#  State of the art for _ab initio_ vs empirical potentials for predicting 6e-
excited state molecular energies: Application to
Li${}_{2}\left(b,1^{3}\Pi_{u}\right)$

Nikesh S. Dattani,${}^{1,2,3,\,}$<EMAIL_ADDRESS>1Quantum Chemistry
Laboratory, Department of Chemistry, Kyoto University, 606-8502, Kyoto, Japan,
2School of Materials Science and Engineering, Nanyang Technological
University, 639798, Singapore,  3Fukui Institute for Fundamental Chemistry,
606-8103, Kyoto, Japan,  Robert J. Le Roy,${}^{3,\,}$<EMAIL_ADDRESS>4Guelph-Waterloo Centre for Graduate Work in Chemistry and Biochemistry,
University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada.

###### Abstract

We build the first analytic empirical potential for the most deeply bound Li2
state: $b\left(1^{3}\Pi_{u}\right)$. Our potential is based on experimental
energy transitions covering $v=0-34$, and very high precision theoretical
long-range constants. It provides high accuracy predictions up to $v=100$
which pave the way for high-precision long-range measurements, and hopefully
an eventual resolution of the age old discrepancy between experiment and
theory for the Li$\left(2^{2}S\right)+\mbox{Li}\left(2^{2}P\right)$ $C_{3}$
value. State of the art _ab initio_ calculations predict vibrational energy
spacings that are all in at most 0.8 cm-1 disagreement with the empirical
potential.

###### 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

There is currently a rather large discrepancy between the best atomic 3e-_ab
initio_ calculation Tang  _et al._ (2010a), and the most current empirical
value Gunton  _et al._ (2013a); Le Roy  _et al._ (2009), for the leading long-
range Li$(2^{2}S)-\mbox{Li$(2^{2}P)$}$ interaction constant ($C_{3}$), despite
the latter being the most precise experimentally determined oscillator
strength for any system, by an order of magnitude Tang  _et al._ (2011).
$\mbox{Li}_{2}(b,1^{3}\Pi_{u})$ is one of the molecular states that
dissociates to Li$(2^{2}S)+\mbox{Li$(2^{2}P)$}$, and therefore its long-range
potential has this $C_{3}$ interaction constant. This “$b$-state” is also the
deepest Li2 potential, and out of the five lowest Li2 states, the $b$-state is
the only one for which an analytic empirical potential has never been made.

Since the highest 2000 cm-1 worth of vibrational levels of the $b$-state still
have not been observed, and part of this region is now accessible by current
ultra-high precision PA (photoassociation) technology Gunton  _et al._
(2013a); Semczuk  _et al._ (2013), an analytic potential would be very useful
for making predictions to assist in observing the missing levels. The
$b$-state also mixes strongly with the $A(1^{1}\Sigma_{u})$-state , which has
by far the most precisely determined excited molecular state potential in all
of chemistry, yet still has a rather large gap of missing data in the middle
of its energy range Gunton  _et al._ (2013a); Le Roy  _et al._ (2009).

Finally, the $b(1^{3}\Pi_{u})$-state has been a key doorway to the triplet
manifold, and was directly involved in the measurements for a vast number of
other triplet states such as $2^{3}\Delta_{g}$ Yi  _et al._ (2001),
$3^{3}\Pi_{g}$ Yiannopoulou  _et al._ (1995a), $3^{3}\Sigma_{g}^{+}$
Yiannopoulou  _et al._ (1995a); Li  _et al._ (2007); Xie and Field (1986),
$2^{3}\Sigma_{g}^{+}$ Yiannopoulou  _et al._ (1995a); Lazarov  _et al._
(2001), $2^{3}\Pi_{g}$ Xie and Field (1986); Yiannopoulou  _et al._ (1995a);
Russier  _et al._ (1997); Li  _et al._ (2007), $1^{3}\Delta_{g}$ Xie and Field
(1986); Linton  _et al._ (1992); Yiannopoulou  _et al._ (1995b); Weyh  _et
al._ (1996a); Russier  _et al._ (1997); Li  _et al._ (2007),
$1^{3}\Sigma_{g}^{-}$, and other undetermined ${}^{3}\Lambda$ states Rai  _et
al._ (1985); Li  _et al._ (1996). Some of these more highly excited triplet
states (namely $3^{3}\Sigma_{g}^{+},$ $2^{3}\Sigma_{g}^{+},$ $2^{3}\Pi_{g},$
and $1^{3}\Delta_{g}$) are so thoroughly covered by these spectroscopic
measurements, that global empirical potentials can be built for them too. For
this, an analytic potential for the $b$-state would be used as a base.

In this work we will build analytic empirical potentials for the $b$-states of
all stable homonuclear isotopologues of Li2. Previous work has shown that
analytic empirical potentials for the $c(1^{3}\Sigma_{g})$-state were able to
predict energies correctly to about 1 cm-1, in the middle of a gap of $>5000$
cm-1 where data were unavailable Dattani and Le Roy (2011a); Semczuk  _et al._
(2013), and this was much better agreement than was obtained with the most
sophisticated Li2 _ab initio_ calculations of the time Halls  _et al._ (2001).

It was recently shown that the best ground-state rotationless _ab initio_
potentials for the 5e- molecules BeH, BeD, and BeT, were able to predict
vibrational energy spacings to within 1 cm-1 for all measured energy levels
except one. The $b$-state of Li2 might be expected to be more challenging _ab
initio_ because it (1) is an excited state, (2) has one more e-, and (3)
involves many more vibrational energies. We will therefore compare our
analytic empirical potentials for the $b$-state of 6,6Li2 and 7,7Li2 with the
most state-of-the-art _ab initio_ calculations, which were published recently
in Musial and Kucharski (2014).

Table 1 summarizes all experiments we could find which provided information on
rovibrational levels of the $b$-state . Unfortunately attempts to recover the
data from Engelke and Hage (1983); Rai  _et al._ (1985); Xie and Field (1986);
Schmidt  _et al._ (1988); Weyh  _et al._ (1996b) were unsuccessful, but we
were still able to include all data from the other experiments in our study.
Furthermore, it is noted that the $b$-state was also involved in various other
studies Yiannopoulou  _et al._ (1995a, c); Li and Lyyra (1999); Yi  _et al._
(2001); Li  _et al._ (2007) but these just made use of rovibrational levels
that were already determined in the studies listed in Table 1, in order to
access levels of other electronic states.

Table 1: Summary of experiments involving Li${}_{2}(b,1^{3}\Pi_{u})$

Isotopes | Year | Type | States Involved | Unc. (cm-1) | $v$ | # Data | Included in dataset | Source
---|---|---|---|---|---|---|---|---
7,7Li2 | 1985 | LIF | $A(1^{1}\Sigma_{u}^{+}),b(1^{3}\Pi_{u})$ | $a$ | $8-25$ | 100s |  | Preuss & Baumgartner Preuss and Baumgartner (1985)
| 1985 | OODR | $(5d\delta^{3}\Delta_{g})\rightarrow b(1^{3}\Pi_{u})$ | 1 | $0$ | 100s | - | Rai et al Rai  _et al._ (1985)
| 1992 | PFOODR | $A(1^{1}\Sigma_{u}^{+}),b(1^{3}\Pi_{u})$ | ? | $19$ | 3 | - | Li et al. Li  _et al._ (1992)
| 1996 | cw PFOODR | $({}^{3}\Lambda_{g})\rightarrow b(1^{3}\Pi_{u})$ | ? | $0,1,4$ | ? | - | Li et al.Li  _et al._ (1996)
| 1996 | CIF | $(1^{3}\Delta_{g})\rightarrow b(1^{3}\Pi_{u})$ | ? | $0-11$ | ? | - | Weyh et al.Weyh  _et al._ (1996b)
| 1997 | PFOODR | $(2^{3}\Pi_{g})\rightarrow b(1^{3}\Pi_{u})$ | 0.005 | $1-27$ | 178 | $\checkmark$ | Russier et al Russier  _et al._ (1997)
| 1997 | PFOODR | $(1^{3}\Delta_{g})\rightarrow b(1^{3}\Pi_{u})$ | 0.005 | $1-25$ | 234 | $\checkmark$ | Russier et al Russier  _et al._ (1997)
| 1997 | CIF | $(1^{3}\Delta_{g})\rightarrow b(1^{3}\Pi_{u})$ | 0.005 | $0-7$ | 314 | $\checkmark$ | Russier et al Russier  _et al._ (1997)
| 2001 | cw PFOODR | $A(1^{1}\Sigma_{u}^{+}),b(1^{3}\Pi_{u})$ | ? | $15,22$ | 2 | - | Lazarov & Lyyra Lazarov  _et al._ (2001)
7,6Li2 | 1985 | LIF | $A(1^{1}\Sigma_{u}^{+}),b(1^{3}\Pi_{u})$ | $a$ | $8-20$ | 100s |  | Preuss & BaumgartnerPreuss and Baumgartner (1985)
6,6Li2 | 1983 | CIF | $(3^{3}\Delta_{g})\rightarrow b(1^{3}\Pi_{u})$ | $0.3$ | $0-10$ | 240 | - | Engelke & Hage
| 1985 | LIF | $A(1^{1}\Sigma_{u}^{+}),b(1^{3}\Pi_{u})$ | $a$ | $8-18$ | 100s |  | Preuss & BaumgartnerPreuss and Baumgartner (1985)
| 1985 |  | $A(1^{1}\Sigma_{u}^{+}),b(1^{3}\Pi_{u})$ | ? | $9,15$ | 2 | - | Xie & Field Xie and Field (1985a, b)
| 1986 | PFOODR | $(2^{3}\Pi_{g})\rightarrow b(1^{3}\Pi_{u})$ | $0.5$ | $0-17$ | ~170 | 32 lines recovered | Xie & Field Xie and Field (1986)
| 1986 | PFOODR | $(1^{3}\Delta_{g})\rightarrow b(1^{3}\Pi_{u})$ | $0.02-0.13$ | $0-11$ | ? |  | Rice, Xie & Field Rice  _et al._ (1986); Rice, Steven F. and Field (1986)
| 1988 | CIF | $(1^{3}\Delta_{g})\rightarrow b(1^{3}\Pi_{u})$ | $0.2$ | $0-11$ | ~200 | - | Schmidt _et al. Schmidt  _et al._ (1988)_
| 1992 | CIF | $(1^{3}\Delta_{g})\leftrightarrow b(1^{3}\Pi_{u})$ | $0.003-0.07$ | $0-9$ | 599 | $\checkmark$ | Linton _et al._ Linton  _et al._ (1992)
TOTAL |  | $0.003-1$ | $0-27$ | ? | 1357 |  Linton  _et al._ (1992)Russier  _et al._ (1997)

aThe measurements were on $v$-levels of the $A$-state, and information about
the $b$-state $v$-levels that perturbed those $A$-state levels was inferred
indirectly.

##  I Hamiltonian

The rovibrational energy levels and wavefunctions for isotopologue $\alpha$
with reduced mass $\mu_{\alpha}$ are treated as the eigenvalues and
eigenfunctions in the effective radial Schroedinger equation:

$\left(-\frac{\hbar}{2\mu_{\alpha}}\frac{{\rm d}^{2}}{{\rm
d}r^{2}}+V_{{\rm\alpha}}(r)+\frac{\hbar
N(N+1)}{2\mu_{\alpha}r^{2}}\left(1+g_{{\rm\alpha}}(r)\right)\right)\psi_{v,N}(r)=E_{v,N}(r)\psi_{v,N}\,.$
(1)

Here $V_{\alpha}(r)$ and $g_{\alpha}(r)$ represent the “adiabatic” potential
and the “non-adiabatic” rotational $g$-factor. The adiabatic potential can be
represented as a “Born-Oppenheimer” potential (which is mass-independent),
plus a (mass-dependent) shift due to the diagonal correction to the Born-
Oppenheimer approximation:

$V_{\alpha}(r)\equiv V_{{\rm BO}}(r)+\Delta V_{\alpha}(r).$ (2)

The $\Delta V_{\alpha}(r)$ correction can be approximated by the expectation
value of the nuclear kinetic energy operator in the molecular electronic
wavefunction basis $\left\langle T_{{\rm nuc,\alpha}}\right\rangle$
McAlexander  _et al._ (1996). For homonuclear diatomics it is given by Van
Vleck (1936); Bunker (1968); McAlexander  _et al._ (1996):

$\displaystyle\Delta V_{\alpha}(r)$ $\displaystyle=$
$\displaystyle\left\langle T_{{\rm
nuc},\alpha}\right\rangle+\Delta_{2}V_{\alpha}(r)$ (3)
$\displaystyle\left\langle T_{{\rm nuc},\alpha}\right\rangle$
$\displaystyle\equiv$ $\displaystyle
Q_{\alpha}(r)+P_{\alpha}(R)+S_{\alpha}(R)$ (4) $\displaystyle Q_{\alpha}(r)$
$\displaystyle\equiv$
$\displaystyle-\frac{\hbar^{2}}{2\mu_{\alpha}}\left\langle\frac{\partial^{2}}{\partial
r^{2}}\right\rangle_{\alpha}$ (5) $\displaystyle P_{\alpha}(r)$
$\displaystyle\equiv$
$\displaystyle\frac{\hbar^{2}}{2\mu_{\alpha}}\left\langle
L_{x}^{2}+L_{y}^{2}\right\rangle_{\alpha}$ (6)

