Newton-X

Newton-X can simulate surface-hopping dynamics with the following programs and quantum-chemical methods:

The surface hopping probability depends on the values of the nonadiabatic couplings between electronic states.

Newton-X can either compute nonadiabatic couplings during the dynamics or read them from an interfaced third-party program. The computation of the couplings in Newton-X is done by finite differences, following the Hammes-Schiffer-Tully approach.[5] In this approach, the key quantity for computation of the surface hopping probability, the inner product between the nonadiabatic couplings (τ_LM) and the nuclear velocities (v) at time t, is given by

${\displaystyle {\boldsymbol {\tau }}_{LM}\cdot \mathbf {v} \approx {\frac {1}{4\Delta t}}\left(3S_{LM}(t)-3S_{ML}(t)-S_{LM}(t-\Delta t)+S_{ML}(t-\Delta t)\right)}$,

where the terms ${\displaystyle S_{LM}(t)\equiv \left\langle \Psi _{L}(t-\Delta t)\mid \Psi _{M}(t)\right\rangle }$ are wavefunction overlaps between states L and M in different time steps.

This method can be generally used for any electronic-structure method, provided that a configuration interaction representation of the electronic wavefunction can be worked out. In Newton-X, it is used with a number of quantum-chemical methods, including MCSCF (Multiconfigurational Self-Consistent Field), MRCI (Multi-Reference Configuration Interaction), CC2 (Coupled Cluster to Approximated Second Order), ADC(2) (Algebraic Diagrammatic Construction to Second Order), TDDFT (Time-Dependent Density Functional Theory), and TDA (Tamm-Dankov Approximation). In the case of MCSCF and MRCI, the configuration interaction coefficients are directly used for computation of couplings. For the other methods, the linear-response amplitudes are used as the coefficients of a configuration interaction wavefunction with single excitations.

Newton-X simulates absorption and emission spectra using the Nuclear Ensemble approach.[4] In this approach, an ensemble of nuclear geometries is built in the initial state and the transition energies and transition moments to the other states are computed for each geometry in the ensemble. A convolution of the results provides spectral widths and absolute intensities.

In the Nuclear Ensemble approach, the photoabsorption cross section for a molecule initially in the ground state and being excited with photoenergy E into N_fs final electronic states is given by

${\displaystyle \sigma (E)={\frac {\pi e^{2}\hbar }{2mc\epsilon _{0}n_{r}E}}\sum _{n}^{N_{fs}}{\frac {1}{N_{p}}}\sum _{l}^{N_{p}}\Delta E_{0,n}(\mathbf {R} _{l})f_{0,n}(\mathbf {R} _{l})g\left(E-\Delta E_{0,n}(\mathbf {R} _{l}),\delta \right)}$,

where e is the elementary charge, ħ is the reduced Planck constant, m is the electron mass, c is the speed of light, ε₀ is the vacuum permittivity, and n_r is the refractive index of the medium. The first summation runs over all target states and the second summation runs over all N_p points in the nuclear ensemble. Each point in the ensemble has nuclear geometry R_p, transition energy ΔE_0,n, and oscillator strength f_0,n (for a transition from the ground state into state n). g is a normalized Gaussian function with width δ given by

${\displaystyle g\left(E-\Delta E_{0,n},\delta \right)={\frac {1}{\left(2\pi (\delta /2)^{2}\right)^{1/2}}}exp\left({\frac {-(E-\Delta E_{0,n})^{2}}{2(\delta /2)^{2}}}\right)}$.

For emission, the differential emission rate is given by

${\displaystyle \Gamma (E)={\frac {e^{2}n_{r}^{3}}{2\pi \hbar mc^{3}\epsilon _{0}}}{\frac {1}{N_{p}}}\sum _{l}^{N_{p}}\Delta E_{1,0}(\mathbf {R} _{l})^{2}\left|f_{1,0}(\mathbf {R} _{l})\right|g\left(E-\Delta E_{1,0}(\mathbf {R} _{l}),\delta \right)}$.

In both absorption and emission, the nuclear ensemble can be sampled either from a dynamics simulation or from a Wigner distribution.

Starting from version 2.0, it is possible to use the nuclear ensemble approach to simulate steady and time-resolved photoelectron spectra.