Convergence of the quartic anharmonic contribution to the Helmholtz free energy in metals

The convergence of the quartic term of the Helmholtz free energy (F₄) in metals is investigated by comparing the numerical results of F₄ by two methods. In the first method, F₄ is calculated by summing over the direct lattice vectors of the product of the fourth-rank tensor derivatives of a two-body potential and correlation functions involving Brillouin-zone and branch-index summations. In the second method, F₄ is calculated from an alternative expression which requires beforehand the knowledge of wave-vector-dependent fourth-rank tensors. The alternative expression for F₄, which is derived in this paper, is more appropriate for the calculation of F₄ in metals where the forces are long range and oscillatory. Calculations are carried out by the two methods, for different volumes, from almost 0 K to the melting temperature range for Na and K using the same first-principle potential as used by Shukla and Taylor in their previous calculations of F₄. Representing the F₄ results of the two methods by F₄(s), where s is the shell index at which the →l summation is truncated in the first method and F₄ (Ewald), we find the two methods giving the same answer for F₄ only in some cases. For example, in Na, F₄ (Ewald) agrees exactly with F₄(s), where $s=18$, at the 90-K volume. The agreement between the two is almost exact at 160-K and 361-K volumes with s=17 and 10, respectively. At the other two volumes referred to T=5 and 293 K, F₄ ({Ewald) and F₄(s) differ by about 1% where s=22 and 21, respectively, for these volumes. In K, F₄ (Ewald) and F₄(s) differ by less than 1% in each case for the three volumes corresponding to temperatures 9, 215, and 299 K where for each of these volumes s=15, 7, and 10, respectively. The largest disagreement between F₄ (Ewald) and F₄(s) ($s=7$) arises for the volume at 99 K. In this case they differ by 3%. It is clear from the above discussion that F₄ (Ewald) and F₄(s) are in agreement with each other for "some" shell index s, but s changes with volume. Thus, without a priori knowledge of an exact value of F₄, such as the one obtained by the Ewald procedure, one cannot decide the shell index (s) at which the summation in the first method should be truncated to obtain a correct answer for F₄.