$\displaystyle S_{\alpha}(r)$ $\displaystyle\equiv$
$\displaystyle-\frac{\hbar^{2}}{8\mu_{\alpha}}\left(\left\langle\sum_{i}\nabla_{i}^{2}\right\rangle+\left\langle\sum_{i\neq
j}\nabla_{i}\nabla_{j}\right\rangle\right)_{\alpha}$ (7) $\displaystyle=$
$\displaystyle\frac{m_{e}}{4\mu_{\alpha}}\left\langle
T_{e,\alpha}\right\rangle-\frac{\hbar^{2}}{8\mu_{\alpha}}\left\langle\sum_{i\neq
j}\nabla_{i}\nabla_{j}\right\rangle_{\alpha}$ (8) $\displaystyle=$
$\displaystyle-\frac{m_{e}}{4\mu_{\alpha}}\left(V_{{\rm
BO}}(r)+r\frac{\partial}{\partial r}V_{{\rm BO}}(r)\right)$ (10)
$\displaystyle\>-\frac{\hbar^{2}}{8\mu_{\alpha}}\left\langle\sum_{i\neq
j}\nabla_{i}\nabla_{j}\right\rangle_{\alpha}$ $\displaystyle\equiv$
$\displaystyle-\frac{m_{e}}{4\mu_{\alpha}}\left(V_{{\rm
BO}}(r)+r\frac{\partial}{\partial r}V_{{\rm BO}}(r)\right)+\Delta
S_{\alpha}(r).$ (11)

where $z$ represents the internuclear axis, $L_{x}$ and $L_{y}$ are then
projections of the total electronic orbital angular momentum, $i,j$ represent
indices for individual electrons of the molecule, and the first term of
$S_{\alpha}(r)$ has been expressed in terms of the average electronic kinetic
energy $\langle T_{e,\alpha}\rangle$ and then re-expressed in terms of
$V_{{\rm BO}}(r)$ using the virial theorem Slater (1933); McAlexander  _et
al._ (1996). We can define a long-range term $P_{{\rm LR,\alpha}}(r)$ by
evaluating $P_{\alpha}(r)$ in the long-range Heitler-London basis, where
electron overlap is zero. $P_{\alpha}(r)$ is then expressed as $P_{{\rm
LR,\alpha}}(r)$ plus a correction $\Delta P_{\alpha}(r)$ McAlexander  _et al._
(1996):

$\displaystyle P_{\alpha}(r)$ $\displaystyle\equiv$ $\displaystyle P_{{\rm
LR},\alpha}(r)+\Delta P_{\alpha}(r)$ (12) $\displaystyle P_{{\rm
LR,\alpha}}(r)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2\mu_{\alpha}r^{2}}\sum_{k}^{2}l_{k}(l_{k}+1),$
(13)

where $l_{k}$ represents the orbital angular momentum of the electrons in
constituent atom $k$ of the molecule. Herein we restrict our attention to the
$b$-state of Li2 which dissociates into Li$(S)$ \+ Li$(P)$:

$\displaystyle P_{{\rm LR,\alpha}}(r)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2\mu_{\alpha}r^{2}}\left(l_{s}(l_{s}+1)+l_{p}(l_{p}+1)\right)$
(14) $\displaystyle=$
$\displaystyle\frac{2\hbar^{2}}{2\mu_{\alpha}r^{2}}\,,\,\mbox{since
}\left(l_{s}=0,l_{p}=1\right).$ (15)

While we know that in the long-range limit, $Q_{\alpha}(r)$ will become a
constant McAlexander  _et al._ (1996), $\Delta P_{\alpha}(r)$ will be zero,
and $\Delta S_{\alpha}(r)$ will be small McAlexander  _et al._ (1996), no
other information about these terms is known. Therefore, we may re-write the
diagonal Born-Oppenheimer correction (DBOC) in terms of what we know, and then
represent these parts that we don’t know, by model functions
$\tilde{S_{k}}(r)$ for each constituent atom $k$ of the molecule:

$\displaystyle\Delta V_{\alpha}(r)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{\mu_{\alpha}r^{2}}-\frac{m_{e}}{4\mu_{\alpha}}\left(V_{{\rm
BO}}(r)+r\frac{\partial}{\partial r}V_{{\rm
BO}}(r)\right)+{\color[rgb]{0,0,1}{\color[rgb]{1,0,0}\left(\Delta
S_{\alpha}(r)+\Delta
P_{\alpha}(r)+Q_{\alpha}(r)+\Delta_{2}V_{\alpha}(r)\right)}}$ (16)
$\displaystyle\equiv$
$\displaystyle\frac{\hbar^{2}}{\mu_{\alpha}r^{2}}-\frac{m_{e}}{4\mu_{\alpha}}\left(V_{{\rm
BO}}(r)+r\frac{\partial}{\partial r}V_{{\rm
BO}}(r)\right)+\sum_{k}\frac{m_{e}}{M_{k}}\tilde{S_{k}}(r),$ (17)

where $m_{e}$ is the electron mass and $M_{k}$ is the mass of the $k^{{\rm
th}}$ constituent nucleus of the molecule. Note that until now, the terms
containing $\tilde{S_{k}}(r)$ represented the entirety of Eq. 17, so less of
$\Delta V_{\alpha}(r)$ was described by theoretically known expressions, and
more was described by empirical fitting functions Watson (2004).

The only part of the Hamiltonian in Eq. 1 that is missing is now the non-
adiabatic term $g_{\alpha}(r)$. This is often represented by model functions
$\tilde{R}_{k}(r)$ for each atom:

$g_{\alpha}(r)\equiv\sum_{k}\frac{m_{e}}{M_{k}}\tilde{R}_{k}(r)\,.$ (18)

Figure 1: When considering the data for all isotopologues, levels up to $v=32$
have been observed. For 6,6Li2, levels $v=33-40$ are accessible by Kirk
Madison’s group in University of British Columbia with the laboratory and
method used in studies of the $A\left(1^{1}\Sigma_{u}^{+}\right)$ Gunton  _et
al._ (2013a) and $c\left(1^{3}\Sigma_{g}^{+}\right)$ Semczuk  _et al._ (2013)
states. The remainder of the levels have not been observed.

##  II Empirical potential and Born-Oppenheimer breakdown (BOB) corrections

We now wish to determine empirical functions for $V_{{\rm
BO}}(r),\,\tilde{S}(r),$ and $\tilde{R}(r)$ that accurately reproduce all
measured energies when using the Hamiltonian of Eq. 1.

There is a gap of more than 2000 cm-1 ($>60$ THz) in experimental information
between the highest observed level of Li${}_{2}\left(b,1^{3}\Pi_{u}\right)$,
and its dissociation energy. This means that when building an empirical
potential that aims to be relevant in the large data gap, it is very important
to take great care in ensuring the potential behaves physically correctly in
the extrapolation region. In 2011 the MLR (Morse/long-range) model was used in
a fit to build empirical potentials from spectroscopic data for the
$c\left(1^{3}\Sigma_{g}^{+}\right)-$states of 6,6Li2 and 7,7Li2, where there
was a gap of more than $5000\>$cm-1 between data near the bottom of the
potential, and data at the very top Dattani and Le Roy (2011a). In 2013
spectroscopic measurements were made in the very middle of this gap Semczuk
_et al._ (2013), and it was found that the vibrational energies predicted by
the MLR potential from Dattani and Le Roy (2011a) were correct to within about
1 cm-1. The present case for the $b$-state is in some sense more interesting
because there is no data at the top helping to anchor the potential with the
right shape near dissociation.

However, as for the $c$-state, the MLR model is still expected to be able to
represent the physics in the extrapolation region faithfully since the correct
theoretical long-range is built into the model. Having this long-range physics
accurately built into the model is almost as helpful as having data in the
long-range region, as was the case of the $c$-state. MLR-type empirical
potentials have now successfully described spectroscopic data for many
diatomic 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 (2011a); 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._
(2013a); Meshkov  _et al._ (2014); Dattani (2014); Coxon and Hajigeorgiou
(2015); Walji  _et al._ (2015); Dattani (2015); Dattani  _et al._ (2014) 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. Therefore, we will proceed to use the MLR model to describe
$V{\rm{}_{{\rm BO}}}(r)$.

The MLR model is defined by

$V_{{\rm
MLR}}(r)\equiv\mathfrak{D}_{e}\left(1-\frac{u(r)}{u(r_{e})}e^{-\beta(r)y_{p}^{r_{e}}(r)}\right)^{2},$
(19)

where $\mathfrak{D}_{e}$ is the dissociation energy, $r_{e}$ is the
equilibrium internuclear distance, and the polynomial $\beta(r)$ is

$\beta(r)\equiv\beta_{p,q}^{r_{{\rm
ref}}}(r)\equiv\beta_{\infty}y_{p}^{r_{{\rm ref}}}(r)+\left(1-y_{p}^{r_{{\rm
ref}}}(r)\right)\sum_{i=0}^{N_{\beta}}\beta_{i}\left(y_{q}^{r_{{\rm
ref}}}(r)\right)^{i},$ (20)

with

$\beta_{\infty}\equiv\lim_{r\rightarrow\infty}\beta(r)=\ln\left(\frac{2\mathfrak{D}_{e}}{u(r_{e})}\right).$
(21)

Equations 19 and 20 also depend on the radial variable

$y_{n}^{r_{{\rm ref}}}(r)=\frac{r^{n}-r_{{\rm ref}}^{n}}{r^{n}+r_{{\rm
ref}}^{n}},$ (22)

where the reference distance $r_{{\rm ref}}$ is simply the equilibrium
distance $r_{e}$ in most cases, but can be adjusted to optimize the fit to
equation 19.

It is well known Le Roy  _et al._ (2009); Dattani and Le Roy (2011a) that for
large $r$ we have

$V(r)\simeq\mathfrak{D}_{e}-u(r)+\cdots,$ (23)

therefore the long-range behavior of the potential is defined by $u(r)$, and
the short to mid-range behavior is defined by $\beta(r).$ In the
$b\left(1^{3}\Pi_{u}\right)$ state, a spin-orbit interaction emerges at large
internuclear distances, which splits the potential into four components.
Therefore, four different potentials $V^{\Omega_{u}}(r)$ can be defined to
have the same $\beta(r)$ defining the short-range behavior where there is no
significant splitting, and to have four different
$u^{{}^{3}\Pi_{u},\Omega_{u}}(r)$ defining the long-range where the splitting
occurs.

For large $r$ where the spin-orbit interaction becomes strong, it is dangerous
to label the spin angular momentum $S$ and orbital angular momentum $\Lambda$
separately, as in the molecular term symbol ${}^{2S+1}\Lambda$. Instead, these
two momenta are combined into a total electronic angular momentum $\Omega$.
For ${}^{3}\Pi_{u}$, $S=1$ and $\Lambda=1$, so there are states with four
possible symmetries in the $\Omega$ representation: $0_{u}^{+}$, $0_{u}^{-}$,
$1_{u}$, and $2_{u}$. Each of these four states has a slightly different
behavior at large internuclear distances, due to coupling with states that
have the same symmetry in the $\Omega$ representation, but different symmetry
in the $\Lambda$ representation. This coupling has been described in Aubert-
Frécon  _et al._ (1998) and has been used for building appropriate analytic
empirical potentials for the $A\left(1^{1}\Sigma_{u}^{+}\right)$ Le Roy  _et
al._ (2009); Gunton  _et al._ (2013a) and $c\left(1^{1}\Sigma_{u}^{+}\right)$
Dattani and Le Roy (2011a); Semczuk  _et al._ (2013) states of Li2. The long-
range function $u(r)$ is defined separately for each spin-orbit state:

$\displaystyle u(r)$ $\displaystyle=$
$\displaystyle\begin{cases}u^{b,0_{u}^{+}}(r)&,\,0_{u}^{+}\\\
u^{b,0_{u}^{-}}(r)&,\,0_{u}^{-}\\\ u^{b,1_{u}}(r)&,\,1_{u}\\\
u^{b,2_{u}}(r)&,\,2_{u}.\end{cases}$ (24)

Each of these functions is an eigenvalue of a matrix $\mathbf{u}^{\Omega_{u}}$
for each $\Omega_{u}$ state. These matrices are given in the subsections
below, in terms of the (positive) spin-orbit splitting energy $\Delta E$, and
neglecting exchange interaction terms.

