Atom-centred basis sets

Table 7 Estimates of total relativistic correction, E , and the first-order Breit energy correction,, obtained by combining the atomic or ionic contributions indicated by the second column. They may be compared with the values of the total relativistic correction, Er. and the first-order Breit interaction, Es, obtained directlyfrom matrix Dirac-Elartree-Fock and Elartree-Fock calculations of the molecular structure using BERTEIA [12], Only the results of the 13s7p2d atom-centred basis sets for Er and Eb are quoted. All energies in atomic units.

The common atom-centred basis sets (ACBS) are designed to be applicable for a range of different geometries, covalent bonds, environments and methods. This generality is both their strength and weakness. The weakness becomes apparent in QMSTAT where, on account of the statistical mechanical procedure (vide infra),... 

While all classical VB-style approaches, even those using variational HAOs, can be formulated only in terms of localised atom-centred basis sets, MO theory can use basis sets of any type, including plane wave basis sets. There are no restrictions on the type of basis that can be used in modern VB (GVB or SC) calculations, and it would be interesting to see what is the size of the plane wave basis that would be required in order to reproduce the usually well-localised orbitals observed when working within atom-centred basis sets. 

In addition to the atom-centred basis sets the DFT calculations on periodic systems can be carried out with the planewave basis set (see section 3.2.1. for details). Since the basis set definition is related to the size and shape of the periodic unit cell and not to particular atoms the BSSE does not occur in planewave calculations. 

Heavy neutral species In order to model the electron density of heavy elements for subsequent use in the construction of pseudopotential representations of the inner-shell regions near heavy nuclei, atomic DHF calculations are often employed as a starting point [158]. These pseudopotential approximations are used in order to reduce the computational cost of molecular or solid-state calculations of extended systems containing heavy elements. Similarly, atomic DHF calculations are used in the design of atom-centred basis sets, either by the direct... 

Sometimes, in particular when atom-centred basis sets are employed, ftee standing (isolated) slabs are considered. 

To solve the Kohn-Sham equations a number of different approaches and strategies have been proposed. One important way in which these can differ is in the choice of basis set for expanding the Kohn-Sham orbitals. In most (but not all) DPT programs for calculating the properties of molecular systems (rather than for solid-state materials) the Kohn-Sham orbitals are expressed as a linear combination of atomic-centred basis functions ... 

The Hartree-Fock limit for the ground state of the hydrogen molecule can be achieved in principle by using a basis set of s-type Gaussian functions distributed along the internuclear axis. Such a basis set will be termed an axis basis set. The atom-centred basis functions, designated (ac), are located on the points... 

The addition of off axis basis functions in the planes perpendicular to the internuclear axis and passing through the atomic centres to an axis basis set containing only atom-centred basis functions. The differences between the calculated and exact energy values are given in parentheses. 

Normally these molecular orbitals are obtained as expansions in a set of atom-centred basis functions (the linear combination of atomic orbitals (LCAO) method), m being the number of such functions. Recently, two-dimensional numerical integration methods have been developed to solve the MCSCF equations for linear molecules. The dimension m is then, in principle, infinite (practice, it is determined by the size of the grid used in the numerical integration). The molecular-orbital space is further divided into three subspaces the inactive, the active and the external orbitals. The inactive and active subspaces constitute the internal (occupied) orbital subspace, while the external orbitals are unoccupied. The CASSCF wavefunction is formed as a linear combination of configuration state functions (CSFs) generated from these orbitals in the following way. 

Most work on the design of molecular basis sets has concentrated on the development of atom-centred polarization sets which can be added to atomic basis sets to describe the molecular environment. 

Secondly, the use of a multi-centre basis set often leads to overestimates of interaction energies between atoms and/or molecules. This is mainly attributable to the basis set superposition ejects and will be discussed further in Section VII. 

In independent-electron models of atomic and molecular electronic structure, such as the Hartree-Fock approximation, only fimctions corresponding to the first few values of the angular momentum quantum number / contribute significantly to the energy, or to other expectation values, when multi-centre basis sets are employed. However, in treatments which take account of electron correlation effects, the higher harmonics are known to be important. A considerable amount of data is available on the convergence properties of the harmonic expansion for atoms and the importance of higher-order terms in the harmonic expansion for molecular systems has also been demonstrated. ... 

Since taking simply ionic or van der Waals radii is too crude an approximation, one often rises basis-set-dependent ab initio atomic radii and constnicts the cavity from a set of intersecting spheres centred on the atoms [18, 19], An alternative approach, which is comparatively easy to implement, consists of rising an electrical eqnipotential surface to define the solnte-solvent interface shape [20],... 

Sometimes it turns out that we need to include a number of polarization functions, not just one of each type. The notation 4-31G(3d, 2p) indicates a standard 4-31G basis set augmented with three d-type primitive Cartesian Gaussians per centre and two p-type primitives on every hydrogen atom. Again, details of the... 

---