# H–He collision-induced satellite in the Lyman-$\alpha$ profile of DBA white
dwarf stars

Nicole F. Allard 1,2, John F. Kielkopf 3, Siyi Xu 4, Grégoire Guillon 5, Bilel
Mehnen 6, Roberto Linguerri 6, Muneerah Mogren Al Mogren 7, Majdi Hochlaf 6,
Ivan Hubeny 8

1GEPI, Observatoire de Paris, Université PSL, UMR 8111, CNRS, 61, Avenue de
l’Observatoire, F-75014 Paris, France
2Sorbonne Université, CNRS, UMR7095, Institut d’Astrophysique de Paris, 98bis
Boulevard Arago, PARIS, France
3Department of Physics and Astronomy, University of Louisville, Louisville,
Kentucky 40292 USA
4Gemini Observatory, 670 N. Aóhoku Place, Hilo, HI 96720 HI, USA
5Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR6303, CNRS, Université
de Bourgogne Franche Comté, 21078 Dijon Cedex, France
6Université Gustave Eiffel, COSYS/LISIS, 5 Bd Descartes 77454, Champs sur
Marne, France
7Chemistry Department, Faculty of Science, King Saud University, PO Box 2455,
Riyadh 11451, Kingdom of Saudi Arabia.
8Department of Astronomy, University of Arizona, 933 N Cherry Ave, Tucson, AZ
85719 USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The spectra of helium-dominated white dwarf stars with hydrogen in their
atmosphere present a distinctive broad feature centered around 1160 Å in the
blue wing of the Lyman-$\alpha$ line. It is extremely apparent in WD 1425+540
recently observed with HST COS. With new theoretical line profiles based on ab
initio atomic interaction potentials we show that this feature is a signature
of a collision-induced satellite due to an asymptotically forbidden
transition. This quasi-molecular spectral satellite is crucial to
understanding the asymmetrical shape of Lyman-$\alpha$ seen in this and other
white dwarf spectra. Our previous work predicting this absorption feature was
limited by molecular potentials that were not adequate to follow the atomic
interactions with spectroscopic precision to the asymptotic limit of large
separation. A new set of potential energy curves and electronic dipole
transition moments for the lowest electronic states of the H–He system were
developed to account accurately for the behaviour of the atomic interactions
at all distances, from the chemical regime within 1 Å out to where the
radiating H atoms are not significantly perturbed by their neighbors. We use a
general unified theory of collision-broadened atomic spectral lines to
describe a rigorous treatment of hydrogen Lyman-$\alpha$ with these potentials
and present a new study of its broadening by radiative collisions of hydrogen
and neutral helium. These results enable ab initio modeling of radiative
transport in DBA white dwarf atmospheres.

###### keywords:

(stars:) white dwarfs < Stars - stars: atmospheres < Stars - atomic data <
Physical Data and Processes - atomic processes < Physical Data and Processes -
line: profiles < Physical Data and Processes - molecular data < Physical Data
and Processes

††pubyear: 2019††pagerange: H–He collision-induced satellite in the
Lyman-$\alpha$ profile of DBA white dwarf stars–7

##  1 Introduction

Theoretical studies of the effects of neutral atom collisions on atomic
spectral lines have often been hindered by our ignorance of the atomic
potentials. Even for systems as simple as H-H or H-He, the interactions and
the electric transition moments are quite difficult to compute with the
accuracy which is needed for evaluating a complete line profile. The
fundamental theory of calculating the spectral line profile (Allard et al.,
1999) requires knowledge of molecular potentials with high accuracy because
the shape and strength of the line profile are very sensitive to the details
of the molecular potential curves describing the atom-atom collisions. In
Allard & Christova (2009) we made an exhaustive study of the red wing of
Lyman-$\alpha$ line perturbed by H–He collisions, where we used the potentials
and electric dipole transition moments of Theodorakopoulos et al. (1984) and
Theodorakopoulos et al. (1987). We considered the high He densities met in
cool DZ white dwarfs and examined the range of validity of the one-perturber
approximation widely used to calculate the line wings. We have shown there
that the extension of the red wing of the Lyman-$\alpha$ line seen in DZ white
dwarf spectra depends strongly on the stellar temperature, while it is not
dependent on the helium density. We also predicted a blue satellite which only
very recently has been observed in Hubble Space Telescope Cosmic Origins
Spectrograph (HST COS) observations (Xu et al., 2017). The importance of a
correct determination of the blue wing of Lyman-$\alpha$ line to interpret the
asymmetrical shape of the Lyman-$\alpha$ line observed with COS is presented
in Sect. 2. An accurate prediction of the satellite and consequently the full
Lyman-$\alpha$ profile requires exacting new ab initio calculations to obtain
the ground and first excited potential energy curves and the corresponding
electric dipole transition moments for the H–He system. The new molecular data
in Sect. 3 corroborate the prediction of a line satellite in the
Lyman-$\alpha$ profile (Allard & Christova, 2009) that is described in Sect.
4. In Allard et al. (1999) we previously derived a classical path expression
for a pressure-broadened atomic spectral line shape that includes the effects
of a radiative electric dipole transition moment that is dependent on the
position of the radiating atom and its dynamic neighbors. Such a comprehensive
unified approach employing the precise molecular data is fundamentally
necessary to obtain an accurate absorption line profile that is valid over the
full breadth of spectral line for the range of densities and temperatures
found in stellar atmospheres.

Figure 1: COS observation of WD 1425+540. The broad distinctive collision-
induced satellite in the blue wing of the Lyman-$\alpha$ line about 1160 Å is
clearly visible (Xu et al., 2017). The strong emission at the center of
Lyman-$\alpha$ is from Earth’s geocoronal hydrogen above the HST orbit.

##  2 COS observation of WD 1425+540

WD 1425+540 (T=14,490 K, log g=7.95) is the prototype of DBA white dwarfs and
it is a helium-dominated white dwarf that also has a large amount of hydrogen
in its atmosphere (Bergeron et al., 2011). It was observed with HST COS under
program 13453, and the details of observation and data reduction strategy were
reported by Xu et al. (2017). Here, we focus on the spectrum of segment B of
the G130M grating, which covers 1130-1270 Å, as shown in Fig. 1. As described
in Xu et al. (2017), there are two unusual features of the Lyman-$\alpha$
profile in WD 1425+540. First, the line profile is very asymmetric exhibiting
an extend blue wing with the satellite feature as noted. Second, previous
white dwarf spectral models cannot reproduce the strength of Lyman-$\alpha$
and Balmer-$\alpha$ simultaneously. The derived hydrogen abundance is more
than a factor of 10 higher from the Lyman-$\alpha$ measurement than from
Balmer-$\alpha$. While WD 1425+540 is the most extreme case so far, these
peculiarities have been observed in other DBA white dwarfs as well, e.g. Jura
et al. (2012).

The asymmetry also could not be produced by white dwarf models of Xu et al.
(2017) because the opacity data used for the Lyman-$\alpha$ profile did not
take into account the quasi-molecular line satellite predicted in Allard &
Christova (2009). Once this feature is included, the observed asymmetry is
reproduced (Gänsicke et al., 2018). The need to have both accurate data for
Lyman-$\alpha$ and for Balmer-$\alpha$ is essential to determine the hydrogen
abundance correctly. The goal of this paper is to develop the foundation of
the atomic and molecular physics needed to compute a complete profile without
making ad hoc assumptions. We emphasize the importance of accurate potentials
and electric dipole transition moment data for this purpose, and here we
provide that data for Lyman-$\alpha$. With new potentials of H-He we also
compute a model DBA white dwarf spectrum that demonstrates their validity.

##  3 H–He diatomic potentials

###  3.1 Methodology and benchmarks

The lowest electronic excited states of hydrogen and helium are at unusually
high energies for neutral atoms (> 10 eV) with respect to their ground states,
and close to the corresponding ionization thresholds. Hydrogen with $n$= 2 or
greater is a Rydberg atom in this sense (Gallagher, 1994).