###  II.1 The $0_{u}^{+}$ states

In addition to the $b($1${}^{3}\Pi_{u})$ state, the other $\Lambda$ state that
can give rise to $0_{u}^{+}$ symmetry is the $A(1^{1}\Sigma_{u}^{+})$ state
Jones  _et al._ (2006). The interstate coupling is therefore given by the
$2\times 2$ matrix Aubert-Frécon  _et al._ (1998):

$\displaystyle\mathbf{u}^{0_{u}^{+}}(r)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\frac{1}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{A\left(1{}^{1}\Sigma_{u}^{+}\right)}+2C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}&\quad\;\frac{\sqrt{2}}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{A\left(1{}^{1}\Sigma_{u}^{+}\right)}-C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{3}}\\\
\frac{\sqrt{2}}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{A\left(1{}^{1}\Sigma_{u}^{+}\right)}-C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}&\qquad\quad\Delta
E+\frac{2}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{A\left(1{}^{1}\Sigma_{u}^{+}\right)}+C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}\end{pmatrix},$
(25)

where the lower energy eigenvalue $u^{A,0_{u}^{+}}(r)$ comes from
$A\left(1^{1}\Sigma_{u}^{+}\right)$ and approaches the dissociation limit of
Li$\left(2^{2}S_{\nicefrac{{1}}{{2}}}\right)+\mbox{Li}\left(2^{2}P{}_{\nicefrac{{1}}{{2}}}\right)$,
and the higher energy eigenvalue $u^{b,0_{u}^{+}}(r)$ comes from
$b\left(1^{3}\Pi_{u}\right)$ and approaches the dissociation limit of
Li$\left(2^{2}S_{\nicefrac{{1}}{{2}}}\right)+\mbox{Li}\left(2^{2}P{}_{\nicefrac{{3}}{{2}}}\right)$
Jones  _et al._ (2006).

###  II.2 The $0_{u}^{-}$ states

In addition to the $b($1${}^{3}\Pi_{u})$ state, the other $\Lambda$ state that
can give rise to $0_{u}^{-}$ symmetry is the $2a(2^{3}\Sigma_{u}^{+})$ state
Jones  _et al._ (2006). The interstate coupling is therefore given by the
$2\times 2$ matrix Aubert-Frécon  _et al._ (1998):

$\displaystyle\mathbf{u}^{0_{u}^{-}}(r)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\frac{1}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{2a\left(2{}^{3}\Sigma_{u}^{+}\right)}+2C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}&\quad\;\frac{\sqrt{2}}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{2a\left(2{}^{3}\Sigma_{u}^{+}\right)}-C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{3}}\\\
\frac{\sqrt{2}}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{2a\left(2{}^{3}\Sigma_{u}^{+}\right)}-C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}&\qquad\quad-\Delta
E+\frac{2}{3}{\displaystyle\sum_{\begin{subarray}{c}m=3,6,8\\\
9,10,11,\ldots\end{subarray}}}\frac{C_{m}^{2a\left(2{}^{3}\Sigma_{u}^{+}\right)}+C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}\end{pmatrix},$
(26)

where the lower energy eigenvalue comes from $b\left(1^{3}\Pi_{u}\right)$ and
approaches the dissociation limit of
Li$\left(2^{2}S_{\nicefrac{{1}}{{2}}}\right)+\mbox{Li}\left(2^{2}P{}_{\nicefrac{{1}}{{2}}}\right)$,
and the higher energy eigenvalue comes from
$2a\left(2^{3}\Sigma_{u}^{+}\right)$ and approaches the dissociation limit of
Li$\left(2^{2}S_{\nicefrac{{1}}{{2}}}\right)+\mbox{Li}\left(2^{2}P{}_{\nicefrac{{3}}{{2}}}\right)$
Jones  _et al._ (2006).

###  II.3 The $1_{u}$ states

In addition to the $b($1${}^{3}\Pi_{u})$ state, the other $\Lambda$ states
that can give rise to $1_{u}$ symmetry are the
$2a\left(2^{3}\Sigma_{u}^{+}\right)$ state and the
$B\left(1{}^{1}\Pi_{u}\right)$state Jones  _et al._ (2006). The interstate
coupling is therefore given by the $3\times 3$ matrix Aubert-Frécon  _et al._
(1998):

$\mathbf{u}^{1_{u}}(r)=$ (27)

$\negthickspace\negthickspace\left(\begin{array}[]{ccc}\frac{1}{3}{\displaystyle\sum_{m}\frac{C_{m}^{2a\left(2{}^{3}\Sigma_{u}^{+}\right)}+C_{m}^{B\left(1{}^{1}\Pi_{u}\right)}+C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}}&\frac{1}{3\sqrt{2}}{\displaystyle\sum_{m}\frac{-2C_{m}^{2a\left(2{}^{3}\Sigma_{u}^{+}\right)}+C_{m}^{B\left(1{}^{1}\Pi_{u}\right)}+C_{m}^{b\left(1,^{3}\Pi_{u}\right)}}{r^{m}}}&\frac{1}{\sqrt{6}}{\displaystyle\sum_{m}\frac{-C_{m}^{B\left(1,^{1}\Pi_{u}\right)}+C_{m}^{b\left(1,^{3}\Pi_{u}\right)}}{r^{m}}}\\\
\frac{1}{3\sqrt{2}}{\displaystyle\sum_{m}\frac{-2C_{m}^{2a\left(2{}^{3}\Sigma_{u}^{+}\right)}+C_{m}^{B\left(1{}^{1}\Pi_{u}\right)}+C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}}&\Delta
E_{{\rm
SO}}+\frac{1}{6}{\displaystyle\sum_{m}\frac{4C_{m}^{2a\left(2{}^{3}\Sigma_{u}^{+}\right)}+C_{m}^{B\left(1{}^{1}\Pi_{u}\right)}+C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}}&\frac{1}{2\sqrt{3}}{\displaystyle\sum_{m}\frac{-C_{m}^{B\left(1{}^{1}\Pi_{u}\right)}+C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}}\\\
\frac{1}{\sqrt{6}}{\displaystyle\sum_{m}\frac{-C_{m}^{B\left(1{}^{1}\Pi_{u}\right)}+C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}}&\frac{1}{2\sqrt{3}}{\displaystyle\sum_{m}\frac{-C_{m}^{B\left(1{}^{1}\Pi_{u}\right)}+C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}}&-\Delta
E+\frac{1}{2}{\displaystyle\sum_{m}\frac{C_{m}^{B\left(1{}^{1}\Pi_{u}\right)}+C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}}\end{array}\right),$
(28)

where the lowest energy eigenvalue comes from $b\left(1^{3}\Pi_{u}\right)$ and
approaches the dissociation limit of
Li$\left(2^{2}S_{\nicefrac{{1}}{{2}}}\right)+\mbox{Li}\left(2^{2}P{}_{\nicefrac{{1}}{{2}}}\right)$,
and the middle and highest energy eigenvalues come from
$B\left(1^{1}\Pi_{u}\right)$ and $2a\left(2^{3}\Sigma_{u}^{+}\right)$
respectively and both approach the dissociation limit of
Li$\left(2^{2}S_{\nicefrac{{1}}{{2}}}\right)+\mbox{Li}\left(2^{2}P{}_{\nicefrac{{3}}{{2}}}\right)$
Jones  _et al._ (2006).

###  II.4 The $2_{u}$ state

The $2_{u}$ state approaching the dissociation limit of $2S+2P$ is alone in
its symmetry, and approaches
Li$\left(2^{2}S_{\nicefrac{{1}}{{2}}}\right)+\mbox{Li}\left(2^{2}P{}_{\nicefrac{{3}}{{2}}}\right)$
Jones  _et al._ (2006). It has the long-range function Tang  _et al._ (2011):

$\displaystyle u^{2_{u}}(r)$ $\displaystyle=$ $\displaystyle-\left(\Delta
E-\sum_{\begin{subarray}{c}m=3,6,8,\\\
9,10,11,\ldots\end{subarray}}\frac{C_{m}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{m}}\right)$
(29) $\displaystyle=$ $\displaystyle-\left(\Delta
E-\frac{C_{3}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{3}}-\frac{C_{6}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{6}}-\frac{C_{8}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{8}}-\frac{C_{9}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{9}}-\frac{C_{10}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{10}}-\frac{C_{11}^{b\left(1{}^{3}\Pi_{u}\right)}}{r^{11}}\cdots.\right)$
(30)

###  II.5 All four $\Omega_{u}$ states combined

One can imagine an experiment which obtains spectroscopic measurements for all
of the four $\Omega_{u}$ states, and fits to all of this data simultaneously
by using the appropriate eigenvalues of the $8\times 8$ matrix below:

$\displaystyle u$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}\mathbf{u^{0_{u}^{+}}}\\\
&\mathbf{u^{0_{u}^{-}}}\\\ &&\mathbf{u^{1_{u}}}\\\
&&&u^{2_{u}}\end{array}\right)$ (35) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccccc}u_{11}^{0_{u}^{+}}&u_{12}^{0_{u}^{+}}\\\
u_{21}^{0_{u}^{+}}&u_{22}^{0_{u}^{+}}\\\
&&u_{11}^{0_{u}^{-}}&u_{12}^{0_{u}^{-}}\\\
&&u_{21}^{0_{u}^{-}}&u_{22}^{0_{u}^{-}}\\\
&&&&u_{11}^{1_{u}}&u_{12}^{1_{u}}&u_{13}^{1_{u}}\\\
&&&&u_{21}^{1_{u}}&u_{22}^{1_{u}}&u_{23}^{1_{u}}\\\
&&&&u_{31}^{1_{u}}&u_{32}^{1_{u}}&u_{33}^{1_{u}}\\\
&&&&&&&u^{2_{u}}\end{array}\right).$ (44)

Figure 2: In the long-range region, the splitting of the
$b\left(1^{3}\Pi_{u}\right)$ state into four spin-orbit components becomes
increasingly obvious. This is well beyond the region where data is available,
so our empirical potential fits to a model with the simplest long-range
potential energy function (the $2_{u}$ state, since there are no other $2_{u}$
states that are nearby in energy, see Eqs. 29-44). The fine-structure
splitting of $0.33532461313$ cm-1 comes from measurements in Brown  _et al._
(2013).

However, Fig. 1 shows the data region, and Fig. 2 shows that the spin-orbit
splitting does not seem to become apparent until well past this region. Since
the measurements that have been done on the $b$-state thus far are far away
from the effect of the spin-orbit splitting, we choose to use the simplest
spin-orbit long-range function: $u^{b,2_{u}}(r)$.

###  II.6 Quadratic corrections and damping functions

Since the leading term not shown in Eq. 23 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 (2011a); Semczuk  _et al._ (2013);
Gunton  _et al._ (2013a), by applying a transformation to all $C_{6}$,
$C_{9}$, and this time also $C_{11}$ terms:

$\displaystyle C_{6}$ $\displaystyle\rightarrow$ $\displaystyle
C_{6}+\frac{C_{3}^{2}}{4\mathfrak{D}_{e}}$ (45) $\displaystyle C_{9}$
$\displaystyle\rightarrow$ $\displaystyle
C_{9}+\frac{C_{3}C_{6}}{2\mathfrak{D}_{e}},$ (46) $\displaystyle C_{11}$
$\displaystyle\rightarrow$ $\displaystyle
C_{11}+\frac{C_{3}C_{8}}{2\mathfrak{D}_{e}}.$ (47)

where the transformation in Eq. 45 has to be made first due to Eq. 46’s
dependence on $C_{6}$.

Additionally, The long-range formulas in terms of $C_{m}$ constants in the
above sub-sections were derived under the assumption that two free atoms are
interacting with each other, and there is no overlap of the electrons’
wavefunctions as 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):

$\displaystyle C_{m}$ $\displaystyle\rightarrow$ $\displaystyle
C_{m}D_{m}^{(s)}\left(r\right)$ (48) $\displaystyle D_{m}^{(s)}\left(r\right)$
$\displaystyle\equiv$ $\displaystyle\left(1-e^{-\left(\frac{b^{(s)}\rho
r}{m}+\frac{c^{(s)}\left(\rho r\right)^{2}}{\sqrt{m}}\right)}\right)^{m+s},$
(49)

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 in Eq. 19 has the physically
desired behavior $V\propto\nicefrac{{1}}{{r^{2}}}$ 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).

###  II.7 Long-range constants

In previous studies of the $A(1^{1}\Sigma_{u}^{+})$ state Le Roy  _et al._
(2009); Gunton  _et al._ (2013b) and $c(1^{3}\Sigma_{g}^{+})$ state Dattani
and Le Roy (2011a); Semczuk  _et al._ (2013), it was found that the most
precise theoretical values of $C_{3}$ known at those times Tang  _et al._
(2010b, 2009a) did not fit as well with the measurements of the high-lying
vibrational levels near the dissociation, as the values of $C_{3}$ obtained by
setting it as a free parameter determined by a least-squares fit to the data.

However, for the $b\left(1^{3}\Pi_{u}\right)$-state, no measurements of such
high-lying vibrational levels have been made, so such an “empirical fit” to
$C_{3}$ is impossible, and we will have to use the most precise theoretical
value known. For 7,7Li2 this is the value from Tang  _et al._ (2010b) and for
6,6Li2 this is an unpublished value from Tang calculated in 2015 Tang (2015).
These values are listed in Table 2, along with the theoretical values for the
higher-order $C_{m}$ constants used in our analysis (it has not yet been
possible to fit these higher-order $m>3$ constants to spectroscopic data in
any direct-potential-fit analysis, so they are held fixed). For $m>8$, no
finite-mass corrections have been calculated yet.

