# Analytic potentials and vibrational energies for Li2 states dissociating to
$\mbox{Li}\left(2S\right)+\mbox{Li}\left(3P\right)$. Part 1: The
${}^{2S+1}\Pi_{\nicefrac{{u}}{{g}}}$ states

Nikesh S. Dattani<EMAIL_ADDRESS>1School of Materials Science and
Engineering, Nanyang Technological University, 639798, Singapore,  2Fukui
Institute for Fundamental Chemistry, 606-8103, Kyoto, Japan,  3Quantum
Chemistry Laboratory, Department of Chemistry, Kyoto University, 606-8502,
Kyoto, Japan,

###### Abstract

Analytic potentials are built for all four ${}^{2S+1}\Pi_{u/g}$ states of Li2
dissociating to Li$(2S)$ \+ Li$(3P)$: $3b(3^{3}\Pi_{u})$, $3B(3^{1}\Pi_{u})$,
$3C(3^{1}\Pi_{g}),$ and $3d(3^{3}\Pi_{g})$. These potentials include the
effect of spin-orbit coupling for large internuclear distances, and include
state of the art long-range constants. This is the first successful
demonstration of fully analytic diatomic potentials that capture features that
are usually considered too difficult to capture without a point-wise
potential, such as multiple minima, and shelves. Vibrational energies for each
potential are presented for the isotopologues 6,6Li2, 6,7Li2, 7,7Li2, and the
elusive ‘halo nucleonic molecule’ 11,11Li2. These energies are claimed to be
accurate enough for new high-precision experimental setups such as the one
presented in [Sebastian _et al._ Phys. Rev. A, 90, 033417 (2014)] to measure
and assign energy levels of these electronic states, all of which have not yet
been explored in the long-range region. Measuring energies in the long-range
region of these electronic states may be significant for studying the _ab
initio_ vs experiment discrepancy discussed in [Tang _et al._ Phys. Rev. A,
84, 052502 (2014)] for the $C_{3}$ long-range constant of Lithium, which has
significance for improving the SI definition of the second.

###### pacs:

02.60.Ed , 31.50.Bc , 82.80.-d , 31.15.ac, 33.20.-t, , 82.90.+j, 97, ,
98.38.-j , 95.30.Ky

Very little is known about the Li2 electronic states dissociating to the
$2S+3P$ asymptote. Out of the first 5 asymptotes (from lowest to highest:
$2S+2S$, $2S+2P$, $2S+3S$, $2P+2P$ and $2S+3P$), the $2S+3P$ is the only one
for which an empirical dissociation energy has not been determined for any of
the electronic states dissociating to it. Furthermore, the only measurements
that have been made for electronic states dissociating to $2S+3P$, were for
the $3c(3^{3}\Sigma_{g}^{+})$ state Xie and Field (1986a); Yiannopoulou  _et
al._ (1995a); Li  _et al._ (2007) the $6X(6^{1}\Sigma_{g}^{+})$ state Bernheim
_et al._ (1983); Song  _et al._ (2002), and the $3d(3^{3}\Pi_{g})$ state
Yiannopoulou  _et al._ (1995b); Li  _et al._ (1996a, b); Ivanov  _et al._
(1999). No measurements have been done on the other states dissociating to
$2S+3P$.

Very recently, a promising experiment has been setup with the ability to use
photoassociation in a magneto-optical trap to make ultra-cold 6Li2 molecules
dissociating to the $2S+3P$ asymptote Sebastian  _et al._ (2014), much like
slightly earlier experiments which have already been successful for creating
ultra-cold 6Li2 molecules dissociating to $2S+2P$ with very similar techniques
Semczuk  _et al._ (2013); Gunton  _et al._ (2013). Measurements of the binding
energies for levels very close to the $2S+3P$ asymptote would allow for an
empirical determination of the long-range constant $C_{3}^{2S+3P}$ which is
the leading interaction constant in the potential energy between Li$(2S)$ and
Li$(3P)$.

$u^{3b}(r)$ | = |  $\begin{cases}\\\ \\\ \\\ \\\ \end{cases}$ | $u^{3b,0_{u}^{+}}(r)$ | $\xrightarrow{\qquad u^{6A,0_{u}^{+}}(r)\qquad}$ | $2S_{1/2}+3P_{1/2}$ |  (higher)
---|---|---|---|---|---|---
| $u^{3b,0_{u}^{-}}(r)$ | $\xrightarrow{\qquad u^{6a,0_{u}^{-}}(r)\qquad}$ | $2S_{1/2}+3P_{3/2}$ |  (lower)
| $u^{3b,1_{u}}(r)$ | $\xrightarrow{\,\,u^{3B,1_{u}}(r),u^{6a,1_{u}}(r)\,\,}$ | $2S_{1/2}+3P_{3/2}$ |  (lowest)
| $u^{3b,2_{u}}(r)$ | $\xrightarrow{\qquad\qquad\qquad\qquad}$ | $2S_{1/2}+3P_{3/2}$ |
$u^{3B}(r)$ | = |  | $u^{3B,1_{u}}(r)$ | $\xrightarrow{\,\,u^{6a,1_{u}}(r),u^{3b,1_{u}}(r)\,\,}$ | $2S_{1/2}+3P_{3/2}$ |  (middle)
$u^{3C}(r)$ | = |  | $u^{3C,1_{g}}(r)$ | $\xrightarrow{\,\,u^{3c,1_{g}}(r),u^{3d,1_{g}}(r)\,\,}$ | $2S_{1/2}+3P_{3/2}$ |  (middle)
$u^{3d}(r)$ | = |  $\begin{cases}\\\ \\\ \\\ \\\ \end{cases}$ | $u^{3d,0_{g}^{+}}(r)$ | $\xrightarrow{\qquad u^{6X,0_{g}^{+}}(r)\qquad}$ | $2S_{1/2}+3P_{1/2}$ |  (lower)
| $u^{3d,0_{g}^{-}}(r)$ | $\xrightarrow{\qquad u^{3c,0_{g}^{-}}(r)\qquad}$ | $2S_{1/2}+3P_{3/2}$ |  (higher)
| $u^{3d,1_{g}}(r)$ | $\xrightarrow{\,\,u^{3c,1_{g}}(r),u^{3C,1_{g}}(r)\,\,}$ | $2S_{1/2}+3P_{3/2}$ |  (highest)
| $u^{3d,2_{g}}(r)$ | $\xrightarrow{\qquad\qquad\qquad\qquad}$ | $2S_{1/2}+3P_{3/2}$ |

At the lower asymptote of $2S+2P$, there is a discrepancy between experiment
and theory for the long-range constant $C_{3}^{2S+3P}$, despite Li only having
3e- and the experimental value being the most precisely determined oscillator
strength ever determined for a molecule, by an order of magnitude Tang  _et
al._ (2011). This has various consequences, reaching as far as limiting
progress towards improving the precision of the SI definition of the second
Mitroy  _et al._ (2010). More precise atomic clocks are needed for various
applications. The current definition of the second is based on a clock
transition frequency in Cs with a relative uncertainty of $\sim 5\times
10^{-16}$, and a commonly quoted target for improved precision is $10^{-18}$
Mitroy  _et al._ (2010). The largest source of uncertainty limiting atomic
clock precision is the blackbody radiation shift, which depends on the static
dipole polarizability of the system being used for the atomic clock Mitroy
_et al._ (2010). Lithium is expected to play a major role in polarizability
metrology, since polarizability ratios can be measured much more precisely
than individual polarizabilities Cronin  _et al._ (2009) and Li is the
preferred choice for the standard in the denominator of such a ratio Mitroy
_et al._ (2010). The discrepancy in $C_{3}$ limits the accuracy of a potential
Li-based standard for polarizabilities Tang  _et al._ (2011), and hence
indirectly impacts progress towards improving the SI definition of the second.

Regarding the empirical value for $C_{3}^{2S+2P}$, for most electronic states,
the mixing of various states towards the $2S+2P$ asymptote significantly
complicates the expressions from which $C_{3}$ is fitted Dattani and LeRoy
(2015); Dattani and Le Roy (2011); Le Roy  _et al._ (2009); Aubert-Frécon  _et
al._ (1998); Bussery and Aubert-Frecon (1985). The complicated expressions for
this mixing are the same at the $2S+3P$ asymptote as they are for the $2S+2P$
asymptote Aubert-Frécon (2015), but the fine structure splitting parameter
which governs the significance of this mixing, is about 3.5 times smaller at
the $2S+3P$ asymptote than at the $2S+2P$ asymptote. For $2S+2P$ the fine
structure splitting parameter for 6Li is $\Delta
E_{2^{2}P_{\nicefrac{{3}}{{2}}}\leftarrow
2^{2}P_{\nicefrac{{1}}{{2}}}}=D_{2}-D_{1}=0.335\,324\,6$ cm-1 Semczuk  _et
al._ (2013); Gunton  _et al._ (2013); Brown  _et al._ (2013); Sansonetti  _et
al._ (2011), while for $2S+3P$ it is only $\Delta
E_{3^{2}P_{\nicefrac{{3}}{{2}}}\leftarrow 3^{2}P_{\nicefrac{{1}}{{2}}}}=0.096$
cm-1 Sansonetti  _et al._ (1995). Therefore, $C_{3}^{2S+3P}$ might be a better
benchmark for an _ab initio_ vs experiment comparison than $C_{3}^{2S+2P}$, as
the effect of this complication is smaller.

Measuring and assigning molecular energy levels using photoassociation
requires reasonably accurate predictions which come from eigenvalues of a
Schrödinger equation, and hence require a reasonably accurate potential energy
surface. Due to the shortage of measurements on the Li2
${}^{2S+1}\Pi_{\nicefrac{{u}}{{g}}}$ states dissociating to $2S+3P$, the most
accurate potentials come from purely _ab initio_ calculations. For the
$3c(3^{3}\Sigma_{g}^{+})$ and $3A(3^{1}\Sigma_{u}^{+})$ states, _ab initio_
calculations were reported in 1985 Schmidt-Mink  _et al._ (1985), 1995 Poteau
and Spiegelmann (1995), 2006 Jasik and Sienkiewicz (2006) and 2014 Musial and
Kucharski (2014); Lee (2014); and for the $3d(3^{3}\Pi_{g})$ state in 1995
Yiannopoulou  _et al._ (1995b) and 2014 Musial and Kucharski (2014); Lee
(2014). But for the rest of the states dissociating to $2S+3P$; namely
$3b(3^{3}\Pi_{u})$, $6X(6^{1}\Sigma_{g}^{+})$, $3B(3^{1}\Pi_{u})$,
$3C(3^{1}\Pi_{g})$, and $6a(6^{3}\Sigma_{u}^{+})$; the only _ab initio_
calculations reported were in Poteau and Spiegelmann (1995); Musial and
Kucharski (2014); Lee (2014). All of these _ab initio_ papers also reported
potentials for states dissociating to lower asymptotes, where plenty of
experimental data is available to gauge the quality of the calculations.

In my very recent paper on comparing experiment to _ab initio_ for the
$b(1^{3}\Pi_{u})$ state Dattani and LeRoy (2015), it was found that the _ab
initio_ potential of Musial and Kucharski (2014) predicted all vibrational
binding energies with a disagreement of $<12$ cm-1 with the corresponding
energies of the empirical potential. Furthermore there was always $<0.8$ cm-1
disagreement between the empirical and _ab initio_ vibrational energy
spacings. Finally, when comparing the dissociation energies
$\mathfrak{\mathfrak{D}}_{e}$ from Musial and Kucharski (2014) to the
corresponding experimental values for all states which have empirical
$\mathfrak{D}_{e}$ values available, the _ab initio_ values from Musial and
Kucharski (2014) were never in disagreement by more than 68 cm-1. Therefore,
the _ab initio_ potentials from Musial and Kucharski (2014) for the states
dissociating to $2S+3P$ are expected to be a good starting point for
predicting energy levels with the precision required for photoassociation
experiments as in Semczuk  _et al._ (2013); Gunton  _et al._ (2013) and as may
be preformed with the new setup in Sebastian  _et al._ (2014) which is capable
of detecting states dissociating to $2S+3P$.

However, the _ab initio_ calculations of Musial and Kucharski (2014) still
have some major drawbacks (including, but not limited to):

  1. 1.

the _ab initio_ points are not on a dense enough mesh to use as the mesh for
solving the effective radial Schrödinger equation for predicting the
vibrational energies (_especially_ for large distances where the energies
become more important for fitting an empirical $C_{3}$ value, and for
experiments such as those potentially resulting from a setup such as in
Sebastian  _et al._ (2014));

  2. 2.

the _ab initio_ points neglect the effect of spin-orbit coupling, which is
particularly important for $nS+n^{\prime}P$ states of Li2, where the effect of
interstate coupling has been shown to be absolutely obligatory for describing
the high vibrational energy measurements Le Roy  _et al._ (2009); Dattani and
Le Roy (2011); Semczuk  _et al._ (2013); Gunton  _et al._ (2013);

  3. 3.

the _ab initio_ points do not go beyond the Born-Oppenheimer approximation (so
do not distinguish between different isotopologues such as 6,6Li${}_{2},$
6,7Li2, 7,7Li2, and the elusive ‘halo nucleonic’ isotopologues containing
11Li), and they are non-relativistic.

Drawback (1) is usually treated by fitting an interpolant through the _ab
initio_ points, but the resulting energies predicted after solving the
Schrödinger equation, will be very sensitive to the type of interpolant used,
especially at the level of precision of photoassociation experiments (the
precision in the Li2 measurements of Semczuk  _et al._ (2013); Gunton  _et
al._ (2013) was $\pm 0.00002$ cm-1 or $\pm 600$ kHz). Also, if for sake of
ease, a spline interpolant is used, it would be defined piecewise and would
have discontinuous first derivatives. The spline also knows nothing about the
physics of nature, and will therefore not know what to do in regions where
fewer _ab initio_ points are available (in this example, for large
internuclear distances).

Rather than interpolating with a spline, we can fit to a fully analytic model
potential that has the correct theoretical behavior incorporated in the long-
range region where fewer _ab initio_ points are available, and this addresses
drawback (2) as well, since the effect of spin-orbit coupling at long-range
can easily be incorporated into the model. Part of drawback (3) can also be
addressed by fitting to a model potential, because the model can also build in
some types of relativistic effects such as QED retardation, as was attempted
in Le Roy  _et al._ (2009); Dattani and Le Roy (2011); Semczuk  _et al._
(2013); Gunton  _et al._ (2013). In 2011 the Morse/long-range (MLR) potential
was fitted to spectroscopic data for the $c(1^{3}\Sigma_{g}^{+})$ state of Li2
where there was a gap of $>5000$ cm-1 between data near the bottom of the
potential’s well, and data at the very top Dattani and Le Roy (2011). In 2013
it was found by experiment that vibrational energies predicted from this MLR
potential in the very middle of this gap were correct to about 1 cm-1 Semczuk
_et al._ (2013). Therefore, fitting the _ab initio_ data to the MLR model can
provide reliable energy predictions in regions where _ab initio_ points are
lacking or are poor in quality.

