The SPAR (Summed Product Analytic Representation) framework, is designed to serve as a general black-box interface through which kinetic energy operators (KEOs), potential energy functions (PEFs), dipole moment functions (DMFs) and possibly other operators in sum-of-products form can be communicated to a varational solver for ro-vibrational calculations of polyatomic molecules.
SPAR is used to represent operators derived in sum-of-products form:
$h(\mathbf{q}) = \sum_{{l}_1,{l}_2,\ldots,l_{3\mathcal{N}-6}}a_{{l}_1,{l}_2,\ldots,l_{3\mathcal{N}-6}} f^{(1)}_{l_1}(q_1) f^{(2)}_{l_2}(q_2) \cdots f^{(3\mathcal{N}-6)}_{l_{3\mathcal{N}-6}}(q_{3\mathcal{N}-6}) \equiv \sum_{\mathbf{l}} a_{\mathbf{l}} \prod^{3\mathcal{N} - 6}_{k = 1} f^{(k)}_{l_k}(q_k).$
Where $h(\mathbf{q})$ is an operator, or component of an operator, that depends on the $3\mathcal{N}-6$ vibrational degrees of freedom $\mathbf{q}=(q_1, q_2,...q_{3\mathcal{N}-6})$. Here $\mathbf{l}$ is a vector of indices $\mathbf{l} = (l_1, l_2,...l_{3\mathcal{N}-6})$ where each $l_k$ refers to a so-called "basic function" $f^{(k)}_{l_k}(q_k)$; the building blocks with which the sum-of-products form is constructed. The sum extends over all terms for which the coefficient $a_{\mathbf{l}} \neq 0$.
For KEOs we also include a mass-dependent form:
$h(\mathbf{q}) = \sum_{{l}_0,{l}_1,l_2,\ldots,} a_{{l}_0,{l}_1,l_2 \ldots,} \frac{1}{m_{{l}_0}} f_{{l}_1}(q_1) f_{{l}_1}(q_1) \cdots \equiv \sum_{\mathbf{l}} a_{\mathbf{l}}\prod^{D}_{k = 0} f^{(k)}_{{l}_k}(q_k),$
where $\mathbf{l}$ is now defined such that index $k=0$ refers to a nuclear mass such that $\mathbf{l} = (l_0, l_1, l_2,...l_{3\mathcal{N}-6})$.