Table 2: The best currently available long-range constants and their sources
(in Hartree atomic units). ∞Li2 represents a Li2 molecule where both nuclei
have infinite mass, since finite mass corrections have not yet been calculated
for $C_{m}$ coefficients with $m\geq 9$.

| $A\left(1^{1}\Sigma_{u}\right)$ | $2a\left(2^{3}\Sigma_{u}\right)$ | $B\left(1^{1}\Pi_{u}\right)$ | $b\left(1^{3}\Pi_{u}\right)$ |
---|---|---|---|---|---
|  6,6Li2 |  7,7Li2 |  6,6Li2 |  7,7Li2 |  6,6Li2 |  7,7Li2 |  6,6Li2 |  7,7Li2 | Ref.
$C_{3}$ |  $11.0009$ Tang (2015) |  $11.0007$ Tang (2015) |  $-11.0009$ Tang (2015) |  $-11.0007$ Tang  _et al._ (2010b) |  $-5.5005$ Tang (2015) |  $-5.5004$ Tang (2015) |  5.5005 Tang (2015) |  5.5004 Tang (2015) | -
$C_{6}$ | 2076.19(7) | 2076.08(7) | 2076.19(7) | 2076.08(7) | 1407.20(2) | 1407.15(5) | 1407.20(2) | 1407.15(5) | Tang  _et al._ (2009b)
$C_{8}$ | 274137(6) | 274128(5) | 991104(5) | 991075(6) | 48566.9(4) | 48566.4(2) | 103053(1) | 103052(1) | Tang  _et al._ (2009b)
|  ∞Li2 |  ∞Li2 |  ∞Li2 |  ∞Li2 |
$C_{9}$ | 2.2880(2)$\times 10^{5}$ |  $-2.2880(2)$$\times 10^{5}$ |  $-5.173(1)$$\times 10^{4}$ | 5.173(1)$\times 10^{4}$ | Tang  _et al._ (2011)
$C_{10}$ | 3.0096$\times 10^{7}$ |  $-1.2113$$\times 10^{8}$ | 8.9295$\times 10^{6}$ | 9.1839$\times 10^{5}$ | Zhang  _et al._ (2007)
$C_{11}$ | $-5.930\times 10^{7}$ |  $-5.0321$$\times 10^{8}$ | $-9.924\times 10^{7}$ | 2.652$\times 10^{7}$ | Tang  _et al._ (2011)

Figure 3: Various $(N_{\beta})_{p,q}^{r_{{\rm ref}}}$ models with the same
quality of fit are in agreement with each other, until large values of $r$ at
which the $N_{\beta}=7$ model dips significantly below the red curve
representing the theoretical long-range behavior. This dip was obvious in the
$-r^{3}V(r)$ vs $\nicefrac{{1}}{{r^{3}}}$ plots for all models mentioned in
the text except for the three $N_{\beta}=11$ cases shown. The inset shows
$-r^{3}V(r)$ for the four mentioned models, subtracted from the red
theoretical curve. All models match the theory for very large $r$ (small
$\nicefrac{{1}}{{r^{3}}}$), but the cases with higher $p$ and/or $q$ match the
theory for a much larger range of $r$. Each green circle represents an outer
classical turning point of the _ab initio_ potential. In the inset, the _ab
initio_ values lie too high above 0 to be seen.

###  II.8 Dissociation energy $\mathfrak{D}_{e}$

At the time of carrying out our analysis, the best experimental value for
$\mathfrak{D}_{e}$ of which we were aware, was the 1983 value from Engelke and
Hage (1983): 12145$\pm$200 cm-1. In a recent study on BeH Dattani (2015), the
gap between the highest observed level and the dissociation asymptote was
$\sim 1000$ cm-1, and the fitted value of $\mathfrak{D}_{e}$ varied by about
$400$ cm-1 as parameters such as $r_{{\rm ref}}$, $p$, and $q$ were changed.
For the present case of the $b$-state of Li2, the data region stops more than
4000 cm-1 below the dissociation asymptote, so we do not expect to be able to
determine $\mathfrak{\mathfrak{D}_{e}}$any more precisely than the 1983
experimental value. However, we still tried, by letting $\mathfrak{D}_{e}$ be
a free parameter, and we indeed found that the fitted values varied by more
than $400$ cm-1. Therefore, it might make sense to use experimental value from
Engelke and Hage (1983) which was claimed to be within 400 cm-1.

However, it is expected that the _ab initio_ value from Musial and Kucharski
(2014) correct to within much less than 400 cm-1. This is because we
systematically checked the _ab initio_ $\mathfrak{D}_{e}$ values for all
electronic Li2 states calculated in Musial and Kucharski (2014), and found
that they were at most 68 cm-1 different from the best experimental value,
even when the experimental values were known to as high of a precision as
0.0023 cm-1 (see Table 3). Furthermore, the _ab initio_ value for the
$b$-state was within the 400 cm-1 confidence interval given by the 1983
experimental value Engelke and Hage (1983) discussed in the previous
paragraph. Therefore, we decided to fix our $\mathfrak{\mathfrak{D}_{e}}$
value at the _ab initio_ value of 12166 cm-1 and to only allow the other
parameters be free parameters for the remainder of the fitting analysis.

After the completion of this work, we discovered that a much less known paper
co-authored by one of the same authors from Engelke and Hage (1983), reported
a more precise $\mathfrak{D}_{e}$ value of (12180.6$\pm 0.6$) cm-1 just over 4
years afterwards Schmidt  _et al._ (1988), but it is not clear in the paper
how this value, nor its uncertainty is obtained. Particularly, it is not clear
whether this is a purely empirical value, or if it also uses the _ab inito_
potential which is part of the subject of the paper.

Table 3: Dissociation energies $\mathfrak{D}_{e}$ in cm-1 of various
electronic states of Li2. “obs - calc” denotes the _ab initio_ value
subtracted from the empirical value. Numbers in parenthesis indicate the
experimental uncertainty in the last digit(s) shown. Where possible, 6,6Li2
values were used. for the empirical values. When empirical values for the
$\mathfrak{D}_{e}$ of 6,6Li2 were not available, the empirical value for
7,7Li2 is given. All _ab initio_ values are for ∞Li2 which is a hypothetical
Li2 molecule with infinite mass.

$2S+2S$ | $X(1^{1}\Sigma_{g}^{+})$ | $a(1^{3}\Sigma_{u}^{+})$ |  |  |  |
---|---|---|---|---|---|---
_ab initio_ Musial and Kucharski (2014) | 8466. |  | 334. |  |  |  |  |
empirical | 8516. | 7800(23) Gunton  _et al._ (2013a) | 333. | 7795(62) Semczuk  _et al._ (2013) |  |  |  |
obs - calc | 51. |  | 0. |  |  |  |  |
$2S+2P$ | $A(1^{1}\Sigma_{u}^{+})$ | $B(1^{1}\Pi_{u})$ | $2X(2^{1}\Sigma_{g}^{+})$ | $C(1^{1}\Pi_{g})$ | $b(1^{3}\Pi_{u})$ | $c(1^{3}\Sigma_{g}^{+})$
_ab initio_ Musial and Kucharski (2014) | 9356. |  | 2930. |  | 3289. |  | 1426. |  |  12166. |  | 7080. |
empirical | 9353. | 1795(28) Gunton  _et al._ (2013a) | 2984. | 444(110) Huang and Le Roy (2003) | 3318(66) He  _et al._ (1991); Barakat  _et al._ (1986) | 1422. | 5(3) Miller  _et al._ (1990) |  12145. | (200) Engelke and Hage (1983) | 7092. | 4926(86) Semczuk  _et al._ (2013)
obs - calc | -3. |  | 54. |  | 30. |  | -3. |  |  -21. |  | 12. |
$2S+3S$ |  $3X(3^{1}\Sigma_{g}^{+})$ (1st min.) |  $2A(1^{1}\Sigma_{u}^{+})$ (1st min.) |  $2A(1^{1}\Sigma_{u}^{+})$ (2nd min.)
_ab initio_ Musial and Kucharski (2014) | 8290. | 5608. | 5389.
empirical | 8317\. Bernheim  _et al._ (1982, 1983) | 5615\. S  _et al._ (2000); Kubkowska  _et al._ (2007) | 5321\. S  _et al._ (2000); Kubkowska  _et al._ (2007)
obs - calc | 27. | 7. | -68.
$2S+3P$ |  $4X(4^{1}\Sigma_{g}^{+})$ (1st min.) | $2B(2^{1}\Pi_{u})$ | $2C(2^{1}\Pi_{g})$ | $D(1{}^{1}\Delta_{g})$ | $2d(2{}^{3}\Pi_{g})$
_ab initio_ Musial and Kucharski (2014) | 8380. | 6481. |  | 7773. |  | 9592. |  | 8505. |
empirical | 8349\. Bernheim (1981a) | 6455. | Bernheim (1981b) | 7773. | 7(3) Kubkowska  _et al._ (2007) | 9579. | Linton  _et al._ (1993) | 8484. | Bernheim (1981b)
obs - calc | -31. | -26. |  | 1. |  | -13. |  | -21. |

###  II.9 Choice of model parameters

Using the Hamiltonian of Eq. 1, we fit the parameters of Eq. 19 to the 1234
data, with the involved energy levels of the upper states $2^{3}\Pi_{g}$ and
$1^{3}\Delta_{g}$ treated as free parameters. All fits to the data were done
using the freely available program ${\tt DPotFit}\,2.0$ Le Roy  _et al._
(2013). Starting parameters for the fits to Eq. 19 were found by fitting to an
RKR potential using the freely available program ${\tt betaFit\,2.1}$ LeRoy
(2012). The RKR potential was made using the program ${\tt RKR1\,2.0}$ LeRoy
(2004) using the Dunham coefficients found in Table IV of Linton  _et al._
(1992).

The quality of a fit was determined by the dimensionless root-mean-square-
deviation ($\overline{dd}$) which scales each deviation between an energy
predicted by the model ($E_{{\rm calc}}$) and the corresponding measurement
($E_{{\rm obs}})$, by the uncertainty of the measurement ($u_{{\rm obs}}$),
for all $N_{{\rm data}}$ measurements:

$\overline{dd}\equiv\sqrt{\frac{1}{N_{{\rm data}}}\sum_{i=1}^{N_{{\rm
data}}}\left(\frac{E_{{\rm calc}}(i)-E_{{\rm obs}}(i)}{u_{{\rm
obs}}(i)}\right)^{2}}.$ (50)

In previous studies of the $A(1^{1}\Sigma_{u}^{+})$ state Gunton  _et al._
(2013b) and $c(1^{3}\Sigma_{g}^{+})$ state Semczuk  _et al._ (2013), it was
determined that there was no benefit in including long-range terms beyond
$C_{8}$, because the data for the high-lying rovibrational energies began to
deviate from the theoretical long-range potential energy curve at distances
shorter than the distance where $C_{9}$ began to give a noticeable effect on
the long-range function $u_{{\rm LR}}(r)$ (see Fig. 6 of Semczuk  _et al._
(2013) for example) .

However, for the $b$-state, no data exists in the long-range region, so it
might make more sense to include more $C_{m}$ terms in $u_{{\rm LR}}(r)$ in
order to anchor the potential somewhat appropriately in the $>$2500 cm-1 gap
at the top of the potential well where no data exists to guide the potential.
Nevertheless, we first followed the $A-$ and $c-$state studies and only used
up to $C_{8}$. We found an excellent fit with $\overline{dd}=0.95$ with only
$N_{\beta}=7$, $p=6,$ $q=2$. However, the long-range behavior of this
potential was in vast disagreement with the long-range behavior expected by
theory (see Fig. 3). This is because with $(p,q)=(6,2)$, the long-range form
of the potential described in Eq. 23 does not “turn on” until too high a value
of $r$ (a larger value of $r$ is needed for $y_{p}^{r_{{}_{{\rm ref}}}}(r)$
and $y_{q}^{r_{{}_{{\rm ref}}}}(r)$ to become sufficiently close to their
limiting values of 1.

We can often encourage the the long-range form to “turn on” earlier by
increasing $p$ and/or $q$, which often comes with the expense of requiring a
higher polynomial degree $N_{\beta}$ to recover the same $\overline{dd}$. We
explored models with $p\in\\{6,7,8,9\\}$ and $q\in\\{2,3\\}$, including the
$C_{9}$ term in $u_{{\rm LR}}$ for $p\geq 7$, $C_{10}$ for $p\geq 8$, and
$C_{11}$ for $p=9$. Ideally we would always use as many $C_{m}$ constants as
are known, but as explained in Le Roy  _et al._ (2009); Dattani and Le Roy
(2011a), the value of $p$ in Eqs. 19 and 20 need to satisfy $p>m_{{\rm
last}}-m_{{\rm first}}$ where $m_{{\rm last}}$ and $m_{{\rm first}}$ represent
respectively the last and first $C_{m}$ terms included in $u_{{\rm LR}}(r)$.
We also note that $C_{12}$ for the
Li$\left(2^{2}S\right)+\mbox{Li}\left(2^{2}P\right)$ asymptote is not
available, as far as we know.

