,

**footnotetext: Equal contribution to this work.

# High-Fidelity Audio Compression
with Improved RVQGAN

Rithesh Kumar*
Descript, Inc.
<EMAIL_ADDRESS>
Prem Seetharaman*
Descript, Inc.
<EMAIL_ADDRESS>
Alejandro Luebs
Descript, Inc.
Ishaan Kumar
Descript, Inc.
Kundan Kumar
Descript, Inc.

###### Abstract

Language models have been successfully used to model natural signals, such as
images, speech, and music. A key component of these models is a high quality
neural compression model that can compress high-dimensional natural signals
into lower dimensional discrete tokens. To that end, we introduce a high-
fidelity universal neural audio compression algorithm that achieves 90x
compression of 44.1 KHz audio into tokens at just 8kbps bandwidth. We achieve
this by combining advances in high-fidelity audio generation with better
vector quantization techniques from the image domain, along with improved
adversarial and reconstruction losses. We compress all domains (speech,
environment, music, etc.) with a single universal model, making it widely
applicable to generative modeling of all audio. We compare with competing
audio compression algorithms, and find our method outperforms them
significantly. We provide thorough ablations for every design choice, as well
as open-source code and trained model weights. We hope our work can lay the
foundation for the next generation of high-fidelity audio modeling.

##  1 Introduction

Generative modeling of high-resolution audio is difficult due to high
dimensionality (~44,100 samples per second of audio) [23, 19], and presence of
structure at different time-scales with both short and long-term dependencies.
To mitigate this problem, audio generation is typically divided into two
stages: 1) predicting audio conditioned on some intermediate representation
such as mel-spectrograms [23, 27, 19, 29] and 2) predicting the intermediate
representation given some conditioning information, such as text [34, 33].
This can be interpreted as a hierarchical generative model, with observed
intermediate variables. Naturally, an alternate formulation is to learn the
intermediate variables using the variational auto-encoder (VAE) framework,
with a learned conditional prior to predict the latent variables given some
conditioning. This formulation, with continuous latent variables and training
an expressive prior using normalizing flows has been quite successful for
speech synthesis [17, 35].

A closely related idea is to train the same varitional-autoencoder with
discrete latent variables using VQ-VAE [37]. Arguably, discrete latent
variables are a better choice since expressive priors can be trained using
powerful autoregressive models that have been developed for modeling
distributions over discrete variables [26]. Specifically, transformer language
models [38] have already exhibited the capacity to scale with data and model
capacity to learn arbitrarily complex distributions such as text[6],
images[12, 43], audio [5, 40], music [1], etc. While modeling the prior is
straightforward, modeling the discrete latent codes using a quantized auto-
encoder remains a challenge.

Learning these discrete codes can be interpreted as a lossy compression task,
where the audio signal is compressed into a discrete latent space by vector-
quantizing the representations of an autoencoder using a fixed length
codebook. This audio compression model needs to satisfy the following
properties: 1) Reconstruct audio with high fidelity and free of artifacts 2)
Achieve high level of compression along with temporal downscaling to learn a
compact representation that discards low-level imperceptible details while
preserving high-level structure [37, 32] 3) Handle all types of audio such as
speech, music, environmental sounds, different audio encodings (such as mp3)
as well as different sampling rates using a single universal model.

While the recent neural audio compression algorithms such as SoundStream [45]
and EnCodec [8] partially satisfy these properties, they often suffer from the
same issues that plague GAN-based generation models. Specifically, such models
exhibit audio artifacts such as tonal artifacts [28], pitch and periodicity
artifacts [24] and imperfectly model high-frequencies leading to audio that
are clearly distinguishable from originals. These models are often tailored to
a specific type of audio signal such as speech or music and struggle to model
generic sounds. We make the following contributions:

  * •

We introduce Improved RVQGAN a high fidelity universal audio compression
model, that can compress 44.1 KHz audio into discrete codes at 8 kbps bitrate
(~90x compression) with minimal loss in quality and fewer artifacts. Our model
outperforms state-of-the-art methods by a large margin even at lower bitrates
(higher compression) , when evaluated with both quantitative metrics and
qualitative listening tests.

  * •

We identify a critical issue in existing models which don’t utilize the full
bandwidth due to codebook collapse (where a fraction of the codes are unused)
and fix it using improved codebook learning techniques.

  * •

We identify a side-effect of quantizer dropout \- a technique designed to
allow a single model to support variable bitrates, actually hurts the full-
bandwidth audio quality and propose a solution to mitigate it.

  * •

We make impactful design changes to existing neural audio codecs by adding
periodic inductive biases, multi-scale STFT discriminator, multi-scale mel
loss and provide thorough ablations and intuitions to motivate them.

  * •

Our proposed method is a universal audio compression model, capable of
handling speech, music, environmental sounds, different sampling rates and
audio encoding formats.

We provide code 111https://github.com/descriptinc/descript-audio-codec,
models, and audio samples 222https://descript.notion.site/Descript-Audio-
Codec-11389fce0ce2419891d6591a68f814d5 that we encourage the reader to listen.

##  2 Related Work

High fidelity neural audio synthesis: Recently, generative adversarial
networks (GANs) have emerged as a solution to generate high-quality audio with
fast inference speeds, due to the feed-forward (parallel) generator. MelGAN
[19] successfully trains a GAN-based spectrogram inversion (neural vocoding)
model. It introduces a multi-scale waveform discriminator (MSD) to penalize
structure at different audio resolutions and a feature matching loss that
minimizes L1 distance between discriminator feature maps of real and synthetic
audio. HifiGAN [18] refines this recipe by introducing a multi-period waveform
discriminator (MPD) for high fidelity synthesis, and adding an auxiliary mel-
reconstruction loss for fast training. UnivNet [16] introduces a multi-
resolution spectrogram discriminator (MRSD) to generate audio with sharp
spectrograms. BigVGAN [21] extends the HifiGAN recipe by introducing a
periodic inductive bias using the Snake activation function [46]. It also
replaces the MSD in HifiGAN with the MRSD to improve audio quality and reduce
pitch, periodicity artifacts [24]. While these the GAN-based learning
techniques are used for vocoding, these recipes are readily applicable to
neural audio compression. Our Improved RVQGAN model closely follows the
BigVGAN training recipe, with a few key changes. Our model uses a new multi-
band multi-scale STFT discriminator that alleviates aliasing artifacts, and a
multi-scale mel-reconstruction loss that better model quick transients.

Neural audio compression models: VQ-VAEs [37] have been the dominant paradigm
to train neural audio codecs. The first VQ-VAE based speech codec was proposed
in [13] operating at 1.6 kbps. This model used the original architecture from
[37] with a convolutional encoder and an autoregressive wavenet [26] decoder.
SoundStream [45] is one of the first universal compression models capable of
handling diverse audio types, while supporting varying bitrates using a single
model. They use a fully causal convolutional encoder and decoder network, and
perform residual vector quantization (RVQ). The model is trained using the VQ-
GAN [12] formulation, by adding adversarial and feature-matching losses along
with the multi-scale spectral reconstruction loss. EnCodec [8] closely follows
the SoundStream recipe, with a few modifications that lead to improved
quality. EnCodec uses a multi-scale STFT discriminator with a multi-scale
spectral reconstruction loss. They use a loss balancer which adjusts loss
weights based on the varying scale of gradients coming from the discriminator.

Our proposed method also uses a convolutional encoder-decoder architecture,
residual vector quantization and adversarial, perceptual losses. However, our
recipe has the following key differences: 1) We introduce a periodic inductive
bias using Snake activations [46, 21] 2) We improve codebook learning by
projecting the encodings into a low-dimensional space [43] 3) We obtain a
stable training recipe using best practices for adversarial and perceptual
loss design, with fixed loss weights and without requiring a sophisticated
loss balancer. We find that our changes lead to a near-optimal effective
bandwidth usage. This allows our model to outperform EnCodec even with 3x
lower bitrate.

Language modeling of natural signals : Neural language models have
demonstrated great success in diverse tasks such as open-ended text generation
[6] with in-context learning capabilities. A key-component of these models is
self-attention [38], which is capable of modeling complex and long-range
dependencies but suffers from a quadratic computational cost with the length
of the sequence. This cost is unacceptable for natural signals such as images
and audio with very high dimensionality, requiring a compact mapping into a
discrete representation space. This mapping is typically learnt using VQ-GANs
[12, 43], followed by training an autoregressive Transformer on the discrete
tokens. This approach has shown success across image [44, 43, 31], audio [5,
40], video and music [9, 1] domains. Codecs like SoundStream and EnCodec have
already been used in generative audio models, like AudioLM [5], MusicLM [1],
and VALL-E [40]. Our proposed model can serve as a drop-in replacement for the
audio tokenization model used in these methods, allowing for highly superior
audio fidelity, and more efficient learning due to our maximum entropy code
representation.

##  3 The Improved RVQGAN Model

Codec |  Sampling rate (kHz) |  Target bitrate (kbps) |  Striding factor |  Frame rate (Hz) |  # of 10-bit codebooks |  Compression factor
---|---|---|---|---|---|---
Proposed | 44.1 | 8 | 512 | 86 | 9 | 91.16
EnCodec | 24 | 24 | 320 | 75 | 32 | 16
48 | 24 | 320 | 150 | 16 | 32
SoundStream | 24 | 6 | 320 | 75 | 8 | 64

Table 1:  Comparison of compression approaches.

Our model is built on the framework of VQ-GANs, following the same pattern as
SoundStream [45] and EnCodec [8]. Our model uses the fully convolutional
encoder-decoder network from SoundStream, that performs temporal downscaling
with a chosen striding factor. Following recent literature, we quantize the
encodings using Residual Vector Quantization (RVQ), a method that recursively
quantizes residuals following an initial quantization step with a distinct
codebook. Quantizer dropout is applied during training to enable a single
model that can operate at several target bitrates. Our model is similarly
trained using a frequency domain reconstruction loss along with adversarial
and perceptual losses.

An audio signal with sampling rate $f_{s}$ (Hz), encoder striding factor $M$,
and $N_{q}$ layers of RVQ produce a discrete code matrix of shape $S\times
N_{q}$, where $S$ is the frame rate defined as $f_{s}/M$. Table 1 compares our
proposed model against baselines to contrast the compression factors and the
frame rate of latent codes. Note that the target bitrate mentioned is an upper
bound, since all models support variable bitrates. Our model achieves a higher
compression factor compared to all baseline methods while outperforming them
in audio quality, as we show later. Finally, a lower frame rate is desirable
when training a language model on the discrete codes, as it results in shorter
sequences.

###  3.1 Periodic activation function

Audio waveforms are known to exhibit high periodicity (especially in voiced
components, music, etc.) While current non-autoregressive audio generation
architectures are capable of generating high fidelity audio, they often
exhibit jarring pitch and periodicity artifacts [24]. Moreover, common neural
network activations (such as Leaky ReLUs) are known to struggle with
extrapolating periodic signals, and exhibit poor out-of-distribution
generalization for audio synthesis [21].

To add a periodic inductive bias to the generator, we adopt the Snake
activation function proposed by Liu et al. [46] and introduced to the audio
domain in the BigVGAN neural vocoding model [21]. It is defined as
$\text{snake}(x)=x+\frac{1}{\alpha}\sin^{2}(\alpha x)$ , where $\alpha$
controls the frequency of periodic component of the signal. In our
experiments, we find replacing Leaky ReLU activations with Snake function to
be an influential change that significantly improves audio fidelity (Table 2).

