Newton-X

Newton-X: A Package for Newtonian Dynamics Close to the Crossing Seam
	Snapshot of Newton-X main menu.
Developer(s)	M. Barbatti, G. Granucci, M. Ruckenbauer, F. Plasser, R. Crespo-Otero, J. Pittner, M. Persico, H. Lischka
Stable release	2.2
Written in	Perl, Fortran, C
Operating system	Linux
Website	www.newtonx.org

Last updated August 14, 2023

Newton-X^[1]^[2] is a general program for molecular dynamics simulations beyond the Born-Oppenheimer approximation. It has been primarily used for simulations of ultrafast processes (femtosecond to picosecond time scale) in photoexcited molecules. It has also been used for simulation of band envelops of absorption and emission spectra.

Newton-X uses the trajectory surface hopping method, a semi-classical approximation in which the nuclei are treated classically by Newtonian dynamics, while the electrons are treated as a quantum subsystem via a local approximation of the Time-dependent Schrödinger Equation. Nonadiabatic effects (the spread of the nuclear wave packet between several states) are recovered by a stochastic algorithm, which allows individual trajectories to change between different potential energy states during the dynamics.

Capabilities

Newton-X is designed as a platform to perform all steps of the nonadiabatic dynamics simulations, from the initial conditions generation, through trajectories computation, to the statistical analysis of the results. It works interfaced to a number of electronic structure programs available for computational chemistry, including Gaussian, Turbomole, Gamess, and Columbus. Its modular development allows to create new interfaces and integrate new methods. Users’ new developments are encouraged and are in due course included into the main branch of the program.

Nonadiabatic couplings, the central quantity in nonadiabatic simulations, can be either provided by a third-party program or computed by Newton-X. When computed by Newton-X, it is done with a numerical approximation based on overlap of electronic wavefunctions obtained in sequential time steps. A local diabatization method is also available to provide couplings in the case of weak nonadiabatic interactions.^[3]

Hybrid combination of methods is possible in Newton-X. Forces computed with different methods for different atomic subsets can be linearly combined to generate the final force driving the dynamics. These hybrid forces may, for instance, be combined into the popular electrostatic-embedding quantum-mechanical/molecular-mechanical method (QM/MM). Important options for QM/MM simulations, such as link atoms, boundaries, and thermostats are available as well.

As part of the initial conditions module, Newton-X can simulate absorption, emission, and photoelectron spectra, using the Nuclear Ensemble approach,^[4] which provides full spectral widths and absolute intensities.

Methods and Interfaces to Third-Party Programs

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

Third-Party Program	Methods
Columbus	MCSCF, MRCI
Turbomole	TDDFT, CC2, ADC(2)
Gaussian	MCSCF, TDDFT, TDA, CIS
Gamess	MCSCF

Nonadiabatic couplings

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

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

where the terms $S_{LM}(t)\equiv \left\langle \Psi _{L}(t-\Delta t)\mid \Psi _{M}(t)\right\rangle$ 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.

Spectrum Simulations

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

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

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

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

For emission, the differential emission rate is given by

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

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.

Development and credits

The development of Newton-X started in 2005 at the Institute for the Theoretical Chemistry of the University of Vienna. It was designed by Mario Barbatti in collaboration with Hans Lischka. The original code used and expanded routines written by Giovanni Granucci and Maurizio Persico from the University of Pisa.^[2]

A modulus for computation of nonadiabatic couplings based on finite differences of either MCSCF or MRCI wavefunctions was implemented by Jiri Pittner (J. Heyrovsky Institute)^[6] and later adapted to work with TDDFT.^[7] A modulus for QM/MM dynamics was developed by Matthias Ruckenbauer.^[8] Felix Plasser implemented the local diabatization method and dynamics based on CC2 and ADC(2).^[3] Rachel Crespo-Otero extended the TDDFT and TDA capabilities.^[3] An interface to Gamess was added by Aaron West and Theresa Windus (Iowa State University).^[9]

Mario Barbatti coordinates new program developments, their integration into the official version, and the Newton-X distribution.

Distribution and training

Newton-X is distributed free of charges for academic usage and with open source. The original paper^[2] describing the program had been cited 190 times by December 22, 2014, according to Google Scholar.

