  The dimensionality of quantum materials strongly affects their physical
properties. Although many emergent phenomena, such as charge-density wave and
Luttinger liquid behavior, are well understood in one-dimensional (1D) systems,
the generalization to explore them in higher dimensional systems is still a
challenging task. In this study, we aim to bridge this gap by systematically
investigating the crystal and electronic structures of molybdenum-oxide family
compounds, where the contexture of 1D chains facilitates rich emergent
properties. While the quasi-1D chains in these materials share general
similarities, such as the motifs made up of MoO6 octahedrons, they exhibit vast
complexity and remarkable tunability. We disassemble the 1D chains in
molybdenum oxides with different dimensions and construct effective models to
excellently fit their low-energy electronic structures obtained by ab initio
calculations. Furthermore, we discuss the implications of such chains on other
physical properties of the materials and the practical significance of the
effective models. Our work establishes the molybdenum oxides as simple and
tunable model systems for studying and manipulating the dimensionality in
quantum systems.
