Least-squares orbital energies

The PNO extrapolations in Fig. 4.8 and Table 4.6 require localization of the occupied SCF orbitals to ensure size-consistency. In order to preserve this size-consistency for the CBS PNO extrapolations, we have restricted these (Zmax + f)-3 extrapolations to a linear form, Eq. (6.2). The new double extrapolation employs this linear extrapolation of pairs of CBS2/cc-pVnZ calculations and thus is rigorously size-consistent. Note that the nonlinear N-parameter (Zmax + a)-" extrapolations using least-squares fits to more than N cc-pVnZ energies are not size-consistent [53,55],... 

The Fermi surface plays an important role in the theory of metals. It is defined by the reciprocal-space wavevectors of the electrons with largest kinetic energy, and is the highest occupied molecular orbital (HOMO) in molecular orbital theory. For a free electron gas, the Fermi surface is spherical, that is, the kinetic energy of the electrons is only dependent on the magnitude, not on the direction of the wavevector. In a free electron gas the electrons are completely delocalized and will not contribute to the intensity of the Bragg reflections. As a result, an accurate scale factor may not be obtainable from a least-squares refinement with neutral atom scattering factors. 

In summary, at least for the frontier orbitals where equation (3.45) is expected to be valid, a net destabilization depending in a complex manner on the square of the overlap integral and proportional to the sum of unperturbed orbital energies ensues from the interaction. 

Calculated from STO-3G orbital energies using equation IP = 0.662 (e)+4.193 which is a least-squares treatment of IP s in footnote a. 

Tab. 11.1. Tight-binding parameters obtained from the least-square-error fit to LMTO band dispersions for the nine ll-VI semiconductors in the sp d basis with the A-B and B-B interactions. The first row lists the interatomic spacings in A, the next eight rows contain the onsite energies for all the orbitals, e.g. the row for dc t2) lists the entries for the t2cl orbital onsite energies for the cation. The subscript a denotes the anion. The last fifteen rows list the Slater Koster parameters. The last column shows the average value of the Slater Koster parameters multiplied by the square of the cation-anion distance, d. ...

The evaluation to the desired numerical accuracy of the density functional total energy has been a major obstacle to such calculations for many years. Part of the difficulty can be related to truncation errors in the orbital representation, or equivalently to basis set limitations, in variational calculations. Another part of the difficulty can be related to inaccuracies in the solution of Poisson s equation. The problem of maximizing the computational accuracy of the Coulomb self-interaction term in the context of least-squares-fitted auxiliary densities has been addressed in [39]. A third part of the difficulty may arise from the numerical integration, which is unavoidable in calculating the exchange and correlation contributions to the total energy in the density functional framework. 

Because of the possibility of cancellation of differences in different ranges of the radius no improvement in the calculation is found. Only marginal improvements in the matches of the functions and their transformations after the application of the kinetic energy operator are found in the least-squares minimization. There is not sufficient flexibility in the approximate Slater function, even when orthogonalized to the Is function, to improve on the best value for the 2s orbital energy, returned by variation of the basic Slater exponent. 

Figure 4.11 Calculation of the 2s orbital energy in hydrogen with the sto-3g ls> basis set, Table 1.6, for the 2s Slater function rendered orthogonal to the sto-3g ls> function. The initial calculation returns a poor estimate of the energy terms and 2 for the minimization condition on the least-squares integral of Chapter 3. Optimization based on the minimization of the energy, using SOLVER on the Slater exponent, returns closer agreement with the exact results.

Figure 4.17 The results found for the calculations on the energy of the 2p orbital in hydrogen using Huzinaga s 2p basis sets listed in Table 1.8. The calculations of the least-squares equivalents of the entries in Figure 4.16 are left as an exercise for the interested reader.

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