We found that if $p<9$ and/or $q<3$, the long-range behavior does not turn on
until about $r=32$ Å (in the very best cases), while the $m$-dependent Le Roy
radius Tsai  _et al._ (1994) calculated from the radial expectation values
found in Zhang  _et al._ (2007) suggests that the long-range behavior should
turn on before $r=10$ Å. With $(p,q)=(9,3)$, we found a fit with
$N_{\beta}=11$ and $\overline{dd}=0.94$, where the long-range behavior turns
on at about $r=20$ Å (see Fig 3). Increasing $q$ to 4 would likely turn the
long-range behavior on at closer to the $m$-dependent Le Roy radius, but no
$(p,q)=(9,4)$ fits with $N_{\beta}\leq 11$ had a $\overline{dd}<1$ and we
needed to push to $N_{\beta}\geq 13$ in order to match the $\overline{dd}$ of
the best $(p,q,N_{\beta})=(9,3,11)$ fits. Using such a high-degree polynomial,
when the data only required $N_{\beta}=7$ for a good fit, can be dangerous in
terms of the potential’s extrapolation in the regions neither constrained by
data nor built-in $C_{m}$ constants. In this respect, we also tried
$(p,q)=(9,3)$ fits with $N_{\beta}=10$ for various $r_{{\rm ref}}$ values, but
no such fit had a $\overline{dd}<1$.

###  II.10 Born-Oppenheimer breakdown (BOB) corrections

With our best MLR model: MLR${}_{p,q}^{r_{{\rm
ref}}}\left(N_{\beta}\right)=\mbox{MLR${}_{9,3}^{3.8}\left(11\right)$}$, we
attempted to add adiabatic ($\tilde{S}(r)$, from Eq. 17) and non-adiabatic
($\tilde{R}(r)$, from Eq. 18) BOB corrections with the same model functions as
used in previous studies of Li2 since these models were improved in 2009 Le
Roy  _et al._ (2009); Dattani and Le Roy (2011b); Semczuk  _et al._ (2013);
Gunton  _et al._ (2013a). It was surprising that despite there being 599 6Li2
data $\left(\mbox{with }v_{{\rm max}}=9,\,N_{{\rm max}}=46\right)$ and 696
7Li2 data $\left(\mbox{with }v_{{\rm max}}=27,\,N_{{\rm max}}=27\right)$,
adding BOB correction functions did not improve the fit. Even when fitting to
3 adiabatic BOB parameters and 3 non-adiabatic BOB parameters, the
$\overline{dd}$ went down by less than 1%. This is unexpected when there is
just as much data for each isotopologue, and there is such a big difference in
the highest $v$ and $N$ levels observed for each isotopologue.

Nevertheless, it seems that the isotopologue shifts due to the kinetic energy
term in the Hamiltonian, and due to the mass-dependent BOB corrections
incorporated from Eqs. 10 and 14, are the only significant sources of energy
difference between 6Li2 and 7Li2 for the $b$-state (within our data’s
precision). This may also explain why the _ab initio_ potentials Musial and
Kucharski (2014) calculated assuming an infinite molecular mass managed to
predict both the 6Li2 and 7Li2 energies so fabulously (see discussion in
Section II.12 and Table IV), while the _ab initio_ BOB correction functions
for the not much lighter molecule BeH, were so crucial in matching the ro-
vibrational energies predicted from the _ab initio_ and empirical potentials
Dattani (2015); Koput (2011). Therefore, the final potential that we
recommend, which is the same for both 6Li2 and 7Li2 in the $b$-state except
for the small mass-dependent contributions coming from the kinetic energy and
the un-colored terms in Eq. 16, does not contain any empirically fitted
$\tilde{S}(r)$ and $\tilde{R}(r)$ BOB correction functions.

###  II.11 Sequential rounding and re-fitting (SRR)

Observing the predicted values for $r_{e}$ yielded by 139 different fits which
had $\overline{dd}<0.957$ (within 1.5% of the optimal fit, which had
$\overline{dd}=0.942$), we see that no fit predicted an $r_{e}$ outside the
range $(2.589\,825<r_{e}<2.589\,871)$ Å, regardless of the values of
$(p,q,r_{{\rm ref}},N_{\beta})$, though the more extreme predictions of
$r_{e}$ within this range corresponded to fits with $N_{\beta}\leq 9$. Based
on this observation, we recommend the value $r_{e}=(2.589\,848\pm 0.000\,023)$
Å, which is the average of these upper and lower bounds, with the uncertainty
being the distance from the average to either bound.

We then re-fitted the potential to the data, but with $r_{e}$ fixed at
2.589848 Å, once with the ${\tt DPotFit}$ setting ${\tt IROUND=-1}$ and once
with ${\tt IROUND=+1}$ in order to implement the SRR procedure described in Le
Roy (1998) and in the ${\tt DPotFit}$ manual Le Roy  _et al._ (2013). Neither
of these cases affected the 3-digit value $\overline{dd}=0.942$. The ${\tt
IROUND=-1}$ fit ended up with 2 more total digits than when ${\tt IROUND=+1}$
was used, but had a lower $\overline{dd}$ in the 4th digit, and has the more
elegant feature that the number of digits in $\beta_{i}$ decreases
monotonically with increasing $i$. Therefore, we recommend the potential with
${\tt IROUND=-1}$, whose parameters are listed in Table 4.

###  II.12 Vibrational energy spacings of the recommended
Li${}_{2}\left(b,^{3}\Pi_{u}\right)$ potential, and comparison to best _ab
initio_ potential

Very recently, a review paper on the 5e- systems BeH, BeD, and BeT Dattani
(2015) revealed that the state of the art _ab initio_ potentials Koput (2011)
(which used MR-ACPF/aug-cc-pCV7Z(i), a further estimate of electron
correlation effects beyond the approximations of MR-ACPF, second-order DKH
scalar relativistic corrections, and mass-dependent BOB corrections),
predicted vibrational energy spacings with up to at most 1.8 cm-1 discrepancy
with the state of the art empirical potential in the region for which
vibrational energies had been measured. The _ab initio_ potential also
predicted the existence of one more vibrational level than the empirical
potential, in the cases of BeH and BeD. This was all for the ground electronic
state $X\left(1{}^{2}\Sigma^{+}\right)$, so it is of interest to see how well
the most state-of-the-art _ab initio_ potential for the 6e- Li2 excited state
$b\left(1{}^{3}\Pi_{u}\right)$ will be.

A Fock space MRCC method based on the (2,0) sector of the Fock space, called
FS-CCSD(2,0) Musial (2012), was recently implemented and used to calculate
potential energy curves for many excited states of Li2 Musial and Kucharski
(2014) with the ANO-RCC basis set Veryazov  _et al._ (2004). While in
principle possible, DKH and BOB corrections have not been made in any Li2 _ab
initio_ calculations to date. However, fortunately we found in Section II.10
that the addition of $\tilde{S}(r)$ or $\tilde{R}(r)$ functions did not
significantly improve the fit to the data, meaning that Born-Oppenheimer
breakdown beyond the effects included from Eqs. 10 and 14 do not seem to have
a big effect in this particular state of Li2, at least in the data region.
Said another way, the _ab initio_ Born-Oppenheimer potential is expected to
give good predictions of the energies of 6,6Li2 and 7,7Li2, with mass-
dependent differences accounted for only by the Hamiltonian’s kinetic energy
operator, as was the case with the empirical MLR potential.

Table 4: Parameters defining the recommended MLR${}_{p,q}^{r_{{\rm ref}}}$
potential for the $b(1^{3}\Pi_{u})$-state of all isotopologues of Li2. The MLR
model is defined in Eqs. 19-49, and has damping parameters $s=-2$ and
$\rho=0.5$. The appropriate long-range constants are presented in Table 2. The
fit’s quality was $\overline{dd}=0.942$.

MLR${}_{9,3}^{3.80}$
---
$\mathfrak{D}_{e}$$\;\;12\,166$ cm-1
$r_{e}$$\;\;2.589\,848(23)$ Å
| $\beta_{0}$ |  $-0.022\,869\,54$ | $\beta_{6}$ |  $-5.586\,9$
| $\beta_{1}$ |  $-2.225\,706$ | $\beta_{7}$ |  $7.787$
| $\beta_{2}$ |  $-6.019\,341$ | $\beta_{8}$ |  $25.55$
| $\beta_{3}$ |  $-10.672\,48$ | $\beta_{9}$ |  $35.64$
| $\beta_{4}$ |  $-13.746\,79$ | $\beta_{10}$ |  $26.5$
| $\beta_{5}$ |  $-12.632\,2$ | $\beta_{11}$ |  $8.4$

Using the _ab initio_ Born-Oppenheimer potential provided to us by the authors
of Musial and Kucharski (2014), and the MLR potential described by Table 4, we
used ${\tt LEVEL}$ to calculate the vibrational energies of both the 6,6Li2
and 7,7Li2 isotopologues. We found that the highest levels had outer classical
turning points of several thousand Angstroms, and therefore we found it useful
to use the recently developed mapping which allows the radial mesh to extend
to $r=\infty$ when numerically solving the Schroedinger equation Meshkov  _et
al._ (2008, 2011), which is also implemented in ${\tt LEVEL}$. With this
method we were able to find up to $v=91$ for 6,6Li2 and $v=98$ for 7,7Li2,
however, when we calculated the scattering wavefunction, the number of nodes
indicated that the highest bound vibrational levels should be $v=92$ and
$v=100$ respectively. Impressively, these results were identical whether we
used the _ab initio_ potential, or the MLR potential.

We used Le Roy-Bernstein theory to predict these missing levels for each
isotopologue: For a $\nicefrac{{C_{3}}}{{r^{3}}}$ potential, the powers
$E_{v}^{\left(\nicefrac{{1}}{{6}}\right)}$ of the binding energies should be
linear in $v$ LeRoy and B. Bernstein (1970). We used the slope calculated from
$v=90$ and $91$ for 6,6Li2, and the slope calculated from $v=97$ and $98$, for
predicting the energies of $v=92$ and $v=99$ levels respectively. We then used
the last two points again to calculate a new slope for predicting the energies
of $v=100$. Interestingly, both the _ab initio_ potential, and the MLR
potential predict the existence of a 6,6Li2 level bound by $<8\times 10^{-8}$
cm-1 ($<3$ kHz) and a 7,7Li2 level bound by $<2\times 10^{-10}\,$cm-1 ($<6$
Hz). Using $C_{3}^{{}^{6}{\rm Li}}/r^{3}$ and $C_{3}^{{}^{7}{\rm Li}}/r^{3}$
we get that the outer classical turning points for the least bound levels of
each isotopologue are predicted to be at least 13 000 Å and 120 000 Å
respectively.

These vibrational energies were then used to calculate the zero point energies
(ZPEs) and vibrational energy spacings $\omega_{i}$, which are presented in
the table below, along with the discrepancy between the _ab initio_ and
empirical potentials. We have compared the vibrational energies (since these
are important for photoassociation experiments) and the vibrational spacings
(since these are important for experiments involving energy transitions). For
both 6,6Li2 and 7,7Li2, the discrepancy for the vibrational energies is less
than 12 cm-1. The agreement for the vibrational spacings is much better than
for the case of BeH discussed in the beginning of this subsection. The largest
discrepancy for a 6,6Li2 vibrational spacing is $<0.8\,$cm-1 and for 7,7Li2
cm-1 is $<0.6\,$cm-1.

##  III Conclusion

The motivation for this work was to build a potential that could predict high-
accuracy vibrational energies for 6,6Li${}_{2}(b)$ in the accessible energy
range of the recently built high-precision experimental setup which has so far
been very successful for photoassociation spectroscopy of
$c\left(1^{3}\Sigma_{g}^{+}\right)$ Semczuk  _et al._ (2013) and
$A\left(1^{1}\Sigma_{u}^{+}\right)$-states Gunton  _et al._ (2013a). A similar
photoassociation apparatus has recently also been setup by Kai Dieckmann’s
group to measure energy levels of ultra-cold 6,6Li2 electronic states
dissociating to the $2S+3P$ asymptote Sebastian  _et al._ (2014). The best _ab
initio_ vs empirical potential comparison for Li2 in the literature Halls  _et
al._ (2001), predicted vibrational levels with a discrepancy of up to 2.04
cm-1 for the $a\left(1^{3}\Sigma_{u}^{+}\right)$, which would have simply not
been good enough for finding the levels in this type of experiment. The
spectroscopic features in this type of experiment are typically around 0.000 2
cm-1; and covering 2 cm-1 with one-minute measurements and a 0.000 2 cm-1 step
size would take about 7 days.