###  3.2 Improved residual vector quantization

Figure 1: Entropy for each codebook, computed using code usage statistics
across a large test set.

While vector quantization (VQ) is a popular method to train discrete auto-
encoder, there are many practical struggles when training them. Vanilla VQ-
VAEs struggle from low codebook usage due to poor initialization, leading to a
significant portion of the codebook being unused. This reduction in effective
codebook size leads to an implicit reduction in target bitrate, which
translates to poor reconstruction quality.

To mitigate this, recent audio codec methods use k-means clustering to
initialize the codebook vectors, and manually employ randomized restarts [9]
when certain codebooks are unused for several batches. However, we find that
the EnCodec model trained at 24kbps target bitrate, as well as our proposed
model with the same codebook learning method (Proposed w/ EMA) still suffers
from codebook under-utilization (Figure 1).

To address this issue, we use two key techniques introduced in the Improved
VQGAN image model[43] to improve codebook usage: factorized codes and
L2-normalized codes. Factorization decouples code lookup and code embedding,
by performing code lookup in a low-dimensional space (8d or 32d) whereas the
code embedding resides in a high dimensional space (1024d). Intuitively, this
can be interpreted as a code lookup using only the principal components of the
input vector that maximally explain the variance in the data. The
L2-normalization of the encoded and codebook vectors converts euclidean
distance to cosine similarity, which is helpful for stability and quality
[43].

These two tricks along with the overall model recipe significantly improve
codebook usage, and therefore bitrate efficiency (Figure 1) and reconstruction
quality (Table 2), while being simpler to implement. Our model can be trained
using the original VQ-VAE codebook and commitment losses [37], without k-means
initialization or random restarts.

###  3.3 Quantizer dropout rate

Figure 2: Effect of quantizer dropout on audio quality vs bitrate.

Quantizer dropout was introduced in SoundStream [45] to train a single
compression model with variable bitrate. The number of quantizers $N_{q}$
determine the bitrate, so for each input example we randomly sample
$n\sim\\{1,2,\ldots,N_{q}\\}$ and only use the first $n_{q}$ quantizers while
training. However, we noticed that applying quantizer dropout degrades the
audio reconstruction quality at full bandwidth (Figure 2).

To address this problem, we instead apply quantizer dropout to each input
example with some probability $p$. Interestingly, we find that dropout
probability $p=0.5$ closely matches the reconstruction quality of baseline at
lower bitrates, while closing the gap to full-bandwidth quality of a model
trained without quantizer dropout ($p=0.0$).

Moreover, we provide additional insight into the practical behavior of
quantizer dropout and it’s interaction with RVQ. Firstly, we find that these
techniques put together lead the quantized codes to learn most-significant to
least significant bits of information with each additional quantizer. When the
codes are reconstructed with $1\ldots N_{q}$ codebooks, we can see each
codebook adds increasing amounts of fine-scale detail. We believe this
interaction is beneficial when training hierarchical generative models on top
of these codes [5, 40, 1], for example to partition the codes into “coarse”
tokens (denoting the most significant codes) and “fine” tokens.

###  3.4 Discriminator design

Like prior work, we use multi-scale (MSD) and multi-period waveform
discriminators (MPD) which lead to improved audio fidelity. However,
spectrograms of generated audio can still appear blurry, exhibiting over-
smoothing artifacts in high frequencies[16]. The multi-resolution spectrogram
discriminator (MRSD) was proposed in UnivNet to fix these artifacts and
BigVGAN [21] found that it also helps to reduce pitch and periodicity
artifacts. However, using magnitude spectrograms discards phase information
which could’ve been otherwise utilized by the discriminator to penalize phase
modeling errors. Moreover, we find that high-frequency modeling is still
challenging for these models especially at high sampling rates.

To address these issues, we use a complex STFT discriminator [45] at multiple
time-scales [8] and find that it works better in practice and leads to
improved phase modeling. Additionally we find that splitting the STFT into
sub-bands slightly improves high frequency prediction and mitigates aliasing
artifacts, since the discriminator can learn discriminative features about a
specific sub-band and provide a stronger gradient signal to the generator.
Multi-band processing was earlier proposed in [42] to predict audio in sub-
bands which are subsequently summed to produce the full-band audio.

###  3.5 Loss functions

Frequency domain reconstruction loss: while the mel-reconstruction loss [18]
is known to improve stability, fidelity and convergence speed, the multi-scale
spectral losses[41, 11, 15] encourage modeling of frequencies in multiple
time-scales. In our model, we combine both methods by using a L1 loss on mel-
spectrograms computed with window lengths of $[32,64,128,256,512,1024,2048]$
and hop length set to $\text{window\\_length}\ /\ 4$. We especially find that
using the lowest hop size of 8 improves modeling of very quick transients that
are especially common in the music domain.

EnCodec [8] uses a similar loss formulation, but with both L1 and L2 loss
terms, and a fixed mel bin size of 64. We find that fixing mel bin size leads
to holes in the spectrogram especially at low filter lengths. Therefore, we
use mel bin sizes $[5,10,20,40,80,160,320]$ corresponding to the above filter
lengths which were verified to be correct by manual inspection.

Adversarial loss: our model uses the multi-period discriminator [18] for
waveform discrimination, as well as the proposed multi-band multi-scale STFT
discriminator for the frequency domain. We use the HingeGAN [22] adversarial
loss formulation, and apply the L1 feature matching loss [19].

Codebook learning: we use the simple codebook and commitment losses with stop-
gradients from the original VQ-VAE formulation [37], and backpropagate
gradients through the codebook lookup using the straight-through estimator
[3].

Loss weighting: we use the loss weightings of $15.0$ for the multi-scale mel
loss, $2.0$ for the feature matching loss, $1.0$ for the adversarial loss and
$1.0$, $0.25$ for the codebook and commitment losses respectively. These loss
weightings are in line with recent works [18, 21] (which use $45.0$ weighting
for the mel loss), but simply rescaled to account for the multiple scales and
$\log_{10}$ base we used for computing the mel loss. We don’t use a loss
balancer as proposed in EnCodec [8].

##  4 Experiments

###  4.1 Data sources

We train our model on a large dataset compiled of speech, music, and
environmental sounds. For speech, we use the DAPS dataset [25], the clean
speech segments from DNS Challenge 4 [10], the Common Voice dataset [2], and
the VCTK dataset [39]. For music, we use the MUSDB dataset [30], and the
Jamendo dataset [4]. Finally, for environmental sound, we use both the
balanced and unbalanced train segments from AudioSet [14]. All audio is
resampled to 44kHz.

During training, we extract short excerpts from each audio file, and normalize
them to -24 dB LUFS. The only data augmentation we apply is to randomly shift
the phase of the excerpt, uniformly. For evaluation, we use the evaluation
segments from AudioSet [14], two speakers that are held out from DAPS [25]
(F10, M10) for speech, and the test split of MUSDB [30]. We extract 3000
10-second segments (1000 from each domain), as our test set.

###  4.2 Balanced data sampling

We take special care in how we sample from our dataset. Though our dataset is
resampled to 44kHz, the data within it may be band-limited in some way. That
is, some audio may have had an original sampling rate much lower than 44kHz.
This is especially prevalent in speech data, where the true sampling rates of
the underlying data can vary greatly (e.g. the Common Voice data is commonly
8-16kHz). When we trained models on varying sampling rates, we found that the
resultant model often would not reconstruct data above a certain frequency.
When investigating, we found that this threshold frequency corresponded to the
average true sampling rate of our dataset. To fix this, we introduce a
balanced data sampling technique.

We first split our dataset into data sources that we know to be full-band \-
they are confirmed to contain energy in frequencies up to the desired Nyquist
frequency (22.05kHz) of the codec - and data sources where we have no
assurances of the max frequency. When sampling batches, we make sure that a
full-band item is sampled. Finally, we ensure that in each batch, there are an
equal number of items from each domain: speech, music, and environmental
sound. In our ablation study, we examine how this balanced sampling technique
affects model performance.

###  4.3 Model and training recipe

Our model consists of a convolutional encoder, a residual vector quantizer,
and a convolutional decoder. The basic building block of our network is a
convolutional layer which either upsamples or downsamples with some stride,
followed by a residual layer consisting of convolutional layers interleaved
with non-linear Snake activations. Our encoder has 4 of these layers, each of
which downsamples the input audio waveform at rates $[2,4,8,8]$. Our decoder
has 4 corresponding layers, which upsample at rates $[8,8,4,2]$. We set the
decoder dimension to $1536$. In total, our model has 76M parameters, with 22M
in the encoder, and 54M in the decoder. We also examine decoder dimensions of
512 (31M parameters) and 1024 (49M parameters).

We use the multi-period discriminator [18], and a complex multi-scale STFT
discriminator. For the first, we use periods of $[2,3,5,7,11]$, and for the
second, we use window lengths $[2048,1024,512]$, with a hop-length that is
$1/4$ the window length. For band-splitting of the STFT, we use the band-
limits $[0.0,0.1,0.25,0.5,0.75,1.0]$. For the reconstruction loss, we use
distance between log-mel spectrograms with window lengths
$[32,64,128,256,512,1024,2048]$, with corresponding number of mels for each of
$[5,10,20,40,80,160,320]$. The hop length is $1/4$ of the window length. We
use feature matching and codebook losses, as described in Section 3.5.

For our ablation study, we train each model with a batch size of $12$ for
$250$k iterations. In practice, this takes about 30 hours to train on a single
GPU. For our final model, we train with a batch size of $72$ for $400$k
iterations. We train with excerpts of duration 0.38s. We use the AdamW
optimizer [adamw] with a learning rate of $1e-4$, $\beta_{1}=0.8$, and
$\beta_{2}=0.9$, for both the generator and the discriminator. We decay the
learning rate at every step, with $\gamma=0.999996$.

###  4.4 Objective and subjective metrics

To evaluate our models, we use the following objective metrics:

  1. 1.

ViSQOL [7]: an intrusive perceptual quality metric that uses spectral
similarity to the ground truth to estimate a mean opinion score.

  2. 2.

Mel distance: distance between log mel spectrograms of the reconstructed and
ground truth waveforms. The configuration of this loss is the same as
described in 3.5.

  3. 3.

STFT distance: distance between log magnitude spectrograms of the
reconstructed and ground truth waveforms. We use window lengths $[2048,512]$.
This metric captures the fidelity in higher frequencies better than the mel
distance.

  4. 4.

Scale-invariant source-to-distortion ratio (SI-SDR) [20]: distance between
waveforms, similar to signal-to-noise ratio, with modifications so that it is
invariant to scale differences. When considered alongside spectral metrics,
SI-SDR indicates the quality of the phase reconstruction of the audio.

  5. 5.

Bitrate efficiency: We calculate bitrate efficiency as the sum of the entropy
(in bits) of each codebook when applied on a large test set divided by the
number of bits across all codebooks. For efficient bitrate utilization this
should tend to 100% and lower percentages indicate that the bitrate is being
underutilized.

