Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules

• Main Authors: Thomas M. Mellor, Sergei N. Yurchenko, Per Jensen
• Format: Article
• Language: English
• Published: MDPI AG 2021-03-01
• Series: Symmetry
• Subjects: ro-vibrational; point groups; molecular symmetry groups; CO₂
• Online Access: https://www.mdpi.com/2073-8994/13/4/548

Linear molecules usually represent a special case in rotational-vibrational calculations due to a singularity of the kinetic energy operator that arises from the rotation about the <i>a</i> (the principal axis of least moment of inertia, becoming the molecular axis at the linear equilibrium geometry) being undefined. Assuming the standard ro-vibrational basis functions, in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mi>N</mi><mo>−</mo><mn>6</mn></mrow></semantics></math></inline-formula> approach, of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∣</mo><msub><mi>ν</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ν</mi><mn>2</mn></msub><mo>,</mo><msubsup><mi>ν</mi><mn>3</mn><msub><mi>ℓ</mi><mn>3</mn></msub></msubsup><mo>;</mo><mi>J</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>m</mi><mo>⟩</mo></mrow></semantics></math></inline-formula>, tackling the unique difficulties of linear molecules involves constraining the vibrational and rotational functions with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><msub><mi>ℓ</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula>, which are the projections, in units of <i>ℏ</i>, of the corresponding angular momenta onto the molecular axis. These basis functions are assigned to irreducible representations (irreps) of the <i><b>C</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mrow><mn>2</mn><mi mathvariant="normal">v</mi></mrow></msub></semantics></math></inline-formula>(M) molecular symmetry group. This, in turn, necessitates purpose-built codes that specifically deal with linear molecules. In the present work, we describe an alternative scheme and introduce an (artificial) group that ensures that the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ℓ</mi><mn>3</mn></msub><mo>=</mo><mi>k</mi></mrow></semantics></math></inline-formula> is automatically applied solely through symmetry group algebra. The advantage of such an approach is that the application of symmetry group algebra in ro-vibrational calculations is ubiquitous, and so this method can be used to enable ro-vibrational calculations of linear molecules in polyatomic codes with fairly minimal modifications. To this end, we construct a—formally infinite—artificial molecular symmetry group <i><b>D</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mrow><mo>∞</mo><mi mathvariant="normal">h</mi></mrow></msub></semantics></math></inline-formula>(AEM), which consists of one-dimensional (non-degenerate) irreducible representations and use it to classify vibrational and rotational basis functions according to <i>ℓ</i> and <i>k</i>. This extension to non-rigorous, artificial symmetry groups is based on cyclic groups of prime-order. Opposite to the usual scenario, where the form of symmetry adapted basis sets is dictated by the symmetry group the molecule belongs to, here the symmetry group <i><b>D</b></i><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mrow><mo>∞</mo><mi mathvariant="normal">h</mi></mrow></msub></semantics></math></inline-formula>(AEM) is built to satisfy properties for the convenience of the basis set construction and matrix elements calculations. We believe that the idea of purpose-built artificial symmetry groups can be useful in other applications.