The electronic excited states of H–He diatomic system of interest in the
present work correlate adiabatically to those of these atoms. Therefore, for
the correct description of the electronic states of the H–He diatomic system
consistent with its isolated atomic fragments one needs the inclusion of
diffuse functions that can flexibly represent the states. In addition to this,
the computation of the possible interactions that may occur between these
electronic states and the subsequent mixing of their wavefunctions that
results in an apparent change in electric dipole transition moments, require
post Hartree-Fock multi-configurational approaches. More specifically, we used
the Complete Active Space Self Consistent Field (CASSCF) (Knowles & Werner,
1985; Werner & Knowles, 1985) followed by the internally contracted Multi-
Reference Configuration Interaction (MRCI) (Knowles & Werner, 1988; Werner &
Knowles, 1988; Shamasundar et al., 2011) methods as implemented in the MOLPRO
2015 package (Werner et al., 2015). In MRCI, the complete CASSCF wave
functions are used as a reference. Furthermore, the Davidson correction
(MRCI+Q) (Langhoff & Davidson, 1974) has been applied to the resulting
energies to account for the lack of size-consistency of the MRCI method. These
computations were performed in the $C_{2v}$ point group, where the $B_{1}$ and
$B_{2}$ representations were treated on equal footing.

Benchmarks on valence-Rydberg electronic states of other molecular systems
(Spelsberg & Meyer, 2001; Ndome et al., 2008; Hochlaf et al., 2010) showed the
need to use a CASSCF active space larger than the full-valence space. The
atomic basis set for the H and He atoms had to be optimized as well. Thus, we
performed a series of benchmark computations at different levels of accuracy
to find the appropriate states for convergence.

Firstly, at the lowest level of accuracy, we adopted a small active space of 3
electrons in 7 molecular orbitals in conjunction with the aug-cc-pV5Z
(Dunning, 1989; Kendall et al., 1992) basis set. With this approach, we found
inconsistencies in the calculated energies, especially in the asymptotic
region. Indeed, with this simplest choice there is a large energy gap of $\sim
0.45$ eV between the two equivalent dissociation limits H($2p\,^{2}P$) +
He($1s^{2}\,{}^{1}S$) and H($2s\,^{2}S$) + He($1s^{2}\,{}^{1}S$). Obviously,
this gap is unphysical since these two asymptotes should be strictly
degenerate because the two H ($n=2$) states have the same energy apart from
Lamb shift and negligibly small fine and hyperfine structure. Moreover, we
found a spurious second potential well ($D_{e}$ $\sim$ 660 cm-1) in the
$C\,\Sigma$ state of H–He at large internuclear separations (for
$R_{\mathrm{H-He}}$ $\sim$ 4.2 Å). Thus, at this level of accuracy, a rather
poor chemical description of the H–He molecule is obtained in spite of the
relatively large size of the MRCI computations with $\sim$ 4.3 x 104
uncontracted configuration state functions (CSFs) per $C_{2v}$ symmetry. This
may be linked to some missing correlation energy in the MRCI wavefunctions
that can be recovered by means of larger active spaces in the reference CASSCF
vector and by adopting more diffuse atomic basis sets.

Secondly, we tried an enlarged CASSCF active space of 3 electrons in 14
molecular orbitals in conjunction with the aug-cc-pV6Z (Dunning, 1989; Kendall
et al., 1992) basis set. In the subsequent MRCI treatment, the multi-
configuration wave functions included $\sim$ 2.1 x 105 uncontracted CSFs per
$C_{2v}$ symmetry. With this ansatz, the energy difference between the above
mentioned asymptotes was reduced to $\sim$ 0.33 eV but still did not vanish.
For modeling based on unified spectral line shape theory an error of this size
would be unacceptable.

Finally, using the same active space as in the second series of computations,
we added a set of diffuse functions to the aug-cc-pV6Z basis set for H and He.
Hereafter, this enlarged set will be denoted as aug-cc-pV6Z⋆. The exponents of
the added Gaussian primitives, which were left uncontracted, are listed in
Table 1 in the Appendix.

This approach, compared to the previous ones, solved all the inconsistencies
mentioned above. That is, it yielded degenerate H($2p\,^{2}P$) +
He($1s^{2}\,{}^{1}S)$ and H($2s\,^{2}S$) + He($1s^{2}\,{}^{1}S)$ dissociation
limits and no spurious potential well in the $C\,\Sigma$ state. We note that
convergence was reached at this step since a further expansion of the aug-cc-
pV6Z⋆ set by inclusion of more diffuse functions led to almost identical
results. In these calculations, the MRCI wave functions included more than
$7.5\times 10^{5}$ uncontracted CSFs per $C_{2v}$ symmetry species. These
relatively large computations for such a small molecular system were necessary
to obtain the precision needed to model the Lyman-$\alpha$ profile accurately.

Figure 2: Top: short-range part of the potential curves of the H–He molecule:
$A$ (red dotted), $B$ (green dashed line) and $C$ (blue solid). Bottom: $X$
(black solid). Note the agreement at short distance with data of
Theodorakopoulos et al. (1984) that are overplotted in dotted cyan.

Figure 3: Top: long range part of the $C\,\Sigma$ potential curve correlated
with $2s$ state. This work (full line), Theodorakopoulos et al. (1984) (dotted
line). Bottom: $\Delta V(R)$ (black) and $\tilde{d}(R)$ (blue) at 14500 K for
the $C-X$ transition. The atomic separation for the maximum in the $C-X$
difference potential is $R_{\mathrm{max}}\approx 2.2\,$Å as shown in Fig. 4
Note that the $C-X$ transition in this work is forbidden asymptotically as it
is a transition between the $2s$ and $1s$ states of the free hydrogen atom at
large $R$.

###  3.2 Potential energy curves and transition moments

The electronic states investigated in the present contribution correlate, at
large internuclear distances, to the  H($1s\;^{2}S$) + He($2s^{2}\;{}^{1}S$),
H($2s\;^{2}P$) + He($2s^{2}\;{}^{1}S$), and  H($2p\;^{2}P$) +
He($2s^{2}\;{}^{1}S$) dissociation limits (see Fig. 2 and Table 2 in the
Appendix). The MRCI+Q/aug-cc-pV6Z⋆ potential energy curves of the four lowest
electronic states of H–He, obtained with the largest active space and basis
set as described in the previous section, are represented in Fig. 2 as a
function of the internuclear distance, $R_{\mathrm{H-He}}$. This figure shows
that the ground state possesses a repulsive potential correlating to the
H($1s\,^{2}S$) + He($1s^{2}\,{}^{1}S$) isolated atom asymptote at large
distances.

The ground $X\,^{2}\Sigma^{+}$ state is repulsive at short range with a
shallow well at $4\,$Å. The excited $A\,^{2}\Sigma^{+}$, $B\,^{2}\Pi$ and
$C\,^{2}\Sigma^{+}$ states have rather deep potential wells in the molecular
region closer than 1 Å, and complex behavior at longer range that can affect
transition probabilities and difference potential energies in subtle ways. We
refer to these as the $X\,\Sigma$, $A\,\Sigma$, $B\,\Pi$, and $C\,\Sigma$
states, or more succinctly by the letter designations $X$, $A$, $B$, and $C$
in the following. They correlate adiabatically to the H($n=2$) +
He($1s^{2}\,{}^{1}S$) dissociation limits at large internuclear separations
(see Table 2 in the Appendix). The ordering of the assignments of labels for
the states is with $A\,\Sigma$ the lowest and $C\,\Sigma$ the highest inside
this close 1 Å region with wells in all the states of the order of
$15\,000\;\mathrm{cm}^{-1}$ deep, with minima located at $R_{\mathrm{H-He}}$ =
0.7407, 0.7686, and 0.8095 Å for the $A$, $B$ and $C$ states, respectively
(see Table 3 in the Appendix). While the $A$ and $B$ states have potentials
with a simple short-range well, the $C$ state also exhibits a potential
maximum of $\approx 0.666$ eV at $R_{\mathrm{H-He}}=2.098$ Å. Its presence
causes a related maximum in the $C-X$ transition difference potential energy
curve which affects the blue wing of Lyman-$\alpha$.

