Basis function atom-centred functions

And what if the basis functions are centred on different atoms The CNDO solution to the problem is to take all possible integrals such as those above to be equal, and to assume that they depend only on the atoms A and B on which the basis functions are centred. This satisfies the rotational invariance requirement. In CNDO theory, we write the two-electron integrals as pab and they are taken to have the same value irrespective of the basis functions on atom A and/or atom B. They are usually calculated exactly, but assuming that the orbital in question is a Is orbital (for hydrogen) or a 2s orbital (for a first row atom). 

The simple orbital basis expansion method which is used in the implementation of most models of molecular electronic structure consists of expanding each R as a linear combination of determinants of a set of (usually) atom-centred functions of one or two standard forms. In particular most qualitative and semi-quantitative theories restrict the terms in this expansion to consist of the (approximate) occupied atomic orbitals of the constituent atoms of the molecule. There are two types of symmetry constraint implicit in this technique. 

We dispose of the simplest problems first. Any orbital-basis theory of molecular electronic structure which purports to be interpretable as a theory of valence is committed to the use of atom-centred functions (or, at least, functions which go over into atomic orbitals for some values of their parameters).7) We therefore wish to stay as... 

The coefficients C/n are the parameters to be determined. The set of functions Xp is known as the basis set and often consists of atom-centred functions obtained from the solution of the Schrodinger equation with some central field potential. 

In using such methods, care must be taken in the choice of basis sets (i.e. the atomic centred functions from which the LCAO-MOs are constructed). Unless good sets are used, then the resulting interatomic potential function will be... 

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT techniques [81]. PAW, however, provides all-electron one-particle waveflmctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in terms of a pseudo-wavefunction (easily expanded in plane waves) term that describes interstitial contributions well, and one-centre corrections expanded in terms of atom-centred functions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential formalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [82. 83]. PAW is also formulated to carry out DFT dynamics, where the forces on nuclei and wavefunctions are calculated from the PAW wavefunctions. (Another all-electron DFT molecular dynamics technique using a mixed-basis approach is applied in [84]. )... 

We will illustrate the stages involved in the Roothaan-Hall approach using the helium hydrogen molecular ion, HeH, as an example. This is a two-electron system. Our objective here is to show how the Roothaan-Hall method can be used to derive the wavefunction, for a fixed internuclear distance of 1 A. We use HeH rather than H2 as our system as the lack of symmetry in HeH makes the procedure more informative. There are two basis functions, Isa (centred on the helium atom) and Isg (on the hydrogen). The numerical values of the integrals that we shall use in our calculation were obtained using a Gaussian series approximation to the Slater orbitals (the STO-3G basis set, which is described in Section 2.6). This detail need not concern us here. Each wavefunction is expressed as a linear combination of the two Is atomic orbitals centred on the nuclei A and B ... 

The atomic orbitals (or basis functions) are centred on the nuclei and so any integrals involving these basis functions will depend on the nuclear positions parametrically. 

These two factors are decisive in fixing the usual basis set not as atomic orbitals but as a set of atom-centred functions which are adapted to the expansion of the AOs of each of the component atoms of the molecule under study. It will also be useful from time to time to augment these basis functions with additional atom-centred functions that allow the description of aspects of the molecular electron distribution which are specific to the molecule. For example, in any satisfactory description of the H2 molecule one would use those sets of spherically symmetric atom-centred functions which are used to expand the Is AOs of the hydrogen atoms. But one might also add to the basis one or more p functions on each atom to allow for the polarisation of the electron distribution on each atom on molecule formation functions which do not take part in the expansion of the AOs of the ground state of the component atoms of the molecule. 

Atomic orbitaJs that is atom-centred functions. These clearly depend on nuclear geometry since, when a nucleus moves, its basis functions move with it. 

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 ... 

For bonded atoms, the off-diagonal terms (where i j) are taken to depend on tjje type and length of the bond joining the atoms on which the basis functions y- and Xj 0 centred. The entire integral is written as a constant, 0ij, which is not the same as the fixY in Hiickel 7r-electron theory. The are taken to be parameters, fixed by calibration against experiment. It is usual to set Pij to zero when the pair of atoms are not formally bonded. 

The diagonal terms (where i = j) need a little more consideration. They are taken to depend on the nature of the atom on which basis function Xi is centred, but they also depend on the namre of the neighbouring atoms. 

In the TT-electron theories, each first-row atom contributes a single basis function. For the all valence electron models there is now an additional complication in at some of the basis functions could be on the same atomic centre. So how should we treat integrals involving basis functions all on the same atomic centre such as... 

The only difference from our discussion about rr-electron systems is that there may be many basis functions on the same atomic centre. 

The diagonal terms (where i = j) are conveniently divided into those contributions that refer to a given atomic centre (atom A), and those that refer to other atomic centres. In the following discussion, assume that basis function x-is centred on nucleus A. We have ... 

The off-diagonal terms (i j) are treated to a similar analysis. Each penetration term involving different basis functions that are on the same atomic centre are given a value of — Vab to maintain invariance. Suppose now that Xi is centred on nucleus A and Xj on nucleus B. We have... 

I have collected together all the electron density terms involving basis functions on atom A into Faa- These expressions are correct even if Xi and Xj tc both centred on the same atom. 

These integrals can be terrifyingly difficult they involve the spatial coordinates of a pair of electrons and so are six-dimensional. They are singular, in the sense that the integrand becomes infinite as the distance between the electrons tends to zero. Each basis function could be centred on a different atom, and there is no obvious choice of coordinate origin in such a case. 

In standard quantum-mechanical molecular structure calculations, we normally work with a set of nuclear-centred atomic orbitals Xi< Xi CTOs are a good choice for the if only because of the ease of integral evaluation. Procedures such as HF-LCAO then express the molecular electronic wavefunction in terms of these basis functions and at first sight the resulting HF-LCAO orbitals are delocalized over regions of molecules. It is often thought desirable to have a simple ab initio method that can correlate with chemical concepts such as bonds, lone pairs and inner shells. A theorem due to Fock (1930) enables one to transform the HF-LCAOs into localized orbitals that often have the desired spatial properties. 

The central assumption of semi-empirical methods is the Zero Differential Overlap (ZDO) approximation, which neglects all products of basis functions depending on the same electron coordinates when located on different atoms. Denoting an atomic orbital on centre A as /ja (it is customary to denote basis functions with /j, u, A and cr in semi-empirical theory, while we are using Xn, xs for ab initio methods), the ZDO... 

---