# Analyzing Robustness of End-to-End Neural Models for Automatic Speech
Recognition

###### Abstract

We investigate robustness properties of pre-trained neural models for
automatic speech recognition. Real life data in machine learning is usually
very noisy and almost never clean, which can be attributed to various factors
depending on the domain, e.g. outliers, random noise and adversarial noise.
Therefore, the models we develop for various tasks should be robust to such
kinds of noisy data, which led to the thriving field of robust machine
learning. We consider this important issue in the setting of automatic speech
recognition. With the increasing popularity of pre-trained models, it’s an
important question to analyze and understand the robustness of such models to
noise. In this work, we perform a robustness analysis of the pre-trained
neural models wav2vec2, HuBERT and DistilHuBERT on the LibriSpeech and TIMIT
datasets. We use different kinds of noising mechanisms and measure the model
performances as quantified by the inference time and the standard Word Error
Rate metric. We also do an in-depth layer-wise analysis of the wav2vec2 model
when injecting noise in between layers, enabling us to predict at a high level
what each layer learns. Finally for this model, we visualize the propagation
of errors across the layers and compare how it behaves on clean versus noisy
data. Our experiments conform the predictions of Pasad et al. [2021] and also
raise interesting directions for future work.

Index Terms: noise robustness, automatic speech recognition, pre-trained
neural models, wav2vec2, HuBERT

##  1 Introduction

Speech recognition has undergone a revolution with the success of pre-trained
models. Pre-trained models such as wav2vec2 [1] amd HuBERT [2] are growing in
popularity and are being used widely for a variety of speech-related tasks.
With this unprecedented growth, natural issues should be considered. One such
issue is the measure of robustness of the model. Robustness of a model can
informally be defined to be the amount of data noise that the model can handle
without dimishing too much in accuracy. The study of robustness has had
important applications in many fields of machine learning especially in
computer vision, since they have safety-critical relevance to downstream tasks
like autonomous driving. In speech recognition, data is almost never noise-
free, so noise is baked in. Indeed, any realistic speech signal has background
noise. Moreover, noise can occur in various other forms stemming from hardware
and software issues, such as white noise, corrupted frames, etc.

In this work, we analyze the performance of popular pre-trained neural speech
models with an eye towards such issues. We perform two classes of experiments.
In the first class, we directly inject noise to the raw waveform input and
study model performance, as quantified by inference time and the Word Error
Rate. In the second class of experiments, we inject noise in between layers of
the neural model during inference and study the model behavior. This
experiment offers deeper insight into the layer-wise behavior of the neural
models. Intuitively, layers that learn higher order semantic information about
the speech should be more robust to noise. For instance, Pasad et al. [3]
predict that some layers of the wav2vec 2.0 model likely encode phonetic
information, while some others likely encode higher level information such as
word content or context. This suggests that the layers that learn higher level
contextual information should be more robust to noise. Our final experiment is
to visualize and compare the activations of the layers on uncorrupted and
corrupted data, to see if and when the noise is eliminated in the neural
network. This enables us to better understand the representations that the
model learns.

We now describe these two classes of experiments in more detail. In the first
class (E1), we study the performance of the models on noisy data, where we
quantify performance by inference time and the standard Word Error Rate
metric. We experiment with various types of simulated noise, including white
noise, speed perturbation and dropped frame chunks. In particular, for all
these kinds of noise, we compare fine-tuned wav2vec2 [1] and HuBERT [2] models
on the LibriSpeech dataset [4]; and also compare fine-tuned wav2vec2 and
DistilHuBERT [5] models on the TIMIT dataset [6].

In the second class of experiments (E2), we focus on wav2vec2 and do a more
detailed layer-wise analysis. In our first experiment of this class (E2A),
during inference on data, we intervene in a specific layer and inject (white)
noise. We then let the inference proceed as usual. We repeat this for other
layers and study how the model performance degrades with our intervention. We
repeat this experiment for both additive and multiplicative noise. In our
second experiment (E2B), we do model inference with the original and noisy
data and compare how the activations differ in each layer.

Our findings are as follows

  1. 1.

From (E1), we conclude that HuBERT is (around $25\%$) slower than wav2vec2 but
on simulated noise, it’s more robust than wav2vec2. Similarly, DistilHuBERT is
more robust than wav2vec2.

  2. 2.

In the additive noise version of (E2A) we observe that layers 6-8 seem very
noise sensitive while other layers seem relatively noise-robust. This suggests
that layers 6-8 learn higher level information such as semantics, context or
meaning. This matches the observations of [3].

  3. 3.

Almost all layers of wav2vec2 are surprisingly robust on multiplicative noise
injection of (E2B), except for layer $11$ which behaves in a unusual manner.

  4. 4.

The experiments (E2B) suggests that the wav2vec2 model “eats” up the noise as
we go up the layers. This is true for all layers except layer 11, where
intriguing things seem to happen.

##  2 Related work

Robust speech recognition has always been an important research avenue in
speech technologies, e.g. the works [7, 8, 9, 10, 11, 12], also see the book
[13]. Perhaps, the work most closely related to ours is [14] where they
analyze model robustness on mismatched domains. Many of these prior modeling
works attempt to build models that are robust to noise. Towards this goal,
they use various techniques such as by having a dedicated denoiser module or a
speech enhancement module or make the model robust enough so that it learns to
discard such noise from the data while it’s processing the data. In the latter
technique, a standard approach is via data augmentation. In this approach, we
augment the dataset with noisy data and then fine-tune it. It’s known that
this reduces generalization error, e.g. see [15]. For prior works studying
this in the context of speech, see e.g. [16, 8]. Finally, some works have also
explored adversarial noise [17, 18, 19].

##  3 Experiments and Findings

To aid our experiments, we use SpeechBrain [20] and open-source versions of
various datasets and pre-trained models from HuggingFace. In particular, we
use the fine-tuned wav2Vec2-Base-960h model and the fine-tuned Hubert-Large-
ls960 model. For LibriSpeech, we use a sub-sampled version of the test-clean
split and for TIMIT, we use a sub-sampled version of the test split.
Experiments were run on an NVIDIA 1080i GPU with 64 GB memory. Our code is
available at https://github.com/weizou52/Robustness_Analysis_ASR.

###  3.1 Experiment (E1)\- Noisy waveform input

In this section, we perturb the raw input with various kinds of noise and
study the behavior of the model. More precisely, let $x\in\mathbb{R}^{n}$ be
the input raw waveform. We then add noise to it to obtain
$x^{\prime}=f(x)\in\mathbb{R}^{n}$. We will explore various kinds of noise.

For the LibriSpeech dataset, we compare the word error rates of wav2vec2 and
HuBERT. We conclude that HuBERT is more robust than wav2vec2 for this kind of
simulated noise. Although it’s worth remarking that inference time for HuBERT
is also slower than wav2vec2. All our plots show the average inference time
per datapoint.

####  3.1.1 White Noise

White noise is when we take the input $x\in\mathbb{R}^{n}$, sample a random
$g\sim\mathcal{N}(0,I_{n})$ and independently for each coordinate $i\leq n$,
with probability $\rho$, we set $x_{i}=x+g_{i}$, otherwise we don’t change
$x_{i}$. Here, $\rho$ is called the mixing probability. This is the simplest
form of noise we could add and helps set a baseline for further experiments.
In Fig. 1, we plot our results.

Figure 1: Word error rate as a function of $\rho$ (White noise)

####  3.1.2 Speed Perturb

Speed perturbation speeds up or slows down the speech. For a given speech
signal $x$ and speed $100/\rho$, $f(x)$ is computed by resampling the audio
signal without changing the sampling rate, using the technique in [21]. See
Fig. 2 for the results. The plot conforms with our intuition that speech that
is sped up or slowed down is harder to predict, giving rise to the convex-
looking plot.

Figure 2: Word error rate as a function of $\rho$ (Speed Perturbation)

####  3.1.3 Chunk drop

In our next experiment, we drop portions of our input signal $x$. We
experiment both with number of chunks dropped $k$ and the length of each chunk
that’s dropped $l$. For the results of WER vs chunk length $l$ when number of
dropped chunks $k$ is fixed to be $100$, see Fig. 3. For the results of WER vs
number of dropped chunks $k$ when chunk length $l$ is fixed to be $100$, see
Fig. 4.

Figure 3: Word error rate as a function of $l$ (Dropping chunks) Figure 4:
Word error rate as a function of $k$ (Dropping chunks)

###  3.2 Experiments on TIMIT

We repeat these experiments on the TIMIT dataset where we compare wav2vec2 and
DistilHuBERT. The plots obtained are in Fig. 5, Fig. 6, Fig. 7 and Fig. 8. In
particular, note that since TIMIT is traditionally a dataset meant for phone
recognition and moreover the diversity in the dictionary isn’t very high, the
word error rate metrics aren’t remarkable. Nevertheless, the trend of
DistilHuBERT outperforming wav2vec2 in robustness can still be seen.