Although the $C\,\Sigma$ H-He molecular state shown in Fig. 2 is correlated
asymptotically with the $2s$ atomic state, we find that at
$R_{\mathrm{H-He}}<7\;$Å the transition probability to the $X\,\Sigma$ ground
state is not zero. Detailed electric dipole transition moments between the
$X\,\Sigma$ ground state and the $A\,\Sigma$, $B\,\Pi$ and $C\,\Sigma$ excited
states as a function of the internuclear distance have been calculated at the
MRCI/aug-cc-pV6Z⋆ level. In this calculation almost all the transition moments
are rather large, particularly for the $C\,\Sigma$ $\leftarrow$ $A\,\Sigma$
and $B\,\Pi$ $\leftarrow$ $A\,\Sigma$ transitions, where corresponding matrix
elements of around -9.2 and -7.5 debye (D or $10^{-18}$ statcoulomb-cm) are
calculated, respectively. Fig. 7 in the Appendix offers a detailed view. These
transition moments correlate to the correct atomic values at dissociation. In
particular, the  $\langle X\,\Sigma|DM|C\,\Sigma\rangle$ matrix element of the
electric dipole transition moment (DM) vanishes at large $R_{\mathrm{H-He}}$
where the $1s-2s$ transition in the isolated hydrogen atom is forbidden to
one-photon electric dipole transitions by parity conservation.

##  4 Lyman-alpha opacity

The theory of spectral line shapes, especially the unified approach we
developed, determines the contributions of specific spectral lines to stellar
opacities and may be incorporated into stellar atmosphere models to make
accurate synthesis of stellar spectra possible. The line shape theory accounts
for neutral atom broadening and shift in both the centers of spectral lines
and their extreme wings with one consistent treatment without ad hoc
assumptions about the line shape or potentials. Complete details and the
derivation of the theory are provided by Allard et al. (1999). The spectrum,
$I(\Delta\omega)$, is the Fourier transform (FT) of a electric dipole
transition autocorrelation function, $\Phi(s)$. For a perturber density
$n_{p}$, we have

$\Phi(s)=e^{-n_{p}g(s)}\;,$ (1)

where the decay of the autocorrelation function with time leads to atomic line
broadening. (See Eq. (121) of Allard et al. (1999).) Our approach introduces
the concept of a modulated electric dipole transition moment
$\tilde{d}_{if}(R(t))$ into the line shape calculation.

$\tilde{d}_{if}[R(t)]=d_{if}[R(t)]e^{-\frac{V_{i}[R(t)]}{2kT}}\;\;,$ (2)

where the potential energy for the initial state is

$V_{i}(R)=E_{i}(R)-E_{i}^{\infty}\;\;.$ (3)

The difference potential energy $\Delta V(R)$ for a transition $if$ is

$\Delta V(R)=V_{if}(R)=V_{f}(R)-V_{i}(R)\;\;.$ (4)

The Boltzmann factor $e^{-\frac{V_{i}(R)}{2kT}}$ in Eq. (2) appears because
the perturbing atoms or ions are in thermal equilibrium with the radiating
atom which affects the probability of finding them initially at a given $R$.
This treatment results in Lyman series line wing profiles that exhibit a
sensitive dependence on temperature. We had to use electric dipole moments
modulated by the Boltzmann factor in the comparison of emission spectra of
Lyman-$\alpha$ (Kielkopf & Allard, 1998) and Balmer $\alpha$ (Kielkopf et al.,
2002) measured in laboratory.

###  4.1 Study of the characteristics of the line satellite

In Allard & Christova (2009) we predicted a line satellite at 1157 Å in
spectra computed for the temperature range of cool DZ white dwarfs with
potentials published in Theodorakopoulos et al. (1984). However, we noticed an
unexpected well of about 150 cm-1 (upper Fig. 3) in the potential energy of
the $C\,\Sigma$ state at $R\sim 8$ Å which may be related to the choice of
basis states and has no clear physical origin. In this work we use the new ab
initio calculations of the potentials over the full range of distances $R$
between the H and He atoms since convergence at large $R$ is now reached. The
long range well of the $C\,\Sigma$ state of Theodorakopoulos et al. (1984) and
Theodorakopoulos et al. (1987) potentials is not found in these new
calculations as we see in Fig. 3.

Figure 4: Top: variation with temperature of the line satellite. The He
density is $1\times 10^{20}$ cm-3, the temperatures are 14500 K (full black
line), $20\,000$ K (blue stars), and $5\,000$ K (red dashed line). Bottom: for
the $C-X$ transition, $\Delta V(R)$ (black solid) and $\tilde{d(R)}$ at 5000 K
(black solid), $10\,000$ K (red dotted), 14500 K (green dashed), and 20000 K
(blue solid). At the highest temperatures the He can reach the inner regions
of the lower state $X\,^{2}\Sigma$ potential and enhance the transition
probability.

The prediction of a satellite in the blue wing of the H–He line profile is
related to a potential maximum at $R=2.1$ Å (see Sect. 3.2) of the $C\,\Sigma$
state. This leads to a maximum of the potential energy difference $\Delta
V(R)$ in Eq. (4) for this transition shown in Fig. 3.

The unified theory predicts that line satellites will be centered periodically
at frequencies corresponding to integer multiples of the extrema of $\Delta
V(R)$. In the quasi-static limit the first satellite on the line would be at
$\Delta\omega=5\,000$ cm-1 corresponding to $\lambda\sim 1150$ Å on the blue
side of Lyman-$\alpha$. In this case the maximum in $\Delta V$ occurs at
rather small internuclear distance, and is quite sharp. The correspondingly
short duration of the close collision leads to a broad satellite centered at
$\lambda$ $\sim$ 1160 Å for T=$14\,500$ K (Fig. 4).

###  4.2 Temperature and density dependence

For a lower temperature, $T=5\,000$ K (Fig. 4), the duration of the collision
is longer, and the line satellite at $\lambda\sim 1153$ Å is sharper and
closer to the predicted quasi-static position than at higher temperatures. The
oscillations which appear on the red side of the quasi-molecular satellite are
due to interference effects described by Royer (1971) and Sando & Wormhoudt
(1973). They depend on the relative velocity and therefore on temperature.
Consequently velocity averaging would moderate their amplitude in observed
spectra. At temperatures below $10\,000$ K the blue wing of Lyman-$\alpha$
shortward of 1150 Å becomes significantly more transparent than at higher
temperature, an order of magnitude effect below 1120 Å. Thus this far blue
wing is a sensitive indicator of temperature in cool helium-rich WD
atmospheres.

The satellite amplitude depends on the value of the electric dipole transition
moment through the region of the potential extremum responsible for the
satellite and on the position of this extremum. The blue line wings shown in
Fig. 4 are unchanged in the range $14\,500$ to $20\,000$ K as there is no
change with $T$ of $\tilde{d}_{if}[R(t)]$ in the internuclear distance where
the potential difference goes through a maximum. $\tilde{d}_{if}[R(t)]$ at
$14\,500$ K for the $C-X$ transition is also plotted in Fig. 3. In the former
work we used electric dipole transition moments of Theodorakopoulos et al.
(1987) where the $C-X$ transition was allowed. Nevertheless the amplitude and
position of the line satellite are unchanged as they are due to a range of
internuclear distance where the potentials and the dipole moments are almost
identical as we see in Fig. 5. The main difference between the two potentials
concerns the red wing which is lowered using dipole moments of
Theodorakopoulos et al. (1987) where the $A-X$ transition was forbidden.

