To accurately solve the general nuclear spin state function in Nuclear Magnetic Resonance (NMR), a rotation wave approach was employed, allowing the reference frame to rotate in sync with the oscillating magnetic field. The spin state system was analogously treated as a Rubik's Cube, ensuring the diagonalization of only the time-dependent part of the state function. Although Gottfried's equation (1966) aligns with transitions between specific spin states m and m′, his second rotation contradicts the conservation of angular momentum, resulting in inaccuracies for spin states with initial phase shifts or entangled states. Contrarily, Schwinger (1937) efficiently computed the coefficients for each spin state in a frequency range opposite to the Larmor frequency, using an unorthodox approach in quantum mechanics, which unfortunately led to the oversight of his work in subsequent citations. This methodology was also applied to derive the general electron spin state function in Nuclear Magnetic Resonance (NMR) and Electron Paramagnetic Resonance (EPR), enabling the construction of a doubly rotated ground state for time-dependent perturbation theory. This was particularly relevant as the Hamiltonians for magnetic dipole, electric quadrupole, and magnetic octupole moments incorporate powers of I · J terms, necessitating the calculation of sub-state energy levels for perturbation, including those of molecules 14N7 and 7Li3. Furthermore, the study expanded to the general Lie group for 3D rotations along three linearly independent axes, resulting in 12 distinct methods to achieve rotations in any arbitrary direction using these axes, yielding wave function with only one spin operator in each exponent. The ongoing research is now concentrated on generating NMR spectra for 14N7 in amino acids, furthering the understanding of nuclear spin dynamics in complex molecular systems.

In the first part of the thesis, the molecular structure and electronic properties of phospholipids at the single molecule level and also for a monolayer structure are investigated via ab initio calculations under different degrees of hydration. The focus of the study is on phosphatidylcholines, in particular dipalmitoylphosphatidylcholine (DPPC), which are the most abundant phospholipids in biological membranes. Upon hydration, the phospholipid shape into a sickle-like structure. The hydration dramatically alters the surface potential, dipole and quadrupole moments of the lipids, and probably guides the interactions of the lipids with other molecules and the communication between cells. The vibrational spectrum of DPPC and DPPC-water complexes are completely assigned and it is shown that water hydrating the lipid head groups enables efficient energy transfer across membrane leaflets on sub-picosecond time scales. Moreover, the vibrational modes and lifetimes of pure and hydrated DPPC lipids, at human body temperature, are estimated by performing ab initio molecular dynamics simulations. The vibrational modes of the water molecules close to the head group of DPPC are active in the frequency range between 0.5 - 55 THz, with a peak at 2.80 THz in the energy spectrum. The computed lifetimes for the high-frequency modes agree well with recent data measured at room temperature, where high-order phonon scattering is not negligible. The structure and auto-ionization of water at the water-phospholipid interface are investigated by ab initio molecular dynamics and ab initio Monte Carlo simulations using local density approximation and generalized gradient approximation for the exchange-correlation energy functional. Depending on the lipid head group, strongly enhanced ionization is observed, leading to dissociation of several water molecules into H+ and OH- per lipid. The results can shed light on the phenomena of the high proton conductivity along membranes that has been reported experimentally. In the second part of the thesis, Monte Carlo simulations of the lipid bilayer, on the basis of a coarse grained model, are performed to gain insight into the mechanical properties of planar lipid bilayers. By using a rescaling method, the Poisson's ratio is calculated for different phases. Additional information on the bending rigidity, determined from height fluctuations on the basis of the Helfrich Hamiltonian, allows for calculation of the Young's modulus for each phase. In addition, the free energy barrier for lipid flip-flop process in the fluid and gel phases are estimated. The main rate-limiting step to complete a flip-flop process is related to a free energy barrier that has to be crossed in order to reach the center of the bilayer. The free energy cost for performing a lipid flip-flop in the gel phase is found to be five times greater than in the fluid phase, demonstrating the rarity of such events in the gel phase. Moreover, an energy barrier is estimated for formation of transient water pores that often precedes lipid translocation events and accounts for the rate-limiting step of these pore-associated lipid translocation processes.