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Cyclic orbital interactions are contained in non-cyclic conjugation as well as cyclic conjugation. For effective interactions,
the orbitals are required to meet simultaneously the phase continuity conditions: (1) out of phase relation between electron-donating
orbitals; (2) in phase relation between electron-accepting orbitals and between electron-donating and -accepting orbitals.
The orbital phase theory is applicable to diverse chemical phenomena of non-cyclic conjugate systems, e.g., relative stabilities
of non-cyclic isomers, and selectivities of the reactions through non-cyclic transition structures. The orbital phase theory
also includes the rules for cyclic systems, i.e., the Wooward–Hoffmann rule for stereoselection of organic reactions and the
Hueckel 4n + 2π electron rule for aromatic molecules. Derivation and applications of the orbital phase theory are reviewed.

Diels-Alder reaction is one of the most fundamental and important reactions for organic synthesis. In this chapter we review
the studies of the π-facial selectivity in the Diels-Alder reactions of the dienes having unsymmetrical π-plane. The theories
proposed as the origin of the selectivity are discussed.

A chemical orbital theory is useful in inorganic chemistry. Some applications are described for understanding and designing
of inorganic molecules. Among the topics included are: (1) valence electron rules to predict stabilities of three- and four-membered
ring metals and for those of regular octahedral M₆ metal clusters solely by counting the number of valence electrons; (2) pentagon stability (stability of five- relative to
six-membered rings in some classes of molecules), predicted and applied for understanding and designing saturated molecules
of group XV elements; (3) properties of unsaturated hydronitrogens N
_m
H
_n
in contrast to those of hydrocarbons C
_m
H
_n
; (4) unusually short nonbonded distances between metal atoms in cyclic molecules.

Over the last three decades the rational design of diradicals has been a challenging issue because of their special features
and activities in organic reactions and biological processes. The orbital phase theory has been developed for understanding
the properties of diradicals and designing new candidates for synthesis. The orbital phase is an important factor in promoting
the cyclic orbital interaction. When all of the conditions: (1) the electron-donating orbitals are out of phase; (2) the accepting
orbitals are in phase; and (3) the donating and accepting orbitals are in phase, are simultaneously satisfied, the system
is stabilized by the effective delocalization and polarization. Otherwise, the system is less stable. According to the orbital
phase continuity requirement, we can predict the spin preference of π-conjugated diradicals and relative stabilities of constitutional
isomers. Effects of the intramolecular interaction of bonds and unpaired electrons on the spin preference, thermodynamic and
kinetic stabilities of the singlet and triplet states of localized 1,3-diradicals were also investigated by orbital phase
theory. Taking advantage of the ring strains, several monocyclic and bicyclic systems were designed with appreciable singlet
preference and kinetic stabilities. Substitution effects on the ground state spin and relative stabilities of diradicals were
rationalized by orbital interactions without loss of generality. Orbital phase predictions were supported by available experimental
observations and sophisticated calculation results. In comparison with other topological models, the orbital phase theory
has some advantages. Orbital phase theory can provide a general model for both π-conjugated and localized diradicals. The
relative stabilities and spin preference of all kinds of diradicals can be uniformly rationalized by the orbital phase property.
The orbital phase theory is applied to the conformations of diradicals and the geometry-dependent behaviors. The insights
gained from the orbital phase theory are useful in a rational design of stable 1,3-diradicals.