Figure 5: Comparison of the unified line profile using the dipole moments of
this work (black line) with the line profile using dipole moments of
Theodorakopoulos et al. (1987) (red dashed line) . The He density is $10^{20}$
cm-3 and the temperature is 14500 K.

In summary the unified line profile calculation leads to a flat blue wing due
to a line satellite. The resulting asymmetry of the Lyman-$\alpha$ line can be
easily appreciated in Fig. 5 the blue side of the line is wider than the red
side. Measured at the strength of the broad collision-induced 1160 Å
satellite, the asymmetry ratio of the width on the blue side to that on the
red is as large as 2.2. Consequently, the near wing is clearly far different
from a symmetric Lorentzian because the satellite is rather close to the
isolated atom line center. This was also the case for the Mg b triplet
perturbed by He (Allard et al., 2016). The existence of the asymmetrical shape
of these line profiles depends strongly on the maximum value of the potential
energy difference $\Delta V(R)$ which predicts the position of the line
satellite and on the atomic collision energies at the temperatures of
interest. These results enable computing atmosphere models and synthetic
spectra which we compare to an HST COS observation of WD 1425+540 in Section
5.

##  5 Model atmosphere and synthetic white dwarf spectrum

To demonstrate the importance of a proper treatment of He perturbers on
hydrogen lines, synthetic spectra of the white dwarf WD 1425+540 were computed
using the stellar atmosphere code TLUSTY (version 207) for computing the
atmospheric structure, and a companion program SYNSPEC (version 53) for
generating detailed synthetic spectra. For a description of the previous
versions (205 and 51) see the works of Hubeny & Lanz (2017) and Hubeny & Lanz
(2011a, b). This procedure allows us to study the effect of the H/He ratio on
the spectrum, and the development of line wings, though it is not fully self-
consistent with the stellar atmosphere model since that would require a
treatment of He I optical lines as well. We have computed a number of H-He
models, with the basic model parameters, $T_{\rm eff}=14,410$ K and $\log
g=7.89$, from Gänsicke et al. (2018), and with varying He/H ratio. For
treating the electron and proton broadening of the hydrogen lines we used
Tremblay & Bergeron (2009) data. The He/H ratio was adjusted to obtain a
reasonable agreement by eye with the observed spectrum, and we found a nominal
ratio of $4\times 10^{3}$ ($\log(N_{\mathrm{H}}/N_{\mathrm{He}})\approx-3.6$)
fitted the observed profile well. Liebert et al. (1979) found 3.7 from a
ground-based H$\beta$ profile, and recently Gänsicke et al. (2018) analyzed
the L$\alpha$ profile and adopted a somewhat larger
$\log(N_{\mathrm{H}}/N_{\mathrm{He}})\approx-4.0\pm 0.20$.

The potential energies for the $n=1$ and $n=2$ electronic states H-He that
were used in our models are the ones described in this paper. Stellar
opacities were computing using H-He electric dipole moments from the previous
work of Theodorakopoulos et al. (1987) in which the $A-X$ transition is
forbidden, and also using new dipole transition moments from this work in
which the $A-X$ transition is allowed. As shown in Fig. 6, the observed red
wing of Lyman-$\alpha$ is consistent with a suppressed $A-X$ transition
probability in the region of atomic separation with difference potential
energy that would contribute.

We conclude that the additional basis states used for the new ab initio
potentials improve the calculation of the potential energy curves, but may not
capture the dipole transition moments of the real H-He system correctly for
the $A-X$ transition. However the combination of this work’s potentials and
the dipole moments of Theodorakopoulos et al. (1987) achieve a remarkable fit
in Fig. 6 to the HST COS spectrum of WD 1425+540 when incorporated into the
unified line shape theory we described here.

Figure 6: The observed spectrum of WD 1425+540 (also see Fig. 1) compared with
a synthetic white dwarf spectrum in the Lyman-$\alpha$ region. The synthetic
spectrum is computed with TLUSTY and SYNSPEC for a temperature of 14 500 K and
a He/H ratio of $4\times 10^{3}$ using the unified line profile with the
potentials of this work. For the dipole moments of Theodorakopoulos et al.
(1987) (red solid line) the $A-X$ transition is forbidden and its
contributions to the opacity are suppressed. For the dipole moments of this
work (blue dashed line), the $A-X$ transition contributes in the red wing of
the model but is absent in the observed spectrum.

##  6 Conclusions

The Lyman-$\alpha$ region of the spectrum of a helium-rich white dwarf with
hydrogen in its atmosphere is determined by the changes in transition energy
and transition probability during the H-He collisions that broaden the atomic
spectral line. We developed new H-He potential energies and transition dipole
moments for the hydrogen $1s$, $2s$, and $2p$ states as input data for a
unified theory calculation of the profile of WD 1425+540 to test the
potentials and dipole moments, and to confirm the origin of the short-
wavelength “blue” satellite. We found that the spectral line profile from the
new molecular data has a satellite feature in the blue wing that agrees with
previous work. These results provide a benchmark implementation of ab initio
atomic and molecular potentials for the most basic neutral non-resonant atom-
atom pair relevant to stellar atmosphere models. The new calculations show how
the profile depends on the variation of the electric dipole transition moment
and interaction potential energy with atomic separation. A comparison with the
observed spectrum of WD 1425+540 was made by using these theoretical opacities
in a stellar atmosphere and spectrum synthesis code. While it was not our goal
to refine the stellar model based on the new theoretical data, the profiles
reproduce the observed spectrum with a reasonable He/H ratio. Further, the
absence of an extended red wing of Lyman-$\alpha$ in the observed spectrum
suggests that the states of the difference potential that could contribute to
that region have the reduced transition dipole moment that was found in
previous molecular models. The new work presented here shows clearly that
there is an opportunity to use stellar spectra to improve the atomic and
molecular physics, ultimately to yield better models for astrophysical
applications. For H–He, the $A-X$ transition dipole moment remains uncertain.

The blue wing of Lyman-$\alpha$ is sensitive to He density and the structure
and temperature of the stellar atmosphere, with a profile that for wavelengths
shortward of $1150\,$Å will have reduced opacity from regions with
temperatures under $10\,000\,$K. Profiles computed with a unified theory of
collision broadening based on accurate data from ab initio molecular physics
take into account the strong dependence of the amplitude of the electric
dipole transition moment on atom-atom separation ($R$) where the potential
energy change $\Delta V(R)$ is an extremum. Incorporated into model
atmospheres, this dependence may be used to probe white dwarf or stellar
atmospheres for density and temperature. This emphasizes the importance of the
accuracy of both the potential energies and the electric dipole transition
moments for the line shape calculations that have traditionally assumed
electric dipole transition moments are constant (Allard & Kielkopf, 1982;
Allard & Koester, 1992; Allard et al., 1994).

