In §1, I transformed the molecular orbitals (MOs) for H₂ into generalized valence bond orbitals (GVBOs) using expansion coefficients from a wavefunction that is slightly more complete than the Hartree-Fock wavefunction. But this small improvement in wavefunction was sufficient to let us see the way that orbitals polarize toward one another when a covalent bond forms. (The GVB wavefunction is inherent better than the HF wavefunction because it is more complete; GVBOs are thus closer to capturing the true n-electron wavefunction.)

In this section, I will present animations of orbitals for H₂, the repulsive triplet H–H interaction, hydrogen fluoride (HF), beryllium hydride (BeH), and the helium hydride cation (HeH⁺). In some cases, the orbital animations will be linked to other information, including the GVB pair overlap discussed in §1 and the potential energy curve for the interaction.

The polarization of the left and right GVBOs toward the other nucleus is a smooth process that is most pronouced near r_e. The GVB overlap increases as the bond forms, then decreases as the animation cycles back toward dissociation. Over the 2 Å change in separation depicted here, the interaction energy increases from near zero to about –110 kcal/mol, the H₂ bond dissociation energy.

The orbitals look quite different when they cannot overlap. There is a nodal plane at the midpoint between the two nuclei that can't be crossed when the electrons on hydrogen have the same spin. The potential energy drives up the repulsive wall when the atoms are pushed together. This animation scans separations from 1 Å to nearly 3 Å, which raises the energy by about +140 kcal/mol.

Fluorine has 9 electrons. Its 1s² 2s² 2p_x² 2p_y² 2p_z¹ configuration can be represented with this glyph:

Remember that the small central circle represents the out-of-plane 2p orbital. I can reorient this glyph so that the singly occupied orbital is aligned with the x, y, or z axes. That means that a hydrogen atom can approach fluorine in these three ways:

Since the singly occupied orbitals on fluorine and hydrogen can have parallel or antiparallel spins, there are six possible combinations. Most of these interactions are repulsive, as the next figure shows.

Bound HF occurs when the two singly occupied orbitals are aligned on the internuclear axis (the z-axis here) and their spins are antiparallel. The coupling diagram for HF as formed from the atoms is:

The next figure shows how some of the orbitals—the H 1s, F 2p_z, and F 2s²—change as the bond forms.

In H₂, the orbitals change symmetrically because the sharing is equal. Here, the hydrogen 1s orbital (top frame) polarizes much more toward the fluorine nucleus than the fluorine 2p_z orbital (middle frame) polarizes toward the hydrogen nucleus. This unequal sharing occurs because the electronegativity of fluorine is much greater than that of hydrogen. HF has a polar covalent bond with substantial (H⁺)-(F^–) character, while H₂ has a non-polar covalent bond. Note that the GVB overlap between the bond orbitals increases in a manner very similiar to H₂, rising to about 0.8 at the minimum.

While it is not too pronounced, the fluorine 2s² orbital (bottom frame) is clearly pushed away from the new bond pair that forms. The bond pair takes precedence, so the 2s² pair moves due to pair-pair repulsion.

A hydrogen atom is able to recouple the 2s² pair on beryllium, but it can't disrupt the 1s² pair on He. Beryllium can undergo recoupling because the 2s orbital can hybridize with the unused 2p orbitals, which lie only slighty higher in energy. Since the resulting orbitals are lobes on either side of the nucleus (as shown in the orbital animation below), a new glyph is needed. The glyph and the reaction to form BeH are shown here.

In BeH, it is energetically favorable for the hydrogen atom to break the coupling between the pair of valence electrons on beryllium to form a new coupling, which leaves an electron left over in the lobe orbital on the far side of beryllium from where the bond forms.

We can anticipate that there will be greater changes in the three GVBOs when BeH forms than when covalent bonds form in H₂ or HF. Here is the animation:

The changes are quite dramatic. At large separations, one sees the lobe orbitals on beryllium in the top two frames and the hydrogen 1s orbital in the bottom frame. As the nuclei approach one another, the inner beryllium lobe orbital polarizes toward hydrogen nucleus. Meanwhile, the other two orbitals are essentially swapping with each other. The top two frames always show which orbitals are coupled together. At long separations, there is an atomic pair on beryllium. At short separations, the inner Be lobe orbital is coupled to the 1s orbital on hydrogen to form a bond pair. Note that the GVB overlap varies between about 0.6 and 0.75: there is always a coupled electron pair—it just changes character with r. The lower frame shows which orbital is not coupled to another orbital. At large separations, it is the hydrogen 1s orbital; at short separations, it is the outer beryllium lobe orbital.

In this case, it may be useful to look at the static orbitals as well.

Recoupling also occurs in carbon, which gives rise to its typical tetravalence. Coupling diagrams for a number of organic compounds that use the linear, planar, and tetrahedral glyphs will be the subject of §4.

The helium hydride cation (HeH⁺) forms when a helium atom combines with a proton. The helium 1s² pair becomes a shared bond pair between the two nuclei as shown in the following animation.

HeH⁺ is an example of a dative bond. More examples of dative bonds are covered in §6.

The next section considers N₂, a compound with multiple bonds.