Simulation Details

A.3.1 Properties of liquid water by ab initio molecular based on SCAN functional

All simulations analyzed in Chapter 3 were performed with the CPMD (Car & Parrinello, 1985) implemented in the Quantum ESPRESSO package (Giannozzi et al., 2017; Giannozzi et al., 2009). We set up a cubic cell with L=12.445 Å and periodic boundary conditions which consists of 64 H2O molecules. The electronic wavefunctions were expanded using a plane wave basis set with an energy cutoff of 85 Ry. The norm-conserving pseudopotentials (Vanderbilt, 1985) in the form of Hamann-Schlüter-Chiang-Vanderbilt (HSCV) were adopted for the O and H atoms. The functionals employed herein include the semi-local PBE GGA functional (Perdew, Burke, et al., 1996), PBE with the non-local vdW/dispersion interactions in the form of the density-dependent Tkatchenko-Scheffler (Tkatchenko & Scheffler, 2009) dispersion correction, i.e., PBE+vdW, and the SCAN meta-GGA functional (J. Sun et al., 2015). The parameters used for the Tkatchenko-Scheffler vdW were adopted from Ref. (DiStasio et al., 2014).

All AIMD simulations were performed in the NVT ensemble at 330 K by following Ref.

(DiStasio et al., 2014). The PBE, PBE+vdW, and SCAN systems were initially equilibrated for

136

about 8 ps, and then continued for 55, 54, and 50 ps, respectively, for data collection. The nuclear mass of deuterium, 2.014 amu, was used for each hydrogen atom, and the mass of

oxygen is 15.999 amu. The ionic temperature was controlled by a single Nosé-Hoover thermostat (Hoover, 1985; Martyna et al., 1992; Nosé, 1984) with a frequency of 60 THz. The CP equations of motion were integrated using the standard Verlet algorithm with a time step of 2.0 a.u.

(~0.0484 fs). To ensure an adiabatic separation between the nuclear and electronic degrees of freedom in the CP dynamics, an appropriate fictitious electronic mass µ should be chosen in CPMD. It was discussed in Refs. (Grossman et al., 2004) that compared to Born-Oppenheimer (BO) dynamics, simulations of liquid water with CP dynamics yield lower maximums and higher minimums in the radial distribution functions. In particular, a larger µ used in CP dynamics further softens the structure of liquid water. In this work, we set the fictitious electronic mass µ to be 100 a.u. with an electronic mass cutoff of 25 a.u. (Tassone, Mauri, & Car, 1994).

Additionally, we compare the NVT data to those obtained with CPMD simulations employing PBE and SCAN functionals in the NpT ensemble, and the parameters of NpT simulations are described in Ref. (M. Chen et al., 2017).

137 A.3.2 X-ray absorption spectra of liquid water

For the simulations on XAS by advanced ab initio methods analyzed in Sec. 4.2, we performed AIMD simulations to generate liquid-water trajectories using a modified version of the Quantum ESPRESSO software package (Giannozzi et al., 2017; Giannozzi et al., 2009). We simulated 128 water molecules in a periodic cubic cell with a cell length of 15.68 Å using the CPMD (Car & Parrinello, 1985) within the canonical ensemble. We employed norm-conserving pseudopotentials in the form of Troullier and Martins (Troullier & Martins, 1991) and set the kinetic-energy cutoff of the electronic wave functions as 71 Ry. We used a hierarchy of XC functionals, including PBE, PBE+vdW, and PBE0+vdW, as mentioned. The hybrid functional PBE0 with a mixing of 25% exact exchange (Perdew, Ernzerhof, et al., 1996) was evaluated in a linear-scaling manner by taking advantage of maximally localized Wannier functions (Wu et al., 2009). The ionic temperatures were controlled through Nosé-Hoover chain thermostats with a chain length of 4 for each ion (Hoover, 1985; Martyna et al., 1992; Nosé, 1984). All AIMD simulations were performed at 330 K, where an increase of 30 K has been found to mimic the NQEs in structural quantities such as the oxygen-oxygen radial distribution function in DFT-based simulations of liquid water (Morrone & Car, 2008). A time step of 4.0 a.u. and a fictitious electronic mass of 300 a.u. were chosen.

For the analyses on XAS signature of quantum nuclei in liquid water in Sec. 4.3, all calculations were performed in a supercell containing 128 water molecules under periodic boundary conditions. Molecular configurations of liquid water were extracted from classical MD and path integral (PI) MD simulations carried out with MB-pol at 270, 298, and 360 K, and were then used in calculations of the associated XAS cross sections using the enhanced static

COHSEX approximation (Kang & Hybertsen, 2010). All XAS spectra were calculated by

138

enforcing the same area from 533 to 546 eV as in the experimental lineshape and aligning the pre-edge features to the experimental value of 535 eV (W. Chen et al., 2010; Kong et al., 2012;

Z. Sun et al., 2017). All classical and PIMD simulations were carried out in the constant number of molecules-constant temperature-constant volume (NVT) ensemble at 270, 298, and 360 K and corresponding experimental densities for a system consisting of 128 H2O molecules in periodic boundary conditions. The equations of motion were propagated using the velocity-Verlet algorithm with a time step ∆t = 0.2 fs and the temperature was controlled via Nosé-Hoover chains of four thermostats coupled to each degree of freedom (Martyna et al., 1992). The PIMD simulations were carried out within the normal mode representation of Feynman’s ring polymers, which were discretized using P = 32 beads (Berne & Thirumalai, 1986). This setup was shown to provide an accurate representation of the quantum properties of liquid water in previous PIMD simulations of water with MB-pol (Medders et al., 2014; Reddy et al., 2017).