The effect of collision broadening is central to understanding the opacity of
stellar atmospheres, yet there have been only a few definitive comparisons
with experimental work for atomic H. (Kielkopf & Allard, 1995, 1998; Kielkopf
et al., 2004). This is because of the difficulty of creating an environment in
a laboratory experiment simulating a stellar atmosphere with accurate
diagnostics. On the theoretical side, the maturing capability of ab initio
methods now offers the possibility of accurately computing the interaction of
H with H (Drira, 1999; Spielfiedel, 2003; Spielfiedel et al., 2004) and H with
He atoms (this work). While an accurate determination of the broadening of
Balmer $\alpha$ with high density atomic hydrogen (that is H–H) has been done
by Allard et al. (2008), nothing comparable exists for H–He. Our calculations
reported in Allard et al. (2008) support the results of Barklem et al. (2000);
Barklem et al. (2002) that the Ali & Griem (1966) theory underestimates the
actual line width. Recent laboratory measurements show a similar result at
high density in environments comparable to white dwarf atmospheres (Kielkopf &
Allard, 2014). It would be possible now to similarly improve the calculation
of Balmer-$\alpha$ broadening and its contribution to the full white dwarf
opacity model. A major improvement to comprehensive theoretical models for DBA
white dwarf spectra is within reach that would determine H-He molecular data
for $n=3$ excited states, and use those to compute accurate Balmer-$\alpha$
profiles under white dwarf atmosphere conditions. Such results would help
understanding the differences in stellar parameters that are found from Balmer
and Lyman line profiles. In conclusion, complete unified line profiles based
on accurate atomic and molecular physics for both the Lyman-$\alpha$ and
Balmer-$\alpha$ lines should be incorporated into the analysis of DBA white
dwarf spectra to derive the hydrogen abundance.

## acknowledgements

The paper was based on observations made with the NASA/ESA Hubble Space
Telescope under program 13453, obtained from the data archive at the Space
Telescope Science Institute. STScI is operated by the Association of
Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.
We thank the COST Action CM1405 MOLecules in Motion (MOLIM) of the European
Community for support. The authors would like to extend their sincere
appreciation to the Deanship of Scientific Research at King Saud University
for funding the research through the Research Group Project No. RGP-333. This
work was supported by the CNRS program Physique et Chimie du Milieu
Interstellaire (PCMI) co-funded by the Centre National d’Etudes Spatiales
(CNES).

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## appendix

Parameters of the H–He molecular potentials are given in Tables 1 and 2.
Figure 7 shows the dependence on $R$ of the radiative transition moments
between the excited states and the perturbations of those states as the H and
He atoms approach from large $R$.

Table 1: Exponents of the diffuse uncontracted Gaussian primitives added to the aug-cc-pV6Z basis set to form the presently used aug-cc-pV6Z* basis sets for the H and He atoms. State | 1 | 2 | 3
---|---|---|---
H(_s_) | 0.00690204 | 0.002520537 | 0.000920468
H(_p_) | 0.026565598 | 0.010533298 | 0.004176468
H(_d_) | 0.055406537 | 0.024364162 | 0.010713761
H(_f_) | 0.106396067 | 0.046204584 | 0.020065249
H(_g_) | 0.168703345 | 0.069928301 | 0.028985598
H(_h_) | 0.175320015 | 0.045069073 | 0.011585793
He(_s_) | 0.017177900 | 0.006596920 | 0.002533450
He(_p_) | 0.050416903 | 0.019858313 | 0.007821833
He(_d_) | 0.094209988 | 0.036827891 | 0.014396494
He(_f_) | 0.151890237 | 0.056684629 | 0.021154402
He(_g_) | 0.232902520 | 0.079072280 | 0.026845675
He(_h_) | 0.248198125 | 0.060632194 | 0.014811808
Table 2: Dissociation fragments, experimental and calculated relative dissociation asymptotic energies, and molecular states for the four lowest electronic states of H–He. Experimental data are from Kramida (2010).  Atomic |  | Observed | Calculated | Molecular
---|---|---|---|---
H | He | (cm${}^{-}1$) | (cm${}^{-}1$) |
$1s\,^{2}S_{g}$ | $1s^{2}\,{}^{1}S_{g}$ | 0a | 0a | $X\,^{2}\Sigma^{+}$
$2p\,^{2}P_{u}$ | $1s^{2}\,{}^{1}S_{g}$ | 82259 | 82308 |  $A\,^{2}\Sigma^{+}$, $B\,^{2}\Pi$
$2s\,^{2}S_{g}$ | $1s^{2}\,{}^{1}S_{g}$ | 82259 | 82308 | $C\,^{2}\Sigma^{+}$
aReference |  |  |  |
Table 3: Spectroscopic constants and dissociation energies for the three lowest excited electronic states of H–He as deduced from the MRCI+Q /aug-cc-pV6Z* potential energy curves. $R_{e}$ corresponds to the equilibrium distance. $\omega_{e}$ and $\omega_{e}x_{e}$ are the vibrational constants. $\beta_{e}$ and $\alpha_{e}$ are the rotational constants. $D_{e}$ is the dissociation energy. State | $R_{e}$ | $\omega_{e}$ | $\omega_{e}x_{e}$ | $\beta_{e}$ | $\alpha_{e}$ | $D_{e}$
---|---|---|---|---|---|---
| Å | cm-1 | cm-1 | cm-1 | cm-1 | eV
$A\,^{2}\Sigma^{+}$ | 0.74074 | 3697.2 | 149.5 | 38.16 | 2.608 | 2.563
$B\,^{2}\Pi$ | 0.76863 | 3313.4 | 149.8 | 35.44 | 2.629 | 2.218
$C\,^{2}\Sigma^{+}$ | 0.80953 | 2906.3 | 144.0 | 31.95 | 2.551 | 1.638

Figure 7: Potential energy differences in cm-1 and electric dipole transition
moments in debye (D or $10^{-18}$ statcoulomb-cm) between the four lowest
electronic states of H–He calculated at the MRCI/aug-cc-pV6Z⋆ level. Note that
the $C\,\Sigma$ $\leftarrow$ $X\,\Sigma$ is asymptotically forbidden, while
transitions between excited states may occur. Upper panel: Energy differences
$A\Sigma-B\Sigma$ (blue) and $A\Sigma-C\Pi$ (red). Lower panel: Electric
dipole transition moments for H in the presence of He for states contributing
to H Lyman-$\alpha$.


# Modernist Materials Synthesis: Finding Thermodynamic Shortcuts with
Hyperdimensional Chemistry

James R. Neilson  Department of Chemistry, Colorado State University, Fort
Collins, CO<EMAIL_ADDRESS>Matthew J. McDermott  Materials
Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA
Kristin A. Persson  Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, CA<EMAIL_ADDRESS>
###### Abstract

Synthesis remains a challenge for advancing materials science. A key focus of
this challenge is how to enable selective synthesis, particularly as it
pertains to metastable materials. This perspective addresses the question: how
can “spectator” elements, such as those found in double ion exchange
(metathesis) reactions, enable selective materials synthesis? By observing
reaction pathways as they happen (_in situ_) and calculating their energetics
using modern computational thermodynamics, we observe transient, crystalline
intermediates that suggest that many reactions attain a local thermodynamic
equilibrium dictated by local chemical potentials far before achieving a
global equilibrium set by the average composition. Using this knowledge, one
can thermodynamically “shortcut” unfavorable intermediates by including
additional elements beyond those of the desired target, providing access to a
greater number of intermediates with advantageous energetics and selective
phase nucleation. Ultimately, data-driven modeling that unites first-
principles approaches with experimental insights will refine the accuracy of
emerging predictive retrosynthetic models for complex materials synthesis.

School of Advanced Materials Discovery, Colorado State University, Fort
Collins, CO Department of Materials Science and Engineering, University of
California, Berkeley, CA Department of Materials Science and Engineering,
University of California, Berkeley, CA

##  1 Introduction: Broader Context and Key Scientific Questions

New materials enable new technology. The most significant material in terms of
economic impact and quantity, concrete, is enabled by a phase that is
metastable at room temperature: Ca3SiO5.1 This metastability is what allows it
to be cured into concrete when mixed with water, CO2, and aggregate. As such,
Ca3SiO5 is synthesized at high temperatures and then quenched rapidly to room
temperature to avoid equilibration. Myriad materials predicted to offer
interesting functional electronic properties are metastable and not
synthesizable under ambient or standard laboratory conditions. Key examples
are the successfully predicted high-temperature superconductors, FeB42 and
LaH10,3 that were subsequently synthesized at extremely high pressures.
Meanwhile, many other functional materials have been predicted to be stable at
ambient conditions.4 For example, the battery electrode Li3FePO4CO3 evaded
traditional synthesis and required alternative hydrothermal and ion exchange
chemistry for synthesis,5 thus demanding an improved understanding of
synthesis itself.

