# Transformer Attractors for Robust and Efficient End-to-end Neural
Diarization

###### Abstract

End-to-end neural diarization with encoder-decoder based attractors (EEND-
EDA) is a method to perform diarization in a single neural network. EDA
handles the diarization of a flexible number of speakers by using an LSTM-
based encoder-decoder that generates a set of speaker-wise attractors in an
autoregressive manner. In this paper, we propose to replace EDA with a
transformer-based attractor calculation (TA) module. TA is composed of a
Combiner block and a Transformer decoder. The main function of the combiner
block is to generate conversational dependent (CD) embeddings by incorporating
learned conversational information into a global set of embeddings. These CD
embeddings will then serve as the input for the transformer decoder. Results
on public datasets show that EEND-TA achieves 2.68% absolute DER improvement
over EEND-EDA. EEND-TA inference is 1.28 times faster than that of EEND-EDA.

Index Terms—  end-to-end neural diarization (EEND), EEND- EDA, Transformer
Attractors, EEND-TA, conversational summary vector

##  1 Introduction

Speaker diarization systems are utilized to identify different speakers and
their speech content in conversations. These conversations are often lengthy,
have an unpredictable number of speakers, and include a significant amount of
overlapping speech. Consequently, an effective diarization system needs to be
able to handle real-life scenarios involving multiple speakers engaged in
overlapping conversations [1, 2, 3]

Traditional diarization methods tend to rely on clustering-based approaches.
However, such methods are often ill-equipped to handle overlapping segments of
speech. This limitation and the reliance on multiple components within a
system increases the complexity associated with their deployment and
maintenance [2, 4, 5, 6].

End-to-end neural diarization (EEND) systems are capable of handling
overlapping speech by addressing it as a multi-label classification problem
[7]. Moreover, EEND systems simplify the deployment process as they only
require a single model, unlike the traditional approaches that involve
multiple components.

By design, the EEND architecture can only predict a predetermined number of
speakers, which is usually fixed to be a relatively small number due to the
combinatorics of Permutation Invariant Training [8, 9]. A natural question was
whether a single model could predict a potentially unlimited number of
speakers. A first breakthrough came with the introduction of the EEND-EDA
model [10]. The Encoder-Decoder Attractor (EDA) module uses an LSTM encoder-
decoder architecture to transform the sequence of frames into a variable-
length sequence of speaker attractors, which are then used to predict each
speaker’s speech activity.

Since the advent of EEND-EDA, most improvements have mostly ignored the EDA
module. These include changes to the encoder [10, 11], loss functions [12,
11], and improved post-processing techniques [13, 14]. Improvements to the EDA
module have mostly been limited to minor tweaks to the LSTM encoder-decoder
architecture. Since the only input to the LSTM decoder is the final state of
the encoder, several methods propose injecting more context into the decoder.
For example, Pan et al. [15] propose a Bahdanau-style attention mechanism [16]
to construct inputs for the LSTM decoder. Similarly, Broughton et al. [17]
propose constructing inputs to the LSTM decoder by using a learned
conversational summary vector.

This work presents EEND-TA, a transformer-based method to replace the LSTM-
based EDA module of EEND-EDA. This follows recent trends of replacing standard
neural network layers by attention mechanisms, as evidenced by the widespread
adoption of the Transformer architecture [18] in a wide range of fields, such
as natural language processing [19, 20, 21], computer vision [22],
reinforcement learning [23], and automatic speech recognition [24, 25].

The proposed Transformer Attractors (TA) module is composed of a Combiner
block and a Transformer decoder. The main function of the combiner block is to
generate conversationally dependent (CD) embeddings by incorporating learned
conversational information into a global set of embeddings. We then use these
CD embeddings as the input for the transformer decoder. Conversational
information is gathered throughout the encoder layers, as proposed in [17]. We
explore several functions for the combiner block. Our experimental results on
publicly available real conversations demonstrate that EEND-TA significantly
outperforms the current state-of-the-art.

There have been previous attempts at incorporating attention-based layers in
neural diarization. In [26], authors propose to use clustered embeddings as
input to a transformer decoder to generate attractors for each speaker. This
puts a lot of responsibility on the clustering method to find the correct
number of speakers and provide reasonable cluster centroids. It is unclear
whether this approach would work in practice, as all experiments are conducted
with a speaker counting oracle on simulated data. Perhaps closest to our setup
is the one presented in [12]. Instead of using a transformer decoder, authors
use a single cross-attention layer to generate a fixed number $N$ of
attractors. This makes the model more akin to an EEND model as opposed to
EEND-EDA, as it is missing a speaker counting component. Furthermore, their
improvements over the EEND-EDA baseline are marginal. In contrast, our method
is truly end-to-end, trained on the same objectives as EEND-EDA, and shows
noticeable improvement over EEND-EDA on real-life datasets. This is all
without any additional “bells and whistles”, such as iterative refinement
[26], intermediate attractors loss or self conditioning [12], each of which
could potentially further boost the performance of our model.

The remainder of the paper is structured as follows: Section 2 provides a
brief review of the related work. Section 3 introduces the proposed EEND-TA
method. Section 4 outlines the experimental setup and Section 5 presents the
results. We conclude our work in Section 6.

##  2 Background

In this section, we review the EEND-EDA model and give a brief introduction to
the conversational summary vector (CSV) enhancement for the EEND-EDA.

###  2.1 EEND-EDA

EEND models are the current state-of-the-art for end-to-end diarization [10,
3, 27, 28]. EEND-EDA is one such architecture that outperforms conventional
EEND methods and allows for diarization of an arbitrary number of speakers
[10, 29, 30, 11].

Given an acoustic input feature sequence $X\in\mathbb{R}^{D^{\prime}\times
T}$, EEND-EDA simultaneously estimates the likelihood of speech activity for
multiple speakers. Here, $X$ is a log-scaled Mel-filterbank feature sequence
with dimension $D^{\prime}$ and length $T$. $X$ is first encoded to produce
hidden embedding $E\in\mathbb{R}^{D\times T}$ by a Transformer-based network
[18, 24]. $E$ is then shuffled in the $T$ dimension for input to an LSTM-based
EDA module that predicts speaker-wise attractors
$A\in\mathbb{R}^{D\times(S+1)}$, where $S$ is the number of speakers.

Predictions $(\textbf{p}_{t})^{T}_{t=1}$ for multiple speakers’ speech
activities are estimated by a sigmoid function acting on the matrix product of
$A$ and $E$. At frame $t$, $\mathbf{p}_{t}:=[p_{t,1},...,p_{t,S}]\in(0,1)^{S}$
are the predicted probabilities for $S$ speakers’ speech activities.

Diarization loss is computed during training with a permutation invariant
objective [8], by taking the minimum loss for all possible speaker assignments
between the predicted probabilities and ground truth labels:

$\mathcal{L}_{\mathrm{diar}}=\frac{1}{TS}\min_{i\text{ }\in\text{
perm}(1,...,S)}\sum^{T}_{t=1}H(\mathbf{p}^{i}_{t},\mathbf{y}_{t}),$ (1)

where $\operatorname{perm}(1,\dotsc S)$ is the symmetric group over $S$
speakers, and $\mathbf{p}^{i}_{t}$ are the permuted predictions at frame $t$
for any $i\in\operatorname{perm}(1,\dotsc,S)$. Ground truth labels are denoted
by $\mathbf{y}_{t}:=[y_{t,1},...,y_{t,S}]\in\\{0,1\\}^{S}$, where $y_{s,t}=1$
and $y_{s,t}=0$ denotes whether speaker $s$ is active or not at frame $t$. The
function $H(\cdot,\cdot)$ denotes binary cross entropy.

The EDA module consists of an encoder $\operatorname{LSTM}^{\text{enc}}$ and a
decoder $\operatorname{LSTM}^{\text{dec}}$. The final hidden and cell states
of the encoder are used to initialize the hidden and cell states of the
decoder. At each iteration $s=1,\dotsc,S+1$,
$\operatorname{LSTM}^{\text{dec}}$ takes an input zero vector and generates a
speaker-wise attractor:

$\displaystyle\textbf{h}^{\text{dec}}_{s},\textbf{c}^{\text{dec}}_{s}$
$\displaystyle=\operatorname{LSTM}^{\text{dec}}(\textbf{0},\textbf{h}^{\text{dec}}_{s-1},\textbf{c}^{\text{dec}}_{s}),$
(2)

where attractor $\mathbf{a}_{s}\in\mathbb{R}^{D}$ for speaker $s$ corresponds
to the hidden state $\textbf{h}^{\text{dec}}_{s}$ at that step.

