# Parity-violating nuclear spin-rotation and NMR shielding tensors in
tetrahedral molecules

I. Agustín Aucar<EMAIL_ADDRESS>Instituto de Modelado e
Innovación Tecnológica (UNNE-CONICET), Facultad de Ciencias Exactas y
Naturales y Agrimensura, Universidad Nacional del Nordeste, Avda. Libertad
5460, Corrientes, Argentina  Yuly Chamorro  Anastasia Borschevsky  Faculty of
Science and Engineering, Van Swinderen Institute for Particle Physics and
Gravity, University of Groningen, 9747 AG Groningen, The Netherlands

###### Abstract

In natural processes involving weak interactions, a violation of spatial
parity conservation should appear. Its effects are expected to be observable
in molecules using different spectroscopic methodologies, but due to the tiny
magnitude of these effects they have never been measured yet. We present a
theoretical analysis of four-component relativistic nuclear-spin-dependent
parity-violating nuclear spin-rotation and NMR shielding tensors in a set of
tetrahedral chiral molecules. This work highlights the crucial role played by
the ligands and the electronic structure of the chiral center in the
enhancement of these effects, leading the way towards targeted design of
promising molecules for measurements.

##  I Introduction

The parity non-conserving nature of physical phenomena affected by weak
interactions is known to be responsible for asymmetric processes with respect
to spatial inversion of the coordinates of the particles in a system. The
existence of parity violation (PV) was first postulated by Lee and Yang[1],
and it was Wu et al.[2] who provided its pioneer experimental verification
using 60Co nuclei and observing nuclear $\beta$-decay processes due to weak
interactions. While the first confirmations of the existence of PV effects
were carried out in nuclear systems[2, 3], since then these phenomena were
also observed in atoms[4, 5, 6, 7, 8, 9, 10, 11].

Parity violating effects due to the weak interactions between electrons and
nucleons are expected to be observable even in molecular systems. In the case
of the two enantiomers of a chiral molecule, for example, a tiny energy
difference is expected to be found due to the PV effects. Many experiments,
using different methods, have been proposed to observe molecular PV
effects[12, 13, 14, 15, 16, 17, 18, 19, 20, 21], but in spite of the ever-
improving precision of such experimental searches none of them was successful
so far. Several theoretical studies addressed the influence of PV effects on
nuclear magnetic resonance (NMR) parameters like shielding or indirect spin-
spin coupling tensors in molecular systems[22, 23, 24, 25, 26, 27, 28, 29, 30,
31], and first such measurements were recently attempted[32, 33].

In this work we focus our attention on a property measured in microwave
spectroscopy, namely, the nuclear spin-rotation (NSR) tensor, where molecular
PV effects could be observed by analyzing the difference between the
measurements of this parameter in two enantiomers of a chiral molecule. When
the nuclear-spin-dependent (NSD) PV Hamiltonian is considered, a difference
between the NSR tensors $\bm{M}$ for the nuclei in the two enantiomers of a
chiral molecule is expected to appear, but if the PV effects are neglected,
the $\bm{M}$ values of the two enantiomers will be equal. The difference
appearing as a PV effect is due to an asymmetry in the electronic environments
around the nuclei, arising from weak interactions. As suggested by Barra and
co-workers, PV effects will contribute not only to the NSR constants, but also
to the NMR shielding and indirect spin-spin coupling tensors[23]. An
experimental observation of PV effects in one of these parameters would give
rise to the first detection of PV interactions in a static system. Atomic
experiments to detect weak interactions, in contrast, need transition
phenomena[34].

In this paper we explore the PV effects on the parity-violating nuclear spin-
rotation (PV-NSR) tensors of the central nuclei of tetrahedral chiral
molecules, focusing on the role of the electronic environment and the atomic
number of the central metal atom. We use the relativistic theoretical
formalism we have recently derived[35] based on previous investigations on
parity-conserving NSR tensors within the relativistic domain[36, 37, 38].

We analyze the 183W and 235U nuclei in the NW$XYZ$ and NU$XYZ$ series of
molecules, respectively (with $X,Y,Z=$ 1H, 19F, 35Cl, 79Br, 127I), and also
the $X$ nuclei in the N$X$HF$Z$ molecules ($X=$ 53Cr, 77Se, 95Mo, 125Te; $Z=$
35Cl, 79Br, 127I). We also report values of four-component (4c) isotropic PV
nuclear shielding constants. Some of the selected systems were used in
previous works to study PV effects in shielding and in other molecular
properties[39, 40, 41, 42].

