Chromium An Example

In Fig.7.3 we show some results from a calculation [7.19] of the properties of anti ferromagnetic chromium. The self-consistent energy bands were calculated as outlined in Sect.7.3, including the spin-polarised exchange-correlation terms mentioned in Sect.7.2.1. The pressure was calculated by means of (7.43) for each spin and the bulk modulus was obtained by numerical differentiation. As indicated in the figure, a few per cent change in the atomic radius produces complete agreement with experiment. 

We shall now discuss the form of the equation of state, i.e. P(n), for Cr in the light of the pressure expressions of the previous sections. To this end, in Fig.7.4 the total pressure has been decomposed into its s, p, and d contributions. We note that the s and p electrons provide repulsive, i.e. positive, pressures while the d electrons provide an attractive or bonding, i.e. negative, pressure. These facts may be understood if we write (7.46) in terms of an average energy E , specified by a volume-independent logarithmic derivative at the atomic sphere and the occupation number, i.e. 

The first-order expressions (7.46-48) provide a more useful decomposition than (7.49). Therefore Fig.7.5 presents some relevant data, and for S =2.684 a.u. data may also be found in Tables 4.1,2, since E is E by choice. The first-order estimates for the partial pressures are included in Fig.7.4 for a few atomic radii. Judging from the figure, they provide accurate estimates, the d contribution being only 10% wrong owing to errors of order higher than (E - C), while the sp contributions are almost correct. 

According to (7.46,47) the partial i pressure may be decomposed into a band-position term proportional to 6V/6ln(S) or SC/6ln(S), and a band-broadening term proportional to ln(TS2)/Sln(S) or 6ln(yS2)/6ln(S). At large volumes in chromium V yxc exc and t 1 for s and p states. The 5V term therefore provides an attractive pressure which is the correction due to exchange and correlation to the repulsive bandwidth or kinetic energy term proportional to 112 

V - exc makes 6ln(TS )/Sln(S) go down. In total, the s and p electrons provide strongly repulsive pressures which are responsible for the incompressibility, i.e. the large bulk modulus, of chromium. 

---