# I still have Time(s): Extending HeidelTime for German Texts

###### Abstract

HeidelTime is one of the most widespread and successful tools for detecting
temporal expressions in texts. Since HeidelTime’s pattern matching system is
based on regular expression, it can be extended in a convenient way. We
present such an extension for the German resources of HeidelTime:
HeidelTimeext. The extension has been brought about by means of observing
false negatives within real world texts and various time banks. The gain in
coverage is 2,7 $\mathrm{\%}$ or 8,5 $\mathrm{\%}$, depending on the admitted
degree of potential overgeneralization. We describe the development of
HeidelTimeext, its evaluation on text samples from various genres, and share
some linguistic observations. HeidelTimeext can be obtained from
https://github.com/texttechnologylab/heideltime.

Keywords: HeidelTime, German, TimeBanks, BIOfid

languageresourceLanguage Resources

I still have Time(s): Extending HeidelTime for German Texts

Andy Lücking, Manuel Stoeckel, Giuseppe Abrami, Alexander Mehler
---
Goethe University Frankfurt, Text Technology Lab
Robert-Mayer-Str. 10, D-60325 Frankfurt am Main, Germany
{luecking, m.stoeckel, abrami<EMAIL_ADDRESS>

Abstract content

##  1\. Motivation

Biodiversity literature exhibits frequent references to times. Hence,
processing biodiversity texts, as is done in the context of specialized
information services [Driller et al., 2020], involves recognizing temponyms,
that is, temporal expressions which have – possibly aided by context
information – a unique interpretation on a time line, like toponyms do in
terms of geo-spatial entities [Kuzey et al., 2016a]. To this end, HeidelTime
[Kuzey et al., 2016b, Strötgen and Gertz, 2013] is used within the BIOfid
project (www.biofid.de) to detect mentions of time-denoting expressions in
mainly German biological texts from the 19th century until today – see
[Lücking et al., 2021] for an overview. These texts, however, contain
temponyms that are outside of the extension of HeidelTime’s rule system and
include spelling variants of mundane temponyms like Herbste ‘fall’111E.g.,
https://sammlungen.ub.uni-frankfurt.de/3728500 and https://sammlungen.ub.uni-
frankfurt.de/4497114, duration-forming constructions such as in den letzten
beiden Jahren ‘in the past two years’222E.g., https://sammlungen.ub.uni-
frankfurt.de/3721623, and set-forming constructions such as in einem
zweijährigen Turnus ‘on a biennial basis’333E.g., https://sammlungen.ub.uni-
frankfurt.de/3748527. Based on such missing patterns of temporal expressions,
among others, we extend the German component of HeidelTime (section 2.). We
want to emphasize that HeidelTime is designed in such a way that such
extensions can be implemented in a comparatively easy manner. The gain in
coverage in assessed on various kinds of corpora (section 3.). A manual
inspection of the novel instances found by HeidelTimeext brought two
overgeneralizing rules to the light, however. Depending on how much potential
overgeneralization is admitted, the relative coverage improvement of
HeidelTimeext is 2.7 $\mathrm{\%}$ (excluding one source of
overgeneralization) or 8.5 $\mathrm{\%}$ (including the source of
overgeneralization). The HeidelTime extension – called HeidelTimeext – is
available at https://github.com/texttechnologylab/heideltime.

##  2\. Procedure

HeidelTime is developed to detect a certain kind of temponyms, namely
temponyms that are categorized as timex3 expressions [ISO, 2012]. timex3
expressions comprise dates (e.g., 10 January, 1999), times (e.g., 12 o’clock),
durations (e.g., two weeks), sets (e.g., every year). To extend HeidelTime, we
first looked for timex3 patterns which are not yet covered.

###  2.1. Manual Exploration

In the context of BIOfid, we manually sampled sentences from the current
BIOfid corpus and processed the sample with HeidelTime. We then inspected the
outcome, focusing on false negatives, that is, timex3 expressions that have
not been detected. Generalizing over these false negatives, missing patterns
or expressions have been identified. In contexts like natural language
processing in biodiversity or the humanities it is important to cover all
these instances irrespective of their frequency (we come back to matters of
frequency as part of the evaluation in Sec. 3.). In general, false negatives
fall in one of four classes:

  * •

Spelling variants. This rather trivial class pertains to typographic
variation. In German, the dative case of masculine singular nouns can be
marked by the suffix -e, as in dem Herbste ‘the-DAT fall’. This spelling
variation, however, is a bit old-fashioned, so that it frequently occurs in
the older texts from the BIOfid corpus, but less so in contemporary writings.
Another example is punctuation in time expressions. Although using a dot
instead of a colon to separate hour and minute – as in 21.30 Uhr ‘9.30 pm’ –
does not comply with international standards [ISO, 2006], it is nonetheless
used in texts and for that reason should be detected.

  * •

Lexical extensions. Time spans are sometimes partitioned according to business
or financial concerns, such as fiscal years. Corresponding nouns (e.g.,
Geschäftsjahr) are a straightforward extension to HeidelTime’s time units.
Further lexical extensions are due to temporal adjectives or adverbs. For
instance, the modifier täglich ‘everyday’ has the same meaning as the
quantified noun phrase jeden Tag ‘every day’, but, in contrast to the noun
phrase is not yet covered by HeidelTime’s lexicon.