Therefore, in this paper MLR models that incorporate the long-range
theoretical effect of spin-orbit coupling are fitted to the _ab initio_ points
from Musial and Kucharski (2014) for the ${}^{2S+1}\Pi_{\nicefrac{{u}}{{g}}}$
states of Li2 dissociating to $2S+3P$. Drawback (3) is not addressed in this
paper. However, Born-Oppenheimer breakdown (BOB) corrections could have been
added to the _ab initio_ points using the molecular electron wavefunction as
described in Dattani and LeRoy (2015). Alternatively, the entire _ab initio_
calculation can be redone using a non-Born-Oppenheimer approach as has been
done for up to 6e- Bubin  _et al._ (2009), but the _a posteriori_ approach of
doing a Born-Oppenheimer calculation and then adding BOB corrections
afterwards has been shown to work better according to the agreement between
experiment and theory for BeH Dattani (2015); Bubin and Adamowicz (2007).
Also, DKH (Douglas-Kroll-Hess) relativistic corrections can be added to the
_ab initio_ potential as in Koput (2011), and QED effects can also be added as
was done for H2 in Komasa  _et al._ (2011) and HeH+ in Pachucki and Komasa
(2012). If any of these answers to drawback (3) were to be addressed by adding
corrections to the _ab initio_ points of Musial and Kucharski (2014), the
procedure applied in the present paper for fitting an MLR function to _ab
initio_ points, could be repeated for even more accurate analytic potentials.

##  I Extending the _ab initio_ calculations

The _ab initio_ calculations in Musial and Kucharski (2014) did not go beyond
22 Å. However, beyond a certain length, analytic expressions for the potential
can be derived from the theory of atom-atom interactions with disregard for
the effect of overlap between each atom’s electronic wavefunction. These
analytic expressions are based on long-range constants that come from atomic
_ab initio_ calculations rather than molecular ones, so for Li2 the
calculations only involve 3e- rather than 6e-. This means, for example, that a
coupled cluster calculation taking account of the full configuration
interaction (FCI) of a basis set only needs to go up to triple excitations
(CCSDT, whose scaling with respect to the number of basis functions $N$ is
$\sim N^{8}$ and has been implemented since 1987 Noga and Bartlett (1987));
whereas a molecular calculation on Li2 would require all the way up to hexuple
excitations (CCSDTQPH, which scales as $\sim N^{14}$, and has been implemented
in only very few studies since 2000 Hirata and Bartlett (2000); Kallay and
Surjan (2001); Kallay and Gauss (2004); Evangelista  _et al._ (2006); DeYonker
and Allen (2012); DeYonker and Shah (2014) with basis sets that have not yet
gone beyond the cc-pVDZ-DK basis set DeYonker and Shah (2014)). Furthermore,
3e- is the limit at which the integrals have been expressed analytically for
explicitly correlated Slater wavefunctions, so treating 6e- would either
require numerically calculating the integrals (which would be too slow even
for small basis sets), or explicitly correlated Gaussian wavefunctions (which
do not necessarily have the correct short- and long-range behavior).
Therefore, beyond a certain distance the analytic expressions ignoring
wavefunction overlap but using long-range constants for Li based on 3e- _ab
initio_ calculations, are expected to be more accurate than the 6e-_ab initio_
calculations of Musial and Kucharski (2014) that include wavefunction overlap.
The distance at which this trade-off begins to lean in favor of the analytic
expressions is heuristically given by the Le Roy radius LeRoy (1973, 1974); Ji
_et al._ (1995).

Figure 1: Point-wise original, and analytic MLR potentials for
$3b(3^{3}\Pi_{u})$ representing the _ab initio_ calculations of Musial and
Kucharski (2014). The inset shows the long-range behavior in Le Roy space: it
demonstrates that the original _ab initio_ points unphysically dip below the
theoretical curve, while the MLR behaves correctly.

Another advantage of using the analytic expressions, is that the _ab initio_
calculations of Musial and Kucharski (2014) do not include the effect of spin-
orbit coupling, but for alkali atoms dissociating to $nS+n^{\prime}P$
asymptotes, the effect of spin-orbit coupling at long-range has been
determined analytically Bussery and Aubert-Frecon (1985); Aubert-Frécon  _et
al._ (1998). Although all papers discussing these analytic expressions to date
only mention $nS+nP$ asymptotes, the expressions are also the same for
$nS+n^{\prime}P$ asymptotes when $n\neq n^{\prime}$ Aubert-Frécon (2015).

###  I.1 Le Roy radii

The $m$-dependent Le Roy radius is given by Ji _et al._ Ji  _et al._ (1995):

$R_{{\rm LR}-m}\equiv 2\sqrt{3}\left(\langle
nlm|z^{2}|nlm\rangle^{\nicefrac{{1}}{{2}}}+\langle
n^{\prime}l^{\prime}m^{\prime}|z^{2}|n^{\prime}l^{\prime}m^{\prime}\rangle^{\nicefrac{{1}}{{2}}}\right),$
(1)

where for hydrogen-like atoms we have Ji  _et al._ (1995):

$\langle
nlm|z^{2}|nlm\rangle^{\nicefrac{{1}}{{2}}}=\left(\frac{1}{3}-\frac{2}{3}\frac{3m^{2}-l(l+1)}{(2l+3)(2l-1)}\right)^{\nicefrac{{1}}{{2}}}\langle
nl|r^{2}|nl\rangle^{\nicefrac{{1}}{{2}}}$ (2)

and for $l=0$ we have (because $m$ is also $0$) Ji  _et al._ (1995):

$\langle nlm|z^{2}|nlm\rangle^{\nicefrac{{1}}{{2}}}=\frac{1}{\sqrt{3}}\langle
r^{2}\rangle^{\nicefrac{{1}}{{2}}}.$ (3)

This means that if both atoms of a diatomic molecule are in $S$ states, the
fact that $l=l^{\prime}=0$ reduces Eq. 1 to the original Le Roy radius of
LeRoy (1973, 1974):

$\displaystyle r_{{\rm LR}}$ $\displaystyle\equiv$ $\displaystyle
2\left(\langle r_{{\rm A}}^{2}\rangle^{\nicefrac{{1}}{{2}}}+\langle r_{{\rm
B}}^{2}\rangle^{\nicefrac{{1}}{{2}}}\right).$ (4)

However for a hydrogenic atom with $l\neq 0$ we have Ji  _et al._ (1995):

$\langle
nl|r^{2}|nl\rangle^{\nicefrac{{1}}{{2}}}=a_{\mu}\frac{n^{2}}{Z}\left(1+\frac{3}{2}\left(1-\frac{l(l+1)-\nicefrac{{1}}{{3}}}{n^{2}}\right)\right)^{\nicefrac{{1}}{{2}}},$
(5)

where $Z$ is the effective nuclear charge, $a_{\mu}=a_{0}\frac{m_{N}}{\mu}$ is
the Bohr radius scaled by the ratio of the mass of the nucleus $m_{N}$ to the
reduced mass $\mu$ of the atom. And for alkali atoms, the principal quantum
number $n$ is replaced by $n-\alpha(l)$, where $\alpha(l)$ is the quantum
defect and can be found in standard references such as Ref [29] of Ji  _et
al._ (1995). Using $\alpha(p)=1.59$ for Li, assuming that $\mu_{{}^{6}{\rm
Li}}=\frac{m_{e}M_{{}^{6}{\rm Li}}}{m_{e}+M_{{}^{6}{\rm Li}}}\approx
m_{e}=1\,\mbox{a.u.}$, and using $\langle r^{2}\rangle$ values calculated in
Zhang  _et al._ (2007), we are able to calculate Eq. 1 for Li2 molecules
dissociating to various asymptotes, and we present these in Table 1.

Table 1: $m$-dependent Le Roy radii for Li2 electronic states that approach $2s+ml$ in Hartree atomic units and spectroscopic units, and the constituent quantities that are used to calculate these radii.  | $\langle r^{2}\rangle$ | $\langle nlm|z^{2}|nlm\rangle^{\nicefrac{{1}}{{2}}}$ | $r_{{\rm LR-m}}(2s+ml)$ | $r_{{\rm LR-m}}(2s+ml)$
---|---|---|---|---
|  a. | u. |  a. | u. |  a. | u. |  Å
$2s$ | 17. | 47 Zhang  _et al._ (2007) | 17. | 47 Zhang  _et al._ (2007) | 16. | 7189 | 8. | 8473
$2p$ | 27. | 06 Zhang  _et al._ (2007) | 0. | 3810 | 27. | 0 | 14. | 0
$3p$ | 168. | 69 Zhang  _et al._ (2007) | 0. | 5910 | 40. | 0 | 20. | 1
Figure 2: Point-wise original, and analytic MLR potentials for
$3B(3^{1}\Pi_{1_{u}})$ representing the _ab initio_ calculations of Musial and
Kucharski (2014). The inset shows the long-range behavior in Le Roy space.

###  I.2 Long-range theory

It is well-known that for large internuclear distances, the MLR model becomes,
Dattani and Le Roy (2011):

$V(r)\simeq\mathfrak{D}_{e}-u(r)+\cdots.$ (6)

Therefore, we can define $u(r)$ to be the analytic expression describing the
theoretical interaction between the constituent atoms of the molecule. Each
${}^{2S+1}\Pi_{u/g}$ state considered in this paper has $\Omega_{u/g}$
daughter states resulting from the spin-orbit coupling that lifts the
degeneracy. For the $2_{u}$ and $2_{g}$ states, which are daughters of
$3b(3^{3}\Pi_{g})$ and $3d(3^{3}\Pi_{g})$ respectively, no other $\Omega=2$
state with the same $u/g$ symmetry approaches the $2S+3P$ asymptote, so the
potential energy curves at long-range are not strongly influenced by other
electronic states. Therefore, these states have the simplest form for $u(r)$:

$\displaystyle u^{2_{\nicefrac{{u}}{{g}}}}(r)$ $\displaystyle=$
$\displaystyle-\left(\Delta E-\sum_{\begin{subarray}{c}m=3,6,8,\\\
9,10,11,\ldots\end{subarray}}\frac{C_{m}^{\left({}^{3}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}\right)$
(7) $\displaystyle=$ $\displaystyle-\left(\Delta
E-\frac{C_{3}^{\left({}^{3}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{3}}-\frac{C_{6}^{\left({}^{3}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{6}}-\frac{C_{8}^{\left({}^{3}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{8}}-\frac{C_{9}^{\left({}^{3}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{9}}-\frac{C_{10}^{\left({}^{3}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{10}}-\frac{C_{11}^{\left({}^{3}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{11}}\cdots\right),$
(8)

where the zero of energy is the Li$(2S_{\nicefrac{{1}}{{2}}})$ \+
Li($3P_{\nicefrac{{1}}{{2}}}$) asymptote, and $\Delta E\equiv\Delta
E_{3^{2}P_{3/2}\leftarrow 3^{2}P_{1/2}}$ is included since the $2_{u}$ and
$2_{g}$ states both dissociate to Li$(2S_{\nicefrac{{1}}{{2}}})$ \+
Li($3P_{\nicefrac{{3}}{{2}}}$). The $3b(3^{3}\Pi_{u})$ state additionally has
a daughter state of $1_{u}$ symmetry, along with $3B(3^{1}\Pi_{u})$; and the
$3d(3^{3}\Pi_{g})$ state additionally has a daughter state of $1_{g}$
symmetry, along with $3C(3^{1}\Pi_{g})$. Since all $1_{\nicefrac{{u}}{{g}}}$
states approaching $2S+3P$ have two other $1_{u/g}$ states of the same
$\nicefrac{{u}}{{g}}$ symmetry approaching $2S+3P$, the $u(r)$ for these
states is defined as the highest, middle, or lowest energy eigenvalue of the
following $3\times 3$ matrix (including the prefactor of $-1$) depending on
whether the state in question is the lowest, middle, or highest in energy
respectively:

$\mathbf{u}^{1_{\nicefrac{{u}}{{g}}}}(r)=$ (9)

$-\left(\begin{array}[]{ccc}\frac{1}{3}{\displaystyle\sum_{m}\frac{C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Sigma_{\nicefrac{{u}}{{g}}}^{+}\right)}+C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}+C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}}&\frac{1}{3\sqrt{2}}{\displaystyle\sum_{m}\frac{-2C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Sigma_{\nicefrac{{u}}{{g}}}^{+}\right)}+C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}+C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}}&\frac{1}{\sqrt{6}}{\displaystyle\sum_{m}\frac{-C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}+C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}}\\\
\frac{1}{3\sqrt{2}}{\displaystyle\sum_{m}\frac{-2C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Sigma_{\nicefrac{{u}}{{g}}}^{+}\right)}+C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}+C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}}&\Delta
E+\frac{1}{6}{\displaystyle\sum_{m}\frac{4C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Sigma_{\nicefrac{{u}}{{g}}}^{+}\right)}+C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}+C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}}&\frac{1}{2\sqrt{3}}{\displaystyle\sum_{m}\frac{-C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}+C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}}\\\
\frac{1}{\sqrt{6}}{\displaystyle\sum_{m}\frac{-C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}+C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}}&\frac{1}{2\sqrt{3}}{\displaystyle\sum_{m}\frac{-C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}+C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}}&-\Delta
E+\frac{1}{2}{\displaystyle\sum_{m}\frac{C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}+C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}}\end{array}\right).$
(10)

Figure 3: Point-wise original, and analytic MLR potentials for
$3C(3^{1}\Pi_{g})$ representing the _ab initio_ calculations of Musial and
Kucharski (2014). The top inset shows the long-range behavior in Le Roy space,
and the bottom inset shows that the MLR successfully captures the tiny second
minimum which has a depth of $\approx 8.5$ cm-1.