We present a systematic analysis of relativistic and electronic correlation
effects on the PV-NSR and PV-shielding tensors, employing the polarization
propagator theory to calculate the corresponding linear response functions,
within the random phase approximation (RPA) theory based on the Dirac-Coulomb
(DC) Hamiltonian. Dirac-Hartree-Fock (DHF) and Dirac-Kohn-Sham (DKS)
methodologies were used to obtain the wave functions. In order to compare
relativistic 4c calculations with their non-relativistic (NR) limit, we also
employed the Lévy-Leblond (LL) Hamiltonian.

This work has the following structure: In Sec. II we give a brief theoretical
background to introduce the relativistic formulation used to calculate the
NSD-PV-NSR and NSD-PV-shielding tensors. In Sec. III, we provide the
computational details for all the calculations presented in this paper,
divided in two main parts: geometry optimizations and linear response
calculations, including a short analysis of the basis set convergence of the
PV-NSR constants in one of the studied molecules. In Sec. IV, we present the
4c computations of the isotropic NSD-PV-NSR constants for all the molecules
studied here. The relativistic and electronic correlation effects are studied
in this Section, as well as the influence of different ligands (keeping the
same metal nucleus or chiral center) and different chiral centers for the same
ligands. Sec. V contains the conclusions of this work and the outlook.

##  II Theory

Within the polarization propagator theory, any static (i.e., zero frequency)
second-order molecular property can be calculated as[43, 44]

$E^{(2)}_{PQ}=\textnormal{Re}\left[\langle\langle\,\hat{H}^{P}\,;\,\hat{H}^{Q}\,\rangle\rangle_{\omega=0}\right],$
(1)

where the operators $\hat{H}^{P}$ and $\hat{H}^{Q}$ are any perturbative
Hamiltonians. The linear response function on the right-hand-side of Eq. (1)
can also be written as the product of the perturbator $\textbf{b}^{P}$ (i.e.,
the property matrix element), the principal propagator $\textbf{M}^{-1}$
(i.e., the inverse of the electronic Hessian), and the perturbator
$\textbf{b}^{Q}$[45, 35]. The linear response function is usually computed by
solving the response equation

$\textbf{M}\;\textbf{X}^{Q}(\omega)=\textbf{b}^{Q},$ (2)

where $\textbf{X}^{Q}(\omega)=\textbf{M}^{-1}\,\textbf{b}^{Q}$ is expanded in
a linear combination of trial vectors, and then it is contracted with the
property matrix $\textbf{b}^{P}$[46].

In the particular case of the nuclear spin-rotation and the NMR shielding
tensors of a nucleus $N$ ($\bm{M}_{N}$ and $\bm{\sigma}_{N}$, respectively) it
is known that they can be obtained as the second order energy derivatives at
zero frequency[36, 47]

$\displaystyle\bm{M}_{N}$ $\displaystyle=$
$\displaystyle-\hslash\frac{\partial^{2}E(\bm{I}_{N},\bm{J})}{\partial\bm{I}_{N}\partial\bm{J}}\bigg{|}_{\bm{I}_{N}=\bm{J}=0}$
(3) $\displaystyle\bm{\sigma}_{N}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}E(\bm{\mu}_{N},\bm{B}_{0})}{\partial\bm{\mu}_{N}\partial\bm{B}_{0}}\bigg{|}_{\bm{\mu}_{N}=\bm{B}_{0}=0}$
(4)

where $\hslash=\frac{h}{2\pi}$ is the reduced Planck constant, $\bm{I}_{N}$ is
the dimensionless spin of nucleus $N$, $\bm{J}$ is the molecular rotational
angular momentum around the molecular center of mass, $\bm{B}_{0}$ is a
uniform external magnetic field, and
$\bm{\mu}_{N}=\gamma_{N}\,\hslash\,\bm{I}_{N}$ the magnetic moment due to the
nuclear spin, where $\gamma_{N}=\frac{e}{2m_{p}}g_{N}$ is the gyromagnetic
ratio of nucleus $N$ and $g_{N}$ is its $g$-factor, and where $e$ is the
fundamental charge and $m_{p}$ is the proton mass. The nuclear spin-rotation
(NSR) tensor $\bm{M}_{N}$ in Eq. (3) is given in units of energy. Therefore,
by dividing it by the Planck constant $h$ the corresponding frequency values
are obtained. SI units are used in the present work.