We also conduct a MUSHRA-inspired listening test, with a hidden reference, but
no low-passed anchor. In it each one of ten expert listeners rated 12 randomly
selected 10-second samples from our evaluation set, 4 of each domain; speech,
music and environmental sounds. We compare our proposed system at 2.67kbps,
5.33kbps and 8kbps to EnCodec at 3kbps, 6kbps and 12kbps.

Ablation on |  Decoder dim. |  Activation |  Multi-period |  Single-scale |  # of STFT bands |  Multi-scale mel. |  Latent dim |  Quant. method |  Quant. dropout |  Bitrate (kbps) |  Balanced samp. |  Mel distance $\downarrow$ |  STFT distance $\downarrow$ |  ViSQOL $\uparrow$ |  SI-SDR $\uparrow$ |  Bitrate efficiency $\uparrow$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| 1536 | snake | ✓ | ✗ | 5 | ✓ | 8 | Proj. | 1.0 | 8 | ✓ | 1.09 | 1.82 | 3.96 | 9.12 | 99%
Architecture | 512 | snake | ✓ | ✗ | 5 | ✓ | 8 | Proj. | 1.0 | 8 | ✓ | 1.11 | 1.83 | 3.91 | 8.72 | 99%
1024 | snake | ✓ | ✗ | 5 | ✓ | 8 | Proj. | 1.0 | 8 | ✓ | 1.07 | 1.82 | 3.96 | 9.07 | 99%
1536 | relu | ✓ | ✗ | 5 | ✓ | 8 | Proj. | 1.0 | 8 | ✓ | 1.17 | 1.81 | 3.83 | 6.92 | 99%
Discriminator | 1536 | snake | ✗ | ✗ | ✗ | ✓ | 8 | Proj. | 1.0 | 8 | ✓ | 1.13 | 1.92 | 4.12 | 1.07 | 62%
1536 | snake | ✓ | ✗ | 1 | ✓ | 8 | Proj. | 1.0 | 8 | ✓ | 1.07 | 1.80 | 3.98 | 9.07 | 99%
1536 | snake | ✗ | ✗ | 5 | ✓ | 8 | Proj. | 1.0 | 8 | ✓ | 1.07 | 1.81 | 3.97 | 9.04 | 99%
| 1536 | snake | ✗ | ✓ | 5 | ✓ | 8 | Proj. | 1.0 | 8 | ✓ | 1.08 | 1.82 | 3.95 | 8.51 | 99%
Reconstruction loss | 1536 | snake | ✓ | ✗ | 5 | ✗ | 8 | Proj. | 1.0 | 8 | ✓ | 1.10 | 1.87 | 4.01 | 7.68 | 99%
Latent dim | 1536 | snake | ✓ | ✗ | 5 | ✓ | 2 | Proj. | 1.0 | 8 | ✓ | 1.44 | 2.08 | 3.65 | 2.22 | 84%
1536 | snake | ✓ | ✗ | 5 | ✓ | 4 | Proj. | 1.0 | 8 | ✓ | 1.20 | 1.89 | 3.86 | 7.15 | 97%
1536 | snake | ✓ | ✗ | 5 | ✓ | 32 | Proj. | 1.0 | 8 | ✓ | 1.10 | 1.84 | 3.95 | 9.05 | 98%
1536 | snake | ✓ | ✗ | 5 | ✓ | 256 | Proj. | 1.0 | 8 | ✓ | 1.31 | 1.97 | 3.79 | 5.09 | 59%
Quantization setup | 1536 | snake | ✓ | ✗ | 5 | ✓ | 8 | EMA | 1.0 | 8 | ✓ | 1.11 | 1.84 | 3.94 | 8.33 | 97%
1536 | snake | ✓ | ✗ | 5 | ✓ | 8 | Proj. | 0.0 | 8 | ✓ | 0.98 | 1.70 | 4.09 | 10.14 | 99%
1536 | snake | ✓ | ✗ | 5 | ✓ | 8 | Proj. | 0.25 | 8 | ✓ | 0.99 | 1.69 | 4.04 | 10.00 | 99%
1536 | snake | ✓ | ✗ | 5 | ✓ | 8 | Proj. | 0.5 | 8 | ✓ | 1.01 | 1.75 | 4.03 | 9.74 | 99%
1536 | snake | ✓ | ✗ | 5 | ✓ | 8 | Proj. | 1.0 | 24 | ✓ | 0.73 | 1.62 | 4.16 | 13.83 | 99%
Data | 1536 | snake | ✓ | ✗ | 5 | ✓ | 8 | Proj. | 1.0 | 8 | ✗ | 1.09 | 1.94 | 3.89 | 8.89 | 99%

Table 2: Results of the ablation study on our proposed codec. The final model
is trained with the same configuration as the baseline (top row), but with a
quantization dropout of $0.5$.

###  4.5 Ablation study

We conduct a thorough ablation study of our model, varying components of our
training recipe and model configuration one-by-one. To compare models, we use
the four objective metrics described in Section 4.4. The results of our
ablation study can be seen in Table 2.

Architecture: We find that varying the decoder dimension has some effect on
performance, with smaller models having consistently worse metrics. However,
the model with decoder dimension $1024$ has similar performance to the
baseline, indicating that smaller models can still be competitive. The change
with the biggest impact was switching out the relu activation for the snake
activation. This change resulted in much better SI-SDR and other metrics.
Similar to the results in BigVGAN [21], we find that the periodic inductive
bias of the snake activation is helpful for waveform generation. For our final
model, we use the largest decoder dimension ($1536$), and the snake
activation.

Discriminator: Next, we removed or changed the discriminators one-by-one, to
see their impact on the final result. First, we find that the multi-band STFT
discriminator does not result in significantly better metrics, except for SI-
SDR, where it is slightly better. However, when inspecting spectrograms of
generated waveforms, we find that the multi-band discriminator alleviates
aliasing of high frequencies. The upsampling layers of the decoder introduce
significant aliasing artifacts [28]. The multi-band discriminator is more
easily able to detect these aliasing artifacts and give feedback to the
generator to remove them. Since aliasing artifacts are very small in terms of
magnitude, their effect on our objective metrics is minimal. Thus, we keep the
multi-band discriminator.

We find that adversarial losses are critical to both the quality of the output
audio, as well as the bitrate efficiency. When training with only
reconstruction loss, the bitrate efficiency drops from 99% to 62%, and the SI-
SDR drops from 9.12 to 1.07. The other metrics capture spectral distance, and
are relatively unaffected. However, the audio from this model has many
artifacts, including buzzing, as it has not learned to reconstruct phase.
Finally, we found that swapping the multi-period discriminator for the single-
scale waveform discriminator proposed in MelGAN [19] resulted in worse SI-SDR.
We retain the multi-period discriminator.

Impact of low-hop reconstruction loss: We find that low-hop reconstruction is
critical to both the waveform loss and the modeling of fast transients and
high frequencies. When replaced with a single-scale high-hop mel
reconstruction (80 mels, with a window length of 512), we find significantly
lower SI-SDR (7.68 from 9.12). Subjectively, we find that this model does much
better at capturing certain sorts of sounds, such as cymbal crashes, beeping
and alarms, and singing vocals. We retain the multi-scale mel reconstruction
loss in our final recipe.

Latent dimension of codebook: the latent dimension of the codebook has a
significant impact on bitrate efficiency, and consequently the reconstruction
quality. If set too low or too high (e.g. 2, 256), quantitative metrics are
significantly worse with drastically lowered bitrate efficiency. Lower bitrate
efficiency results in effectively lowered bandwidth, which harms the modeling
capability of the generator. As the generator is weakened, the discriminator
tends to “win”, and thus the generator does not learn to generate audio with
high audio quality. We find $8$ to be optimal for the latent dimension.

Quantization setup: we find that using exponential moving average as the
codebook learning method, as in EnCodec[8], results in worse metrics
especially for SI-SDR. It also results in poorer codebook utilization across
all codebooks (Figure 1). When taken with its increased implementation
complexity (requiring K-Means initialization and random restarts), we retain
our simpler projected lookup method for learning codebooks, along with a
commitment loss. Next, we note that the quantization dropout rate has a
significant effect on the quantitative metrics. However, as seen in Figure 2,
a dropout of $0.0$ results in poor reconstruction with fewer codebooks. As
this makes usage of the codec challenging for downstream generative modeling
tasks, we instead use a dropout rate of $0.5$ in our final model. This
achieves a good trade-off between audio quality at full bitrate as well as
lower bitrates. Finally, we show that we can increase the max bitrate of our
model from 8kbps to 24kbps and achieve excellent audio quality, surpassing all
other model configurations. However, for our final model, we train at the
lower bitrates, in order to push the compression rate as much as possible.

Balanced data sampling: When removed, this results in worse metrics across the
board. Empirically, we find that without balanced data sampling, the model
produces waveforms that have a max frequency of around 18kHz. This corresponds
to the max frequency preserved by various audio compression algorithms like
MPEG, which make up the vast majority of our datasets. With balanced data
sampling, we sample full-band audio from high-quality datasets (e.g. DAPS)
just as much as possibly band-limited audio from datasets of unknown quality
(e.g. Common Voice). This alleviates the issue, allowing our codec to
reconstruct full-band audio, as well as band-limited audio.

###  4.6 Comparison to other methods

We now compare the performance of our final model with competitive baselines:
EnCodec [8], Lyra [45], and Opus [36], a popular open-source audio codec. For
EnCodec, Lyra, and Opus, we use publicly available open-source implementations
provided by the authors. We compare using both objective and subjective
evaluations, at varying bitrates. The results are shown in Table 3. We find
that the proposed codec out-performs all competing codecs at all bitrates in
terms of both objective and subjective metrics, while modeling a much wider
bandwidth of 22kHz.

Figure 3: MUSHRA scores, with 95% confidence intervals vs bitrate for EnCodec,
our proposed approach, and the reference.

Codec |  Bitrate (kbps) |  Bandwidth (kHz) |  Mel distance $\downarrow$ |  STFT distance $\downarrow$ |  ViSQOL $\uparrow$ |  SI-SDR $\uparrow$
---|---|---|---|---|---|---
Proposed | 1.78 | 22.05 | 1.39 | 1.95 | 3.76 | 2.16
2.67 | 22.05 | 1.28 | 1.85 | 3.90 | 4.41
5.33 | 22.05 | 1.07 | 1.69 | 4.09 | 8.13
8 | 22.05 | 0.93 | 1.60 | 4.18 | 10.75
EnCodec | 1.5 | 12 | 2.11 | 4.30 | 2.82 | -0.02
3 | 12 | 1.97 | 4.19 | 2.94 | 2.94
6 | 12 | 1.83 | 4.10 | 3.05 | 5.99
12 | 12 | 1.70 | 4.02 | 3.13 | 8.36
24 | 12 | 1.61 | 3.97 | 3.16 | 9.59
Lyra | 9.2 | 8 | 2.71 | 4.86 | 2.19 | -14.52
Opus | 8 | 4 | 3.60 | 5.72 | 2.06 | 5.68
14 | 16 | 1.23 | 2.14 | 4.02 | 8.02
24 | 16 | 0.88 | 1.90 | 4.15 | 11.65

Table 3: Objective evaluation of the proposed codec at varying bitrates, along
with results from competing approaches.