The notation ${}^{\nicefrac{{1}}{{3}}}\Lambda_{u/g}$ means $(^{1}\Lambda_{u}$
or ${}^{3}\Lambda_{g})$. The zero of energy is once again the
Li$(2S_{\nicefrac{{1}}{{2}}})$ \+ Li($3P_{\nicefrac{{1}}{{2}}}$) asymptote.
Finally, the $3b$ state additionally has $0_{u}^{+}$ and $0_{u}^{-}$ daughter
states, and the $3d$ state additionally has $0_{g}^{+}$ and $0_{g}^{-}$
daughter states. Since all $0_{\nicefrac{{u}}{{g}}}^{\nicefrac{{+}}{{-}}}$
states approaching $2S+3P$ have one other
$0_{\nicefrac{{u}}{{g}}}^{\nicefrac{{+}}{{-}}}$ state of the same
$\nicefrac{{u}}{{g}}$ symmetry and the same $\nicefrac{{+}}{{-}}$ symmetry
approaching $2S+3P$, the $u(r)$ for these states is defined as the higher, or
lower energy eigenvalue of the following $2\times 2$ matrix (including the
prefactor of $-1$) depending on whether the state in question is lower, or
higher in energy respectively:

$\displaystyle\mathbf{u}^{0_{\nicefrac{{u}}{{g}}}^{\nicefrac{{+}}{{-}}}}(r)$
$\displaystyle=-$
$\displaystyle\begin{pmatrix}\frac{1}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Sigma_{\nicefrac{{u}}{{g}}}^{+}\right)}+2C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}&\quad\;\frac{\sqrt{2}}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Sigma_{\nicefrac{{u}}{{g}}}^{+}\right)}-C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{3}}\\\
\frac{\sqrt{2}}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Sigma_{\nicefrac{{u}}{{g}}}^{+}\right)}-C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}&\qquad\quad-\Delta
E+\frac{2}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{\left({}^{\nicefrac{{3}}{{1}}}\Sigma_{\nicefrac{{u}}{{g}}}^{+}\right)}+C_{m}^{\left({}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}\right)}}{r^{m}}\end{pmatrix}.$
(11)

The zero of energy is once again the Li$(2S_{\nicefrac{{1}}{{2}}})$ \+
Li($3P_{\nicefrac{{1}}{{2}}}$) asymptote.

Figure 4: Point-wise original, and analytic MLR potentials for
$3d(3^{3}\Pi_{g})$ representing the _ab initio_ calculations of Musial and
Kucharski (2014). The top inset shows the long-range behavior in Le Roy space,
and the bottom inset shows that the MLR successfully captures the tiny second
minimum which has a depth of $\approx 13.5$ cm-1.

Since the leading term not shown in Eq. 6 is
$\frac{u(r)^{2}}{4\mathfrak{D}_{e}}$, the contribution of the $C_{3}$ terms to
the long-range form of the potential, will interfere with the desired $C_{6}$
and $C_{8}$ terms, and all $C_{9}$ and $C_{11}$ terms will therefore have
spurious contributions from the cross-terms formed by the products of the
$C_{3}$ terms with the $C_{6}$ and $C_{8}$ terms respectively. We fix this in
the same way as was done for $C_{6}$ and $C_{9}$ in Dattani  _et al._ (2008);
Le Roy  _et al._ (2009); Dattani and Le Roy (2011); Semczuk  _et al._ (2013);
Gunton  _et al._ (2013); Dattani and LeRoy (2015), by applying a
transformation to all $C_{6}$, $C_{9}$, and this time also $C_{11}$ terms:

$\displaystyle C_{6}$ $\displaystyle\rightarrow$ $\displaystyle
C_{6}+\frac{C_{3}^{2}}{4\mathfrak{D}_{e}}$ (12) $\displaystyle C_{9}$
$\displaystyle\rightarrow$ $\displaystyle
C_{9}+\frac{C_{3}C_{6}}{2\mathfrak{D}_{e}},$ (13) $\displaystyle C_{11}$
$\displaystyle\rightarrow$ $\displaystyle
C_{11}+\frac{C_{3}C_{8}}{2\mathfrak{D}_{e}}.$ (14)

where the transformation in Eq. 12 has to be made first due to Eq. 13’s
dependence on $C_{6}$.

Additionally, the long-range formulas in terms of $C_{m}$ constants in Eqs.
8,10,11 were derived under the assumption that two free atoms are interacting
with each other, and there is no overlap of the electrons’ wavefunctions as
would be in a bound molecule. To take into account the effect of electron
overlap, we use the damping function form from Le Roy  _et al._ (2011):

$\displaystyle C_{m}$ $\displaystyle\rightarrow$ $\displaystyle
C_{m}D_{m}^{(s)}\left(r\right)$ (15) $\displaystyle D_{m}^{(s)}\left(r\right)$
$\displaystyle\equiv$ $\displaystyle\left(1-e^{-\left(\frac{b^{(s)}\rho
r}{m}+\frac{c^{(s)}\left(\rho r\right)^{2}}{\sqrt{m}}\right)}\right)^{m+s},$
(16)

where for interacting atoms A and B, $\rho\equiv\rho_{{\rm
AB}}=\frac{2\rho_{{\rm A}}\rho_{{\rm B}}}{\rho_{{\rm A}}\rho_{{\rm B}}},$ in
which $\rho_{{\rm X}}\equiv\left(\nicefrac{{I^{{\rm X}}}}{{I^{{\rm
H}}}}\right)^{\nicefrac{{2}}{{3}}}$ is defined in terms of the ionization
potentials of atom X, denoted $\left(I^{{\rm X}}\right)$, and hydrogen
$\left(I^{{\rm H}}\right)$. We use $s=-1$, which as shown in Le Roy  _et al._
(2011), means that the MLR potential has the physically desired behavior
$V(r)\simeq\nicefrac{{1}}{{r^{2}}}$ in the limit as $r\rightarrow 0$. For
$s=-1$, the system independent parameters take the values $b^{(-1)}=3.30$, and
$c^{(-1)}=0.423$ Le Roy  _et al._ (2011).

###  I.3 Long-range constants

For electronic states of Li2 approaching the $2S+2P$ asymptote, the
$C_{3,6,8}$ constants for all electronic state symmetries have been calculated
with finite-mass corrections for 6Li and 7Li Tang  _et al._ (2009), and even
an attempt at relativistic corrections has been made for the $C_{3}$ constants
Tang (2015); Tang  _et al._ (2010). Furthermore, for $2S+2P$, third-order
perturbation theory has been used to calculate non-relativistic infinite-mass
values for $C_{9}$ and $C_{11}$ Tang  _et al._ (2011) , meaning that it was
possible to also include the non-relativistic infinite-mass value of $C_{10}$
calculated in Zhang  _et al._ (2007).

The situation is much less convenient for $2S+3P$. No third-order perturbation
theory calculation has been done for $C_{9}$ or $C_{11}$, and without $C_{9}$
it does not make sense to include the $C_{10}$ value, which was calculated in
the same study as for the $2S+2P$ asymptote Zhang  _et al._ (2007). Also, no
finite-mass or relativistic corrections have been calculated for the
$C_{3,6,8}$ values associated with $2S+3P$. Nevertheless, we have available
the non-relativistic infinite-mass values for $C_{3,6,8}$ that were calculated
in Zhang  _et al._ (2007), and these were reported with an order of magnitude
higher precision than in the very highly cited 1995 paper of Marinescu and
Dalgarno Marinescu and Dalgarno (1995), and only one order of magnitude lower
precision than the $2S+2P$ values which are known (see Table 2 in Dattani and
LeRoy (2015) for a list of the best known $C_{m}$ constants for each symmetry
approaching $2S+2P$). All $C_{m}$ constants that are used in this study for
$2S+3P$ are given in Table 2.

Table 2: The best currently available long-range constants for Li2 electronic
states that dissociate to $2S+3P$ (in Hartree atomic units). All values come
from Zhang  _et al._ (2007) and were calculated without relativistic
corrections, and under the assumption that both Li nuclei have infinite mass.

| ${}^{\nicefrac{{1}}{{3}}}\Sigma_{\nicefrac{{u}}{{g}}}$ | ${}^{\nicefrac{{3}}{{1}}}\Sigma_{\nicefrac{{u}}{{g}}}$ | ${}^{\nicefrac{{1}}{{3}}}\Pi_{\nicefrac{{u}}{{g}}}$ | ${}^{\nicefrac{{3}}{{1}}}\Pi_{\nicefrac{{u}}{{g}}}$
---|---|---|---|---
$C_{3}$ | 0. | 0033314 |  $-$0. | 0033314 |  $-$0. | 0016657 | 0.0016657
$C_{6}$ | 3. | 8236$\times 10^{4}$ | 3. | 8236$\times 10^{4}$ | 2. | 0282$\times 10^{4}$ | 2.0282$\times 10^{4}$
$C_{8}$ | 2. | 4870$\times 10^{7}$ | 2. | 3183$\times 10^{7}$ | 7. | 8976$\times 10^{5}$ | 3.7222$\times 10^{5}$

Table 3: Parameters defining the MLR${}_{p,q}^{r_{{\rm ref}}}(N_{\beta})$
potentials fitted to _ab initio_ points from Musial and Kucharski (2014), and
with long-range functions $u(r)$ defined according to the descriptions in
section I.3, and with long-range constants presented in Table 2. Numbers in
parentheses are 95% confidence limit uncertainties in the last digit(s) shown,
calculated from the least-squares fitting procedure.

$3b(3^{3}\Pi_{u})$ | $3B(3^{1}\Pi_{u})$ | $3C(3^{1}\Pi_{g})$ | $3d(3^{3}\Pi_{g})$
---|---|---|---
MLR${}_{6,7}^{5.9}(17)$ | MLR${}_{6,6}^{6.2}(4)$ | MLR${}_{6,8}^{7.0}(8)$ | MLR${}_{6,9}^{7.2}(4)$
$\mathfrak{D}_{e}$ |  5765.593 cm-1 |  |  | $\mathfrak{D}_{e}$ |  5368.8(38) cm-1 | $\mathfrak{D}_{e}$ |  $4066.0467$ cm-1 | $\mathfrak{D}_{e}$ |  $6045.135$ cm-1
$r_{e}$ |  3.984 527(23) Å |  |  | $r_{e}$ |  3.165 8(18) Å | $r_{e}$ |  3.184 402(23) Å | $r_{e}$ | 3. | 137 142(23) Å
$\beta_{0}$ |  0.378369938 | $\beta_{10}$ |  -3213.8074 | $\beta_{0}$ |  -0.8509 | $\beta_{0}$ |  0.78516117 | $\beta_{0}$ | 0. | 78516117
$\beta_{1}$ |  0.918196351 | $\beta_{11}$ |  1673.5923 | $\beta_{1}$ |  -0.592 | $\beta_{1}$ |  2.3951022 | $\beta_{1}$ | 2. | 39510218
$\beta_{2}$ |  2.08782371 | $\beta_{12}$ |  6482.429 | $\beta_{2}$ |  -0.038 | $\beta_{2}$ |  6.950787 | $\beta_{2}$ | 6. | 9507868
$\beta_{3}$ |  0.96103485 | $\beta_{13}$ |  -1705.759 | $\beta_{3}$ |  0.74 | $\beta_{3}$ |  -4.68625 | $\beta_{3}$ | -4. | 686246
$\beta_{4}$ |  -46.0405588 | $\beta_{14}$ |  -7329.75 | $\beta_{4}$ |  0.57 | $\beta_{4}$ |  -5.17796 | $\beta_{4}$ | -5. | 17796
$\beta_{5}$ |  -117.75187 | $\beta_{15}$ |  822.19 |  |  | $\beta_{5}$ |  5.22029 |  |
$\beta_{6}$ |  271.27645 | $\beta_{16}$ |  4297.08 |  |  | $\beta_{6}$ |  7.5253 |  |
$\beta_{7}$ |  880.09337 | $\beta_{17}$ |  -122.54 |  |  | $\beta_{7}$ |  -2.100 |  |

##  II MLR potentials

It has been suggested that fully analytic potentials Huang and Le Roy (2003),
and specifically the MLR Pashov (2008) may not have the flexibility required
to capture some features such as multiple minima and shelves (see examples of
these features appearing in $2S-3P$ potentials of Li2 in Figs 1-4). While no
attempt (as far as I am aware) has thus far been made to use a fully analytic
potential to capture such features, an increasing number of applications of
the MLR potential after the publications of Huang and Le Roy (2003); Pashov
(2008) has made it a strong case for a “universal” potential form. MLR-type
potentials have successfully described spectroscopic data for many electronic
states of many diatomic molecules Le Roy  _et al._ (2006); Roy and Henderson
(2007); Salami  _et al._ (2007); Shayesteh  _et al._ (2007); Le Roy  _et al._
(2009); Coxon and Hajigeorgiou (2010); Stein  _et al._ (2010); Piticco  _et
al._ (2010); Le Roy  _et al._ (2011); Ivanova  _et al._ (2011); Dattani and Le
Roy (2011); Xie  _et al._ (2011); Yukiya  _et al._ (2013); Knöckel  _et al._
(2013); Semczuk  _et al._ (2013); Wang  _et al._ (2013); Li  _et al._ (2013a);
Gunton  _et al._ (2013); Meshkov  _et al._ (2014); Dattani (2014); Coxon and
Hajigeorgiou (2015); Walji  _et al._ (2015); Dattani (2015); Dattani and LeRoy
(2015); Dattani  _et al._ (2014). It has also become customary to fit _ab
initio_ data for diatomic Xiao  _et al._ (2013a, b); Kedziera  _et al._
(2015); Teodoro  _et al._ (2015); You  _et al._ (2015a, b, c) and polyatomic
Li and Le Roy (2008); Li  _et al._ (2010); Tritzant-Martinez  _et al._ (2013);
Wang  _et al._ (2013); Li  _et al._ (2013b); Ma  _et al._ (2014) systems to
MLR models . Therefore, we use the MLR model in this study, and the results
here support the idea of the MLR model being a strong candidate for a
“universal” model for potential energy curves and surfaces.

All MLR potentials were made by fitting to the _ab initio_ points of Ref.
Musial and Kucharski (2014) with the program ${\tt DPotFIT}$ Le Roy  _et al._
(2013). Since this is a non-linear least-squares fitting, ‘starting
parameters’ are needed in order to allow ${\tt DPotFit}$ to achieve reasonable
fits. Starting parameters were obtained from the program ${\tt BetaFIT}$ Le
Roy  _et al._ (2013). When fitting to _ab initio_ points, ${\tt DPotFIT}$ aims
to minimize the dimensionless root mean square deviation:

$dd\equiv\sqrt{\frac{1}{N_{{\rm data}}}\sum_{i=1}^{N_{{\rm
data}}}\left(\frac{V_{{\rm MLR}}(i)-V_{{\rm\textit{ab
initio}}}(i)}{u_{\textit{ab initio}}(i)}\right)^{2}},$ (17)

where $V_{{\rm{\rm MLR}}}(i)$ and $V_{{\rm\textit{ab initio}}}(i)$ are the
values of the respective potentials at the $i^{{\rm th}}$ internuclear
distance value (the order of course does not matter) and $N_{{\rm data}}$ is
the total number of _ab initio_ points to which the MLR potential is being
fitted. $u_{\textit{ab initio}}(i)$ is the uncertainty in the $i^{{\rm th}}$
_ab initio_ point, so that the MLR potential is likely to lie more closely to
_ab initio_ points at distances where the _ab initio_ calculation is expected
to be more reliable, and the requirement for the MLR potential to match the
_ab initio_ is less harsh in areas where the _ab initio_ calculation is
expected to be less accurate.

