The Gaussian basis set

About the same time that NDDO approximations were first used, S. F. Boys (1950) showed that the integrals in Eq. (3.10) can be evaluated if the basis set is taken as 

It is easy to see that a single Gaussian doesn t represent an s-type atomic orbital as well as the STO because the Gaussian is rounded at r = 0 where the atomic orbital has a cusp point. A linear combination of many Gaussians can be made to approach the atomic orbital as closely as we please near (but not precisely at) the nucleus depending on software efficiency, the power of the computer, and the amount of CPU time we wish to expend. This idea led to programs in which 3 Gaussians 

In split-valence basis sets, inner or core atomic orbitals are represented by one basis function and valence atomic orbitals are represented by two. The carbon atom in methane is represented by one Is inner orbital and 2(2,s, 2px, 2py, 2pz) = 8 valence orbitals. Each hydrogen atom is represented by 2 valence orbitals, hence the number of orbitals is 

In the 6-31G basis, the inner shell of carbon is represented by 6 primitives and the 4 valence shell orbitals are represented by 2 contracted orbitals each consisting of 4 primitives, 3 contracted and 1 uncontracted (hence the designation 6-31). That gives us 4(4) = 16 primitives in the valence shell. The single hydrogen valence shells are represented by 2 orbitals of 2 primitives each. That gives us a total of 

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The picture here is of uncoupled Gaussian functions roaming over the PES, driven by classical mechanics. The coefficients then add the quantum mechanics, building up the nuclear wavepacket from the Gaussian basis set. This makes the treatment of non-adiabatic effects simple, as the coefficients are driven by the Hamiltonian matrices, and these elements couple basis functions on different surfaces, allowing hansfer of population between the states. As a variational principle was used to derive these equations, the coefficients describe the time dependence of the wavepacket as accurately as possible using the given... 

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. 

Besides the Gaussian basis set for the wavefunction, an additional set of nuclear centered Gaussians, gk, can be used for expanding the electronic density [25], which can be written as... 

Thus, to the best of our knowledge, there is a lack of embedding cluster studies on the yttrium ceramics where with a sufficient precision both aspects of the ECM were taken in account. In the study [44], we attempted to fill such a gap and carried out the electronic structure calculations of the YBa2Cu307 ceramics at the Moller-Plesset level with a self-consistent account of the infinite crystal surrounding to the quantum cluster. The Gaussian basis set employed (6-31IG) was larger than those used in previous cluster calculations [16,20,22,29]. 

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. 

The choice of the value of the coherent state width 7 is arbitrary, since this parameter does not affect the mathematical properties of the Gaussian basis set which determine the form of the semi-classical propagator. In fact, the value of this quantity is usually chosen in practical implementations so as to facilitate the numerical convergence. In the following, we shall set 7 = 1/2 since with this choice (25) simplifies considerably and becomes... 

The grid basis set requires no orthogonalization. On the other hand, canonical orthogonalization with the Gaussian basis set requires overlap matrix diagonalization and one matrix multiplication. 

Additionally, the breadth of the study is restricted by considering only a few QC methods and a few standard sets of basis functions. Since Gaussian 94 [5] and Gaussian 98 [6] have been used, the Gaussian basis sets normally used by... 

Still more of the programming complexities vanish if another constraint is applied to the Gaussian basis set, namely that its exponents be infinite, which yields a basis composed entirely of Dirac delta functions... 

Fig. 1. The Hirshfeld electron densities (Hh) of bonded hydrogen atoms obtained from the molecular density (H2). The free hydrogen densities (H°) and the resulting electron density of the promolecule (H2) are also shown for comparison. The density values and inter-nuclear distances are in a.u. The zero cusp at nuclear positions is the artifact of the Gaussian basis set used in DFT calculations.

These facilities belong to a standard equipment of ab initio programs in the Gaussian basis set.) The localisation is applied separately for the set of inner shells (K, L,. . . ) and the valence MOs. For multiple bonds the localisation leads to banana bonds [22]. 

Table 1.5 The Gaussian basis sets proposed by Reeves to represent the hydrogenic radial functions. The table entries, for each basis set, are the exponents, a, of the primitive Gaussians and then in the second columns the complete normalized coefficients, d, of the linear combinations.

The general problem is set out schematically in Figure 5.5 and in equations 5.51 and 5.53. We know that the two-electron terms over the Gaussian primitives require the evaluation of equation 5.55 over the Gaussian basis set. 

EXAMPLE Use the Gaussian Basis Set Order Form on the Internet to find the 3-21G basis functions for the oxygen atom. 

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