In Figure 3, we show the result of our MUSHRA study, which compares EnCodec to
our proposed codec at various bitrates. We find that our codec achieves much
higher MUSHRA scores than EnCodec at all bitrates. However, even at the
highest bitrate, it still falls short of the reference MUSHRA score,
indicating that there is room for improvement. We note that the metrics of our
final model are still lower than the 24kbps model trained in our ablation
study, as can be seen in Table 2. This indicates that the remaining
performance gap may be closed by increasing the maximum bitrate.

##  5 Conclusion

We have presented a high-fidelity universal neural audio compression algorithm
that achieves remarkable compression rates while maintaining audio quality
across various types of audio data. Our method combines the latest
advancements in audio generation, vector quantization techniques, and improved
adversarial and reconstruction losses. Our extensive evaluation against
existing audio compression algorithms demonstrates the superiority of our
approach, providing a promising foundation for future high-fidelity audio
modeling. With thorough ablations, open-source code, and trained model
weights, we aim to contribute a useful centerpiece to the generative audio
modeling community.

Broader impact and limitations: our model has the capability to make
generative modeling of full-band audio much easier to do. While this unlocks
many useful applications, such as media editing, text-to-speech synthesis,
music synthesis, and more, it can also lead to harmful applications like
deepfakes. Care should be taken to avoid these applications. One possibility
is to add watermarking and/or train a classifier that can detect whether or
not the codec is applied, in order to enable the detection of synthetic media
generated based on our codec. Also, our model is not perfect, and still has
difficulty reconstructing some challenging audio. By slicing the results by
domain we find that, even though the proposed codec outperforms competing
approaches across all of the domains, it performs best for speech and has more
issues with environmental sounds. Finally, we notice that it does not model
some musical instruments perfectly, such as glockenspeil, or synthesizer
sounds.

## References

  * Agostinelli et al. [2023] Andrea Agostinelli, Timo I Denk, Zalán Borsos, Jesse Engel, Mauro Verzetti, Antoine Caillon, Qingqing Huang, Aren Jansen, Adam Roberts, Marco Tagliasacchi, et al.  Musiclm: Generating music from text.  _arXiv preprint arXiv:2301.11325_ , 2023.
  * Ardila et al. [2019] Rosana Ardila, Megan Branson, Kelly Davis, Michael Henretty, Michael Kohler, Josh Meyer, Reuben Morais, Lindsay Saunders, Francis M Tyers, and Gregor Weber.  Common voice: A massively-multilingual speech corpus.  _arXiv preprint arXiv:1912.06670_ , 2019.
  * Bengio et al. [2013] Yoshua Bengio, Nicholas Léonard, and Aaron Courville.  Estimating or propagating gradients through stochastic neurons for conditional computation.  _arXiv preprint arXiv:1308.3432_ , 2013.
  * Bogdanov et al. [2019] Dmitry Bogdanov, Minz Won, Philip Tovstogan, Alastair Porter, and Xavier Serra.  The mtg-jamendo dataset for automatic music tagging.  In _Machine Learning for Music Discovery Workshop, International Conference on Machine Learning (ICML 2019)_ , Long Beach, CA, United States, 2019\.  URL http://hdl.handle.net/10230/42015.
  * Borsos et al. [2022] Zalán Borsos, Raphaël Marinier, Damien Vincent, Eugene Kharitonov, Olivier Pietquin, Matt Sharifi, Olivier Teboul, David Grangier, Marco Tagliasacchi, and Neil Zeghidour.  Audiolm: a language modeling approach to audio generation.  _arXiv preprint arXiv:2209.03143_ , 2022.
  * Brown et al. [2020] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al.  Language models are few-shot learners.  _Advances in neural information processing systems_ , 33:1877–1901, 2020.
  * Chinen et al. [2020] Michael Chinen, Felicia SC Lim, Jan Skoglund, Nikita Gureev, Feargus O’Gorman, and Andrew Hines.  Visqol v3: An open source production ready objective speech and audio metric.  In _2020 twelfth international conference on quality of multimedia experience (QoMEX)_ , pages 1–6. IEEE, 2020.
  * Défossez et al. [2022] Alexandre Défossez, Jade Copet, Gabriel Synnaeve, and Yossi Adi.  High fidelity neural audio compression.  _arXiv preprint arXiv:2210.13438_ , 2022.
  * Dhariwal et al. [2020] Prafulla Dhariwal, Heewoo Jun, Christine Payne, Jong Wook Kim, Alec Radford, and Ilya Sutskever.  Jukebox: A generative model for music.  _arXiv preprint arXiv:2005.00341_ , 2020.
  * Dubey et al. [2022] Harishchandra Dubey, Vishak Gopal, Ross Cutler, Sergiy Matusevych, Sebastian Braun, Emre Sefik Eskimez, Manthan Thakker, Takuya Yoshioka, Hannes Gamper, and Robert Aichner.  Icassp 2022 deep noise suppression challenge.  In _ICASSP_ , 2022.
  * Engel et al. [2020] Jesse Engel, Lamtharn Hantrakul, Chenjie Gu, and Adam Roberts.  Ddsp: Differentiable digital signal processing.  _arXiv preprint arXiv:2001.04643_ , 2020.
  * Esser et al. [2021] Patrick Esser, Robin Rombach, and Bjorn Ommer.  Taming transformers for high-resolution image synthesis.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 12873–12883, 2021.
  * Gârbacea et al. [2019] Cristina Gârbacea, Aäron van den Oord, Yazhe Li, Felicia SC Lim, Alejandro Luebs, Oriol Vinyals, and Thomas C Walters.  Low bit-rate speech coding with vq-vae and a wavenet decoder.  In _ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pages 735–739. IEEE, 2019.
  * Gemmeke et al. [2017] Jort F Gemmeke, Daniel PW Ellis, Dylan Freedman, Aren Jansen, Wade Lawrence, R Channing Moore, Manoj Plakal, and Marvin Ritter.  Audio set: An ontology and human-labeled dataset for audio events.  In _2017 IEEE international conference on acoustics, speech and signal processing (ICASSP)_ , pages 776–780. IEEE, 2017.
  * Gritsenko et al. [2020] Alexey Gritsenko, Tim Salimans, Rianne van den Berg, Jasper Snoek, and Nal Kalchbrenner.  A spectral energy distance for parallel speech synthesis.  _Advances in Neural Information Processing Systems_ , 33:13062–13072, 2020.
  * Jang et al. [2021] Won Jang, Dan Lim, Jaesam Yoon, Bongwan Kim, and Juntae Kim.  Univnet: A neural vocoder with multi-resolution spectrogram discriminators for high-fidelity waveform generation.  _arXiv preprint arXiv:2106.07889_ , 2021.
  * Kim et al. [2021] Jaehyeon Kim, Jungil Kong, and Juhee Son.  Conditional variational autoencoder with adversarial learning for end-to-end text-to-speech.  In _International Conference on Machine Learning_ , pages 5530–5540. PMLR, 2021.
  * Kong et al. [2020] Jungil Kong, Jaehyeon Kim, and Jaekyoung Bae.  Hifi-gan: Generative adversarial networks for efficient and high fidelity speech synthesis.  _Advances in Neural Information Processing Systems_ , 33:17022–17033, 2020.
  * Kumar et al. [2019] Kundan Kumar, Rithesh Kumar, Thibault De Boissiere, Lucas Gestin, Wei Zhen Teoh, Jose Sotelo, Alexandre de Brébisson, Yoshua Bengio, and Aaron C Courville.  Melgan: Generative adversarial networks for conditional waveform synthesis.  _Advances in neural information processing systems_ , 32, 2019.
  * Le Roux et al. [2019] Jonathan Le Roux, Scott Wisdom, Hakan Erdogan, and John R Hershey.  Sdr–half-baked or well done?  In _ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pages 626–630. IEEE, 2019.
  * Lee et al. [2022] Sang-gil Lee, Wei Ping, Boris Ginsburg, Bryan Catanzaro, and Sungroh Yoon.  Bigvgan: A universal neural vocoder with large-scale training.  _arXiv preprint arXiv:2206.04658_ , 2022.
  * Lim and Ye [2017] Jae Hyun Lim and Jong Chul Ye.  Geometric gan.  _arXiv preprint arXiv:1705.02894_ , 2017.
  * Mehri et al. [2016] Soroush Mehri, Kundan Kumar, Ishaan Gulrajani, Rithesh Kumar, Shubham Jain, Jose Sotelo, Aaron Courville, and Yoshua Bengio.  Samplernn: An unconditional end-to-end neural audio generation model.  _arXiv preprint arXiv:1612.07837_ , 2016.
  * Morrison et al. [2021] Max Morrison, Rithesh Kumar, Kundan Kumar, Prem Seetharaman, Aaron Courville, and Yoshua Bengio.  Chunked autoregressive gan for conditional waveform synthesis.  _arXiv preprint arXiv:2110.10139_ , 2021.
  * Mysore [2014] Gautham J Mysore.  Can we automatically transform speech recorded on common consumer devices in real-world environments into professional production quality speech?—a dataset, insights, and challenges.  _IEEE Signal Processing Letters_ , 22(8):1006–1010, 2014.
  * Oord et al. [2016] Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu.  Wavenet: A generative model for raw audio.  _arXiv preprint arXiv:1609.03499_ , 2016.
  * Ping et al. [2017] Wei Ping, Kainan Peng, Andrew Gibiansky, Sercan O Arik, Ajay Kannan, Sharan Narang, Jonathan Raiman, and John Miller.  Deep voice 3: Scaling text-to-speech with convolutional sequence learning.  _arXiv preprint arXiv:1710.07654_ , 2017.
  * Pons et al. [2021] Jordi Pons, Santiago Pascual, Giulio Cengarle, and Joan Serrà.  Upsampling artifacts in neural audio synthesis.  In _ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pages 3005–3009. IEEE, 2021.
  * Prenger et al. [2019] Ryan Prenger, Rafael Valle, and Bryan Catanzaro.  Waveglow: A flow-based generative network for speech synthesis.  In _ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pages 3617–3621. IEEE, 2019.
  * Rafii et al. [2017] Zafar Rafii, Antoine Liutkus, Fabian-Robert Stöter, Stylianos Ioannis Mimilakis, and Rachel Bittner.  The musdb18 corpus for music separation, 2017.
  * Ramesh et al. [2021] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever.  Zero-shot text-to-image generation.  In _International Conference on Machine Learning_ , pages 8821–8831. PMLR, 2021.
  * Razavi et al. [2019] Ali Razavi, Aaron Van den Oord, and Oriol Vinyals.  Generating diverse high-fidelity images with vq-vae-2.  _Advances in neural information processing systems_ , 32, 2019.
  * Ren et al. [2020] Yi Ren, Chenxu Hu, Xu Tan, Tao Qin, Sheng Zhao, Zhou Zhao, and Tie-Yan Liu.  Fastspeech 2: Fast and high-quality end-to-end text to speech.  _arXiv preprint arXiv:2006.04558_ , 2020.
  * Shen et al. [2018] Jonathan Shen, Ruoming Pang, Ron J Weiss, Mike Schuster, Navdeep Jaitly, Zongheng Yang, Zhifeng Chen, Yu Zhang, Yuxuan Wang, Rj Skerrv-Ryan, et al.  Natural tts synthesis by conditioning wavenet on mel spectrogram predictions.  In _2018 IEEE international conference on acoustics, speech and signal processing (ICASSP)_ , pages 4779–4783. IEEE, 2018.
  * Tan et al. [2022] Xu Tan, Jiawei Chen, Haohe Liu, Jian Cong, Chen Zhang, Yanqing Liu, Xi Wang, Yichong Leng, Yuanhao Yi, Lei He, et al.  Naturalspeech: End-to-end text to speech synthesis with human-level quality.  _arXiv preprint arXiv:2205.04421_ , 2022.
  * Valin et al. [2012] Jean-Marc Valin, Koen Vos, and Timothy Terriberry.  Definition of the opus audio codec.  Technical report, 2012.
  * Van Den Oord et al. [2017] Aaron Van Den Oord, Oriol Vinyals, et al.  Neural discrete representation learning.  _Advances in neural information processing systems_ , 30, 2017.
  * Vaswani et al. [2017] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  _Advances in neural information processing systems_ , 30, 2017.
  * Veaux et al. [2017] Christophe Veaux, Junichi Yamagishi, Kirsten MacDonald, et al.  Cstr vctk corpus: English multi-speaker corpus for cstr voice cloning toolkit.  _University of Edinburgh. The Centre for Speech Technology Research (CSTR)_ , 2017.
  * Wang et al. [2023] Chengyi Wang, Sanyuan Chen, Yu Wu, Ziqiang Zhang, Long Zhou, Shujie Liu, Zhuo Chen, Yanqing Liu, Huaming Wang, Jinyu Li, et al.  Neural codec language models are zero-shot text to speech synthesizers.  _arXiv preprint arXiv:2301.02111_ , 2023.
  * Yamamoto et al. [2020] Ryuichi Yamamoto, Eunwoo Song, and Jae-Min Kim.  Parallel wavegan: A fast waveform generation model based on generative adversarial networks with multi-resolution spectrogram.  In _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pages 6199–6203. IEEE, 2020.
  * Yang et al. [2021] Geng Yang, Shan Yang, Kai Liu, Peng Fang, Wei Chen, and Lei Xie.  Multi-band melgan: Faster waveform generation for high-quality text-to-speech.  In _2021 IEEE Spoken Language Technology Workshop (SLT)_ , pages 492–498. IEEE, 2021.
  * Yu et al. [2021] Jiahui Yu, Xin Li, Jing Yu Koh, Han Zhang, Ruoming Pang, James Qin, Alexander Ku, Yuanzhong Xu, Jason Baldridge, and Yonghui Wu.  Vector-quantized image modeling with improved vqgan.  _arXiv preprint arXiv:2110.04627_ , 2021.
  * Yu et al. [2022] Jiahui Yu, Yuanzhong Xu, Jing Yu Koh, Thang Luong, Gunjan Baid, Zirui Wang, Vijay Vasudevan, Alexander Ku, Yinfei Yang, Burcu Karagol Ayan, et al.  Scaling autoregressive models for content-rich text-to-image generation.  _arXiv preprint arXiv:2206.10789_ , 2022.
  * Zeghidour et al. [2021] Neil Zeghidour, Alejandro Luebs, Ahmed Omran, Jan Skoglund, and Marco Tagliasacchi.  Soundstream: An end-to-end neural audio codec.  _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , 30:495–507, 2021.
  * Ziyin et al. [2020] Liu Ziyin, Tilman Hartwig, and Masahito Ueda.  Neural networks fail to learn periodic functions and how to fix it.  _Advances in Neural Information Processing Systems_ , 33:1583–1594, 2020.