Speaker count is modeled by a probability vector
$q_{s}=\sigma(\operatorname{FF}(a_{s}))$, obtained from each attractor
$a_{s}$. Here $\operatorname{FF}$ is a fully connected feed-forward layer, and
$\sigma$ is the sigmoid function. During inference, we generate a new
attractor $a_{i}$ whenever $q_{i-1}>0.5$. Therefore, when training on an $S$
speaker recording, we want that $q_{1},\dotsc,q_{S}=1$, and $q_{S+1}=0$. This
translates to the training objective

$\displaystyle\mathcal{L}_{\mathrm{exist}}$
$\displaystyle=\frac{1}{S+1}H(\mathbf{l},\textbf{q}),$ (3)

where $\mathbf{l}:=[1,...,1,0]^{\top}$ is a vector with $S$ ones and 1 zero.

The model is optimized during training with the full objective:

$\mathcal{L}=\mathcal{L}_{\mathrm{diar}}+\lambda\mathcal{L}_{\mathrm{exist}},$
(4)

where $\lambda$ is a hyperparameter.

###  2.2 Conversational Summary Vector (CSV) for EEND-EDA

One of the limitations of the EEND-EDA is that the LSTM decoder fully relies
on the LSTM encoder to create a summary of the entire recording as in the form
of hidden and cell states. The main idea behind the CSV is to enhance the
information flow to the LSTM decoder by providing an additional summary.

Drawing on inspiration from the BERT classification token [CLS] in natural
language processing [19], a method to generate a CSV for EEND-EDA was recently
suggested [17]. Here, a CSV is learned in the encoder similar to BERT, and is
seen by the $\operatorname{LSTM}^{\text{dec}}$ during attractor generation.
This was shown to outperform conventional EEND-EDA methods.

The Conformer encoder responsible for generating output embedding $E$ is
modified to additionally output a CSV $\hat{\textbf{u}}$ from the original
input feature sequence:

$\hat{\textbf{u}},E=\text{enc}(X),$ (5)

where $\hat{\textbf{u}}\in\mathbb{R}^{D\times 1}$ and $E\in\mathbb{R}^{D\times
T}$. Zero vectors input to $\operatorname{LSTM}^{\text{dec}}$, as shown in
equation 2, are replaced with $\hat{\textbf{u}}$ such that each speaker-wise
attractor is conditioned on the CSV:

$\displaystyle\textbf{h}^{\text{dec}}_{s},\textbf{c}^{\text{dec}}_{s}$
$\displaystyle=\operatorname{LSTM}^{\text{dec}}(\hat{\textbf{u}},\textbf{h}^{\text{dec}}_{s-1},\textbf{c}^{\text{dec}}_{s}),$
(6)

for $s=1,\dotsc,S+1$.

The CSV is constructed in a modified Conformer encoder by prepending a
learnable special token to the feature sequence $X$. Thus the first frame of
every input sequence to the encoder corresponds to the CSV, and the special
token has its embedding vector updated by the training objective. To maintain
a global representation, the CSV bypasses convolutional modules of the
Conformer encoder.

In [17], authors show that incorporating a CSV to EEND-EDA delivers superior
results. Most of the improvements come from recordings with four or more
speakers. Furthermore, when using CSV, one can train an EEND-EDA model more
robustly with increasing lengths of input utterances. This suggests that the
vanilla EEND-EDA model has a tendency of forgetting certain speakers, and that
the LSTM encoder alone is not enough to capture a rich enough representation
of the input recording. The addition of the CSV as an input to the LSTM
decoder alleviates that issue, as it injects context learned inside the
Conformer encoder, bypassing the LSTM encoder.

Fig. 1: Illustration of the EEND-TA high-level architecture. EEND-TA contains
a Conformer encoder and Transformer Attractors (TA) module. TA consists of a
Combiner and a Transformer Decoder.

##  3 EEND-TA : Transformer Attractors for EEND

In this work, we introduce an EEND model that is free from LSTMs. Concretely,
we propose replacing the LSTM-based EDA with a module called Transformer
Attractors (TA) that is based on transformers. The TA module consists of a
Combiner block and a Transformer decoder. The role of the combiner block is to
prepare a set of inputs for the transformer decoder. This is done by injecting
conversational information gathered in the Conformer encoder into a learnable
global set of embeddings. The inputs are then fed to the transformer decoder,
using cross-attention with the framewise speech embeddings to output speaker-
wise attractors.

Here we present the architecture in detail. The architecture of EEND-TA is
depicted in Figure 1. The TA module takes as an input framewise speaker
embeddings $E\in\mathbb{R}^{D\times T}$ and a CSV
$\hat{\mathbf{u}}\in\mathbb{R}^{D\times 1}$, and outputs $S+1$ attractors
$A=(\mathbf{a}_{i})_{i=1}^{S+1}$, and $S+1$ speaker existence probabilities
$q_{1},\dotsc,q_{S+1}$. As in EEND-EDA, if there are $S^{\prime}$ speakers in
the conversation, we keep only the first $S^{\prime}$ attractors and
$S^{\prime}+1$ existence probabilities in the loss computations during
training. During inference, we estimate the number of speakers $\hat{S}$, and
output the first $\hat{S}$ attractors
$\mathbf{a}_{1},\dotsc,\mathbf{a}_{\hat{S}}$.

###  3.1 Combiner

As show in Figure 1, the transformer decoder’s first layer receives inputs
$I_{0}$ that are computed using a combiner function. This function takes the
CSV $\hat{\mathbf{u}}$ and the learnable global embeddings
$G\in\mathbb{R}^{D\times(S+1)}$ as inputs. Its purpose is to transform a set
of global embeddings into conversationally dependent embeddings. We conduct
experiments with various combiner functions.

The first combiner function is

$\displaystyle\phi_{\text{add}}(\hat{\mathbf{u}},G)=\hat{\mathbf{u}}+G,$ (7)

where the CSV is added to each global embedding.

Another intuitive function is the element-wise multiplication between CSV and
global embeddings

$\displaystyle\phi_{\text{mult}}(\hat{\mathbf{u}},G)=\hat{\mathbf{u}}*G,$ (8)

where $*$ denotes element-wise multiplication.

Similar to some well-known speaker adaptation mechanisms [31, 32], we can
learn conversational contributions for global embeddings by treating
$\hat{\mathbf{u}}$ as a gating function for the entries of $G$:

$\displaystyle\phi_{\text{amp}}(\hat{\mathbf{u}},G)=\alpha\cdot\sigma(\hat{\mathbf{u}})*G,$
(9)

where $\alpha>0$ is the amplitude parameter and $\sigma$ is the sigmoid
function.

Finally, to evaluate the importance of conversation-dependent inputs to the
transformer decoder, we experiment with removing the combiner altogether,
therefore ignoring the conversational summary vector. Results for these
experiments can be found in Table 4.

###  3.2 Transformer decoder

We use a standard transformer decoder architecture [18] with $N_{D}$ blocks
and no positional encoding. Each block is fed a set of $S+1$ inputs
$I_{i}\in\mathbb{R}^{D\times(S+1)}$, which — together with the encoder output
$E\in\mathbb{R}^{D\times T}$ — produce the $S+1$ attractors
$A\in\mathbb{R}^{D\times(S+1)}$. To summarize, each transformer decoder block
performs the following computation

$\displaystyle\begin{split}Z&=\operatorname{LN}(\operatorname{MHA}(I_{i},I_{i},I_{i})+I_{i})\\\
Z^{\prime}&=\operatorname{LN}(\operatorname{MHA}(Z,E,E)+Z)\\\
I_{i+1}&=\operatorname{LN}(\operatorname{FF}(Z^{\prime})+Z^{\prime}),\end{split}$
(10)

where $\operatorname{MHA}(Q,K,V)$ is a multi-head attention layer with query
$Q$, key $K$, value $V$, $\operatorname{LN}$ is Layer Norm [33], and
$\operatorname{FF}$ is a feedforward network. The output attractors
$\mathbf{a}_{1},\dotsc,\mathbf{a}_{S+1}$ are obtained from the columns of
$I_{N_{D}}$. Speaker existence probabilities are computed as
$q_{i}=\sigma(\operatorname{FF}(a_{i}))\in(0,1)$, where is the sigmoid
function.

###  3.3 Discussion

We posit that the transformer architecture is a more natural choice for
attractor generation tasks than the LSTM-based EDA. Since LSTM layers are
inherently order dependent, special care needs to be taken when presenting the
embeddings to the EDA module. The EDA module performs much better when the
framewise embeddings are input in shuffled order [10]. We hypothesize that
this is due to the LSTM cell forgetting speakers appearing earlier in the
sequence, and shuffling makes all speakers appear more uniformly throughout
the sequence. The transformer decoder overcomes these limitations because it
takes into account all framewise embeddings simultaneously and remains
unaffected by their order. There are also computational benefits, as the
transformer decoder generates all attractors in parallel, compared to the
LSTM’s sequential frame encoding and speaker decoding, see Table 7.