It is extremely hard to determine accurate estimates on the uncertainties for
_ab initio_ points. The _ab initio_ points we are using from Musial and
Kucharski (2014) were all calculated with the same basis set (which the
authors denoted by ANO-RCC+), therefore there is no indication of the size of
the basis set error. Furthermore, all of their calculations were done with the
same number of excitations included in their coupled cluster method: FS-
CCSD(2,0) only includes 1- and 2-electron excitations, so it would be
extremely unlikely to estimate the deviation from the full 6-electron (FCI)
limit. Perhaps even more importantly, the calculations of Musial and Kucharski
(2014) neglected relativistic, spin-orbit, and non-Born-Oppenheimer effects,
so accurately estimating $u_{{\rm\textit{ab initio}}}(i)$ might seem
impossible.

However, in my recent benchmark paper Dattani and LeRoy (2015), it was shown
that none of the vibrational energies associated with the _ab initio_
potential from Musial and Kucharski (2014) for the $b(1^{3}\Pi_{u})$ state of
Li${}_{2},$ deviated from the empirical potential’s vibrational energies by
more than 12 cm-1. Since that _ab initio_ potential used the same basis set
and method as their potentials for the electronic states approaching $2S+3P$,
I aimed to make $V_{{\rm MLR}}(i)-V_{{\rm\textit{ab initio}}}(i)$ less than 15
cm-1 for all $i$ except at very small internuclear distances near the $r=0$
singularity where the inner wall of the potential rapidly increases, crosses
the dissociation limit, and then attains extremely large energy values. The
exact values used for $u_{\textit{ab initio}}(i)$ that were used are presented
in Tables 4, 6, 8 and 10. Furthermore, there are places in which it was
desirable to make $u_{\textit{ab initio}}(i)$ smaller than 15 cm-1. This was
in places where the potentials from Musial and Kucharski (2014) had features
such as tiny second minima, tiny shelves, or any other type of abrupt change.
The subsections (below) for each electronic state will describe in detail the
nature of these features and how this affected the choice of $u_{\textit{ab
initio}}(i)$ (once again the exact values are given in Tables 4, 6, 8 and 10).

The MLR model was fitted to the points from Musial and Kucharski (2014) with
various manually adjusted values of the MLR parameters
$(N_{{\rm\beta}},p,q,r_{{\rm ref}})$ in search for the lowest $dd$ according
to Eq. 17 such that increasing $N_{{\rm\beta}}$ no longer reduced $dd$
significantly further than the best $dd$ obtained with the previous increase
in $N_{\beta}$. Details for each electronic state are described in the
subsections below which focus on each state.

###  II.1 The $3b(3^{3}\Pi_{u})$ state

The first $b$ state of Li2 dissociates to the $2S+2P$ asymptote. A very
detailed analysis of theory vs experiment for the first $b$ state of Li2 was
recently reported Dattani and LeRoy (2015), which summarized 14 different
experiments providing new information on the $b(1^{3}\Pi_{u})$ state since
1983 Engelke and Hage (1983); Preuss and Baumgartner (1985); Rai  _et al._
(1985); Xie and Field (1985a, b, 1986a); Rice  _et al._ (1986); Rice and Field
(1986); Li  _et al._ (1992); Linton  _et al._ (1992); Li  _et al._ (1996c);
Weyh  _et al._ (1996); Russier  _et al._ (1997); Lazarov  _et al._ (2001a),
and mentioned several other papers that involved this state without providing
information on any new levels. Due to the spin-orbit coupling between $b$
states of alkali dimers and their respective $A(^{1}\Sigma_{u}^{+})$ states,
recent experimental and theoretical papers have studied the lowest $b$ state
of Rb2, Salami  _et al._ (2007); Drozdova  _et al._ (2013); Tomza  _et al._
(2013), NaCs Zaharova  _et al._ (2009), KCs Kruzins  _et al._ (2010), RbCs
Docenko  _et al._ (2010), Cs2 Bai  _et al._ (2011) and NaK Harker  _et al._
(2015).

It is thus surprising that no experiments have been reported for the second
$b$ state of Li2, which dissociates to the $2P+2P$ asymptote. The present
paper is concerned with the third $b$ state, which dissociates to $2S+3P$. The
_ab initio_ potential for $3b(3^{3}\Pi_{u})$ from Musial and Kucharski (2014)
has a small shelf-like feature before the minimum, and another much longer one
closer to dissociation (see Fig. 1). The first shelf is located between $v=0$
and $v=1$, and it lasts from about $3.2-3.6$ Å. The second shelf starts after
$v=27$ and lasts from about $6.4-8$ Å. Despite these fairly pronounced shelf
features, at the resolution of the _ab initio_ points (which is about
$0.1\,\mbox{\AA}$), the $3b$ state only has one minimum!

In Section II it was mentioned that, with the exception of points at very
small values of $r$, the goal was to match all _ab initio_ points of Musial
and Kucharski (2014) to within $\pm$15 cm-1 (and points at larger values of
$r$ even better), since this was about the level of accuracy found when
comparing _ab initio_ points Musial and Kucharski (2014) to an empirical
potential for $b(1^{3}\Pi_{u})$ in Dattani and LeRoy (2015). Preliminary fits
used such a weighting scheme for the least-squares fitting, except with points
comprising the two shelves mentioned in the previous paragraph, weighted with
much smaller uncertainties. This was especially important for the second
shelf, which only spanned a range of $<100$ cm-1: Because if the discrepancy
between the MLR and _ab initio_ was $-15$ cm-1 at one point, and $+15$ cm-1 at
another point, the $30$ cm-1 range of discrepancy would constitute a
significant portion of the range of the entire shelf itself. After these
preliminary fits, it was found that in order to get the MLR matching the
original data with the desired precision, it helped to decrease the
uncertainties on the non-shelf _ab initio_ points to slightly below $\pm 15$
cm-1. The best fits were found with the weights shown in Table 4. Once these
final weights were chosen, fits were performed for 252 different combinations
of the MLR parameters $(N_{\beta},p,q,r_{{\rm ref}})$, with $4\leq
N_{\beta}\leq 17$, $6\leq p\leq 8$, $2\leq q\leq 8$, $(5\leq r_{{\rm ref}}\leq
6.5)$ Å, though not every point in the convex hull formed by these ranges was
used. For example, some fits were done with $(p,q)=(6,2)$ and some fits were
done with $(p,q)=(8,6)$ but it did not seem necessary to do fits with
$(p,q)=(8,2)$. The best fit was found with $(N_{\beta},p,q,r_{{\rm
ref}})=(17,6,7,5.9\,$Å), and had $dd=1.548$, while the best fit with
$N_{\beta}=16$ was with $(N_{\beta},p,q,r_{{\rm
ref}})=(16,6,6,5.3\,\text{\AA})$ and had $dd=2.790$. Apart from in the inner
wall of the potential, the worst discrepancy between the MLR and the original
points for this $N_{\beta}=16$ case it was $>30$ cm-1, while for this
$N_{\beta}=17$ case was $<11$ cm-1 so it was quite easy to select the
$N_{\beta}=17$ fit. Since this $N_{\beta}=17$ fit satisfied all of our
desiderata, $N_{\beta}=18$ fits were not explored.

The final MLR parameters for the chosen case are given in Table 3. The inset
of Fig. 1 shows the long-range behavior of the MLR potential and the original
_ab initio_ points of Musial and Kucharski (2014) in Le Roy space, and
compares them to the theoretical long-range potential based on Eq. 8 and the
long-range constants in Table 2. The agreement is surprisingly excellent,
however after about 17.7 Å $(1/r^{3}\approx 0.001\,8\,\mbox{\AA}^{-3})$ we see
that the original points dip below the theoretical curve, which should not
happen because $C_{8}$ is attractive (see Section 4.3 of Roy and Henderson
(2007), for example). The fact that the MLR potential matches the theoretical
curve in this regard, is yet another advantage of using an MLR to represent
the _ab initio_ points. Furthermore, while not shown in Fig. 1, the
theoretical long-range curve _without_ damping, matches the damped curve shown
in the figure to graphical accuracy at least in the range $14\,\mbox{\AA}\leq
r\leq\infty$ $(0\leq 1/r^{3}=0.000\,36\,\mbox{\AA}^{-3})$, so this conclusion
about the original points spuriously dipping below the theoretical curve is
true whether or not long-range damping is considered.

###  II.2 The $3B(3^{1}\Pi_{u})$ state

The first $B$ state has a barrier before dissociating to the $2S+2P$
asymptote, and has therefore been the subject of many empirical studies Hessel
and Vidal (1979); Russier  _et al._ (1994); Bouloufa  _et al._ (1999);
Cacciani and Kokoouline (2000); Bouloufa  _et al._ (2001a, b); Huang and Le
Roy (2003). It has also been used to study other states, such as in Bernheim
(1982); Barakat  _et al._ (1986); Linton  _et al._ (1990); Kubkowska  _et al._
(2007); Linton  _et al._ (1992). The second $B$ state potential (sometimes
called the “$C^{1}\Pi_{u}$ state” rather than the $2B$ state) dissociates to
$2P+2P$ and hugs the inside of the first minimum of the
$2A(1^{1}\Sigma_{u}^{+})$ potential, and due to the perturbations between
these $2B$ and $2A$ states, there have been many empirical studies of the $2B$
state Barrow  _et al._ (1960); Ishikawa  _et al._ (1991); Weyh  _et al._
(1996); Ross  _et al._ (1998); Kasahara  _et al._ (2000); Kubkowska  _et al._
(2007).

The present paper is concerned with third $B$ state. The $3B$ state
dissociates to $2S+3P$ and there has only been one experiment which studied
the third $B$ state (sometimes called the “$D^{1}\Pi_{u}$ state”) Barrow  _et
al._ (1960), which was over 55 years ago! The authors of that work mentioned
in their paper that they were not able to confidently assign vibrational
quantum numbers to their data, and therefore they were only able to conclude
that $T_{e}<34\,140$ cm-1 and $\omega_{e}\approx 250$ cm-1. In that study, the
anharmonic values $x_{e}\omega_{e}$, equilibrium rotational constants $B_{e}$,
and dissociation energies $\mathfrak{D}_{e}$ were determined for the $2B$
state of Li2 and for the $2B$ and $3B$ states of Na2, but not for the $3B$
state of Li2.

It is no surprise that none of the experiments on the $2B$ state showed any
indication of a barrier in the potential, because the leading long-range term
$(C_{5})$ is attractive Zhang  _et al._ (2007). However, it is perhaps
surprising that the _ab initio_ potential of Musial and Kucharski (2014) for
the $3B$ state does not have a barrier (at least before the calculations
stopped at about 21 Å), because the leading long-range interaction term
$(C_{3})$ for this state is repulsive Zhang  _et al._ (2007). There is however
a good theoretical explanation for the lack of barrier in the $3B$ state. The
long-range potential for $3B$ is not just a simple sum of inverse powers
($-C_{3}/r^{3}-C_{6}/r^{6}\cdots$), but it is the middle eigenvalue of the
$3\times 3$ spin-orbit interaction matrix of Eq. 10, which involves the
$6a(6^{3}\Sigma_{u})$ and $3b(3^{3}\Pi_{u})$ states. For large internuclear
distances, the $6a$ state pushes down on the $3B$ state enough to remove the
barrier. Indeed, numerical calculations of the eigenvalues of the $3\times 3$
matrix show that the potential is attractive at all distances (beyond the
repulsive inner wall for $r\ll r_{e}$). This $3\times 3$ matrix for $3B$ is
the exact same as the one for the first $B$ state, which _does_ have a
barrier, but for the $3B$ state the $C_{3}$ is three orders of magnitude
smaller than for the first $B$ state, and the $C_{6}$ is one order of
magnitude bigger than for the first $B$ state Zhang  _et al._ (2007). This
highlights the importance of using the $3\times 3$ coupling matrix because a
simple inverse power sum with a negative leading long-range term (such as the
negative $C_{3}$ in the present case) is guaranteed to have a barrier, and
there was even a barrier when using the middle eigenvalue of the $3\times 3$
coupling matrix for the $2S+2P$ values of $C_{m}$, but the specific values of
$C_{m}$ at $2S+3P$ seem to be past a bifurcation point at which the barrier is
lost.

Since alkali parent states of $B(^{1}\Pi_{u})$ symmetry only have one spin-
orbit daughter state ($\Omega_{u/g}=1_{u})$, we do not need to worry about
defining the MLR long-range function $u(r)$ in a piece-wise manner, and it is
simply defined as the middle eigenvalue of the $3\times 3$ interaction matrix
of Eq. 10. Also, since there are no features such as barriers, multiple minima
or shelves, the weighting strategy was straightforward. Preliminary fits were
done with all points from Musial and Kucharski (2014) being weighted with
uncertainty $\pm 15$ cm-1 if $V(r)<-100\,$cm-1, with $\pm 5$ cm-1 if
$V(r)<-20$ cm-1, with $\pm 1$ cm-1 if $V(r)<1$ cm-1 and $\pm 0.5$ cm-1 for the
one point for which $-1$ cm${}^{-1}<V(r)<0$ cm-1. It was then found that most
uncertainties could be even further reduced without making the fitting too
difficult, and that for very small values of $r$ it was very difficult to
achieve $\pm 15$ cm-1 agreement in the fit. In the end, the points from Musial
and Kucharski (2014) for the smallest values of $r$ (with $V(r)>-4000\,$cm-1)
were weighted with uncertainty $\pm 100$ cm-1, and all other points were
weighted with $\pm 1$ cm-1, except for the last point which was weighted with
$\pm 0.5$ cm-1 because $V(r)$ itself was only $\approx-0.7$ cm-1. These final
weights are shown in Table 6.