A.3.3 Properties of hydroxide and hydronium in liquid water

Simulations in Chapter 5 are generated by the Quantum ESPRESSO (Giannozzi et al., 2017; Giannozzi et al., 2009) software. We used a method (Wu et al., 2009) based on Wannier functions to compute efficiently exact exchange in PBE0 calculations, and evaluated self-consistently (DiStasio et al., 2014) the TS dispersion contribution. The pure water system is composed of 128 H2O. The hydronium system consists of 63 H2O with one excess proton (127 H atoms and 63 O atoms while the hydroxide system consists of 63 H2O with one hydroxide ion (127 H atoms and 64 O atoms). In order to reproduce the experimental density of liquid water at ambient conditions, the cubic cells used for ion and pure water simulations have the cell lengths being 12.4 and 15.7 Å, respectively. Only the gamma point was used to sample the Brillouin

139

zone of the supercell. The periodic boundary conditions were utilized with the energy cutoff of plane wave basis being 72 Ry. The Troullier-Martins (Hassanali et al., 2014) norm conserving pseudopotentials were employed.

We performed the CPMD (Car & Parrinello, 1985) with the standard Verlet algorithm to propagate nuclear and electronic degrees of freedom. We used a fictitious electronic mass of 150 a.u. to ensure the adiabatic separation between the nuclear and electronic degrees of freedom, and the mass pre-conditioning with a kinetic energy cutoff of 6 Ry was applied to all Fourier components of electronic wave functions (Tassone et al., 1994). All simulations were performed in the NVT ensemble at 330 K (DiStasio et al., 2014). The ionic temperature was controlled using the Nosé-Hoover chain thermostats (Martyna et al., 1992) with one Nosé-Hoover chain per atom and four thermostats in each chain. The time step was set to be 3.5 a.u. (~0.08 fs). The nuclear mass of deuterium (2.0135 amu) was set for each hydrogen atom in order to accelerate the convergence, while the nuclear mass of oxygen was set to 15.9995 amu. We generated 28, 45, and 32 ps trajectories for the hydronium systems using the PBE (Perdew, Burke, et al., 1996), PBE with the vdW interactions in the form of Tkatchenko and Scheffler (Tkatchenko &

Scheffler, 2009) (PBE-TS), and PBE-TS with a mixing of 25% exact exchange (Perdew,

Ernzerhof, et al., 1996) (PBE0-TS) functionals, respectively; we also generated 54, 55, and 38 ps for the hydroxide systems using the PBE, PBE-TS, and PBE0-TS functionals, respectively. For the pure liquid water system, we have trajectories of 14, 14, and 25 ps for PBE, PBE-TS, and PBE0-TS trajectories, respectively. We defined the H-bond within a cutoff of 3.5 Å for O-O distance and an H-O-O angle less than 30° (Luzar & Chandler, 1996b). We also used a cutoff of 1.24 Å for the O-H covalent bond.

140

A.3.4 Proton transfer in liquid water by ab initio molecular dynamics based on SCAN functional

All simulations were performed with the CPMD (Car & Parrinello, 1985) implemented in the Quantum ESPRESSO package (Giannozzi et al., 2017; Giannozzi et al., 2009). In order to reproduce the experimental density of liquid water at ambient conditions, we set up a cubic cell with L=12.445 Å. The hydronium system consists of 63 H2O with one excess proton (127 H atoms and 63 O atoms) whereas the hydroxide system consists of 63 H2O with one hydroxide ion (127 H atoms and 64 O atoms). The periodic boundary conditions were utilized, with the

electronic wavefunctions being expanded using a plane wave basis set with an energy cutoff of 85 Ry. The norm-conserving pseudopotentials (Vanderbilt, 1985) in the form of HSCV were adopted for the O and H atoms. The functionals employed herein include the semi-local PBE GGA functional (Perdew, Burke, et al., 1996) and the SCAN meta-GGA functional (J. Sun et al., 2015).

All AIMD simulations were performed in the NVT ensemble at 330 K (M. Chen et al., 2018; DiStasio et al., 2014; Zheng et al., 2018). For data collection, we generated 106- and 84-ps trajectories for the hydroxide systems using the PBE and SCAN functionals, respectively; we also generated 113- and 110-ps trajectories for the hydronium systems using the PBE and SCAN functionals, respectively. The nuclear mass of deuterium, 2.014 amu, was used for each

hydrogen atom, and the mass of oxygen is 15.999 amu. The ionic temperature was controlled by a single Nosé-Hoover thermostat (Hoover, 1985; Martyna et al., 1992; Nosé, 1984) with a frequency of 60 THz. The CP equations of motion were integrated using the standard Verlet algorithm with a time step of 2.0 a.u. (~0.0484 fs). To ensure an adiabatic separation between the nuclear and electronic degrees of freedom in the CP dynamics, we set the electronic fictitious

141

mass µ to be 100 a.u. with an electronic mass cutoff of 25 a.u. (DiStasio et al., 2014; Zheng et al., 2018).

142 APPENDIX B