  * •

Compounds. German is well-known for its “tapeworm words” [Twain et al., 2016].
However, HeidelTime is not concerned with compounds, and presumably for a good
reason: the risk of overgeneration is large. For instance, Jugendzeit ‘young
days’/‘adolescence’/‘youth’, although ending on -zeit ‘-time’, is not a timex3
expression but denotes a developmental stage. However, there are a couple of
“well behaved” compounds. We count compounds where the modifying noun is a
known temponym among them, such as Winterzeit ‘wintertime’ or Sommermonate
‘months of summer’. Arguably, the head nouns do not contribute much in these
cases so that, for instance, a combination of a season term and -zeit ‘-time’
can be normalized to the value of the season term in a straightforward way.
Compounding also underlies the formation of temporal expressions of a set
type. An example is -basis: Wochenbasis ‘on a weekly basis’ means every week.

  * •

Rule extensions. Some of HeidelTime’s rules are restricted to a certain class
of expression. For instance, while quantifying over years is covered,
quantifying over seasons is not. For that reason, the duration denoting
expression viele Winter ‘for many winters’ is not recognized as such, but has
become recognizable by adding corresponding duration rules. A related
observation can be made with regard to relative times. There are rules that
capture times such as letzten Freitag ‘last Friday’, but the synonymous
expression vorheriger Freitag ‘previous Friday’ had to be licensed by an
additional rule.

All extensions are marked as such within the source files making up
HeidelTimeext. It should be mentioned that not much emphasis is put on
grammatical well-formedness or common usage: seldom or questionable compounds
are recognized as well as phrases that lack morpho-syntactic agreement. After
all, HeidelTime, as HeidelTimeext, is an annotator, not a grammar.

###  2.2. Populating Negative Rules

In writing regular expressions, care has to be taken to not to overgenerate.
To this end, HeidelTime employs so-called _negative rules_ , that is, rules
which, when apply, remove their matched expressions from the output. To give
an example: while season names are welcome temponyms, they can also figure as
family names such as in Herr Sommer ‘Mister Summer’. Such instances can be
excluded by a negative rule that says that if a season term follows Herr
‘Mister’ or Frau ‘Miss’, then remove it. We added such a negative rule.
However, one cannot stop here: the BIOfid example Assistent Sommer ‘assistant
Summer’ (file https://sammlungen.ub.uni-frankfurt.de/3673151) still
circumvents our new negative rule. Obviously, in addition to addressing
particles, also profession terms can mark a season term as a proper name.
Therefore, we collected a list of profession terms from the German agency for
employment and added them to HeidelTimeext’s pattern files.444To be more
precise, we applied a string manipulation on the profession terms first: we
removed subdomain classifications (for instance, distinguishing farmers for
different agricultural sectors), and extended each entry into a separate
masculine and feminine form. To avoid running into errors due to too long
regular expressions, we restricted the list to single-word profession terms.
Addressing particles and profession terms still fall short of capturing
Ehepaar Sommer ‘the married couple Summer’, however. This example is, of
course, covered by HeidelTimeext, but it illustrates that time recognition
seems to be an open-ended task. For this reason, we followed a more dynamic
approach and used BERT, a transformer-based language model trained for
contextual embeddings of words [Devlin et al., 2019]. BERT can be used like a
cloze test: suggestions for a masked item can be obtained from left and right
context information. In this manner, we used the sentence containing Assistent
Sommer as input, masked the noun Assistent, and collected the 30,000 words (in
fact, BERT also suggests non-word character sequences) which according to BERT
fit best into the placeholder position. We then removed all suggestions that
are shorter than 4 characters and excluded fragmentary items (starting with
“##”). Finally, we selected the first 5,000 suggestions (using [much] more
leads to regular expressions which are too long to handle for the system). The
BERT list provides an immediate benefit for the negative rule: since it
includes typical given names, full names ending on a season or weekday term
are rightly excluded from HeidelTimeext’s temponym recognition.

###  2.3. Harvesting Time Banks

To obtain indications of further extensions of HeidelTime, we looked at
instances of expressions that have been marked as timex3 expressions in
several time banks. We extracted the content of timex3 tags from the French
TimeBank [Bittar et al., 2011] FrenchTimeBank, the Basque TimeBank [Altuna et
al., 2020] EusTimeML:resource, and the MEANTIME newsreader corpus [Minard et
al., 2016] Newsreader:resource (Dutch, English, Italian, Spanish). We then
used www.deepl.com to translate the time expressions from different languages
into German ones. We fed the list into HeidelTime and inspected the outcome,
most notably lines that lacked a HeidelTime tag.555Since the temporal
expression-based input lacks sentential context, ambiguity due to conflicting
pattern scopes arises. For instance, if a day term precedes a date, then both
are taken to be a single time expression. If, however, a second day term
follows, a conflict arises whether the first or the second day mention
constitutes a time expression together with the date. Such ambiguities do not
do harm to our approach, since the output is manually inspected anyway. This
procedure resulted in 83 sample pattern which underlie the coverage extension
of HeidelTimeext. The examples have been chosen manually. False negatives
which do not make up clear temponyms have been ignored. This includes event-
denotating expressions (e.g., l’heure de l’Europe et de la mondialisation ‘in
the age of Europe and globalization’ or temps de guerre contre le terrorisme
‘in times of the war on terrorism’), “vacuous” temporal expressions (such as
any time), and temporal modifiers which typically are used to modify not-
temporal nouns (e.g., recent).