Once these final weights were chosen, fits were performed for 150 different
combinations of the MLR parameters $(N_{\beta},p,q,r_{{\rm ref}})$, with
$3\leq N_{\beta}\leq 5$, $6\leq p\leq 7$, $2\leq q\leq 7$, $(5\leq r_{{\rm
ref}}\leq 10)$ Å, though not every point in the convex hull formed by these
ranges was used. The best fit with $N_{\beta}=4$ was with $(p,q,r_{{\rm
ref}})=(6,6,6.2\,\text{\AA})$ which had $dd=4.307$ while the best fit found
using $N_{\beta}=5$ was only marginally better ($dd=4.128)$ and no fits were
found with $N_{\beta}=3$ that had $dd<10$. Therefore, the choice of MLR model
for this electronic state was easy to make, and the final parameters are
listed in Table 3.

###  II.3 The $3C(3^{1}\Pi_{g})$ state

The first $C(^{1}\Pi_{g})$ state dissociates to $2S+2P$ and was not studied in
detail until 1990 Miller  _et al._ (1990). This was 11 years after the second
state of $C(^{1}\Pi_{g})$ symmetry (sometimes called the “$G^{1}\Pi_{g}$
state” since it was given this name in Bernheim (1981)) was studied in detail
in 1979 Bernheim  _et al._ (1979). This $2C$ state dissociates to $2P+2P$ and
was studied again in a series of follow-up papers by Bernheim _et al._
Bernheim (1981); Bernheim  _et al._ (1982, 1983). Impressively, empirical
spectroscopic constants have also been reported for all Rydberg states in the
series $nd\pi^{1}\Pi_{g}$ for $n=3-15$ (!) Bernheim  _et al._ (1982, 1983). In
the same paper, it was determined that the $n=3$ state in this series is in
fact the $2C$ state.

The third $C$ state is the subject of the present work, since it dissociates
to $2S+3P$. It only has one spin-orbit daughter state, which has $1_{g}$
symmetry and couples to the $1_{g}$ daughters of the $3c(3^{3}\Sigma_{g}^{+})$
and $3d(3^{3}\Pi_{g})$ states. The potential energy of the $3C$ state’s
$1_{g}$ daughter is given by the middle eigenvalue of the appropriate $3\times
3$ matrix. Preliminary fits were done with the same weighting scheme as for
the $3B$ state, except with the points surrounding the second minimum weighted
more strongly (with $\pm 0.5$ cm-1), since the depth of this well is $<10$
cm-1 and therefore it would not be satisfactory to fit to these points with an
agreement of only $\pm 15$ cm-1! It was then found that many points could be
weighted more strongly without making the fitting too difficult. The weights
were adjusted to $\pm 10$ cm-1 for points at the smallest values of $r$
($V(r)>-3000$ cm-1), to $\pm 5$ cm-1 for the rest of the points with
$V(r)<-1000$ cm-1, to $\pm 1$ cm-1 for points with $V(r)<-1$ cm-1, except for
the points near the second minimum, and the very last point for which
$V(r)\approx-0.75$ cm-1. The final weights are presented in Table 8.

With these final weights, fits were performed for 302 different combinations
of the MLR parameters $(N_{\beta},p,q,r_{{\rm ref}})$, with $2\leq
N_{\beta}\leq 11$, $6\leq p\leq 11$, $(4\leq r_{{\rm ref}}\leq 12)$ Å, though
not every point in the convex hull formed by these ranges was used. The best
fit with $N_{\beta}=8$ was with $(p,q,r_{{\rm ref}})=(6,5,7.5\,\text{\AA})$
which had $dd=1.612$, but this potential did not capture the second minimum
very well (particularly, it approaches the barrier leading to that minimum
with the _ab initio_ point of $V(7.408\,\text{\AA})=-88.76$ cm-1 being
represented with a discrepancy of $>5.5$ cm-1). The best fit with
$N_{\beta}=8$ which captured $V(7.408\,\text{\AA})$ with a discrepancy of $<2$
cm-1 was with $(p,q,r_{{\rm ref}})=(6,8,7.0\,\text{\AA})$ which had
$dd=3.536$. None of the $N_{\beta}=7$ models that reproduced
$V(7.408\,\text{\AA})$ with discrepancy $<2$ cm-1 had an overall $dd<4$, and
while $N_{\beta}=9$ fits were found with discrepancies for this point $<2$
cm-1 and overall $dd$ as low as $2.593$, there were no points beyond $r=2.6$ Å
for which the $N_{\beta}=8$ case with $dd=3.536$ misrepresented an original
point by $>18$ cm-1 (the highest discrepancy was $17.37$ cm-1 at 3.175 Å, and
among these $N_{\beta}=9$ cases, the lowest discrepancy for this same point
was 12.70 cm-1). While deciding not to go beyond $N_{\beta}=8$ was not an easy
choice, there is not much reason to believe that the calculation in Musial and
Kucharski (2014) for $V(3.175\text{}\text{\AA})$ is so precise that
representing it more closely by $\approx 5$ cm-1 is worth adding an extra
parameter. Here it is mentioned that while the comparison against the
empirical potential in Dattani and LeRoy (2015) for the lowest $b$ state
showed no discrepancy of $>12$ cm-1, that paper also noted the surprisingly
small effect of Born-Oppenheimer breakdown in that system, meaning that it is
likely that the potentials in Musial and Kucharski (2014) for other electronic
states (especially ones approaching $2S-3P$, which seem to interact with each
other more than the ones approaching the $2S-2P$ state) will be accurate to
slightly less precision than $\pm$12 cm-1. The final MLR parameters for the
chosen model are listed in Table 3.

###  II.4 The $3d(3^{3}\Pi_{g})$ state

The first $d$ state has a potential energy curve which approaches the $2S+2P$
asymptote, but the _ab initio_ calculations of Musial and Kucharski (2014)
indicate that it has no bound states. Therefore, it is no surprise that no
bound levels have been found in experiments on this state, though it was
indeed involved in some experiments Li  _et al._ (1998); Lazarov  _et al._
(2001b); Dai  _et al._ (2005). While the prediction in Musial and Kucharski
(2014) that the first $d$ state has no bound levels is likely to be true, it
should be noted that _ab initio_ predictions of this sort are not always
reliable. The 1995 _ab initio_ study of Poteau and Spiegelmann (1995)
predicted that the $1^{1}\Sigma_{u}^{-}$ state would have no bound levels, but
the 2014 calculations of Musial and Kucharski (2014) found there to be a
dissociation energy of $\mathfrak{D}_{e}=$14 cm-1 and an equilibrium harmonic
frequency of $\omega_{e}=10$ cm-1, indicating the existence of at least two
bound vibrational levels! Likewise, the 2006 _ab initio_ study of Jasik and
Sienkiewicz (2006) found the $1^{3}\Delta_{u}$ state to not have any bound
levels, but the earlier 1995 study of Poteau and Spiegelmann (1995) and the
2014 study of Musial and Kucharski (2014) both predicted $\mathfrak{D}_{e}\geq
3430$ cm-1 and $\omega_{e}\leq 255$ cm-1.

The second $d$ state has been studied extensively. Spectroscopic measurements
for $2d$ were made in Xie and Field (1986a); Yiannopoulou  _et al._ (1995b);
Li  _et al._ (1996b, a); Weyh  _et al._ (1996); Russier  _et al._ (1997); Li
_et al._ (2007), and $2d$ was also used in various other experimental studies
such as Xie and Field (1985c, 1986b); Rice  _et al._ (1986); Rice and Field
(1986); Linton  _et al._ (1992, 1999); Lazarov  _et al._ (2001b); Dai  _et
al._ (2005).

The focus of this paper is on the third $d$ state, which is the only
$\Lambda=\Pi$ state dissociating to $2S+3P$ for which rovibrationally resolved
spectra have been measured, but only 13 lines were observed (with
$v=6,7,8,10$) Yiannopoulou  _et al._ (1995b). Hyperfine structure was also
studied experimentally for $3d$ in Li  _et al._ (1996b), but it was only for
the $N=6,8$ levels of $v=8$, which had already been studied without focus on
hyperfine structure in Yiannopoulou  _et al._ (1995b). Finally, the $3d$ state
was involved in the experiments of Li  _et al._ (1996a), but the focus of that
study was not the $3d$ state.

Since the $3d$ state has four spin-orbit daughter states
($0_{g}^{+},0_{g}^{-},1_{g},2_{g}$, analogous with the $3b$ state), we treat
the $2_{g}$ symmetry in this paper, since there is only one state dissociating
to $2S+3P$ with $2_{g}$ symmetry and therefore the long-range potential is a
simple sum of inverse-power terms rather than a complicated $2\times 2$ or
$3\times 3$ interaction matrix. The fitting strategy was very similar to what
it was for the $3C$ state, except the $3d$ state seemed to require stronger
weighting of the points near the second minimum, and weaker weighting of other
points. These final weights are shown in Table 6.

Once these final weights were chosen, fits were performed for 238 different
combinations of the MLR parameters $(N_{\beta},p,q,r_{{\rm ref}})$, with
$3\leq N_{\beta}\leq 11$, $5\leq p\leq 8$, $2\leq q\leq 10$, $(5\leq r_{{\rm
ref}}\leq 7.5)$ Å, though not every point in the convex hull formed by these
ranges was used, and it is noted that $p$ must be $\geq 6$ in order to ensure
the correct long-range behavior Dattani and Le Roy (2011), but fits with $p=5$
were still instructive to better understand the model dependence for this
potential. The best fit with $N_{\beta}=4$ and $p\geq 6$ was with
$(p,q,r_{{\rm ref}})=(6,7,7.2\,\text{\AA})$ which had $dd=4.292$, only 0.043
higher than the best fit found with $p=5$. Fits with $N_{\beta}=5$ had $dd$
values as low as $2.972$ but the mentioned $N_{\beta}=4$ case did not
misrepresent any of the original points beyond $2.434\,\text{\AA})$ by $>13$
cm-1, so all desiderata were satisfied without resorting to $N_{\beta}=5$. No
$N_{\beta}=3$ cases had $dd<10$, so the choice of MLR model for this
electronic state was easy to make, and the final parameters are listed in
Table 3.

##  III Conclusion

Analytic MLR potentials were fitted to the _ab initio_ points from Musial and
Kucharski (2014) and with correct long-range behavior incorporated according
to effects of spin-orbit coupling described in Eqs. 8, 10, 11 and the long-
range constants in Table 2. Despite the potentials from Musial and Kucharski
(2014) having unusual features such as multiple minima, barriers, and shelves,
which have never been described by an MLR-type model before, all of these
features were successfully captured with the MLR model. This answers an age-
old question of whether or not fully analytic potentials can have the
flexibility needed in order to capture such features. Pashov’s 2008 paper
“Pointwise and analytic potentials for diatomic molecules. An attempt for
critical comparison” Pashov (2008) described lack of flexibility as one of the
three drawbacks of analytic potentials, and suggested that the MLR model may
not be able to capture double minima or shelf-like features. Five years
earlier in 2003, Huang and Le Roy suggested in Huang and Le Roy (2003) that
Pashov’s pointwise approach would be the method of choice for potentials such
as those described in this paper:

> “A particular strength of [Pashov’s pointwise] model is the fact that it has
> more local flexibility than do fully analytical potential function forms, in
> that a shift of one potential point has only a modest effect on the function
> outside its immediate neighborhood. This would tend to make [Pashov’s
> pointwise] model the method of choice for cases where the potential has
> substantial local structure or undergoes an abrupt change of character on a
> small fraction of the overall interval, such as occurs near an avoided curve
> crossing. In contrast, a change in one of the parameters defining a [fully
> analytic] such as our DELR function will in general affect the potential
> across the whole domain. This makes the parameters defining [fully analytic]
> potentials very highly correlated and can give rise to difficulty in
> achieving full unique convergence in a fit.”

At the time when this quote was written, the MLR model did not exist yet, and
at the time of Pashov’s paper Pashov (2008), only a primitive (less flexible)
form of the MLR existed, which was used in just four simple cases of ground
electronic states Le Roy  _et al._ (2006); Roy and Henderson (2007); Salami
_et al._ (2007); Shayesteh  _et al._ (2007). It is possible that the notion
that analytic potentials cannot capture such special features may be
attributed to the lack of diversity in attempted applications at that early
stage in time, and the lack of some of the newer MLR features which were
introduced in Le Roy  _et al._ (2009) and Le Roy  _et al._ (2011) for
increasing flexibility, and in Dattani  _et al._ (2008) for correcting the
long-range behavior.

This paper also represents, to the best of my knowledge, the most detailed
study of analytic potentials for excited electronic molecular states.

## Acknowledgments

I am indebted to Monique Aubert-Frécon for taking the time to look through her
old notes to answer my request for advice on the analytic form for the long-
range potentials at the $nS-n^{\prime}P$ asymptotes of alkali dimers. I also
thank Michael Bromley of The University of Queensland (Australia) for advice
about the long-range constants at the $nS-n^{\prime}P$ asymptote, and Robert
J. Le Roy of University of Waterloo (Canada) for many helpful discussions.