# Möbius boron–nitride nanobelts interacting with heavy metal nanoclusters

C. Aguiar1 N. Dattani2,3<EMAIL_ADDRESS>I. Camps1,3<EMAIL_ADDRESS>1Laboratório de Modelagem Computacional - _La_ Model, Instituto de Ciências
Exatas - ICEx. Universidade Federal de Alfenas - UNIFAL-MG, Alfenas, Minas
Gerais, Brasil  2HPQC College, Waterloo, Canada  3HPQC Labs, Waterloo, Canada

###### Abstract

How do nickel, cadmium, and lead nanoclusters interact with boron-nitride and
Mobius-type boron-nitride nanobelts? To answer this question, we used the
semiempirical tight binding framework, as implemented in the xTB software, to
determine the lowest energy geometries, binding energy, complexes stability,
and electronic properties. Our calculations show that heavy metal nanoclusters
favorably bind to both boron-nitride nanobelts, although the interaction is
stronger with the Mobius-type nanobelt. The calculations show that the nickel
nanocluster has the lowest binding energy and the greatest charge transfer
with the nanobelts, followed by the cadmium and lead nanoclusters. During the
simulation time, the molecular dynamic simulation showed that all complexes
were stable at 298 K. Following the nanobelt’s symmetry, the frontier orbitals
are distributed homogeneously throughout the structure. This distribution
changed when the nanobelt was twisted to create the Mobius-type nanobelt. The
topological study indicated that the number of bonds between the metal
nanoclusters and the Mobius-type nanobelt doubled and that the bonds formed
with the nickel nanocluster were stronger than those formed with the cadmium
and lead metals. Combining all the results, we conclude that the nickel
nanoclusters are chemisorbed, whereas the cadmium and lead nanoclusters are
physisorbed in both nanobelts.

###### keywords:

heavy metals , boron nitride nanobelt , Möbius belt , Nickel , Cadmium , Lead

##  1 Introduction

Industrialization and population growth have significantly contributed to the
increase in pollution [1; 2]. Heavy metal ions, when released into the
environment, can cause numerous diseases such as cancer, hepatitis,
spontaneous abortions, anemia, among others [3; 4]. Lead (Pb), for example, is
a carcinogenic element and its toxicity can cause mental retardation,
congenital defects, brain damage, and death [5]. Nickel ions (Ni) can cause
skin diseases [6]. Cadmium (Cd) is not only carcinogenic but can also cause
damage to the kidneys and respiratory system [7; 8]. This is because these
metals have no biological function, but can accumulate and interfere with
metabolism and physiological processes [8]. Therefore, prevention, as well as
the identification and treatment of pollutants, are essential for the
maintenance and preservation of the environment. In this sense, nanotechnology
can play a fundamental role in identifying and treating effluents [8; 9].

Nanocarbon materials, due to their large surface area, have mesopores that
make them ideal for removing heavy metal ions through adsorption [10; 11; 8].
Moreover, these nanomaterials can be functionalized with numerous molecules
that can make them specific to adsorb certain materials [12; 8]. Carbon
nanostructures have already been reported for use in heavy metal adsorption
and are also used in purifying water contaminated with heavy metals [13; 14;
11; 8; 15; 16]. In addition, it is reported that the adsorbed metal ions can
be easily desorbed, and the nanomaterials can be recycled and reused [8].
Furthermore, carbon-based nanomaterials are highly biocompatible with living
organisms and the environment [17].

When carbon atoms are replaced by boron and nitrogen atoms, boron nitride
nanotubes (BNNTs) are obtained [18]. With an identical structure to carbon
nanotubes, BNNTs have similar mechanical properties [19; 18] and they are
electrically insulating, with a forbidden band between 5.0–6.0 eV, independent
of chirality [20; 18]. In addition, the thermal properties are improved due to
boron nitride (BN) having high atmospheric stability, especially at high
temperatures [21]. This makes BNNT an excellent candidate for applications in
electronics [22], sensors [23], hydrogen storage [24], medicine [25; 26], and
also water purification [15; 27; 28; 16; 29; 30] among others.

Due to these particularities, research on a new structure has been
investigated, the boron nitride nanobelts (BN-nanobelts), an inorganic analog
of cyclofenacene synthesized in 2017 [31; 32]. Aromatic BN-nanobelts have
radially oriented p orbitals with photoluminescent properties and excellent UV
absorber, suggesting that this molecule can be used as a UV detector [32; 33;
34]. In addition, BN-nanobelts have high chemical stability, thermal stability
with positive vibrational frequencies and insulating character with an
estimated gap of around 5 eV [35; 32]. Considering this, properties
originating from these different topologies can help solve current problems
such as pollution caused by undesirable chemical contaminants that cause
adverse effects on nature and living organisms [8; 9]. Contamination by heavy
metal ions, mainly in water resources, is a serious environmental problem.

In this study, the interaction of Cadmium (Cd), Nickel (Ni), and Lead (Pb),
with boron–nitride and Möbius-type boron–nitride nanobelts were investigated
using the semiempirical tight binding theory. Several methods were used to
characterize the systems: best interaction region detection, geometry
optimization, molecular dynamics, electronic property calculations, and
topology studies.

##  2 Materials and Methods

In this work, three heavy metals (Cd, Ni, and Pb) were used, together with two
different types of boron-nitride (BN) nanostructures.

The metals considered here are in the form of four–atom nanoclusters (M4). We
considered a one-dimensional linear chain (1DL, Figure 1(a)), a one-
dimensional zigzag chain (1DZ, Figure 1(b)), a two-dimensional plane (2D,
1(c)), and a three-dimensional tetrahedron (3D, 1(d)).

Two types of BN nanostructures were considered: one consisting of a nanobelt
(NB) and the other consisting of a Möbius nanobelt (twisted nanobelt). The
structures were generated using the Virtual NanoLab Atomistix Toolkit software
[36] as follows. We started with 2 unit cells of (10,0) boron-nitride
nanosheet repeated 10 times in the z direction and then wrapped 360 degrees.
After that, the periodicity was removed, and the border atoms were passivated
with hydrogen. In the case of Möbius nanobelts, after the initial repetition
of the cells, the nanobelt was twisted 180 degrees and then wrapped. Both
nanostructures are shown in Figure 2.

In order to clearly identify the systems, the following nomenclature was used:
BNNB for boron-nitride nanobelt, MBNNB for Möbius boron-nitride nanobelt, and
BNNB+M4 (MBNNB+M4) for the complexes formed by the BNNB (MBNNB) and the metal
cluster, M4.

|
---|---
|

(a) 1DL

(b) 1DZ

(c) 2D

(d) 3D

Figure 1:  Nanoclusters geometry: (a) one dimensional linear chain (1DL), (b)
one dimensional zigzag chain (1DZ), (c) two dimensional plane (2D), and (d)
three dimensional tetrahedron (3D).

|
---|---

(a) BNNB

(b) MBNNB

Figure 2:  (a) Boron–nitride nanobelt (BNNB). (b) Möbius boron–nitride
nanobelt (MBNNB).

To perform the calculations, we used the semiempirical tight binding method,
as implemented in the xTB program [37; 38]. The calculations were done
following a series of steps described below.

First, we optimized the structures of each individual system (two nanobelts
and four nanoclusters for each metal). Then, using the automated Interaction
Site Screening (aISS) [39], we generated different intermolecular geometries
(e.g., BNNB+M4 and MBNNB+M4) and subjected them to further genetic
optimization. We used the interaction energy (xTB-IFF) [40] of each newly
generated structure for ranking, and the genetic step was repeated ten times
until the best complex was obtained. The best–ranked complexes were further
subjected to structural optimization. In order to study the complexes’
stability, each structure obtained in the aISS step was subjected to a
molecular dynamic (MD) simulation for a period of 100 ps.