However, the empirical MLR potential of Dattani and Le Roy (2011a) for the
$c$-state predicted energies were accurate enough to cut the experiment’s
duration to under 2 days, since the first level in the laser’s range turned
out to be predicted correctly to within 0.525 cm-1, despite this energy being
right in the middle of a 5000 cm-1 gap in available experimental data to guide
the empirical potential. In our table comparing the _ab initio_ and empirical
MLR energies for the $b$-state we see that the vibrational energies predicted
by the _ab initio_ potential are sometimes in $>10$ cm-1 disagreement with the
empirical values in the region where the energies have in fact been measured.
However, the _ab initio_ seems to predict all vibrational energy spacings
correctly within less than 1 cm-1 which is much better than the result in the
current best ground state 5e- BeH study Dattani (2015); Koput (2011).

The reason we are interested in measuring more levels of the $b$-state with
high-precision, is because it is surprising that the best _ab initio_
calculation of the first Li$(2^{2}S)-\mbox{Li$(2^{2}P)$}$ interaction term
($C_{3}$) is still in vast disagreement with the empirically fitted values
from the studies of the $A$-state Le Roy  _et al._ (2009); Semczuk  _et al._
(2013) and $c$-state Dattani and Le Roy (2011a); Gunton  _et al._ (2013a),
despite lithium only having 3e-, and this $C_{3}$ value having significance
for atomic clocks Tang  _et al._ (2011). Lithium is also 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). But this discrepancy in $C_{3}$ limits
the accuracy of a potential Li-based standard for polarizabilities Tang  _et
al._ (2011). Consolingly, in this study we have found that the $b$-state is
predicted to have levels bound by $<8\times 10^{-8}$ cm-1 ($<3$ kHz) which
would imply an outer classical turning point of $>13\,000\,$Å, which is larger
than any case in our awareness. Since the less bound the level measured, the
more precisely $C_{3}$ can be determined from a fit, these extremely weakly
bound energies are promising for resolving the discrepancy. While the
technology to measure these extremely weakly bound energies may still be years
away, many of the very high vibrational levels predicted in our analysis _are_
indeed accessible with today’s photoassociation technology.

The least bound levels for the $A$-state which have been measured have binding
energies of $E_{v=83}\approx 6$ cm-1, $E_{v=88}\approx 0.7$ cm-1 and
$E_{v=97}\approx 0.4$ cm-1 for 6.7Li2, 6,6Li2 and 7,7Li2 respectively, and the
least-squares fit to the data gave a $C_{3}$ value with a 95% confidence limit
uncertainty of about $\pm$8 cm${}^{-1}/\text{\AA}^{3}$ Le Roy  _et al._
(2009), which is currently the most precise experimentally determined
oscillator strength for any system, by an order of magnitude Tang  _et al._
(2011). The _ab initio_ and empirical MLR potentials for the $b$-state
compared in this work, both give predictions that are in great agreement for
energy levels that are several orders of magnitude less deeply bound than the
least deeply bound $A$-state measurements, making it therefore possible to
obtain an empirical $C_{3}$ value far more precise than in Le Roy  _et al._
(2009). Hopefully, this would resolve the age-old discrepancy between
experiment and theory for this $C_{3}$ value, which was first measured
experimentally by Loomis and Nusbaum in 1931 Loomis and Nusbaum (1931).

Empirical potentials have recently been built for the $b$\- and $A$-states of:
Rb2 in 2009 Salami  _et al._ (2007) and again in 2013Drozdova  _et al._
(2013), NaCs in 2009 Zaharova  _et al._ (2009), KCs in 2010 Kruzins  _et al._
(2010), RbCs in 2010 Docenko  _et al._ (2010), Cs2 in 2011 Bai  _et al._
(2011) and NaK in 2015 Harker  _et al._ (2015), however, this is to our
knowledge, the frst empirical potential built for the $b$-state of Li2.

Table IV. Comparison of the binding energies, denoted $G(v_{i})$; zero-point
energies (ZPE); and vibrational energy spacings, denoted $\omega_{i}\equiv
G(v_{i+1})-G(v_{i})$; for 6,6Li2 and 7,7Li2. The last column is the difference
between the two columns directly prior. Discrepancies of $\geq$ 0.5 cm-1 are
marked by one star (two stars if it was for vibrational level within the data
range). Lines with blue font are for unobserved levels, lines with bold green
font are for unobserved levels which are accessible by the UBC lab.All
energies were calculated by the program LEVEL 8.2 with atomic masses and
NUSE=0, IR2=1, ILR=3, NCN=6, and $\texttt{CNN=}C_{3}$. All numbers were
converged with respect to the radial mesh parameters, to at least to the
number of digits shown.