Table 4: Quality of MLR${}_{6,7}^{5.9}(17)$ fit to original _ab initio_ energies of Musial and Kucharski (2014) for the $3b(3^{3}\Pi_{2_{u}})$ state. When an isotopologue listed in Table 5 has at least one vibrational energy level beyond the range of _ab initio_ data available, approximate distances are given for the outer classical turning points of the corresponding vibrational wavefunctions.  $r$ | Original | Weight | MLR - Original | MLR
---|---|---|---|---
[Å] | [cm-1] | [cm-1] | [cm-1] | [cm-1]
2. | 328 380 | -1 501. | 30 | 40.00 | 195. | 839 | -1305. | 461
2. | 434 215 | -2 661. | 44 | 40.00 | 9. | 542 | -2651. | 898
2. | 540 051 | -3 556. | 46 | 40.00 | -79. | 794 | -3636. | 254
2. | 645 886 | -4 232. | 00 | 40.00 | -104. | 149 | -4336. | 149
2. | 751 721 | -4 727. | 14 | 40.00 | -89. | 498 | -4816. | 638
2. | 857 557 | -5 075. | 00 | 20.00 | -57. | 523 | -5132. | 523
2. | 963 392 | -5 304. | 35 | 10.00 | -25. | 034 | -5329. | 384
3. | 069 228 |  -5 440. | 87 | 0.70 |  -2. | 753 |  -5443. | 623
3. | 175 063 |  -5 508. | 47 | 0.70 |  5. | 647 |  -5502. | 823
3. | 280 899 |  -5 530. | 85 | 0.70 |  3. | 179 |  -5527. | 671
3. | 386 734 |  -5 533. | 92 | 0.70 |  -1. | 971 |  -5535. | 891
3. | 492 570 |  -5 543. | 14 | 0.70 |  -2. | 801 |  -5545. | 941
3. | 598 405 |  -5 576. | 94 | 0.70 |  1. | 583 |  -5575. | 357
3. | 704 240 |  -5 634. | 66 | 0.70 |  3. | 546 |  -5631. | 114
3. | 810 076 |  -5 700. | 51 | 0.70 |  0. | 373 |  -5700. | 137
3. | 915 911 |  -5 750. | 55 | 0.70 |  -2. | 666 |  -5753. | 216
4. | 021 747 |  -5 761. | 08 | 0.70 |  -0. | 047 |  -5761. | 127
4. | 127 582 | -5 708. | 41 | 0.90 | 2. | 634 | -5705. | 776
4. | 233 418 | -5 578. | 48 | 0.90 | -0. | 382 | -5578. | 862
4. | 339 253 | -5 376. | 12 | 1.00 | -3. | 307 | -5379. | 427
4. | 445 089 | -5 120. | 65 | 2.00 | 1. | 698 | -5118. | 952
4. | 550 924 | -4 832. | 92 | 4.00 | 9. | 143 | -4823. | 777
4. | 656 759 | -4 528. | 51 | 4.00 | 7. | 006 | -4521. | 504
4. | 762 595 | -4 217. | 95 | 4.00 | -4. | 138 | -4222. | 088
4. | 868 430 | -3 908. | 28 | 4.00 | -9. | 716 | -3917. | 996
4. | 974 266 | -3 603. | 65 | 1.00 | -1. | 519 | -3605. | 169
5. | 080 101 | -3 307. | 57 | 3.00 | 10. | 262 | -3297. | 308
5. | 185 937 | -3 022. | 26 | 3.00 | 9. | 62 | -3012. | 64
5. | 291 772 | -2 749. | 45 | 1.00 | -2. | 696 | -2752. | 146
5. | 609 278 | -2 018. | 82 | 1.00 | 1. | 364 | -2017. | 456
5. | 820 949 | -1 611. | 25 | 1.00 | -2. | 927 | -1614. | 177
6. | 138 456 | -1 127. | 31 | 1.00 | 4. | 595 | -1122. | 715
6. | 350 127 |  -958. | 32 | 0.30 |  -0. | 912 |  -959. | 232
6. | 667 633 |  -935. | 05 | 0.30 |  0. | 886 |  -934. | 164
6. | 879 304 |  -931. | 76 | 0.30 |  -1. | 386 |  -933. | 146
7. | 196 810 |  -925. | 40 | 0.30 |  1. | 252 |  -924. | 148
7. | 408 481 |  -917. | 28 | 0.30 |  -1. | 012 |  -918. | 292
7. | 725 987 | -895. | 77 | 1.50 | 3. | 616 | -892. | 154
7. | 937 658 | -872. | 50 | 1.50 | 5. | 113 | -867. | 387
8. | 255 164 | -817. | 85 | 1.50 | -2. | 201 | -820. | 051
8. | 466 835 | -764. | 30 | 4.00 | -6. | 928 | -771. | 228
8. | 784 342 | -656. | 98 | 4.00 | -5. | 076 | -662. | 056
8. | 996 013 | -572. | 26 | 4.00 | 9. | 21 | -563. | 05
9. | 260 601 | -464. | 06 | 4.00 | 8. | 64 | -455. | 42
9. | 525 190 | -364. | 42 | 4.00 | 8. | 00 | -356. | 42
9. | 789 778 | -280. | 36 | 4.00 | 7. | 29 | -273. | 07
10. | 054 367 | -213. | 64 | 4.00 | 6. | 54 | -207. | 1
10. | 318 956 | -163. | 16 | 4.00 | 5. | 78 | -157. | 38
10. | 583 544 | -125. | 85 | 2.00 | 5. | 04 | -120. | 81
11. | 112 721 | -78. | 44 | 2.00 | 4. | 35 | -74. | 09
11. | 641 899 | -52. | 11 | 2.00 | 3. | 69 | -48. | 42
12. | 171 076 | -36. | 52 | 2.00 | 3. | 16 | -33. | 36
12. | 700 253 | -26. | 43 | 2.00 | 2. | 72 | -23. | 71
13. | 229 430 | -19. | 84 | 2.00 | 2. | 37 | -17. | 47
13. | 758 607 | -15. | 24 | 5.50 | 2. | 09 | -13. | 15
14. | 287 785 | -11. | 94 | 5.50 | 1. | 88 | -10. | 06
14. | 816 962 | -9. | 53 | 5.50 | 1. | 71 | -7. | 82
15. | 346 139 | -7. | 55 | 5.00 | 1. | 53 | -6. | 02
15. | 875 316 | -6. | 24 | 5.00 | 1. | 37 | -4. | 87
16. | 933 671 | -4. | 04 | 5.00 | 1. | 21 | -2. | 83
17. | 992 025 | -2. | 73 | 5.00 | 1. | 08 | -1. | 65
19. | 050 380 | -1. | 85 | 5.00 | 0. | 96 | -0. | 89
20. | 108 734 | -1. | 19 | 5.00 | 0. | 86 | -0. | 33
21. | 167 088 | -0. | 75 | 0.50 | 0. | 73 | -0. | 02
65. |  |  $-$ | $-$ | $-$ | -0. | 003
70. |  |  $-$ | $-$ | $-$ | -0. | 002
Table 5: Vibrational energies in cm-1 for the $3b(3^{3}\Pi_{2_{u}})$ state predicted by the MLR${}_{6,7}^{5.9}(17)$ from Table 3. $v$ |  6,6Li2 |  6,7Li2 |  7,7Li2 |  11,11Li2
---|---|---|---|---
$\mathfrak{D}_{e}$ | $5\,765.593$ | $5\,765.593$ | $5\,765.593$ | $5\,765.593$
$0$ | $-5\,654.932$ | -5658.488 | -5662. | 222 | -5661. | 726
$1$ | $-5\,493.765$ | -5500.003 | -5506. | 570 | -5505. | 696
$2$ | $-5\,385.710$ | -5397.683 | -5356. | 570 | -5408. | 260
$3$ | $-5\,241.269$ | -5260.427 | -5280. | 073 | -5277. | 492
$4$ | $-5\,083.976$ | -5110.717 | -5138. | 177 | -5134. | 567
$5$ | $-4\,914.852$ | -4980.180 | -4984. | 997 | -4980. | 318
$6$ | $-4\,737.691$ | -4780.180 | -4823. | 968 | -4818. | 203
$7$ | $-4\,554.829$ | -4605.151 | -4657. | 110 | -4650. | 263
$8$ | $-4\,368.240$ | -4426.227 | -4486. | 207 | -4478. | 297
$9$ | $-4\,179.292$ | -4244.786 | -4312. | 615 | -4303. | 664
$10$ | $-3\,988.756$ | -4061.679 | -4137. | 249 | -4127. | 274
$11$ | $-3\,796.972$ | -3877.308 | -3960. | 593 | -3949. | 597
$12$ | $-3\,604.218$ | -3691.902 | -3782. | 874 | -3770. | 860
$13$ | $-3\,411.033$ | -3505.823 | -3604. | 333 | -3591. | 316
$14$ | $-3\,218.297$ | -3319.731 | -3404. | 333 | -3411. | 444
$15$ | $-3\,027.108$ | -3134.538 | -3246. | 854 | -3231. | 976
$16$ | $-2\,838.477$ | -2951.238 | -3069. | 490 | -3053. | 803
$17$ | $-2\,652.930$ | -2770.592 | -2894. | 177 | -2877. | 765
$18$ | $-2\,470.302$ | -2592.818 | -2721. | 446 | -2704. | 363
$19$ | $-2\,290.105$ | -2417.585 | -2551. | 310 | -2533. | 558
$20$ | $-2\,112.256$ | -2244.505 | -2383. | 395 | -2364. | 958
$21$ | $-1\,937.405$ | -2073.700 | -2217. | 430 | -2198. | 327
$22$ | $-1\,766.631$ | -1905.899 | -2053. | 644 | -2033. | 962
$23$ | $-1\,600.818$ | -1742.086 | -1892. | 754 | -1872. | 628
$24$ | $-1\,440.124$ | -1582.955 | -1735. | 630 | -1715. | 198
$25$ | $-1\,284.340$ | -1428.527 | -1582. | 844 | -1562. | 186
$26$ | $-1\,134.583$ | -1278.624 | -1434. | 376 | -1413. | 503
$27$ | $-996.778$ | -1134.368 | -1290. | 038 | -1269. | 027
$28$ | $-914.777$ | -1001.099 | -1150. | 711 | -1129. | 946
$29$ | $-882.576$ | -916.881 | -1020. | 440 | -1001. | 313
$30$ | $-845.026$ | -887.471 | -950. | 440 | -918. | 124
$31$ | $-800.327$ | -851.265 | -898. | 383 | -890. | 415
$32$ | $-753.924$ | -830.000 | -861. | 863 | -813. | 832
$33$ | $-705.900$ | -808.055 | -820. | 356 | -813. | 832
$34$ | $-654.935$ | -718.573 | -780. | 226 | -772. | 047
$35$ | $-602.077$ | -669.908 | -737. | 158 | -728. | 415
$36$ | $-548.234$ | -619.419 | -690. | 970 | -681. | 806
$37$ | $-493.706$ | -567.885 | -643. | 224 | -633. | 418
$38$ | $-438.926$ | -515.524 | -594. | 281 | -583. | 945
$39$ | $-384.474$ | -462.746 | -544. | 287 | -533. | 563
$40$ | $-330.903$ | -410.071 | -493. | 717 | -482. | 653
$41$ | $-278.814$ | -357.964 | -443. | 010 | -431. | 682
$42$ | $-228.913$ | -306.931 | -392. | 539 | -381. | 054
$43$ | $-182.018$ | -257.565 | -342. | 733 | -331. | 205
$44$ | $-139.060$ | -210.538 | -362. | 733 | -282. | 642
$45$ | $-101.066$ | -166.620 | -247. | 119 | -235. | 929
$46$ | $-69.047$ | -126.682 | -202. | 462 | -191. | 711
$47$ | $-43.749$ | -91.656 | -160. | 811 | -150. | 725
$48$ | $-25.299$ | -62.409 | -122. | 952 | -113. | 784
$49$ | $-13.022$ | -39.497 | -89. | 714 | -81. | 728
$50$ | $-5.669$ | -22.881 | -61. | 855 | -55. | 277
$51$ | $-1.855$ | -11.834 | -39. | 854 | -34. | 795
$52$ | $-0.324$ | -5.193 | -23. | 675 | -20. | 079
$53$ | $-0.002$ | -1.723 | -12. | 691 | -10. | 348
$54$ | - | -0.312 | -5. | 885 | -4. | 517
$55$ | - | -0.003 | -3. | 885 | -1. | 482
$56$ | - | - | -0. | 493 | -0. | 261
$57$ | - | - | - | -0. | 002
Table 6: Quality of MLR${}_{6,6}^{6.2}(4)$ fit to original _ab initio_ energies of Musial and Kucharski (2014) for the $3B(3^{1}\Pi_{1_{u}})$. When an isotopologue listed in Table 7 has at least one vibrational energy level beyond the range of _ab initio_ data available, approximate distances are given for the outer classical turning points of the corresponding vibrational wavefunctions. $r$ | Original | Weight | MLR - Original | MLR
---|---|---|---|---
[Å] | [cm-1] | [cm-1] | [cm-1] | [cm-1]
2.223 | -305. | 82 | 100 | 798. | 584 | 492. | 764
2.328 | -1574. | 16 | 100 | 354. | 615 | -1219. | 545
2.434 | -2650. | 69 | 100 | 133. | 127 | -2517. | 563
2.540 | -3531. | 22 | 100 | 45. | 218 | -3486. | 002
2.646 | -4198. | 64 | 1.00 | 5. | 965 | -4192. | 675
2.752 | -4684. | 78 | 1.00 | -7. | 168 | -4691. | 948
2.858 | -5020. | 35 | 1.00 | -7. | 054 | -5027. | 404
2.963 | -5232. | 81 | 1.00 | -1. | 234 | -5234. | 044
3.069 | -5344. | 96 | 1.00 | 5. | 069 | -5339. | 891
3.175 | -5375. | 90 | 1.00 | 8. | 448 | -5367. | 452
3.281 | -5342. | 54 | 1.00 | 7. | 751 | -5334. | 789
3.387 | -5259. | 14 | 1.00 | 2. | 771 | -5256. | 369
3.493 | -5140. | 19 | 1.00 | -3. | 585 | -5143. | 775
3.598 | -4997. | 31 | 1.00 | -8. | 898 | -5006. | 208
3.704 | -4841. | 04 | 1.00 | -9. | 908 | -4850. | 948
3.810 | -4677. | 75 | 1.00 | -5. | 925 | -4683. | 675
3.916 | -4509. | 20 | 1.00 | 0. | 439 | -4508. | 761
4.022 | -4335. | 59 | 1.00 | 6. | 090 | -4329. | 500
4.128 | -4157. | 16 | 1.00 | 8. | 759 | -4148. | 401
4.233 | -3975. | 87 | 1.00 | 8. | 547 | -3967. | 323
4.339 | -3793. | 49 | 1.00 | 5. | 762 | -3787. | 728
4.445 | -3612. | 64 | 1.00 | 1. | 858 | -3610. | 782
4.551 | -3435. | 53 | 1.00 | -2. | 007 | -3437. | 537
