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They are slightly excited orbitals obtained by solving the valence electron problem for the isolated atom with the same pseudopotential and LDA approximations as used in the system Hamiltonian but with the boundary condition that the orbitals vanish beyond a cut-o radius rc . The construction of the KS Hamiltonian matrix and overlap matrix with respect to { } can be proceeded as follows. The matrix elements of the overlap matrix (S = | ), the kinetic 2 ˜ ˜ energy ( |p =2m| ), Vnl ( |Vnl (˜r − Ri )| ), and Vna ( |Vna (˜r − Ri )| ) are calculated only once beforehand and tabulated as functions of the relative positions of the “atomic” centers.
The function f(x) = 3x2 − 2x3 vs. x, reproduced from Fig. 1 in Ref. [21]. will approach ∞, while above the Fermi level will approach −∞. This is certainly not the expected result of = 1 for states below the Fermi level and = 0 for states above the Fermi level. Since the idempotency of ˆ is responsible for being 1 for the occupied states and zero for the unoccupied states, Li et al. 4) to control the situation. From Eq. 4), it can be seen that an idempotent matrix is invariant under the puriÿcation transformation.
It can be seen that the LDOS of an isolated C dimer at and below the Fermi energy (∼− 6:9 eV) is very small, whereas there is a broad pronounced peak above the Fermi energy that is centered around −4:9 eV. Hence the C dimer will appear dark in the ÿlled-state and very bright in the empty-state STM images. To understand larger growth structures, they examined possible three-adatom structures. Using the C dimer as the seed, they placed the third adatom at sites 3, 4, or 5 in Fig. 8 and allowed the system to relax.