The PV contribution to the NSR tensor was recently derived within a
relativistic framework, and it has been shown that it can be written as the
linear response function[35]

$\displaystyle\bm{M}_{N}^{PV}$ $\displaystyle=$
$\displaystyle\frac{\hslash\,G_{F}}{2\sqrt{2}\,c_{0}}\kappa_{N}\langle\langle\;\rho_{N}(\bm{r})\;c\bm{\alpha}\;;\;\bm{J}_{e}\;\rangle\rangle\;\cdot\bm{I}^{-1},$
(5)

where $G_{F}$ is the Fermi coupling constant, whose most recent value is
$G_{F}/(\hslash\,c_{0})^{3}=1.1663787\times 10^{-5}$ GeV-2, or equivalently
$G_{F}\simeq 2.222516\times 10^{-14}\,E_{h}\,a_{0}^{3}$[48].

Besides,
$\kappa_{N}=-2\lambda_{N}\left(1-4\,\textnormal{sin}^{2}\theta_{W}\right)$,
where $\lambda_{N}$ is a nuclear state dependent parameter. It can be noted
that the constant factor in the NSD-PV Hamiltonian is found to be written in
different ways in the literature[49, 23, 24, 31, 50, 51, 42, 35]. There are
three predominant contributions to $\lambda_{N}$ due to the following
interactions[52, 53]: the weak coupling between neutral electronic vectors and
nucleon axial-vector currents[54], the electromagnetic interactions between
electrons and nuclear anapole moments (which become the dominant contributions
for heavy nuclei)[55, 56, 57], and the nuclear-spin-independent electron
axial-vector and nucleon vector current weak interactions combined with
hyperfine interactions[58].

As $\lambda_{N}$ is a factor with a nuclear structure origin and to facilitate
the comparison with previous works, we have set $\lambda_{N}=1$ in our
calculations. Therefore, all the reported values of $\bm{M}_{N}^{PV}$ and
$\bm{\sigma}_{N}^{PV}$ must be scaled by the true value of $\lambda_{N}$ in
order to get results that can be compared with measurable physical quantities.
For heavy nuclei it is expected that $1<\lambda_{N}<10$[55, 56].

While the most recent value of the sine-squared weak mixing angle $\theta_{W}$
is 0.23857(5)[59], we use $\textnormal{sin}^{2}\theta_{W}=0.2319$[60] as the
Weinberg parameter throughout this work for ease of comparison with earlier
investigations[35].

In addition, in Eq. (5) $\bm{\alpha}$ are the $4\times 4$ Dirac matrices given
in the standard representation based on the Pauli spin matrices, $\bm{r}$ is
the position of the electrons with respect to the coordinate origin,
$\rho_{N}(\bm{r})$ is the normalized nuclear electric charge density of
nucleus $N$ at the position of electron (given in units of the inverse of cube
distances), $\frac{1}{c_{0}}$ is linearly proportional to the fine structure
constant (in SI units, the fine structure constant is
$\frac{1}{4\pi\epsilon_{0}}\frac{e^{2}}{\hslash\,c_{0}}$), $c$ is the speed of
light in vacuum, scalable to infinity at the NR limit, $\bm{I}^{-1}$ is the
inverse molecular inertia tensor with respect to the molecular center of mass
(CM) in the equilibrium geometry, and $\bm{J}_{e}=\bm{L}_{e}+\bm{S}_{e}$ is
the $4\times 4$ total electronic angular momentum operator.

In the present work we neglect the contributions to Eq. (5) due to the Breit
electron-nucleus interaction. These were shown to be very small for the
parity-conserving NSR tensors[38].

Similarly to Eq. (5), the PV contribution to the NMR shielding tensor is given
by[24, 30]

$\displaystyle\bm{\sigma}_{N}^{PV}$ $\displaystyle=$
$\displaystyle\frac{m_{p}\,G_{F}}{2\,\sqrt{2}\,\hslash\,c_{0}}\;\frac{\kappa_{N}}{g_{N}}\langle\langle\;\rho_{N}(\bm{r})\;c\bm{\alpha}\;;\;\bm{r}_{GO}\times
c\bm{\alpha}\;\rangle\rangle,$ (6)

where $\bm{r}_{GO}=\bm{r}-\bm{R}_{GO}$ is the electronic position relative to
the gauge origin position for the external magnetic potential.