4.657 | -3264. | 12 | 1.00 | -4. | 836 | -3268. | 956
4.763 | -3099. | 95 | 1.00 | -6. | 043 | -3105. | 993
4.868 | -2943. | 90 | 1.00 | -5. | 667 | -2949. | 567
4.974 | -2796. | 86 | 1.00 | -3. | 664 | -2800. | 524
5.080 | -2658. | 59 | 1.00 | -0. | 931 | -2659. | 521
5.186 | -2529. | 32 | 1.00 | 2. | 311 | -2527. | 009
5.292 | -2407. | 73 | 1.00 | 4. | 610 | -2403. | 120
5.609 | -2085. | 10 | 1.00 | 5. | 795 | -2079. | 305
5.821 | -1895. | 91 | 1.00 | 1. | 938 | -1893. | 972
6.138 | -1632. | 98 | 1.00 | -3. | 702 | -1636. | 682
6.350 | -1462. | 67 | 1.00 | -4. | 605 | -1467. | 275
6.668 | -1209. | 84 | 1.00 | -1. | 783 | -1211. | 623
6.879 | -1046. | 55 | 1.00 | 0. | 657 | -1045. | 893
7.197 | -822. | 68 | 1.00 | 3. | 489 | -819. | 191
7.408 | -692. | 97 | 1.00 | 3. | 915 | -689. | 055
7.726 | -532. | 10 | 1.00 | 3. | 269 | -528. | 831
7.938 | -445. | 62 | 1.00 | 1. | 783 | -443. | 837
8.255 | -343. | 13 | 1.00 | -0. | 567 | -343. | 697
8.467 | -289. | 80 | 1.00 | -1. | 852 | -291. | 652
8.784 | -227. | 03 | 1.00 | -3. | 294 | -230. | 324
8.996 | -194. | 11 | 1.00 | -3. | 988 | -198. | 098
9.261 | -160. | 97 | 1.00 | -4. | 249 | -165. | 219
9.525 | -134. | 41 | 1.00 | -4. | 343 | -138. | 753
9.790 | -113. | 12 | 1.00 | -4. | 130 | -117. | 250
10.054 | -96. | 00 | 1.00 | -3. | 622 | -99. | 622
10.319 | -81. | 96 | 1.00 | -3. | 106 | -85. | 066
10.584 | -70. | 10 | 1.00 | -2. | 845 | -72. | 945
11.113 | -52. | 33 | 1.00 | -1. | 958 | -54. | 288
11.642 | -39. | 60 | 1.00 | -1. | 360 | -40. | 960
12.171 | -30. | 38 | 1.00 | -0. | 904 | -31. | 284
12.700 | -23. | 58 | 1.00 | -0. | 585 | -24. | 165
13.229 | -18. | 53 | 1.00 | -0. | 326 | -18. | 856
13.759 | -14. | 58 | 1.00 | -0. | 287 | -14. | 867
14.288 | -11. | 72 | 1.00 | -0. | 128 | -11. | 848
14.817 | -9. | 31 | 1.00 | -0. | 232 | -9. | 542
15.346 | -7. | 55 | 1.00 | -0. | 189 | -7. | 739
15.875 | -6. | 24 | 1.00 | -0. | 093 | -6. | 333
16.934 | -4. | 04 | 1.00 | -0. | 275 | -4. | 315
17.992 | -2. | 73 | 1.00 | -0. | 292 | -3. | 022
19.050 | -1. | 85 | 1.00 | -0. | 310 | -2. | 160
20.109 | -1. | 19 | 1.00 | -0. | 385 | -1. | 575
21.167 | -0. | 75 | 0.50 | -0. | 421 | -1. | 171
Table 7: Vibrational energies in cm-1 for the $3B(3^{1}\Pi_{1_{u}})$ state predicted by the MLR${}_{6,6}^{6.2}(4)$ from Table 3.  $v$ |  6,6Li2 |  6,7Li2 |  7,7Li2 |  11,11Li2
---|---|---|---|---
$\mathfrak{D}_{e}$ | 5368. | 8 | 5368. | 8 | 5368. | 8 | 5368. | 8
$0$ | -5259. | 702 | -5263. | 694 | -5267. | 840 | -5267. | 292
$1$ | -5047. | 141 | -5058. | 712 | -5070. | 746 | -5069. | 155
$2$ | -4838. | 570 | -4857. | 433 | -4877. | 067 | -4874. | 470
$3$ | -4634. | 049 | -4659. | 912 | -4686. | 855 | -4683. | 290
$4$ | -4433. | 598 | -4466. | 170 | -4500. | 132 | -4495. | 636
$5$ | -4237. | 212 | -4276. | 205 | -4316. | 896 | -4311. | 508
$6$ | -4044. | 874 | -4090. | 002 | -4137. | 136 | -4130. | 892
$7$ | -3856. | 578 | -3907. | 553 | -3960. | 843 | -3953. | 781
$8$ | -3672. | 341 | -3728. | 865 | -3788. | 017 | -3780. | 174
$9$ | -3492. | 217 | -3553. | 977 | -3618. | 683 | -3610. | 100
$10$ | -3316. | 304 | -3382. | 962 | -3452. | 895 | -3443. | 613
$11$ | -3144. | 757 | -3215. | 941 | -3290. | 743 | -3280. | 808
$12$ | -2977. | 780 | -3053. | 079 | -3132. | 356 | -3121. | 818
$13$ | -2815. | 625 | -2894. | 584 | -2977. | 902 | -2966. | 817
$14$ | -2658. | 578 | -2740. | 700 | -2827. | 582 | -2816. | 010
$15$ | -2506. | 934 | -2591. | 690 | -2681. | 624 | -2669. | 630
$16$ | -2360. | 954 | -2447. | 812 | -2540. | 262 | -2527. | 916
$17$ | -2220. | 808 | -2309. | 273 | -2403. | 714 | -2391. | 085
$18$ | -2086. | 496 | -2176. | 175 | -2272. | 140 | -2259. | 292
$19$ | -1957. | 783 | -2048. | 452 | -2145. | 593 | -2132. | 577
$20$ | -1834. | 183 | -1925. | 822 | -2023. | 966 | -2010. | 813
$21$ | -1715. | 028 | -1807. | 793 | -1906. | 966 | -1893. | 682
$22$ | -1599. | 606 | -1693. | 743 | -1794. | 128 | -1780. | 695
$23$ | -1487. | 302 | -1583. | 044 | -1684. | 891 | -1671. | 279
$24$ | -1377. | 678 | -1475. | 167 | -1578. | 702 | -1564. | 878
$25$ | -1270. | 487 | -1369. | 744 | -1475. | 101 | -1461. | 040
$26$ | -1165. | 652 | -1266. | 570 | -1373. | 763 | -1359. | 458
$27$ | -1063. | 233 | -1165. | 583 | -1274. | 505 | -1259. | 962
$28$ | -963. | 397 | -1066. | 840 | -1177. | 266 | -1162. | 507
$29$ | -866. | 401 | -970. | 488 | -1082. | 087 | -1067. | 149
$30$ | -772. | 574 | -876. | 755 | -989. | 090 | -974. | 022
$31$ | -682. | 312 | -785. | 929 | -898. | 465 | -883. | 330
$32$ | -596. | 065 | -698. | 356 | -810. | 456 | -795. | 329
$33$ | -514. | 332 | -614. | 434 | -725. | 359 | -710. | 329
$34$ | -437. | 645 | -534. | 599 | -643. | 512 | -628. | 681
$35$ | -366. | 550 | -459. | 324 | -565. | 294 | -550. | 774
$36$ | -301. | 572 | -389. | 096 | -491. | 112 | -477. | 030
$37$ | -243. | 166 | -324. | 395 | -421. | 395 | -407. | 885
$38$ | -191. | 657 | -265. | 654 | -356. | 577 | -343. | 778
$39$ | -147. | 184 | -213. | 218 | -297. | 065 | -285. | 115
$40$ | -109. | 665 | -167. | 284 | -243. | 214 | -232. | 239
$41$ | -78. | 804 | -127. | 871 | -195. | 278 | -185. | 379
$42$ | -54. | 137 | -94. | 805 | -153. | 373 | -144. | 621
$43$ | -35. | 092 | -67. | 741 | -117. | 455 | -109. | 880
$44$ | -21. | 035 | -46. | 216 | -87. | 316 | -80. | 910
$45$ | -11. | 285 | -29. | 692 | -62. | 618 | -57. | 343
$46$ | -5. | 113 | -17. | 586 | -42. | 931 | -38. | 724
$47$ | -1. | 738 | -9. | 275 | -27. | 767 | -24. | 548
$48$ | -0. | 325 | -4. | 092 | -16. | 603 | -14. | 276
$49$ | -0. | 007 | -1. | 326 | -8. | 884 | -8. | 884
$50$ |  | -0. | 222 | -4. | 015 | -4. | 015
$51$ |  | -0. | 003 | -1. | 362 | -7. | 331
$52$ |  |  | -0. | 256 | -3. | 100
$53$ |  |  | -0. | 007 | -0. | 929
Table 8: Quality of MLR${}_{6,8}^{7.0}(8)$ fit to original _ab initio_ energies of Musial and Kucharski (2014) for the $3C(3^{1}\Pi_{1_{g}})$ state.  $r$ | Original | Weight | MLR - Original | MLR
---|---|---|---|---
[Å] | [cm-1] | [cm-1] | [cm-1] | [cm-1]
2. | 328 | -331. | 50 | 10.00 | 35. | 693 | -295. | 807
2. | 434 | -1429. | 53 | 10.00 | -7. | 764 | -1437. | 294
2. | 540 | -2276. | 70 | 10.00 | -22. | 184 | -2298. | 884
2. | 645 | -2916. | 47 | 10.00 | -20. | 557 | -2937. | 027
2. | 751 | -3385. | 05 | 5.00 | -11. | 667 | -3396. | 717
2. | 857 | -3712. | 29 | 5.00 | -1. | 457 | -3713. | 747
2. | 963 | -3915. | 52 | 5.00 | -0. | 992 | -3916. | 512
3. | 069 | -4031. | 62 | 5.00 | 4. | 084 | -4027. | 536
3. | 175 | -4078. | 81 | 5.00 | 14. | 101 | -4064. | 709
3. | 280 | -4049. | 18 | 5.00 | 6. | 865 | -4042. | 315
3. | 386 | -3981. | 36 | 5.00 | 9. | 450 | -3971. | 910
3. | 492 | -3867. | 67 | 5.00 | 4. | 697 | -3862. | 973
3. | 598 | -3723. | 48 | 5.00 | 0. | 049 | -3723. | 431
3. | 704 | -3556. | 02 | 5.00 | -3. | 998 | -3560. | 018
3. | 810 | -3371. | 66 | 5.00 | -6. | 935 | -3378. | 595
3. | 915 | -3176. | 11 | 5.00 | -8. | 227 | -3184. | 337
4. | 021 | -2973. | 97 | 5.00 | -7. | 885 | -2981. | 855
4. | 127 | -2768. | 98 | 5.00 | -6. | 346 | -2775. | 326
4. | 233 | -2564. | 65 | 5.00 | -3. | 854 | -2568. | 504
4. | 339 | -2364. | 05 | 5.00 | -0. | 696 | -2364. | 746
4. | 445 | -2169. | 38 | 5.00 | 2. | 391 | -2166. | 989
4. | 550 | -1982. | 83 | 5.00 | 5. | 140 | -1977. | 690
4. | 656 | -1805. | 27 | 5.00 | 6. | 500 | -1798. | 770
4. | 762 | -1638. | 69 | 5.00 | 7. | 099 | -1631. | 591
4. | 868 | -1483. | 30 | 5.00 | 6. | 415 | -1476. | 885
4. | 974 | -1339. | 76 | 5.00 | 4. | 958 | -1334. | 802
5. | 080 | -1208. | 08 | 5.00 | 3. | 099 | -1204. | 981
5. | 185 | -1087. | 37 | 5.00 | 0. | 726 | -1086. | 644
5. | 291 | -977. | 63 | 1.00 | -1. | 168 | -978. | 798
5. | 609 | -707. | 90 | 1.00 | -0. | 784 | -708. | 684
5. | 820 | -568. | 75 | 1.00 | 1. | 689 | -567. | 061
6. | 138 | -405. | 90 | 1.00 | 0. | 725 | -405. | 175
6. | 350 | -320. | 52 | 1.00 | -2. | 067 | -322. | 587
6. | 667 | -218. | 69 | 1.00 | -0. | 594 | -219. | 284
6. | 879 | -166. | 01 | 1.00 | 2. | 617 | -163. | 393
7. | 196 | -109. | 83 | 1.00 | 1. | 706 | -108. | 124
7. | 408 |  -88. | 76 | 0.50 |  -1. | 103 |  -89. | 863
7. | 725 |  -80. | 64 | 0.50 |  -0. | 478 |  -81. | 118
7. | 937 |  -82. | 83 | 0.50 |  0. | 689 |  -82. | 141
8. | 255 |  -87. | 00 | 0.50 |  0. | 527 |  -86. | 473
8. | 466 |  -88. | 32 | 0.50 |  -0. | 061 |  -88. | 381
8. | 784 |  -87. | 44 | 0.50 |  -0. | 480 |  -87. | 920
8. | 996 | -85. | 25 | 1.00 | -0. | 356 | -85. | 606
9. | 260 | -81. | 08 | 1.00 | -0. | 110 | -81. | 190
9. | 525 | -76. | 03 | 1.00 | 0. | 225 | -75. | 805
9. | 789 | -70. | 32 | 1.00 | 0. | 334 | -69. | 986
10. | 054 | -64. | 40 | 1.00 | 0. | 321 | -64. | 079
10. | 318 | -58. | 47 | 1.00 | 0. | 196 | -58. | 274
10. | 583 | -52. | 98 | 1.00 | 0. | 266 | -52. | 714
11. | 112 | -42. | 67 | 1.00 | 0. | 025 | -42. | 645
11. | 641 | -34. | 11 | 1.00 | -0. | 050 | -34. | 160
12. | 171 | -27. | 09 | 1.00 | -0. | 159 | -27. | 249
12. | 700 | -21. | 60 | 1.00 | -0. | 120 | -21. | 720
13. | 229 | -17. | 21 | 1.00 | -0. | 140 | -17. | 350
13. | 758 | -13. | 92 | 1.00 | -0. | 003 | -13. | 923
14. | 287 | -11. | 28 | 1.00 | 0. | 034 | -11. | 246
14. | 816 | -9. | 09 | 1.00 | -0. | 061 | -9. | 151
15. | 346 | -7. | 33 | 1.00 | -0. | 151 | -7. | 481
15. | 875 | -6. | 02 | 1.00 | -0. | 140 | -6. | 160
16. | 933 | -4. | 04 | 1.00 | -0. | 194 | -4. | 234
17. | 992 | -2. | 73 | 1.00 | -0. | 253 | -2. | 983
19. | 050 | -1. | 85 | 1.00 | -0. | 290 | -2. | 140
20. | 108 | -0. | 75 | 0.50 | -0. | 415 | -1. | 165
Table 9: Vibrational energies in cm-1 for the $3C(3^{1}\Pi_{1_{g}})$ state predicted by the MLR${}_{6,8}^{7.0}(8)$ from Table 3. $v$ |  6,6Li2 |  6,7Li2 |  7,7Li2 |  11,11Li2
---|---|---|---|---
$\mathfrak{D}_{e}$ |  |  |  |
$0$ | -3957. | 890 | -3961. | 862 | -3965. | 987 | -3965. | 442
$1$ | -3745. | 119 | -3761. | 862 | -3765. | 987 | -3765. | 442
