Self-consistent phonons revisited. II. A general and efficient method for computing free energies and vibrational spectra of molecules and clusters

Abstract

The self-consistent phonons (SCP) method provides a consistent way to include anharmonic effects when treating a many-body quantum system at thermal equilibrium. The system is then described by an effective temperature-dependent harmonic Hamiltonian, which can be used to estimate the system's properties, such as its free energy or its vibrational spectrum. The numerical bottleneck of the method is the evaluation of Gaussian averages of the potential energy and its derivatives. Several algorithmic ideas/tricks are introduced to reduce the cost of such integration by orders of magnitude, e.g., relative to that of the previous implementation of the SCP approach by Calvo et al. [J. Chem. Phys. 133, 074303 (2010)]. One such algorithmic improvement is the replacement of standard Monte Carlo integration by quasi-Monte Carlo integration utilizing low-discrepancy sequences. The performance of the method is demonstrated on the calculation of vibrational frequencies of pyrene. It is then applied to compute the free energies of five isomers of water hexamer using the WHBB potential of Bowman and co-workers [J. Chem. Phys. 134, 094509 (2011)]. The present results predict the hexamer prism being thermodynamically most stable, with the free energy of the hexamer cage being about 0.2 kcal mol(-1) higher at all temperatures below T = 200 K.

Second-order structural phase transitions, free energy curvature, and temperature-dependent anharmonic phonons in the self-consistent harmonic approximation: Theory and stochastic implementation

“…This problem can be overcome by pathintegral molecular dynamics 32 , but the even greater computational cost that the method needs to incorporate the quantum character of atomic vibrations makes it challenging. To surmount these difficulties, several methods 3,4,[33][34][35][36][37][38] have been developed, mainly inspired by the self-consistent harmonic approximation (SCHA) devised by Hooton 39 . The main idea of the SCHA is to use a variational principle, the Gibbs-Bogoliubov (GB) principle, in order to approximate the free energy of the true ionic Hamiltonian with the free energy calculated with a trial harmonic density matrix for the same system.…”

Anharmonic stabilization and band gap renormalization in the perovskiteCsSnI3

“…We perform our ab initio investigation using a stochastic implementation of self-consistent phonon theory [24][25][26][27]. We show that the SnI 6 octahedra are stabilized against tilts and rotations by interacting with the renormalized vibrations of the Cs ions.…”

“…[26,27]) also minimizes F with respect to phonon eigenvectors and equilibrium ionic positions, but in the current study we keep these quantities fixed at their harmonic values. The reasons for performing this simplification are (i) for the high-symmetry B-α phase, many of the phonon eigenvectors (including the soft modes) are fixed by the crystal symmetry, and (ii) the large unit cells and low symmetry of the B-β and B-γ phases render a full minimization of F impractical [44], even after performing the symmetrization techniques of Ref.…”

“…Different approaches to this problem have been developed, including methods based on parametrizations and perturbative expansions of the PES [36][37][38], molecular dynamics [39,40], and self-consistent phonons [24][25][26][27]41]. Here we follow the selfconsistent phonon approach and calculate a fictitious free energy F(ω,T ) as…”

“…[27]. Then, as has been done previously for calculating free energies [26,27], absorption spectra [45], and magnetic spectroscopies [46], we evaluate the thermal averages of Eq. (3) stochastically from an ensemble of configurations with ionic displacements distributed according…”