We note that contrary to models in the EEND-EDA family, the maximum number of
speakers that EEND-TA can predict is fixed to $S$. While this might seem like
a limitation, in practice the situation is slightly more nuanced. Despite
being able to predict an unlimited number of speakers, EEND-EDA is empirically
shown to be able to predict at most the number of speakers seen in training,
with diarization performance dropping dramatically when predicting higher
numbers [10]. Thus, due to the scarcity of diarization training data for a
high number of speakers, EEND-EDA based models are essentially capped to a
certain number of speakers. This is further evidenced by the use of other
techniques such as iterative inference [10, 14] or clustering [13] when the
number of speaker grows too large. These techniques can be readily used with
EEND-TA when inferencing for a higher number of speakers.

##  4 Experiments

###  4.1 Datasets

We use mixtures from the LibriSpeech corpus [34] for all the pre-training
phases. These mixtures are simulated using Algorithm 1 outlined in [10].
Models are first pre-trained with 100,000 simulated mixtures of 2 speakers
with average interval parameter $\beta=2$. This $\beta$ is a parameter that
controls the silence duration and overlap ratio [10]. The second phase of the
pre-training uses 400,000 simulated mixtures of 1,2,3,4 speakers with
$\beta=2,2,5,9$, respectively.

Table 1: Summary of datasets used during finetuning and inference.

Dataset |  No. of speakers |  Finetuning Files |  Inference Files
---|---|---|---
| DIHARD III
---
VoxConverse
MagicData-RAMC
AMI MIX
AMI SDM1
| 1 - 10
---
1 - 21
2
3 - 5
3 - 5
| 203
---
216
289
136
135
| 259
---
232
43
16
16

During the fine-tuning stage, we use a number of real public datasets: DIHARD
III [35], VoxConverse [2], MagicData-RAMC [36], AMI MIX [37] and AMI SDM1
[37]. This creates a combined training set with a total number of 979 files.
For our evaluations, test sets from each of the above corpora are used. The
details of datasets used during fine-tuning and inference are summarized in
Table 1.

###  4.2 Experimental Setup

Our EEND-TA and EEND-EDA models consist of 4 Conformer [24] blocks with 256
hidden units. Each Conformer block has four attention heads and feed-forward
layers with 1024 hidden units. No positional encoding is used. The input
features are 23-dim log Mel-filterbanks extracted with a window size of 25 ms
and a window shift of 10 ms. Unless specified, the input length of the
training utterances are set to 5000 frames. Throughout we have $D=256$ and
$S=4$, i.e. our model can predict at most 4 speakers.

All the models are first pre-trained with 2-speaker simulated mixtures for 100
epochs, followed by mixtures up to 4 speakers for 25 epochs. The Noam
scheduler [18] with 100,000 warm-up steps are used during pre-training. During
fine-tuning, models are updated for 1000 epochs. We use a fixed learning rate
of $5\cdot 10^{-5}$ with the Adam optimizer [38]. The gradient cutting for
attractor existence loss is applied before the EDA to avoid back-propagation
to the preceding Conformer encoder. All the models are trained to output four
dominant speakers due to time complexity of the permutation-invariant loss
(PIT) [10].

We evaluate all the models with a diarization threshold of 0.5 and attractor
existence threshold of 0.5. The reported DER is based on a collar of 0.0. To
obtain the inference model, we average the 10-best checkpoints based on
validation DER.

##  5 Results

Table 2:  Diarization Error Rate (DER) % for various EEND baselines across all
test datasets combined.

Model | NS1 | NS2 | NS3 | NS4 | NS1 to NS4
---|---|---|---|---|---
EDA | 7.81 | 12.83 | 19.67 | 27.21 | 17.45
EDA + CSV | 10.37 | 12.78 | 20.35 | 25.62 | 17.13

In Table 2, we report the results for EEND-EDA baseline and EEND-EDA with CSV.
EEND-EDA performance is significantly better than the EEND-EDA with CSV for
single speaker test recordings (NS1). However, EEND-EDA with CSV outperforms
the EEND-EDA baseline by 0.32% absolute DER improvement for recordings from
one to four active speakers (NS1 to NS4). Most of the gains of EDA with CSV
are coming from four active speaker test recordings (NS4) which is a 1.59%
absolute DER improvement. This shows that CSV enhancement to EDA facilitates
remembering more speakers.

Table 3:  DER (%) for different numbers of Transformer Decoder layers.

# Layers | NS1 | NS2 | NS3 | NS4 | NS1 to NS4
---|---|---|---|---|---
1 | 7.13 | 11.87 | 17.29 | 25.98 | 16.30
2 | 7.75 | 11.82 | 18.55 | 24.34 | 15.93
3 | 8.61 | 11.47 | 17.80 | 22.98 | 15.30
4 | 8.72 | 11.55 | 18.90 | 22.83 | 15.40

Next, we conduct experiments using EEND-TA. Specifically, we investigate the
number of decoder layers for the Transformer decoder. In all the experiments
mentioned in Table 3, we utilize the combiner function $\phi_{\text{add}}$ as
stated in Equation 7. It shows that the optimal performance (15.30%) is
achieved when we employ 3 decoder layers. When compared to the EEND-EDA
baseline, there is a 2.15% absolute DER improvement. Moreover, the EEND-TA
with 3 decoder layers performs better than the EEND-EDA with CSV, with a 1.83%
absolute DER improvement. Therefore, for the remainder of our experiments, we
will use 3 decoder layers with EEND-TA.

Table 4:  DER (%) for various Combiner functions. The function “None” means
the combiner is skipped.

Combiner Func. | NS1 | NS2 | NS3 | NS4 | NS1 to NS4
---|---|---|---|---|---
None | 8.59 | 11.40 | 17.69 | 23.67 | 15.46
$\phi_{\text{add}}$ | 8.61 | 11.47 | 17.80 | 22.98 | 15.30
$\phi_{\text{multi}}$ | 7.51 | 11.40 | 17.97 | 23.83 | 15.48
$\phi_{\text{amp}},_{\alpha=0.8}$ | 8.66 | 11.55 | 17.64 | 22.92 | 15.31
$\phi_{\text{amp}},_{\alpha=1.0}$ | 8.87 | 11.42 | 17.76 | 21.25 | 14.77
$\phi_{\text{amp}},_{\alpha=3.0}$ | 7.96 | 11.43 | 18.61 | 24.02 | 15.62
$\phi_{\text{amp}},_{\alpha=5.0}$ | 8.48 | 11.40 | 18.16 | 22.80 | 15.23
$\phi_{\text{amp}},_{\alpha=10.0}$ | 8.00 | 11.47 | 17.28 | 22.47 | 15.07
$\phi_{\text{amp}},_{\alpha=15.0}$ | 8.75 | 11.41 | 17.88 | 23.25 | 15.36

We present the results for all the combining functions in Table 4. The best
performance is achieved with the $\phi_{\text{amp}},_{\alpha=1.0}$ function.
That combiner function will be used for all subsequent experiments. We also
study the significance of incorporating the CSV into the inputs to the
transformer decoder by comparing with a model where the CSV is ignored. We
observe a decline in performance when ignoring the CSV for the NS4 test sets,
with the error rate increasing to 23.67%, higher than almost every other
combiner function. This highlights the importance of conversational summaries
for recordings that involve more than just a few speakers. These findings
align with the conclusions drawn in [17].

Table 5:  Summary of the results for the best models for each technique. Attractor | NS1 to NS4 | NS1 to NS9
---|---|---
EDA | 17.45 | 21.68
EDA + CSV | 17.13 | 21.34
TA | 14.77 | 18.78

In Table 5, we summarize the results for each model. Clearly EEND-TA
outperforms by a significant margin both EEND-EDA and EEND-EDA with CSV. When
considering the combined test sets from NS1 to NS4, EEND-TA demonstrates
absolute DER improvements of 2.68% and 2.36% over EEND-EDA and EEND-EDA with
CSV, respectively.

For completeness, we also present the results for the combined test sets up to
9 speakers. For EEND-EDA (+ CSV), we let the model predict as many speakers as
necessary until the speaker existence probability drops below 50%. The EEND-TA
model however is fixed to predicting a maximum of 4 speakers. Remarkably, we
observe considerably better overall performance of the EEND-TA, despite having
a hard limit on the number of predicted speakers. This observation brings
further evidence to the “empirical upper limit” of the EEND-EDA family of
models: while the number of speakers predicted is a priori uncapped, it is
more beneficial _not_ to make the extra speaker prediction if they are of poor
quality.

