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There have been many developments since Lang and Kohn to extend the free electron approach, first to s-p bonded metals and then to the complications of transition metals involving d-electrons, and in the case of the rare earths, f-electrons as well. The d-electrons give an angular character to the bonding, often resulting in structures which are not close-packed, e.g. b.c.c (Fe, Mo, W, etc.) or complex structures like a-Mn. This is in contrast to s-p bonded metals which typically are either f.c.c or h.c.p. There are many challenges left for quantum mechanical models of metals.

To start we need a few names of the methods, to supplement the NFE methods discussed in Lecture 3 and Tight-binding, to be discussed in the next lecture. Some of these are pseudopotentials, orthogonalized plane waves (OPW), augmented plane waves (APW), Korringa-Kohn-Rostoker (KKR), etc. These long-standing methods are described by Ashcroft and Mermin, chapter 11, with experimentally determined band structures in chapter 15. There are descriptions of these methods with a more modern "feel" to them in the books by Kaxiras, chapter 4 and by Martin, chapter 16. A shorter version is given by Marder, chapter 10. These books, especially Richard Martin's, almost amount to "how-to" manuals, and are backed up with detailed computational examples. Such computations would be suitable for a detailed project, but are not to be undertaken lightly.

Typically tight-binding (where interatomic overlap integrals are thought of as small) is taken as the opposite extreme to the nearly-free electron model (where Fourier coefficients of the lattice potential are thought of as small). However, this is more apparent than real, in that both pictures can work for arbitrarily large overlap integrals or lattice potentials; the only requirement is that the basis sets are complete for the problem being studied. This of course can lead to some semantic problems: methods which sound different may not in fact be so different; in particular, when additional effects are included they are almost certainly not simply additive.

The basic feature caused by including the ions via any of these methods is that the electron density is now modulated in x, y and z with the periodicity of the lattice. So there are now two length scales in the problem which compete. When defects of any kind are present, there is no need for the two length scales to bear any relationship to each other. For example, when translational invariance in the z-direction is lost due to the presence of a surface, the resulting surface states have oscillation periods with no simple relation to the lattice period in the z-direction.

One of the first such calculations was to metallic Lithium, which, with only
three electrons per atom, is amenable to full first principles calculations.
An example of a surface-related calculation is shown in
figure 5.1, taken
from an early review article by Appelbaum & Hamann (1976). Large scale
calculations of the band structures for many metals using similar methods
have been performed by Moruzzi *et al.* (1978) and by Papaconstantopoulos (1986).

Figure 5.1: Valence electron density at several x// points for Li(001) in a pseudo-potential calculation (from Alldredge & Kleinman 1974, after Appelbaum & Hamann 1976).

The cohesive function E_{c}(n) is a function of the homogeneous
electron gas density n in which the atom is embedded
(Jacobsen *et al.* 1987, Jacobsen 1988, Nørskov et al. 1993). A major effect of
these models is to show clearly that metallic binding is strongly non-linear
with coordination number. The first ‘bonds’ to form are strong, and get
progressively weaker as extra metal atoms are added to the first coordination
shell. Some examples will be shown and discussed in the lecture.

Increasingly what counts is the speed of the computer code; if this speed
scales with a lower power of the number, N, of electrons in the system,
then more complex/ larger problems can be tackled; O(N) methods are
discussed later in Lecture 8.
For example, because the interactions between atoms and the electron gas
are parameterised initially, EMT calculations are fast enough that they
can be used to simulate dynamic processes such as adsorption, nucleation
or melting on metal surfaces; here an approximate electronic structure
calculation is being done for each set of positions of the nuclei, i.e.
at each time step (Jacobsen *et al.* 1987, Stolze 1994, 1997). This
requires computer speeds that would have been inconceivable just a
few years ago. It is now feasible to download EMT programs from
Per Stoltze's website
in Denmark in order to run them anywhere. There are
real possibilities for experiment-theory collaborations here which were
impractical just a few years ago. If this is of interest as a project, let
me know.