All geometry optimizations were performed using the GFN2–xTB method, which is
an accurate self–consistent method that includes multipole electrostatics and
density–dependent dispersion contributions [41]. Extreme optimization level
was ensured, with a convergence energy of $5\times 10^{-8}$ Eh and gradient
norm convergence of $5\times 10^{-5}$ Eh/a0 (where a0 is the Bohr radius). MD
simulations were also conducted using the GFN2–xTB method.

The electronic properties calculated included the system energy, the energy of
the highest occupied molecular orbital (HOMO) ($\varepsilon_{H}$), the energy
of the lowest unoccupied molecular orbital (LUMO) ($\varepsilon_{L}$), the
energy gap between HOMO and LUMO orbitals
($\Delta\varepsilon=\varepsilon_{H}-\varepsilon_{L}$, and the atomic charges
using the CM5 scheme [42].

From the calculated charges, we estimated the charge transfer between the
isolated metal nanocluster and the nanobelts using the expression

$\Delta{Q_{M4}}=Q_{M4}^{ads}-Q_{M4}^{iso},$ (1)

where $Q_{M4}^{ads}$ is the total charge on the metal nanocluster after
adsorption, and $Q_{M4}^{iso}$ is the total charge for the isolated metal
nanocluster.

The binding energies ($E_{b}$) of the adsorbed metals on the nanobelts were
calculated using the following expression

$E_{b}=E_{NB+M4}-E_{NB}-E_{M4}.$ (2)

In equation 2, $E_{NB}$ and $E_{M4}$ are the energies for the isolated
nanobelts and metal nanoclusters, respectively, and $E_{NB+M4}$ is the energy
of the NB+M4 complex (BNNB+M4 and MBNNB+M4 systems).

Finally, the topological characterization was carried out using the MULTIWFN
[43] software,where the topological properties such as critical points,
critical path, basins, etc. were determined for each complex.

##  3 Results and discussion

###  3.1 Nanoclusters adsorption at the BN nanobelts

Figure 3 shows the fully relaxed structures, with the point of view being the
same as in Figure 2. Only the complexes with the lowest final energy for each
metal are displayed. For BNNB, the Cd1DL, Ni1DZ, and Pb2D nanoclusters had the
lowest energy complexes. The MBNNB behaved similarly for cadmium and lead, but
for nickel, the optimal structure was with the Ni2D cluster. At first sight,
the interaction with nickel nanoclusters deforms the nanobelts to a greater
extent than the other metals and also forms more bonds. Both behaviors may
indicate greater binding between BNNB and MBNNB with Ni.

|  |
---|---|---
|  |

(a) BNNB+Cd1DL

(b) BNNB+Ni1DZ

(c) BNNB+Pb2D

(d) MBNNB+Cd1DL

(e) MBNNB+Ni2D

(f) MBNNB+Pb2D

Figure 3:  Fully relaxed complexes with lowest energies. Image rendered with
Jmol software [44].

A significant contrast between chemisorption and physisorption is the effect
on the electronic states of both the adsorbate and the adsorbent [45]. We will
observe in sections 3.2 and 3.3 that the electronic properties of the systems
are altered upon adsorption, confirming this phenomenon.

Table 1: Data from geometry optimization and electronic calculation†.

Complex | $E_{b}^{aISS}$ | $E_{b}$ | $\Delta Q_{M}$ | Gap ($\Delta\varepsilon$) | Distances
---|---|---|---|---|---
BNNB+Cd1DL | -95.33 | -82.87 | 0.0622 | 0.999 | 2.48/3.03/3.18/3.23/3.32/3.54
BNNB+Ni1DZ | -142.77 | -116.01 | -0.1195 | 0.144 | 2.24/2.27/2.32/2.44
BNNB+Pb2D | -93.36 | -54.31 | 0.0014 | 1.226 | 3.51/3.53/3.71
MBNNB+Cd1DL | -109.25 | -96.78 | 0.0085 | 0.826 | 2.59/2.68/2.85/2.88/2.90/3.38
|  |  |  |  | 3.39/3.41/3.58/3.69/3.83/3.92
MBNNB+Ni2D | -163.68 | -149.83 | -0.2223 | 0.109 | 2.15/2.21/2.52/2.57/2.70/2.71/3.60
MBNNB+Pb2D | -99.00 | -59.98 | 0.0248 | 1.449 | 2.81/2.92/3.31/3.37/3.61/4.41

† $E_{b}^{aISS}$ and $E_{b}$ are in units kcal/mol, $\Delta Q_{M}$ are in
units of $e$, $\Delta\varepsilon$ is in units of $eV$ and the distances are in
Å, respectively.

Table 1 displays the data for the complexes with the lowest energy. Two
binding energies are reported: $E_{b}^{aISS}$ and $E_{b}$. The former is
obtained from the aISS step, while the latter is calculated using equation 2
for the fully optimized individual structures and final complex. Comparing
both binding energies for each system, the Ni clusters have the lowest binding
energies, indicating stronger adsorption, followed by Cd and Pb. As all the
binding energies are negative, all the adsorption processes are favorable.

The graphical representation of the complexes in Figure 3 suggests that only
Cd and Ni formed bonds with the nanobelts. In the case of Cd, it made one bond
with BNNB and two bonds with MBNNB. The Ni cluster formed ten bonds with BNNB
and eight with MBNND, whereas Pb did not make any bonds with either BNNB or
MBNNB. Since the bond information from Figure 3 is only based on geometrical
data, the Quantum Theory of Atoms in Molecule (QTAIM) [46] was used in Section
3.3 as a more accurate method to study bond formation.

Geometry optimization involves using an algorithm to obtain a local minimum
structure on the potential energy surface (PES), which allows us to determine
the lowest energy conformers of a system. However, this method provides no
information about the system’s stability over time. Molecular dynamics
simulation, on the other hand, analyzes the movement of atoms and molecules at
a specific temperature (here, 298.15 K) and provides a way to explore the PES.
We used this method to perform simulations on each complex, starting with the
structures obtained from the aISS step as initial conformations. The
simulations were run for a production time frame of 100 ps with a time step of
2 fs and an optional dump step of 50 fs, at which the final structure was
written to a trajectory file.

Figures 4 and 5 show the system frames at several simulation times (0 ps, 25
ps, 50 ps, 75 ps, and 100 ps). In all cases, the metal nanoclusters remained
bound to their respective BN nanobelts, indicating system stability. By
comparing the snapshots from figures 4 and 5 with Figure 2, we confirmed that
the Ni nanocluster caused the most significant modifications to the nanobelt.
The full molecular dynamics movies can be downloaded from the Zenodo server
[47].

|  |  |  |
---|---|---|---|---
|  |  |  |
|  |  |  |

(a) 0 ps (b) 25 ps

(c) 50 ps (d) 75 ps (e) 100 ps

(f) 0 ps (g) 25 ps

(h) 50 ps (i) 75 ps

(j) 100 ps (k) 0 ps

(l) 25 ps (m) 50 ps

(n) 75 ps (o) 100 ps

Figure 4:  Molecular dynamics snapshots at different simulation times for BNNB
complexes. Simulation times (from left to right): 0 ps, 25 ps, 50 ps, 75 ps
and 100 ps, respectively. And systems (from top to bottom): BNNB+Cd1DL,
BNNB+Ni1DZ and BNNB+Pb2D, respectively.

|  |  |  |
---|---|---|---|---
|  |  |  |
|  |  |  |

(a) 0 ps (b) 25 ps

(c) 50 ps (d) 75 ps (e) 100 ps

(f) 0 ps (g) 25 ps

(h) 50 ps (i) 75 ps

(j) 100 ps (k) 0 ps

(l) 25 ps (m) 50 ps

(n) 75 ps (o) 100 ps

Figure 5:  Molecular dynamics snapshots at different simulation times for
MBNNB complexes. Simulation times (from left to right): 0 ps, 25 ps, 50 ps, 75
ps and 100 ps, respectively. And systems (from top to bottom): MBNNB+Cd1DL,
MBNNB+Ni2D and MBNNB+Pb2D, respectively.

###  3.2 Electronic properties

Figures 6 and 7 show the calculated highest occupied/lowest unoccupied
molecular orbitals (HOMO/LUMO) for all optimized systems.

The molecular orbitals for the BNNB system (Figures 6(a) and 7(a)) are
homogeneously distributed over the entire structure, as expected due to the
belt symmetry. These distributions change when the nanobelts are twisted,
breaking their symmetry (figures 6(e) and 7(e)). In the Möbius nanobelt, there
is a redistribution of both molecular orbital surfaces, with more volume
concentrated around the regions with the twist, especially for the HOMO. The
electronic gap for the BNNB is equal to 3.973 eV, which is slightly decreased
to 3.859 eV for the MBNNB.

In the case of the complexes, the interaction with the heavy metal
nanoclusters greatly modifies the HOMO for all systems, with the orbital
volume becoming more concentrated around the metal adsorption regions. A
similar behavior occurs for the LUMO, but to a greater extent for Ni and Pb
than for Cd. The gap for the complexes is also modified after adsorption, with
the greatest change observed for Ni, followed by Cd and Pb. The interaction of
both nanobelts with all the metals decreases the gap, as shown in Table 1.
Again, the interaction with Ni induced the greatest change in the electronic
gap.

Another indicator of adsorption strength is the charge transfer
($\Delta{Q_{M4}}$) between the adsorbent and adsorbate. The $\Delta{Q_{M4}}$
was calculated using the equation 1 and its value is shown in Table 1 for all
systems. In both cases (BNNB and MBNNB), the greatest charge transfer occurs
between the nanobelts and the Ni nanocluster.

Based on the intensity of the modifications in the electronic properties of
the BNNB and MBNNB after interaction with metals, we can conclude that
chemisorption occurs for all systems. This conclusion from the analysis of
electronic properties will be confirmed by the topological studies presented
in the next section.

|  |  |
---|---|---|---
|  |  |

(a) BNNB

(b) BNNB+Cd1DL

(c) BNNB+Ni1DZ

(d) BNNB+Pb2D

(e) MBNNB

(f) MBNNB+Cd1DL

(g) MBNNB+Ni2D

(h) MBNNB+Pb2D

Figure 6:  Highest occupied molecular orbital (HOMO) for all systems. Red
(green) color represents negative (positive) values. Orbital surfaces rendered
with with isovalue equal to 0.001 and with Jmol software [44].

|  |  |
---|---|---|---
|  |  |

(a) BNNB

(b) BNNB+Cd1DL

(c) BNNB+Ni1DZ

(d) BNNB+Pb2D

(e) MBNNB

(f) MBNNB+Cd1DL

(g) MBNNB+Ni2D

(h) MBNNB+Pb2D

Figure 7:  Lowest unoccupied molecular orbital (LUMO) for all systems. Red
(green) color represents negative (positive) values. Orbital surfaces rendered
with with isovalue equal to 0.001 and with Jmol software [44].

###  3.3 Topological analysis

In topological analysis, a key objective is to identify critical points. These
critical points (CPs) are points where the gradient norm of the function value
is zero, and they are classified into four types based on the negative
eigenvalues of the Hessian matrix of the real function [46].