Table 6: DER (%) breakdown for each test set with EEND-TA’s relative
improvements over EEND-EDA.

Dataset | Attractor | NS1 to NS4 | Rel. Improvement
---|---|---|---
DIHARD III | EDA | 14.07 | -
TA | 12.93 | 8.10%
VoxConverse | EDA | 15.75 | -
TA | 9.89 | 37.2%
MagicData-RAMC | EDA | 14.45 | -
TA | 13.58 | 6.02%
AMI Mix | EDA | 19.85 | -
TA | 17.88 | 9.92%
AMI SDM1 | EDA | 32.24 | -
TA | 24.64 | 23.55%

In Table 6 we show that EEND-TA shows significant improvement across all test
sets, demonstrating the generalizability of our approach. It is worth noting
the substantial relative improvements of 37.2% and 23.55% on VoxConverse and
AMI SDM1 respectively. The smaller relative gains in the MagicData-RAMC
dataset are due to it only consisting of conversations between 2 speakers. In
contrast, VoxConverse includes more recordings with higher number of speakers,
and EEND-TA shows the highest relative improvement on that dataset. It is
worth noting that the AMI SDM1 data is recorded in far-field conditions. This
suggests the possibility that EEND-TA is more robust to far-field conditions
compared to EEND-EDA.

Table 7:  Comparison of inference throughput of the baselines and EEND-TA. Attractor | Speed Up | # Params(M)
---|---|---
EDA | 1.00x | 8.1
EDA + CSV | 0.94x | 8.1
TA | 1.28x | 10.2

Next, we examine the inference efficiency of the EEND-TA compared to the
baselines. Table 8 reveals that EEND-TA can process audio 1.28 times faster
than EEND-EDA. This behavior is notable considering that EEND-TA has more
parameters (10.2 Million) than EEND-EDA (8.1 Million). We attribute the
increase in performance to the TA module being able to predict all attractors
in parallel, whereas the LSTM needs to process each frame and each speaker
sequentially. We further note that the addition of the transformer decoder
does not increase computational or space complexity. The framewise self-
attention layers in the Conformer encoder each have complexity
$\mathcal{O}(T^{2})$. In the transformer decoder, the self-attention layers
are only $\mathcal{O}(S^{2})$, whereas the cross-attention layers are
$\mathcal{O}(TS)$. Since $T$ is usually orders of magnitude larger than $S$
(in our case $T=500$, $S=4$), the computational and space complexity are
dominated by the encoder part.

Table 8: DER (%) results when various lengths of input is used during fine-
tuning.

Model | Length | NS1 | NS2 | NS3 | NS4 | NS1 to NS4
---|---|---|---|---|---|---
EEND-EDA | 50s | 7.81 | 12.83 | 19.67 | 27.21 | 17.45
100s | 8.11 | 12.56 | 20.61 | 24.93 | 16.72
150s | 7.89 | 13.03 | 19.76 | 25.55 | 17.08
200s | 7.59 | 11.88 | 19.41 | 25.51 | 16.37
EEND-TA | 50s | 8.87 | 11.42 | 17.76 | 21.25 | 14.77
100s | 7.23 | 11.35 | 17.49 | 20.47 | 14.39
150s | 9.26 | 11.49 | 16.84 | 20.29 | 14.46
200s | 6.68 | 11.44 | 17.87 | 18.91 | 13.99

Finally, in Table 8, we compare the performance of EEND-TA and EEND-EDA when
trained with long utterances. Both EEND-EDA and EEND-TA show consistent
improvements as we increase the training length. Specifically, EEND-EDA’s
performance improves from 17.45% to 16.37%. However, even with only 50 seconds
of training, EEND-TA outperforms EEND-EDA with 200 seconds of training. For
EEND-TA, performance improves from 14.77% to 13.99% as we increase the
training utterance length. Notably, when we increase the training utterance
length, EEND-EDA and EEND-TA report relative improvements over NS4 test sets
of 6.25% and 11.0%, respectively. This further demonstrates that EEND-TA
extracts speakers more accurately. The best EEND-TA model achieves a 2.38%
absolute DER reduction compared to the best EEND-EDA model. Increasing the
training utterance length further is impractical due to hardware limitations.

##  6 Conclusion

In this work we propose EEND-TA, a novel end-to-end diarization model. The
model uses a transformer decoder to predict speaker attractors, combining
conversationally dependent inputs with the encoded framewise memory in a
cross-attention layer. This is in stark contrast with traditional, LSTM-based
EEND-EDA models, whose only memory of the acoustic information is a collapsed
representation obtained from the LSTM encoder. Similarly to the EEND-EDA
model, we show that performance of the EEND-TA model can be improved by
injecting a conversational summary vector, which aggregates context from deep
in the Conformer encoder. We show that our approach outperforms EEND-EDA, with
major improvements in recordings with higher number of speakers and far-field
recordings. EEND-TA also enjoys improved accuracy when trained with longer
sequences. Despite our best model having slightly more parameters than the
best EEND-EDA model, it is computationally faster due to the inherently
parallel nature of attention mechanisms.

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# Exomol line list - NaO

G. B. Mitev,1 S. Taylor,1 Jonathan Tennyson,1 S. N. Yurchenko,1 A. A.
Buchachenko,2 A. V. Stolyarov3
1 Department of Physics and Astronomy, University College London, Gower St,
London WC1E 6BT, UK
2 Skolkovo Institute of Science and Technology, Skolkovo Innovation Center,
Moscow 121205, Russia
3 Department of Chemistry, Lomonosov Moscow State University, Moscow, 119991,
Russia

(Accepted XXX. Received YYY; in original form ZZZ)

# ExoMol molecular line lists – XLIII: Rovibronic transitions corresponding to
the close-lying X ${}^{2}\Pi$ and A ${}^{2}\Sigma^{+}$ states of NaO

G. B. Mitev,1 S. Taylor,1 Jonathan Tennyson,1 S. N. Yurchenko,1 A. A.
Buchachenko,2 A. V. Stolyarov3
1 Department of Physics and Astronomy, University College London, Gower St,
London WC1E 6BT, UK
2 Skolkovo Institute of Science and Technology, Skolkovo Innovation Center,
Moscow 121205, Russia
3 Department of Chemistry, Lomonosov Moscow State University, Moscow, 119991,
Russia

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The sodium monoxide radical (NaO) is observed in night glow in the Earth’s
mesosphere and likely has astronomical importance. This study concerns the
optical transitions within the ground X ${}^{2}\Pi$ state and to the very low-
lying ($T_{\rm e}\approx 2000$ cm-1) excited A ${}^{2}\Sigma^{+}$ state. A
line list consisting of rovibronic term values, allowed electric dipole
transitions, Einstein coefficients, and partition functions for varying
temperature are produced due to a variational solution of the coupled-channel
Schrödinger equations using the program Duo. MRCI ab initio calculations
characterising the potential energy curves of the both states, spin-orbit and
$L$-uncoupling non-adiabatic matrix elements, as well as permanent and
transition dipole moments were integral in formation of the final
deperturbation model. Ab initio potential energy curves are represented in the
analytical Extended-Morse-Oscillator form and refined, along with the spin-
orbit and $L$-uncoupling functions, by least-squares fitting to the available
spectroscopic data. The input experimental data consisted of pure rotational
transitions within the fine structure components of the X ${}^{2}\Pi$ state
for $v^{\prime\prime}\in[0,3]$ vibrational levels as well as the rovibronic A
${}^{2}\Sigma^{+}(v^{\prime}=0)\leftarrow$ X ${}^{2}\Pi(v^{\prime\prime}=0)$
transitions, both with limited coverage over rotational excitation. The lack
of data detailing the vibrational structure of the X and A states prompts a
request for further experimental study of higher excited levels which would
provide a robust spectroscopic model. The NaO NaOUCMe line list is available
via www.exomol.com and the CDS database.

