Weak interaction plays a significant role in many fundamental processes in materials science, chemistry, physics, and biology. Although the energies of weak interactions are several orders of the magnitude weaker than the typical covalent bonds or ionic interactions, they are qualitatively very important and their cumulative effect can be profound. In a molecular assembly with many weak interactions, the energetic sum might exceed that of a covalent bond. Therefore, the weak interactions, including van der Waals force, H-bonding and π electron interactions, are of great importance in functional material growth, solution, organic light-emitting diodes, protein folding, and nucleic acids. Theoretical investigations of these weakly interacting systems should include the dispersion interaction, which is hard to treat using the conventional Hartree-Fock or the density functional theory (DFT). Benefiting from the development of efficient computational methods and using representative examples from molecular biology to materials science, in this thesis we have investigated a number of important weak interactions, including hydrogen bonds, π–π interactions, and metal–π interactions. At the quantum mechanical level, our calculations provide a more realiable and improved understanding of the role and feature of weak interactions, which cannot be accurately predicted by conventional methods with classical potentials which ignore the descriptions of electron behaviors. The hydrogen bonding is a unique phenomenon and an extremely important interaction in almost all natural processes and plays a ubiquitous role in determining the structures of biomolecules [e.g. deoxyribonucleic acid (DNA), ribonucleic acid (RNA), and proteins] and supramolecular constructs. The Watson-Crick nucleic acid base pairs adenine-thymine (AT) and guanine-cytosine (GC) were studied using an economic basis set in density functional calculations. The economic basis set, in which the polarization function is added only to oxygen and nitrogen atoms of strong electronegativity, can predict reliable geometric structures and dipole moment of nucleic acid base pairs, comparable to those obtained from the basis set of 6-31G* in B3LYP calculations. Combining single point calculations with the standard basis set on the geometric structures optimized by the economic basis set, the present approach has predicted reliable natural bond orbital (NBO) charge, binding energy, electronegativity, hardness, softness, and electrophilicity index. The principle for basis set selection presented in this study can be regarded as a general guideline in the computation of large biological systems with considerably high accuracy and low computational expense. H–π interactions are weaker hydrogen bonds and have received much attention in organic crystals, protein folding, and molecular recognition. H–π interactions include the “conventional” XH–π interactions (X is N, O, etc) and “unconventional” CH–π interactions. To reveal the binding features of such systems, in this thesis we have studied the interaction between single-walled carbon nanotubes (SWCNTs) and protein amino arenes, including indole (tryptophan), imidazole (histidine), and phenol (tyrosine), using a reliable and efficient self-consistent-charge density-functional tight-binding approach augmented by an empirical London dispersion correction (SCC-DFTB-D). Our calculations show that these three arenes can be adsorbed perpendicularly to the armchair (5,5) carbon nanotube (CNT) surface by H–π interactions with the binding energy ranging from -0.20 to -0.31 eV, and the indole molecule demonstrates the strongest binding to CNT surface. Of the three kinds of H–π bonds considered, including NH–π, OH–π, and CH–π, NH–π is the strongest. The distances of N–π, O–π, and C–π increase from 3.1 to 3.3 Å, with all the values within the sum of van der Waals radii. The structure and electronic properties of the metallic (5,5) tube are unchanged. In addition, the molecular orbital surfaces also confirm the inherently weak interaction of these H–π bonded systems. In particular, a single NH–π bond formed between imidazole and CNT has a binding energy of -0.21 eV, which is as large as that of classical hydrogen bonding. This atypically large interaction energy is expected to play a significant role for a better understanding of NH–π interactions in biological systems. Our quantum mechanical quantification of these weakly polar H–π interactions provides deeper insights into the binding between proteins and SWCNTs. Different from the polar H described above, the non-polar H in molecular hydrogen can also be attracted to the π electron systems, as is shown in molecular hydrogen physisorption on CNTs. The involved weak H–π interaction is important for hydrogen physisorption which has great potential as