Figure 5: Word error rate as a function of $\rho$ (White noise) Figure 6: Word
error rate as a function of $\rho$ (Speed Perturbation) Figure 7: Word error
rate as a function of $l$ (Dropping chunks) Figure 8: Word error rate as a
function of $k$ (Dropping chunks)

###  3.3 Experiment (E2)\- Layerwise analysis

In this section, we analyze the layers of wav2vec2 in different ways to gain
more insight into the robustness properties of the model.

####  3.3.1 Experiment (E2A)\- Injecting noise into the layers

While performing inference on the test dataset, we inject noise in between
various layers and let inference proceed as usual. Specifically, let the
output of layer $i$ be $out_{i}\in\mathbb{R}^{d_{i}}$ where $d_{i}$ is the
total number of activations of the $i$-th layer. For a fixed value of $i$, we
modify this output

$out_{i}^{\prime}=out_{i}+\rho\cdot g$ (1)

where $g\sim\mathcal{N}(0,I_{d_{i}})$ and $\rho$ is the standard deviation. In
each run, we only noise one layer and the other layers are unaffected. By
layer $0$, we mean the CNN feature extractor, just as in [3]. In Fig. 9, we
show the changes in WER with respect to layers (where the noise $\rho\cdot g$
is injected). The same data is shown in Fig. 10 but as a function of WER with
respect to $\rho$ for every other layer.

Because layers 6-8 seem fairly sensitive to noise compared to other layers
which are noise robust, we conclude that layers 6-8 learn higher level
information about the speech signal, such as context, meaning and semantics.
Whereas other layers learn local information such as phone information or task
specific information. The intuition is that local information or lower level
information is robust to noise because surrounding contexts can help denoise
but on the other hand, higher level information is sensitive to noise. This
conforms with the findings of [3].

Figure 9: WER against layer (where additive noise is injected) Figure 10: WER
against $\rho$ (additive noise)

The kind of noise in Eq. 1 may not be the right kind of noising since
different layers may have different scales. Therefore, we also consider the
following kind of noising which takes scaling into account. Let $out_{i,j}$ be
the $j$th coordinate of $out_{i}$ for $j\leq d^{\prime}$. Then, we noise as
follows

$out_{i,j}^{\prime}=out_{i,j}(1+\rho\cdot g_{j})$ (2)

where $g_{j}$ are iid sampled from $\mathcal{N}(0,1)$. For this kind of
multiplicative noise, the corresponding outputs are shown in Fig. 11 and Fig.
12. In particular, note that layer $11$ is highly sensitive to noise. Similar
unusual behavior of layer $11$ was observed in [3].

Figure 11: WER against layer (where multiplicative noise is injected) Figure
12: WER against $\rho$ (multiplicative noise)

####  3.3.2 Experiment (E2B)\- Evolution of the activations on noisy data

In our next experiment, we compare how the model inference propogates across
layers when the inputs are $x^{(1)}$ and $x^{(2)}=x^{(1)}+\rho\cdot g$ where
$g\sim\mathcal{N}(0,I_{n})$. In particular, for layer $i$, we compute the
normalized L2 loss

$dist_{i}=\frac{1}{\sqrt{d_{i}}}\lVert out_{i}^{(1)}-out_{i}^{(2)}\rVert_{2}$
(3)

where $out_{i}^{(1)}$ and $out_{i}^{(2)}$ are the activations of layer $i$ on
inputs $x^{(1)}$ and $x^{(2)}$ respectively, and $d_{i}$ is the number of
neurons in the $i$th layer. To compute the L2 loss per neuron, we divide by
the scaling factor $\sqrt{d_{i}}$. Finally, we take the average over the
samples from the test dataset. In Fig. 13, we plot the loss as we go higher up
in the layers. In this plot, we fix $\rho=0.1$. As we can see, the layers are
essentially nullifying the loss incurred as we go higher up in the model,
suggesting that when the model performs inference, it’s simultaneously
denoising.

In Fig. 14, we repeat the experiment for different values of $\rho$. We notice
a similar trend except for the unusual layer $11$. Strange behavior for layer
$11$ was also observed in [3] and it’s possible that they are related, whose
investigation we leave for future work.

Figure 13: Average L2 loss across layers with $\rho=0.1$ Figure 14: Average L2
loss across layers for various $\rho$

##  4 Potential future directions

  1. 1.

Background noise: Background noise is an interesting direction for further
study, which perhaps maybe a more realistic kind of noise. To generate
realistic background noise, some works, e.g. [9] mixed other speech datasets
with the main one, using various techniques. So it would be interesting to
repeat our experiments in this setting.

  2. 2.

Adversarial noise: Compared to a field like Computer Vision, there is very
limited research exploring adversarial noise in speech (e.g. [17, 18]) but all
these works are in the pre-transformer era. But it seems like in recent
months, the Speech community is slowly starting to focus their attention on
adversarial noise, e.g. [19]. So this is a deeply fascinating area for further
research.

##  5 Acknowledgements

We thank Ju-Chiech Chou and Karen Livescu for useful comments and suggestions.
We thank Toyota Technological Institute at Chicago for their compute clusters.

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# Correlated reference-assisted variational quantum eigensolver

Nhan Trong Le  University of Science, Vietnam National University, Ho Chi Minh
City 700000, Vietnam  Lan Nguyen Tran<EMAIL_ADDRESS>National Institute
of Applied Mechanics and Informatics, Vietnam Academy of Science and
Technology, Ho Chi Minh City 700000, Vietnam

###### Abstract

We propose an active-space approximation to reduce the quantum resources
required for variational quantum eigensolver (VQE). Starting from the unitary
coupled-cluster (UCC) ansatz, we divide the total excitation operator into
internal and external contributions accordingly to the active-inactive
partitioning of the orbital space. The inactive space is treated using a
correlated mean-field theory called one-body second-order Møller-Plesset
perturbation theory (OBMP2), which is derived using the canonical
transformation and the cumulant approximation. Employing the downfolding
technique, we arrive at the active-space Hamiltonian solved by VQE composing
of the bare Hamiltonian and a potential caused by the internal-external
interaction. Considering different systems with singlet and doublet ground
states, we examine the accuracy in predicting both energy and density matrix
(by evaluating dipole moment). Our approach can dramatically outperform the
active-space VQE with uncorrelated Hartree-Fock reference due to the dynamical
correlation outside the active space. Our framework is general and can be used
for different types of UCC ansatz.

##  I Introduction

Quantum computing is highly promising for simulating challenging molecules and
materials Bauer  _et al._ (2020); McArdle  _et al._ (2020). It is most likely
beneficial for systems with the presence of strong correlations where
perturbative techniques failCao  _et al._ (2019); Motta and Rice (2021).
Unfortunately, current quantum hardware that is called noisy intermediate-
scale quantum (NISQ) is limited due to noise and decoherencede Leon  _et al._
(2021). The variational quantum eigensolver (VQE) has been proposed as a low-
depth quantum algorithm Peruzzo  _et al._ (2014); McClean  _et al._ (2016);
Kandala  _et al._ (2017) to take advantage of NISQ computers. It is a hybrid
quantum-classical method that must be carried out on quantum and classical
computers. On the quantum computer, quantum states depending on a set of
variational parameters are prepared, and the expectation value of the
Hamiltonian is then measured. Since most operations on a quantum computer are
unitary, unitary coupled-cluster (UCC) wavefunction Bartlett  _et al._ (1989);
Taube and Bartlett (2006); Romero  _et al._ (2018); Anand  _et al._ (2022) has
been proposed as a low-circuit-depth state-preparation ansatz for VQE. Next,
the set of variational variables is optimized on classical computers, and the
loop is repeated until converged.

However, the VQE applicability is limited by the dimensionality of many-body
systems associated with the number of variational variables and the circuit
depths. Recently, many methodologies have been developed to reduce the
resources required in VQE. Grimsley and co-workers proposed the adaptive-VQE
(ADAPT-VQE) ansatzGrimsley  _et al._ (2019) that is progressively built by
subsequently incorporating into it the operators that contribute most to
minimizing the VQE energy towards the ground-state energy. Later on, several
groups have extended ADAPT-VQE to make the algorithm more efficientTang  _et
al._ (2021); Zhang  _et al._ (2021). While ADAPT-VQE was shown to outperform
standard ansatz, its iterative nature makes calculations more costly.
Freericks and co-workers, on the other hand, have devised the factorized UCC
ansatz for VQEChen  _et al._ (2022); Xu  _et al._ (2022) that employs a Taylor
expansion in the small amplitudes, trading off circuit depth for additional
measurements. Strong correlations were considered by performing the expansion
about a small set of UCC factors that are treated exactly.