##  3\. Evaluation

The gain in coverage of HeidelTime and HeidelTimeext (Sec. 3.3.) is assessed
and compared on various resources (Sec. 3.1.), which have been pre-processed
as described in Sec. 3.2..

###  3.1. Evaluation Corpus

Texts have been sampled from five different sources to balance potential
effects of text type to the frequency of temporal expression use:

  * •

10 randomly collected protocols of the German Bundestag
(https://www.bundestag.de/services/opendata).

  * •

10 books from the _German Text Archive_ (DTA,
https://www.deutschestextarchiv.de, namely Dickens, Weihnachtsabend (1844),
Fontane, Effi Briest (1896), Goethe, Faust 1 (1808), von Humboldt, Kosmos,
vol. 1 (1845), Kafka, Die Verwandlung (1915), Lessing, Nathan der Weise
(1779), Marx, Das Kapital, vol. 1 (1867), Nietzsche, Homer und die klassische
Philologie (1869).

  * •

10 randomly selected tests from the _Zoologisch-Botanische Datenbank_
(Zobobat, https://www.zobodat.at). The texts have been converted to plain text
files from OCR PDFs and hence contain some token errors.

  * •

766 articles from the _Süddeutsche Zeitung_ (SZ), collected within three TEI
files with the following time stamps: 15. 06. 1996, 04. 09. 2002 and 07. 12.
2013.

  * •

10,000 randomly selected sentences from Wikipedia (WP) from the _Leipzig
Wortschatz_ 666https://wortschatz.uni-leipzig.de/en/download, dump from the
year 2012 [Goldhahn et al., 2012].

The size of the samples is summarized in Tab. 1.

Sample | # sentences | # tokens
---|---|---
Bundestag | 188,768 | 3,682,370
DTA | 27,687 | 810,582
SZ | 18,938 | 359,706
Zobodat | 6,231 | 92,003
WP | 10,000 | 176,775
sum | 251,624 | 5,121,436
Table 1: Number of tokens (incl. punctuation) and sentences within the
evaluation samples.

###  3.2. Pipeline

Since HeidelTime requires single white spaces and, at least for some rules,
part-of-speech information, all texts described in the previous section have
been pre-processed using a TextImager pipeline [Hemati et al., 2016] as
follows:

  1. 1.

Normalization: All white spaces have been normalized to single spaces.

  2. 2.

Segmentation: Sentences have been segmented using the OpenNLP Max Entropy
Model.777opennlp-de-ud-gsd-sentence-1.0-1.9.3.bin, Apache 2.0 License
https://opennlp.apache.org/index.html

  3. 3.

Tokenization: Word forms have been tokenized by using the Stanford CoreNLP
[Manning et al., 2014] via DKPro [Eckart de Castilho and Gurevych, 2014].

  4. 4.

Part-of-speech Tagging: Parts-of-speech (POS) have been assigned by using the
POS tagger from MateTools [Bohnet and Nivre, 2012] via DKPro [Eckart de
Castilho and Gurevych, 2014]

HeidelTime and HeidelTimeext are run on the pre-processed texts and compared
by means of two views in a UIMA CAS.

###  3.3. Results

HeidelTimeext found 4,458 more timex3 expressions than the original
HeidelTime, a gain of 8.5 $\mathrm{\%}$, as summarized in Tab. 2. However, as
discussed below in Sec. 4., the bare gain in coverage has to put into
perspective.

Sample | HeidelTime | HeidelTimeext
---|---|---
Bundestag | 36,068 | 37,507
DTA | 5,827 | 7,568
SZ | 4,399 | 5,305
Zobodat | 2,110 | 2,235
WP | 3,966 | 4,213
sum | 52,370 | 56,828
Table 2: Number of timex3 expressions found by original HeidelTime and
HeidelTimeext (see also Sec. 4.).

The coverage is detailed in Tab. 3. Column “novel” lists the number of
temporal expressions newly found by HeidelTimeext, whereas the column
“missing” counts the expressions only found by the original HeidelTime. Since
there are 298 missing timex3 in total (assuming that these are true
positives), this means that the newly added rules in HeidelTimeext interfere
with the application of some of the original rules. The extended rule system
of HeidelTimeext also generally covers larger token spans, as expressed in the
column “extended”. Smaller token spans – column “reduced” – can be ignored due
to their little frequency of occurrence.

Sample | novel | extended | reduced | missing
---|---|---|---|---
Bundestag | 267 | 70 | 2 | 97
DTA | 258 | 37 | — | 59
SZ | 436 | 75 | 1 | 103
Zobodat | 86 | 5 | — | 8
WP | 133 | 28 | — | 31
sum | 1,180 | 215 | 3 | 298
Table 3: Coverage details. Novel: newly found temporal expressions; extended:
“longer” matches; reduced: “shorter” matches; missing: failed detection by
HeidelTimeext.

The following examples provide an impression of the kinds of temporal
expressions which HeidelTimeext is designed for: Vorjahr ‘preceding year’,
übermorgen ‘the day after tomorrow’, Wintermonate ‘winter months’, eine halbe
Stunde ‘half an hour’. Furthermore, durations as property denotations like
dreitägig ‘three-day’ as in dreitägige Exkursion ‘three-day excursion’ are
also taken to be timex3 expressions. However, a manual inspection of the
outcome of HeidelTimeext revealed two kinds of expressions which need to be
discussed.

