Chemistry Reference
In-Depth Information
A10.1 Chemical Bond Formation
The formation of a molecule from atomic hydrogen and an H
+
cation can be thought of as
the reaction
H
+
H
+
→
H
2
+
(A10.1)
The chemical bond formation energy is just the energy change for this reaction. We would
expect a stable bond to have a lower energy than that of the atoms or ions that are involved.
So the chemical bond energy here should be negative.
As discussed in the main text, Equation (A10.1) is the reverse of the usual experimental
situation in which the energy required to dissociate from the molecular state is usually
measured. However, for calculations, the association of the reactants is easier to consider,
as at large separation the interaction terms that we will outline below all tend to zero.
In the following sections we will use the Schrödinger equation to obtain the energy
of the reactant and product side of Equation (A10.1)and so estimate the bond formation
energy. In this analysis we ignore the zero-point vibrational energy of the molecular ion
and so are calculating the molecular energy as if the structure could be frozen at the bot-
tom of the interaction potential-well plotted in Figure 7.5. The addition of the zero-point
energy based on the ground state of the harmonic oscillator (
h
ν/
2) is a straightforward and
relatively small correction.
As H
2
+
has only one electron, the potential energy can be analysed using only nuclear-
electron and nuclear-nuclear interaction terms, which can be calculated reasonably easily.
In more complex molecules the electron-electron interaction must also be accounted for,
which is a more difficult task. Indeed, the quest for methods to evaluate the electron-
electron interaction accurately is still an active area of research in theoretical chemistry.
A10.2 H Atom and H
+
Cation
The reactant side of Equation (A10.1) has the H atom and an H
+
cation. The H atom con-
sists of a negatively charged electron moving in the electrostatic potential of the nucleus,
a single positively charged proton. The total energy of the H atom is referenced to the
electron and proton separated, so that they no longer interact, and most of this section is
concerned with obtaining the atom's total energy in the lowest energy state. The cation on
the reactant side of Equation (A10.1) is a completely isolated proton, and so its energy will
be taken as zero.
The H AOs and their related orbital energy values have been discussed in some detail in
Appendix 9 (see Tables A9.1 and A9.2 and Equations (A9.51) and (A9.52)).
The discussion of Appendix 9 also introduces the atomic unit (au) system, and we will
use that again for our discussion of bonding. The atomic unit system takes the Bohr radius
(1 bohr = 0.529 177 Å) as the unit of length and uses the electron mass and charge to define
the mass and charge fundamental units. The energy unit in this system is the hartree (1 Ha
= 4.359 744
10
−
18
J) and we will see how this arises in the solution of the H atom ground
state. Chemical energies are often quoted in per mole units; for example, the bond energies
given in Tables 7.1 and 7.3 are given in kilojoules per mole units. It is useful to note that
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