Hydrogen molecule, calculations

In our hydrogen molecule calculation in Section 2.4.1 the molecular orbitals were provided as input, but in most electronic structure calculations we are usually trying to calculate the molecular orbitals. How do we go about this We must remember that for many-body problems there is no correct solution we therefore require some means to decide whether one proposed wavefunction is better than another. Fortunately, the variation theorem provides us with a mechanism for answering this question. The theorem states that the... 

Curve 1 shows the electronic energy of the hydrogen molecule neglecting interelectronic interaction (from Burrau s solution for the molecule-ion) curve 2, the electronic energy empirically corrected by Condon s method and curve 3, the total energy of the hydrogen molecule, calculated by Condon s method. 

Fig. 6 11 Total energy (E(R) + e2/4ne0R) of the hydrogen molecule, calculated by the method of approximation embodied in (6 1)...

Balandin (2) shortly afterwards began the publication of a series of papers developed from the theory that the catalytic decomposition of a relatively large molecule could only take place by simultaneous adsorption at several points this was known as the Multiplet Theory. With cyclohexane, for example, attachment at six centers was supposed to be necessary before benzene could be formed by the loss of three hydrogen molecules. Calculations along these lines for metallic catalysts, whose lattice dimen-... 

The example considered above is illuminating in many ways. It is true that nobody would do a hydrogen-molecule calculation this way but the results expose some of the basic difficulties of any perturbation approach. First of all, it is sometimes necessary to work quite hard to reproduce a very trivial result. Secondly, in spite of apparent size consistency in all orders, the perturbation method does not always have the property of separability. In fact, no amount of infinite summation can give the fuIl-CI energy (i.e. the basis-set limit) once the distance between the hydrogen atoms increases beyond about twice the equilibrium bond length—simply because the expansion has a very limited radius of convergence. 

V. Hydrogen Molecule Minimum Basis Set Calculation VI. Conclusions Appendix A Useful Integrals Acknowledgments References... 

In the case of hydrogen molecule, the term (ri l/ r + R ri2), which involves three centers, does not show up in the calculation. We will not discuss this integral in the present work. 

Calculating the Energy from the Wavefunction the Hydrogen Molecule... 

Tie hydrogen molecule is such a small problem that all of the integrals can be written out in uU. This is rarely the case in molecular orbital calculations. Nevertheless, the same irinciples are used to determine the energy of a polyelectronic molecular system. For an ([-electron system, the Hamiltonian takes the following general form ... 

We shall examine the simplest possible molecular orbital problem, calculation of the bond energy and bond length of the hydrogen molecule ion Hj. Although of no practical significance, is of theoretical importance because the complete quantum mechanical calculation of its bond energy can be canied out by both exact and approximate methods. This pemiits comparison of the exact quantum mechanical solution with the solution obtained by various approximate techniques so that a judgment can be made as to the efficacy of the approximate methods. Exact quantum mechanical calculations cannot be carried out on more complicated molecular systems, hence the importance of the one exact molecular solution we do have. We wish to have a three-way comparison i) exact theoretical, ii) experimental, and iii) approximate theoretical. 

Even though the problem of the hydrogen molecule H2 is mathematically more difficult than, it was the first molecular orbital calculation to appear in the literature (Heitler and London, 1927). In contrast to Hj, we no longer have an exact result to refer to, nor shall we have an exact energy for any problem to be encountered from this point on. We do, however, have many reliable results from experimental thermochemistry and spectroscopy. 

A very important difference between H2 and molecular orbital calculations is electron correlation. Election correlation is the term used to describe interactions between elections in the same molecule. In the hydrogen molecule ion, there is only one election, so there can be no election correlation. The designators given to the calculations in Table 10-1 indicate first an electron correlation method and second a basis set, for example, MP2/6-31 G(d,p) designates a Moeller-Plesset electron coiTclation extension beyond the Hartiee-Fock limit canied out with a 6-31G(d,p) basis set. 

Carry out a series of calculations comparable to those in Computer Project 10-1 on the hydrogen molecule. Estimate the conelation energy from the GAUSSIAN calculations. 

For both types of orbitals, the coordinates r, 0, and (j) refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. Although Slater-type orbitals (STOs) are preferred on fundamental grounds (e.g., as demonstrated in Appendices A and B, the hydrogen atom orbitals are of this form and the exact solution of the many-electron Schrodinger equation can be shown to be of this form (in each of its coordinates) near the nuclear centers), STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals < XaXbl g I XcXd > (each... 

If a covalent bond is broken, as in the simple case of dissociation of the hydrogen molecule into atoms, then theRHFwave function without the Configuration Interaction option (see Extending the Wave Function Calculation on page 37) is inappropriate. This is because the doubly occupied RHFmolecular orbital includes spurious terms that place both electrons on the same hydrogen atom, even when they are separated by an infinite distance. 

In order to calculate the total probability (which comes to 1), we have to integrate over both space dr and spin ds. In the case of the hydrogen molecule-ion, we would write LCAO wavefunctions... 

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