$v$ | 2015 Empirical | 2014 |  |  | 2015 Empirical | 2014 |
---|---|---|---|---|---|---|---
[Present work] | _ab initio_ | $E_{v}(2015)-E_{v}(2014)$ | [Present work] | _ab initio_ | $\omega(2015)-\omega(2014)$
|  6,6Li2
$\mathfrak{D}_{e}$ | -12166 | -12166 | - |  | - | - | - |  - |
$0$ | -11977.92296303 | -11978.33396628 | 0.00 |  | ZPE | 188.07703700 | 187.66603 370 |  0.41 |
$1$ | -11608.64518318 | -11609.63836960 | 0.41 |  | $\omega_{0}$ | 369.27777985 | 368.69559 668 |  0.58 |  **
$2$ | -11244.06766785 | -11245.60583257 | 0.99 | ** | $\omega_{1}$ | 364.57751533 | 364.03253 703 |  0.54 |  **
$3$ | -10884.17443003 | -10886.23056706 | 1.54 | ** | $\omega_{2}$ | 359.89323782 | 359.37526 551 |  0.52 |  **
$4$ | -10528.95288123 | -10531.55359827 | 2.06 | ** | $\omega_{3}$ | 355.22154880 | 354.67696 879 |  0.54 |  **
$5$ | -10178.39360662 | -10181.59083078 | 2.60 | ** | $\omega_{4}$ | 350.55927461 | 349.96276 749 |  0.60 |  **
$6$ | -9832.49064407 | -9836.21009569 | 3.20 | ** | $\omega_{5}$ | 345.90296255 | 345.38073 509 |  0.52 |  **
$7$ | -9491.24176614 | -9495.44752764 | 3.72 | ** | $\omega_{6}$ | 341.24887793 | 340.76256 805 |  0.49 |
$8$ | -9154.64865155 | -9159.33297339 | 4.21 | ** | $\omega_{7}$ | 336.59311459 | 336.11455 425 |  0.48 |
$9$ | -8822.71699888 | -8827.85266438 | 4.68 | ** | $\omega_{8}$ | 331.93165267 | 331.48030 900 |  0.45 |
$10$ | -8495.45667267 | -8501.03662308 | 5.14 | * | $\omega_{9}$ | 327.26032621 | 326.81604 131 |  0.44 |
$11$ | -8172.88194455 | -8178.90484561 | 5.58 | * | $\omega_{10}$ | 322.57472812 | 322.13177 747 |  0.44 |
$12$ | -7855.01184367 | -7861.48242495 | 6.02 | * | $\omega_{11}$ | 317.87010088 | 317.42242 065 |  0.45 |
$13$ | -7541.87058935 | -7548.78199962 | 6.47 | * | $\omega_{12}$ | 313.14125432 | 312.70042 533 |  0.44 |
$14$ | -7233.48805823 | -7240.83072224 | 6.91 | * | $\omega_{13}$ | 308.38253112 | 307.95127 738 |  0.43 |
$15$ | -6929.90023995 | -6937.67179854 | 7.34 | * | $\omega_{14}$ | 303.58781828 | 303.15892 370 |  0.43 |
$16$ | -6631.14965407 | -6639.34414481 | 7.77 | * | $\omega_{15}$ | 298.75058588 | 298.32765 373 |  0.42 |
$17$ | -6337.28572765 | -6345.89198438 | 8.19 | * | $\omega_{16}$ | 293.86392641 | 293.45216 042 |  0.41 |
$18$ | -6048.36515810 | -6057.36911882 | 8.61 | * | $\omega_{17}$ | 288.92056956 | 288.52286 556 |  0.40 |
$19$ | -5764.45230323 | -5773.82634065 | 9.00 | * | $\omega_{18}$ | 283.91285486 | 283.54277 818 |  0.37 |
$20$ | -5485.61964752 | -5495.32152707 | 9.37 | * | $\omega_{19}$ | 278.83265571 | 278.50481 358 |  0.33 |
$21$ | -5211.94838867 | -5221.93208012 | 9.70 | * | $\omega_{20}$ | 273.67125885 | 273.38944 695 |  0.28 |
$22$ | -4943.52917649 | -4953.75231611 | 9.98 | * | $\omega_{21}$ | 268.41921218 | 268.17976 401 |  0.24 |
$23$ | -4680.46301756 | -4690.88456534 | 10.22 | * | $\omega_{22}$ | 263.06615893 | 262.86775 077 |  0.20 |
$24$ | -4422.86233998 | -4433.44560079 | 10.42 | * | $\omega_{23}$ | 257.60067758 | 257.43896 455 |  0.16 |
$25$ | -4170.85219392 | -4181.56792748 | 10.58 | * | $\omega_{24}$ | 252.01014606 | 251.87767 331 |  0.13 |
$26$ | -3924.57154815 | -3935.39380033 | 10.72 | * | $\omega_{25}$ | 246.28064577 | 246.17412 715 |  0.11 |
$27$ | -3684.17463083 | -3695.08166990 | 10.82 | * | $\omega_{26}$ | 240.39691732 | 240.31213 043 |  0.08 |
$28$ | -3449.83225353 | -3460.80662810 | 10.91 | * | $\omega_{27}$ | 234.34237731 | 234.27504 179 |  0.07 |
$29$ | -3221.73304927 | -3232.76002819 | 10.97 | * | $\omega_{28}$ | 228.09920426 | 228.04659 991 |  0.05 |
$30$ | -3000.08454633 | -3011.14856240 | 11.03 | * | $\omega_{29}$ | 221.64850293 | 221.61146 579 |  0.04 |
$31$ | -2785.11398599 | -2796.19367267 | 11.06 | * | $\omega_{30}$ | 214.97056035 | 214.95488 973 |  0.02 |
$32$ | -2577.06877209 | -2588.13740983 | 11.08 | * | $\omega_{31}$ | 208.04521390 | 208.05626 284 |  -0.01 |
$33$ | -2376.21640980 | -2387.24930291 | 11.07 | * | $\omega_{32}$ | 200.85236229 | 200.88810 692 |  -0.04 |
$34$ | -2182.84374651 | -2193.80704981 | 11.03 | * | $\omega_{33}$ | 193.37266329 | 193.44225 309 |  -0.07 |
$35$ | -1997.25526804 | -2008.09026394 | 10.96 | * | $\omega_{34}$ | 185.58847847 | 185.71678 588 |  -0.13 |
$36$ | -1819.77012567 | -1830.41842846 | 10.83 | * | $\omega_{35}$ | 177.48514237 | 177.67183 548 |  -0.19 |
$37$ | -1650.71747620 | -1661.11794182 | 10.65 | * | $\omega_{36}$ | 169.05264947 | 169.30048 664 |  -0.25 |
$38$ | -1490.42961601 | -1500.50577996 | 10.40 | * | $\omega_{37}$ | 160.28786019 | 160.61216 186 |  -0.32 |
$39$ | -1339.23230071 | -1348.91624693 | 10.08 | * | $\omega_{38}$ | 151.19731531 | 151.58953 302 |  -0.39 |
$40$ | -1197.43160448 | -1206.63990375 | 9.68 | * | $\omega_{39}$ | 141.80069623 | 142.27634 318 |  -0.48 |
$41$ | -1065.29676111 | -1073.93938637 | 9.21 | * | $\omega_{40}$ | 132.13484336 | 132.70051 738 |  -0.57 |  *
$42$ | -943.03875490 | -951.03657844 | 8.64 | * | $\omega_{41}$ | 122.25800621 | 122.90280 794 |  -0.64 |  *
$43$ | -830.78514629 | -838.07561047 | 8.00 | * | $\omega_{42}$ | 112.25360861 | 112.96096 797 |  -0.71 |  *
$44$ | -728.55286334 | -735.09419468 | 7.29 | * | $\omega_{43}$ | 102.23228295 | 102.98141 579 |  -0.75 |  *
$45$ | -636.22246270 | -642.00353247 | 6.54 | * | $\omega_{44}$ | 92.33040065 | 93.09066 221 |  -0.76 |  *
$46$ | -553.51928840 | -558.55927311 | 5.78 | * | $\omega_{45}$ | 82.70317429 | 83.44425 935 |  -0.74 |  *
$47$ | -480.00806566 | -484.35824966 | 5.04 | * | $\omega_{46}$ | 73.51122274 | 74.20102 345 |  -0.69 |  *
$48$ | -415.10634051 | -418.79881220 | 4.35 | * | $\omega_{47}$ | 64.90172515 | 65.55943 746 |  -0.66 |  *
$49$ | -358.11786906 | -361.27983059 | 3.69 | * | $\omega_{48}$ | 56.98847145 | 57.51898 161 |  -0.53 |  *
$50$ | -308.28056166 | -311.00095472 | 3.16 | * | $\omega_{49}$ | 49.83730740 | 50.27887 587 |  -0.44 |
$51$ | -264.81823461 | -267.18508104 | 2.72 | * | $\omega_{50}$ | 43.46232706 | 43.81587 369 |  -0.35 |
$52$ | -226.98471189 | -229.07543942 | 2.37 | * | $\omega_{51}$ | 37.83352271 | 38.10964 161 |  -0.28 |
$53$ | -194.09315392 | -195.97050546 | 2.09 | * | $\omega_{52}$ | 32.89155797 | 33.10493 396 |  -0.21 |
$54$ | -165.52972015 | -167.24471115 | 1.88 | * | $\omega_{53}$ | 28.56343376 | 28.72579 431 |  -0.16 |
$55$ | -140.75515255 | -142.28518172 | 1.71 | * | $\omega_{54}$ | 24.77456760 | 24.95952 943 |  -0.18 |
$56$ | -119.29925480 | -120.66881943 | 1.53 | * | $\omega_{55}$ | 21.45589775 | 21.61636 229 |  -0.16 |
$57$ | -100.75242791 | -101.96013470 | 1.37 | * | $\omega_{56}$ | 18.54682690 | 18.70868 473 |  -0.16 |
$58$ | -84.75687062 | -85.83380145 | 1.21 | * | $\omega_{57}$ | 15.99555729 | 16.12633 326 |  -0.13 |
$59$ | -70.99870070 | -71.96307286 | 1.08 |  | $\omega_{58}$ | 13.75816993 | 13.87072 859 |  -0.11 |
$60$ | -59.20138815 | -60.06259503 | 0.96 |  | $\omega_{59}$ | 11.79731255 | 11.90047 783 |  -0.10 |
$61$ | -49.12045392 | -49.85521457 | 0.86 |  | $\omega_{60}$ | 10.08093423 | 10.20738 046 |  -0.13 |
$62$ | -40.53922142 | -41.20212282 | 0.73 |  | $\omega_{61}$ | 8.58123250 | 8.65309 175 |  -0.07 |
$63$ | -33.26538336 | -33.81150557 | 0.66 |  | $\omega_{62}$ | 7.27383806 | 7.39061 725 |  -0.12 |
$64$ | -27.12817939 | -27.62889501 | 0.55 |  | $\omega_{63}$ | 6.13720397 | 6.18261 056 |  -0.05 |
$65$ | -21.97602831 | -22.37297713 | 0.50 |  | $\omega_{64}$ | 5.15215108 | 5.25591 788 |  -0.10 |
$66$ | -17.67450246 | -18.02313545 | 0.40 |  | $\omega_{65}$ | 4.30152586 | 4.34984 168 |  -4.8$\times 10^{-2}$ |
$67$ | -14.10456638 | -14.42865043 | 0.35 |  | $\omega_{66}$ | 3.56993607 | 3.59448 502 |  -2.5$\times 10^{-2}$ |
$68$ | -11.16102583 | -11.45198368 | 0.32 |  | $\omega_{67}$ | 2.94354055 | 2.97666 675 |  -3.3$\times 10^{-2}$ |
$69$ | -8.75114886 | -9.00454330 | 0.29 |  | $\omega_{68}$ | 2.40987697 | 2.44744 038 |  -3.8$\times 10^{-2}$ |
$70$ | -6.79343135 | -7.00872682 | 0.25 |  | $\omega_{69}$ | 1.95771751 | 1.99581 648 |  -3.8$\times 10^{-2}$ |
$71$ | -5.21648541 | -5.39528263 | 0.22 |  | $\omega_{70}$ | 1.57694594 | 1.61344 419 |  -3.7$\times 10^{-2}$ |
$72$ | -3.95803381 | -4.10349318 | 0.18 |  | $\omega_{71}$ | 1.25845160 | 1.29178 945 |  -3.3$\times 10^{-2}$ |
$73$ | -2.96399643 | -3.08008273 | 0.15 |  | $\omega_{72}$ | 0.99403738 | 1.02341 045 |  -2.9$\times 10^{-2}$ |
$74$ | -2.18765712 | -2.27859975 | 0.12 |  | $\omega_{73}$ | 0.77633931 | 0.80148 298 |  -2.5$\times 10^{-2}$ |
$75$ | -1.58890132 | -1.65885334 | 9.1$\times 10^{-2}$ |  | $\omega_{74}$ | 0.59875580 | 0.61974640 |  -2.1$\times 10^{-2}$ |
$76$ | -1.13351626 | -1.18633787 | 7.0$\times 10^{-2}$ |  | $\omega_{75}$ | 0.45538506 | 0.47251547 |  -1.7$\times 10^{-2}$ |
$77$ | -0.79254699 | -0.83167981 | 5.3$\times 10^{-2}$ |  | $\omega_{76}$ | 0.34096927 | 0.35465806 |  -1.4$\times 10^{-2}$ |
$78$ | -0.54170263 | -0.57011432 | 3.9$\times 10^{-2}$ |  | $\omega_{77}$ | 0.25084436 | 0.26156549 |  -1.1$\times 10^{-2}$ |
$79$ | -0.36080824 | -0.38098790 | 2.8$\times 10^{-2}$ |  | $\omega_{78}$ | 0.18089439 | 0.18912642 |  -8.2$\times 10^{-3}$ |
$80$ | -0.23329861 | -0.24728478 | 2.0$\times 10^{-2}$ |  | $\omega_{79}$ | 0.12750962 | 0.13370312 |  -6.2$\times 10^{-3}$ |
$81$ | -0.14575097 | -0.15517771 | 1.4$\times 10^{-2}$ |  | $\omega_{80}$ | 0.08754764 | 0.09210707 |  -4.6$\times 10^{-3}$ |
$82$ | -0.08745419 | -0.09360471 | 9.4$\times 10^{-3}$ |  | $\omega_{81}$ | 0.05829678 | 0.06157300 |  -3.3$\times 10^{-3}$ |
$83$ | -0.05001277 | -0.05387381 | 6.2$\times 10^{-3}$ |  | $\omega_{82}$ | 0.03744142 | 0.03973090 |  -2.3$\times 10^{-3}$ |
$84$ | -0.02698408 | -0.02929721 | 3.9$\times 10^{-3}$ |  | $\omega_{83}$ | 0.02302869 | 0.02457659 |  -1.6$\times 10^{-3}$ |
$85$ | -0.01354795 | -0.01485584 | 2.3$\times 10^{-3}$ |  | $\omega_{84}$ | 0.01343614 | 0.01444138 |  -1.0$\times 10^{-3}$ |
$86$ | -0.00620764 | -0.00689477 | 1.3$\times 10^{-3}$ |  | $\omega_{85}$ | 0.00734031 | 0.00796107 |  -6.2$\times 10^{-4}$ |
$87$ | -0.00252196 | -0.00284979 | 6.9$\times 10^{-4}$ |  | $\omega_{86}$ | 0.00368568 | 0.00404498 |  -3.6$\times 10^{-4}$ |
$88$ | -0.00086788 | -0.00100493 | 3.3$\times 10^{-4}$ |  | $\omega_{87}$ | 0.00165408 | 0.00184486 |  -1.9$\times 10^{-4}$ |
$89$ | -0.00023358 | -0.00028081 | 1.4$\times 10^{-4}$ |  | $\omega_{88}$ | 0.00063430 | 0.00072412 |  -9.0$\times 10^{-5}$ |
$90$ | -0.00004176 | -0.00005366 | 4.7$\times 10^{-5}$ |  | $\omega_{89}$ | 0.00019182 | 0.00022715 |  -3.5$\times 10^{-5}$ |
$91$ | -0.00000312 | -0.00000473 | 1.2$\times 10^{-5}$ |  | $\omega_{90}$ | 0.00003863 | 0.00004893 |  -1.0$\times 10^{-5}$ |
$92$ | -0.000 000 03 | -0.000 000 08 | 1.6$\times 10^{-6}$ |  | $\omega_{91}$ | 0.00000309 | 0.00000466 |  -1.6$\times 10^{-6}$ |
|  7,7Li2
$\mathfrak{D}_{e}$ | -12 166 | -12 166 | - |  | - | - | - |  - |
$0$ | -11 991.93478510 | -11 992.31616802 | 0.38 |  | ZPE | 173.68383198 | 174.06521490 |  0.38 |
$1$ | -11 649.69907905 | -11 650.60378579 | 0.90 |  | $\omega_{0}$ | 341.71238223 | 342.23570605 |  0.52 |  **
$2$ | -11 311.49276634 | -11 312.89921455 | 1.41 | ** | $\omega_{1}$ | 337.70457125 | 338.20631270 |  0.50 |  **
