### Abstract

We have examined in detail the construction of state-dependent dynamic potentials (SDPs) based on an exact separation scheme of the wavefunction as a product of coupled conditional and marginal probability amplitudes. For particles of equal mass and for rather general forms of the kinetic energy operator in curvilinear coordinates, it is proven that the SDP becomes the expectation value of the total hamiltonian with respect to the conditional amplitude obtained from an assumed real wavefunction. The use of these SDPs for the description of localized wavefunctions is also discussed. As an example we have undertaken the analysis of SDPs arising from various separation schemes of Hylleraas coordinates for two simple wavefunctions approximating the ground state of helium. The separation of s r_{1} + r_{2}, t r_{2} r_{1} and r_{12} leads to SDPs exhibiting minima in the regions where localization of the wavefunction takes place. The possibility of an adiabatic breakup, Born-Oppenheimer type, of the problem is also considered. Our results show that strong "non-adiabaticity" is a distinct feature of all the separations we considered. The separation of r_{1} is treated too as an example of a non-collective variable and the corresponding SDP resembles an effective Hartree-like one-electron potential. We finally sketch some possible applications of the present approach for the analysis of localization and non-adiabaticity in more complex situations in atomic and molecular cases.

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

Pages (from-to) | 159-172 |

Number of pages | 14 |

Journal | Chemical Physics |

Volume | 112 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 1 1987 |

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### ASJC Scopus subject areas

- Physical and Theoretical Chemistry
- Spectroscopy
- Atomic and Molecular Physics, and Optics

### Cite this

*Chemical Physics*,

*112*(2), 159-172. https://doi.org/10.1016/0301-0104(87)80158-1

**Dynamic coordinate separation and state-dependent potentials. An analysis of localization in helium.** / Mujica, Vladimiro; Li, Yin; Goscinski, Osvaldo.

Research output: Contribution to journal › Article

*Chemical Physics*, vol. 112, no. 2, pp. 159-172. https://doi.org/10.1016/0301-0104(87)80158-1

}

TY - JOUR

T1 - Dynamic coordinate separation and state-dependent potentials. An analysis of localization in helium

AU - Mujica, Vladimiro

AU - Li, Yin

AU - Goscinski, Osvaldo

PY - 1987/4/1

Y1 - 1987/4/1

N2 - We have examined in detail the construction of state-dependent dynamic potentials (SDPs) based on an exact separation scheme of the wavefunction as a product of coupled conditional and marginal probability amplitudes. For particles of equal mass and for rather general forms of the kinetic energy operator in curvilinear coordinates, it is proven that the SDP becomes the expectation value of the total hamiltonian with respect to the conditional amplitude obtained from an assumed real wavefunction. The use of these SDPs for the description of localized wavefunctions is also discussed. As an example we have undertaken the analysis of SDPs arising from various separation schemes of Hylleraas coordinates for two simple wavefunctions approximating the ground state of helium. The separation of s r1 + r2, t r2 r1 and r12 leads to SDPs exhibiting minima in the regions where localization of the wavefunction takes place. The possibility of an adiabatic breakup, Born-Oppenheimer type, of the problem is also considered. Our results show that strong "non-adiabaticity" is a distinct feature of all the separations we considered. The separation of r1 is treated too as an example of a non-collective variable and the corresponding SDP resembles an effective Hartree-like one-electron potential. We finally sketch some possible applications of the present approach for the analysis of localization and non-adiabaticity in more complex situations in atomic and molecular cases.

AB - We have examined in detail the construction of state-dependent dynamic potentials (SDPs) based on an exact separation scheme of the wavefunction as a product of coupled conditional and marginal probability amplitudes. For particles of equal mass and for rather general forms of the kinetic energy operator in curvilinear coordinates, it is proven that the SDP becomes the expectation value of the total hamiltonian with respect to the conditional amplitude obtained from an assumed real wavefunction. The use of these SDPs for the description of localized wavefunctions is also discussed. As an example we have undertaken the analysis of SDPs arising from various separation schemes of Hylleraas coordinates for two simple wavefunctions approximating the ground state of helium. The separation of s r1 + r2, t r2 r1 and r12 leads to SDPs exhibiting minima in the regions where localization of the wavefunction takes place. The possibility of an adiabatic breakup, Born-Oppenheimer type, of the problem is also considered. Our results show that strong "non-adiabaticity" is a distinct feature of all the separations we considered. The separation of r1 is treated too as an example of a non-collective variable and the corresponding SDP resembles an effective Hartree-like one-electron potential. We finally sketch some possible applications of the present approach for the analysis of localization and non-adiabaticity in more complex situations in atomic and molecular cases.

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UR - http://www.scopus.com/inward/citedby.url?scp=7044261621&partnerID=8YFLogxK

U2 - 10.1016/0301-0104(87)80158-1

DO - 10.1016/0301-0104(87)80158-1

M3 - Article

VL - 112

SP - 159

EP - 172

JO - Chemical Physics

JF - Chemical Physics

SN - 0301-0104

IS - 2

ER -