###### keywords:

keyword1 – keyword2 – keyword3

††pubyear: YYYY††pagerange: Exomol line list - NaO–Exomol line list - NaO

##  1 Introduction

Air-glow, sometimes referred to as night-glow, was first reported by Slipher
(1929) and has been identified as the emission of Na D-lines. It is proposed
that these lines arise from the Chapman mechanism (Chapman, 1939; Sarkhel et
al., 2010; Plane et al., 2012) with some modifications suggested in light of
modern measurements (Slanger et al., 2005; Sarkhel et al., 2010), as given by

$\textrm{Na}+\textrm{O}_{3}\xrightarrow{k_{1}}\textrm{NaO}+\textrm{O}_{2}$ (1)
$\displaystyle\textrm{NaO}+\textrm{O}$ $\displaystyle\xrightarrow{\alpha
k_{2}}\textrm{Na}(^{2}P)+\textrm{O}_{2}$ (2)
$\displaystyle\xrightarrow{(1-\alpha)k_{2}}\textrm{Na}(^{2}S)+\textrm{O}_{2}$
(3)
$\textrm{Na}(^{2}P)\rightarrow\textrm{Na}(^{2}S)+h\nu(\textrm{D}_{1},\textrm{D}_{2}),$
(4)

where $k_{1,2}$ are the temperature-dependent rate constants, $\alpha$ is the
branching ratio, and $\textrm{D}_{1,2}$ are the D-line transition energies.

This mechanism suggests that Na atoms, which are assumed to be ablated from
meteors, react with O3 in the atmosphere to form NaO, re-reacting with O atom
to reform Na. Long lasting visible meteor trails have been attributed to the
production of electronically excited Na by in the said reaction chain (Kolb &
Elgin, 1976; Joo et al., 1999). In the Chapman mechanism, the rate limiting
reaction is the formation of NaO, see Eq. (1). As this reaction takes place in
the mesosphere, the variation of mesospheric ozone concentration affects Na
air-glow intensity. As altitude profiles of ozone are available, ozone
concentration variation can be used for deeper study into Sodium air-glow and
subsequently NaO. (Plane, 2003; Sarkhel et al., 2010).

The experimental spectroscopic data related to optical transitions of the gas-
phase NaO molecules is very sparse. Yamada et al. (1989) provide comprehensive
microwave data on the pure rotational transitions of the ground X${}^{2}\Pi$
electronic state for the lowest vibrational $v^{\prime\prime}=0,1,2,3$ levels.
However, no experimental data on ro-vibrational transitions are available, and
hence, the separations between vibrational levels are not yet experimentally
determined. Some information is available on the B ${}^{2}\Pi\rightarrow$ X
${}^{2}\Pi$ emission spectrum, however this is minimal, unassigned with no
characterisation of the B state (Pfeifer & Gole, 1984; Pugh et al., 1996),
which sometimes is assigned as C ${}^{2}\Pi$ state (Langhoff et al., 1991).
The only experimental observation of the lowest excited A ${}^{2}\Sigma^{+}$
state is due to Joo et al. (1999) who could assign a handful of the low
rotational levels $J=[4.5,19.5]$ belonging to the rovibronic A
${}^{2}\Sigma^{+}(v^{\prime}=0)-$ X ${}^{2}\Pi(v^{\prime\prime}=0)$
transitions.

The electronic structure of NaO demonstrates pronounced ionic inter-atomic
bond leading to large permanent dipoles as well as the close-lying ground X
${}^{2}\Pi$ and first excited A ${}^{2}\Sigma^{+}$ state converging to
different dissociation thresholds. The low excitation energy of the A
${}^{2}\Sigma^{+}$ state means that its spin-allowed A–X electronic transition
lies in the infrared, largely in the 3 to 5 $\mu$m region. From the
theoretical viewpoint, the A and X states are expected to be strongly coupled
to each other due to spin-orbit and electronic-rotational intramolecular
perturbations. Ab initio potential energy curves (PECs) and permanent dipole
moments are available for both lowest electronic states (Buenker & Liebermann,
2009; Langhoff et al., 1991; Soldán et al., 1999). However, the corresponding
non-adiabatic coupling matrix elements are not known so far.

In this paper we present new ab initio PECs for both X ${}^{2}\Pi$ and A
${}^{2}\Sigma^{+}$ states of NaO accompanied by spin-orbit and $L$-uncoupling
non-adiabatic matrix elements which we tune to the limited available
spectroscopic data. Finally, a rovibronic line list for NaO is simulated to be
suitable for modeling spectra up to temperature 2500 K. This line list is
constructed as part of the ExoMol project (Tennyson & Yurchenko, 2012). We
note that a previous ExoMol line list for CaO (Yurchenko et al., 2016b),
constructed using similar methodology to that employed here, was used by
Berezhnoy et al. (2018) to assign a spectrum of CaO in the wake of the
Beneshov bolide at the height of 29 km above the Earth’s surface.

##  2 Method

###  2.1 Ab initio electronic structure calculation

The initial set of required potential energy curves (PECs), permanent and
transition dipole moments for the ground X${}^{2}\Pi$ and near-lying excited
A${}^{2}\Sigma^{+}$ states of NaO were obtained in the framework of ab initio
electronic structure calculations which were accompanied by the evaluation of
the relevant non-adiabatic matrix elements for the spin-orbit and Coriolis
(electron-rotational) coupling. Scalar relativistic effects are introduced by
means of the second-order Douglas-Kroll Hamiltonian (Reiher & Wolf, 2004). All
calculations were performed in the range of internuclear distance
$R\in[1.0,5.0]$ Å using the MOLPRO program suite (Werner et al., 2010).

For both Na and O atoms, aug-cc-pVQZ all electrons basis sets was used. The
optimized molecular orbitals (MO’s) were generated using the state-averaged
self-consistent field (SA-CASSCF) method (Werner & Knowles, 1985), taking the
lowest doublet (1-3)$\Pi$, (1)$\Sigma^{+}$, (1)$\Sigma^{-}$, and (1)$\Delta$
states with equal weights. Internally-contracted multi-reference configuration
interaction (MR-CISD) calculations (Knowles & Werner, 1992) followed, in which
impact of higher excitation on the correlation energy was taken into account
implicitly using the Davidson correction (Langhoff & Davidson, 1974). For both
SA-CASSCF and MR-CISD steps the active space consisted of $7a_{1}$, $3b_{1}$
and $3b_{2}$ orbitals corresponding to the point group symmetry $C_{2v}$,
while the lowest two $a_{1}$ orbitals were kept to be doubly occupied. The
full counterpoise correction (Boys & Bernardi, 1970) was accounted for each
state individually with the residual size-consistency error eliminated at
$R=20$ Å.

Both permanent and transition dipole moments as well as non-adiabatic
$L$-uncoupling matrix element were computed using the MR-CISD wave functions.
The same wave functions and the full Breit-Pauli operator were used to compute
the SOCs (Berning et al., 2000).

###  2.2 Fitting procedure

The adiabatic PECs ab initio calculated above in Sec. 2.1 were combined with
the relevant spin-orbit coupling (SOC) and $L$-uncoupling functions in order
to evaluate non-adiabatic rovibronic term values and corresponding multi-
channel wavefunctions for the mutually perturbed X ${}^{2}\Pi$ and A
${}^{2}\Sigma^{+}$ states of NaO. However, the pure ab initio estimates are a
priori and are not accurate enough to predict the rovibronic energy with the
spectroscopic accuracy required. For better agreement between predicted and
observed transitions, PECs and coupling curves are refined by making
constrained adjustments to the parameters which describe them (see below).
This is done by a least-squares fitting to experimental data and is performed
in Duo, an open-source Fortran 2009 program which provides variational
solutions to the coupled rovibronic Schrodinger equations for a general open-
shell diatomic molecule (Yurchenko et al., 2016a).

The input set of the experimental data taken from (Yamada et al., 1989)
consisted of pure rotational transition frequencies (with $\Delta v_{X}=0$ and
$v_{X}=0,1,2,3$) within up to $J^{\prime\prime}=16.5$. Joo et al. (1999)
provided rovibronic frequencies for the A – X $(0,0)$ band with up to
$J^{\prime\prime}=15.5$ together with relative absorption intensities. Joo et
al. also observed the $v^{\prime}-v^{\prime\prime}=1$ band suggesting an A
${}^{2}\Sigma^{+}$ state vibrational spacing of 498.9 cm-1. This spacing was
included in the present fit using an artificial
$A(v^{\prime}=1)-A(v^{\prime\prime}=0)$ line for
$J^{\prime}=J^{\prime\prime}=0.5$.

The ab initio point-wise PECs of both states treated were approximated using
the fully analytical Extended Morse Oscillator (EMO) form (Lee et al., 1999;
Yurchenko et al., 2016a; Le Roy, 2017):

$\displaystyle V_{\textrm{EMO}}(r)=T_{\rm e}+D_{\rm
e}\left[1-e^{-\beta(r)(r-r_{\rm e})}\right]^{2}$ (5)

where $T_{\rm e}$ is the electronic term, $D_{\rm e}$ is the dissociation
energy and $r_{\rm e}$ is the equilibrium distance. In contrast to the
conventional Morse potential, the $r$-dependent exponent coefficient
$\beta(r)$ in Eq. (5) is defined as the polynomial series

$\displaystyle\beta(r)=\sum^{N}_{i=0}\beta_{i}[y_{p}(r)]^{i};\quad
N=\begin{cases}N_{+}&\textrm{for}~{}~{}r>r_{\rm e}\\\
N_{-}&\textrm{for}~{}~{}r\leq r_{\rm e}\end{cases}$ (6)

with respect to the reduced coordinate

$\displaystyle y_{p}(r)=\frac{r^{p}-r_{\rm e}^{p}}{r^{p}+r_{\rm e}^{p}},$ (7)

which was first introduced by Šurkus et al. (1984).

