Tunneling.h File Reference

#include "Matter.h"
#include <array>
#include <functional>
#include <limits>
#include <memory>
#include <string>
#include <vector>

Go to the source code of this file.

Classes
class	eonc::tunneling::Profile
	Energy along the path, interpolated as a monotone cubic (Fritsch and Carlson), flat at both ends because the ends of a band are minima. More...
struct	eonc::tunneling::Splitting
struct	eonc::tunneling::InstantonOptions
struct	eonc::tunneling::Instanton
struct	eonc::tunneling::RateInstantonOptions
struct	eonc::tunneling::RingSpectrum
	Spectrum of a closed ring's Hessian without forming it. More...
struct	eonc::tunneling::RateInstanton

Namespaces
namespace	eonc
	RAII resource manager for the ARTn C library with global synchronization.
namespace	eonc::tunneling

Typedefs
using	eonc::tunneling::BatchPotential
	V (eV) and dV/dq (eV / (amu^0.5 Angstrom)) at every point of q, all in one call so a potential can spread the beads over its calculators.
using	eonc::tunneling::BeadHessian = std::function<MatrixXd(long j, const VectorXd &q)>
	The mass-weighted Hessian d2V/dq2 at interior bead j (1..P-1).
using	eonc::tunneling::RingBeadHessian = std::function<MatrixXd(long j, const VectorXd &q)>
	The mass-weighted Hessian d2V/dq2 at ring bead j (0..N-1).

Functions
double	eonc::tunneling::massWeightedDistance (const Matter &a, const Matter &b)
	sqrt(sum_i m_i |b_i - a_i|^2) under the minimum image of a's cell.
std::vector< double >	eonc::tunneling::massWeightedPath (const std::vector< std::shared_ptr< Matter > > &band)
	Cumulative mass-weighted arc length at each image of a band.
double	eonc::tunneling::wellCurvature (const Profile &p, bool leftEnd)
	d2V/ds2 at one end of the path, in eV / (amu Angstrom^2), from a least squares fit of a s^2 + b s^3 to the points within half the barrier above that end.
double	eonc::tunneling::hbarOmega (double curvature)
	hbar omega in eV for a mass-weighted curvature.
double	eonc::tunneling::wkbAction (const Profile &p, double energy, int points=4001)
	(1/hbar) integral sqrt(2 (V(s) - E)) ds over the path where V > E.
Splitting	eonc::tunneling::wkbSplitting (const Profile &p, double hwReactant, double hwProduct)
	delta0 = (hbar omega / pi) exp(-S) with the Landau and Lifshitz prefactor.
Splitting	eonc::tunneling::bandSplitting (const std::vector< std::shared_ptr< Matter > > &band, double referenceEnergy)
	The splitting of a converged band, with the well frequencies from the band's curvature at each end.
std::vector< VectorXd >	eonc::tunneling::ringFromPath (const std::vector< VectorXd > &path, const std::vector< double > &energies, double betaHbar, long beads)
	Closed ring of beads samples of path whose imaginary-time period is betaHbar.
double	eonc::tunneling::wkbLogRateAlongPath (const Profile &profile, double beta, double hwReactant)
	ln(k), k in 1/time, for the one-dimensional thermal rate along profile.
double	eonc::tunneling::pathOmega (const MatrixXd &hessStart, const MatrixXd &hessEnd, const VectorXd &start, const VectorXd &end)
	omega along the straight line between the minima from the curvature of each well there, the larger of the two; in 1 / time.
Instanton	eonc::tunneling::optimizeInstanton (const VectorXd &start, const VectorXd &end, double betaHbar, std::vector< VectorXd > guess, const BatchPotential &potential, const InstantonOptions &options)
	Minimises the action from guess (P + 1 beads, ends at the minima, or empty for a tanh kink along the straight line).
void	eonc::tunneling::instantonSplitting (Instanton &inst, const BeadHessian &hessian, const MatrixXd &hessStart, const MatrixXd &hessEnd)
	Fills delta0, zeroMode and modeSeparation from the bead Hessians and the Hessians of the two minima.
double	eonc::tunneling::crossoverTemperature (const MatrixXd &hessSaddle)
	T_c = hbar omega_b / (2 pi kB) from the mass-weighted Hessian at the saddle, in K; throws when the Hessian has no negative eigenvalue.
double	eonc::tunneling::parabolicFactor (double temperature, double crossover)
	(pi T_c / T) / sin(pi T_c / T).
double	eonc::tunneling::harmonicTstLogRate (const MatrixXd &hessReactant, const MatrixXd &hessSaddle, double beta, double barrier, long rigidModes)
	ln(k) for classical harmonic transition-state theory, k in 1/time.
double	eonc::tunneling::quantumHarmonicTstLogRate (const MatrixXd &hessReactant, const MatrixXd &hessSaddle, double beta, double barrier, long rigidModes)
	ln(k) for quantum harmonic transition-state theory, k in 1/time: (1 / (2 pi beta hbar)) prod_r 2 sinh(beta hbar omega_r / 2) / prod'_s 2 sinh(beta hbar omega_s / 2) exp(-beta barrier), the zero-point and quantised partition functions of every bound mode; the saddle's unstable mode leaves the product.
RingSpectrum	eonc::tunneling::ringSpectrum (const std::vector< MatrixXd > &beadHessians, double c, const std::vector< VectorXd > &tau)
	The ring Hessian of bead Hessians beadHessians (d2V/dq2 at each of the N beads) and spring constant c, with the normalised direction tau (N beads) projected out through the determinant lemma det(J + tau tau^T) = det' J when J tau = 0.
RateInstanton	eonc::tunneling::optimizeRateInstanton (const VectorXd &saddle, const MatrixXd &hessSaddle, double beta, std::vector< VectorXd > guess, const BatchPotential &potential, const RateInstantonOptions &options)
	Finds the rate instanton at inverse temperature beta (1 / eV).
void	eonc::tunneling::instantonRate (RateInstanton &inst, const RingBeadHessian &hessian, const MatrixXd &hessReactant, double vReactant, const MatrixXd &hessSaddle=MatrixXd(), double vSaddle=0.0, long rigidModes=0, long denseLimit=4096)
	Fills the rate from the bead Hessians, the reactant minimum's Hessian and energy, and optionally the saddle's Hessian and energy for the classical comparison (pass an empty matrix to skip it).
double	eonc::tunneling::cyclicRingLogAbsDet (double c, const std::vector< MatrixXd > &diag)
	log|det| of the cyclic block-tridiagonal ring Hessian.
std::vector< VectorXd >	eonc::tunneling::cyclicRingSolve (double c, const std::vector< MatrixXd > &diag, const std::vector< VectorXd > &rhs)
	Solves that same cyclic ring Hessian.

Variables
constexpr double	eonc::tunneling::kHbar = 0.06465415129579072
	hbar in eV^0.5 amu^0.5 Angstrom, correctly rounded from the exact SI values (h = 6.62607015e-34 J s, e = 1.602176634e-19 C) and the CODATA 2022 dalton, 1.66053906892e-27 kg.
constexpr double	eonc::tunneling::kBoltzmann = 8.617333262145177e-5
	Boltzmann constant in eV / K, correctly rounded from the exact 1.380649e-23 J / K.
constexpr double	eonc::tunneling::kTimeUnitSeconds = 1.0180505717871193e-14
	One unit of time, sqrt(amu Angstrom^2 / eV), in seconds.