James and Coolidge

The best orbitals for this simple system are those which exploit its symmetry. James and Coolidge (1935) found that a good approximation to the lOg orbital was... 

Taking up the idea of "correlated wave functions containing r12, James and Coolidge (1933) made a careful study of the H2 problem and, after a great deal of numerical work, they obtained finally an energy value in complete agreement with experience. This was another successful test of the validity of the Schrodinger... 

We note that the power series expansion III. 119is a direct generalization of the Hylleraas form III. 114 to which it should go over in the limiting case Rab = 0. James and Coolidge obtained a value of the electronic energy, —1.17347 at.u., in excellent agreement with the experimental results available, and their work forms even today the best basis for our understanding of the electronic structure of the chemical bond. 

Evett, A. A., /. Chem. Phys. 24, 150, Ground state of (HeH)+." Method of James and Coolidge 20-term function. 

Because of the interelectronic repulsion term l/ri2, the electronic Hamiltonian is not separable and only approximate solution of the wave equation can be considered. The obvious strategy would be to use Hj wave functions in a variation analysis. Unfortunately, these are not known in functional form and are available only as tables. A successful parameterization, first proposed by James and Coolidge [89] and still the most successful procedure, consists of expressing the Hamiltonian operator in terms of the four elliptical coordinates 1j2 and 771 >2 of the two electrons and the variable p = 2ri2/rab. The elliptical coordinates 4> 1 and 2, as in the case of Hj, do not enter into the ground-state wave function. The starting wave function for the lowest state was therefore taken in the power-series form... 

Although the calculated molecular parameters De = 3.15 eV, re = 1.64 a0 do not compare well with experiment the simplicity of the method is the more important consideration. Various workers have, for instance, succeeded to improve on the HL result by modifying the simple Is hydrogenic functions in various ways, and to approach the best results obtained by variational methods of the James and Coolidge type. It can therefore be concluded that the method has the correct symmetry to reproduce the experimental results if atomic wave functions of the correct form and symmetry are used. The most important consideration will be the effect of the environment on free-atom wave functions. 

The simplest molecular system exhibiting effects of electron correlation is the hydrogen molecule. For this molecule the explicitly correlated wave function has been applied in the early days of quantum mechanics (James and Coolidge, 1933), It was later generalised by Kolos and Wolniewicz (Kcrfos and Wolniewicz, 1965) and successfully used to solve variety of problems in the ground and excited states of the hydrogen molecule. This wave function, called there the Kolos-Wolniewicz function (Kolos and Wolniewicz, 1965) is assumed in the form of an expansion ... 

One can use as many terms as one might wish with this function. In 1933 James and Coolidge [20] used 13 terms and obtained an energy only 0.02 eV less than the full value of 4.75 eV, whilst Kolos and Roothaan [21] extended the basis set to 50 terms and obtained a calculated energy in exact agreement with experiment. 

James and Coolidge, 138 Jibben, 361 Jones, 317, 320 Jong, de, see De Jong... 

It is well known, of course, that the n and cr orbitals that are used in these calculations are built from atomic orbitals which are rough approximations to the accurate wave functions. Thus the electronic density calculated from the James and Coolidge function for the hydrogen molecule is very different from that obtained from the usual functions built from atomic orbitals.63 It is thus at least possible that a very accurate wave function for an aromatic hydrocarbon would give an electronic density which could resemble Fig. 4a. If this should prove not to be the case, then scheme 4b could be used where the delocalized bond of naphthalene now contains 32 electrons. 

The results of the various calculations described in this section are collected in Table 43-1, together with the final results of James and Coolidge (see following section). 

James and Coolidge chose as the variation function the expression... 

In a later note, J. Chem. Phy. 3, 129 (1936), James and Coolidge state that these values are about 0.05 per cent too large. 

H. M. James and A. S. Coolidge, J. Chern. Phys. 3, 129 (1935). We are indebted to Dra. James and Coolidge for the personal communication of this and other results of their work. 

Heitler and London using the perturbation method, but the best solution has more recently been obtained by James and Coolidge ... 

It must be remembered that, fundamental as this phenomenon of exchange is, it appears only as a result of the necessity and the methods of approximation. James and Coolidge, by changing the system of co-ordinates and including the terms of electron repulsion, obtained the best solution of the H2 molecule. Their curve corresponds almost exactly with the experimental one, and no exchange integrals of the type of Hiu appear in their calculation. 

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