When three eigenvalues of the Hessian matrix are negative, the CPs are known
as nuclear critical points (NCPs) and their positions are almost identical to
atomic positions. They are represented as $\bf(3,-3)$. If two eigenvalues of
the Hessian matrix are negative, the CPs are represented as $\bf(3,-1)$ and
are referred to as bond critical points (BCPs). These BCPs generally appear
between atom pairs in electron density analysis and are related to the
strength and type of bonds formed. The value of the electron density ($\rho$)
and the sign of its Laplacian ($\nabla^{2}\rho$) at the $\bf(3,-1)$ CPs can be
related to the strength and type of the bonds [48]. CPs with one negative
eigenvalue are represented as $\bf(3,+1)$ and are called ring critical points
(RCPs) because they usually appear in the center of a ring system. Finally,
CPs with no negative eigenvalues are represented as $\bf(3,+3)$ and are called
cage critical points (CCPs) because they typically appear in the center of a
cage system during electron density analysis.

Diagrams with the 3D distribution of critical points for each complex are
shown in Figure 8. The orange dots represent the bond critical points (BCPs),
the yellow dots represent the ring critical points (RCPs), and the green dots
represent the cage critical points. All the complexes show several critical
points, indicating a favorable interaction between the metals and the belts.
In all cases, the number of critical points made between the metal nanocluster
and the Möbious nanobelts is greater than with the nanobelts alone. This can
be associated with the fact that Möbious belts form like small pockets where
the metal nanocluster can be docked. The interaction between the Ni2D
nanocluster with the MBNNB is so intense that it pulled both sides of the belt
in such a way that new bonds are formed between both sides (see Figure 8(e)).
For better visualization of the formed critical points, movies with spinning
structures can be downloaded from the Zenodo server [47]. Another confirmation
that the strongest interactions are between the Ni nanoclusters and the boron-
nitride nanobelts is the lowest bond distances shown in Table 1.

|  |
---|---|---
|  |

(a) BNNB+Cd1DL

(b) BNNB+Ni1DZ

(c) BNNB+Pb2D

(d) MBNNB+Cd1DL

(e) MBNNB+Ni2D

(f) MBNNB+Pb2D

Figure 8:  Detected critical points: BCPs (orange dots), RCPs (yellow dots),
and CCPs (green dots). Image rendered with VMD software [49].

The strength of a bond, whether it is covalent or non–covalent, can be
classified by examining the electron density ($\rho$) and the sign of its
Laplacian ($\nabla^{2}\rho$). A value of $\rho$ greater than 0.20 a.u.
indicates a covalent bond, while a value less than 0.10 a.u. indicates a
non–covalent bond. Moreover, a bond can be classified as covalent if
$\nabla^{2}\rho$ is less than 0, and as non–covalent if $\nabla^{2}\rho$ is
greater than 0 [48]. The confinement of electron movement is related to the
ELF index, which takes values in the range of 0 to 1 [50; 51]. Large values of
the ELF index indicate that electrons are highly localized, which is
indicative of the presence of a covalent bond. The LOL index is another
function that is used to identify regions of high localization [52]. Values of
the LOL index are also in the range of 0 to 1, with smaller (larger) values
typically appearing in the boundary (inner) regions.

Figure 9 shows the electron density ($\rho$), Laplacian of the electron
density ($\nabla^{2}\rho$), electron localization function (ELF) index, and
localized orbital locator (LOL) index at all the detected bond critical
points. The maximum number of BCPs for MBNNB is twice that for BNNB,
indicating that using Möbius boron–nitride nanobelts to capture heavy metal
nanoclusters is a better choice. According to the classification criteria
above, higher values of $\rho$ indicate stronger bonds. Figures 9(a), 9(b),
9(e), and 9(f) show that MBNNB made stronger bonds with Ni clusters (green
regions) than with the other metals (orange regions for Cd clusters and gray
regions for Pb clusters). Even though Cd nanocluster forms more bonds with
MBNNB than Ni nanocluster, they have lower values of $\rho$ and
$\nabla^{2}\rho$. The ELF (figures 9(c), and 9(g)) and LOL (figures 9(d), and
9(h)) indexes show that some of the BCPs for Cd nanoclusters have greater
values than for Ni and Pb clusters. This higher electron localization of Cd
cluster is associated with the HOMO redistribution shown in figures 6(b), and
6(f). The topological analysis confirms that both boron–nitride nanobelts are
capable of adsorbing the three heavy metal nanoclusters studied here.
Nevertheless, the MBNNB presented greater values for all the descriptors used.
In all cases, the Ni nanoclusters are chemisorbed, whereas Cd and Pb
nanoclusters are physisorbed.

|  |  |
---|---|---|---
|  |  |

(a) BNNB+M4 ($\rho$)

(b) BNNB+M4 ($\nabla^{2}\rho$)

(c) BNNB+M4 (ELF)

(d) BNNB+M4 (LOL)

(e) MBNNB+M4 ($\rho$)

(f) MBNNB+M4 ($\nabla^{2}\rho$)

(g) MBNNB+M4 (ELF)

(h) MBNNB+M4 (LOL)

Figure 9:  Topology results. The data for Ni clusters is in green, for Cd
clusters is in orange and for Pb clusters is in gray.

##  4 Conclusions

In this work, the interactions of Ni, Cd, and Pb nanoclusters with boron-
nitride nanobelts with different geometries were studied within the
semiempirical tight binding framework using several methods: best interaction
region, geometry optimization, molecular dynamics, electronic property
calculations, and topology studies.

The more favorable interaction regions were determined using the automated
Interaction Site Screening (aISS) and used as a starting point for the other
simulations. All the structures (individual clusters, nanobelts, and
complexes) were optimized under extreme optimization level to ensure good
convergence. The final energy of the optimized structures was used to
calculate the binding energy for each complex. From the energy analysis, the
Ni nanocluster shows the lower binding energy (most favorable interaction)
followed by Cd and Pb nanoclusters. The final geometry for the MBNNB+Ni2D
system shows that the Ni nanocluster pulled the nanobelt sides in such a way
that both sides start interacting, creating a closed pocket around the metal
nanocluster. To study the complexes’ time stability, molecular dynamics
simulations were run for a production time of 100 ps, showing that, in all
cases, the heavy metals remain bound to the nanobelts.

The electronic calculations showed that the topology of the MBNNB changed the
HOMO/LUMO distribution when compared to BNNB as there is a break in symmetry
(for BNNB) that induced a redistribution of the orbitals around the twisted
region (for MBNNB). The interaction of heavy metals nanocluster produced a
further modification of the HOMO/LUMO surfaces, being now redistributed around
the region where the metal was located. Due to the metal/nanobelt interaction,
the charges of the nanocluster were also modified to a greater extent for the
Ni nanocluster than for the other metals.

The topological analysis detected the critical points used to better
characterize the type and strength of the interactions. The MBNNB has twice
the number of bond critical points than the BNNB system. This is an indicator
that using MBNNB to adsorb the heavy metals is a better choice. Comparing the
values of the descriptors used (electron density, Laplacian of the electron
density, ELF, and LOL indexes) for all the systems, we can conclude that the
Ni nanocluster is better adsorbed than the Cd and Pb nanoclusters.

Combining the results from the geometry optimization, the binding energy
calculation, and the topological analysis, we can conclude that the Ni
nanoclusters are chemisorbed, whereas Cd and Pb nanoclusters are physisorbed
in both nanobelts, but this adsorption is more favorable for the Möbius boron-
nitride nanobelts.

## CRediT authorship contribution statement

C. Aguiar: Investigation, Formal analysis, Writing–original draft,
Writing–review & editing. N. Dattani: Investigation, Resources, Formal
analysis, Writing–original draft, Writing–review & editing. I. Camps:
Conceptualization, Methodology, Software, Formal analysis, Resources,
Writing–review & editing, Supervision, Project administration.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data availability

Data will be made available on request.

## Acknowledgements

We would like to acknowledge financial support from the Brazilian agencies
CNPq, CAPES and FAPEMIG. Part of the results presented here were developed
with the help of a CENAPAD-SP (Centro Nacional de Processamento de Alto
Desempenho em São Paulo) grant UNICAMP/FINEP–MCT, CENAPAD–UFC (Centro Nacional
de Processamento de Alto Desempenho, at Universidade Federal do Ceará), and
Digital Research Alliance of Canada (via project bmh-491-09 belonging to Dr.
Nike Dattani), for the computational support.