Alternatively, one can reduce the dimensionality of the many-body Hamiltonian
used in VQE by partitioning the whole system into smaller active spaces that
can be handled by quantum computing. Karol and co-workers have employed the
downfolding framework based on the double UCC (DUCC) to construct effective
active-space Hamiltonians Bauman  _et al._ (2019); Kowalski (2021) that
integrate out high-energy Fermionic degrees of freedom while being capable of
reproducing exact energy of quantum systems. To this end, one needs to define
subsets of excitations either entirely within the subsets considered or
involving some external orbitals. The approach can capture the effect of the
whole orbital space in small-size active spacesMetcalf  _et al._ (2020);
Chládek  _et al._ (2021). Inspired by the divide-and-conquer technique in
classical quantum chemistry, Nakagawa and co-workers proposed a method called
deep VQEFujii  _et al._ (2022); Mizuta  _et al._ (2021). In the first step of
this method, the whole system is divided into much smaller subsystems, each of
which is solved independently using VQE. In the next step, the ground states
of subsystems are used as a basis with reduced degrees of freedom to construct
an effective Hamiltonian considering the inter-subsystem interactions. The
resulting effective Hamiltonian is finally solved using VQE.

Quantum embedding frameworks, such as density matrix embedding theory
(DMET)Mineh and Montanaro (2022); Tilly  _et al._ (2021); Ralli  _et al._
(2022), dynamical mean-field theory (DMFT)Keen  _et al._ (2020); Bauer  _et
al._ (2016), and density functional embedding theoryRalli  _et al._ (2022);
Gujarati  _et al._ (2022), have been employed to make quantum computation
feasible for real molecules and materials. Recently, Rossmannek et al. have
demonstrated the performance of the VQE embedding into classical mean-field
methods, including Hartree-Fock (HF) and density functional theory
(DFT)Rossmannek  _et al._ (2021). Those authors restricted the quantum
computation to a critical subset of molecular orbitals, whereas the remaining
electrons provide the embedding potential computed using classical mean-field
theories. The proposed embedding schemes obtained significant energy
corrections to the HF and DFT reference for several simple molecules in their
strongly correlated regime and larger systems of the oxirane size. However,
most of those calculations were limited to a minimal basis. It may be
interesting to explore the performance of the active-space VQE approach on
larger basis sets.

In the present work, we propose an active-space VQE approximation where VQE is
naturally embedded in a correlated mean-field reference. To this end, we start
from the UCC ansatz and divide the total excitation operator into internal and
external contributions accordingly to the active-inactive partitioning of the
orbital space. For the inactive space, instead of HF and DFT, we employ our
recently-developed correlated mean-field theory called one-body second-order
Møller-Plesset perturbation theory (OBMP2)Tran and Yanai (2013); Tran (2021).
Unlike standard MP2, OBMP2 is self-consistent, meaning that it can bypass
challenges caused by the non-iterative nature of standard MP2. For the active
space, we employ VQE to solve the effective Hamiltonian composing of the bare
Hamiltonian and a potential caused by the internal-external interaction.
Details of the procedure are given in Section II. We demonstrate the
performance of our approach by considering different systems with singlet and
doublet ground states in Section III. We examine the accuracy in predicting
both energy and density matrix (by evaluating dipole moment). We show that the
active-space VQE with the correlated reference outperforms the standard
active-space VQE.

##  II Theory

###  II.1 Variational quantum eigensolver: UCC ansatz

VQE relies on the variational principle, which states that the ground-state
energy $E_{0}$ is always less than or equal to the expectation value of
Hamiltonian $\hat{H}$ calculated with the trial wavefunction
$\left|\psi\right>$

$\displaystyle
E_{0}\leq\frac{\left<\psi\right|\hat{H}\left|\psi\right>}{\left<\psi\right|\left|\psi\right>}$
(1)

with the molecular Hamiltonian as

$\displaystyle\hat{H}=\hat{h}+\hat{v}=\sum_{pq}h^{p}_{q}\hat{a}_{p}^{q}+\tfrac{1}{2}\sum_{pqrs}g^{pr}_{qs}\hat{a}_{pr}^{qs}$
(2)

where $\left\\{p,q,r,\ldots\right\\}$ indices refer to general ($all$) sin
orbitals. The objective of the VQE is to minimize the expectation value of the
Hamiltonian with respect to $\left|\psi\right>$.

To implement this optimization problem on the quantum computer, one has to
start by defining a wavefunction ansatz that can be expressed as a series of
quantum gates. To this end, we express $\left|\psi\right>$ as the application
of a parametrized unitary operator $U(\bm{\theta})$ to an initial state
$\left|\bm{0}\right>$ for $N$ qubits, with $\bm{\theta}$ representing a set of
parameters varying values in $\left(-\pi,\pi\right]$. Given that trial
wavefunctions, $\left|\psi\right>$, are necessarily normalized, we can now
write the VQE optimization problem as follows.

$\displaystyle
E_{\text{VQE}}=\min_{\bm{\theta}}\left<\bm{0}\right|U^{\dagger}(\bm{\theta})\hat{H}U(\bm{\theta})\left|\bm{0}\right>$
(3)

The unitary coupled cluster (UCC) ansatz is perhaps the most widely-used
ansatz for VQE and given as.

$\displaystyle\left|\psi_{\text{UCC}}\right>=e^{\hat{A}}\left|\bm{0}\right>,$
(4)

where $\left|\bm{0}\right>$ is the HF reference and $\hat{A}$ is an anti-
Hermitian combination of particle-hole excitation and de-excitation:

$\displaystyle\hat{A}$ $\displaystyle=\hat{T}-\hat{T}^{\dagger}$ (5)
$\displaystyle\hat{T}$
$\displaystyle=\sum_{i}^{occ}\sum_{a}^{vir}T_{i}^{a}\hat{a}_{a}^{i}+\sum_{ij}^{occ}\sum_{ab}^{vir}T_{ij}^{ab}\hat{a}_{ab}^{ij}+...$
(6)

where $\left\\{i,j,k,\ldots\right\\}$ indices refer to occupied ($occ$) spin
orbitals and $\left\\{a,b,c,\ldots\right\\}$ indices refer to virtual ($vir$)
spin orbitals. The amplitudes $T_{i}^{a}$ and $T_{ij}^{ab}$ are parameterized
into rotation angles $\bm{\theta}$ that are the variationally optimized.
Because the computational cost scales exponentially with the system size, the
excitation operator is usually truncated at single and double excitations,
resulting in UCC singles and doubles (UCCSD).

VQE employs HF wavefunction as a reference, and orbitals are fixed during
calculation. However, it is well-known that HF orbitals are not optimal for
correlated methods. Recently, several works have proposed the orbital-
optimized VQE (OO-VQE) method, in which orbitals are optimized by making the
energy stationary with respect to orbital rotation parametersSokolov  _et al._
(2020); Mizukami  _et al._ (2020); Yalouz  _et al._ (2021); Ratini  _et al._
(2022). This approach requires the orbital gradient of VQE energy, demanding
additional computational costs.

###  II.2 Correlated mean-field theory: OBMP2

Let us recap the OBMP2 theory, whose formulation details are presented in
Refs. 32 and 31. The OBMP2 approach was derived through the canonical
transformation Yanai and Chan (2006, 2007); Chan and Yanai (2007); Neuscamman
_et al._ (2009, 2010a, 2010b), in which an effective Hamiltonian that includes
dynamic correlation effects is achieved by a similarity transformation of the
molecular Hamiltonian $\hat{H}$ using a unitary operator $e^{\hat{A}}$ :

$\displaystyle\hat{\bar{H}}=e^{\hat{A}^{\dagger}}\hat{H}e^{\hat{A}},$ (7)

with the anti-Hermitian excited operator $\hat{A}$ defined as in Eq 5. In
OBMP2, the cluster operator $\hat{A}$ is modeled such that including only
double excitation.

