Knowledge Bank

Using a projector formulation $\mathrm{<mrow\; data-mjx-texclass="ORD">1,2,3,4</mrow>}$ for the Van Vleck transform action, it is possible to go to tenth order without excessive effort. This is the $\kappa$-order needed to include sixth order centrifugal distortion in the effective vib-rot Hamiltonian for a state v. The Wilson-Howard Hamiltonian is first formulated as a Sum of ``scalar products”: \begin{equation} H=h_{2}{^{(0)}}+\kappa h_{3}{^{(0)}}+\kappa^{2}(h_{4}{^{(0)}}+h_{2}{^{(1)}}\cdot J{^{(1)}}+h_{0}{^{(2)}})+\kappa^{3}\ldots \end{equation} where denotes (n indices). The effective Hamiltonian is written similarly as: \begin{eqnarray} A_{v}&=&a_{2}{^{(0)}}+\kappa^{2}(a_{4}{^{(0)}}+a_{2}{^{(1)}}\cdot J{^{(1)}}+a_{0}{^{(2)}}\cdot J{^{(2)}})+\kappa^{4}(a_{6}{^{(0)}}+a_{4}{^{(1)}}\cdot J{^{(1)}}+a_{2}{^{(2)}}\cdot J{^{(2)}})\nonumber\ &&+\kappa^{6}(..+a_{4}{^{(2)}}\cdot J{^{(2)}}+a_{2}{^{(3)}}\cdot J{^{(3)}}+a_{0}{^{(4)}}\cdot J{^{(4)}})\nonumber\ &&+ \kappa^{8}(\ldots +a_{2}{^{(4)}}\cdot J{^{(4)}}+a_{0}{^{(5)}}\cdot J{^{(5)}})\nonumber\ &&+\kappa{^{(10)}}(\ldots+ a_{0}{^{(6)}}\cdot J{^{(6)}})+.. \end{eqnarray} As an example $(a2\mathrm{<mrow\; data-mjx-texclass="ORD">(2)</mrow>})fg$ has the form: (with symbols to be explained) \begin{equation} (a_{2}{^{(2)}}){fg}={x(h{2}{^{(1)}}){f}: (h{2}{^{(1)}}){g}x +2xh{3}{^{(0)}}: (h_{1} {^{(2)}}.){fg}x+x(h_{2}{^{(2)}}){fg}x}{H} \end{equation} This operator leads to $\alpha -$ and q-constants.