$3$ | -10 977.30289732 | -10 979.17703538 | 1.87 | ** | $\omega_{2}$ | 333.72217916 | 334.18986902 |  0.47 |
$4$ | -10 647.11910591 | -10 649.47046247 | 2.35 | ** | $\omega_{3}$ | 329.70657292 | 330.18379142 |  0.48 |
$5$ | -10 320.93332504 | -10 323.83503748 | 2.90 | ** | $\omega_{4}$ | 325.63542498 | 326.18578086 |  0.55 |  **
$6$ | -9 998.73993844 | -10 002.14677658 | 3.41 | ** | $\omega_{5}$ | 321.68826091 | 322.19338661 |  0.51 |  **
$7$ | -9 680.53599058 | -9 684.39645127 | 3.86 | ** | $\omega_{6}$ | 317.75032530 | 318.20394786 |  0.45 |
$8$ | -9 366.32133328 | -9 370.63111765 | 4.31 | ** | $\omega_{7}$ | 313.76533363 | 314.21465729 |  0.45 |
$9$ | -9 056.09871321 | -9 060.83607551 | 4.74 | ** | $\omega_{8}$ | 309.79504214 | 310.22262007 |  0.43 |
$10$ | -8 749.87385050 | -8 755.02510535 | 5.15 | ** | $\omega_{9}$ | 305.81097016 | 306.22486271 |  0.41 |
$11$ | -8 447.65555774 | -8 453.21503238 | 5.56 | ** | $\omega_{10}$ | 301.81007297 | 302.21829276 |  0.41 |
$12$ | -8 149.45592611 | -8 155.42591892 | 5.97 | ** | $\omega_{11}$ | 297.78911345 | 298.19963163 |  0.41 |
$13$ | -7 855.29057880 | -7 861.67464296 | 6.38 | ** | $\omega_{12}$ | 293.75127597 | 294.16534731 |  0.41 |
$14$ | -7 565.17897189 | -7 571.97154020 | 6.79 | ** | $\omega_{13}$ | 289.70310276 | 290.11160691 |  0.41 |
$15$ | -7 279.14471358 | -7 286.33906354 | 7.19 | ** | $\omega_{14}$ | 285.63247666 | 286.03425831 |  0.40 |
$16$ | -6 997.21587514 | -7 004.81028500 | 7.59 | ** | $\omega_{15}$ | 281.52877854 | 281.92883844 |  0.40 |
$17$ | -6 719.42527720 | -6 727.41708681 | 7.99 | ** | $\omega_{16}$ | 277.39319819 | 277.79059793 |  0.40 |
$18$ | -6 445.81074981 | -6 454.19261209 | 8.38 | ** | $\omega_{17}$ | 273.22447472 | 273.61452739 |  0.39 |
$19$ | -6 176.41537833 | -6 185.17792610 | 8.76 | ** | $\omega_{18}$ | 269.01468598 | 269.39537148 |  0.38 |
$20$ | -5 911.28775836 | -5 920.41199797 | 9.12 | ** | $\omega_{19}$ | 264.76592814 | 265.12761997 |  0.36 |
$21$ | -5 650.48228761 | -5 659.93548890 | 9.45 | ** | $\omega_{20}$ | 260.47650906 | 260.80547075 |  0.33 |
$22$ | -5 394.05952269 | -5 403.80140979 | 9.74 | ** | $\omega_{21}$ | 256.13407911 | 256.42276492 |  0.29 |
$23$ | -5 142.08662331 | -5 152.07806776 | 9.99 | ** | $\omega_{22}$ | 251.72334203 | 251.97289938 |  0.25 |
$24$ | -4 894.63789792 | -4 904.84167206 | 10.20 | ** | $\omega_{23}$ | 247.23639569 | 247.44872539 |  0.21 |
$25$ | -4 651.79545438 | -4 662.17593685 | 10.38 | ** | $\omega_{24}$ | 242.66573521 | 242.84244353 |  0.18 |
$26$ | -4 413.64994883 | -4 424.17762178 | 10.53 | ** | $\omega_{25}$ | 237.99831508 | 238.14550555 |  0.15 |
$27$ | -4 180.30141612 | -4 190.95182023 | 10.65 | ** | $\omega_{26}$ | 233.22580155 | 233.34853271 |  0.12 |
$28$ | -3 951.86015751 | -3 962.61210740 | 10.75 | * | $\omega_{27}$ | 228.33971282 | 228.44125861 |  0.10 |
$29$ | -3 728.44765475 | -3 739.28322072 | 10.84 | * | $\omega_{28}$ | 223.32888668 | 223.41250275 |  0.08 |
$30$ | -3 510.19747508 | -3 521.10198215 | 10.90 | * | $\omega_{29}$ | 218.18123857 | 218.25017968 |  0.07 |
$31$ | -3 297.25612707 | -3 308.21675935 | 10.96 | * | $\omega_{30}$ | 212.88522280 | 212.94134801 |  0.06 |
$32$ | -3 089.78382285 | -3 100.78764610 | 11.00 | * | $\omega_{31}$ | 207.42911325 | 207.47230422 |  0.04 |
$33$ | -2 887.95509525 | -2 898.98514722 | 11.03 | * | $\omega_{32}$ | 201.80249888 | 201.82872760 |  0.03 |
$34$ | -2 691.95920890 | -2 702.99252739 | 11.03 | * | $\omega_{33}$ | 195.99261984 | 195.99588635 |  0.00 |
$35$ | -2 502.00028970 | -2 513.01470900 | 11.01 | * | $\omega_{34}$ | 189.97781839 | 189.95891920 |  -0.02 |
$36$ | -2 318.29707659 | -2 329.27183801 | 10.97 | * | $\omega_{35}$ | 183.74287099 | 183.70321311 |  -0.04 |
$37$ | -2 141.08217128 | -2 151.97922220 | 10.90 | * | $\omega_{36}$ | 177.29261581 | 177.21490531 |  -0.08 |
$38$ | -1 970.60062518 | -1 981.36772751 | 10.77 | * | $\omega_{37}$ | 170.61149469 | 170.48154610 |  -0.13 |
$39$ | -1 807.10765720 | -1 817.69937842 | 10.59 | * | $\omega_{38}$ | 163.66834908 | 163.49296798 |  -0.18 |
$40$ | -1 650.86524375 | -1 661.22516252 | 10.36 | * | $\omega_{39}$ | 156.47421590 | 156.24241345 |  -0.23 |
$41$ | -1 502.13726684 | -1 512.20192623 | 10.06 | * | $\omega_{40}$ | 149.02323630 | 148.72797691 |  -0.30 |
$42$ | -1 361.18285956 | -1 370.89527684 | 9.71 | * | $\omega_{41}$ | 141.30664939 | 140.95440727 |  -0.35 |
$43$ | -1 228.24756934 | -1 237.53719262 | 9.29 | * | $\omega_{42}$ | 133.35808422 | 132.93529022 |  -0.42 |
$44$ | -1 103.55200378 | -1 112.34139839 | 8.79 | * | $\omega_{43}$ | 125.19579423 | 124.69556556 |  -0.50 |  *
$45$ | -987.27778440 | -995.49678888 | 8.22 | * | $\omega_{44}$ | 116.84460951 | 116.27421938 |  -0.57 |  *
$46$ | -879.55097974 | -887.13736900 | 7.59 | * | $\omega_{45}$ | 108.35941989 | 107.72680466 |  -0.63 |  *
$47$ | -780.42379337 | -787.33772386 | 6.91 | * | $\omega_{46}$ | 99.79964514 | 99.12718637 |  -0.67 |  *
$48$ | -689.85617046 | -696.06733906 | 6.21 | * | $\omega_{47}$ | 91.27038480 | 90.56762291 |  -0.70 |  *
$49$ | -607.70005708 | -613.21286576 | 5.51 | * | $\omega_{48}$ | 82.85447329 | 82.15611337 |  -0.70 |  *
$50$ | -533.68996174 | -538.53759852 | 4.85 | * | $\omega_{49}$ | 74.67526725 | 74.01009535 |  -0.67 |  *
$51$ | -467.44362166 | -471.64704573 | 4.20 | * | $\omega_{50}$ | 66.89055279 | 66.24634008 |  -0.64 |  *
$52$ | -408.47529876 | -412.10026696 | 3.62 | * | $\omega_{51}$ | 59.54677876 | 58.96832289 |  -0.58 |  *
$53$ | -356.22132835 | -359.35248572 | 3.13 | * | $\omega_{52}$ | 52.74778125 | 52.25397041 |  -0.49 |
$54$ | -310.07388045 | -312.80033778 | 2.73 | * | $\omega_{53}$ | 46.55214793 | 46.14744790 |  -0.40 |
$55$ | -269.41626626 | -271.81289323 | 2.40 | * | $\omega_{54}$ | 40.98744456 | 40.65761420 |  -0.33 |
$56$ | -233.65308035 | -235.78701838 | 2.13 | * | $\omega_{55}$ | 36.02587484 | 35.76318591 |  -0.26 |
$57$ | -202.23098465 | -204.14973196 | 1.92 | * | $\omega_{56}$ | 31.63728642 | 31.42209570 |  -0.22 |
$58$ | -174.64940848 | -176.41999456 | 1.77 | * | $\omega_{57}$ | 27.72973740 | 27.58157617 |  -0.15 |
$59$ | -150.46306385 | -152.05839795 | 1.60 | * | $\omega_{58}$ | 24.36159661 | 24.18634463 |  -0.18 |
$60$ | -129.27919847 | -130.72668975 | 1.45 | * | $\omega_{59}$ | 21.33170820 | 21.18386537 |  -0.15 |
$61$ | -110.75225736 | -112.04396517 | 1.29 | * | $\omega_{60}$ | 18.68272458 | 18.52694112 |  -0.16 |
$62$ | -94.57781357 | -95.73004711 | 1.15 | * | $\omega_{61}$ | 16.31391806 | 16.17444379 |  -0.14 |
$63$ | -80.48680790 | -81.52898488 | 1.04 | * | $\omega_{62}$ | 14.20106223 | 14.09100567 |  -0.11 |
$64$ | -68.24053974 | -69.18328816 | 0.94 |  | $\omega_{63}$ | 12.34569672 | 12.24626816 |  -0.10 |
$65$ | -57.62650023 | -58.46432033 | 0.84 |  | $\omega_{64}$ | 10.71896782 | 10.61403951 |  -0.10 |
$66$ | -48.45497030 | -49.19025319 | 0.74 |  | $\omega_{65}$ | 9.27406714 | 9.17152993 |  -0.10 |
$67$ | -40.55624814 | -41.21340141 | 0.66 |  | $\omega_{66}$ | 7.97685178 | 7.89872217 |  -0.08 |
$68$ | -33.77836857 | -34.32796868 | 0.55 |  | $\omega_{67}$ | 6.88543273 | 6.77787957 |  -0.11 |
$69$ | -27.98519806 | -28.49667773 | 0.51 |  | $\omega_{68}$ | 5.83129095 | 5.79317050 |  -0.04 |
$70$ | -23.05481516 | -23.47124957 | 0.42 |  | $\omega_{69}$ | 5.02542816 | 4.93038290 |  -0.10 |
$71$ | -18.87811036 | -19.23410495 | 0.36 |  | $\omega_{70}$ | 4.23714462 | 4.17670480 |  -6.0$\times 10^{-2}$ |
$72$ | -15.35755817 | -15.69365239 | 0.34 |  | $\omega_{71}$ | 3.54045255 | 3.52055219 |  -2.0$\times 10^{-2}$ |
$73$ | -12.40612792 | -12.71214017 | 0.31 |  | $\omega_{72}$ | 2.98151222 | 2.95143024 |  -3.0$\times 10^{-2}$ |
$74$ | -9.94630919 | -10.21898077 | 0.27 |  | $\omega_{73}$ | 2.49315940 | 2.45981873 |  -3.3$\times 10^{-2}$ |
$75$ | -7.90923390 | -8.14661426 | 0.24 |  | $\omega_{74}$ | 2.07236651 | 2.03707529 |  -3.5$\times 10^{-2}$ |
$76$ | -6.23388145 | -6.43647544 | 0.20 |  | $\omega_{75}$ | 1.71013883 | 1.67535246 |  -3.5$\times 10^{-2}$ |
$77$ | -4.86635576 | -5.03619187 | 0.17 |  | $\omega_{76}$ | 1.40028356 | 1.36752569 |  -3.3$\times 10^{-2}$ |
$78$ | -3.75922538 | -3.89914383 | 0.14 |  | $\omega_{77}$ | 1.13704804 | 1.10713038 |  -3.0$\times 10^{-2}$ |
$79$ | -2.87091886 | -2.98425889 | 0.11 |  | $\omega_{78}$ | 0.91488494 | 0.88830653 |  -2.7$\times 10^{-2}$ |
$80$ | -2.16516910 | -2.25549313 | 9.0$\times 10^{-2}$ |  | $\omega_{79}$ | 0.72876576 | 0.70574976 |  -2.3$\times 10^{-2}$ |
$81$ | -1.61050129 | -1.68134448 | 7.1$\times 10^{-2}$ |  | $\omega_{80}$ | 0.57414865 | 0.55466780 |  -1.9$\times 10^{-2}$ |
$82$ | -1.17975971 | -1.23444662 | 5.5$\times 10^{-2}$ |  | $\omega_{81}$ | 0.44689786 | 0.43074159 |  -1.6$\times 10^{-2}$ |
$83$ | -0.84966949 | -0.89120346 | 4.2$\times 10^{-2}$ |  | $\omega_{82}$ | 0.34324316 | 0.33009022 |  -1.3$\times 10^{-2}$ |
$84$ | -0.60043025 | -0.63144125 | 3.1$\times 10^{-2}$ |  | $\omega_{83}$ | 0.25976221 | 0.24923924 |  -1.1$\times 10^{-2}$ |
$85$ | -0.41533866 | -0.43807249 | 2.3$\times 10^{-2}$ |  | $\omega_{84}$ | 0.19336875 | 0.18509159 |  -8.3$\times 10^{-3}$ |
$86$ | -0.28043783 | -0.29677187 | 1.6$\times 10^{-2}$ |  | $\omega_{85}$ | 0.14130062 | 0.13490084 |  -6.4$\times 10^{-3}$ |
$87$ | -0.18419163 | -0.19566591 | 1.2$\times 10^{-2}$ |  | $\omega_{86}$ | 0.10110596 | 0.09624620 |  -4.9$\times 10^{-3}$ |
$88$ | -0.11718257 | -0.12503825 | 7.9$\times 10^{-3}$ |  | $\omega_{87}$ | 0.07062767 | 0.06700906 |  -3.6$\times 10^{-3}$ |
$89$ | -0.07183187 | -0.07705183 | 5.2$\times 10^{-3}$ |  | $\omega_{88}$ | 0.04798641 | 0.04535070 |  -2.6$\times 10^{-3}$ |
$90$ | -0.04214098 | -0.04548936 | 3.4$\times 10^{-3}$ |  | $\omega_{89}$ | 0.03156247 | 0.02969089 |  -1.9$\times 10^{-3}$ |
$91$ | -0.02345378 | -0.02551255 | 2.1$\times 10^{-3}$ |  | $\omega_{90}$ | 0.01997681 | 0.01868720 |  -1.3$\times 10^{-3}$ |
$92$ | -0.01223887 | -0.01344075 | 1.2$\times 10^{-3}$ |  | $\omega_{91}$ | 0.01207180 | 0.01121492 |  -8.6$\times 10^{-4}$ |
$93$ | -0.00589161 | -0.00654915 | 6.6$\times 10^{-4}$ |  | $\omega_{92}$ | 0.00689160 | 0.00634726 |  -5.4$\times 10^{-4}$ |
$94$ | -0.00255567 | -0.00288656 | 3.3$\times 10^{-4}$ |  | $\omega_{93}$ | 0.00366259 | 0.00333593 |  -3.3$\times 10^{-4}$ |
$95$ | -0.00096382 | -0.00111272 | 1.5$\times 10^{-4}$ |  | $\omega_{94}$ | 0.00177384 | 0.00159185 |  -1.8$\times 10^{-4}$ |
$96$ | -0.00029784 | -0.00035508 | 5.7$\times 10^{-5}$ |  | $\omega_{95}$ | 0.00075764 | 0.00066598 |  -9.2$\times 10^{-5}$ |
$97$ | -0.00006752 | -0.00008483 | 1.7$\times 10^{-5}$ |  | $\omega_{96}$ | 0.00027026 | 0.00023032 |  -4.0$\times 10^{-5}$ |
$98$ | -0.00000869 | -0.00001212 | 3.4$\times 10^{-6}$ |  | $\omega_{97}$ | 0.00007270 | 0.00005883 |  -1.4$\times 10^{-5}$ |
$99$ | -0.00000023 | -0.00000043 | 2.1$\times 10^{-7}$ |  | $\omega_{98}$ | 0.00001169 | 0.00000846 |  -3.2$\times 10^{-6}$ |
$100$ | -0.000 000 000 004 | -0.000 000 000 1 | 1.2$\times 10^{-10}$ |  | $\omega_{99}$ | 0.00000043 | 0.00000023 |  -2.1$\times 10^{-7}$ |

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