$\displaystyle\hat{A}=\hat{A}_{\text{D}}=\tfrac{1}{2}\sum_{ij}^{occ}\sum_{ab}^{vir}T_{ij}^{ab}(\hat{a}_{ab}^{ij}-\hat{a}_{ij}^{ab})\,,$
(8)

with the MP2 amplitude

$\displaystyle
T_{ij}^{ab}=\frac{g_{ij}^{ab}}{\epsilon_{i}+\epsilon_{j}-\epsilon_{a}-\epsilon_{b}}\,,$
(9)

where $\epsilon_{i}$ is the orbital energy of the spin-orbital $i$. Using the
Baker–Campbell–Hausdorff transformation, the OBMP2 Hamiltonian is defined as

$\displaystyle\hat{H}_{\text{OBMP2}}=\hat{H}_{\text{HF}}+\left[\hat{H},\hat{A}_{\text{D}}\right]_{1}+\tfrac{1}{2}\left[\left[\hat{F},\hat{A}_{\text{D}}\right],\hat{A}_{\text{D}}\right]_{1}.$
(10)

with

$\displaystyle\hat{H}_{\text{HF}}$
$\displaystyle=\hat{F}+C=\hat{h}+\hat{v}_{\text{HF}}+C$ (11)

Where $\hat{h}$ is the one-electron Hamiltonian defined in Eq 2. The HF
potential $\hat{v}^{\text{HF}}$ and the constant $C$ is given as:

$\displaystyle\hat{v}_{\text{HF}}$
$\displaystyle=\sum_{pq}^{all}\sum_{i}^{occ}\left(g^{pi}_{qi}-g^{pi}_{iq}\right)$
(12) $\displaystyle C$
$\displaystyle=\,\,\sum_{ij}^{occ}\left(g^{ij}_{ji}-g_{ij}^{ij}\right)\,.$
(13)

In Eq.10, commutators with the subscription 1, $[\ldots]_{1}$, involve one-
body operators and constants that are reduced from many-body operators using
the cumulant approximationKutzelnigg and Mukherjee (1997); Mazziotti (1998a,
b); Kutzelnigg and Mukherjee (1999). Doing some derivation, we eventually
arrive at the OBMP2 Hamiltonian as follows

$\displaystyle\hat{H}_{\text{OBMP2}}=$
$\displaystyle\,\,\hat{H}_{\text{HF}}+\hat{V}_{\text{OBMP2}}$ (14)

where $\hat{v}_{\text{OBMP2}}$ is a correlated potential composing of one-body
operators. The working expression is given as

$\displaystyle\hat{V}_{\text{OBMP2}}=$
$\displaystyle\overline{T}_{ij}^{ab}\left[f_{a}^{i}\,\hat{\Omega}\left(\hat{a}_{j}^{b}\right)+g_{ab}^{ip}\,\hat{\Omega}\left(\hat{a}_{j}^{p}\right)-g^{aq}_{ij}\,\hat{\Omega}\left(\hat{a}^{b}_{q}\right)\right]$
$\displaystyle-2\overline{T}_{ij}^{ab}g^{ij}_{ab}+\,f_{a}^{i}\overline{T}_{ij}^{ab}\overline{T}_{jk}^{bc}\,\hat{\Omega}\left(\hat{a}_{c}^{k}\right)$
$\displaystyle+f_{c}^{a}T_{ij}^{ab}\overline{T}_{il}^{cb}\,\hat{\Omega}\left(\hat{a}^{l}_{j}\right)+f_{c}^{a}T_{ij}^{ab}\overline{T}_{kj}^{cb}\,\hat{\Omega}\left(\hat{a}^{k}_{i}\right)$
$\displaystyle-f^{k}_{i}T_{ij}^{ab}\overline{T}_{kl}^{ab}\,\hat{\Omega}\left(\hat{a}_{l}^{j}\right)-f^{p}_{i}T_{ij}^{ab}\overline{T}_{kj}^{ab}\,\hat{\Omega}\left(\hat{a}^{p}_{k}\right)$
$\displaystyle+f^{k}_{i}T_{ij}^{ab}\overline{T}_{kj}^{ad}\,\hat{\Omega}\left(\hat{a}_{b}^{d}\right)+f_{k}^{i}T_{ij}^{ab}\overline{T}_{kj}^{cb}\,\hat{\Omega}\left(\hat{a}_{a}^{c}\right)$
$\displaystyle-
f_{c}^{a}T_{ij}^{ab}\overline{T}_{ij}^{cd}\,\hat{\Omega}\left(\hat{a}^{b}_{d}\right)\,-f_{p}^{a}T_{ij}^{ab}\overline{T}_{ij}^{cb}\,\hat{\Omega}\left(\hat{a}^{p}_{c}\right)$
$\displaystyle-2f_{a}^{c}{T}_{ij}^{ab}\overline{T}_{ij}^{cb}+2f_{i}^{k}{T}_{ij}^{ab}\overline{T}_{kj}^{ab}.$
(15)

with $\overline{T}_{ij}^{ab}={T}_{ij}^{ab}-{T}_{ji}^{ab}$, the symmetrization
operator
$\hat{\Omega}\left(\hat{a}^{p}_{q}\right)=\hat{a}^{p}_{q}+\hat{a}^{q}_{p}$,
and the Fock matrix

$\displaystyle
f_{p}^{q}=h_{p}^{q}+\sum_{i}^{occ}\left(g^{pi}_{qi}-g^{pi}_{iq}\right).$ (16)

Note that, for convenience, we have used Einstein’s convention in Eq. 15 to
present the summations over repeated indices. We rewrite
$\hat{H}_{\text{OBMP2}}$ (Eqs. 10 and 14) in a similar form to Eq. 11 for
$\hat{H}_{\text{HF}}$ as follows:

$\displaystyle\hat{H}_{\text{OBMP2}}=$ $\displaystyle\hat{\bar{F}}+\bar{C}$
(17)

with $\hat{\bar{F}}=\bar{f}^{p}_{q}\hat{a}_{p}^{q}$. $\bar{f}^{p}_{q}$ is so-
called correlated Fock matrix and written as

$\displaystyle\bar{f}^{p}_{q}$ $\displaystyle=f^{p}_{q}+v^{p}_{q}.$ (18)

$v^{p}_{q}$ is the matrix representation of the one-body operator
$\hat{V}_{\text{OBMP2}}$, serving as the correlation potential altering the
uncorrelated HF picture. We update the MO coefficients and energies by
diagonalizing the matrix $\bar{f}^{p}_{q}$, leading to orbital relaxation in
the presence of dynamic correlation effects. The OBMP2 method is implemented
within a local version of PySCFSun  _et al._ (2018).

###  II.3 VQE with OBMP2 reference

We can see that both UCC and OBMP2 are formulated using a unitary exponential
operator $e^{\hat{A}}$ (Eqs 5 and 8), implying that one can combine these two
naturally. Partitioning the whole orbital space into active and inactive
spaces, one can decompose excitation operators into internal and external
contributions as:

$\displaystyle\hat{A}=\hat{A}_{\text{int}}+\hat{A}_{\text{ext}}.$ (19)

Here, the internal (int) defines excitations within active space, and the
external (ext) is the remaining excitations involving at least one inactive
space orbital.

The total energy can be written as:

$\displaystyle E$
$\displaystyle=\left<\bm{0}\right|e^{\hat{A}^{\dagger}}\hat{H}e^{\hat{A}}\left|\bm{0}\right>$
(20)
$\displaystyle=\left<\bm{0}\right|e^{\hat{A}_{\text{int}}^{\dagger}}e^{\hat{A}_{\text{ext}}^{\dagger}}\hat{H}e^{\hat{A}_{\text{ext}}}e^{\hat{A}_{\text{int}}}\left|\bm{0}\right>$
(21)

In the standard active space approximation, the external contribution is zero.
Here, we treat it using OBMP2 described in the subsection II.2. We arrive then
at

$\displaystyle E$
$\displaystyle=\left<\bm{0}\right|e^{\hat{A}_{\text{int}}^{\dagger}}\hat{H}^{\text{act}}_{\text{eff}}e^{\hat{A}_{\text{int}}}\left|\bm{0}\right>+E_{\text{OBMP2}}^{\text{ext}}$
(22)

where $\hat{H}^{\text{act}}_{\text{eff}}$ is the effective Hamiltonian within
the active space including the bare Hamiltonian $\hat{H}^{\text{act}}$ and an
effective one-body potential $v^{\text{act}}_{\text{eff}}$ from the remaining
electrons outside this space:

$\displaystyle\hat{H}^{\text{act}}_{\text{eff}}$
$\displaystyle=\hat{H}^{\text{act}}+\hat{v}^{\text{act}}_{\text{eff}},$ (23)
$\displaystyle\hat{v}^{\text{act}}_{\text{eff}}$
$\displaystyle=\hat{v}^{\text{act}}_{\text{HF}}+\hat{V}^{\text{act}}_{\text{OBMP2}}.$
(24)

In the present implementation, we have not included
$\hat{V}_{\text{OBMP2}}^{\text{act}}$ in the active-space Hamiltonian.
Instead, the active-inactive interaction is buried in the external
contribution determined by:

$\displaystyle
E_{\text{OBMP2}}^{\text{ext}}=E_{\text{OBMP2}}^{\text{tot}}-E_{\text{OBMP2}}^{\text{act}}$
(25)

with $E_{\text{OBMP2}}^{\text{tot}}$ the total OBMP2 energy of the whole
system and $E_{\text{OBMP2}}^{\text{act}}$ the OBMP2 energy of the active
space.

