Bond order potentials for covalent elements. This chapter will be dedicated to a short history of the so called \bond order" potentials [1, 2, 3, 4, 5, 6, 7] (BOPs), that represent an interesting class of the rather scattered family of analytic potentials. Analytic interatomic potentials (sometimes referred to as empirical, semi-empirical or clas- sical potentials) are used for a variety of purposes, ranging from the estimation of minimum energy structures for surface reconstruction, grain boundaries or related defects, to the descrip- tion of the liquid structure. All analytic potentials are de¯ned through a functional form of the interatomic interactions, whose parameters are ¯tted to a selected database. According to Brenner [8], an analytic potential needs to be: ² Flexible: the function should be flexible enough to accommodate the inclusion of a relatively wide range of structures in a ¯tting database. ² Accurate: the potential function must be able to accurately reproduce quantities such energies, bond lengths, elastic constant, and related properties entering a ¯tting database. ² Transferable: the functional form of the potential should be able to reproduce related prop- erties that are not included in the ¯tting database. In practise the potential should be able to give a good description of the energy landscape for any possible realistic con¯guration characterized by the set of atomic positions frig. ² Computationally e±cient: the function should be of such a form that it is tractable for a desired calculation, given the available computing resources. We note that one of the paradigms that has led science through its development, i.e. the Occam razor (\Entities should not be multiplied beyond necessity"), is not mentioned here. This is intentional: as it will become clearer through the chapter, even if bond order potentials are based on the beauty and simplicity of ab initio theories, they achieve computational e±ciency and transferability paying the fee of a proliferation of somewhat ad-hoc adjustments, any of them implying propter-hoc parameters. In other words, on one hand bond order potentials increase transferability, with respect to simple two-body potentials, increasing the number of parameters. On the other hand, they increase computational e±ciency, with respect to ab initio calculations, at the same price of adding parameters. It has to be noted that the focus of bond order potentials is put on their performance, i.e. on the fact that they work. Thus, thinking at the Occam razor, bond order potentials give to \necessity" a meaning that is di®erent from the traditional one. It is indeed necessary to introduce new entities (viz. parameters), for ful¯lling the four requirements stated above. BOPs are all expressed by giving the binding energy of a system as a sum over all pairs: 1 X X £ ¤ E = V R(r ) ¡ b V A(r ) (1.1) b 2 ij ij ij i j6=i R A where V (rij) and V (rij ) are pair-additive interactions representing all interatomic repulsions and all those attractions coming from valence electrons, respectively; rij is the distance between 2 Bond order potentials for covalent elements. atoms i and j and the (scalar) quantity bij is a function of the local environment of atoms i and j, called \bond order". With a meaning indeed close to that used for up-to-date bond-order potentials, the term \bond order" was introduced in 1939 by Coulson [9] in treating the strength of ¼-bonds in polyenes and aromatic molecules. Following this author, if a given molecular orbital of a molecule (ª(n)) is expressed as a linear combination of atomic orbitals (Á(n)) of the individual ¼-electrons: P i ª(n) = a(n)Á(n), with ai a complex number, then the partial mobile bond order (b(n)) of a i i i h i ij (n) 1 (n) (n)¤ (n)¤ (n) ¼-bond between atoms i and j is de¯ned as bij = 2 ai aj + ai aj . The total mobile bond order (bij) of the ¼-bond, between atom i and j, is the sum of the partial bond orders P (n) over all the molecular orbitals involved: bij = (n) bij . The bond order is the total mobile bond order plus one, to account for the σ bond. With this de¯nition, a bond order of value 1 or 2, would correspond to the intuitive concepts of single and double bonds, respectively. In a a benzene molecule the total mobile bond order of each bond is 2=3, and the bond order is 5/3, closer to a double than to a single bond. The form of Eq. 1.1 can be justi¯ed on the basis of the Local Orbital approach (LO) that Abell [10] developed in