The Combined Correction Term

Although the errors introduced by the atomic-sphere approximation are unimportant for many applications, e.g. self-consistency procedures, there are cases where energy bands of high accuracy are needed, and where one should include the perturbation (6.2) in some form. Below, we derive an expression which accounts to first order for the differences between the sphere, atomic or muffin tin, and the atomic polyhedron, re-establishes the correct kinetic energy in the region between the sphere and the polyhedron, and corrects for the neglect of higher partial waves. The extra terms added to the LMTO matrices which accomplish these corrections are called the combined correction terms [6.2]. 

We evaluate the perturbation (6.2) over the interstitial region (I) as a difference between an integral over the cell (C), i.e. the polyhedron, and an integral over the sphere (S). Hence, 

With this definition, xL is equal to the proper orbital in the interstitial region only, while inside the spheres it is derived from a constant (pseudo) potential v(r) = E - k. For that reason the integral over the sphere appearing in (6.44) is simply the LMTO overlap matrix (5.47) evaluated for the free-electron potential parameters from Sect.4.4 corresponding to = D j (kS). Hence, the contribution from the second term in (6.44) may be included in the LMTO equations (5.45) by subtracting 

We proceed by evaluating the integral over the cell as a single sum in 

The Fourier transform of the free-electron muffin-tin orbital (6.45) is now found by taking the transforms of j , , and n separately. For p we 

---

With the combined correction term included in an LMTO calculation, one... 

The basic output from STR is the canonical structure constants used in CANON and LMTO to calculate band structures. In addition, STR produces a file with real and reciprocal space vectors which is used by the combined correction term programme COR. This file may also be read by STR next time the same crystal structure is encountered, thus saving the time used to generate these vectors. 

---