Newton-X counts with a comprehensive documentation and a public discussion forum. A tutorial is also available on line, showing how to use the main features of the program step-by-step. Examples of simulations are shown at a YouTube channel. The program itself is distributed with a collection of input and output files of several worked-out examples.

A number of workshops on nonadiabatic simulations using Newton-X have been organized in Vienna (2008), Rio de Janeiro (2009), Sao Carlos (2011), Chiang Mai (2011, 2015), and Jeddah (2014).^[10]

Program philosophy and architecture

A main concept guiding the Newton-X development is that the program should be simple to use, but still providing as many options as possible to customize the jobs. This is achieved by a series of input tools that guide the user through the program options, providing context-dependent variable values always that possible.

Newton-X is written as a combination of independent programs. The coordinated execution of these programs is done by drivers written in Perl, while the programs dealing with integration of the dynamics and other mathematical aspects are written in Fortran 90 and C. Memory is dynamically allocated and there are no formal limits for most of variables, such as number of atoms or states.

Newton-X works in a three-level parallelization: the first level is a trivial parallelization given by the Independent-Trajectories approach used by the program. Complete sets of input files are redundantly written to allow each trajectory to be executed independently. They can be easily merged for final analysis in a later step. In a second level, Newton-X takes advantage of the parallelization of the third-party programs with which it is interfaced. Thus, a Newton-X simulation using the interface with Gaussian program can be first distributed over a cluster in terms of independent trajectories and each trajectory runs parallelized version of Gaussian. In the third level, the coupling computations in Newton-X are parallelized.

Starting with version (1.3, 2013), Newton-X uses meta-codes to control the dynamics simulation behavior. Based on a series of initial instructions provided by the user, new codes are automatically written and executed on-the-fly. These codes allow, for instance, checking specific conditions to terminate the simulations.

Drawbacks

To keep a modular architecture for easy inclusion of new algorithms, Newton-X is organized as a series of independent programs connected by general program drivers. For this reason, a large amount of input/output is required during the program's execution, reducing its efficiency. When dynamics is based on ab initio methods, this is normally not a problem, as the time bottleneck is in the electronic structure calculation. Low efficiency due to input/output can, however, be relevant with semiempirical methods.

Other problems with the current implementation are the lack of parallelization of the code, especially of the couplings computation, and the restriction of the program to Linux systems.

Related Research Articles

In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. Due to the larger relative mass of a nucleus compared to an electron, the coordinates of the nuclei in a system are approximated as fixed, while the coordinates of the electrons are dynamic. The approach is named after Max Born and his 23-year-old graduate student J. Robert Oppenheimer, the latter of whom proposed it in 1927 during a period of intense ferment in the development of quantum mechanics.

Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.

In plasma physics, the particle-in-cell (PIC) method refers to a technique used to solve a certain class of partial differential equations. In this method, individual particles in a Lagrangian frame are tracked in continuous phase space, whereas moments of the distribution such as densities and currents are computed simultaneously on Eulerian (stationary) mesh points.

Verlet integration is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics. The algorithm was first used in 1791 by Jean Baptiste Delambre and has been rediscovered many times since then, most recently by Loup Verlet in the 1960s for use in molecular dynamics. It was also used by P. H. Cowell and A. C. C. Crommelin in 1909 to compute the orbit of Halley's Comet, and by Carl Størmer in 1907 to study the trajectories of electrical particles in a magnetic field . The Verlet integrator provides good numerical stability, as well as other properties that are important in physical systems such as time reversibility and preservation of the symplectic form on phase space, at no significant additional computational cost over the simple Euler method.

Vibronic coupling in a molecule involves the interaction between electronic and nuclear vibrational motion. The term "vibronic" originates from the combination of the terms "vibrational" and "electronic", denoting the idea that in a molecule, vibrational and electronic interactions are interrelated and influence each other. The magnitude of vibronic coupling reflects the degree of such interrelation.