At the first step, the EMO potentials (5) were least-squares fit to the
original ab initio PECs in Python. The electronic energy of the ground state
was fixed to zero, and both PECs were fit to fourth order in the $\beta(r)$
polynomial expansion (6). The parameter $p$ in Eq.(7) was fixed to 6 and 4 for
the ground and excited state, respectively.

At the second step, the resulting ab initio EMO PECs were refined by adjusting
to the experimental transition frequencies (Joo et al., 1999; Yamada et al.,
1989) where only $\beta_{0}$(X, A), $T_{\rm e}$(A), $r_{\rm e}$(X) and $r_{\rm
e}$(A) fitting parameters were varied. Furthermore, both diagonal, $A_{\rm
SO}(r)$, and off-diagonal, $\xi_{\textrm{\rm AX}}(r)$, spin-orbit coupling
(SOCs) functions as well as $L$-uncoupling function, $L_{\textrm{\rm AX}}(r)$,
between A ${}^{2}\Sigma^{+}$ and X ${}^{2}\Pi$ states were also fitted
simultaneously with the trial EMO PECs above. This was particularly done using
the facility in Duo to adjust curves using a morphing function, $f_{\rm{C}}$:

$\displaystyle f_{\textrm{C}}^{\textrm{morphed}}(r)=f_{\rm
C}(r)f^{ab}_{\textrm{c}}(r),$ (8)

where C is the coupling curve in question. $f^{ab}_{\textrm{c}}(r)$ are the
original ab initio point-wise functions interpolated by ordinary cubic splines
with so-called natural boundary conditions. The morphing function, $f_{\rm
C}(r)$, in Eq.(8), is tacitly assumed to be an analytical function of the
internuclear distance $r$, and it was modelled by the following linear
function

$\displaystyle f_{\rm C}(r)=B_{0}(1-y_{p}(r))+y_{p}(r)B_{\infty}$ (9)

of the Šurkus-like variable, $y_{p}(r)$, (see Eq. (7)), where the equilibrium
distance $r_{\rm e}$ is substituted for the fixed parameter $r_{0}=2.053$ Å .
The parameter $p=6$ and $B_{\infty}=1$ are also fixed for all morphing
functions while only $B_{0}$ parameter is kept to be variable.

Morphing (Eq: 8) was used as it avoids the need for an analytical
representation of the ab initio point-wise coupling curves by changing their
original shapes directly. This approach is, however, not normally applied for
PECs for the reason of difficulty to control the equilibrium parameter $r_{\rm
e}$, which is a value of interest not explicitly derivable from this morphing
form.

##  3 Results and Discussion

###  3.1 Potential energy curves, spin-orbit and $L$-uncoupling matrix
elements

The resulting ab initio adiabatic PECs obtained for the X- and A-states of NaO
are depicted on Fig. 1 together with their empirical (adjusted) counterparts.
The fitted parameters of the EMO potentials (Eq. (5)) for the both states
treated are collected in Table 2. The corresponding ab initio diagonal spin-
orbit function, $A_{\rm SO}(r)$, of the ground X-state as well as the off-
diagonal SOC, $\xi_{\rm AX}(r)$, and the $L$-uncoupling, $L_{\rm AX}(r)$,
functions are shown in Fig. 2. The relevant morphing parameters for these
functions are given in Table 2.

The fitted EMO PECs and coupling functions reproduce the input experimental
data with an overall root-mean-squares (RMS) error of $\approx
0.086~{}\textrm{cm}^{-1}$. This RMS is deceptively large due to the pronounced
discrepancy of the A ${}^{2}\Sigma^{+}$ artificial line, skewing the mean RMS,
see Table 1. Indeed, for the ground X ${}^{2}\Pi$ state, there is agreement
with the experiment on order of $\sim 0.006$ cm-1and for the
$\rm{A}(v^{\prime}=0)\leftarrow\rm{X}(v^{\prime\prime}=0)$ case there is
agreement on order $\sim 0.02$ cm-1.

The ab initio $L_{\rm AX}(r)$ function obeys Van Vleck’s hypothesis (Mulliken
& Christy, 1931) of _pure precession_ in the interval $r\in[1.3,3.7]$ Å :

$\displaystyle L_{\rm AX}(r)\approx\sqrt{L(L+1)-|\Lambda|(|\Lambda|\pm
1)}=\sqrt{2},$ (10)

where the total electronic angular momentum of molecule $L$ is equal to 1
while its projection on the internuclear axis is zero. Moreover, the
relationship

$\displaystyle\xi_{\rm AX}(r)\approx A_{\rm SO}(r)L_{AX}(r)/2\approx A_{\rm
SO}(r)/\sqrt{2}$ (11)

between diagonal and off-diagonal spin-orbit functions is valid as well.

The ab initio SO splitting of the ground X-state is found to be close to the
empirical $A_{\rm SO}$ value determined by Yamada et al. (1989) (see Table 3).
The reliability of the ab initio off-diagonal spin-orbit $L$-uncoupling
functions is indirectly confirmed by a good agreement of the
$\Lambda$-doubling constants ($p$, $q$) empirically obtained for the regular
perturbed levels of the ground state by Yamada et al. (1989) with their
theoretical counterparts roughly estimated for the lowest vibrational levels,
$v_{X}$, according to the relations (Lefebvre-Brion & Field, 2004; Pazyuk et
al., 1994):

$\displaystyle p=2\langle v_{X}|B\xi_{\rm AX}L_{\rm AX}/\Delta
U_{XA}|v_{X}\rangle$ (12) $\displaystyle q=2\langle v_{X}|[BL_{\rm
AX}]^{2}/\Delta U_{XA}|v_{X}\rangle$ (13)

where $B=1/2\mu R^{2}$, $\mu$ is the reduced molecular mass, $\Delta
U_{XA}=U_{X}-U_{A}$ is the difference of PECs, while $|v_{X}\rangle_{R}$ are
the vibrational wavefunctions of the ground state.

Figure 1: Empirical EMO (fitted) and original ab initio point-wise potential energy curves obtained for the X ${}^{2}\Pi$ and A ${}^{2}\Sigma^{+}$ states of NaO (upper panel); Distance dependent exponent coefficient for X ${}^{2}\Pi$ and A ${}^{2}\Sigma^{+}$ state EMO fits as normalised by $\beta_{0}=A_{0}$ (lower panel). Table 1: Root-mean squares errors $\Delta$(RMS) in cm-1 comparing fitted model to a: Yamada et al. (1989) and b: Joo et al. (1999) rotational transitions. $v$ | $\Delta_{\textrm{a}}$ | $\Delta_{\textrm{b}}$
---|---|---
0 | 0.002664 | 0.022501
1 | 0.002846 | 0.679901a
2 | 0.002758 |
3 | 0.017115 |
Ave. | 0.006345 | 0.351201
Table 2: PEC and morphing parameters obtained for the X ${}^{2}\Pi$ and A
${}^{2}\Sigma^{+}$ states of NaO. For the definition of the EMO parameters see
Eq. (5), and for morphing parameters see Eq. (9).
PEC Parameterisation
---
EMO Param. | X ${}^{2}\Pi$ | A ${}^{2}\Sigma^{+}$
$T_{\rm e}$ | 0 | 2029.147a
$r_{\rm e}$ | 2.05189a | 1.96723a
$D_{\rm e}$ | 21735.7 | 35496.4
${p}$ | 6 | 4
${N}_{-}^{b}$ | 2 | 2
${N}_{+}^{b}$ | 4 | 4
${\beta}_{0}$ | 1.50708327083484a | 0.989061691136872a
${\beta}_{1}$ | -0.04809897091909 | 0.181504940347642
${\beta}_{2}$ | 0.16743487156786 | 0.424100841906937
${\beta}_{3}$ | -0.95627676523677 | -2.926458322108940
${\beta}_{4}$ | 1.16624139895144 | 2.522897112151440
$B_{0}^{a}$ parameters used for morphed coupling functionsc
Function | $B_{0}$ |
$A_{\textrm{so}}$ | 1.0098 |
$\xi_{\textrm{AX}}$ | 0.6277 |
$L_{\textrm{AX}}$ | -0.7455 |