##  4\. Discussion

To assess the outcome of HeidelTimeext not only in quantitative terms, we
manually inspected 1,198 timex3 expressions only found by
HeidelTimeext.888Bundestag: 220, Zobodat: 228, DTA: 312, WP: 155, SZ: 283.
Each instance is labeled as being a true or a false positive – see Tab. 4 for
an overview. The relatively high number of false positives, however, is mainly
due to two kinds of overgeneralizations.

The first overgeneralization is bound up with two time rules which wrongly
apply not only to times of day expressions but to any measure terms formed
with digits and a separating punctuation symbol. From 168 examples of this
type, which have been part of manual inspection, 139 are regarded false, and
29 true positives. This overgeneralization is attenuated in the release of
HeidelTimeext as of 2022-04-19 and following ones.

Sample | true | false
---|---|---
Bundestag | 185 | 35
DTA | 217 | 95
SZ | 210 | 73
Zobodat | 64 | 64
WP | 125 | 30
sum | 801 | 297
Table 4: Instances of true and false positives in newly detected timex3
expressions.

The second kind of overgeneralization is due to a significant proportion of
newly detected occurrences of German nun, which has not been part of the
original HeidelTime. HeidelTimeext found 2,530 instances in
total.999Bundestag: 1,010, DTA: 1,119, SZ: 289, Zobodat: 28, WP: 84. This
particle word, however, has two broad uses which translate into English as
‘now’ respectively “well” (that is, the adverb typically used sentence
initial). Obviously, only the first of these uses is a temporal one, namely
present_ref.101010Actually, things are more complicated than that: as is well
known, even indexical expressions are evaluated with reference to three
temporal indices, namely event time, reference time, and utterance time
[Reichenbach, 1947]. For a semantics of temporal reference see [Kamp, 1979].
During manual inspection, 455 instances of nun (irrespective of
capitalization) have been checked: 333 of them correspond to a temporal, 122
to a discourse use. Since there does not seem to be typical contexts which
distinguish between temporal and discourse-functional nun, overgeneration
cannot simply be prevented by a negative rule. This example therefore
exemplifies a limit of temponym recognition based on regular expressions. The
question therefore arises whether neural network-based approaches such as CNNs
[Lin et al., 2017], which have been trained on date expressions, fare better
with regard to temporal particles.111111Neural networks can be used for
temponym _detection_ , however, but not so easily for time _normalization_.
For the time being, a user of HeidelTimeext can choose how to proceed by
commenting out the rule in question (rule date_r8a-explicit).

Removing the counts for nun and for the overgeneralizing time rules from the
figures given in Tab. 2, we get the more “cautious” overview in Tab. 5. There
is now a gain of 1,416 timex3 expressions, or 2.7 $\mathrm{\%}$.

Sample | HeidelTime | HeidelTimeext
---|---|---
Bundestag | 36,068 | 36,386
DTA | 5,827 | 6,429
SZ | 4,399 | 4,748
Zobodat | 2,110 | 2,136
WP | 3,966 | 4,087
sum | 52,370 | 53,786
Table 5: Number of timex3 expressions found by original HeidelTime and
HeidelTimeext with problematic cases removed.

Note finally that we still found examples that circumvented our highly
extended negative rules, cf. Sec. 2.2.. For instance, the given name Carl, as
opposed to the form variant Karl, is not part of any black list, so that the
proper name Carl Winter still triggers a season of the year rule.

##  5\. Conclusion

Based on manually collected false negatives from various time banks and
biological texts, we developed HeidelTimeext, a German extension of
HeidelTime. We constructed an evaluation corpus to quantify the gain in
coverage of the extension. A manual inspection of novel instances found by
HeidelTimeext identified two kinds of rules which tend to overgeneralize to
non-temporal instances, however. Depending on how much potential
overgeneralization is admitted, the relative coverage improvement of
HeidelTimeext is between 2.7 $\mathrm{\%}$ and 8.5 $\mathrm{\%}$. At this
point, one could object that our approach mainly discovers rarely occurring
expressions of time. But especially these cases are interesting for
disciplines of the humanities or biodiversity (as outlined in Secs. 1. and
2.1.), which also deal with what is rare rather than with what is frequent.

Future work might deal with a compositional approach to written numbers such
as einhundertdreiundfünfzig ‘one hundred and fifty-three’. A difficulty here
of course is to generate a corresponding norm value. Another issue is the
improvement of negative rules to prevent overgeneralization. We used a list-
based approach, making use of governmental material and material generated by
a BERT model. HeidelTimeext is available from
https://github.com/texttechnologylab/heideltime.

##  6\. Acknowledgements

This work is funded by the DFG, grant ME2746/5-2.

