Path‐Integral Approaches to Non‐Adiabatic Dynamics

Saller, Maximilian A. C.; Runeson, Johan E.; Richardson, Jeremy O.

doi:10.1002/9781119417774.ch20

Cited by 10 publications

(7 citation statements)

References 81 publications

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“…This is a significant improvement compared to the MMST based NRPMD dynamics. 36,37,57 Compared to the previous NRPMD approach with the MMST formalism, the SM-NRPMD approach provides an additional advantage that the dynamics is in-variant with respect to the splitting between the stateindependent potential U 0 ( R) and the state-dependent potential. This is because the spin-mapping formalism explicitly enforces the total population to be 1, such that [ Î] s (u) = 1.…”

Section: Resultsmentioning

confidence: 99%

“…In addition, the mapping dynamics of the spin variables θ and ϕ are bounded on the Bloch sphere of radius r s , as opposed to un-bounded phase space variables (in the mapping oscillator phase space) in the MMST formalism (see Appendix A). Together, these advantages of the spin-mapping variables make it a more accurate and convenient mapping representation for developing nonadiabatic dynamics methods, 53,57 and we extend it to the NRPMD dynamics in this work.…”

Section: Resultsmentioning

confidence: 99%

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Non-Adiabatic Ring Polymer Molecular Dynamics with Spin Mapping Variables

Bossion,

Chowdhury,

Huo

2021

Preprint

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We present a new non-adiabatic ring polymer molecular dynamics (NRPMD) method based on the spin mapping formalism, which we refer to as the spin-mapping NRPMD (SM-NRPMD) approach. We derive the path-integral partition function expression using the spin coherent state basis for the electronic states and the ring polymer formalism for the nuclear degrees of freedom (DOFs). This partition function provides an efficient sampling of the quantum statistics. Using the basic property of the Stratonovich-Weyl transformation, we derive a Hamiltonian which we propose for the dynamical propagation of the coupled spin mapping variables and the nuclear ring polymer. The accuracy of the SM-NRPMD method is numerically demonstrated by computing nuclear position and population auto-correlation functions of non-adiabatic model systems. The results from SM-NRPMD agree very well with the numerically exact results. The main advantage of using the spin mapping variables over the harmonic oscillator mapping variables is numerically demonstrated, where the former provides nearly time-independent expectation values of physical observables for systems under thermal equilibrium, the latter can not preserve the initial quantum Boltzmann distribution. We also explicitly demonstrate that SM-NRPMD provides invariant dynamics upon various ways of partitioning the state-dependent and state-independent potentials.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Non-Adiabatic Ring Polymer Molecular Dynamics with Spin Mapping Variables

Bossion,

Chowdhury,

Huo

2021

Preprint

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“…28b means that the SW transform preserves the identity in the electronic Hilbert subspace, contrarily to the Wigner transform 47,48 of the identity operator in the MMST formalism 22,49 and hence, does not introduce any ambiguity of the identity expression. 50 Thus, Eq. 23 performs a mapping of an operator in the electronic subspace onto a phase space of continuous variables Ω as follows…”

Section: B Basic Properties Of the Stratonovich-weyl Transformmentioning

confidence: 98%

Non-adiabatic Dynamics using the Generators of the su(N) Lie Algebra

Bossion

Chowdhury

Huo

2022

Preprint

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We present the rigorous theoretical framework of the generalized spin mapping representation for non- adiabatic dynamics. This formalism is based on the generators of the su(N) Lie algebra to represent N discrete electronic states, thus preserving the size of the original Hilbert space in the state representation. We use the generalized spin coherent states representation and the Stratonovich-Weyl transform to describe these electronic spin-mapping variables in the continuous variables space. Wigner representation is used to de- scribe the nuclear degrees of freedom. Using the above representations, we derived an exact expression of the time-correlation function as well as the exact quantum Liouvillian. Making the linearization approximation, this exact Liouvillian is reduced to the Liouvillian of the recently proposed spin-Linearized Semi-Classical (spin-LSC) dynamics. These expressions lead to a self-consistent trajectory-based method to simulate non- adiabatic dynamics, which is based entirely on the generalized spin mapping formalism to treat the electronic states without the necessity of converting back to the cartesian mapping variables of the bosonic DOF (which are commonly referred to as the Meyer-Miller-Stock-Thoss mapping variables). The accuracy of this approach is tested with several challenging non-adiabatic model systems.

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“…30b means that the S-W transform preserves the identity in the electronic Hilbert subspace, contrarily to the Wigner transform 65,66 of the identity operator in the MMST formalism 22,67 and hence, does not introduce any ambiguity of the identity expression. 68 To conveniently evaluate any operator Â( R) under a S-W transformation, one starts by decomposing it on the GGM basis (Eqs 2-4) as follows…”

Section: A Spin Coherent Statesmentioning

confidence: 99%

Non-adiabatic Mapping Dynamics in the Phase Space of the SU(N) Lie Group

Bossion

Ying

Chowdhury

et al. 2022

Preprint

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We present the rigorous theoretical framework of the generalized spin mapping representation for non- adiabatic dynamics. This formalism is based on the generators of the su(N) Lie algebra to represent N discrete electronic states, thus preserving the size of the original Hilbert space in the state representation. The Stratonovich-Weyl transform is then used to map an operator in the Hilbert space to a continuous func- tion on the SU(N) Lie Group manifold which is a phase space of continuous variables. Wigner representation is used to describe the nuclear degrees of freedom. Using the above representations, we derived an exact expression of the time-correlation function as well as the exact quantum Liouvillian. Making the linearization approximation, this exact Liouvillian is reduced to the Liouvillian of the several recently proposed meth- ods. These expressions lead to a self-consistent trajectory-based method to simulate non-adiabatic dynamics, which is based entirely on the generalized spin mapping formalism to treat the electronic states without the necessity of converting back to the cartesian Meyer-Miller-Stock-Thoss mapping variables. We envision that the theoretical work presented in this work provides a rigorous and unified framework to formally derive non-adiabatic quantum dynamics approaches with continuous variables.

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Path‐Integral Approaches to Non‐Adiabatic Dynamics

Cited by 10 publications

References 81 publications

Non-Adiabatic Ring Polymer Molecular Dynamics with Spin Mapping Variables

Non-Adiabatic Ring Polymer Molecular Dynamics with Spin Mapping Variables

Non-adiabatic Dynamics using the Generators of the su(N) Lie Algebra

Non-adiabatic Mapping Dynamics in the Phase Space of the SU(N) Lie Group

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