Nonadiabatic coupling matrix

The current implementation of the nonadiabatic coupling is based on: Plasser, F.; Granucci, G.; Pittner, j.; Barbatti, M.; Persico, M.; Lischka. Surface hopping dynamics using a locally diabatic formalism: Charge transfer in the ethylene dimer cation and excited state dynamics in the 2-pyridone dimer. J. Chem. Phys. 2012, 137, 22A514.

The total time-dependent wave function \(\Psi(\mathbf{R}, t)\) can be expressed in terms of a linear combination of N adiabatic electronic eigenstates \(\phi_{i}(\mathbf{R}(t))\),

\[\Psi(\mathbf{R}, t) = \sum^{N}_{i=1} c_i(t)\phi_{i}(\mathbf{R}(t)) \quad \mathbf(1)\]

The time-dependent coefficients are propagated according to

\[\frac{dc_j(t)}{dt} = -i\hbar^2 c_j(t) E_j(t) - \sum^{N}_{i=1}c_i(t)\sigma_{ji}(t) \quad \mathbf(2)\]

where \(E_j(t)\) is the energy of the jth adiabatic state and \(\sigma_{ji}(t)\) the nonadiabatic matrix, which elements are given by the expression

\[\sigma_{ji}(t) = \langle \phi_{j}(\mathbf{R}(t)) \mid \frac{\partial}{\partial t} \mid \phi_{i}(\mathbf{R}(t)) \rangle \quad \mathbf(3)\]

that can be approximate using three consecutive molecular geometries

\[\sigma_{ji}(t) \approx \frac{1}{4 \Delta t} (3\mathbf{S}{ji}(t) - 3\mathbf{S}{ij}(t) - \mathbf{S}{ji}(t-\Delta t) + \mathbf{S}{ij}(t-\Delta t)) \quad \mathbf(4)\]

where \(\mathbf{S}_{ji}(t)\) is the overlap matrix between two consecutive time steps

\[\mathbf{S}{ij}(t) = \langle \phi{j}(\mathbf{R}(t-\Delta t)) \mid \phi_{i}(\mathbf{R}(t)) \rangle \quad \mathbf(5)\]

and the overlap matrix is calculated in terms of atomic orbitals

\[\mathbf{S}{ji}(t) = \sum{\mu} C^{*}{\mu i}(t) \sum{\nu} C_{\nu j}(t - \Delta t) \mathbf{S}_{\mu \nu}(t) \quad \mathbf(6)\]

Where :math:C_{mu i} are the Molecular orbital coefficients and \(\mathbf{S}_{\mu \nu}\) The atomic orbitals overlaps.

\[\mathbf{S}{\mu \nu}(\mathbf{R}(t), \mathbf{R}(t - \Delta t)) = \langle \chi{\mu}(\mathbf{R}(t)) \mid \chi_{\nu}(\mathbf{R}(t - \Delta t)\rangle \quad \mathbf(7)\]

Nonadiabatic coupling algorithm implementation

The figure belows shows schematically the workflow for calculating the Nonadiabatic coupling matrices from a molecular dynamic trajectory. The uppermost node represent a molecular dynamics trajectory that is subsequently divided in its components andfor each geometry the molecular orbitals are computed. These molecular orbitals are stored in a HDF5. binary file and subsequently calculations retrieve sets of three molecular orbitals that are use to calculate the nonadiabatic coupling matrix using equations 4 to 7. These coupling matrices are them feed to the PYXAID package to carry out nonadiabatic molecular dynamics.

The Overlap between primitives are calculated using the Obara-Saika recursive scheme and has been implemented using the C++ libint2 library for efficiency reasons. The libint2 library uses either OpenMP or C++ threads to distribute the integrals among the available CPUs. Also, all the heavy numerical processing is carried out by the highly optimized functions in NumPy.

The nonadiabaticCoupling package relies on QMWorks to run the Quantum mechanical simulations using the [CP2K]( package. Also, the noodles is used to schedule expensive numerical computations that are required to calculate the nonadiabatic coupling matrix.