The rovibrational energy levels of AlF

For singlet states, Hunds' cases (a) and (b) are the same, with S = \(\Sigma=0\), J = N [1]. The rotational energy level can be calculated by [1]

\begin{equation} F(J) = E(v) + B(J(J+1)-\Lambda^2) - D(J(J+1)-\Lambda^2)^2 \end{equation} where D is the centrifugal distortion constant. For the \(A^1\Pi\) state, the \(\Lambda\)-doubling parameter \(q\) has been determined. The splitting can be calculated by \(F_f - F_e = -qJ(J+1)\) [6].

For triplet states, Hund's cases (a) and (b) rotational energy levels can be obtained by solving [1]

\begin{align} \left. \begin{array}{c} \;\;\Lambda-1 \\ \Lambda \\ \Lambda + 1 \\ \end{array} \right. \begin{vmatrix} T_1^0 & H_{12}^0 & 0 \\ H_{21} & T_2^0 & H_{23}^0 \\ 0 & H_{32}^0 & T_3^0 \end{vmatrix} = 0 \end{align}

which is written with the basis of \( \Omega=[\Lambda-1, \Lambda, \Lambda+1] \).

For Hund's case (a) states, for example the \(a^3\Pi\), \(d^3\Pi\) states, the diagonal terms \(T_1^0\), \(T_2^0\), \(T_3^0\) and the non-diagonal terms can be calculated by [1]

\begin{align} T_1^0 &= E(v_0) - A\Lambda + B[J(J+1) - \Lambda(\Lambda-2)] + \frac{2}{3}\lambda\\ T_2^0 &= E(v_0) + B[J(J+1) - (\Lambda-1)(\Lambda+1) + 1] - \frac{4}{3} \lambda \\ T_3^0 &= E(v_0) + A\Lambda + B[J(J+1) - \Lambda(\Lambda+2)] + \frac{2}{3} \lambda\\ H_{12}^0 &= H_{21}^0 = B \sqrt{2[J(J+1) - (\Lambda-1)\Lambda]} \\ H_{23}^0 &= H_{32}^0 =B \sqrt{2[J(J+1) - \Lambda(\Lambda+1)]} \\ \end{align}

with the spin-spin interaction terms expressed in terms of \(\lambda\).

In Ref. [3], for the \(a^3\Pi\) state, there is an extra \(\lambda\)-doubling term that doubles the \(\Omega=0\) level, defined as \(c_v = 4/3 \lambda + 590\) MHz. However, here we follow the derivations in Ref. [6], in particular the matrix elements listed in Table 1 of Ref. [6] for the \(^3\Pi\) states.

For Hund's case (b) states, for example the \(b^3 \Sigma^+\), \(f^3 \Sigma^+\), \(e^3\Delta\) states, these terms become [1]

\begin{align} T_1^0 &= T(N=J+1, J) \\ T_2^0 &= T(N=J, J) \\ T_3^0 &= T(N=J-1, J) \\ H_{12}^0 &= H_{21}^0 = H(J, J) \\ H_{23}^0 &= H_{32}^0 = H(J - 1, J) \\ \end{align}

where

\begin{align} T(N,J) &= E(v_0) + B[N(N+1) - \Lambda^2] + \Lambda^2 A[J(J+1) - N(N+1) - 2] / [2N(N+1)] \\ H(N,J) &= \Lambda A \sqrt{[(N+1)^2 - \Lambda^2] [(J + N + 1)(J+N+2)-2]} \frac{\sqrt{2-(J-N)(J-N-1)} }{ 2(N+1) \sqrt{(2N+1)(2N+3)}} \\ \end{align}

The couplings between the vibrational and rotational components are expressed in terms of vibrational-dependent rotational constant \(B\) and spin-orbit interaction constant \(A\). In particular, for each vibrational state \(v\), \begin{align} B_v &= B_0 - \alpha_e (v + 1/2) + \beta_e(v+1/2)^2 \\ A_v &= A_0 - \zeta_e (v + 1/2) + \eta_e(v+1/2)^2 \\ \end{align}

The vibrational energy levels of AlF are calculated by the Dunham expansion

\begin{equation} E(v) = \omega_e (v + 1/2) - \omega_ex_e (v + 1/2) ^ 2 + \omega_ey_e (v + 1/2) ^ 3 + \omega_ez_e (v + 1/2) ^ 4 \end{equation}

References

[1] István Kovács and William Lichten. Rotational structure in the spectra of diatomic molecules, 1972.

[2] John M. Brown and Alan Carrington. Rotational spectroscopy of diatomic molecules. Cambridge university press, 2003.

[3] Nicole Walter, Maximilian Doppelbauer, Silvio Marx, Johannes Seifert, Xiangyue Liu, Jesús Pérez-Ríos, Boris G. Sartakov, Stefan Truppe, and Gerard Meijer. Spectroscopic characterization of the a³Π state of aluminum monofluoride. The Journal of Chemical Physics, 156(12), 2022.

[4] Nicole Walter, Maximilian Doppelbauer, Sascha Schaller, Xiangyue Liu, Russell Thomas, Sidney Wright, Boris G. Sartakov, and Gerard Meijer. Triplet rydberg states of aluminum monofluoride. The Journal of Physical Chemistry A, 128(14):2752-2762, 2024.

[5] Maximilian Doppelbauer, Nicole Walter, Simon Hofsäss, Silvio Marx, Christian H. Schewe, Sebastian Kray, Jesús Pérez-Ríos, Boris G. Sartakov, Stefan Truppe, and Gerard Meijer. Characterisation of the b³Σ⁺, v=0 state and its interaction with the A¹Π state in aluminium monofluoride. Molecular Physics, 119(1-2):e1810351, 2021.

[6] John M. Brown and A. J. Merer. Lambda-type doubling parameters for molecules in \(\Pi\) electronic states of triplet and higher multiplicity. Journal of Molecular Spectroscopy, 74, 488-494, 1979

[7] S. Truppe, S. Marx, S. Kray, M. Doppelbauer, S. Hofsäss, H. C. Schewe, N. Walter, J. Pérez-Ríos, B. G. Sartakov, and G. Meijer. Spectroscopic characterization of aluminum monofluoride with relevance to laser cooling and trapping. Phys. Rev. A 100, 052513, 2019.