Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2207.09377 · Physical Review Research

Solutions of the Schrödinger equation for anisotropic dipole-dipole interaction plus van der Waals interaction

Ruijie Du, Ran Qi, Peng Zhang

Published May 19, 2025• 2 citations
arXivHTMLPDF

Abstract

By generalizing Gao's approach [] for solving the Schrödinger equation for an isotropic van der Waals (vdW) potential to the systems with a multiscale anisotropic long-range interaction, we derive the solutions for the Schrödinger equation for an anisotropic dipole-dipole interaction plus an isotropic attractive vdW potential, i.e., C d ( 1 − 3 cos 2 θ ) / r 3 − C 6 / r 6 , which is projected to the subspace with angular momentum l ≤ l cut , with l cut being an arbitrary angular-momentum cutoff. Here θ is the polar angle of the coordinate r and r = | r | . The asymptotic behaviors of these solutions for r → 0 and r → ∞ are obtained. The generalization of our approach for the case of an anisotropic vdW potential is also discussed. These results can be used in the research of collisions and chemical reactions between ultracold polar molecules in a static electric field. Our approach to derive the solutions can be applied to the systems with a general long-range potential ∑ λ = 2 λ max V λ ( θ , φ ) / r λ , with φ being the azimuthal angle of r , and thus can be used in collision problems between various kinds of molecules, ions, or atoms (e.g., the collisions between ultracold dipolar atoms).

DOI
10.1103/physrevresearch.7.023162

Related papers

Dispersion interactions and reactive collisions of ultracold polar molecules

Svetlana Kotochigova

Progress in ultracold experiments with polar molecules requires a clear understanding of their interactions and reactivity at ultralow collisional energies. Two important theoretical steps in this process are the characterization of interaction potentials between molecules and the modeling of the reactive scattering mechanism. Here, we report on the ab initio calculation of isotropic and anisotropic van der Waals interaction potentials for polar KRb and RbCs colliding with each other or with ultracold atoms. Based on these potentials and two short-range scattering parameters, we then develop a single-channel scattering model with flexible boundary conditions. Our calculations show that, at low temperatures (and in the absence of an external electric field), the reaction rates between molecules or between molecules and atoms have a resonant character as a function of the short-range parameters. We also find that both the isotropic and anisotropic van der Waals coefficients have significant contributions from dipole coupling to excited electronic states. Their values can differ dramatically from those solely obtained from the permanent dipole moment. A comparison with recently obtained reaction rates of fermionic 40 K 87 Rb shows that the experimental data cannot be explained by a model where the short-range scattering parameters are independent of the relative orbital angular momentum or partial wave.

arXiv:1003.2672•New Journal of Physics•Jul 29, 2010•74 citations

van der Waals coefficients for systems with ultracold polar alkali-metal molecules

Piotr Szymon Żuchowski, Maciej B. Kosicki, Monika Kodrycka +1 more

A systematic study of the leading isotropic van der Waals coefficients for the alkali-metal atom+molecule and molecule+molecule systems is presented. Dipole moments and static and dynamic dipole polarizabilities are calculated employing high-level quantum chemistry calculations. The dispersion, induction, and rotational parts of the isotropic van der Waals coefficient are evaluated. The known van der Waals coefficients are then used to derive characteristics essential for simple models of the collisions involving the corresponding ultracold polar molecules.

arXiv:1302.2746•Physical Review A•Feb 19, 2013•52 citations

Quantum reactive scattering in the long-range ion-dipole potential

Tomasz Wasak, Zbigniew Idziaszek

An ion and a polar molecule interact by an anisotropic ion-dipole potential scaling as $\ensuremath{-}\ensuremath{\alpha}\phantom{\rule{0.16em}{0ex}}cos(\ensuremath{\theta})/{r}^{2}$ at large distances. Due to its long-range character, it modifies the properties of angular wave functions, which are no longer given by spherical harmonics. In addition, an effective centrifugal potential in the radial equation can become attractive for low angular momenta. In this paper, we develop a general framework for an ion-dipole reactive scattering, focusing on the regime of large $\ensuremath{\alpha}$. We introduce modified spherical harmonics as solutions of the angular part of the Schr\"odinger equation and derive several useful approximations in the limit of large $\ensuremath{\alpha}$. We present a formula for the scattering amplitude expressed in terms of the modified spherical harmonics and we derive expressions for the elastic and reactive collision rates. The solutions of the radial equation are given by Bessel functions, and we analyze their behavior in two distinct regimes corresponding, basically, to attractive and repulsive long-range centrifugal potentials. Finally, we study reactive collisions in the universal regime, where the short-range probability of loss or reaction is equal to unity.

arXiv:2007.12460•Physical Review A•Feb 23, 2021•1 citation

Exact solutions and quantum defect theory for van der Waals potentials in ultracold molecular systems