Calculations start by running OBMP2 for the whole system and selecting active
space using OBMP2 orbitals. VQE is then used for the active space with the
effective Hamiltonian. We use UCC with doubles (UCCD) as the ansatz in the
present work. The total double-excitation amplitude for the whole system
$T_{2}$ is the summation of internal and external contributions:

$\displaystyle T_{2}=T_{2}^{\text{int}}+T_{2}^{\text{ext}}.$ (26)

The density matrix needed for properties is then evaluated using the total
amplitude. It is essential to remind that, unlike standard MP2, the OBMP2
amplitude is relaxed through a self-consistency, bypassing issues caused by
the non-self-consistent nature of standard MP2. The classical calculation is
carried out using PySCFSun  _et al._ (2018), and the quantum part is done
using the Qiskit packageANIS  _et al._ (2021).

##  III Results and discussion

###  III.1 Full-space VQE with OBMP2 orbitals

Figure 1: Potential energy curves of H4 chain in STO-6G. VQE was performed for
the full orbital space. Figure 2: Potential energy curves of LiH in STO-6G.
VQE was performed for the full orbital space.

Several authors have shown that orbital relaxation is important to reduce VQE
errors Sokolov  _et al._ (2020); Mizukami  _et al._ (2020); Yalouz  _et al._
(2021); Ratini  _et al._ (2022). In these studies, the energy of VQE is
minimized concerning both cluster amplitudes and orbitals, resulting in a
self-consistency that demands higher computational costs than standard VQE. It
is thus interesting to examine whether correlated orbital reference
preoptimized using a lower-level method can improve the accuracy of “single-
shot” VQE. Here, we performed VQE only once on OBMP2 orbitals that are relaxed
in the presence of dynamical correlation at the MP2 level. In fact, since MP2
is the first-order amplitude truncation of coupled-cluster doubles (CCD), one
can consider OBMP2 is an approximation to orbital-optimized CCD (OO-CCD)
Mizukami  _et al._ (2020); Sokolov  _et al._ (2020).

In Figure 1, we plot the potential energy curves of the H4 chain in the STO-6G
basis. VQE with the UCCD solver for the entire orbital space was performed
using HF and OBMP2 orbitals. While both cases are almost identical and close
to the FCI reference when $R<1.5$Å, errors become more prominent as stretching
the distance. Noticeably, VQE with the OBMP2 orbitals is better than that with
the HF orbitals, and its error is twice smaller than the HF counterpart. We
consider another example, LiH in the STO-6G basis. As shown in Figure 2, VQE
again benefits from correlated orbital reference. In general, OBMP2 orbital
reference can help to reduce VQE errors without any additional costs. Using
OBMP2 orbitals as a basis may be a good approximation to OO-CCD.

###  III.2 Active-space VQE with restricted OBMP2

Figure 3: Potential energy curves of H4 chain in cc-pVDZ. VQE was performed in
the active space of four orbitals. Figure 4: Potential energy curves of H4
square in cc-pVDZ. VQE was performed in the active space of four orbitals .

This subsection considers the active-space approximation for molecules with
the ground state singlet. The restricted OBMP2 is used as the reference for
VQE.

Figure 5: Potential energy curves of LiH in cc-pVDZ with different active
spaces for VQE.

Figure 3 represents potential energy curves of H4 chain in cc-pVDZ. VQE was
performed within an active space of four orbitals. For comparison, we also
plot HF and OBMP2 curves. Due to the lack of dynamic correlation outside the
active space, VQE is far from the FCI reference. While OBMP2 is better than
VQE around equilibrium, its errors are significantly larger than VQE errors at
long distances. Also, the VQE curve is much more parallel to FCI than OBMP2,
resulting from a proper description of static correlation in VQE. Capturing
dynamical correlation outside the active space, VQE-in-OBMP2 dramatically
outperforms VQE and yields errors relative to FCI much smaller than those of
VQE. Interestingly, the non-parallelity error (NPE), defined as the difference
between the minimum and maximum errors, is smaller for VQE-in-OBMP2 than VQE.

The next system we consider is the H4 ring as depicted in the inset of Figure
4. We analyze the change of energy when the four atoms move along the
circumference with radius $R=1.8$Å by varying the angle $\theta$. The square
geometry corresponds to $\theta=90^{\circ}$. We perform VQE calculation within
the active space of four valence orbitals. For comparison, we also present the
CCSD result. As we can see in Figure 4, while CCSD agrees very well with the
FCI reference for $\theta$ far from $90^{\circ}$, it fails to predict the
convexity of the energy around $90^{\circ}$. In contrast, VQE with UCC ansatz
can properly describe the energy profile, which is consistent with the work of
Sokolov et al. Sokolov  _et al._ (2020). Harsha et al. showed that the
variational UCC is superior to standard CC, particularly when strong electron
correlation is involvedHarsha  _et al._ (2018). Due to the lack of dynamical
correlation outside the active space, there is a large up-shift of the VQE
curve with a maximum error of 40 mHa. When the correlation outside the active
space is taken into account, VQE-in-OBMP2 can significantly improve upon
standard VQE with a maximum error four times smaller than VQE.

We now examine the systematic improvement with respect to the size of the
active space. To this end, we consider LiH in cc-pVDZ and gradually enlarge
the active space by adding $\sigma$-type orbitals. Here, the core orbital Li
$1s$ is not included in the active space and treated at the (correlated) mean-
field level. All results are summarized in Figure 5. We can see a systematic
improvement when the size of the active space increases. However, the largest
active space of seven orbitals is still insufficient for standard VQE to
achieve satisfying accuracy. While retaining the systematic improvement with
respect to the size of active space, VQE-in-OBMP2 can dramatically reduce
errors. Its NPEs are also smaller than those of standard VQE. With the largest
active space considered here, VQE-in-OBMP2 can approach closer to the FCI
reference with a maximum error of 6 mHa.

###  III.3 Active-space VQE with unrestricted OBMP2

Figure 6: Left: potential energy curves of CH with the ground state doublet.
Right: The change of CH dipole moment. The basis set is cc-pVDZ. VQE was
performed in the active space of five orbitals.

This subsection considers two systems with the ground state doublet (e.g.,
having one unpaired electron): CH and H2O+. The unrestricted HF (UHF) and
OBMP2 (UOBMP2) are used as the reference for VQE. In addition to potential
energy curves, we also calculate dipole moments that are a direct measure of
the density matrix. The cc-pVDZ basis set is used for all calculations.

Figure 6 represents the potential energy curves and dipole moments of CH from
different methods and their errors relative to the CCSD reference. VQE is
performed in the active space of five orbitals, composed of C $2s2p$ and H
$1s$. Although unrestricted HF (UHF) can describe the dissociation adequately,
a large NPE is observed due to a bump at the unrestricted point 1.5Å. Standard
VQE performed on UHF can reduce errors and parallel the curve to the FCI
reference. When VQE is performed with the UOBMP2 reference, the errors in
energy dramatically decrease. We plot the change of dipole moments by
stretching the C–H bond in the right panel of Figure 6. All the methods yield
curves that behave similarly to the CCSD reference. VQE-in-OBMP2 predicts
dipole moment closer to the CCSD reference than standard VQE for $R<1.5\r{A}$,
indicating the importance of dynamic correlation in the accurate prediction of
density-related properties. However, its errors are still significant in the
stretched regime, demanding the enlargement of the active space to get more
accurate dipole moments.

Figure 7: Left: potential energy curves of H2O+ with the ground state doublet.
Right: The change of H2O+ dipole moment. The basis set is cc-pVDZ. VQE was
performed in the active space of six orbitals.

Let us now consider H2O+ for which VQE is performed in the active space of six
orbitals composed of O $2s2p$ and H $1s$. Its potential energy curves and
dipole moments evaluated using different methods are depicted in Figure 7. As
for the potential energy curves, although VQE with six orbitals in the active
space does not significantly improve UHF, it makes the curve more parallel to
the CCSD reference, in particular, at the unrestricted point ($R\approx
1.3\r{A}$). When combined with UOBMP2, VQE yields the curve much closer to the
reference and further reduces NPE. As for dipole moments, the VQE-in-OBMP2
curve is overall closest to the CCSD reference. It agrees very well with CCSD
for short distances ($R<1.5\r{A}$). However, it deviates from the reference at
long distances, requiring a larger active space for VQE.

##  IV Conclusion

We have proposed an active space approximation in which VQE is naturally
embedded in a correlated mean-field reference derived from the UCC ansatz. By
partitioning the whole orbital space into active and inactive spaces, we
divided the total excitation operator in the UCC ansatz into internal and
external contributions. The inactive space is treated using OBMP2, a
correlated mean-field theory recently developed by us. The effective
Hamiltonian for the active space is derived as the sum of the bare Hamiltonian
in the active space and a potential describing the internal-external
interaction. Considering different systems with singlet and doublet ground
states in the minimal and larger basis sets, we demonstrated the accuracy of
our approach in predicting energies and dipole moments. We show that the VQE
with the OBMP2 reference significantly improves upon the standard active-space
VQE with the uncorrelated HF reference.

