrix o f the many-electron system under study. It is interesting to note that, as a result, the amount o f information contained in the multi-electronic (let say n-electron) wave-function (that, in principle, allow n-th order density matrices to be evaluated) is more than that needed to determine the

mean value of the observables in that system. This result is explained if we take into account the ‘statistical’ nature (mean value) o f the information that is provided.

A number o f properties can be evaluated using CRYSTAL. We will briefly comment on those that are used in Chapter 5 to characterize the electronic structure o f silicates.

3.3.4.2.1 Band structure and Density o f States (DOS)

The eigenvalues o f the one-electron Fock Hamiltonian are only an approximation for the electronic spectrum o f the real crystal; in particular, states that belong to the conduction band are very poorly described; band gaps and band widths are systematically overestimated. A further unsatisfactory feature is the sharp fall to zero o f the density of states at the Fermi energy (Monkhorst, 1979; Delhalle and Calais 1986, 1987).

In spite of these limitations, useful information can be obtained from band structures and DOSs; projected DOSs into sets o f orbitals characterize the crystalline orbitals associated with a particular band, and allow, for example, contributions to the bonds and hybridation processes to be analyzed. Projected density o f states can also be compared to experimental X-ray and UV emission spectra.

The total density of states is defined as:

P (e) = Z P , P® (e)

(3.3.31)

with the summation j extended over all the AOs.

By integrating the individual contributions p® (e) one can obtain Mulliken band populations or total Mulliken populations (if the integration is performed up to the Fermi level).

It is well known that Mulliken populations can sometimes be misleading, being strongly dependent on the basis set and on the partition scheme adopted. However, they provide useful preliminary information about the nature o f the bonding, as will be shown and discussed throughout this thesis.