Hydrogen molecule, bond orbitals

To get the molecular orbital of the hydrogen molecule, the orbital equations of the two atoms are combined. When the orbital equations are added together, the result is a bonding molecular orbital that extends over both atoms. Subtracting the orbital equations of the atoms produces an antibonding molecular orbital. This process is called the linear combination of atomic orbitals or LCAO. 

The hydrogen molecule molecular orbital and valence bond treatments... 

We will now describe the bonding in the hydrogen molecule using the MO model. The first step is to obtain the hydrogen molecule s orbitals, a process that is greatly simplified if we assume that the MOs can be constructed from the hydrogen Is orbitals. 

Dipole-dipole forces among HCl molecules Hydrogen bonds between water molecules Eondon forces between hydrogen molecules Molecular orbitals in the hydrogen molecule Molecular orbitals in a helium molecule ... 

As described previously for the hydrogen molecule, molecular orbital (MO) theory takes the atomic orbitals of the atoms, and mathematically combines the wave functions that represent these atomic orbitals (using an approach known as the linear combination of atomic orbitals). This combination produces new molecular orbitals that describe the regions of space occupied by the bonding electrons. The number of new molecular orbitals formed is the same as the number of atomic orbitals combined. The wave functions that represent the new molecular orbitals can be used to calculate the energy of an electron in those molecular orbitals. 

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. 

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. 

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. 

Now consider the alkynes, hydrocarbons with carbon-carbon triple bonds. The Lewis structure of the linear molecule ethyne (acetylene) is H—O C- H. To describe the bonding in a linear molecule, we need a hybridization scheme that produces two equivalent orbitals at 180° from each other this is sp hybridization. Each C atom has one electron in each of its two sp hybrid orbitals and one electron in each of its two perpendicular unhybridized 2p-orbitals (43). The electrons in the sp hybrid orbitals on the two carbon atoms pair and form a carbon—carbon tr-bond. The electrons in the remaining sp hybrid orbitals pair with hydrogen Ls-elec-trons to form two carbon—hydrogen o-bonds. The electrons in the two perpendicular sets of 2/z-orbitals pair with a side-by-side overlap, forming two ir-honds at 90° to each other. As in the N2 molecule, the electron density in the o-bonds forms a cylinder about the C—C bond axis. The resulting bonding pattern is shown in Fig. 3.23. 

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