Our proposed approach is generally applicable to different types of UCC
ansatz, such as generalized UCCLee  _et al._ (2018), paired UCCLee  _et al._
(2018); Stein  _et al._ (2014), and pair-natural orbital-UCCKottmann  _et al._
(2021). We believe it is useful to study real chemistry and materials on
quantum computers. Further work is planned to develop more sophisticated
schemes of active-space selection to treat systems with large active spaces.
For example, one can split the active space into smaller subspaces and treat
them independently using VQE as has been done in quantum embedding methods
Welborn  _et al._ (2016); Wouters  _et al._ (2016); Tran  _et al._ (2016);
Tran and Zgid (2017). In the current work, orbitals are only optimized in
OBMP2, and VQE is performed as a ”single-shot” calculation. We are now
implementing a fully self-consistent scheme, in which the total amplitude (Eq.
26) is used to construct correlated Fock (Eq. 18). Molecular orbitals and
orbital energies are then relaxed in the presence of both UCC and MP2
correlations. We hope to report these results in near-future publications.

## Acknowledgments

This work is supported by the Vietnam Academy of Science and Technology (VAST)
under the grant number CSCL14.01/22-23.

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# Collective enhancement in nuclear level density of 72Ga and its effect on
71Ga(n, $\gamma$)72Ga capture cross-section

Rajkumar Santra<EMAIL_ADDRESS>Balaram Dey  Subinit Roy  R.
Palit  Md. S. R. Laskar  H. Pai  S. Rajbanshi  Sajad Ali  Saikat Bhattacharjee
F. S. Babra  Anjali Mukherjee  S. Jadhav  Balaji S Naidu  Abraham T.
Vazhappilly  Sanjoy Pal  Department of Nuclear and Atomic Physics, Tata
Institute of Fundamental Research, Mumbai-400005, India.  Department of
Physics, Bankura University, Bankura-722155, India.  Nuclear Physics Division,
Saha Institute of Nuclear Physics, Kolkata-700064, India.  Extreme Light
Infrastructure - Nuclear Physics, Bucharest-Magurele, 077125, Romania.
Department of Physics, Presidency University, Kolkata-700073, India.
Government General Degree College at Pedong, Kalimpong, 734311, India.

###### Abstract

The $\gamma$-gated proton spectra measured in the reactions 64Ni(9Be, p2n)70Ga
and 64Ni(9Be, pn)71Ga, have been utilized to obtain the nuclear level density
(NLD) of 71Ga and 72Ga nuclei by using the statistical model (SM)
calculations. It is seen that the $\gamma$-gated proton spectrum are
reasonably explained by using the large value of the inverse level density
parameter ($k$ = 11.2 MeV) in the NLD prescription of the Fermi gas (FG)
model. The large value of $k$ is indicative of the rotational enhancement,
which is consistent with the earlier results in other mass regions.
Furthermore, a rotational enhancement factor has been included in the NLD and
used in the SM calculation keeping the systematic value of $k$=8.6 MeV and it
explains the $\gamma$-gated proton spectrum nicely. The result clearly
indicates the presence of collective enhancement in NLD. Subsequently, the NLD
with collective enhancement has been utilized in the TALYS calculation, for
the first time, to calculate the 71Ga(n, $\gamma$)72Ga capture cross-section.
It is observed that, while the FG model without the collective enhancement in
the NLD for 72Ga under predicts the capture data, with the rotational
enhancement correction the FG model over predicts the data by similar amount
at higher energies. However, in the energy range of 0.01 MeV to 0.1 MeV, the
FG model corrected for rotational enhancement describes the data quite well.
Thus, the present work indicates that collective enhancement, whenever
required, should be taken into account fro proper description of low energy
capture cross section data.

###### keywords:

Nuclear level density, Collective enhancement , neutron capture cross section

The basic understanding of a quantum many-body system such as an atomic
nucleus is highly crucial to unfold the chemical evolution of the Universe in
its present state. To perceive the characteristics of the atomic nucleus,
usually it is brought under small external perturbations. Nuclear level
density (NLD) is one of the important physical quantities in order to
understand the response of an atomic nucleus to such external perturbations
[1, 2]. Level density, defined as the number of excited levels per unit of
excitation energy for a nucleus, plays an important role as one of the key
ingredients in the statistical model description of low energy nuclear
reactions. Consequently, the knowledge of NLD is also important in the
modeling stellar nucleosynthesis and evolution [3]. However, NLD provides the
highest uncertainties in estimating the reaction cross-section for production
of stable and unstable nuclei in the nucleosynthesis process. Reliable nuclear
level densities are therefore very crucial in the determination of nuclear
reaction rates in the field of nuclear astrophysics [4, 5]. In addition, the
enhancement in the NLD due to collective degrees of freedom (such as rotation
and vibration) i.e collective enhancement [6, 7, 8, 9] is another puzzling
topic [8]. The collective enhancement present in the NLD, especially around
the particle separation energy, could have a significant role in the particle
capture cross-sections. Therefore, an accurate extraction of the NLD using a
reliable experimental technique is essential to constrain the quantity in
estimating the important low energy capture reaction cross-section as
precisely as possible.

Neutron capture on 71Ga is one of the important astrophysical reactions in the
s-process, which not only determines the abundances of gallium isotopes but
also affects the abundances of the elements up to zirconium [10]. Gallium is
mostly produced by the weak s-process that takes place in massive stars of
more than eight solar masses during convective helium-burning (at temperatures
T $\sim$ 30 keV) and shell carbon burning (at T $\sim$ 90 keV) [10]. A recent
simulation indicates that a 50$\%$ change in the neutron capture cross section
on one gallium isotope changes the s-abundances of all the following isotopes
in shell carbon burning by up to 20$\%$ [11]. In addition, there are three
discrepant experimental results on the neutron capture cross-section on 71Ga
at 25 keV [12, 13, 14].

The primary goal of the present work is clearly two fold. First, to
investigate the collective enhancement in the NLDs of deformed nuclei 72Ga and
71Ga by using the evaporation particle-gamma coincidence technique. Secondly,
to use the extracted NLD of 72Ga isotope in the statistical model description
of 71Ga(n,$\gamma$)72Ga capture cross sections by upgrading the statistical
model code TALYS [36] to include the effect of collective enhancement in the
NLD prescription.

Experimentally, the precise measurement of NLD is a very difficult task due to
the lack of suitable techniques. The density of low lying excited levels of
the nucleus (small in number, well separated, and simple in nature) is
generally measured directly by counting the levels [15] and from neutron
resonance studies [16]. Whereas, the Oslo technique based on particle-gamma
coincidence [17], the two-step cascade method [18], the $\beta$-Oslo method
[19], the gamma-ray calorimetry [20], etc, are used to estimate the level
density especially up to the particle separation energy. Another approach to
estimate the level density below as well as above the particle threshold
energy is the particle evaporation technique [21, 22, 23, 24, 25]. In this
technique, the evaporated particles (neutron or proton or alpha) are measured
and compared with the statistical model calculations using the different
formalisms for level density such as FG (Fermi gas), Gilbert-Cameron, etc. The
parameters characterizing the level density of each formalism are varied to
explain the evaporated particle spectrum and the study yields the level
density of the corresponding residual nucleus.The main concern with the
particle evaporation technique is to measure the particles from compound
nuclear reactions, which can be confirmed by measuring the particles at
backward angles and studying the angular distribution of the emitted particles
in light-ion induced reactions [21, 22, 23, 24, 25]. Very recently, we adopted
a new approach to select the evaporated particles only from compound nuclear
reactions by using the gamma-gated particle spectra [26]. The same approach,
as discussed in Ref. [26] has been used in the present work.

Figure 1:  (Color online) [Top panel] p-$\gamma$ coincidence matrix extracted
from raw particle-$\gamma$ matrix. [Botom panel] Projected gamma energy
spectrum. Symbols with $\gamma$-energy indicate $\gamma$-lines of different
residual nuclei associated with proton emitting channels.

In this work, we have measured the $\gamma$-gated evaporated protons coming
from the reactions 64Ni(9Be, p2n$\gamma$)70Ga and 64Ni(9Be, pn$\gamma$)71Ga.
Particle-gamma coincidence has been performed to eliminate the events from
target contamination and pre-equilibrium emission. The $\gamma$-gated proton
spectra have been utilized to obtain the level densities of residual nuclei.
The obtained level density for 72Ga has further been used to calculate the
71Ga(n,$\gamma$) capture reaction cross-section as a function of neutron
energy.