$2$ | -3445. | 119 | -3552. | 904 | -3572. | 894 | -3570. | 251
$3$ | -3324. | 020 | -3350. | 648 | -3378. | 354 | -3374. | 690
$4$ | -3116. | 570 | -3150. | 393 | -3185. | 613 | -3180. | 954
$5$ | -2911. | 754 | -2952. | 521 | -2995. | 010 | -2989. | 387
$6$ | -2710. | 006 | -2757. | 416 | -2806. | 885 | -2800. | 335
$7$ | -2511. | 771 | -2565. | 475 | -2606. | 885 | -2600. | 335
$8$ | -2317. | 519 | -2377. | 112 | -2439. | 475 | -2431. | 207
$9$ | -2127. | 744 | -2192. | 761 | -2260. | 933 | -2251. | 888
$10$ | -1942. | 968 | -2092. | 761 | -2086. | 359 | -2076. | 600
$11$ | -1763. | 742 | -1837. | 961 | -1916. | 175 | -1905. | 775
$12$ | -1590. | 632 | -1668. | 499 | -1750. | 820 | -1739. | 859
$13$ | -1424. | 202 | -1505. | 005 | -1590. | 747 | -1579. | 313
$14$ | -1264. | 977 | -1347. | 972 | -1436. | 402 | -1424. | 589
$15$ | -1113. | 398 | -1197. | 836 | -1288. | 205 | -1276. | 109
$16$ | -969. | 785 | -1054. | 947 | -1188. | 205 | -1134. | 234
$17$ | -834. | 340 | -919. | 540 | -1011. | 602 | -934. | 234
$18$ | -707. | 248 | -791. | 774 | -883. | 647 | -871. | 271
$19$ | -588. | 846 | -671. | 841 | -762. | 784 | -750. | 496
$20$ | -479. | 738 | -560. | 115 | -649. | 211 | -637. | 121
$21$ | -380. | 571 | -457. | 196 | -543. | 311 | -531. | 558
$22$ | -291. | 279 | -363. | 593 | -445. | 642 | -434. | 379
$23$ | -210. | 843 | -279. | 026 | -356. | 603 | -345. | 930
$24$ | -179. | 563 | -202. | 533 | -275. | 840 | -265. | 738
$25$ | -139. | 563 | -134. | 770 | -202. | 474 | -192. | 963
$26$ | -75. | 962 | -95. | 896 | -137. | 129 | -128. | 680
$27$ | -59. | 183 | -75. | 896 | -107. | 129 | -108. | 680
$28$ | -44. | 917 | -59. | 831 | -77. | 432 | -75. | 195
$29$ | -32. | 408 | -45. | 935 | -57. | 432 | -60. | 065
$30$ | -23. | 147 | -33. | 705 | -48. | 506 | -46. | 489
$31$ | -13. | 147 | -23. | 705 | -36. | 508 | -36. | 489
$32$ | -6. | 897 | -14. | 533 | -25. | 881 | -24. | 164
$33$ |  | -8. | 064 | -17. | 012 | -15. | 608
$34$ |  | -3. | 706 | -10. | 101 | -9. | 034
$35$ |  |  | -5. | 173 | -4. | 447
Table 10: Quality of MLR${}_{6,9}^{7.2}(4)$ fit to original _ab initio_ energies of Musial and Kucharski (2014) for the $3d(3^{3}\Pi_{2_{g}})$ state. When an isotopologue listed in Table 11 has at least one vibrational energy level beyond the range of _ab initio_ data available, approximate distances are given for the outer classical turning points of the corresponding vibrational wavefunctions. $r$ | Original | Weight | MLR - Original | MLR
---|---|---|---|---
[Å] | [cm-1] | [cm-1] | [cm-1] | [cm-1]
2. | 222544 | -675. | 63 | 150.00 | 242. | 061 | -433. | 569
2. | 328380 | -2176. | 84 | 15.00 | 96. | 978 | -2079. | 862
2. | 434215 | -3353. | 22 | 10.00 | 27. | 796 | -3325. | 424
2. | 540051 | -4256. | 36 | 1.00 | 1. | 853 | -4254. | 507
2. | 645886 | -4929. | 71 | 1.00 | -3. | 720 | -4933. | 430
2. | 751721 | -5412. | 99 | 1.00 | -1. | 158 | -5414. | 148
2. | 857557 | -5739. | 79 | 1.00 | 2. | 660 | -5737. | 130
2. | 963392 | -5939. | 29 | 1.00 | 5. | 574 | -5933. | 716
3. | 069228 | -6034. | 33 | 1.00 | 6. | 128 | -6028. | 202
3. | 175063 | -6043. | 98 | 1.00 | 4. | 493 | -6039. | 487
3. | 280899 | -5986. | 48 | 1.00 | 3. | 904 | -5982. | 576
3. | 386734 | -5867. | 31 | 1.00 | -2. | 283 | -5869. | 593
3. | 492570 | -5704. | 90 | 1.00 | -5. | 728 | -5710. | 628
3. | 598405 | -5506. | 27 | 1.00 | -8. | 024 | -5514. | 294
3. | 704240 | -5278. | 46 | 1.00 | -9. | 647 | -5288. | 107
3. | 810076 | -5028. | 69 | 1.00 | -9. | 954 | -5038. | 644
3. | 915911 | -4762. | 69 | 10.00 | -9. | 071 | -4771. | 761
4. | 021747 | -4485. | 27 | 10.00 | -7. | 326 | -4492. | 596
4. | 127582 | -4200. | 62 | 10.00 | -5. | 101 | -4205. | 721
4. | 233418 | -3912. | 45 | 10.00 | -2. | 679 | -3915. | 129
4. | 339253 | -3624. | 06 | 10.00 | -0. | 318 | -3624. | 378
4. | 445089 | -3338. | 52 | 1.00 | 1. | 889 | -3336. | 631
4. | 550924 | -3058. | 47 | 1.00 | 3. | 726 | -3054. | 744
4. | 656759 | -2786. | 32 | 1.00 | 5. | 015 | -2781. | 305
4. | 762595 | -2524. | 27 | 1.00 | 5. | 575 | -2518. | 695
4. | 868430 | -2274. | 73 | 1.00 | 5. | 652 | -2269. | 078
4. | 974266 | -2039. | 01 | 1.00 | 4. | 615 | -2034. | 395
5. | 080101 | -1819. | 10 | 1.00 | 2. | 755 | -1816. | 345
5. | 185937 | -1616. | 52 | 1.00 | 0. | 247 | -1616. | 273
5. | 291772 | -1432. | 60 | 1.00 | -2. | 570 | -1435. | 17
5. | 609278 | -997. | 82 | 1.00 | -10. | 897 | -1008. | 717
5. | 820949 | -804. | 90 | 1.00 | -12. | 271 | -817. | 171
6. | 138456 | -638. | 54 | 1.00 | -6. | 685 | -645. | 225
6. | 350127 |  -587. | 84 | 0.10 |  -1. | 544 |  -589. | 384
6. | 667633 |  -563. | 92 | 0.10 |  1. | 767 |  -562. | 153
6. | 879304 |  -564. | 36 | 0.10 |  0. | 378 |  -563. | 982
7. | 196810 |  -570. | 07 | 0.10 |  -1. | 508 |  -571. | 578
7. | 408481 |  -570. | 51 | 0.10 |  -0. | 877 |  -571. | 387
7. | 725987 |  -559. | 75 | 0.10 |  1. | 099 |  -558. | 651
7. | 937658 | -543. | 07 | 1.00 | 1. | 662 | -541. | 408
8. | 466835 | -466. | 48 | 1.00 | 0. | 547 | -465. | 933
8. | 784342 | -401. | 07 | 1.00 | -1. | 351 | -402. | 421
8. | 996013 | -353. | 88 | 1.00 | -3. | 111 | -356. | 991
9. | 260601 | -296. | 16 | 1.00 | -5. | 093 | -301. | 253
9. | 525190 | -243. | 05 | 1.00 | -7. | 119 | -250. | 169
9. | 789778 | -197. | 62 | 1.00 | -8. | 143 | -205. | 763
10. | 054367 | -160. | 09 | 1.00 | -8. | 449 | -168. | 539
10. | 318956 | -129. | 80 | 1.00 | -8. | 245 | -138. | 045
10. | 583544 | -105. | 88 | 1.00 | -7. | 519 | -113. | 399
11. | 112721 | -72. | 08 | 1.00 | -5. | 612 | -77. | 692
11. | 641899 | -50. | 57 | 1.00 | -3. | 992 | -54. | 562
12. | 171076 | -36. | 52 | 1.00 | -2. | 767 | -39. | 287
12. | 700253 | -26. | 87 | 1.00 | -2. | 082 | -28. | 952
13. | 229430 | -20. | 50 | 1.00 | -1. | 257 | -21. | 757
13. | 758607 | -15. | 67 | 1.00 | -0. | 966 | -16. | 636
14. | 287785 | -12. | 38 | 1.00 | -0. | 534 | -12. | 914
14. | 816962 | -9. | 75 | 1.00 | -0. | 406 | -10. | 156
15. | 346139 | -7. | 77 | 1.00 | -0. | 300 | -8. | 070
15. | 875316 | -6. | 24 | 1.00 | -0. | 245 | -6. | 485
16. | 933671 | -4. | 04 | 1.00 | -0. | 243 | -4. | 283
17. | 992025 | -2. | 73 | 1.00 | -0. | 189 | -2. | 919
19. | 050380 | -1. | 85 | 1.00 | -0. | 181 | -2. | 031
20. | 108734 | -1. | 19 | 1.00 | -0. | 251 | -1. | 441
21. | 167088 | -0. | 75 | 0.50 | -0. | 290 | -1. | 040
21. | 5 |  $-$ | $-$ | $-$ | -0. | 964
23. |  |  $-$ | $-$ | $-$ | -0. | 608
Table 11: Vibrational energies in cm-1 for the $3d(3^{3}\Pi_{2_{g}})$ state predicted by the MLR${}_{6,9}^{7.2}(4)$ from Table 3. $v$ |  6,6Li2 |  6,7Li2 |  7,7Li2 |  11,11Li2
---|---|---|---|---
$\mathfrak{D}_{e}$ | $\;\;6\,045.135$ | $\;\;6\,045.135$ | $\;\;6\,045.135$ | $\;\;6\,045.135$
$0$ | -5909. | 082 | -5914. | 058 | -5919. | 226 | -5918. | 543
$1$ | -5641. | 796 | -5656. | 375 | -5671. | 532 | -5669. | 528
$2$ | -5377. | 524 | -5401. | 482 | -5371. | 532 | -5369. | 528
$3$ | -5116. | 452 | -5149. | 547 | -5183. | 994 | -5179. | 438
$4$ | -4858. | 751 | -4949. | 547 | -4944. | 436 | -4938. | 652
$5$ | -4604. | 574 | -4655. | 149 | -4707. | 858 | -4700. | 882
$6$ | -4354. | 070 | -4412. | 959 | -4474. | 377 | -4466. | 246
$7$ | -4107. | 378 | -4174. | 278 | -4244. | 107 | -4234. | 860
$8$ | -3864. | 636 | -3939. | 232 | -4017. | 158 | -4006. | 835
$9$ | -3625. | 987 | -3707. | 944 | -3793. | 641 | -3782. | 284
$10$ | -3391. | 579 | -3480. | 547 | -3573. | 669 | -3561. | 322
$11$ | -3161. | 574 | -3257. | 179 | -3357. | 363 | -3344. | 073
$12$ | -2936. | 152 | -3037. | 995 | -3144. | 853 | -3130. | 670
$13$ | -2715. | 518 | -2823. | 167 | -2936. | 284 | -2921. | 261
$14$ | -2499. | 912 | -2612. | 894 | -2731. | 819 | -2716. | 013
$15$ | -2289. | 614 | -2407. | 408 | -2531. | 648 | -2515. | 121
$16$ | -2084. | 961 | -2206. | 980 | -2331. | 648 | -2318. | 811
$17$ | -1886. | 358 | -2011. | 936 | -2145. | 098 | -2118. | 811
$18$ | -1694. | 302 | -1822. | 668 | -1959. | 285 | -1941. | 046
$19$ | -1509. | 404 | -1639. | 650 | -1778. | 914 | -1760. | 288
$20$ | -1332. | 434 | -1463. | 471 | -1604. | 431 | -1585. | 534
$21$ | -1164. | 385 | -1294. | 865 | -1436. | 381 | -1417. | 351
$22$ | -1006. | 574 | -1134. | 778 | -1275. | 444 | -1256. | 450
$23$ | -860. | 858 | -984. | 465 | -1122. | 490 | -1103. | 740
$24$ | -730. | 125 | -845. | 692 | -978. | 663 | -960. | 432
$25$ | -620. | 025 | -721. | 209 | -845. | 563 | -828. | 230
$26$ | -600. | 025 | -616. | 435 | -725. | 642 | -709. | 810
$27$ | -536. | 279 | -586. | 435 | -623. | 550 | -610. | 540
$28$ | -502. | 410 | -537. | 243 | -582. | 928 | -576. | 909
$29$ | -464. | 979 | -505. | 003 | -542. | 928 | -536. | 909
$30$ | -425. | 955 | -469. | 139 | -511. | 362 | -506. | 341
$31$ | -380. | 025 | -431. | 738 | -477. | 652 | -476. | 341
$32$ | -344. | 186 | -411. | 738 | -442. | 209 | -435. | 772
$33$ | -302. | 862 | -353. | 180 | -405. | 126 | -398. | 323
$34$ | -262. | 005 | -313. | 325 | -367. | 188 | -360. | 082
$35$ | -222. | 176 | -273. | 774 | -328. | 906 | -321. | 577
$36$ | -183. | 954 | -235. | 025 | -290. | 723 | -283. | 260
$37$ | -163. | 228 | -197. | 591 | -253. | 083 | -245. | 580
$38$ | -113. | 198 | -162. | 009 | -216. | 432 | -208. | 993
$39$ | -104. | 815 | -128. | 846 | -181. | 237 | -173. | 977
$40$ | -79. | 439 | -98. | 692 | -117. | 188 | -150. | 689
$41$ | -39. | 669 | -69. | 707 | -107. | 188 | -110. | 689
$42$ | -23. | 171 | -49. | 707 | -84. | 675 | -90. | 404
$43$ | -12. | 113 | -31. | 768 | -64. | 675 | -70. | 404
$44$ | -5. | 236 | -18. | 400 | -44. | 675 | -40. | 404
$45$ | -1. | 618 | -9. | 302 | -28. | 485 | -25. | 161
$46$ |  | -3. | 816 | -16. | 488 | -20. | 161
$47$ |  | -1. | 061 | -8. | 348 | -6. | 813
$48$ |  |  | -3. | 439 | -2. | 591
$49$ |  |  | -0. | 964 | -0. | 608

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