a source of energy for the future. Due to the high surface to volume ratio, CNTs have been proposed to be one of the promising ways to store hydrogen. Our results show that molecular hydrogen could be non-covalently bonded both inside and outside CNT surface at physisorption distances. The tube’s curvature is crucial in the binding strength while the chirality plays a negligible effect. For external adsorptions, the binding energy of H2 on CNTs increases as the tube diameter increases, approaching but always being smaller than that of H2 on planar graphene. For internal adsorption, the binding energy for armchair and zigzag CNTs with diameter larger than 6Å will always be larger than that of external adsorption and that of H2 on graphene. There are binding maxima at (5,5) and (8,0) CNTs for armchair and zigzag CNTs, respectively. For chiral (6,3) CNT with diameter similar to (8,0) tube, the binding energy is almost the same as that of (8,0) tube. The binding energies of those three tubes are around -0.2 eV, which is 3 times as large as that of H2 on graphene. We propose that CNTs with diameter around 6 - 7 Å are energetically optimum candidates for physisorption of molecular hydrogen. Our results are expected to partly elucidate the currently challenging issues of physisorption of molecular hydrogen to SWCNTs. The π–π electron attraction plays a key role in the determining the geometry structures of molecules composed of aromatic groups, such as folded protein. Recent years, mainly based on π–π interactions, the non-covalent modifications of carbon nanotubes with organic molecules, proteins, and DNA/RNA, have attracted much interest, since such modifications can not only overcome the major hurdles of polydispersity and poor solubility of CNTs, but also show interesting and useful applications, such as DNA sensors and enzyme immobilization. For the π–π bonded systems, we have systematically studied the interactions of SWCNTs with 1-Pyrenebutanoic Acid, Succinimidyl Ester (PSE, organic), peptides (mini-proteins), and base pair GC (nucleic acid), respectively, using SCC-DFTB-D. Our calculations demonstrated that for all the three kinds of large complexes, the isolated PSE, peptides, and base pair GC can spontaneously be physisorbed on the CNT surface due to π–π interactions. The electronic structures of both metallic and semiconducting nanotubes are undamaged after adsorption of these functional molecules, and no significant hybridization between the respective orbitals of the two entities takes place. We also considered other important properties for those different π–π bonded systems, respectively. Specifically, (1) in the case of PSE, increasing the CNTs diameter leads to a higher binding energy. CNTs contribute to the main peak of the electron excitation procedure in the uv/vis spectrum, with a slight red shift after adsorption of PSE, and the high intensity of the main peak is almost unchanged; (2) for the three mini-proteins with 12 residues [NB1 (sequence LPPSNASVADYS), B1 (sequence HWKHPWGAWDTL), and B3 (sequence HWSAWWIRSNQS)], our computation reveals that the aromatic residues such as His and Trp are keys in determination of the peptide/CNTs binding, which is consistent with the experimental observation. The competition between the intra-molecular hydrogen bonding and the intermolecular π–π interaction determines the final structures of peptides/CNT complexes. Our calculations also predict that the non-covalent modification of CNTs by these active peptides increases the electron transfer capabilities of CNTs; and (3) for nucleic acid base pair GC, the binding strength shows insensitivity to the tube curvature and chirality, which is consistent with experimental findings. After adsorption on CNT surface, the three hydrogen bonds between guanine and cytosine merely change. Metal–π interactions have attracted intense attention for the wide applications in devices based on functional materials, such as the luminescent diodes. A clear understanding of how metal atoms interact with π– bonding electronic structures is greatly needed but still challenging. We have studied the interactions between polycyclic aromatic hydrocarbons (PAHs) and one metal atom (Al and Ca) using second-order Møller–Plesset perturbation (MP2) theory. The structural and electronic properties show that the metal atoms interact weakly with the π electrons of aromatic hydrocarbons. The NBO charge analysis shows that there is negligible charge transfer between metal and PAHs. The bonding nature between the metal–π systems depends on the valence orbital type of the metal atom and the π–bonding distribution of the conjugated aromatic hydrocarbon. Our results indicate that as the size of PAHs increases, the interactions between metal and PAHs generally increase, due to the enhanced π conjugation of PAHs.