##  7\. Bibliographical References

## References

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##  8\. Language Resource References

lrec languageresource


# Predicted Complex Lithium Phases at Terapascal Pressures

Jack Whaley-Baldwin<EMAIL_ADDRESS>TCM Group, Cavendish Laboratory,
University of Cambridge  Miguel Martinez-Canales  Scottish Universities
Physics Alliance (SUPA), School of Physics and Astronomy and Centre for
Science at Extreme Conditions, University of Edinburgh, Edinburgh EH9 3FD, UK
Chris J. Pickard  Department of Materials Science and Metallurgy, University
of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom
Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira,
Aoba, Sendai 980-8577, Japan

###### Abstract

We investigate the pressure-temperature ($p$-$T$) phase diagram of elemental
lithium (Li) up to multiterapascal (TPa) pressures using ab-initio random
structure search (AIRSS) and density functional theory (DFT). At zero
temperature, beyond the high-pressure $Fd\bar{3}m$ diamond structure predicted
in previous studies, we find eleven solid-state phase transitions to
structures of greatly varying complexity, in addition to two structures that
we calculate will become stable with sufficient temperature. The full $p$-$T$
dependence of the phase boundaries are computed within the vibrational
harmonic approximation, and the solid-liquid melting line is calculated using
ab-initio molecular dynamics simulations. Notably, between $39.1$ TPa and
$55.7$ TPa, Li adopts an elaborate monoclinic structure with 46 atoms in the
primitive unit cell, and between $71.9$ TPa and $103$ TPa, an incommensurate
host-guest phase of the Ba-IV type. We find that Li, hitherto predicted to be
an electride at TPa pressures, abruptly loses its electride character above
$16$ TPa, reverting back to normal metallic behaviour with a corresponding
rise in the Fermi-level electronic density of states (eDOS) and broadening of
the electronic bands.

##  I Introduction

Under compression, lithium (Li) transforms through a multitude of crystal
structures with varying structural complexity and markedly different
electronic properties, which in-turn exhibit metallic Shimizu  _et al._
(2002), superconducting Shimizu  _et al._ (2002) and semiconducting Marqués
_et al._ (2011); Lv  _et al._ (2011) behaviour. At zero temperature and
pressure, the ground state structure of Li is face-centered cubic (fcc)
Ackland  _et al._ (2017), and it is a simple metal well-described by the
nearly-free electron (NFE) model. With an increase in pressure, experiment has
shown that Li transitions to the hR1 phase (space group $R\bar{3}m$) at 39
GPa, and the cI16 phase (space group $I\bar{4}3d$) at $42$ GPa Hanfland  _et
al._ (2000); Guillaume  _et al._ (2011); Frost  _et al._ (2019). Further
transitions to the exceptionally complex oC88, oC40 and oC24 structures occur
at $61$ GPa, $70$ GPa and $115$ GPa respectively Marqués  _et al._ (2011);
Guillaume  _et al._ (2011); Lv  _et al._ (2011); Frost  _et al._ (2019), with
oC24 being a narrow- or zero- gap semiconductor Marqués  _et al._ (2011); Lv
_et al._ (2011); Matsuoka and Shimizu (2009), and oC40 being, remarkably, a
wide-gap semiconductor Marqués  _et al._ (2011); Lv  _et al._ (2011). In
addition to exhibiting a rich solid-state phase transition sequence, the
melting line of Li also displays intriguing behaviour with increasing
pressure; namely, a pronounced drop in melting temperature at $50$ GPa to just
$190$ K, implying that Li can be brought into a liquid state upon compression
even at temperatures well below room temperature (‘cold melting’) Guillaume
_et al._ (2011); Hernández  _et al._ (2010); Wang  _et al._ (2022); Frost  _et
al._ (2019).

First-principles computational structure searches have further extended the
high-pressure phase diagram of Li, with predicted transitions to a
$P4_{2}/mbc$ structure at $247$ GPa, $R\bar{3}m$ structure at $449$ GPa, and
eventually the $Fd\bar{3}m$ diamond structure ($483$ GPa) Lv  _et al._ (2011);
Pickard and Needs (2009).

Whilst the very highest static laboratory pressures are typically only several
hundred GPa, dynamic compression techniques have allowed pressures of several
terapascal (TPa) to be accessed Lazicki  _et al._ (2021); Fratanduono  _et
al._ (2021, 2020); Lazicki  _et al._ (2015); Wang  _et al._ (2016); Han  _et
al._ (2021); Henderson  _et al._ (2021); Jeanloz  _et al._ (2007); Millot  _et
al._ (2020); Hansen  _et al._ (2021); Fratanduono  _et al._ (2011); Smith  _et
al._ (2018); Gorman  _et al._ (2022), with one exceptional experiment
exceeding $5$ TPa Smith  _et al._ (2014). Concomitant with this progress in
TPa-pressure generation, advances in x-ray diffraction techniques have allowed
the crystal structures of such TPa-compressed matter to be calculated Rygg
_et al._ (2012, 2020); Coppari  _et al._ (2021), and there is a particular
interest in determining the structure of pure elements under pressure,
especially in using their equation of state (EoS) as a reference standard
Smith  _et al._ (2014); Lazicki  _et al._ (2021); Fratanduono  _et al._ (2021,
2020); Lazicki  _et al._ (2015); Wang  _et al._ (2016); Han  _et al._ (2021);
Henderson  _et al._ (2021). In determining such EoSs, computational structure
search provides invaluable insight which complements the experimental results,
and is especially important when the target system is difficult to access
experimentally. Li itself is such an example; having just three electrons, it
is a difficult diffraction target and possesses an extremely small x-ray
absorption cross-section, and its two main isotopes show neutron absorption
cross-sections that differ by several orders of magnitude Overhauser (1984);
Smith (1987); Schaeffer  _et al._ (2015). Li is also highly reactive with
materials commonly used to achieve multi-TPa pressures Hanfland  _et al._
(1999), and therefore a computational search may, for the time being, provide
the only insight into its behaviour at TPa pressures.

