Two centres, problem

Another curvilinear coordinate system of importance in two-centre problems, such as the diatomic molecule, derives from the more general system of confo-cal elliptical coordinates. The general discussion as represented, for instance by Margenau and Murphy [5], will not be repeated here. Of special interest is the case of prolate spheroidal coordinates. In this system each point lies at the intersection of an ellipsoid, a hyperboloid and and a cylinder, such that... 

Exact forms (83,81,82) for the integrals, J, K and S for the two-centre problems of dihydrogen and other homogeneous diatomic molecular species were achieved in early calculations by transforming to the elliptical coordinates system defined in Figure 6.3. In this coordinate system the nuclear positions are the two fixed points of the coordinate system the foci of the ellipses separated by the bond length R with a point P defined by the three coordinates... 

As is well-documented [12] the two-centre problem is separable in the co-ordinates and correspondingly molecular orbital states... 

They are different from a similar case recently investigated by P. Hund, Z. Phys. 40, 742 (1927), namely the two-centre problem with one electron, where solutions of the forms and —... 

It is shown how the eigenvalues and eigenfunctions of the hydrogen molecule can be obtained with a relatively simple method. FVom these solutions of the two-centre problem we construct the eigenfunctions of the molecule in first order and obtain the energy of the molecule by a perturbation calculation. If we introduce fractional nuclear charges for the outer electron, then the perturbation becomes very small in the excited states. Therefore the complicated perturbational calculations do not have to be performed very strictly it is sufficient to take some main terms of the eigenfunctions into account. 

We can therefore view the totality of problems originating from the theory of the rotational and vibrational states of a molecule as being satisfactorily solved. The problem of a complete quantitative control of the molecular problem from the theoretical point of view is therefore dependent on the calculation of the electronic terms with sufficient accuracy. For each state of the molecule these terms define a potential energy, and as a matter of fact we should know the complete form of this potential curve, even for infinite distance between the two atoms. In most cases we know the decay products, either from experimental data or from theoretical considerations, so that in general only the calculation of the small region in the neighbourhood of the minimum is necessary. For such small or moderate distances between the nuclei the method described here, which starts from the two-centre problem, is especially suitable. 

In our special two-centre problem the internucleax distance is called 2ii. R is therefore half of the internuclear distance in o/f/2 or the full internucleax distance in units a/f. We indicate the two nuclei by a and 6, and take the Z-axis as the internucleax axis with the origin of the coordinates in its centre. The nuclei-electron distance is called Xa and r, and the angle (p. 

Fig. 1. Orbit types in the two-centre problem of classical mechanics.

This difficulty, which appears in the classical theory, disappears when quantum-mechanics is employed. Since it does not only appear in a two-centre problem, but also for much simpler problems, those with one degree of freedom, we will study the quantum-theoretical behaviour with the help of a one-dimensional example. The essential thing is clearly that the potential function U (x) can assume certain values more than twice, others only twice, so that the number of zero points of V x) — W can change suddenly by changing one parameter. We will use Schrodinger s method of wave mechanics, since this allows a visual description of the stationary states through the eigenfunctions. 

After the study of such simple cases we will return to the two-centre problem and apply it to the term orderings in two-atomic molecules. 

The two-centre problem (two fixed centres and one electron) leads to a Schrodinger equation that is separable in elliptic coordinates The sta-... 

The most simple molecule is the positive ion of the hydrogen molecule (Hj ) Its model, when we neglect the angular momentum s, is distinct from the two-centre problem only in the respect that the internuclear distance is not fixed, but vibrates around an equilibrium configuration which is also determined by the electronic configuration. Instead of each term in the two-centre problem a function of the distance r between the nuclei appears, of which the minimum determines the equilibrium configuration and the value of each of its terms transforms in the value of the terms of the separated systems. For small r the functions behave like 1/r, their distances like the distances between the terms in the case of a united nucleus. 

We now proceed with a qualitative consideration of the hydrogen molecule. The term scheme lies on the boundary between the systems H-l-H and H" "H-H on the one hand and the He atom on the other hand. The electron affinity of the H atom, even if it is positive, is only a few volts the lowest terms of the H2 molecule will transform into the terms of the system H-l-H when the molecule is pulled apart. The term scheme is therefore roughly represented by Fig. 12 (however, probably the term Isis of H" "-fH has a small influence on the term scheme). Instead of a term in the corresponding two-centre problem we obtain a function of the internuclear... 

First we consider the possible energy levels of one electron in the presence of a two-nuclear system. An accurate anal3dical solution of the two-centre-problem is not yet possible, but certain approximate methods can now be used to indicate the relative position of the various energy levels - in each case as a function of internuclear distance. We use this system of energy levels for the many-electron problem just as was done in atoms. We then suppose a molecule built up in the following way. We add one charge at a time to each nucleus and then, supposing the nuclei held fixed, add to the system two electrons successively. The system is then allowed to take up its equilibrium value adiabatically. 