Figure 2:  (Color Online) Filled symbols represent the experimental
$\gamma$-gated proton spectrum. Lines represent the statistical model
calculated proton (SMCP) spectrum. [Top panel] Red solid and blue dashed lines
represent SMCP spectrum considering 1st chance proton emission from CN, with
k= 11.2 MeV and k=8.6 MeV, respectively. [Botom panel] Red continuous, blue
dashed and pink dash-dotted lines represent SMCP spectrum considering 1st ,
2nd and 3rd chance proton emission from CN, respectively, with k=11.2 MeV.

The 9Be beam of 30 MeV energy with an average current of 5 nA from BARC-TIFR
Pelletron Linac Facility, Mumbai, was used to bombard a self-supporting 64Ni
target of thickness 500 $\mu$g/cm2, populating the 73Ge compound nucleus (CN).
The outgoing protons were detected by using CsI(Tl) detectors with the angle
of coverage from 22∘ to 67∘. The 14 Compton-suppressed Clover detectors were
used to detect the de-exiting discrete $\gamma$-rays coming from the residual
nuclei. XIA-LLC-based digital data acquisition system was used to store the
data in list mode. Tantalum foil of thickness 30 mg/cm2 was used to stop the
elastically scattered events from entering the detectors. The proton spectrum
is calibrated using 229Th source with pulse height defect correction based on
light output vs. energy data of CsI(Tl) in Ref.[27]. Multi parameter time
stamped based coincidence search program (MARCOS) [28] was used to construct
the p-$\gamma$ matrix as shown in Fig. 1. From this matrix, proton yield
spectra were extracted gated by characteristic 154.9 (5- $\rightarrow$ 4-),
187.6 (4- $\rightarrow$ 2-), 198 keV $\gamma$-rays from the residual 70Ga
nucleus and 488 keV $\gamma$-ray from 71Ga nucleus [29]. Finally, the
$\gamma$-gated proton spectra are added after efficiency correction ( YTotal=
$\frac{Y_{154}}{\epsilon_{154}}$ \+ $\frac{Y_{187}}{\epsilon_{187}}$ \+
$\frac{Y_{198}}{\epsilon_{198}}$ \+ $\frac{Y_{488}}{\epsilon_{488}}$ ) to
visualize the shape of the final proton spectrum with higher statistics as
shown in Fig. 2. The quantities, $\epsilon_{i}$ and Yi, are the detection
efficiency of ith $\gamma$-ray and the yield of proton that are in coincidence
with the $i^{th}$ $\gamma$-ray.

Figure 3: (Color online) Filled symbols represent the $\gamma$-gated proton
spectrum. Lines represent the statistical model calculated proton (SMCP)
spectrum. [Top panel] Red continuous line represents SMCP with k=8.6 MeV
considering 1st chance proton emission from CN 73Ge and including collective
enhancement factor in NLD of 72Ga residual nucleus. [Botom panel] Purple
dashed line represents SMCP with k=8.6 MeV considering 1st chance proton
emission from CN 73Ge and including collective enhancement factor in NLD of
72Ga residual nucleus. Pink dashed line represents SMCP with k=8.6 MeV
considering 2nd chance proton emission from CN 73Ge and including collective
enhancement factor in NLD of 71Ga residual nucleus. Red continuous line
represents the total spectrum with weitage factor as Ytotal =Y(1st chance) +
0.3$\times$Y(2nd chance).

In order to investigate the NLD of residual nuclei, the $\gamma$-gated proton
spectrum has been compared with the statistical model calculations using the
code CASCADE [30] with the Fermi-Gas (FG) model of NLD given by

$\rho(E^{*},J)=\frac{2J+1}{12\theta^{3/2}}\sqrt{a}\frac{exp(2\sqrt{aU})}{U^{2}}$
(1)

where, U$=$E∗$-$$\frac{J(J+1)}{\theta}$$-$Sp$-$$\Delta$P, E∗ and J are the
initial excitation energy and the angular momentum of the nucleus,
respectively, $\theta=\frac{2I_{eff}}{h^{2}}$, with Ieff, Sp and $\Delta$P
being the effective rigid-body moment of inertia, the proton separation energy
and the pairing energy respectively. The quantity $a$ in Eq.1 is the level
density parameter. Ignatyuk prescription [31] of level density parameter $a$,
which takes into account the shell effects as a function of excitation energy
is adopted and it is expressed as

$a=\tilde{a}[1+\frac{\delta S}{U}[1-\exp(-\gamma U)]]$ (2)

where, asymptotic level density parameter $\tilde{a}$ = A/$k$ and $k$ is the
inverse density parameter. $\delta S$ is the ground-state shell correction
defined as the difference of the experimental and theoretical (liquid drop)
masses. The inverse of $\gamma$ in the exponent of Eq.2, given by
$\gamma^{-1}$ = $\frac{0.4A^{4/3}}{\tilde{a}}$ is the rate at which the shell
effect is damped with the increase in excitation energy. The optical model
potential parameters for proton transmission coefficient are taken from Refs.
[32]. The moment of inertia of the CN is taken as
$I_{eff}=I_{0}(1+\delta_{1}J^{2}+\delta_{2}J^{4}$), where $I_{0}$(=
$\frac{2}{5}MA^{5/3}r_{0}^{2}$) is the moment of inertia of a spherical
nucleus, $\delta_{1}$ (= 1.0 $\times$ 10-4) and $\delta_{2}$ (= 2.0 $\times$
10-6) are the deformability parameters, r0=1.25 fm is the radius parameter and
$J$ is the total spin of the nucleus. It is seen that the choice of proton
optical potential has negligible effect on the slope of the spectrum. The
deformability parameters have also negligible effect on slope ($\delta_{1}$
change from 10-6 to 10-4 and $\delta_{2}$ change from 10-8 to 10-6). The slope
of the spectrum mainly depends on the inverse level density parameter $k$,
which is generally varied to explain the evaporated particle spectrum by using
$\chi^{2}$ minimization technique.

The $\gamma$-gated proton spectrum along with the statistical model
calculations is shown in Fig. 2. It is observed that the systematic value of
$k$ (8.6 MeV) is unable to explain the experimental data above 10 MeV as shown
in Fig. 2(a). Indeed, a large value of $k$ (11.2 MeV) is required to explain
the experimental data considering the emission of first chance protons from CN
as shown in Fig. 2(a). This observation is consistent with the previous
results obtained in refs. [6, 7]. The Ground-state quadrupole deformations of
71Ga and 72Ga are -0.207 and -0.215, respectively as have been reported by P.
Moller et. al [33]. Moreover, rotational bands have also been observed up to
the 4.2 MeV excitation energy in 71Ga [34]. One can expect also a rotational
band structure at high spin in 72Ga.Therefore, a rotational enhancement in the
Gallium isotopes is in agreement with the structural study. So the collective
enhancement due to ground state deformation of residual nucleus could be the
possible reason for this large $k$ value. On the other hand, as the evaporated
proton spectrum is gated with the $\gamma$-rays of 70Ga and 71Ga residual
nuclei, it could also contain the contributions of protons coming out after
one or two neutrons (1n or 2n) are emitted from the CN. Therefore, a
statistical model calculation has been carried out by using $k$=11.2 MeV to
obtain the proton spectrum after 1n and 2n emissions and compared with the
experimental data as shown in Fig.2(b). It should be mentioned that the same
normalization has been used in each case (protons from the first chance,
protons after 1n emission, and protons after 2n emission) while comparing with
the experimental data. It is seen that there could have been some
contributions of protons after 1n emission. However, the proton after 2n
emission is negligibly small as shown in Fig. 2(b).

A large value of $k$ = 11.2 MeV from the previous statistical model analysis
with FG model for NLD and the bumps around the centre of mass energies of 12
MeV and 15 MeV, could be a possible reason for the existence of collective
enhancement in NLD due to the deformation of the residual nuclei. In a
subsequent analysis, the $\gamma$-gated proton spectrum has been compared with
the statistical model prediction of proton (SMCP) spectrum by including
collective enhancement factor in NLD of 72Ga residual nucleus as shown in
Fig.3. The model calcualtion has been carried out by considering 1st chance
proton emission with the systematic value of $k$=8.6 MeV (which best explains
the low energy part of the proton spectra below 10 MeV) and a rotational
enhancement factor ($K_{rot}$) has been inlcuded as
$\rho$=$\rho_{int}K_{rot}$, where $\rho_{int}$ is intrisic level density. The
factor Krot is introduced with the intrinsic single particle level density to
consider the contribution of rotational enhancement factor in the NLD.
Microscopic shell model studies [6] have predicted that for nuclei with finite
ground state deformation, rotational collectivity causes large enhancement of
NLD (K${}_{rot}\approx$ 100). With increasing excitation energy, the
deformation effect gradually dies down due to the intrinsic motion. The
deformed shape of the residual nucleus changes into spherical shape and
rotational levels die out. Due to this shape transition, fade out of
collective enhancement in NLD is observed after a certain excitation energy.
The energy dependent Fermi function has been used in the rotational
enhancement factor given as