Figure 1: Phase diagram of elemental Li up to $140$ TPa. Each coloured region
denotes the structure with the lowest Gibbs free energy, with
$G(p,T)=U_{elec}+U_{phonon}-TS_{elec}-TS_{phonon}+pV$. Solid-state vibrational
effects have been included at the level of the harmonic approximation (HA),
and solid circles on the phase boundaries denote explicitly calculated $(p,T)$
points according to the HA. The melting line, as calculated using ab-initio
molecular dynamics simulations (red crosses), is fitted to a fifth-order
polynomial and is shown as a solid blue line, with a thickness according to
its uncertainty. Dashed vertical lines indicate second-order soft-phonon
transitions.

As a high-pressure element of interest, Li is an extremely important case that
lies between the quantum nuclear effect dominated hydrogen Pickard  _et al._
(2012); Drummond  _et al._ (2015) and helium Monserrat  _et al._ (2014)
systems, and the rest of the periodic table. In the TPa regime, computational
structure searches have been performed on neighbouring group-I elements sodium
(Na) Li  _et al._ (2015) and potassium (K) Whaley-Baldwin (2023), as well as
other first-row elements carbon (C) Martinez-Canales  _et al._ (2012),
Nitrogen (N) Sun  _et al._ (2013), Oxygen (O) Sun  _et al._ (2012) and
Fluorine (F) Duan  _et al._ (2021), but we are unaware of any such work on Li.
In addition to laying the groundwork for any future TPa-experimental work on
Li, a first-principles search in Li is also of great interest to determine
whether the remarkable structural complexity of Li persists at multi-TPa
pressures.

In this work, we extend the pressure-temperature ($p$-$T$) phase diagram of Li
up to multiterapascal pressures, proposing eleven solid-state phase
transitions beyond the diamond $Fd\bar{3}m$ phase discovered in previous
searches Pickard and Needs (2009), in addition to two structures that are
stabilized at higher temperatures. We further calculate the full $p$-$T$
dependence of the solid-liquid melting line using ab-initio molecular dynamics
simulations, and provide a detailed analysis of the nature of bonding in Li
through extensive calculations of the electronic charge density at different
pressures.

##  II Computational Details

We used the ab-initio Random Structure Searching (AIRSS) package Pickard and
Needs (2006, 2011) to perform our structure search, with the first-principles
Density Functional Theory (DFT) code `CASTEP` Clark  _et al._ (2005)
performing the underlying electronic structure calculations. We performed
structure searches at $1$, $2.5$, $5$, $7.5$, $10$, $15$, $25$, $50$, $100$
and $150$ TPa, using randomly constructed unit cells containing between 1 and
64 atoms, and possessing between 1 and 48 randomly chosen symmetry operations.
Each search generated approximately $6,000$ relaxed structures, giving $\sim
60,000$ structures in total. Due to the appearance of a large number of
complex structures around $50$ TPa, we further constructed a machine-learned
potential specifically at this pressure using the Ephemeral Data Derived
Potential (EDDP) package Pickard (2022), which allowed us to search in larger
cells containing up to 256 atoms, and which generated $\sim 500,000$
structures. For both the structure searches and our final results presented
here, we constructed two all-electron ultrasoft pseudopotentials with cutoff
radii of $0.40$ Bohr (for calculations at $100$ TPa and above) and $0.60$ Bohr
(for all other calculations), requiring plane-wave cutoffs of $4.50$ keV and
$2.25$ keV respectively to converge relative energies to better than $0.1$ meV
per atom. We used Monkhorst-Pack k-point grids with a sampling spacing of
$2\pi\times 0.025$ Å-1. Full details of our calculations can be found in the
Supplemental Material sup .

##  III Results and Discussion

The $p$-$T$ phase diagram of Li, as calculated using the PBE exchange-
correlation functional Perdew  _et al._ (1996) and with ionic vibrations
included at the level of the harmonic approximation, is shown in Fig 1. We
find that vibrational contributions to the free energy are very important in
Li, observing a substantial variance in the vibrational energies of the
different structures - much larger than the static lattice enthalpy
differences between these structures - which are sufficient to substantially
change the phase transition sequence when compared with the static-lattice
picture. We note that similar observations have been made in Li before Ackland
_et al._ (2017); Ashcroft (1989), and it is known that the neglect of
vibrational effects in Li leads to an incorrect phase diagram Gorelli  _et
al._ (2012). Several structures that are the lowest in enthalpy at the static
lattice level become unfavourable when vibrations are included. Below $1.30$
TPa, at zero temperature, we find that Li adopts the diamond structure
($Fd\bar{3}m$ symmetry), in agreement with previous works Lv  _et al._ (2011);
Pickard and Needs (2009). At $1.30$ TPa, Li undergoes a symmetry-lowering
distortion via a soft optical phonon mode at the $\Gamma$-point to a structure
of $Imma$ symmetry with 2 atoms in the primitive cell. We then find further
transitions to structures of gradually increasing complexity, with phases of
$P4_{1}32$, $I\bar{4}2m$, $Fmm2$, $Cmce$ and $P\bar{6}2m$ symmetry at $2.32$
TPa, $6.43$ TPa, $8.50$ TPa, $15.9$ TPa and $18.5$ TPa respectively. The
$P4_{1}32$ structure has 4 atoms in its primitive cell, the $I\bar{4}2m$ and
$Fmm2$ structures have 10 atoms, and the $Cmce$ and $P\bar{6}2m$ phases
possess 14 atoms in their primitive cells.