The Born–Huang approximation is an approximation closely related to the Born–Oppenheimer approximation. It takes into account diagonal nonadiabatic effects in the electronic Hamiltonian better than the Born–Oppenheimer approximation. Despite the addition of correction terms, the electronic states remain uncoupled under the Born–Huang approximation, making it an adiabatic approximation.

In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, the Lagrangian, which may contain all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories.

In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten.

Car–Parrinello molecular dynamics or CPMD refers to either a method used in molecular dynamics or the computational chemistry software package used to implement this method.

The Gross–Pitaevskii equation describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.

Stokesian dynamics is a solution technique for the Langevin equation, which is the relevant form of Newton's 2nd law for a Brownian particle. The method treats the suspended particles in a discrete sense while the continuum approximation remains valid for the surrounding fluid, i.e., the suspended particles are generally assumed to be significantly larger than the molecules of the solvent. The particles then interact through hydrodynamic forces transmitted via the continuum fluid, and when the particle Reynolds number is small, these forces are determined through the linear Stokes equations. In addition, the method can also resolve non-hydrodynamic forces, such as Brownian forces, arising from the fluctuating motion of the fluid, and interparticle or external forces. Stokesian Dynamics can thus be applied to a variety of problems, including sedimentation, diffusion and rheology, and it aims to provide the same level of understanding for multiphase particulate systems as molecular dynamics does for statistical properties of matter. For $rigid particles of radius suspended in an incompressible Newtonian fluid of viscosity and density, the motion of the fluid is governed by the Navier-Stokes equations, while the motion of the particles is described by the coupled equation of motion:$

In computational chemistry, a constraint algorithm is a method for satisfying the Newtonian motion of a rigid body which consists of mass points. A restraint algorithm is used to ensure that the distance between mass points is maintained. The general steps involved are: (i) choose novel unconstrained coordinates, (ii) introduce explicit constraint forces, (iii) minimize constraint forces implicitly by the technique of Lagrange multipliers or projection methods.

In numerical analysis, Broyden's method is a quasi-Newton method for finding roots in $k$ variables. It was originally described by C. G. Broyden in 1965.

Differential dynamic programming (DDP) is an optimal control algorithm of the trajectory optimization class. The algorithm was introduced in 1966 by Mayne and subsequently analysed in Jacobson and Mayne's eponymous book. The algorithm uses locally-quadratic models of the dynamics and cost functions, and displays quadratic convergence. It is closely related to Pantoja's step-wise Newton's method.

The cluster-expansion approach is a technique in quantum mechanics that systematically truncates the BBGKY hierarchy problem that arises when quantum dynamics of interacting systems is solved. This method is well suited for producing a closed set of numerically computable equations that can be applied to analyze a great variety of many-body and/or quantum-optical problems. For example, it is widely applied in semiconductor quantum optics and it can be applied to generalize the semiconductor Bloch equations and semiconductor luminescence equations.

Surface hopping is a mixed quantum-classical technique that incorporates quantum mechanical effects into molecular dynamics simulations. Traditional molecular dynamics assume the Born-Oppenheimer approximation, where the lighter electrons adjust instantaneously to the motion of the nuclei. Though the Born-Oppenheimer approximation is applicable to a wide range of problems, there are several applications, such as photoexcited dynamics, electron transfer, and surface chemistry where this approximation falls apart. Surface hopping partially incorporates the non-adiabatic effects by including excited adiabatic surfaces in the calculations, and allowing for 'hops' between these surfaces, subject to certain criteria.

Hans Lischka is an Austrian computational theoretical chemist specialized on development and application of multireference methods for the study of molecular excited states. He is the main developer of the software package Columbus for ab initio multireference calculations and co-developer of the Newton-X program.

<span class="mw-page-title-main">Mixed quantum-classical dynamics</span> Computational chemistry methods to simulate non-adiabatic processes

Mixed quantum-classical (MQC) dynamics is a class of computational theoretical chemistry methods tailored to simulate non-adiabatic (NA) processes in molecular and supramolecular chemistry. Such methods are characterized by:

Propagation of nuclear dynamics through classical trajectories;
Propagation of the electrons through quantum methods;
A feedback algorithm between the electronic and nuclear subsystems to recover nonadiabatic information.

