Atom-centred functions

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]. )... 

The term orbital is used extremely loosely in quantum chemistry sometimes unintentionally through long-established abbreviations and acronyms. Thus LCAO originally meant Linear Combination of Atomic Orbitals and STO, Slater-Type Orbital but, in each of these acronyms, orbital means atom-centred function with a family resemblance to an orbital . This usage is clarified here. 

In addition to the fact that the Hamiltonian clearly depends on the positions of the nuclei parametrically, the basis functions are usually atom-centred functions and so they depend parametrically on the nuclear coordinates. 

Again, if i = j and k = the electron-repulsion integral takes on the simple point-charge asymptotic form of 1/R, where this time R is the distance between the centroids of the two distributions. Since the basis functions are always atom-centred functions, the centroid of the diagonal charge distributions pu) are the relevant atoms and so the distances R are actually inter-atomic distances. 

Mathematically, the set of Atomic Orbitals (atom-centred functions) may be extended indefinitely so that the approximation may be made arbitrarily accurate in principle. 

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. 

These considerations lead naturally to the decision to use, as basis functions for the expansion of MOs, sets of atom-centred functions which are ... 

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 ... 

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... 

Like formulae for contributions to total adiabatic corrections from individual atomic centres above, the corresponding coefficients for the linear term have the same sign and comparable magnitude, hut for subsequent coefficients agreement is lacking. The maximum region of validity of the experimental functions is the same as for the vibrational g factor, specified above, whereas the region for which the calculated points define the contributions and total adiabatic correction [122] is i /10 °m=[l,3]. 

Computational spectrometry, which implies an interaction between quantum chemistry and analysis of molecular spectra to derive accurate information about molecular properties, is needed for the analysis of the pure rotational and vibration-rotational spectra of HeH in four isotopic variants to obtain precise values of equilibrium intemuclear distance and force coefficient. For this purpose, we have calculated the electronic energy, rotational and vibrational g factors, the electric dipolar moment, and adiabatic corrections for both He and H atomic centres for intemuclear distances over a large range 10 °m [0.3, 10]. Based on these results we have generated radial functions for atomic contributions for g g,... 

For a molecular ion with charge number Q a transformation between isotopic variants becomes complicated in that the g factors are related directly to the electric dipolar moment and irreducible quantities for only one particular isotopic variant taken as standard for this species these factors become partitioned into contributions for atomic centres A and B separately. For another isotopic variant the same parameters independent of mass are still applicable, but an extra term must be taken into account to obtain the g factor and electric dipolar moment of that variant [19]. The effective atomic mass of each isotopic variant other than that taken as standard includes another term [19]. In this way the relations between rotational and vibrational g factors and and its derivative, equations (9) and (10), are maintained as for neutral molecules. Apart from the qualification mentioned below, each of these formulae applies individually to each particular isotopic variant, but, because the electric dipolar moment, referred to the centre of molecular mass of each variant, varies from one cationic variant to another because the dipolar moment depends upon the origin of coordinates, the coefficients in the radial function apply rigorously to only the standard isotopic species for any isotopic variant the extra term is required to yield the correct value of either g factor from the value for that standard species [19]. 

Equations (9) and (11) indicate how the auxiliary radial function for the rotational factor becomes separable into contributions from atomic centres of types A and B. An analogous separation is practicable for both the vibrational g factor and the total adiabatic corrections for the latter quantity this separation is effected in the original quantum-chemical calculations. Accordingly we express these calculated values of rotational and vibrational g factors, presented in Table 1, and adiabatic corrections, presented in Table 3, of He H" to generate coefficients of radial functions for atomic centres of either type. He or H. The most useful variable for these functions is z, defined in terms of instantaneous R and equihbrium R internuclear distances as... 

The adiabatic corrections for the individual atomic centres Fie and H, listed as a function of R in Table 3, are combined in the final column of that table into a total correction according to this formula... 

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