At $39.1$ TPa, Li undergoes a transition to a remarkably complex structure of
$C2/c$ symmetry, with 46 (92) atoms in its primitive (conventional) cell. This
structure (denoted as $C2/c\ [46]$) comprises interpenetrating networks of Li
atoms, similar to phases adopted by Li at much lower pressures of several
hundred GPa Lv  _et al._ (2011); Pickard and Needs (2009). In fact, we find
that several highly complex polymorphs become very close to being lowest in
enthalpy around 50 TPa, and two of these structures, 22-atom $P6_{3}/m$ and
43-atom $P\bar{1}$, possess sufficient vibrational entropy to become
stabilised at finite temperatures within relatively small regions of the
$p$-$T$ plane; above 1500 K and in the pressure range 25-37 TPa (see Fig 1).
Given the appearance of these complex structures, we constructed a machine-
learned potential at $50$ TPa using the Ephemeral Data Derived Potential
(EDDP) package Pickard (2022). The use of a machine-learned potential allowed
us to perform structure searches in much larger cells and with a computational
expense orders of magnitude smaller than the equivalent DFT calculations. The
resulting search generated over half a million structures with up to 256 atoms
in the unit cell, but none were found to be lower in free energy than the
46-atom $C2/c$ in any part of the $p$-$T$ plane. We note that Wang et. al.
also used a machine-learned potential to investigate highly complex structures
of Li, albeit at much lower pressures Wang  _et al._ (2023). Similar to our
work here, the authors found many competitive structures but were unable to
find any $p$-$T$ region in which the new structures became lowest in free
energy. Eventually, Li transitions from $C2/c\ [46]$ to a 22-atom $C2/c$
structure ($C2/c\ [22]$) at $55.7$ TPa.

Structure | Pressure (TPa) | Interstitial Charges ($|e|$)
---|---|---
$Imma$ | 2.00 |  $-0.456$ (4e)
$P4_{1}32$ | 5.00 |  $-0.100$ (8c)
$I\bar{4}2m$ | 7.50 |  $-0.443$ (4c)
|  |  $-0.292$ (8i)
$Fmm2$ | 12.0 |  $-0.419$ (8b)
|  |  $-0.283$ (8c)
Table 1: Values and locations of the interstitial electronic charges for the
electride phases discovered in this work, as obtained from a Bader charge
analysis. Wyckoff positions are shown in brackets. Figure 2: Fermi-level
electronic density of states (eDOS) in units of states eV-1 Å-3 for the phases
discussed in this work. The grey vertical dashed lines mark the structural
phase transitions, and the dashed orange line is the corresponding free-
electron eDOS.

(a) Electronic density of states (eDOS) for the $I\bar{4}2m$ ($7.50$ TPa) and
$Cmce$ ($17.5$ TPa) structures.

(b) $Cmce$ bandstructure at $17.5$ TPa.

(c) $I\bar{4}2m$ bandstructure at $7.50$ TPa.

Figure 3: Electronic DOS and bandstructures for the $Cmce$ (non-electride) and
$I\bar{4}2m$ (electride) structures. In Figs. 3(b) and 3(c), the bands are
coloured red (blue) if they are below (above or cross) the Fermi level. Figure
4: The cubic $I\bar{4}2m$ structure, with the two interstitial charge species
shown as small red (4c) and blue (8i) spheres. Li atoms are shown in green.