Mario Barbatti is a Brazilian physicist, computational theoretical chemist, and writer. He is specialized in the development and application of mixed quantum-classical dynamics for the study of molecular excited states. He is also the leading developer of the Newton-X software package for dynamics simulations. Mario Barbatti held an A*Midex Chair of Excellence at the Aix Marseille University between 2015 and 2019, where he is a professor since 2015.

<span class="mw-page-title-main">Nuclear ensemble approach</span> Semiclassical approach for molecular spectrum simulations.

The Nuclear Ensemble Approach (NEA) is a general method for simulations of diverse types of molecular spectra. It works by sampling an ensemble of molecular conformations in the source state, computing the transition probabilities to the target states for each of these geometries, and performing a sum over all these transitions convoluted with shape function. The result is an incoherent spectrum containing absolute band shapes through inhomogeneous broadening.

References

↑ Barbatti, Mario; Ruckenbauer, Matthias; Plasser, Felix; Pittner, Jiri; Granucci, Giovanni; Persico, Maurizio; Lischka, Hans (January 2014). "Newton-X: a surface-hopping program for nonadiabatic molecular dynamics". Wiley Interdisciplinary Reviews: Computational Molecular Science. 4 (1): 26–33. doi:10.1002/wcms.1158.
1 2 3 Barbatti, Mario; Granucci, Giovanni; Persico, Maurizio; Ruckenbauer, Matthias; Vazdar, Mario; Eckert-Maksić, Mirjana; Lischka, Hans (August 2007). "The on-the-fly surface-hopping program system Newton-X: Application to ab initio simulation of the nonadiabatic photodynamics of benchmark systems". Journal of Photochemistry and Photobiology A: Chemistry. 190 (2–3): 228–240. doi:10.1016/j.jphotochem.2006.12.008.
1 2 3 Plasser, Felix; Granucci, Giovanni; Pittner, Jiri; Barbatti, Mario; Persico, Maurizio; Lischka, Hans (2012). "Surface hopping dynamics using a locally diabatic formalism: Charge transfer in the ethylene dimer cation and excited state dynamics in the 2-pyridone dimer". The Journal of Chemical Physics. 137 (22): 22A514. Bibcode:2012JChPh.137vA514P. doi: 10.1063/1.4738960 .
1 2 Crespo-Otero, Rachel; Barbatti, Mario (9 June 2012). "Spectrum simulation and decomposition with nuclear ensemble: formal derivation and application to benzene, furan and 2-phenylfuran". Theoretical Chemistry Accounts. 131 (6). doi:10.1007/s00214-012-1237-4.
↑ Hammes-Schiffer, Sharon; Tully, John C. (1994). "Proton transfer in solution: Molecular dynamics with quantum transitions". The Journal of Chemical Physics. 101 (6): 4657. Bibcode:1994JChPh.101.4657H. doi:10.1063/1.467455.
↑ Pittner, Jiri; Lischka, Hans; Barbatti, Mario (February 2009). "Optimization of mixed quantum-classical dynamics: Time-derivative coupling terms and selected couplings". Chemical Physics. 356 (1–3): 147–152. Bibcode:2009CP....356..147P. doi:10.1016/j.chemphys.2008.10.013.
↑ Barbatti, Mario; Pittner, Jiří; Pederzoli, Marek; Werner, Ute; Mitrić, Roland; Bonačić-Koutecký, Vlasta; Lischka, Hans (September 2010). "Non-adiabatic dynamics of pyrrole: Dependence of deactivation mechanisms on the excitation energy". Chemical Physics. 375 (1): 26–34. Bibcode:2010CP....375...26B. doi:10.1016/j.chemphys.2010.07.014.
↑ Ruckenbauer, Matthias; Barbatti, Mario; Müller, Thomas; Lischka, Hans (July 2010). "Nonadiabatic Excited-State Dynamics with Hybrid ab Initio Quantum-Mechanical/Molecular-Mechanical Methods: Solvation of the Pentadieniminium Cation in Apolar Media". The Journal of Physical Chemistry A. 114 (25): 6757–6765. Bibcode:2010JPCA..114.6757R. doi:10.1021/jp103101t.
↑ West, Aaron C.; Barbatti, Mario; Lischka, Hans; Windus, Theresa L. (July 2014). "Nonadiabatic dynamics study of methaniminium with ORMAS: Challenges of incomplete active spaces in dynamics simulations". Computational and Theoretical Chemistry. 1040–1041: 158–166. doi:10.1016/j.comptc.2014.03.015.
↑ "Newton-X webpage".