$K_{rot}=(\sigma^{2}_{\perp}-1)\times\it{f(E)}+1$ (3)
$f(E)=\frac{1}{1+\exp(\frac{E-E_{cr}}{d_{cr}})}$ (4)

The spin cut off parameter, $\sigma_{\perp}^{2}$ is replaced by $\lambda$T,
where T [=$\sqrt{(}U/a)$] is nuclear temperature and $\lambda$ is treated as
the magnitude of the enhancement parameter. The parameters $\lambda$, Ecr and
dcr are generally varied to explain the experimental data. It is observed that
the the experimental data below 10 MeV and above 13 MeV are nicely described
by the result of the statistical model calcualtion considering the first
chance protons and including the rotational enhancement in NLD of 72Ga, as
shown in Fig. 3(a). However, the experimental data in between 10 to 13 MeV
centre of mass energy cannot be explained. The conjecture is the presence of
protons from 2nd chance emission in that energy domain. Therefore, the proton
spectrum has also been calculated considering the level density and including
the rotational enhancement factor of 71Ga residual nucleus and shown in Fig.
3(b). Finally, the 1st and 2nd chance statistical model calculated proton
(SMCP) spectra are added with weightage obtaining the total yield as Ytotal=
Y(1st chance) + $w$$\times$ Y(2nd chance), which nicely explains the
$\gamma$-gated proton spectrum as shown in Fig. 3(b). A $\chi^{2}$
minimization procedure, with parameters from two rotational enhancement
factors corresponding to the NLDs of residual nuclei 72Ga and 71Ga and the
weight factor $w$ in the total yield, has been carried out to fit the
experimental spectrum. The best fit parameters of the rotational enhancement
factor in NLD are $\lambda$=4.0 $\pm$ 0.5, Ecr=14 $\pm$ 0.6 MeV and dcr=0.8
$\pm$ 0.1 MeV for 72Ga, and $\lambda$=5.0 $\pm$ 0.5, Ecr=19 $\pm$ 0.7 MeV and
dcr=0.7 $\pm$ 0.12 MeV for 71Ga and the weight factor $w$=0.3. The fit clearly
indicates that the collective enhancement in NLD due to the deformation of
residual nucleaus is required to explain the experimental data. The nuclear
level densities of the residual nuclei 72Ga and 71Ga including the collective
enhancements used in the statistical model calculation for explaining the
$\gamma$-gated spectrum are shown in Fig. 4. The NLD of 72Ga is normalized
with the experimental NLD at Sn obtained from neutron resonance spacing [35].

In Fig. 4, two different model nuclear level densities have been plotted to
compare with the present level densities of 72Ga and 71Ga and also compared
with the cumulative level distribution computed from the levels given in the
RIPL-3 [15] at low excitation energy. The constant temperature (CT) model,
which is known to best describe the level densities in the low excitation
energy domain up to the particle threshold energy, is represented by solid
curve. Also, shown the prediction of FG model with the systematic value of the
parameter $k$ = 8.6 MeV and not including the collective enhancement
contribution (dashed line). The CT model level density is given as $\rho_{\rm
CT}$ = $\frac{1}{T}$e${}^{(E-E_{0})/T}$, where T and E0 are free parameters.
It should be mentioned that the systematic values of CT model parameters
(E0=-2.44 MeV, T=0.92 MeV), obtained from the fitting of the NLD data from
RIPL-3 and the data at the neutron separation energy (Sn [35] are used while
comparing with the level density of 72Ga from the present work. As the
experimental NLD of 71Ga at Sn (from neutron resonance studies) is not
available, the present NLD of 71Ga is normalized with the results obtained
from CT model by using the value of E0=-2.5 MeV, T=0.99 MeV based on RIPL-3
data. The important aspects of the comparisons in Fig. 4 should be emphasized
here. It is seen that the extracted density that includes the enhancement
effect follows the trend of the CT model prediction locally around the
normalization point at Sn. However, with increasing excitation energy beyond
the neutron separation energy, as the collective enhancement effect wanes, the
NLD from the present work deviates from CT model prediction and gradually
coincides with the prediction of FG Model without enhancement using $k$=8.6
MeV. Thus, the excitation energy dependence of the extracted NLD, $\it{i.e.}$,
matching the CT prediction at lower excitations and that of the FG model
prediction without the deformation effect at higher excitations, indicates
that the consideration of rotational enhancement in NLD can play a significant
role in the low energy neutron capture process.

Figure 4:  (Color online) The enhanced Nuclear level densities as a function
of excitation energy are shown as used in the CASCADE. Histogram represents
the cumulative level distribution computed from the levels

given in RIPL-3 [15]

Figure 5:  (Color online) 71Ga(n, $\gamma$)72Ga capture cross-section as
function of neutron energy. Reported experimental data are compared with the
results obtained from TALYS calculation using the present enhanced level
density parameter.

Furthermore, the collective enhancement in NLD has been included in TALYS 1.95
code [36], for the first time, to investigate the effect of collective
enhancement in NLD on neutron capture cross-section. A rotational enhancement
function for 72Ga nucleus has been included in the FG model prescription of
NLD $\rho(E_{x})$=$\rho_{int}(E_{x})K_{rot}$ in TALYS 1.95 code to estimate
the 71Ga(n, $\gamma$)72Ga capture cross-section by using the systematic value
of inverse level density parameter (k = 8.6 MeV). In the TALYS code, default
global neutron optical model potential has been used for 71Ga+n and a
generalized Lorentzian form [42] of E1 $\gamma$-ray strength function has been
considered for 72Ga. It is found that (Fig. 5), the NLD without the collective
enhancement underpredicts the neutron capture cross-section data, while the
inclusion of collective enhancement in NLD can explain the data much better,
especially in the low excitation energy domain. The lower and upper limit (red
dashed line in Fig. 5) of the present cross-section arises due to the
uncertainties of parameters characterizing the collective enhancement function
as discussed earlier. As can be seen from Fig. 5, the neutron capture cross-
section with including collective enhancement is $\sim$2.1 times greater than
that of FG model with systematic LD parameter in the energy range of 25 - 90
keV and matches the data quite well in the low energy region. At higher
energies the capture data, however, fall within the lower limit of the curve
including the enhancement effect. It should be mentioned here that there
exists some difference in the relative normalization between the different
capture data sets. If the normalization is properly taken care of, the
matching with the present calculation will improve further. The present result
indicates that with the inclusion of collective enhancement in the NLD, the
calculated 71Ga(n, $\gamma$)72Ga cross-sections are closer to the direct
measurement data. In the low excitation energy domain, there are three
discrepant results for 71Ga(n, $\gamma$)72Ga cross-sections at 25 keV, which
are 104 $\pm$ 14 mb [12], 140 $\pm$ 30 mb [13] and 75 $\pm$ 10 mb [14].
Interestingly, the result at 25 keV from the present work is consistent with
the result obtained in Ref. [13]. It is worthwhile to mention that overall in
the low energy region, which is of astrophysical importance, the NLD with the
collective enhancement that explains the proton evaporation spectrum, better
describes the available neutron capture data. It should be added that the
$\gamma$-ray strength function could also play an important role to estimate
the neutron capture cross-section. Therefore, level density and $\gamma$-ray
strength function are both essential to estimate the astrophysical reaction
cross-section as precisely as possible.

In summary, the measured $\gamma$-gated proton spectrum is predominantly from
the compound nuclear process, which is ensured by the coincidence measurement
with selected $\gamma$ rays from the residues of p2n and pn channels following
the 9Be+64Ni reaction. The $\gamma$ gated proton spectrum has been utilized to
obtain the level densities of residual nuclei (72Ga and 71Ga). It is observed
that collective enhancement factors due to the deformation of residual nuclei
are required to be included in the NLD prescription of the Fermi-Gas (FG)
model to explain the experimental $\gamma$-gated proton spectrum. Furthermore,
the obtained level density for 72Ga has been used in the TALYS reaction code
to calculate the 71Ga(n, $\gamma$) 72Ga capture cross-section. It is seen that
NLD with collective enhancement explains the neutron capture data better than
the NLD without collective enhancement. The present results indicate that
collective enhancement in NLD could play an important role in order to
understand the production of nuclei through the neutron capture process
precisely. Therefore, a systematic and exclusive investigation is required in
near future to understand the effect of collective enhancement on neutron
capture cross-section to provide a reliable prediction of astrophysical
abundances.

Acknowledgements

We would like to thank Dr. D. Pandit, VECC for his help and advice in this
project. Authors thank the BARC-TIFR PLF staff for uninterrupted, steady beam
during the experiment.This work is supported by the Department of Atomic
Energy, Government of India (Project Identification No. RTI 4002), and the
Department of Science and Technology, Government of India (Grant
No.IR/S2/PF-03/2003-II).

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