Hydrogen molecule simple calculations

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. 

Let s assume that we want to find the structure of an unknown hydrocarbon. A molecular weight determination on the unknown yields a value of 82, which corresponds to a molecular formula of CfcHjo Since the saturated Q alkane (hexane) has the formula C61-114, the unknown compound has two fewer pairs of hydrogens (H]4 - H l() = H4 = 2 H2), and its degree of unsaturation is two. The unknown therefore contains two double bonds, one ring and one double bond, two rings, or one triple bond. There s still a long way to go to establish structure, but the simple calculation has told us a lot about the molecule. 

Curves showing these two quantities as functions of rAB are given in Figure 42-2. It is seen that 1ps corresponds to attraction, with the formation of a stable molecule-ion, whereas pA corresponds to repulsion at all distances. There is rough agreement between observed properties of the hydrogen molecule-ion in its normal state and the values calculated in this simple way. The dissociation energy, calculated to be 1.77 v.e., is actually 2.78 v.e., and the equilibrium value of rAB, calculated as 1.32 A, is observed to be 1.06 A. 

In order to answer these questions, accurate experimental and theoretical results were needed for representative molecular systems. Theoreticians, for obvious reasons, have favored very simple systems, such as the hydrogen molecular ion (Hj) for their calculations. However, with only one electron, this system did not provide a proper test case for the molecular quantum mechanical methods due to the absence of the electron correlation. Therefore, the two-electron hydrogen molecule has served as the system on which the fundamental laws of quantum mechanics have been first tested. 

MCIs are calculated from the hydrogen suppressed skeleton of a molecule. First, each non-hydrogen atom is assigned a delta value (8). For simple indices, 8 is equal to the number of atoms to which it is bonded for valence indices, 8 values are based upon the number of valence electrons not involved in bonds to hydrogen atoms. Simple and valence indices of different orders and types can be calculated for a given molecule. 

The chemist is accustomed to think of the chemical bond from the valence-bond approach of Pauling (7)05), for this approach enables construction of simple models with which to develop a chemical intuition for a variety of complex materials. However, this approach is necessarily qualitative in character so that at best it can serve only as a useful device for the correlation and classification of materials. Therefore the theoretical context for the present discussion is the Hund (290)-Mulliken (4f>7) molecular-orbital approach. Nevertheless an important restriction to the application of this approach must be emphasized at the start viz. an apparently sharp breakdown of the collective-electron assumption for interatomic separations greater than some critical distance, R(. In order to illustrate the theoretical basis for this breakdown, several calculations will be considered, the first being those for the hydrogen molecule. 

In Chapter 3 the covalent bond has been discussed and the question now arises whether this is the only possible type of bond between atoms. Let us consider the gaseous molecule of sodium chloride. The sodium and chlorine atoms each have one unpaired electron, 35 in sodium and 3/> in chlorine, so that in principle the formation of a covalent bond is possible. The calculation of bond energies presents considerable difficulties even in a simple molecule such as hydrogen and the calculation for more complicated molecules is impossible except by an approximate method such as that introduced by Pauling In this method, it is assumed that the energy of the covalent bond A—B, is equal to one half of the sum of the bond energies of the homopolar molecules A—A and B—B, i.e. 

The simple expression or the more complicated ones can be used to describe solvent perturbations on infrared spectra. They do not explain solvent shifts, since their explanation requires a priori calculations of 17, C7",. This kind of calculation demands a detailed quantum mechanical examination of the intricate many-body interactions between the electrons and nuclei of the solvent and those of the dissolved molecule. Such calculations may be just barely possible for, say, a system of hydrogen atoms dissolved in liquid helium. They are not tractable for most solvent-solute systems unless drastic approximations are made. 

As previously stated, truncated Cl s are neither size extensive nor size consistent. A simple (and often used) example makes this clear. Consider two noninteracting hydrogen molecules. If the CISD method is used, then the energy of the two molecules at large separation will not be the same as the sum of their energies when calculated separately. For this to be the case, one would have to include quadruple excitations in the supermolecule calculation, since local double excitations could happen simultaneously on A and B. 

From the calculations on such simple molecules as the hydrogen molecule and from the experimental results, we know that U(r) for a stable diatomic molecule is similar to the function plotted in Figure 34-2. When the atoms are very far apart (r large), the energy is just the sum of the energies of the two individual atoms. As the atoms approach one another there... 

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