Optimization, exponent

Selected results are presented in Table 6.3 for typical olefinic carbon atoms. The Mulliken net charges were obtained [40] from full (geometry and exponents) optimizations in the STO-3G basis. 

A comparison of minimal-basis perfect-pairing and MO wavefunctions for HF and HF+ has appeared,198 in which complete exponent optimization was carried out. 

Further comparative calculations by both VB and MO methods on BeEfe have been reported, using a minimal STO basis, with exponents optimized for various r zsz With choice of particular VB configurations, it is shown that a model based on the resonance valence state, including ionic configurations, is particularly useful. Both methods show a localized-pair bond description to be useful. 

Lehn and Munsch582 have also reported SCF calculations on PF3 using extensive basis sets, and have compared the results with work on NH3. Walker683 has recently computed force constants for PH3, using a large basis set, with rf-orbital-exponent optimization. Table 10 summarizes the recent work on the NH3 molecule. 

According to Hurley,103 the matrix elements Hab.ca, Aab.cd are to be calculated with all orbital exponents optimized so as to achieve the best possible molecular energy. This takes into account to some extent the deformations of the atoms when the molecule is formed, so that the expansion... 

As we have shown in Section 2.A., the extension of the minimum to DZ basis set brings about the decrease in energy. For this extension, of course, a new set of exponents is required. Exponent optimization of STO basis sets, particularly extended basis sets, is not simple, though all one-center integrals appearing in the SCF atomic problem are expressible in closed analytical forms. It involves the optimization of a number of nonlinear parameters, in addition to the same number of linear parameters in eqn. (2.1), Excellent mathematical a-... 

For molecules the optimization of exponents is very important for small basis sets but it can never alone absorb the deficiency due to a lack of expansion functions. The number and secondly the kind of STO basis set functions (i.e. s, p, d,. .. type) are the most important considerations. The decreasing importance of the exponent optimization as the basis set grows is observable from Table 2.4,... 

Use of minimum STO basis sets with exponents optimized... 

With barriers to internal rotation in C2Hg and H2O2, the exponent optimization noted in Table 2,5 brought about no improvement. Actually, use of optimum exponents for H2O instead of Slater exponents resulted in a worse agreement with experimental results for H2O2, Difficulties in accounting for the barrier to internal rotation in H2O2 are not compatible with the problems of minimum basis set optimization and they will be discussed later. We note it here just to point out... 

Examine now the determination of exponents for polarization functions. Obviously, the atomic ground state calculations that are so useful in the optimization of valence shell exponents cannot help us. There is a possibility of performing calculations for excited states of atoms. This approach is, however, not appropriate. The role of polarization functions is to polarize valence orbitals in bonds so that the excited atomic orbitals are not very suitable for this purpose. Chemically, more well-founded polarization functions are obtained by direct exponent optimization in molecules. Actually, this was done for a series of small molecules in both Slater and Gaussian basis sets. Among the published papers, we cite. Since expo-... 

A thorough d-exponent optimization of Cartesian GTF s with respect to "chemical environment" of atoms in molecules was performed by the... 

Magnasco, V. (2008) Orbital exponent optimization in elementary VB calculations of the chemical bond in the ground state of simple molecular systems. J. Chem. Edu., 85, 1686-1691. 

Dunning s DZP basis, 6-31G, etc. Cations require higher and anions require lower d exponents . Optimizing the correlation energy itself also yields higher exponents. ... 

A very similar discussion applies to quadrupole properties as well. The f functions needed for these properties should have their exponents optimized with respect to the contribution of quadrupole transitions to the dispersion energy or with respect to the quadrupole polarizability a, which relates the induced quadrupole moment to an inducing field gradient. Optimal single f exponents have been reported as a° = 0.18, af = 0.275 and aP = 0.28 . These values are 6-8 times lower than the exponents which maximize the contribution of f functions to the total correlated monomer energyand so, as for d exponents, energy-optimized values are unsuitable. Static quadrupole moments are changed by added f func-... 

Electron affinity of sulfur hexafluoride was studied using various density functionals and systematically expanded basis sets with polarization function exponents optimized in molecular environment, /-type polarization functions were found necessary to ensure convergence of both geometry and electron affinity. The ion was found to have octahedral symmetry, and the smallest value of electron affinity is only 1.40 eV (1.6 eV with zero-point energy correction). 

As several density functionals became available in the Gaussian program (19, 20), it became possible to study their effect on the computed value of electron affinity with systematically enlarged basis sets. The plan of the present work is as follows (a) initial calculations were done using the Hartree-Fock methods (RHF, the restricted Hartree-Fock ROHF, the restricted open-shell Hartree-Fock UHF, the unrestricted open-shell Hartree-Fock), and various non-local density functionals, defined in Table 2, with several standard basis sets (Section 2) (b) a large, saturated sp basis set was systematically enlarged by adding polarization functions with exponents optimized in molecular Hartree-Fock and density-functional cal-... 

As in the closed-shell case, we estimate the correlation energy involved in the lowest triplet state of the confined helium atom. In Table 6 we report the total energy obtained from Table IV of Ref. [19], also the HF results from this work, and results obtained from correlated methods. We see that the results obtained with the HF method, coupled with exponent optimization, compete with the other correlated methods, since for some confinement radii our results are below those reported by the other authors. Comparing our Hartree-Fock results with those obtained with 40-terms Hylleraas wave... 

In this work we show the importance of exponent optimization on the Roothaan approach to the solution of the Hartree-Fock equations for the lowest singlet and triplet state of the confined helium atom. By using this procedure, we found for the lowest triplet state the Hartree-Fock method as an appropriate method to describe it. By contrasting the exchange-only... 

Calculations were originally carried out (2 ) for the lowest states of Hz, LiH, BH, NH, and HF and the corresponding deuterated molecules with the LCAO-MO wavefunction of Coulson for Hz (11) and the LCAO-MO-SCF functions of Hansil (12) for the other molecules. These wavefunctions contain a minimum basis set of inner and valence shell Slater-type orbitals with the orbital exponents optimized at the experimental equilibrium intemuclear distance. The S states are the ground states of the respective molecules except in the case of NH. 

---