In the Hiickel TT-electron model, ethene is a two-electron problem. 1 have numbered the carbon atoms C and Cj, and X is centred on Ci with X2 on Cj. The HF matrix becomes... 

Aove Advanced Example Lactone (25) was needed in the synthesis of pcderamldc, an inhibitor of protein synthesis found in a beetle. Disconnection of the lactone reveals two stereochemical problems a c-ir, double bond is required and two adjacent chiral centres ( in 26) must be set up correctly. 

Despite these modifications there remain a number of well-documented problems with the AM1/PM3 core-repulsion function [37] which has resulted in further refinements. For example, Jorgensen and co-workers have developed the PDDG (pair-wise distance directed Gaussian) PM3 and MNDO methods which display improved accuracy over standard NDDO parameterisations [38], However, for methods which include d-orbitals (e.g. MNDO/d [23,24], AMl/d [25] and AMI [39,40]) it has been found that to obtain the correct balance between attractive and repulsive Coulomb interactions requires an additional adjustable parameter p (previously evaluated using the one-centre two-electron integral Gss, Eq. 5-7), which is used in the evaluation of the two-centre two-electron integrals (Eq. 5-8). 

With more than two unspecified chiral centres, problems multiply rapidly - three chiral centres yield eight stereoisomers, and thus four possible sets of signals and so on. From this, it follows that n chiral centres give rise to 2" chiral entities of which 2"/2 will be distinguishable by NMR. 

As an example of the first type, consider the linear XeF2 molecule. A Lewis structure with two-centre, electron-pair bonds offends the octet rule. This is not a problem if we allow the use of the Xe 5d orbitals. However, as discussed in Section 6.1, many authors prefer to avoid such use. If we restrict our basis set to the 5p(Xe) and 2p(F) orbitals, and if we consider only a overlap to be important, we need consider only three AOs 2p.(F,), 2p2(F2) and 5pz(Xe), where the molecular axis is labelled z. Thus we will obtain three MOs -vp, ij>2 and t )3 as shown below ... 

Instead of expressing the scattering problem in terms of the one-centre polar coordinates rq and 0, it is more appropriate to use the two-centre prolate spheroidal coordinates defined by... 

At the very centre of biology there are two complementary problems How does an organism produce an egg (the problem of generation), and How does an egg produce an organism (the problem of embryonic development). These questions have been debated since... 

By referring the motion to the centre of mass, the two-body problem has been reduced to a one-body problem of the vibrational motion of a particle of mass p against a fixed point, under the restraining influence of a spring of length R with a force constant k. 

We are faced with two interconnected problems related to the intelligibility of the presentation. The first one concerns the nomenclature of the centres other than isolated atoms and the second the labelling of the optical transitions. These problems are not trivial, [5], but not as severe for H-like centres as for deep centres. The different notations for the shallow thermal donor complexes in silicon, discussed in Sect. 6.4.2, are however, a counter-example of this statement. In this book, on the basis of the present knowledge, names of centres, in direct relation with their atomic structure, have been privileged, but the usual label has however been indicated. When the exact structure is not simple and when there exist an acronym, like TDD for thermal double donor , it has been used. The labelling by their excited states of the transitions of the shallow donor centres and of similar species, whose spectra... 

If we consider the various structure types adopted by metals and then try to provide a model for localized metal-metal bonding, we run into a problem there are not enough valence shell orbitals or electrons for each metal atom to form two-centre two-electron bonds with all its neighbours. For example, an alkali metal has eight near-neighbours (Table 5.2), but only one valence electron. We must therefore use a bonding model with multi-centre orbitals (see Sections... 

It may be shown quite generally, that such a two-body problem may be reduced to a one-body problem. We choose the centre of gravity of the two particles as the origin of co-ordinates 0 and determine the direction of the line joining m2 and m1 by the polar coordinates 9, tf). If then jq and r2 are the distances of the particles from 0, their polar co-ordinates will be rlt 9, and r2, tt—9, ir+ and further, r1+r2=r. The Hamiltonian function becomes... 

So far as the calculation is concerned it is immaterial whether we consider our problem as a one-body or as a two-body problem. In the first case we have a fixed centre of force, and the potential of the field of force is a function U(r) of the distance from the centre. In the second case we have two masses, whose mutual potential energy U(r) depends only on their distance apart they move about the common centre of gravity. As wc have shown generally in 20, the Hamiltonian function in polar co-ordinates is precisely the same for the two cases, if, in the one-body problem, the mass /x of the moving... 

We shall show that the problem of two centres referred to above is separable in the co-ordinates f, tj, . The potential energy of an electric charge — e attracted by two positively charged points is... 

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