Accurate prediction of reaction kinetics from first principles requires
knowledge of the complex potential energy landscape defined by all atoms in
materials, as well as their defects, surfaces, and interfaces. Therefore, if
we assume that reaction kinetics are too challenging to predict accurately for
an arbitrary reaction, then how do we take advantage of what we can calculate
– thermodynamics – to predict useful synthesis reactions? A key enabling
hypothesis explored in this perspective is that reactions between solids often
achieve local thermodynamic equilibria at interfaces. Several approaches can
predict phase equilibria under different intensive and experimentally
controllable thermodynamic variables, such as temperature and pressure.6
However, we have a relatively poor understanding of how intensive _chemical_
variables (e.g., chemical potential, $\mu$) influence the formation of solids,
especially when we desire materials that are characterized as metastable.7 As
a thought experiment illustrated in Figure 1, consider the stability and
synthesis reactions for various isomers of molecular C3H8O. To make the ground
state 2-propanol, one can combine the essential ingredients for a
stoichiometric reaction, CH3CHCH2 + H2O. The preparation of other isomers is
facilitated by the inclusion of additional elements (e.g., Ag and I in the
case of 1-propanol; Na and Br in the case of methoxyethane). We call these
additional elements “spectators” because they do not appear in the final
product. In reality, however, they have a profound influence over reaction
selectivity.

In the context of molecular chemistry, we think about those elements as
facilitating specific bond exchanges, since the transformations can occur one
bond or one individual molecule at a time. This contrasts the formation of
crystalline solids, during which many bonds exchange in concert. As such, it
is not so straightforward to imagine how selective changes in the chemical
formula or chemical structure can manifest at the atomistic scale. Yet, in
some cases, specific elements can be selectively inserted and extracted in
topochemical reactions,8 like those enabling the lithium-ion battery chemistry
of our modern personal electronics era.9 There, the lithium chemical potential
can be changed electrochemically to provide a selective driving force for
lithium intercalation. What about the other elements? How can the presence (or
absence) of some elements and different chemical potentials be used for
selective synthesis?

Figure 1: Standard enthalpies of formation of different isomers of C3H8O:
2-propanol, 1-propanol, and methoxyethane.10 Also indicated are balanced
reactions that include additional species needed for selective reactions to
produce the metastable isomers 1-propanol and methoxyethane.

Double ion exchange reactions, also known as “metathesis” reactions,11 provide
an avenue to address these questions, where the addition of non-essential
elements influences the local elemental chemical potentials and the reaction
progress. In the generic reaction scheme to make a target compound $AB$, the
simplest reaction could be written as $A$ \+ $B$ -¿ $AB$. In a metathesis
reaction, $AC$ \+ $BD$ -¿ $AB$ \+ $CD$, the addition of $C$ and $D$ in the
reaction modify the chemical potentials of the species $A$ and $B$. This is
because the compounds $AC$ and $BD$ are stable over different ranges of
chemical potentials than $A$ and $B$. Unfortunately, it has remained
challenging to predict which species $C$ and $D$ are most suitable for
synthesizing an arbitrary target $AB$, as it has historically required
knowledge of the full experimental thermodynamic landscape.12 Today, modern
thermodynamic databases fueled by accurate calculation of phase energies using
density functional theory (e.g., Materials Project,13 OQMD,14 aflowlib15),
have since provided the needed infrastructure for enabling _chemical_ control
of solid synthesis, as elaborated below. Furthermore, modern analytical
methods, such as synchrotron X-ray and time-of-flight neutron scattering
methods, now allow us to interrogate reaction progress as it happens in a so-
called “panoramic synthesis.” 16 By “cooking and looking,” we test the
thermodynamic hypothesis and can identify the breaking points of the model to
move toward a modernist paradigm of predictive retrosynthesis, a “holy grail”
of computational chemistry.17

##  2 Background: Existing approaches for synthesis prediction

Two overarching hypotheses allow us to understand and predict the progress of
solid state reactions: (a) reactions tend to be dictated by the interfaces
between _pairs_ of reactants 18 and (b) the total composition of the reacting
system is locally undefined at the interface (i.e., the interface does not
“see” how far the reactants extend). Several different studies 19, 20, 21 have
incorporated one or both of these concepts to deliver new insights.

Pairwise reactions have been observed directly with _in situ_ electron
microscopy during the formation of YBa2Cu3O_6+$x$, thus enabling understanding
of correlated changes in bulk phase fractions determined by _in situ_ X-ray
diffraction.18 Simply, the pairwise interaction between reactants derives from
the geometric constraints of a locally planar interface. As illustrated in
Figure 2(a), the interface between three solid components has a very limited
probability of occurrence (note that it is still possible, however, to have a
three-component solid state reaction with an “open” component such as a liquid
or gas).22 Controlling the nature of these interfaces can influence the
reaction products. For example, engineered planar interfaces in “modulated
elemental reactants” 23 and sputtered thin films of reactants24 provide a
geometric control over which pairwise reactions occur first. In bulk powder
reactions, providing a higher density of interfaces directly improves the
reaction progress.25 By focusing on pairwise reactions, it is possible to
understand why and how those partners evolve together.

Figure 2:  Approaches for predicting and understanding materials synthesis
using local thermodynamics. (a) Pairwise reactions at interfaces tend to
represent the pathway of complex reactions involving solids. Reproduced with
permission from Ref. 18. (b) For a given set of phases, one can compute the
minimum (most negative) possible reaction energy across a series of
compositions.19 A central hypothesis is that the reaction with the most
negative free energy, unconstrained by the overall composition, will be the
first to occur at the interface (Reaction 1), as the nucleation barrier
depends strongly on the reaction energy (Eqn. 1). However, if two phases have
a high degree of structural similarity, one can hypothesize that the surface
energy contribution of the nucleating phase will be low (e.g., topochemistry)
and also result in a low nucleation barrier, $\Delta G^{\dagger}$ (Reaction
2).21 (c) Schematic reaction coordinate diagram for a simple serial reaction
pathway is represented on the left. Actual solid-state reactions take place by
multiple, parallel steps and can be better modeled with a reaction network, as
shown on the right. Reproduced with permission from Ref. 26.

Since the interface cannot “see” how much of each reactant there is, the first
phase to form may not be an equilibrium phase for the composition of the total
system. The uncertainty around first-phase formation has created a long-
standing problem brought to technological relevance with the development of
Si-based microelectronics, where the understanding of metal silicide formation
is crucial. Initial hypotheses proposed that the most competitive phase to
form at an interface is the congruently melting phase with the highest melting
point next to the lowest-temperature eutectic.27 Nowadays, access to large
thermodynamic databases enabled by computational chemistry allows us to
calculate the most negative reaction energy for the _compositionally-
unconstrained_ mixture of the two reactants, considering all available
mixtures of products (Figure 2(b)).19 In the context of synthesis, the first-
phase-to-form often establishes the reaction pathway and the subsequent series
of steps, as observed during the formation of MgZrN2 by avoiding unreactive
ZrN 28 or during the formation of NaCoO2 in which the most favorable
interfacial reaction between CoO and Na2O2 yields O3-NaCoO2, even when the
reaction is deficient in Na2O2.20 Relating to classical nucleation theory, the
phase with the lowest (most negative) reaction energy should have the smallest
activation barrier. This activation barrier scales as

$\Delta G^{\dagger}\propto\frac{\gamma^{3}}{\Delta G_{rxn}^{2}},$ (1)

where $\gamma$ is a term denoting the energy cost of forming a surface of the
nucleating phase, $\Delta G_{rxn}$ is the free energy of phase formation per
volume, and shape effects are ignored.29 Thus, there is a fundamental
relationship that links thermodynamic quantities (reaction energy, surface
energy) to the relative rates of nucleation for different competing
intermediates, with the assumption that a higher nucleation rate yields a
dominant reaction rate. However, it is challenging to assert that such
reactions are truly limited by nucleation when reactions occur in a diffusion-
limited regime. Regardless, if one assumes that diffusion is universally
limiting, a minimal nucleation barrier can provide an advantage. A
complementary approach for assessing the relative rates of phase formation
without explicit calculation of surface energies is to estimate relative
reaction barriers from structural self-similarity based on the principle of
heterogeneous nucleation. In other words, reactants that are more structurally
similar to their products ought to have a lower barrier.21 This is
schematically illustrated in Figure 2(b): while Reaction (1) has the most
negative reaction energy, Reaction (2) may have a comparably low activation
barrier for formation due to the structural similarity between precursors and
products. Building on the notion of “phase selection through nucleation,” 21
these physical principles allow the data-enabled synthetic scientist to start
making predictions.

