Basis-set-free calculations

Calculations of IIq(O) are very sensitive to the basis set. The venerable Clementi-Roetti wavefunctions [234], often considered to be of Hartree-Fock quality, get the sign of IIq(O) wrong for the sihcon atom. Purely numerical, basis-set-free, calculations [232,235] have been performed to establish Hartree-Fock limits for the MacLaurin expansion coefficients of IIo(p). The effects of electron correlation on IIo(O), and in a few cases IIq(O), have been examined for the helium atom [236], the hydride anion [236], the isoelectronic series of the lithium [237], beryllium [238], and neon [239] atoms, the second-period atoms from boron to fluorine [127], the atoms from helium to neon [240], and the neon and argon atoms [241]. Electron correlation has only moderate effects on IIo(O). 

Dickson, R. M., Becke, A. D., 1993, Basis-set-free Local Density-Functional Calculations of Geometries of... 

However, most wave function based calculations also contain a semiempirical component. For example, the primitive Gaussian functions in all commonly used basis sets (e.g., the six Gaussian functions used to represent a li orbital on each first row atom in the 6-3IG basis set) are contracted into sums of Gaussians with fixed coefficients and each of these linear combinations of Gaussians is used to represent one of the independent basis functions that contribute to each AO. The sizes of the primitive Gaussians (compact versus diffuse) and the coefficient of each Gaussian in the contracted basis functions, are obtained by optimizing the basis set in calculations on free atoms or on small molecules." ... 

Figure 7.2 Quasi-chemical contributions of the hydration free energy of Be (aq). Cluster geometries were optimized using the B3LYP hybrid density functional (Becke, 1993) and the 6-31- -G(d, p) basis set. Frequency calculations confirmed a true minimum, and the zero point energies were computed at the same level of theory. Single-point energies were calculated using the 6-311- -G(2d, p) basis set. A purely inner-shell n = 5 cluster was not found the optimization gave structures with four (4) inner-sphere water molecules and one (1) outer-sphere water molecule. For n = 6 both a purely inner-shell configuration, and a structure with four (4) inner-shell and two (2) outer-shell water molecules were obtained. The quasi-chemical theory here utilizes only the inner-shell structure. O - rin [/ff -f (left ordinate) vs. n. A ...

There are alternatives to the most commonly used Boys-Bemardi counterpoise scheme. One approach that shows promise, for example, is a chemical Hamiltonian approach (CHA), pioneered by Mayer " ", which attempts to isolate the superposition error directly in the Hamiltonian operator. The Schrodinger equation that is solved is hence a modified one, which yields a wave function that is hopefully free of superposition error. In the case of (HF)2, it was found that this approach mimics rather closely the results of the standard counterpoise scheme for a series of small to moderate sized basis sets ". Later calculations " extended these tests to other small H-bonded systems as well, again limiting their testing to basis sets no larger than 6-31G. A recent test " has extended the method s use-... 

In fact, very large basis set, correlated calculations suggest that the CH tautomer is favored over the NH in the gas phase by only 6.2 kcal/mol. Assuming an approximately equal mixture of the two tautomers in aqueous solution,3i3 the AMI-SMIa model is within about 1 kcal/mol of correctly predicting the relative free energies of solvation. The more general AM1-SM2 model is not quite as accurate in its consideration of heteroatom-bound protons, and it thus provides a somewhat poorer prediction in this instance, although it remains closer than any of the other models. 

R. M. Dickson and A. D. Becke (1993) Basis-set-free local density-functional calculations of geometries of polyatomic-molecules. J. Chem. Phys. 99(5), pp. 3898-3905... 

The KS energy levels and KS orbitals for the LDA functional are shown in Table 7. The orbitals are calculated with two different numerical methods, the first is fully numerical basis set free (i.e., solved on a real space grid) while the other uses the Sadlej (52 orbitals) basis set [the OEP results for the EXX (KLI) approximation shown in Table 7 are also calculated basis set free]. Note that the eigenvalues for the higher unoccupied states are positive. This is due to the LDA potential being too shallow and not having the correct asymptotic... 

Secondly, the ultimate properties of polymers are of continuous interest. Ultimate properties are the properties of ideal, defect free, structures. So far, for polymer crystals the ultimate elastic modulus and the ultimate tensile strength have not been calculated at an appropriate level. In particular, convergence as a function of basis set size has not been demonstrated, and most calculations have been applied to a single isolated chain rather than a three-dimensional polymer crystal. Using the Car-Parrinello method, we have been able to achieve basis set convergence for the elastic modulus of a three-dimensional infinite polyethylene crystal. These results will also be fliscussed. 

The free energy of activation at the QCISD(T)/6-31 H-- -G(d,p) level amounts to 21.1 kcal/mol. According to the authors, the large electron density redistribution arising upon cyclization makes it necessary to use extended basis sets and high-order electron correlation methods to describe the gas-phase thermodynamics, which indicates clearly the gas-phase preference of the azido species. However, the equilibrium is shifted toward the tetrazole as the polarity of a solvent is increased. For instance, SCRF calculations (e = 78.4) yield a relative free energy of solvation with respect to the cw-azido isomer of —2.4 kcal/mol for the tmns-zziAo compound and of —6.8 kcal/mol for the tetrazole isomer. At a much lower level, the... 

For the calculation of the LDA ground-state one can proceed either via the direct" methods, i.e. via the glocal minimization of the total free energy with respect to the electronic degrees of freedom, or via the the diagonalization (for large PW basis-sets necessarily iterative diagonalization) of the KS Hamiltonian in combination with an iterative update of chai ge-density and potential. 

Figure 3.12. Optimized structure of the free triplet state (a) and the triplet of the carbonyl H-bond complex (b) calculated from the DPT calculations using the UB3LYP method with a 6-311G basis set. (Reprinted with permission from reference [42]. Copyright (2005) American Chemical Society.)...

We should also mention that basis sets which do not actually comply with the LCAO scheme are employed under certain circumstances in density functional calculations, i. e., plane waves. These are the solutions of the Schrodinger equation of a free particle and are simple exponential functions of the general form... 

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