Figure 2: Upper: Diagonal, $A_{\rm SO}(r)$, and off-diagonal, $\xi_{\rm AX}(r)$, spin-orbit coupling curves ab initio calculated for the X-state and A-states. Lower: $L_{\rm AX}(r)$ is the corresponding non-adiabatic matrix element of electron-rotational interaction between the X-state and A-state. Table 3: Comparison of ab initio calculated spin-orbit splitting ($A_{\textrm{so}}$) and $\Lambda$-doubling constants (p, q) of the ground NaO state with their empirical counterparts taken from Yamada et al. (1989). All values in cm-1. Parameter | Empirical | Ab initio
---|---|---
$A_{\textrm{so}}$ | -107 | -120
p | +0.044 | +0.04
q | +0.0003 | +0.0003

###  3.2 Permanent and Transition Dipole Moments

The ab initio permanent dipole moments $d_{\rm XX}(r)$, $d_{\rm AA}(r)$
calculated for the X${}^{2}\Pi$ and A${}^{2}\Sigma^{+}$ states of the NaO
molecule are given in Fig. 3 together with the corresponding A–X transition
dipole moment $d_{\rm AX}(r)$. The present dipole moments are in a good
agreement with their previous theoretical counterparts computed by Langhoff et
al. (1991).

According to the expected ion-pair character of the interatomic bond, the
permanent dipole moments of both A and X states are rapidly increasing linear
functions of $r$ while the corresponding A–X electronic transition dipole
moment is very small, but not negligible (see Fig. 3). The maximal values of
the present A–X transition moment are about 0.1-0.15 a.u. which agree well
with the estimate $|d_{\rm AX}|<0.2$ D given in Joo et al. (1999). At $r>3.5$
Å, the ground state permanent dipole decreases due to the avoided crossing
effect with the excited $C$ ${}^{2}\Pi$ state. An even more abrupt drop in the
permanent dipole is observed for the A-state but at larger internuclear
distances, presumably also due to an avoided crossing.

O’Hare & Wahl (1972) give an equilibrium dipole moment for the X ${}^{2}\Pi$
state of 8.71 D which was adopted by the JPL Molecular Spectroscopy Database
(Pickett et al., 1998) to provide transition intensities. Our ab initio
equilibrium value is $|d_{\rm XX}(r_{\rm e})|=8.43$ D, while the vibrationally
averaged dipole moment extracted from the Duo the matrix element $|\langle
v_{X}=0|d_{\rm XX}|v_{X}=0\rangle|$ is 8.44 D, which is very similar to the
equilibrium dipole moment. Our results suggest that JPL predicted transitions
are slightly (about 6%) too strong.

Figure 3: Permanent, $d_{\rm XX}(r)$, $d_{\rm AA}(r)$, and transition, $d_{\rm
AX}(r)$, dipole moment functions ab initio calculated for the X-state and
A-states of NaO molecule.

###  3.3 Partition Function

The temperature dependent partition function for NaO molecule was calculated
as the sum:

$\displaystyle Q(T)=\sum_{i}g_{i}e^{\frac{-E_{i}}{k_{B}T}},$ (14)

where $g_{i}=g_{\rm ns}(2J+1)$ is the total degeneracy of the state, $g_{\rm
ns}$ is the nuclear statistical weight factor, and $k_{B}$ is the Boltzmann
constant. As $Q(T)$ sums over $J$, the value of $Q(T)$ at some $T$ depends on
the value of $J_{\textrm{max}}$ used in the sum. To ensure completeness of the
line list, it is a requirement that, up to a given temperature, in the case of
this study, 2500 K, the value of $Q(2500)$ converges with $J_{\textrm{max}}$.
As shown in Fig. 4, use $J_{\textrm{max}}=200.5$ converges $Q$ at $=2500$ K.

Figure 4: Partition function $Q(T)$ convergence of NaO as dependent on maximum
value of rotational quantum number, $J_{\textrm{max}}$ for temperatures up to
2500 K.

The JPL database provides partition function values up to $T$=300 K. Barklem &
Collet (2016) provide numerical values for the partition functions of NaO up
to 10 000 K which we multiply by four to account for different treatments of
the Na nuclear spin degeneracy; we (and JPL) include $g_{\rm ns}=2I_{\rm
Na}+1=4$ (which is appropriate for $I_{\rm Na}=\frac{3}{2}$) into $Q(T)$,
while Barklem & Collet (2016) does not. A comparison between these partition
functions is given in Fig. 5. The JPL values agree well with ours (see Table
4), with level of agreement decreasing as temperature goes up which likely due
to a lack of higher vibrational levels contribution in JPL partition sum.
However, there is poor coincidence with Barklem & Collet (2016) at low
temperature with agreement improving for higher temperature. The reason for
the low temperature disagreement between Barklem & Collet (2016) and the
results of JPL and our own is unclear but we believe our values are correct in
this region. It should be noted that the data from Barklem & Collet (2016) is
based on calculated rather than experimental molecular constants taken from
Huber & Herzberg (1979).

Figure 5: Comparison between, (a): calculated partition function based on Duo states and available data for the partition function of NaO (b): Pickett et al. (1998), (c): Barklem & Collet (2016). Note, $Q(T)$ by Barklem & Collet (2016) has been multiplied by four to account for the convention of including the hyperfine structure degeneration number $g_{\rm{ns}}$ in the partition function calculation, see Eq. 14. Upper: Close-up of calculated and reference partition functions for $T\in[0,300]$ K. Lower: Log scaled representation of calculated and reference partition functions for $T\in[0,2500]$ K. Table 4: Partition Function, $Q(T)$ as calculated using the Duo computed energies compared to partition functions from JPL database. T / K | Q${}_{\textrm{Duo}}$ | Q${}_{\textrm{JPL}}$ |  $\Delta$ /%
---|---|---|---
300 | 7027.4727 | 6633.1886 | 5.61
225 | 4694.9814 | 4491.5223 | 4.33
150 | 2734.0243 | 2645.0047 | 3.26
75 | 1134.4648 | 1109.3245 | 2.22
37.5 | 518.8865 | 516.1226 | 0.53
18.75 | 262.9106 | 262.9886 | -0.030
9.375 | 139.2536 | 139.3226 | -0.050
Table 5: Uncertainty estimates (in cm-1) are applied for the rovibronic energy levels of NaO as a function of state quantum numbers. State | X ${}^{2}\Pi$ | A ${}^{2}\Sigma^{+}$
---|---|---
Case |  $J_{X}<16.5$, $v_{X}=0$ |  $J_{X}>16.5$, $v_{X}=0$ | $v_{X}>0$ | all $v_{A}$ and $J$ levels
Uncertainty | 0.003 | $0.001J_{X}^{2}+0.01J_{X}+0.01$ | 20-40 | 10-30

###  3.4 Final line list and absorption spectra simulation

The Duo package was used to produce a NaO line list which we call NaOUCMe.
This line list is given in ExoMol format (Tennyson et al., 2014, 2020) as
"*.states" file containing term values for the lowest 36 120 rovibronic levels
of the $A\sim X$ complex up to $J=200.5$ and $v_{\rm X}=29$ and a "*.trans"
file consisting of the corresponding Einstein coefficients and frequencies for
4 726 137 dipole-allowed $X-X$, $A-A$ and $X-A$ transitions. In line with the
updated ExoMol standard (Tennyson et al., 2020), the "*.states" files also
provide approximate uncertainties for the energies which allow the accuracy
with which we predict each transition to be estimated. For fitted regions, we
recommend transition uncertainties equal to those in Table 1. For the ground
electronic state, for the case where $J<16.5$ and $v_{X}=0$, an uncertainty of
$0.003~{}\rm{cm}^{-1}$ is recommended. Typically, energy level uncertainties
scale with $J$ quadratically and the following functional form for $J>15.5$ is
recommended: $\delta E\approx 0.001J^{2}-0.01J+0.01$. For all other
(extrapolated) cases in the X state, the ab initio vibrational uncertainty
takes precedence at 20-40 cm-1.

The vibrational spacing between the $v_{\rm A}=0$ and $v_{\rm A}=1$ states
were fit to a satisfactory level of agreement (artificial line from sec. 2.2).
However, it is recommended that this line list only be used for high
resolution observational studies at long wavelengths spectral range covering
pure rotational transitions. Overall energy level uncertainties are summarised
in Table 5.

Samples from the "*.states" and "*.trans" files are given in Tables 6 and 7.
These were processed in ExoCross (Yurchenko et al., 2018) to produce Fig. 6,
an absolute absorption spectrum for the region studied by Joo et al. (1999),
and Fig. 7, the absorbance cross section for NaO as a function of temperature
computed at a resolution of 1 cm-1 with a half-width at half-maximum (HWHM) of
1 cm-1.