## References

  * [1] P. Mehndiratta, A. Jain, S. Srivastava, N. Gupta, Environmental pollution and nanotechnology, Environment and Pollution 2 (2013) 1 (2013).  doi:10.5539/ep.v2n2p49.
  * [2] X. Zhang, Q. Huang, F. Deng, H. Huang, Q. Wan, M. Liu, Y. Wei, Mussel-inspired fabrication of functional materials and their environmental applications: Progress and prospects, Appl. Mater. Today 7 (2017) 222–238 (2017).  doi:10.1016/j.apmt.2017.04.001.
  * [3] J. Sardans, F. Montes, J. Peñuelas, Electrothermal atomic absorption spectrometry to determine As, Cd, Cr, Cu, Hg, and Pb in soils and sediments: A review and perspectives, Soil Sediment Contam. 20 (2011) 447–491 (2011).  doi:10.1080/15320383.2011.571526.
  * [4] M. Bali, H. Tlili, Removal of heavy metals from wastewater using infiltration-percolation process and adsorption on activated carbon, Int. J. Environ. Sci. Te. 16 (2018) 249–258 (2018).  doi:10.1007/s13762-018-1663-5.
  * [5] G. A. Engwa, P. U. Ferdinand, F. N. Nwalo, M. N. Unachukwu, Mechanism and health effects of heavy metal toxicity in humans, IntechOpen, 2019 (2019).  doi:10.5772/intechopen.82511.
  * [6] C. Moreno-Castilla, M. A. Álvarez-Merino, M. V. López-Ramón, J. Rivera-Utrilla, Cadmium ion adsorption on different carbon adsorbents from aqueous solutions. effect of surface chemistry, pore texture, ionic strength, and dissolved natural organic matter, Langmuir 20 (2004) 8142–8148 (2004).  doi:10.1021/la049253m.
  * [7] M. Jaishankar, T. Tseten, N. Anbalagan, B. B. Mathew, K. N. Beeregowda, Toxicity, mechanism and health effects of some heavy metals, Interdiscip. Toxicol. 7 (2014) 60–72 (2014).  doi:10.2478/intox-2014-0009.
  * [8] R. Baby, B. Saifullah, M. Z. Hussein, Carbon nanomaterials for the treatment of heavy metal-contaminated water and environmental remediation, Nanoscale Res. Lett. 14 (2019) 341 (2019).  doi:10.1186/s11671-019-3167-8.
  * [9] Y. Wu, H. Pang, Y. Liu, X. Wang, S. Yu, D. Fu, J. Chen, X. Wang, Environmental remediation of heavy metal ions by novel-nanomaterials: A review, Environ. Pollut. 246 (2019) 608–620 (2019).  doi:10.1016/j.envpol.2018.12.076.
  * [10] V. K. Gupta, S. Agarwal, T. A. Saleh, Chromium removal by combining the magnetic properties of iron oxide with adsorption properties of carbon nanotubes, Water Res. 45 (2011) 2207–2212 (2011).  doi:10.1016/j.watres.2011.01.012.
  * [11] A. E. Burakov, E. V. Galunin, I. V. Burakova, A. E. Kucherova, S. Agarwal, A. G. Tkachev, V. K. Gupta, Adsorption of heavy metals on conventional and nanostructured materials for wastewater treatment purposes: A review, Ecotox. Environ. Safe. 148 (2018) 702–712 (2018).  doi:10.1016/j.ecoenv.2017.11.034.
  * [12] M. Bastos, I. Camps, Interactions of lead with carboxyl and hydroxyl-decorated (10, 0) single-walled carbon nanotubes: First-principle calculations, Appl. Surf. Sci. 285P (2013) 198–204 (2013).  doi:10.1016/j.apsusc.2013.08.036.
  * [13] K. Kaneko, C. Ishii, T. Arai, H. Suematsu, Defect-associated microporous nature of fullerene C60 crystals, J. Phys. Chem. 97 (1993) 6764–6766 (1993).  doi:10.1021/j100128a002.
  * [14] M. Bastos, I. Camps, First-principles calculations of nickel, cadmium, and lead adsorption on a single-walled (10,0) carbon nanotube, J. Mol. Model. 20 (2014) 2094 (2014).  doi:10.1007/s00894-014-2094-y.
  * [15] H. Cho, J. H. Kim, J. H. Hwang, C. S. Kim, S. G. Jang, C. Park, H. Lee, M. J. Kim, Single- and double-walled boron nitride nanotubes: Controlled synthesis and application for water purification, Sci. Rep. 10 (2020) 7416 (2020).  doi:10.1038/s41598-020-64096-z.
  * [16] A. O. Maselugbo, H. B. Harrison, J. R. Alston, Boron nitride nanotubes: A review of recent progress on purification methods and techniques, J. Mater. Res. 37 (2022) 4438–4458 (2022).  doi:10.1557/s43578-022-00672-5.
  * [17] V. Dutta, R. Verma, C. Gopalkrishnan, M. H. Yuan, K. M. Batoo, R. Jayavel, A. Chauhan, K. Y. A. Lin, R. Balasubramani, S. Ghotekar, Bio-inspired synthesis of carbon-based nanomaterials and their potential environmental applications: a state-of-the-art review, Inorganics 10 (2022) 169 (2022).  doi:10.3390/inorganics10100169.
  * [18] T. Li, Z. Tang, Z. Huang, J. Yu, A comparison between the mechanical and thermal properties of single-walled carbon nanotubes and boron nitride nanotubes, Phys. E: Low-Dimens. Syst. Nanostructures 85 (2017) 137–142 (2017).  doi:10.1016/j.physe.2016.08.012.
  * [19] J. Yin, J. Li, Y. Hang, J. Yu, G. Tai, X. Li, Z. Zhang, W. Guo, Boron nitride nanostructures: Fabrication, functionalization and applications, Small 12 (2016) 2942–2968 (2016).  doi:10.1002/smll.201600053.
  * [20] L. Vaccarini, C. Goze, L. Henrard, E. Hernández, P. Bernier, A. Rubio, Mechanical and electronic properties of carbon and boron–nitride nanotubes, Carbon 38 (2000) 1681–1690 (2000).  doi:10.1016/S0008-6223(99)00293-6.
  * [21] R. Ansari, F. Sadeghi, S. Ajori, Oscillation characteristics of carbon nanotori molecules along carbon nanotubes under various system parameters, Europ. J. Mech. A/Solids 62 (2017) 67–79 (2017).
  * [22] A. M. Patki, R. K. Goyal, High performance polyetherketone-hexagonal boron nitride nanocomposites for electronic applications, J. Mater. Sci.: Mater. Electron. 30 (2019) 3899–3908 (2019).  doi:10.1007/s10854-019-00675-9.
  * [23] H. Oh, J. Kim, Fabrication of polymethyl methacrylate composites with silanized boron nitride by in-situ polymerization for high thermal conductivity, Compos. Sci. Technol. 172 (2019) 153–162 (2019).  doi:10.1016/j.compscitech.2019.01.021.
  * [24] R. N. Muthu, S. Rajashabala, R. Kannan, Hexagonal boron nitride (h-BN) nanoparticles decorated multi-walled carbon nanotubes (MWCNT) for hydrogen storage, Renew. Energ. 85 (2016) 387–394 (2016).  doi:10.1016/j.renene.2015.06.056.
  * [25] M. Jedrzejczak-Silicka, M. Trukawka, M. Dudziak, K. Piotrowska, E. Mijowska, Hexagonal boron nitride functionalized with Au nanoparticles–properties and potential biological applications, Nanomaterials 8 (2018) 605 (2018).  doi:10.3390/nano8080605.
  * [26] D. Maiti, X. Tong, X. Mou, K. Yang, Carbon-based nanomaterials for biomedical applications: A recent study, Front. Pharmacol. 9 (2019) 01401 (2019).  doi:10.3389/fphar.2018.01401.
  * [27] D. Gonzalez-Ortiz, C. Salameh, M. Bechelany, P. Miele, Nanostructured boron nitride–based materials: synthesis and applications, Mater. Today Adv. 8 (2020) 100107 (2020).  doi:10.1016/j.mtadv.2020.100107.
  * [28] R. Madeira, I. Camps, First-principles calculations of nickel, cadmium, and lead nanoclusters adsorption on single-wall (10,0) boron-nitride nanotube, Appl. Surf. Sci. 573 (2022) 151547 (2022).  doi:10.1016/j.apsusc.2021.151547.
  * [29] E. A. Turhan, A. E. Pazarçeviren, Z. Evis, A. Tezcaner, Properties and applications of boron nitride nanotubes, Nanotechnology 33 (2022) 242001 (2022).  doi:10.1088/1361-6528/ac5839.
  * [30] A. Khalid, P. Ahmad, A. Khan, M. U. Khandaker, I. Kebaili, M. Alam, I. U. Din, S. Muhammad, Z. Razzaq, I. U. Rehman, H. A. Abbasi, D. Hayat, Cytotoxic and photocatalytic studies of hexagonal boron nitride nanotubes: a potential candidate for wastewater and air treatment, RSC Advances 12 (2022) 6592–6600 (2022).  doi:10.1039/d2ra00300g.
  * [31] B. L. Merner, L. N. Dawe, G. J. Bodwell, 1,1,8,8-tetramethyl[8](2,11)teropyrenophane: Half of an aromatic belt and a segment of an (8,8) single-walled carbon nanotube, Angew. Chem. Int. Ed. 48 (2009) 5487–5491 (2009).  doi:10.1002/anie.200806363.
  * [32] L. S. Barbosa, B. C. de Almeida, E. Moreira, D. L. Azevedo, First-principle investigation of boron nitride nanobelt, Comput. Theor. Chem. 1208 (2022) 113571 (2022).  doi:10.1016/j.comptc.2021.113571.
  * [33] G. Povie, Y. Segawa, T. Nishihara, Y. Miyauchi, K. Itami, Synthesis of a carbon nanobelt, Science 356 (2017) 172–175 (2017).  doi:10.1126/science.aam8158.
  * [34] K. Y. Cheung, S. Gui, C. Deng, H. Liang, Z. Xia, Z. Liu, L. Chi, Q. Miao, Synthesis of armchair and chiral carbon nanobelts, Chem 5 (2019) 838–847 (2019).  doi:10.1016/j.chempr.2019.01.004.
  * [35] Z. Xia, S. H. Pun, H. Chen, Q. Miao, Synthesis of zigzag carbon nanobelts through scholl reactions, Angew. Chemie. 133 (2021) 10399–10406 (2021).  doi:10.1002/ange.202100343.
  * [36] Virtual NanoLab - Atomistix ToolKit. QuantumWise. v2017.1 (2017).
  * [37] C. Bannwarth, E. Caldeweyher, S. Ehlert, A. Hansen, P. Pracht, J. Seibert, S. Spicher, S. Grimme, Extended tight-binding quantum chemistry methods, WIREs Comput. Mol. Sci. 11 (2020) e1493 (2020).  doi:10.1002/wcms.1493.
  * [38] S. Grimme, C. Bannwarth, P. Shushkov, A robust and accurate tight-binding quantum chemical method for structures, vibrational frequencies, and noncovalent interactions of large molecular systems parametrized for all spd-block elements (Z=1–86), J. Chem. Theory Comput. 13 (2017) 1989–2009 (2017).  doi:10.1021/acs.jctc.7b00118.
  * [39] C. Plett, S. Grimme, Automated and efficient generation of general molecular aggregate structures, Angew. Chem. Int. Ed. 62 (2022).  doi:10.1002/anie.202214477.
  * [40] S. Grimme, C. Bannwarth, E. Caldeweyher, J. Pisarek, A. Hansen, A general intermolecular force field based on tight-binding quantum chemical calculations, The Journal of Chemical Physics 147 (2017) 161708 (2017).  doi:10.1063/1.4991798.
  * [41] C. Bannwarth, S. Ehlert, S. Grimme, GFN2–xTB–An accurate and broadly parametrized self-consistent tight-binding quantum chemical method with multipole electrostatics and density-dependent dispersion contributions, J. Chem. Theory Comput. 15 (2019) 1652–1671 (2019).  doi:10.1021/acs.jctc.8b01176.
  * [42] A. V. Marenich, S. V. Jerome, C. J. Cramer, D. G. Truhlar, Charge Model 5: an extension of Hirshfeld population analysis for the accurate description of molecular interactions in gaseous and condensed phases, Journal of Chemical Theory and Computation 8 (2012) 527–541 (2012).  doi:10.1021/ct200866d.
  * [43] T. Lu, F. Chen, Multiwfn: A multifunctional wavefunction analyzer, J. Comput. Chem. 33 (2012) 580–592 (2012).  doi:10.1002/jcc.22885.
  * [44] Jmol: An open-source Java viewer for chemical structures in 3D. http://www.jmol.org/.
  * [45] D. H. Everett, Manual on Definitions, Terminology and Symbols in Colloid and Surface Chemistry (internet version), Vol. 31, IUPAC. Division of Physical Chemistry, 2001 (2001).  doi:10.1351/pac197231040577.
  * [46] R. F. W. Bader, Atoms in Molecules: A Quantum Theory, Clarendon Press, 1994 (1994).
  * [47] C. Aguiar, N. Dattani, I. Camps, (VIDEOS) Möbius boron–nitride nanobelts interacting with heavy metal nanoclusters (2023).  doi:10.5281/zenodo.7662326.
  * [48] C. F. Matta, R. J. Boyd, The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design, John Wiley & Sons, 2007 (2007).
  * [49] W. Humphrey, A. Dalke, K. Schulten, VMD: Visual molecular dynamics, Journal of Molecular Graphics 14 (1996) 33–38 (fe 1996).  doi:10.1016/0263-7855(96)00018-5.
  * [50] A. D. Becke, K. E. Edgecombe, A simple measure of electron localization in atomic and molecular systems, The Journal of Chemical Physics 92 (1990) 5397–5403 (1990).  doi:10.1063/1.458517.
  * [51] K. Koumpouras, J. A. Larsson, Distinguishing between chemical bonding and physical binding using electron localization function (ELF), J. Phys.: Condens. Matter 32 (2020) 315502 (2020).  doi:10.1088/1361-648X/ab7fd8.
  * [52] H. L. Schmider, A. D. Becke, Chemical content of the kinetic energy density, J. Mol. Struct.: THEOCHEM 527 (2000) 51–61 (2000).  doi:10.1016/S0166-1280(00)00477-2.