bond defn | delocalized bond | induced dipole bond | realistic bond
The simplest molecule is a one electron diatomic molecule. This molecule contains one electron and two nuclei, and is necessarily a cation, such as H2+, HHe+2, etc. Also, since the molecule contains an odd number of electrons it is also a radical. Such molecules are referred to as radical cations, and are commonly observed in mass spectrometers.
The calculation of an exact wavefunction for a one-electron molecule is a difficult task. Therefore, I will show how one can use "model" wavefunctions to investigate the bonding in these molecules. A "model" wavefunction is one that makes useful predictions about molecular properties (such as the existence of a bond of a given strength and length) even though it does not exactly satisfy the molecule's Schrödinger equation
The next few pages will consider several questions including:
Let's begin by considering again what we mean by a "bond". You are probably accustomed to thinking of a bond as something associated with electron pairs (covalent bond) or salts (ionic bond), and a "one electron" bond does not belong to either category.
"Bond" in this (or any other) context simply means the "thing" that holds the two nuclei close together. That is, the system has a "bond" if its energy is lower when the two nuclei are close together, and if its energy is higher when the nuclei move apart. The necessary energy relationships are shown in the following diagram.
One appealing bond model is shown by the following diagram.
The Lewis structure on the left corresponds to a wavefunction in which the electron is assigned to the lowest energy orbital (1s orbital) on atom L, øL. The Lewis structure on the right has an analogous meaning, but uses the 1s orbital centered on nucleus R. The resonance arrow indicates that our bonding model consists of a resonance hybrid between the two Lewis structures.
This resonance picture can be expressed in the mathematical language of wavefunctions. We define a "delocalized" wavefunction, ødel, as:
where each Lewis structure is represented by a particular orbital. cL and cR are just numbers and are referred to as "orbital coefficients". One part of our analysis will be to determine the optimal values for these coefficients (this will turn out to depend on the distance between the two nuclei when the nuclei are different).
Another bonding model can be constructed by asking how the electron density distribution around atom L responds as cation R approaches. One expects the electron to be attracted and the nucleus to be repelled by the approaching cation. This induces a dipole in L and the resulting ion-induced dipole interaction might yield a stable molecule.
In terms of wavefunctions, the electron in the "dipolar" atom can be thought of as occupying an s-p hybrid orbital directed at the approaching cation. Since a hybrid orbital is a mixture of s and p atomic orbitals, this bond model can be thought of as a resonance hybrid between two resonance structures consisting of different orbitals (see orbital drawing). Alternatively, we can write an "induced dipole" wavefunction, øind, as a linear combination of two orbitals: øS and øP (both centered on nucleus L).
Now that we have two different bonding models or hypotheses it is reasonable to ask if one is right and the other wrong. This does not turn out to be the case. We will see that the most realistic bonding model incorporates features of both models (and others as well).
(last updated 6/8/97)