Gaussian basis sets approximation

Consider the example of the boron atom and the making of comparisons of the Herman-Skillman numerical radial functions for 1 s, 2s and 2p with the Slater functions and possible Gaussian basis set approximations. 

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

In hybrid DET-Gaussian methods, a Gaussian basis set is used to obtain the best approximation to the three classical or one-election parts of the Schroedinger equation for molecules and DET is used to calculate the election correlation. The Gaussian parts of the calculation are carried out at the restiicted Hartiee-Fock level, for example 6-31G or 6-31 lG(3d,2p), and the DFT part of the calculation is by the B3LYP approximation. Numerous other hybrid methods are currently in use. 

Minimal basis sets use fixed-size atomic-type orbitals. The STO-3G basis set is a minimal basis set (although it is not the smallest possible basis set). It uses three gaussian primitives per basis function, which accounts for the 3G in its name. STO stands for Slater-type orbitals, and the STO-3G basis set approximates Slater orbitals with gaussian functions. ... 

At this stage we are at the very beginning of development, implementation, and application of methods for quantum-mechanical calculations of molecular systems without assuming the Born-Oppenheimer approximation. So far we have done several calculations of ground and excited states of small diatomic molecules, extending them beyond two-electron systems and some preliminary calculations on triatomic systems. In the non-BO works, we have used three different correlated Gaussian basis sets. The simplest one without r,y premultipliers (4)j = exp[—r (A t (8> Is) "]) was used in atomic calculations the basis with premultipliers in the form of powers of rj exp[—r (Aj (8> /sjr])... 

Illas et al. compared in detail the MEP maps calculated by ab initio method using a variety of Gaussian basis sets [36] with those obtained by the MNDO method [37], Since the MNDO method is based on the ZDO approximation, in accordance with the previously published results [22, 27], the ZDO molecular electron density function was used for generating the MEP maps after inverse Lowdin s transformation (deorthogonalization). It was found that, whereas the MNDO MEP maps could reproduce the main characteristics of the HF/6-31G MEP maps, the position of the MNDO minima were too close to the molecules and they were deeper than the corresponding ab initio ones. Fortunately, the ratios of the HF/6-31G and MNDO MEP minima energies are constant, and therefore the MNDO MEP minima energies can be scaled to reproduce the... 

The term STO-3G denotes a minimal Gaussian basis set in which each Slater-type atomic orbital (s, p or d) is approximated by a fixed block of three Gaussian functions189,... 

The Ni atoms are treated as one-electron systems in which the effects of the Ar-like core and the nine 3d electrons are replaced by a modified effective potential (MEP) as suggested by Melius et al./165/ A contracted gaussian basis set is used for Ni, which includes two functions to describe the 4s and one to describe the 4p atomic orbitals. Since a previous study/159/ found the O-Ni spacing and the vibrational frequency we insensitive to correlations in this open-shell system, the authors have adopted the SCF calculation scheme. To check the approximation of treating the Ni atoms as a one-electron system, they performed both the MEP and an all electron SCF calculation for the NisO cluster. They found that the MEP spacing is 0.37 A or 35% smaller than the all-electron value the MEP we value is 90 cm-1 or 24% smaller. Since the 3d... 

Kitaura, K., C. Satoko, and K. Morokuma (1979). Total energies of molecules with the local density functional approximation and Gaussian basis sets. Chem. Phys. Lett. 65, 206-11. 

In order to test such an application we have calculated the spin and charge structure factors from a theoretical wave function of the iron(III)hexaaquo ion by Newton and coworkers ( ). This wave function is of double zeta quality and assumes a frozen core. Since the distribution of the a and the B electrons over the components of the split basis set is different, the calculation goes beyond the RHF approximation. A crystal was simulated by placing the complex ion in a lOxIOxlOA cubic unit cell. Atomic scattering factors appropriate for the radial dependence of the Gaussian basis set were calculated and used in the analysis. 

For different reasons, STO-NG basis sets give total energies inferior to those given by other Gaussian basis sets with approximately the same number of primitive Gaussians, though some other molecular properties are reproduced well (see for example the first two rows in... 

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