Mathematical expressions of carbon sp³, sp² and sp hybrid orbitals were visualized by various techniques such as orbital picture, real 3-dimensional probability density picture in a glass block, and isosurface representation. Orbital picture and probability density picture were especially useful for recognizing equivalency and directional characteristics of the chemical bonds. Isosurface representation uniquely showed the difference of s or p characters between these three hybrid orbitals.

When each of the four identical carbon sp³ hybrid orbitals overlaps with the hydrogen 1s orbital, four identical C-H σ bonds are formed and methane (CH₄) results. Because the four sp³ hybrid orbitals have a specific geometry, the angle formed by each H-C-H is 109.47°, the so-called tetrahedral angle. The structure of ethane (C₂H₆) is similarly explained. The bonds in methane and ethane are called single bonds because they result from the sharing of one electron pair between bonded atoms. Carbon atoms can also form double bonds by sharing two electron pairs or triple bonds by sharing three electron pairs. Ethylene (H₂C=CH₂) contains a carbon-carbon double bond. When two carbons with sp² hybrid approach each other, they form a strong σ bond by sp²-sp² overlap. The unhybridized p orbitals interact by sideways overlap to form a π bond, resulting the formation of a carbon-carbon double bond. Four hydrogen atoms form σ bonds with the remaining four sp² orbitals to complete the structure of ethylene. The H-C=C or H-C-H bond angle in ethylene molecule is 121.3°or 117.4°respectively. Acetylene (HC ≡ CH) contains a carbon-carbon triple bond. When two carbons with sp hybrid approach each other, they form a strong σ bond by sp-sp overlap. The unhybridized p_y or p_z orbitals interact by sideways overlap to form two π bonds (p_y − p_y and p_z − p_z), resulting the formation of a carbon-carbon triple bond. The two remaining sp hybrid orbitals each form a σ bond with hydrogen to complete the acetylene molecule. Acetylene is a linear molecule with H-C ≡ C bond angle of 180°. Difference electron density of HOMO or LUMO in ethylene molecule is visualized. The effect of π electron ring current for the chemical shift measurement in NMR (Nuclear Magnetic Resonance) spectroscopy is also discussed.

We investigate optimized structures of a cyclic methanol cluster (CH₃OH)₄ by starting different-sized planar symmetric structures, where all the H atoms of the OH groups direct to the next O atom, and all the COH parts are in-plane. At the HF/STO-3G level, initial O-O distances at 2.7 Å and more lead to the most stable structure, whereas those less than 2.7 Å result in two other stable conformers. The border distance indicates the efficient range of hydrogen bonding forces of the OH groups. HF/6-31G (d) results in a planar structure with negative frequencies, whereas B3LYP/6-31G (d) gives a conformer whose methyl groups face the same direction. These structures result from the balance between the stabilization due to the orbital broadening of unshared electron pairs of O and the repulsive force of the methyl groups, reflecting the tendency for transformation among the conformers. To calculate force constants with the calcall option is effective in obtaining the most stable structure regardless of the initial O-O distance.

A dimeric titanium planar structure conventionally explains surface reactions of TiO₂ photocatalysis. Although it has been assumed to be planar, we found that the hydroxyl-terminated cluster model (Ti₂O₆H₄) essentially has a non-planar structure calculated with B3LYP/6-311G. The hydrogen atom positions shift the HOMO and LUMO energy levels, relating to photocatalytic oxidation and reduction abilities. The results indicate that settling a new larger model to fit many phenomena quantitatively is highly required since the planar structure potentially involves a severe electronic problem.