In modern materials synthesis planning, we identify connections and
correlations within the large landscape of databases that link together
composition, structure, and thermodynamic potentials. When we consider a
simplistic notion of a reaction coordinate diagram, we can consider the energy
balance progressing in a serial and linear fashion, as illustrated in Figure
2(c). However, a more realistic representation is to consider the parallel but
pairwise reactions with different reactants and intermediates in a graph-based
network, in which each node is a phase (or mixture of phases), and each edge
is an individual reaction weighted by its “cost”. In its most simple form, the
cost of a reaction is often a function of its energy, $\Delta G_{\text{rxn}}$
(Figure 2(d)).26 The modernist era provides us with access to a very large
available landscape in this representation, with numerous approaches to weight
the edge lengths and algorithms to identify the lowest cost pathways through
the network.30 With brute-force enumeration of all possible reactions using
these large databases, selective reactants have been identified as those with
the fewest competing reaction products that also have a favorable reaction
energy.21 Altogether, there is a powerful quiver at our disposal for enabling
a new paradigm of materials synthesis.

##  3 Phase selection from local chemical potentials

With the conceptual stage set – unconstrained reservoirs, large thermodynamic
databases, cooking-and-looking – it is now possible to rationalize unusual
observations from synthesis studies, such as the use of “assisted” metathesis
to produce different Y-Mn-O ternary phases. In the assisted metathesis
reaction (cf., Ref. [ 31]), 3 $A$2CO3 + 2 YCl3 + Mn2O3 + 1 atm O2, where $A$
is an alkali, each alkali yields different products when the reaction
temperature is below 850 ℃ (see Fig 3). For $A$ = Li, the reaction yields the
_perovskite_ YMnO3 polymorph ($o$-YMnO3), yet when $A$ = Na, the reaction
yields the _pyrochlore_ Y2Mn2O7 polymorph under otherwise identical synthesis
conditions. For $A$ = K, the reaction yields a mixture of phases below
$T<850$℃, and exclusively the _hexagonal_ YMnO3 polymorph for $T\geq 850$℃
($h$-YMnO3). When the reaction temperature exceeds 850 ℃, all three alkali
species yield $h$-YMnO3. These observations raise the questions: (1) which of
the three ternary Y-Mn-O phases are stable under what conditions, and (2)
how/why does the alkali “spectator” ion produce such distinct changes in
product formation?

Figure 3: Observed reaction products for different $A$ = Li, Na, vs K in the
reaction, 3 $A$2CO3 + 2 YCl3 + Mn2O3 + O2 (1 atm): $o$-YMnO_3+$\delta$,
Y2Mn2O7, and $h$-YMnO3,32 their crystal structures, and known stability
conditions. Adapted with permission from Ref. 32. Copyright 2019 American
Chemical Society

From the preceding experimental phase diagram and other studies of high-
temperature ceramic reactions,33, 34 $h$-YMnO3 was reported to be the bulk
stable phase above $T=850$ ℃. Synthesis of $o$-YMnO3 typically requires quite
oxidizing conditions, such as 35 kbar ($\approx$3.5 GPa) of oxygen pressure at
$T_{\text{rxn}}$ = 1000 ℃,35 or curiously, a lower-temperature reaction at 800
℃ using an amorphous precursor formed by the decomposition of citrate salts.36
In the modernist approach, we illustrated that the incorporation of metal
vacancies from oxidation (e.g., Y_$1-x$Mn_$1-y$O3, also referred to as
YMnO_$3+\delta$) thermodynamically stabilizes the $o$-YMnO3 polymorph 37 in
agreement with $T$\- and $p$O2-dependent metathesis reactions.38 Y2Mn2O7
typically requires rather extreme conditions for synthesis, such as 30 kbar O2
reactions with KIO4 at $T=$ 1000 ℃,39 or 500 ℃ hydrothermal reactions with
ClO3^-.40 In the modernist approach, with density functional theory
calculations in conjunction with $T$-dependent elemental references and a
machine-learned expression for vibrational entropy,41 we find that Y2Mn2O7 is
thermodynamically stable below $T\leq$ 1000 ℃ and favored over YMnO3 when
$p$O2 $\geq$ 0.7 atm.42. It follows that Y2Mn2O7 should actually be formed in
all of these assisted metathesis reactions, given that they are performed at
$p$O2 $\approx$ 0.85 atm. Therefore, the alkali “spectators” must somehow
influence the reaction pathway to shift the local equilibrium. It is sensible
that a lower-temperature reaction can yield a more oxidized product; however,
it is not yet clear why Li2CO3 and Na2CO3 precursors arrive at different
reaction products even when performed at an identical temperature and $p$O2.
The fact that the Na-based precursor leads to the more oxidized product,
Y2Mn2O7, is even more perplexing given that the Na-based reaction is more
exothermic and could yield more reducing conditions than the Li-based reaction
if the local interface experiences quasi-adiabatic self-heating.

Figure 4: (a) Generalized chemical potential diagram of the Y-Mn-O system.
Reaction of Y2O3 + 2 MnO2 should result in the initial formation of some
amount of YMn2O5. (b) Y-Mn-O chemical potential diagram with a projection of
the LiMnO2 domain.42 The illustrated overlap between this domain and YMnO3
suggests a lower effective activation barrier for LiMnO2 to form YMnO3. (c)
Similarly, the location of the NaMnO2 overlapping region42 allows for a lower
effective activation barrier in forming Y2Mn2O7. Adapted with permission from
Ref. 42. Copyright 2021 American Chemical Society

To understand the interface between reactants, the intensive variable of
chemical potential, $\mu$, best describes the local equilibrium between
effectively unrestricted reservoirs of reacting elemental species. All
interfaces evolve towards a local equilibrium where atomic chemical potentials
equalize across the interface, as the rate of mass transfer is proportional to
the chemical potential gradient. While a large difference in chemical
potential can provide a high rate of diffusion (and thus reaction), if the
difference is too large, it may also necessitate the formation of another
phase that is stable over intermediate ranges of chemical potential. This fact
also explains why synthesis reactions from pure elemental precursors (mixed in
the proper ratios) do not always result in the expected products despite
possessing the most significant thermodynamic advantage (i.e., the most
negative reaction energy possible). The difficulty in synthesizing a target
product is especially true when multiple polymorphs may be accessible, such as
with the isomers in Figure 1. A thermodynamic explanation is that the
elemental precursor and desired target often do not form a stable interface
due to competition from other phases that form over intermediate chemical
potentials. By adding spectator ions to our elemental precursors, we can
modify the chemical potentials of the reacting species such that a stable
interface should be formed with our desired target, supporting direct and
selective reaction to a product that is stable over a particular range of
chemical potentials. The addition of spectator elements also increases the
dimensionality of the chemical potential space, allowing one to bypass
undesired intermediates. Finally, if the spectator ions together form an
energetically favorable byproduct (e.g., an alkali halide), then it is
possible to “re-claim” some of the reaction driving force that is lost to
using a more stable precursor (i.e., one containing bonds with the spectator
species).