Jianwen Jie, Shiyou Chen, Yue Chen +1 more

Abstract In this paper, we have provided exact two-body solutions to the 2D and 3D Schrödinger equations with isotropic van der Waals potentials of the form ± 1 / r 6 . Based on these solutions, we developed an analytical quantum defect theory (QDT) applicable to both quasi-2D and 3D geometries, and applied it to study the scattering properties and bound-state spectra of ultracold polar molecules confined in these geometries. Interestingly, we find that in the attractive (repulsive) van der Waals potential case, the short-range interaction can be effectively modeled by an infinite square barrier (finite square well), which leads to narrow and dense (broad and sparse) resonance structures in the quantum defect parameter. In the quasi-2D attractive case, shape resonances can appear in an ordered fashion across different partial waves, characterized by sharp phase jumps as the scattering energy is varied. Furthermore, the low-energy analytical expansions derived from QDT show excellent agreement with the exact numerical results, validating the accuracy and usefulness of our analytical approach in describing two-body physics governed by long-range van der Waals interactions.

arXiv:2202.08694•Journal of Physics B Atomic Molecular and Optical Physics•Sep 2, 2025•0 citations

Four-body long-range interactions between ultracold weakly-bound diatomic molecules

Maxence Lepers, Goulven Quéméner, Eliane Luc-Koenig +1 more

Abstract Using the multipolar expansion of electrostatic and magnetostatic potential energies, we characterize the long-range interactions between two weakly-bound diatomic molecules, taking as an example the paramagnetic Er 2 Feshbach molecules which were produced recently. Since inside each molecule individual atoms conserve their identity, the intermolecular potential energy can be expanded as the sum of pairwise atomic potential energies. In the case of Er 2 Feshbach molecules, we show that the interaction between atomic magnetic dipoles gives rise to the usual term of the multipolar expansion, where R is the intermolecular distance, but also to additional terms scaling as and so on. Those terms are due to the interaction between effective molecular multipole moments, and are strongly anisotropic with respect to the orientation of the molecules. Similarly, the atomic pairwise van der Waals interaction results in ... terms in the intermolecular potential energy. By calculating the reduced electric-quadrupole moment of erbium ground level a.u., we also demonstrate that the electric–quadrupole interaction energy is negligible with respect to the magnetic dipole and van der Waals interaction energies. The general formalism presented in this article can be applied to calculate the long-range potential energy between arbitrary charge distributions composed of almost free subsystems.

arXiv:1508.06066•Journal of Physics B Atomic Molecular and Optical Physics•Nov 24, 2015•6 citations

Quantum defect theory for the van der Waals plus dipole-dipole interaction

Gao‐Ren Wang, Ting Xie, Yin Hui Huang +2 more

We investigate the scattering dynamics governed by the long-range van der Waals plus dipole-dipole interaction potential, $\ensuremath{-}{C}_{6}/{R}^{6}\ensuremath{-}{C}_{3}/{R}^{3}$, which describes the long-range interaction between two polar molecules in an electric field. In the spirit of quantum defect theory, a set of parameters which are nearly constants in the threshold regime is defined to characterize the scattering process. Using appropriate boundary conditions for the scattering wave functions and relevant parameters, we explore the quantum reflection by and quantum tunneling through the long-range potential. As a sample application, the reactive collision rates of ${}^{40}$K${}^{87}$Rb $+$ ${}^{40}$K${}^{87}$Rb are calculated.

