Two-centre approximation

To rationalize the two-centre approximation , the effective potential is written as... 

As shown in [6.6] the LMTO-ASA Hamiltonian matrix may be transformed into the two-centre form [6.7] where the hopping integrals are products of potential parameters and the canonical structure constants. This result was already stated in Sect.2.5. A less accurate two-centre approximation based upon the KKR-ASA equations will be presented in Sect.8.1.2. The canonical structure constants which, after multipiication by the appropriate potential parameters, form the two-centre hopping integrals are 1 isted in Table 6.1. The... 

Three-centre terms, i.e. I m n. These are frequently neglected, in what is called the two-centre approximation, based on the assumed strong localization of the orbitals... 

The calculation on electrostatic potential distribution was carried out in accordance with the procedure described in [132] in two-centred approximation without a preliminary orthogonalization of atomic basis. 

Referring to figure BLIP. 7 consider electrons from the event under study as well as from other events all arriving at the two detectors. The electrons from the event under study are correlated in time and result in a peak in the time spectrum centred approximately at the delay time. There is also a background level due to events that bear no fixed time relation to each other. If the average rate of tlie background events in each detector is R and i 2> then the rate that two such events will be recorded within time Ax is given by i g, where... 

In the Complete Neglect of Differential Overlap (CNDO) approximation only the Coulomb one-centre and two-centre two-electron integrals remain (eq. (3.78)). 

Other than gas velocity and the physical properties of the feed, particle size is a parameter which has a significant effect. Smaller bed particles are more likely to form permanent bonds, and to quench, because of their smaller inertia. The force tending to pull two particles apart is equal to the product of the particle mass and the distance between the two centres of mass. For the case of two spherical particles joined together at their surfaces, this force is proportional to the particle diameter raised to the fourth power. Other cases approximate to this relationship. 

An interesting mixed-basis-set method for use in SCF calculations has been described by Billingsley and Trindle,52 with application to LiC>2. One-centre and most two-centre integrals are evaluated analytically, whilst less tractable integrals are approximated by a gaussian expansion of the STO s. Examination of portions of the... 

Of those methods that aim to mimic ab initio results some are very accurate indeed. The disadvantage is that they may be only marginally more economical than the full ah initio calculation. Such methods are PDDO (Projectors of Diatomic Differential Overlap), developed by Newton and co-workers,56-58 and LEDO (Limited Expansion of Diatomic Overlap), devised by Billingsley and Bloor.59 Both methods are essentially ways of accurately approximating some of the multi-centre integrals. Both approximate a two-centre distribution by one-centre distribution according to... 

When extending the molecular orbital concept developed for the monoelec-tronic species H2 to polyelectronic diatomic molecules, we start by acknowledging the role of two fundamental approximations (a) one associated with the existence of two nuclei as attractive centres, namely the Born-Oppenheimer approximation, as already encountered in H2" and (b) the other related to the concept of the orbital when two or more electrons are present, that is the neglect of the electron coulomb correlation, as already discussed on going from mono- to polyelectronic atoms. Within the orbital approach, an additional feature when comparing to H2" is the exchange energy directly associated with the Pauli principle. 

The integral J can be simplified to the double sum of two-centre Coulombic repulsion integrals -yM (Equation 4.29) by adopting the zero differential overlap (ZDO) approximation, that is, we assume that atomic orbitals located on different atoms do not overlap, 0 for pffv. 

Pauli1 and Nicssen 2 have endeavoured to treat the quantum theory of the problem of two centres, and to apply it to the hydrogen molecule positive ion, which consists of two nuclei with charges +e (i.e. Z1=Za=l), and one electron. To a first approximation, the motion of the nuclei can be neglected on account of their large mass. The first step is to calculate the motion of the electrons when the... 

Whereas the computation required for an INDO calculation is little more than for the analogous CNDO calculation, in NDDO the number of two-electron, two-centre integrals is increased by a factor of approximately 100 for each pair of heavy atoms in the system. 

---