Hydrogen molecule, orbitals

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 ... 

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... 

One widely used valence bond theory is the generalised valence bond (GVB) method of Goddard and co-workers [Bobrowicz and Goddard 1977]. In the simple Heitler-London treatment of the hydrogen molecule the two orbitals are the non-orthogonal atomic orbitals on the two hydrogen atoms. In the GVB theory the analogous wavefunction is written ... 

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. 

Approximate Theoretical. The simplest molecular orbital problem is that of the hydrogen molecule ion (Pig KJ-3), is a preliminary example of all molecular orbital problems to come, w hich, although they may be very complicated, are elaborations on this simple example. 

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. 

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. 

Asimple example is the formation of the hydrogen molecule from two hydrogen atoms. Here the original atomic energy levels are degenerate (they have equal energy), but as the two atoms approach each other, they interact to form two non degenerate molecular orbitals, the lowest of which is doubly occupied. 

The simplest example of covalent bonding is the hydrogen molecule. The proximity of the two nuclei creates a new electron orbital, shared by the two atoms, into which the two electrons go (Fig. 4.5). This sharing of electrons leads to a reduction in energy, and a stable bond, as Fig. 4.6 shows. The energy of a covalent bond is well described by the empirical equation... 

The concepts of directed valence and orbital hybridization were developed by Linus Pauling soon after the description of the hydrogen molecule by the valence bond theory. These concepts were applied to an issue of specific concern to organic chemistry, the tetrahedral orientation of the bonds to tetracoordinate carbon. Pauling reasoned that because covalent bonds require mutual overlap of orbitals, stronger bonds would result from better overlap. Orbitals that possess directional properties, such as p orbitals, should therefore be more effective than spherically symmetric 5 orbitals. 

T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

The most important molecular- orbitals are the so-called frontier molecular- orbitals. These are the highest (energy) occupied molecular- orbital (HOMO), and lowest (energy) unoccupied molecular- orbital (LUMO). The following picture shows the LUMO surface for the hydrogen molecule, H2. The LUMO consists of two separate surfaces, a red... 

Orbital Surfaces. Molecular orbitals provide important clues about chemical reactivity, but before we can use this information we first need to understand what molecular orbitals look like. The following figure shows two representations, a drawing and a computer-generated picture, of a relatively high-energy, unoccupied molecular orbital of hydrogen molecule, H2. 

Imagine a model hydrogen molecule with non-interacting electrons, such that their Coulomb repulsion is zero. Each electron in our model still has kinetic energy and is still attracted to both nuclei, but the electron motions are completely independent of each other because the electron-electron interaction term is zero. We would, therefore, expect that the electronic wavefunction for the pair of electrons would be a product of the wavefunctions for two independent electrons in H2+ (Figure 4.1), which I will write X(rO and F(r2). Thus X(ri) and T(r2) are molecular orbitals which describe independently the two electrons in our non-interacting electron model. 

Coulson, C. A., and Fischer, I., Phil. Mag. 40, 386, Notes on the molecular orbital treatment of the hydrogen molecule."... 

Kotani, M., Proc. Shelter Island Conference on Quantum Mechanical Methods in Valence Theory, p. 139. Best orbital for the hydrogen molecule."... 

Mueller, C. R., and Eyring, H., J. Chan. Phys. 19, 1495, Semi-localized orbitals. I. The hydrogen molecule. Combination of Inui (1941) and Coulson-Fischer (1949). 

Wallis, R. F., J. Chem. Phys. 23, 1256, "Molecular orbitals for the hydrogen molecule ground state." In-out effect for H2. 

The original VB wave function was introduced in the treatment of the hydrogen molecule by Heitler and London in 1932. This treatment considered only the one Is orbital on each hydrogen atom and assumed that the best wave function for a system of two electrons on two different atoms is a product of the two atomic Is orbitals i/ — XisXis- This wave function needs to be modified, however, to accommodate the antisymmetry of the wave function and to take into account the spin of the two electrons. 

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