The NR limits for the PV-NSR and PV-NMR-shielding tensors given in Eqs. (5)
and (6), respectively, were derived by some of us in Ref. 35 by applying the
linear response within the elimination of small components (LRESC) approach.
In the same work, it was shown that the NR limits of the linear response
functions involved in Eqs. (5) and (6) are exactly equal each other in the
cases where $\bm{R}_{GO}$ is placed at the molecular CM. Following the LRESC
model[36, 61, 62], it is possible to expand these two relativistic second
order properties in terms of the fine structure constant and to get their
leading order relativistic corrections. Work in this line is currently in
progress[63], and it can be shown that a close relationship between these
properties also appears in the relativistic regime.

When the two enantiomers of a chiral molecule are analyzed, the isotropic
nuclear spin-rotation constants
$M_{X,iso}=\frac{1}{3}\textnormal{Tr}(\bm{M}_{X})$ of their chiral centers $X$
will be given as the sum of a parity-conserving and a parity-violating
contributions: $M_{X,iso}=M^{PC}_{X,iso}\pm M^{PV}_{X,iso}$, where the PV term
of each of the two enantiomers have an opposite sign with respect to the
other. This means that there will be a difference between the isotropic NSR
constants of the same nuclei in the left(S)- and right(R)-handed enantiomers,
which will be $|\Delta
M_{X,iso}|=|M^{S}_{X,iso}-M^{R}_{X,iso}|=2\,|M^{PV}_{X,iso}|$. Similarly,
$|g_{X}\,\Delta\sigma_{X,iso}|=|g_{X}\,\sigma^{S}_{X,iso}-g_{X}\,\sigma^{R}_{X,iso}|=2\,|g_{X}\,\sigma^{PV}_{X,iso}|$.
In the following, we will report the $|\Delta M_{X,iso}|$ and the
$\Delta\sigma_{X,iso}|$ quantities.

##  III Computational details

In this work we studied the PV-NSR and PV-NMR shielding tensors in the
N183W$XYZ$ and N235U$XYZ$ series of molecules (with $X,Y,Z=$ 1H, 19F, 35Cl,
79Br, 127I), as well as in the N$X$HF$Z$ (with $X=$ 53Cr, 77Se, 95Mo, 125Te;
$Z=$ 35Cl, 79Br, 127I) systems, schematically shown in Fig. 1. This selection
of molecules allows us to analyze the effects produced by both the ligands and
by the chiral centers on the studied properties.

Figure 1: Schematic representation of the molecules studied in this work.

We optimized all the molecular geometries using density functional theory
(DFT), with the PBE0 functional[64]. Furthermore, due to the fact that
specifically scalar relativistic effects are crucial for obtaining reliable
molecular geometries[65, 66, 67], we used scalar relativistic pseudo-
potentials for the heavy atoms.

In the optimizations, Dunning’s correlation consistent basis sets (aug-cc-
pV5Z) were used on the light elements H, N, F[68, 69], Cl[70], and Cr[71], and
the Stuttgart-Koeln pseudo-potentials and correlation consistent basis sets
(aug-cc-pV5Z-PP) on the heavy atoms, Br[72], Mo[73], I[72, 74], and W[75].
These correlation consistent basis sets are not available for U, and thus we
used the atomic natural orbital valence basis set in conjunction with the
Stuttgart pseudo-potentials[76, 77, 78]. The obtained structural parameters
are shown in the Supplementary Material.

We performed all the optimizations using the default settings in the energy
minimization scheme of the Gaussian program package[79]. Additionally, we
performed a frequency analysis on the obtained geometries to confirm these as
true minima. We computed the frequencies corresponding to the final geometries
with the same method employed for the optimization and using the Gaussian
code.

Regarding the parity-violation contributions to the properties analyzed in
this work, to calculate $\bm{\sigma}^{PV}$ we used the Dirac program
package[80, 81], whereas a locally modified version of the same code was used
to obtain the values of $\bm{M}^{PV}$. We used the DC and the LL Hamiltonians
to obtain the relativistic and the NR results, respectively[82]. The standard
procedure to avoid the explicit calculation of (SS$\mid$SS) integrals in both
the self-consistent field and the linear response blocks was followed in all
the DC calculations, replacing these integrals by an energy correction. This
choice is the default one in the Dirac code[83].