Physical Review A•Dec 14, 2012•15 citations
By generalizing Gao's approach [] for solving the Schrödinger equation for an isotropic van der Waals (vdW) potential to the systems with a multiscale anisotropic long-range interaction, we derive the solutions for the Schrödinger equation for an anisotropic dipole-dipole interaction plus an isotropic attractive vdW potential, i.e., C d ( 1 − 3 cos 2 θ ) / r 3 − C 6 / r 6 , which is projected to the subspace with angular momentum l ≤ l cut , with l cut being an arbitrary angular-momentum cutoff. Here θ is the polar angle of the coordinate r and r = | r | . The asymptotic behaviors of these solutions for r → 0 and r → ∞ are obtained. The generalization of our approach for the case of an anisotropic vdW potential is also discussed. These results can be used in the research of collisions and chemical reactions between ultracold polar molecules in a static electric field. Our approach to derive the solutions can be applied to the systems with a general long-range potential ∑ λ = 2 λ max V λ ( θ , φ ) / r λ , with φ being the azimuthal angle of r , and thus can be used in collision problems between various kinds of molecules, ions, or atoms (e.g., the collisions between ultracold dipolar atoms).
Progress in ultracold experiments with polar molecules requires a clear understanding of their interactions and reactivity at ultralow collisional energies. Two important theoretical steps in this process are the characterization of interaction potentials between molecules and the modeling of the reactive scattering mechanism. Here, we report on the ab initio calculation of isotropic and anisotropic van der Waals interaction potentials for polar KRb and RbCs colliding with each other or with ultracold atoms. Based on these potentials and two short-range scattering parameters, we then develop a single-channel scattering model with flexible boundary conditions. Our calculations show that, at low temperatures (and in the absence of an external electric field), the reaction rates between molecules or between molecules and atoms have a resonant character as a function of the short-range parameters. We also find that both the isotropic and anisotropic van der Waals coefficients have significant contributions from dipole coupling to excited electronic states. Their values can differ dramatically from those solely obtained from the permanent dipole moment. A comparison with recently obtained reaction rates of fermionic 40 K 87 Rb shows that the experimental data cannot be explained by a model where the short-range scattering parameters are independent of the relative orbital angular momentum or partial wave.
A systematic study of the leading isotropic van der Waals coefficients for the alkali-metal atom+molecule and molecule+molecule systems is presented. Dipole moments and static and dynamic dipole polarizabilities are calculated employing high-level quantum chemistry calculations. The dispersion, induction, and rotational parts of the isotropic van der Waals coefficient are evaluated. The known van der Waals coefficients are then used to derive characteristics essential for simple models of the collisions involving the corresponding ultracold polar molecules.
An ion and a polar molecule interact by an anisotropic ion-dipole potential scaling as $\ensuremath{-}\ensuremath{\alpha}\phantom{\rule{0.16em}{0ex}}cos(\ensuremath{\theta})/{r}^{2}$ at large distances. Due to its long-range character, it modifies the properties of angular wave functions, which are no longer given by spherical harmonics. In addition, an effective centrifugal potential in the radial equation can become attractive for low angular momenta. In this paper, we develop a general framework for an ion-dipole reactive scattering, focusing on the regime of large $\ensuremath{\alpha}$. We introduce modified spherical harmonics as solutions of the angular part of the Schr\"odinger equation and derive several useful approximations in the limit of large $\ensuremath{\alpha}$. We present a formula for the scattering amplitude expressed in terms of the modified spherical harmonics and we derive expressions for the elastic and reactive collision rates. The solutions of the radial equation are given by Bessel functions, and we analyze their behavior in two distinct regimes corresponding, basically, to attractive and repulsive long-range centrifugal potentials. Finally, we study reactive collisions in the universal regime, where the short-range probability of loss or reaction is equal to unity.
Abstract In this paper, we have provided exact two-body solutions to the 2D and 3D Schrödinger equations with isotropic van der Waals potentials of the form ± 1 / r 6 . Based on these solutions, we developed an analytical quantum defect theory (QDT) applicable to both quasi-2D and 3D geometries, and applied it to study the scattering properties and bound-state spectra of ultracold polar molecules confined in these geometries. Interestingly, we find that in the attractive (repulsive) van der Waals potential case, the short-range interaction can be effectively modeled by an infinite square barrier (finite square well), which leads to narrow and dense (broad and sparse) resonance structures in the quantum defect parameter. In the quasi-2D attractive case, shape resonances can appear in an ordered fashion across different partial waves, characterized by sharp phase jumps as the scattering energy is varied. Furthermore, the low-energy analytical expansions derived from QDT show excellent agreement with the exact numerical results, validating the accuracy and usefulness of our analytical approach in describing two-body physics governed by long-range van der Waals interactions.
Abstract Using the multipolar expansion of electrostatic and magnetostatic potential energies, we characterize the long-range interactions between two weakly-bound diatomic molecules, taking as an example the paramagnetic Er 2 Feshbach molecules which were produced recently. Since inside each molecule individual atoms conserve their identity, the intermolecular potential energy can be expanded as the sum of pairwise atomic potential energies. In the case of Er 2 Feshbach molecules, we show that the interaction between atomic magnetic dipoles gives rise to the usual term of the multipolar expansion, where R is the intermolecular distance, but also to additional terms scaling as and so on. Those terms are due to the interaction between effective molecular multipole moments, and are strongly anisotropic with respect to the orientation of the molecules. Similarly, the atomic pairwise van der Waals interaction results in ... terms in the intermolecular potential energy. By calculating the reduced electric-quadrupole moment of erbium ground level a.u., we also demonstrate that the electric–quadrupole interaction energy is negligible with respect to the magnetic dipole and van der Waals interaction energies. The general formalism presented in this article can be applied to calculate the long-range potential energy between arbitrary charge distributions composed of almost free subsystems.
We investigate the scattering dynamics governed by the long-range van der Waals plus dipole-dipole interaction potential, $\ensuremath{-}{C}_{6}/{R}^{6}\ensuremath{-}{C}_{3}/{R}^{3}$, which describes the long-range interaction between two polar molecules in an electric field. In the spirit of quantum defect theory, a set of parameters which are nearly constants in the threshold regime is defined to characterize the scattering process. Using appropriate boundary conditions for the scattering wave functions and relevant parameters, we explore the quantum reflection by and quantum tunneling through the long-range potential. As a sample application, the reactive collision rates of 40{}^{40}40K87{}^{87}87Rb +++ 40{}^{40}40K87{}^{87}87Rb are calculated.