External links

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Barbatti, Mario; Ruckenbauer, Matthias; Plasser, Felix; Pittner, Jiri; Granucci, Giovanni; Persico, Maurizio; Lischka, Hans (January 2014). "Newton-X: a surface-hopping program for nonadiabatic molecular dynamics". Wiley Interdisciplinary Reviews: Computational Molecular Science. 4 (1): 26–33. doi:10.1002/wcms.1158.

[JPPA-2] 1 2 3 Barbatti, Mario; Granucci, Giovanni; Persico, Maurizio; Ruckenbauer, Matthias; Vazdar, Mario; Eckert-Maksić, Mirjana; Lischka, Hans (August 2007). "The on-the-fly surface-hopping program system Newton-X: Application to ab initio simulation of the nonadiabatic photodynamics of benchmark systems". Journal of Photochemistry and Photobiology A: Chemistry. 190 (2–3): 228–240. doi:10.1016/j.jphotochem.2006.12.008.

[LocDiab-3] 1 2 3 Plasser, Felix; Granucci, Giovanni; Pittner, Jiri; Barbatti, Mario; Persico, Maurizio; Lischka, Hans (2012). "Surface hopping dynamics using a locally diabatic formalism: Charge transfer in the ethylene dimer cation and excited state dynamics in the 2-pyridone dimer". The Journal of Chemical Physics. 137 (22): 22A514. Bibcode:2012JChPh.137vA514P. doi: 10.1063/1.4738960 .

[ReferenceA-4] 1 2 Crespo-Otero, Rachel; Barbatti, Mario (9 June 2012). "Spectrum simulation and decomposition with nuclear ensemble: formal derivation and application to benzene, furan and 2-phenylfuran". Theoretical Chemistry Accounts. 131 (6). doi:10.1007/s00214-012-1237-4.

[HST-5] Hammes-Schiffer, Sharon; Tully, John C. (1994). "Proton transfer in solution: Molecular dynamics with quantum transitions". The Journal of Chemical Physics. 101 (6): 4657. Bibcode:1994JChPh.101.4657H. doi:10.1063/1.467455.

[6] Pittner, Jiri; Lischka, Hans; Barbatti, Mario (February 2009). "Optimization of mixed quantum-classical dynamics: Time-derivative coupling terms and selected couplings". Chemical Physics. 356 (1–3): 147–152. Bibcode:2009CP....356..147P. doi:10.1016/j.chemphys.2008.10.013.

[7] Barbatti, Mario; Pittner, Jiří; Pederzoli, Marek; Werner, Ute; Mitrić, Roland; Bonačić-Koutecký, Vlasta; Lischka, Hans (September 2010). "Non-adiabatic dynamics of pyrrole: Dependence of deactivation mechanisms on the excitation energy". Chemical Physics. 375 (1): 26–34. Bibcode:2010CP....375...26B. doi:10.1016/j.chemphys.2010.07.014.

[8] Ruckenbauer, Matthias; Barbatti, Mario; Müller, Thomas; Lischka, Hans (July 2010). "Nonadiabatic Excited-State Dynamics with Hybrid ab Initio Quantum-Mechanical/Molecular-Mechanical Methods: Solvation of the Pentadieniminium Cation in Apolar Media". The Journal of Physical Chemistry A. 114 (25): 6757–6765. Bibcode:2010JPCA..114.6757R. doi:10.1021/jp103101t.

[9] West, Aaron C.; Barbatti, Mario; Lischka, Hans; Windus, Theresa L. (July 2014). "Nonadiabatic dynamics study of methaniminium with ORMAS: Challenges of incomplete active spaces in dynamics simulations". Computational and Theoretical Chemistry. 1040–1041: 158–166. doi:10.1016/j.comptc.2014.03.015.

[10] "Newton-X webpage".

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]