In the second-moment approximation to tight bind­ing, the cohesive energy is proportional to the square root of the bandwidth, which can be approximated as a sum of pairwise potentials representing squared hopping integrals. Assuming atomic charge neutral­ity, this argument can be extended to all band occu­pancies and shapes22 (Figure 7).

The computational simplicity of FS and EAM fol­lows from the formal division of the energy into a sum of energies per atom, which can in turn be evaluated locally. Within tight binding, we should consider a local density of states projected onto each atom. The preceding discussion of FS potentials concentrates solely on the d-electron binding, which dominates transition metals. However, good potentials are diffi­cult to make for elements early in d-series (e. g., Sc, Ti) where the s-band plays a bigger role. An extension to the second-moment model, which keeps the idea of

Figure 7 In second-moment tight binding, the band shape is assumed constant at all atoms, the effect of changing environments being a broadening of the band.

locality and pairwise functions, is to consider two separate bands, for example, s and d.

This was first considered for the alkali and alka­line earth metals, where s-electrons dominate. These appear at first glance to be close-packed metals, forming fcc, hcp, or bcc structures at ambient pres­sures. However, compared with transition metals, they are easily compressible, and at high pressures adopt more complex ‘open’ structures (with smaller interatomic distances). The simple picture of the physics here is of a transfer of electrons from an s — to a d-band, the d-band being more compact but higher in energy. Hence, at the price of increasing their energy (U), atoms can reduce their volumes (V). Since the stable structure at 0 K is determined by minimum enthalpy H = U + PV at high pressure, this s—d transfer becomes energetically favorable. The net result is a metal-metal phase transformation characterized by a large reduction in volume and often also in conductivity, since the s-band is free electron like while the d-band is more localized. Two-bands potentials capture this transition, which is driven by electronic effects, even though the crystal structure itself is not the primary order parameter.

Materials such as cerium have isostructural tran­sitions. It was thought for many years that Cs also had such a transition, but this has recently been shown to be incorrect,23 and the two-band model was origi­nally designed with this misapprehension in mind.24

For systems in which electrons change, from an s-type orbital to a d-type orbital as the sample is pressurized, one considers two rectangular bands of widths W1 and W2 as shown in Figure 8 with widths evaluated using
eqn [3]. The bond energy of an atom may be written as the sum of the bond energies of the two bands on that atom as in eqn [4], and a third term giving the energy of promotion from band 1 to band 2 (see eqn [8]):

Ubond ni1(ni1 N1 )

2Ni

і 1

+ ni2(ni2 — N2) + Eprom [6]

2 N2

where N1 and N2 are the capacities of the bands (2 and 10 for s and d respectively) and ni1 and ni2 are the occupa­tion of each band localized on the ith atom.

For an ion with total charge T, assuming charge neutrality,

Пі1 + na = T [7]

The difference between the energies of the band cen­ters a1 and a2 is assumed to be fixed. The values of a correspond to the appropriate energy levels in the isolated atom. Thus, a2 — a1 is the excitation energy from one level to another. For alkali and alkaline earth metals, the free atom occupies only s-orbitals; the promotion energy term is therefore simply

Eprom = n2(a2 — a1) = n2Eo [8]

where E0 = a2 — a1.

Thus, the band energy can be written as a function of ni1, ni2, and the bandwidths (evaluated at each atom as a sum of pair potentials, within the second — moment approximation). Defining,

Vi = n,1 — n,2 [9]

and using eqn [7], we can write as follows:

D(E)

N/W1 + N/W2

N/W, —

s-band

Figure 8 Schematic picture of density of electronic states in rectangular two-band model. Shaded region shows those energy states actually occupied.

Ubond = — Wa) — T (Wn + W,2)

‘ v2 + T2 /Wn WA

8 N1 + N2

ViT Wn_ Wi2 + 4 N1 N2

+ T—1 Eo [10]

Although this expression looks unwieldy, it is com­putationally efficient, requiring only two sums of pair potentials for Wj and a minimization at each site independently with respect to vi, which can be done analytically.

In addition to the bonding term, a pairwise repul­sion between the ions, which is primarily due to the screened ionic charge and orthogonalization of the valence electrons, is added. In general, this pair potential should be a function of vi and Vj. But to maintain locality, one has to write this pairwise con­tribution to the energy in the intuitive form, as the sum of two terms, one from each ‘band,’ proportional to the number of electrons in that band:

V(r, j ) = (nil + nj 1) Vi (rjj) + (ni2 + nj2) V2 (rjj) [11]

The variational property expressed in eqn [14] can be exploited to derive the force on the ith atom:

r dUtot

j drj

_ dUtot _ 9Utot @V = dr, lv dr,

Hence, the force is simply the derivative of the energy at fixed v. Basically, this is the Hellmann- Feynman theorem25 which arises here because v is essentially a single parameter representation of the electronic structure.

This result means that, like the energy, the force can be evaluated by summing pairwise potentials. Hence, the two-band second-moment model is well suited for large-scale MD. The force derivation itself is somewhat tedious, and the reader is referred to the original papers. There is no Hellman-Feynman type simplification for the second derivative, so analytic expressions for the elastic constants in two-band models are long ranged and complicated. Conseq­uently, elastic constants are best evaluated numerically.

the mechanism may involve shearing rather than isostructural collapse, particularly if a continuous interface between the two phases exists, as in a shock­wave. With the two-band model, the transition is first order, the volume collapse occurring before the bulk modulus becomes negative in the unstable region. Although the shear and tetragonal shear decrease in the unstable region, neither actually goes negative.

The two-band model is applicable to transition metals, but since the d-band is occupied at all pres­sures, electron transfer is continuous and there is no phase transition. This makes the empirical division of the energy into s and d components challenging. However, once appropriately scaled for ionic charge and number of electrons T in principle, the method could be extended to alloys with noninteger T The results of such an extrapolation are extraordinarily good (Figure 10), considering that there is no fitting to any material other than Cs. The extrapolation breaks down at high Z where the amount of sp hybri­dization is not fully captured in the parameterization. As with FS, no information about band shape is included, and so the sequence of crystal structures cannot be reproduced.

While the extrapolated potentials do not represent the optimal parameterization for specific transition
metals, the recovery of the trends across the group lends weight to the idea that the two-band model correctly reproduces the physics of this series.

The s-d two-band approach has also been applied with considerable success by considering the s-band as an alloying band.26 This has been applied to the FeCr system, which we discuss in more detail later.