Approximate symmetries of long-range Rydberg molecules including spin effects

An operator that generates an approximate symmetry of long-range Rydberg molecules (LRRMs) formed by two alkali atoms, one in a Rydberg state and the other in the ground state, is identified. This is first done by evaluating the natural orbitals associated with a variational calculation of the binding wave function within the Born-Oppenheimer description of the molecule including <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>s</a:mi></a:math> and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>p</b:mi></b:math> Fermi pseudopotential and the hyperfine structure energy terms. The resulting orbitals with the highest occupation number are shown to be identical to those obtained by a perturbative model for high angular momentum—trilobite and butterfly—LRRMs. Whenever the slight dependence of the quantum defects of the Rydberg electron on its total momentum <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mover accent="true"><c:mi>j</c:mi><c:mo>⃗</c:mo></c:mover><c:mo>=</c:mo><c:mover accent="true"><c:mi>ℓ</c:mi><c:mo>⃗</c:mo></c:mover><c:mo>+</c:mo><c:msub><c:mover accent="true"><c:mi>s</c:mi><c:mo>⃗</c:mo></c:mover><c:mn>1</c:mn></c:msub></c:mrow></c:math> can be neglected, the symmetry operator of the high angular momentum LRRMs orbitals is identified as the sum of the spin of the Rydberg electron <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mover accent="true"><g:mi>s</g:mi><g:mo>⃗</g:mo></g:mover><g:mn>1</g:mn></g:msub></g:math>, spin of the valence electron <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msub><i:mover accent="true"><i:mi>s</i:mi><i:mo>⃗</i:mo></i:mover><i:mn>2</i:mn></i:msub></i:math>, and the spin of nucleus <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mover accent="true"><k:mi>i</k:mi><k:mo>⃗</k:mo></k:mover></k:math> of the ground-state atom, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mover accent="true"><m:mi>N</m:mi><m:mo>⃗</m:mo></m:mover><m:mo>=</m:mo><m:mover accent="true"><m:msub><m:mi>s</m:mi><m:mn>1</m:mn></m:msub><m:mo>⃗</m:mo></m:mover><m:mo>+</m:mo><m:mover accent="true"><m:msub><m:mi>s</m:mi><m:mn>2</m:mn></m:msub><m:mo>⃗</m:mo></m:mover><m:mo>+</m:mo><m:mover accent="true"><m:mi>i</m:mi><m:mo>⃗</m:mo></m:mover></m:mrow></m:math>. The spin orbitals that diagonalize <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:mover accent="true"><r:mi>N</r:mi><r:mo>⃗</r:mo></r:mover></r:math> define compact basis sets for the description of LRRMs beyond the aforementioned approximations. The matrix elements of the Hamiltonian in these basis sets have simple expressions, so that the relevance of triplet and singlet contributions can be directly estimated. The expected consequences of this approximate spin-symmetry on the spectra of LRRMs are briefly described.