For the general reader, we should bear in mind that that both EAM and EMT
*are* approximations, which are aimed at increasing broad-brush
understanding, i.e. getting most of the story, most of the time. There are
examples where these approaches simply don't work, despite the best efforts of
the specialists involved. Detailed studies of such cases, and comparison of
methods have then been used to find out where they went wrong. An example
is that both EAM and EMT generally get too low a surface energy, and
an extreme case is that of Pd. This metal has become a test case in that
it can have two *chemical* configurations. The ground state is
4d^{10}, but the 4d^{9}5s state is quite low lying
(Balasubramanian 1988). Thus, at a surface or other special configuration,
we can have changes in the amount, or even the type, of hybridisation;
such effects disturb the continuous variation of energies with
coordination number which are a feature of these approximate methods.

A chemist might wonder what the fuss is all about, since ideas of hybridisation are central to their world view. We explore these ideas in the context of silicon in Lecture 7. But first we survey the effects of d-electrons in relation to magnetism in the next section, and then we explore the tight-binding and related methods from a more chemical viewpoint in the next lecture.

The 3d, 4d and 5d series have a major contribution to cohesion from both
s-d and d-d interactions. In the case of 3d, the overall cohesion peaks
before and after the middle of the transition series, unlike the 4d and 5d
series, where cohesion from the d-bands peaks in the middle. An example of
a comparison of EMT with KKR methods (Morruzzi *et al.* 1978,
Jacobsen *et al.* 1987, Jacobsen 1988) is given in
figure 5.2. Note that these
particular calculations do not include spin-correlation effects, which are
discussed later.

Figure 5.2: Calculated cohesive energies and the
equilibrium radius
r_{s} for the 3d transition series, comparing EMT (open circles)
with KKR methods (Morruzzi *et al.* 1978, closed circles). The modified
EMT corresponds to EMT applied at the density given by the KKR method
(after Jacobsen *et al.* 1987 and Jacobsen 1988).

The magnetism of the parent atoms is a result of Hund’s rule, which asserts that the first 5 d-electrons are populated with parallel spins, and the remaining 5 then fill up the band with antiparallel alignment. This is due to the reduced electron-electron Coulomb interaction between pairs with parallel spins, because the exchange-correlation hole which accompanies each electron keeps these electrons further apart on average. The rare earth elements are an important class of magnetic materials based on 4f electrons, but are not discussed here.

When these atoms are assembled into solids, several effects occur which
we should not try to oversimplify. The d-band is very important for
cohesion, and the simplest model is that due to Friedel (1969), which
predicts a parabolic dependence of the bond energy as the number of
d-electrons N_{d} is increased across the series. This model
leads to the contribution of d-d bonding to the pair-bond energy,
E_{b}

_{b} = 2ò^{EF}
(E - e_{d})(5/W)dE
= -(W/20)N_{d}(10 - N_{d}),
(5.1) |

where e_{d} is the unperturbed atomic
d-level energy and W is the d-band width in the solid.
In terms of the second moment of the energy distribution
m_{2}, the overlap integrals between
d-orbitals of strength b, and the band width are
related by W = (12z)^{1/2}|b|,
with z nearest neighbors; this can be derived for a rectangular d-band,
where the second moment
m_{2} = W^{2}/12,
as explained by Sutton on pages 174-175. This parabolic behaviour with
N_{d} is quite closely obeyed by the 4d and 5d series, leading
to surface energies displaying similar trends (Skriver and Rosengaard 1992).

However, when magnetic effects are considered, the shape of the d-band is
also very important, and ferromagnetism only results when both the d-d
nearest neighbor overlap is strong and the density of states near the Fermi
energy is large. These conditions are fulfilled towards the end of the 3d series,
aided by the two-peaked character of the density of states, sketched in
figure 5.3; this energy distribution has a
large fourth moment m_{4},
which is also implicated in the discussion of why Fe has the b.c.c
structure, points which can be explored further via project 5.4.2.

Figure 5.3: Schematic distribution of s-d band
overlap with the d-band having a double-peaked density of states, and hence a
large fourth moment m_{4}.

When detailed band structure calculations are done including magnetic interactions, we have to account separately for the majority spin-up (r) and minority spin-down (r¯) densities. By analogy to LDA, there is a corresponding local spin density (LSD) approximation, which can be explored further via project 5.4.3.

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