Table 6: Extract from "*.states" file produced by Duo for NaO. Lifetimes calculated with ExoCross, for uncertainties see Sec.4 and Table 5 NN | Energy | gtot | $J$ | uncertainty | lifetime | Landé | $\tau$ | e/f | State | $v$ | $\Lambda$ | $\Sigma$ | $\Omega$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 118.305521 | 8 | 0.5 | 0.003000 | 5.1528E+03 | 0.001493 | + | e | X2Pi | 0 | 1 | -0.5 | 0.5
2 | 704.551592 | 8 | 0.5 | 30.000000 | 2.8981E+00 | 0.001932 | + | e | X2Pi | 1 | 1 | -0.5 | 0.5
3 | 1282.294758 | 8 | 0.5 | 30.000000 | 1.4807E+00 | 0.002994 | + | e | X2Pi | 2 | 1 | -0.5 | 0.5
4 | 1850.69127 | 8 | 0.5 | 30.000000 | 1.0028E+00 | 0.012246 | + | e | X2Pi | 3 | 1 | -0.5 | 0.5
5 | 2045.653545 | 8 | 0.5 | 20.000000 | 1.0894E-01 | 1.988261 | + | e | A2Sigma | 0 | 0 | 0.5 | 0.5
6 | 2409.940124 | 8 | 0.5 | 30.000000 | 6.5698E-01 | 0.041505 | + | e | X2Pi | 4 | 1 | -0.5 | 0.5
7 | 2544.609918 | 8 | 0.5 | 20.000000 | 6.4721E-02 | 1.957978 | + | e | A2Sigma | 1 | 0 | 0.5 | 0.5
8 | 2960.425378 | 8 | 0.5 | 30.000000 | 3.7063E-01 | 0.112844 | + | e | X2Pi | 5 | 1 | -0.5 | 0.5
9 | 3042.140902 | 8 | 0.5 | 20.000000 | 4.9272E-02 | 1.887345 | + | e | A2Sigma | 2 | 0 | 0.5 | 0.5
10 | 3501.549396 | 8 | 0.5 | 30.000000 | 1.4379E-01 | 0.388061 | + | e | X2Pi | 6 | 1 | -0.5 | 0.5

Table 7: Extract from the "*.trans" file produced by Duo for NaO. n${{}_{\textrm{f}}}$ | n${}_{\textrm{i}}$ | A${}_{\textrm{fi}}$ | $\nu_{\textrm{fi}}$
---|---|---|---
2097 | 2007 | 7.7832E-17 | 0.001009
843 | 933 | 2.4392E-16 | 0.001035
1799 | 1889 | 9.3118E-17 | 0.001043
1883 | 1793 | 9.4292E-17 | 0.001055
1219 | 1309 | 1.6125E-16 | 0.001066
3848 | 3758 | 2.7618E-17 | 0.001137
1520 | 1430 | 1.7928E-16 | 0.001148
1595 | 1685 | 1.3851E-16 | 0.001153
3146 | 3056 | 4.9277E-17 | 0.001177
1688 | 1598 | 1.5019E-16 | 0.001179
766 | 676 | 5.6326E-16 | 0.001185

###  3.5 Comparison with the experimental X$\to$ A spectra

The experimental A ${}^{2}\Sigma^{+}(0,1)\leftarrow$ X ${}^{2}\Pi(0)$
absorption bands were observed by Joo et al. (1999) but their measurements
were only sensitive in the very narrow spectral regions: 2015–2095 cm-1 and
2646–2697 cm-1. The lines in these regions were assigned to the $\Delta v=0$
and $\Delta v=+1$ bands respectively. An absolute intensity absorption
spectrum was calculated as a stick spectrum to compare to this, see Fig. 6.
The calculated absorption lines result in a line profile significantly broader
than those visible from measurements of Joo et al. (1999). The observed
spectrum (relative intensities) has been scaled by matching the intensity of
the strongest line [A $(J^{\prime}=7.5,v^{\prime}=0)\leftarrow$ X
$(J^{\prime\prime}=7.5,v^{\prime\prime}=0,\Omega^{\prime\prime}=-1.5)$] from
experiment to that of its conjugate calculated line; it can be seen that the
observed $\Delta v=+1$ transitions lie on the tail end, $R$-branch of the
calculated absorption spectrum with a matching downward slope.

We note that Joo et al. (1999) used their measurements to determine the
vibronic term value for the A(0)–X(0) band, $T_{0}$(A), of 1992.905 cm-1.
Using approximate $J=0$ zero-point energies (ZPEs) generated by Duo, and
$T_{0}({\rm A})=T_{\rm e}({\rm A})-{\rm ZPE}({\rm X})+{\rm ZPE}({\rm A})$
shows that our $T_{0}(A)$ from our deperturbation model is 1982.3 cm-1, which
is less than that of Joo et al. (1999) by 10 cm-1. An attempt was made to
reproduce the experimental $T_{0}$(A) value above by manually adjusting
$T_{\rm e}$(A) and refitting. However, this results in significantly worse
fits in the transitions (around an order of magnitude) without improving
agreement of $T_{0}$(A) with experiment.

Figure 6: Absolute absorption intensity (arbitrary units) spectrum of NaO
computed at $T=400$ K. Black lines represent the experimental transitions
containing (from left to right) the observed A
${}^{2}\Sigma^{+}(0,1)\leftarrow$ X ${}^{2}\Pi(0)$ transition bands from Joo
et al. (1999). Black vertical lines indicate start and end of the observed
band transition regions. Figure 7: Spectrum of NaO at varying temperatures.
Absorption cross sections were modelled with a Gaussian line profile with HWHM
at 1 cm-1 with an output on a grid with 1 cm-1spacing.

##  4 Conclusions

An optical line list called NaOUCMe is presented for the alkali monoxide
radical, NaO covering transitions in the 0–10000 cm-1 energy range up to
$J=200.5$ and $v_{X}=29$ with applicability up to the equilibrium temperature
$T=2500$ K. The experimental lines for pure rotational transitions in the
ground electronic state (Yamada et al., 1989) and rovibronic A
${}^{2}\Sigma^{+}(0,1)\leftarrow$ X ${}^{2}\Pi(0)$ transitions (Joo et al.,
1999) were used to refine the deperturbation model applied. An average RMS of
$\sim 0.006$ cm-1 was achieved for the X ${}^{2}\Pi$ state rotational
transitions and $0.35~{}\rm{cm}^{-1}$ for the A ${}^{2}\Sigma^{+}$
$\leftarrow$ X ${}^{2}\Pi$ transitions, with $\sim 0.023~{}\rm{cm}^{-1}$ RMS
for the A$(v^{\prime}=0)\leftarrow$ X$(v^{\prime\prime}=0)$ transitions and
$\sim 0.680~{}\rm{cm}^{-1}$ RMS for the
A$(v^{\prime}=1,J^{\prime}=0.5)\leftarrow$
A$(v^{\prime\prime}=0,J^{\prime\prime}=0.5)$ artificial line (see sec. 2.2).
Uncertainties of the reproduced and predicted lines rapidly increase as the
vibrational and rotational quantum numbers increase for both electronic
states, and, hence, further experimental data on higher excited transitions is
required to create a more robust line list.

The success of the CaO line list (Yurchenko et al., 2016c) in a spectrum
assignment at 29 km above the Earth’s surface by Berezhnoy et al. (2018) is
indicative of potential importance of s-block metal monoxides like NaO. NaO
similarly is a molecule that can be studied in a terrestrial environment at
approximately 90 km above the Earth’s surface (Slanger et al., 2005) and has
been recently studied as a transient species in airglow by Sarkhel et al.
(2009, 2010) and Plane et al. (2012). As such, we strongly encourage further
experimental study into NaO for refinement of the spectroscopic model and
production of a higher accurate line list.

## Acknowledgements

We thank Dr. A. A. Berezhnoy for bringing the problem to our attention and
Prof. I. A. Osterman for taking part in the preliminary ab initio
calculations. This work was supported by ERC Advanced Investigator Project
883830 and by UK STFC under grant ST/R000476/1. A.V.S. is grateful for the
support by the Russian Science Foundation (RSF), Grant No.18-13-00269.

## Data Availability

The Duo model input file and calculated partition function file given as a
supplementary materials to this article. The NaO NaOUCMe states and transition
files of can be downloaded from www.exomol.com and the CDS data centre
cdsarc.u-strasbg.fr. The open access programs ExoCross and Duo are available
from github.com/exomol.

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