the framework of Chemical Pseudopotential (CPP) theory [11, 12, 13]. This theory in turn relies on the HÄuckel Molecular-Orbital (HMO) theory [14]. The derivation of Abell is rather instructive: a series of non trivial assumptions lead to two important results. The ¯rst is a motivation of the unexpected alikeness [15] of the shape of the binding energy curves in rather di®erent environments, ranging from pure covalent to metallic bonds; the relevant parameter that identi¯es the type of bonding of a reference atom is the number of its ¯rst neighbours Z. The second is a justi¯cation of the initial guess on the functional form of the ¡1=2 term bij as used in successful BOPs: b = Z . In the following section we will give a brief exposition of the derivation of Eq. 1.1 from LO. The subsequent section (1.2) will suggest an alternative and much more intuitive argument to derive 1.1, based on the moments theorem [16]. In the remainder of the chapter (sections 1.4 and 1.5), a brief overview of BOPs will be given. Focus will be put on those features of the BOPs that are recurrent in all the formulations, underlining analogies and di®erences. Thus, we will often skip the details and we will never give the values of the ¯tted parameters. We will provide a full description of a BOP for the Long range Carbon Bond Order Potential (LCBOP) [7] in section 1.6. The full description is necessary, since an improvement of this potential has been one the early targets of the present thesis work: this modi¯cation is illustrated in section 1.7. 1.1 Localized orbitals model Abell [10], following Anderson [11, 12, 13], writes the Hamiltonian for a system of atoms as: X H = K + Va (1.2) a where K is the total kinetic-energy operator and Va is the e®ective potential due to the presence of atom a in the system and the sum extends over all the atoms. The crucial hypothesis of this approach is that the Va's are short ranged and strongly localized about the corresponding atoms. It is also assumed that only one atomic species is involved. Molecular orbitals (Ãi) end their energies ("i) are the eigenfunctions and eigenvalues of the SCF Hamiltonian: H j Ãi >= "i j Ãi > (1.3) Abell shows that it is possible to ¯nd a suitable basis set, that spans the molecular-orbital sub- space of fÃig, and whose elements are atomic-like orbitals, that allows to write the Hamiltonian as a sum of (environment dependent) pairwise interaction terms. Calling 'm the elements of this searched basis, the Hamiltonian can be rewritten: X X H = [²0 + VR(rkm)] j 'm >< 'm j + m k6=m X X + VA(rkm) j 'm >< 'k j (1.4) m k6=m 1.1 { Localized orbitals model 3 and X Ãi = Cmi'm (1.5) m where the index m runs all over the atoms. The explicit expressions for VR and VA, coming from the CPP theory [13], need the intro- duction of V~m { as the SCF one-electron atomic potential of atom m (including exchange and correlation) appropriated to the isolated atom { and of V~k { as the di®erence from the atomic ~ 0 potential Vm due to the presence of atom k. Introducing the unperturbed atomic orbitals Ám, the energy ²0 of the isolated atom is easily written: 0 ~ 0 ²0 =< Ám j (T + Vm) j Ám > (1.6) and 0 ~ 0 VA(rkm) = < Ák j VkjÁm > 0 ~ 0 0 0 ~ 0 VR(rkm) = < Ám j Vk j Ám > ¡Skm < Ák j Vk j Ám > (1.7) 0 0 0 with Skm =< Ák j Ám >. In the above expressions for VA and VR, the crucial approximation 0 of using the unperturbed atomic orbitals Ám has been adopted . The Hamiltonian in Eq. 1.4 is therefore approximated by the introduction of these VA and VR. The binding energy Eb of the system is de¯ned as the total energy, expressed as the sum of ¤ the energies "i of the occupied orbitals, minus the sum of the energies ²0 of the isolated atoms : X Eb = ni("i ¡ ²0) (1.8) i where ni = 0;1 is the occupancy of the molecular orbital Ãi. De¯ning X : 2 qm = nijCmij (1.9) i X : ¤ bkm = niCkiCmi (1.10) i and using Eqs. 1.4 and 1.5, yields X X X Eb = [qm VR(rkm) + bkmVA(rkm)] (1.11) m k6=m k6=m Note that the de¯nition of bkm explicitly resembles the de¯nition of bond order as given by Coulson [9]. The term qm represents the net electron density on site m. Eq. 1.11 can be particularized for regular structures (in which each atom `sees' the same environment): E X E = b = Z (qV + b V ) (1.12) b N k Rk k Ak k where Eb is the bond energy per atom, Zk is the number of atoms in the k-th shell, bk is the bond order between the reference atom and the atoms in the k-th shell, and q is the number of (valence) electrons per atom.