### Abstract

For a two state system coupled to each other by a nonzero matrix element Δ and to the bath arbitrarily, the generalized master equation is derived by applying the well-known projection operator techniques to the quantum Liouville equation. The time-dependent rate kernel is expressed by an infinite summation of the perturbative terms in Fourier-Laplace space. The Schwinger's stationary variation principle in Hilbert space is extended to Liouville space and then applied to the resummation of the rate kernel. The Cini-Fubini-type trial state vector in Liouville space is used to calculate the variational parameters. It is found that the resulting stationary value for the rate kernel in Fourier-Laplace space is given by the [N,N-1]-Padé approximants, in the N-dimensional subspace constructed by the N perturbatively expanded Liouville space vectors. The (first-order) simplest approximation satisfying the variational principle turns out to be equal to the [1,0] Padé approximant instead of the second-order Fermi golden rule expression. Two well-known approximations, the noninteracting blip approximation (NIBA) and nonadiabatic approximation, are discussed in the context of the [1,0] Padé approximants, based on the variational principle. A higher-order approximation, [2,1] Padé approximant, is also briefly discussed.

Original language | English |
---|---|

Pages (from-to) | 2654-2661 |

Number of pages | 8 |

Journal | Journal of Chemical Physics |

Volume | 106 |

Issue number | 7 |

Publication status | Published - 1997 Feb 15 |

### Fingerprint

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*Journal of Chemical Physics*,

*106*(7), 2654-2661.

**On the transition from nonadiabatic to adiabatic rate kernel : Schwinger's stationary variational principle and Padé approximation.** / Cho, Minhaeng; Silbey, Robert J.

Research output: Contribution to journal › Article

*Journal of Chemical Physics*, vol. 106, no. 7, pp. 2654-2661.

}

TY - JOUR

T1 - On the transition from nonadiabatic to adiabatic rate kernel

T2 - Schwinger's stationary variational principle and Padé approximation

AU - Cho, Minhaeng

AU - Silbey, Robert J.

PY - 1997/2/15

Y1 - 1997/2/15

N2 - For a two state system coupled to each other by a nonzero matrix element Δ and to the bath arbitrarily, the generalized master equation is derived by applying the well-known projection operator techniques to the quantum Liouville equation. The time-dependent rate kernel is expressed by an infinite summation of the perturbative terms in Fourier-Laplace space. The Schwinger's stationary variation principle in Hilbert space is extended to Liouville space and then applied to the resummation of the rate kernel. The Cini-Fubini-type trial state vector in Liouville space is used to calculate the variational parameters. It is found that the resulting stationary value for the rate kernel in Fourier-Laplace space is given by the [N,N-1]-Padé approximants, in the N-dimensional subspace constructed by the N perturbatively expanded Liouville space vectors. The (first-order) simplest approximation satisfying the variational principle turns out to be equal to the [1,0] Padé approximant instead of the second-order Fermi golden rule expression. Two well-known approximations, the noninteracting blip approximation (NIBA) and nonadiabatic approximation, are discussed in the context of the [1,0] Padé approximants, based on the variational principle. A higher-order approximation, [2,1] Padé approximant, is also briefly discussed.

AB - For a two state system coupled to each other by a nonzero matrix element Δ and to the bath arbitrarily, the generalized master equation is derived by applying the well-known projection operator techniques to the quantum Liouville equation. The time-dependent rate kernel is expressed by an infinite summation of the perturbative terms in Fourier-Laplace space. The Schwinger's stationary variation principle in Hilbert space is extended to Liouville space and then applied to the resummation of the rate kernel. The Cini-Fubini-type trial state vector in Liouville space is used to calculate the variational parameters. It is found that the resulting stationary value for the rate kernel in Fourier-Laplace space is given by the [N,N-1]-Padé approximants, in the N-dimensional subspace constructed by the N perturbatively expanded Liouville space vectors. The (first-order) simplest approximation satisfying the variational principle turns out to be equal to the [1,0] Padé approximant instead of the second-order Fermi golden rule expression. Two well-known approximations, the noninteracting blip approximation (NIBA) and nonadiabatic approximation, are discussed in the context of the [1,0] Padé approximants, based on the variational principle. A higher-order approximation, [2,1] Padé approximant, is also briefly discussed.

UR - http://www.scopus.com/inward/record.url?scp=0000676275&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000676275&partnerID=8YFLogxK

M3 - Article

VL - 106

SP - 2654

EP - 2661

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 7

ER -