Fig. 2 presents a summary of the key electronic properties of the structures
discovered in this work, and Fig. 3 specifically compares the electronic
properties of the $I\bar{4}2m$ and $Cmce$ phases. Notably, whilst the Fermi-
level eDOS $\cal{N}(\epsilon_{F})$ of Li is small and relatively constant up
to $\sim 16$ TPa, it nearly doubles in value upon transition to the $Cmce$
structure at $15.9$ TPa and increases almost linearly with pressure
thereafter. We find that this is not because the $Cmce$ structure has a
particularly large eDOS; in fact, it has $\cal{N}(\epsilon_{F})$ within $3\%$
of the corresponding free electron value. Rather, the phases below $15.9$ TPa
have unusually low $\cal{N}(\epsilon_{F})$ values. As an example, the
$I\bar{4}2m$ phase at $7.50$ TPa has $\cal{N}(\epsilon_{F})$ $=0.0270$ states
eV-1 Å-3, whereas the equivalent free electron value is
$\cal{N}(\epsilon_{F})$ $=0.0515$ states eV-1 Å-3 (a reduction of $48\%$).
This significant eDOS reduction has been noted before in lower-pressure phases
of Li Lv  _et al._ (2011); Pickard and Needs (2009), where Li effectively
becomes semiconducting. Whilst the $Fd\bar{3}m$, $Imma$, $P4_{1}32$,
$I\bar{4}2m$ and $Fmm2$ structures should not be described as semiconductors,
they should be considered as poor conductors relative to the $Cmce$ and
higher-pressure phases. This difference is highlighted in Fig. 3(a), which
shows the eDOS as a function of energy for the $I\bar{4}2m$ and $Cmce$ phases;
all structures below $15.9$ TPa exhibit a significant dip in the Fermi-level
eDOS similar to the $I\bar{4}2m$, whereas the phases above this pressure
feature no such dip and have relatively flat $\cal{N}(\epsilon_{F})$ near the
Fermi level. This behaviour can be clearly linked to a sharp change in the
bonding topology of Li with increasing pressure; phases below $15.9$ TPa, such
as $I\bar{4}2m$, feature significant interstitial accumulations of electronic
charge in spaces between atoms, forming ‘electrides’ with localized charge
that is unable to contribute to conduction. Several elements in the periodic
table also form electride structures under sufficient compression Miao and
Hoffmann (2014). These electride structures exhibit electronic bandstructures
with substantially flattened bands when compared to non-electride structures
(compare Figs. 3(b) and 3(c)). Conversely, the phases above $15.9$ TPa are not
electrides; they have no interstitial charge maxima, are metallically bonded,
and exhibit much more dispersive bands as would be expected for a crystal
under such substantial compression (again, see Fig. 3(b)). We find that even
up to pressures of $10$ petapascal (PPa), Li does not exhibit re-entrant
electride behaviour and remains metallically bonded. Table 1 shows the
magnitudes and Wyckoff positions of the interstitial electronic charges in the
electride phases as determined using a Bader analysis Tang  _et al._ (2009);
the $Fd\bar{3}m$, $Imma$, $P4_{1}32$, $I\bar{4}2m$ and $Fmm2$ phases all show
similar interstitial accumulations of charge (see Fig. 4). The amount of
localised interstitial charge per site ranges from $-0.456\ |e|$ of charge (4e
sites of the $Imma$ structure) to $-0.100\ |e|$ of charge (8c sites of the
$P4_{1}32$ structure).

In the pressure range $71.9$-$103$ TPa, we find that Li adopts the
incommensurate Ba-IV host-guest (HG) structure, which is assumed by several
elements across the periodic table under compression Chris J. Pickard (2010);
Nelmes  _et al._ (1999); Brown  _et al._ (2018); Arapan  _et al._ (2008);
McMahon  _et al._ (2006); Tsuppayakorn-aek  _et al._ (2018); Whaley-Baldwin
_et al._ (2021). The Ba-IV HG structure comprises a ‘host’ framework within
which chains of ‘guest’ atoms lie, and the $c$-axis periodicities of the host
and guest structures are incommensurate. We calculated the enthalpy as a
function of the HG ratio $\gamma=\frac{c_{H}}{c_{G}}$ at several pressures, in
order to determine the ideal ratio $\gamma_{0}$ as a function of pressure (see
Supplemental Material sup ). We find that the ideal HG ratio in Li decreases
very slowly with increasing pressure, changing by only $\sim~{}0.2\%$ over the
entire stability range of the HG phase; in contrast, the HG phase in high-
pressure elemental sulfur changes its ideal ratio over the stability range by
($\sim~{}12\%$) Whaley-Baldwin  _et al._ (2021). Beyond the HG phase, Li
transitions to a $C2$ structure with 12 atoms in its primitive cell at $103$
TPa, and eventually to the hexagonal close-packed (hcp) structure at $116$
TPa.

##  IV Conclusions

We have performed an extensive structure search in elemental Li in the
terapascal regime, uncovering 11 phase transitions beyond the previously-
predicted $Fd\bar{3}m$ diamond structure. The $p$-$T$ dependence of the phase
boundaries has been calculated at the level of the harmonic approximation, and
the solid-liquid melting line has been determined using ab-initio MD
simulations. We have found that in the TPa regime, Li adopts the phase
transition sequence $Fd\bar{3}m$ $\rightarrow$ $Imma$ $\rightarrow$ $P4_{1}32$
$\rightarrow$ $I\bar{4}2m$ $\rightarrow$ $Fmm2$ $\rightarrow$ $Cmce$
$\rightarrow$ $P\bar{6}2m$ $\rightarrow$ $C2/c\ [46]$ $\rightarrow$ $C2/c\
[22]$ $\rightarrow$ HG $\rightarrow$ $C2$ $\rightarrow$ hcp, with further
transitions to structures of $P6_{3}/m$ and $P\bar{1}$ symmetry induced by
temperatures of a few thousand K at pressures $25$-$37$ TPa. The $Fd\bar{3}m$,
$Imma$, $P4_{1}32$, $I\bar{4}2m$ and $Fmm2$ structures are poorly conducting
electrides, whereas the $Cmce$, $P\bar{6}2m$, $C2/c\ [46]$, $C2/c\ [22]$, HG,
$C2$ and hcp phases are normal metals with a Fermi-level eDOS within a few
percent of the corresponding free electron value.

##  V Acknowledgements

The computational resources for this project were provided by the Cambridge
Service for Data Driven Discovery (CSD3), and by the UK national high
performance computing service, ARCHER2, for which access was obtained via the
UKCP consortium and funded by EPSRC grant refs. EP/F036884 and EP/K013564/1.

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