The Rayleigh theorem for eigenvalues, as discussed below, enables the energy minimization that is required in many self-consistent calculations of electronic and related properties of materials, from atoms, molecules, and nanostructures to semiconductors, insulators, and metals.
i.e., the ones of the form HѰ = λѰ, where H is an operator, Ѱ is a function and λ is number called the eigenvalue.
Let the same eigenvalue equation be solved using a basis set of dimension N + 1 that comprises the previous N functions plus an additional one.
Then, the Rayleigh theorem for eigenvalues states that λ′i ≤ λi for i = 1 to N. A subtle point about the above statement is that the smaller of the two sets of functions must be a subset of the larger one.
In density functional theory (DFT) calculations of electronic energies of materials, the eigenvalue equation, HѰ = λѰ, has a companion equation that gives the electronic charge density of the material in terms of the wave functions of the occupied energies.
Using only the new wave functions of the occupied energies, one repeats the cycle of constructing the charge density and of generating the potential and the Hamiltonian.
The characteristics and number[1][2] of the known functions utilized in the expansion of Ѱ naturally have a bearing on the quality of the final, self-consistent results.
The selection of atomic orbitals that include exponential or Gaussian functions, in additional to polynomial and angular features that apply, practically ensures the high quality of self-consistent results, except for the effects of the size[1][2] and of attendant characteristics (features) of the basis set.
A priori, there is no known mechanism for selecting a single basis set so that, after self consistency, the charge density it generates is that of the ground state.
A larger basis set that contains the first one generally leads self consistent eigenvalues that are lower than or equal to their corresponding values from the previous calculation.
Let us first recall that a self-consistent density functional theory calculation, with a single basis set, produces a stationary solution which cannot be claimed to be that of the ground state.
To find the DFT ground state of a material, one has to vary[5][6] the basis set (in size and attendant features) in order to minimize the energy content of the Hamiltonian, while keeping the number of particles constant.
Depending on the s, p, d, or f character of this orbital, the size of the new basis set (and the dimension of the Hamiltonian matrix) will be larger than that of the initial one by 2, 6, 10, or 14, respectively, taking the spin into account.
Naturally, one cannot affirm that the results from Calculation II describe the ground state of the material, given the absence of any proof that the occupied energies cannot be lowered further.
This paragraph described how successive augmentation of the basis set solves one aspect of the conundrum, i.e., a generalized minimization of the energy content of the Hamiltonian to reach the ground state of the system under study.
In the current literature, the only calculations that have reproduced[8][9][10] or predicted [11][12][13] the correct, electronic properties of semiconductors have been the ones that (1) searched for and reached the true ground state of materials and (2) avoided the utilization of over complete basis sets as described above.
These accurate DFT calculations did not invoke the self-interaction correction (SIC)[14] or the derivative discontinuity[15][16][17] employed extensively in the literature to explain the woeful underestimation of the band gaps of semiconductors[16] and insulators.