Mapping the Y-Mn-O phase equilibria in terms of the elemental chemical
potentials on a generalized chemical potential diagram (e.g., a 3-dimensional
diagram with the axes, $\mu_{\text{Y}}$, $\mu_{\text{Mn}}$, and
$\mu_{\text{O}}$, Figure 4(a))12 allows us to directly observe which phases
exhibit stable interfaces with each other. This map of stable interfaces
rationalizes the observation of specific intermediate compounds. For example,
a reaction of Y2O3 + 2 MnO2 or Y2O3 + Mn2O3 should necessarily produce YMn2O5
on route to forming Y2Mn2O7 or YMnO3. Indeed, previous work32, 38 has shown
that for $T\leq 850$ ℃, the reaction halts at YMn2O5 and does not yield
stoichiometric completion of the reaction which would be the global
thermodynamic equilibrium.33

However, the assisted metathesis reactions proceed through $A_{x}$MnO2 and
YOCl intermediates, as learned from _in situ_ synchrotron X-ray diffraction
experiments. To visualize this high-dimensional phase equilibria, it is
possible to extract lower-dimensional slices of the full $N=6$ high-
dimensional phase space, as illustrated by the intersection of the different
$A_{x}$MnO2 phases onto the Y-Mn-O chemical subspace (Figure 4). Focusing on
the role of LiMnO2 in Figure 4(b), it shares a stable interface with YMnO3 but
not Y2Mn2O7. Therefore, to maintain a local equilibrium, the LiMnO2
intermediate would have to subsequently form Li2MnO3, LiMn2O4, and “Li5Mn7O16”
(each with distinct crystal structures) to produce Y2Mn2O7.42 As the chemistry
proceeds at a lower temperature through a defective LixMnO2 intermediate, the
higher $\mu_{\text{O}}$ stabilizes the $o$-YMnO3 polymorph.37 In other words,
we infer that LiMnO2 has a lower effective activation barrier to form YMnO3
than Y2Mn2O7 because the reaction does not need to proceed through subsequent
intermediates.

In contrast, the assisted metathesis reaction with Na proceeds through NaxMnO2
intermediates. Visualized in Figure 4(c), the projection of NaMnO2 into the
Y-Mn-O subspace reveals a stable interface with Y2Mn2O7. Hence, the Na-
containing reaction exhibits a lower effective activation barrier to nucleate
Y2Mn2O7 and does not necessitate the formation of additional intermediates. In
the case of KMnO2, it forms stable interfaces with both YMnO3 and YMn2O5 and
indeed, a mixture of both of these products forms after 24 h reactions at
$T_{\text{rxn}}$ = 850 ℃.32 These _in situ_ observations, coupled with a map
of the high-dimensional chemical space, illustrate that different “spectator”
elements modify the chemical potentials of the precursors differently, thereby
changing which intermediates and/or products are accessible. This
understanding enables a chemistry-based control of phase selection.

Figure 5: As illustrated for the nearly 3000 enumerated reaction pathways to
synthesize Y2Mn2O7 in the Na-Cl-Y-Mn-O-C chemical system,42 (a) including
additional elements in a reaction tends not to significantly shift the mean
grand potential reaction energy, $\Delta\Phi_{\text{rxn}}$), when examining
all enumerated pathways to form Y2Mn2O7. (b) However, increasing the number of
elements in the system does significantly decrease the total change in
chemical potential (e.g., chemical potential “distance”), thus illustrating
the presence of thermodynamic shortcuts. Reprinted with permission from Ref.
42. Copyright 2021 American Chemical Society

More generally, the addition of non-essential elements into the reaction
facilitates the reaction progress with thermodynamics alone. For example, when
considering over 3000 different enumerated reactions to produce Y2Mn2O7, the
average change in local chemical potentials (i.e., the minimum chemical
potential “distance” as measured on a chemical potential diagram) decreases
systematically with the number of elements while not significantly affecting
the mean reaction energy available (Figure 5). This can be intuited: as you
add more “spectators,” you increase the likelihood that there will be an
intermediate phase that forms stable interfaces between reactant(s) and
product(s), or between other intermediates. This also matches geometric
intuition – by increasing the number of dimensions of our chemical potential
space, we increase the available number of phase boundaries shared with the
desired product(s). The spectator approach is a way of replacing the need to
form two or more intermediates with only one, or possibly none at all. This
is, in effect, a shortcut through the now larger accessible thermodynamic
phase space and a means to affect selectivity.

##  4 Outlook

When we consider “hyperdimensional” chemical reactions – reactions that
include elements beyond those contained in the product – the reaction design
space increases dramatically. In a previous perspective concerning “kinetic
control” in materials synthesis,43 we voiced the challenge of how to
selectively control reaction barriers in a general manner. Here, we effect
this notion by using the local chemical potential to minimize the number of
phases that need to nucleate. The chemical control provides thermodynamic
shortcuts as a means for “phase selection through nucleation.”

Looking forward, a significant challenge is finding other non-trivial
reactions that test these principles. In the synthesis of YMnO3 phases, it was
recognized many years ago that “kinetic factors” played a role in reaction
outcomes.44 Here we argue that these kinetic factors are intuited by the
formation of local chemical equilibrium intermediates that effectively halt
the reaction. To explore to what extent the broader term kinetic factors can
be cast into a more quantitative framework of local thermodynamics, we need to
test our hypothesis with additional systems. We note that our hypothesis
encompasses the notion of “remnant metastability,”7 such that the phases we
call “metastable” often form under conditions where that phase is indeed,
locally or globally, the most stable phase that can be formed at an interface.
Knobs that can be turned to affect the thermodynamic landscape include e.g.
strain, dimensionality, external fields, and chemical potential.45 Given the
use of databases and natural language processing techniques,46, 47, 48, 49
modernist materials chemists have many tools at their disposal.

If our goal is to arrive at a fully predictive model for synthesis planning,
many fundamental knowledge gaps remain. Examining thermodynamic variables
alone (e.g., reaction energies, minimum changes in chemical potentials), we do
not yet understand how to accurately and efficiently include the role of
atomistic disorder when assessing energy or diffusion. In reactivity models
using graph networks, the “kinetic” barrier imagined to contribute to the
distance between nodes is still unknown30 Furthermore, we acknowledge that in
many real-life cases, there is a true time dependence to the reaction outcome
(e.g., the specific 17$\pm$0.5 h reaction required to make superconducting
Bi3O2S3 50). In fact, there is a clear variability in the formation of
Y2Mn2O7, such that some reactions, unexpectedly, produce a mixture of Y2O3 and
Mn2O3 binaries.32, 51 Given the immense complexity of the true potential
energy landscape and the time-dependent kinetic processes that influence the
movement across that landscape, new modernist approaches beyond reaction-
diffusion kinetics 52 will enable significant advances. Different communities
with different tools and approaches united by fundamental synthesis questions
– be it for a specific application or scientific curiosity – have made
significant strides and still have many exciting discoveries to be made at the
next interface.

##  5 Acknowledgements

This work was supported as part of GENESIS: A Next Generation Synthesis
Center, an Energy Frontier Research Center funded by the U.S. Department of
Energy, Office of Science, Basic Energy Sciences under Award Number DE-
SC0019212.

##  6 Conflict of Interest Statement

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

##  7 Data Availability

The full set of enumerated chemical reactions, as well as all experimental raw
data, processed data, data processing scripts, and figure plotting scripts,
are available at:
https://github.com/GENESISEFRC/y2mn2o7-selectivity, and
https://github.com/GENESIS-EFRC/reaction-network.

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