Unless otherwise stated, we have employed the Dyall’s relativistic cv3z
uncontracted basis sets (dyall.cv3z) for all the elements analyzed in this
work[84, 85, 86, 87, 88, 89, 90, 91]. An analysis of the basis set convergence
of $M^{PV}_{iso}$ was performed for the tungsten nucleus in the NWHFI molecule
and in Table 1 we show the values of $M^{PV}_{W,iso}$ obtained using double-,
triple-, and quadruple-zeta valence, core-valence, and all-electron (i.e.,
including correlating functions for all shells) uncontracted Dyall’s basis
sets (dyall.v$Y$z, dyall.cv$Y$z, and dyall.ae$Y$z, with $Y=$ 2, 3, 4)[92, 84,
85, 86, 89, 93, 91]. The value obtained using the dyall.cv3z basis set shows a
good convergence and therefore this is the basis set we use across this work.

Table 1:  $M^{PV}_{iso}$ (in $\mu$Hz) for the 183W nucleus in the NWHFI molecule for different uncontracted Dyall’s basis sets, using the DC Hamiltonian and the DFT/PBE0 approach. $X$ | dyall.v$X$z | dyall.cv$X$z | dyall.ae$X$z
---|---|---|---
2 | 33.29 | 33.40 | 33.40
3 | 34.80 | 34.88 | 34.89
4 | 35.02 | 35.07 | 35.08

The common-gauge-origin approach was used in all the calculations, and the
gauge origin for the external magnetic potential has been placed at the
molecular center of mass. The small component basis sets were generated from
the large component basis sets in all cases by applying the unrestricted
kinetic balance (UKB) prescription[94].

In the self-consistent-field calculations and for describing the NSD-PV
perturbed Hamiltonian, a spherically symmetric Gaussian-type nuclear charge
density distribution model was used[95], as it has been shown that use of
finite nuclear model is important to adequately describe the parity-conserving
nuclear spin-rotation tensors[96]. However, a point-like nuclear magnetic
moment $\bm{\mu}_{N}$ was employed in the term arising from the vector
potential
$\bm{A}_{N}=\frac{\mu_{0}}{4\pi}\bm{\mu}_{N}\times\frac{\bm{r}_{N}}{|\bm{r}_{N}|^{3}},$
where $\mu_{0}$ is the vacuum permeability, and $\bm{r}_{N}=\bm{r}-\bm{R}_{N}$
is the electronic position relative to nucleus $N$.

The response calculations were carried out at the 4c polarization propagator
RPA level of theory employing the DHF and DKS-DFT wave functions. DFT
calculations were performed to study the influence of electronic correlation
effects, and they were based on both the 4c-DC and NR-LL Hamiltonians. We used
the NR exchange-correlation hybrid PBE0 functional[64], because of its good
performance in the 4c calculations of parity-conserving NSR constants (in
comparison with experimental values)[96, 97]. The same functional was recently
used to calculate the PV-NSR tensors in a different series of molecules[35].

A deeper analysis of the electronic correlation effects was performed for the
NWHF$X$ and NUHF$X$ series of molecules (with $X=$ Cl, Br, I). For these, we
calculated the PV-NSR tensors using the following DFT functionals: (i) The
local density approximation (LDA) functional[98, 99], (ii) the generalized
gradient approximation (GGAs) PBE functional[100], and (iii) the hybrid
functionals PBE0[64] and CAM-B3LYP[101].

The response of Eq. (2) was solved with respect to the property gradient
associated with the following operators: (i) the total electronic orbital and
spin angular momenta operators, to calculate $\bm{M}_{N}^{PV}$; and (ii) the
external magnetic field, for $\bm{\sigma}_{N}^{PV}$.

The employed nuclear $g$-factors used to calculate $\bm{\sigma}_{N}^{PV}$ were
taken from Ref. 102 and their values are displayed in Table 2.

Table 2:  Nuclear $g$-factors used in the calculations of $\bm{\sigma}_{X}^{PV}$. $X$ | $g_{X}$
---|---
53Cr | $-0.31636$
77Se | $1.070084$
95Mo | $-0.36568$
125Te | $-1.77701$
183W | $0.235569$
209Po | $1.376$
235U | $-0.1085714286$

##  IV Results and discussions

We report 4c relativistic calculations of isotropic PV-NSR and PV-NMR
shielding constants. These linear response properties were obtained by
applying the RPA approach of the polarization propagator theory, combined with
DHF and DKS wave functions based on the DC Hamiltonian. The influence of
electron correlation effects (taken as the difference between linear response
calculations using DKS and DHF wave functions) as well as of relativistic
effects on the calculated PV-NSR constants is investigated. We then proceed to
analyze the effects of the chiral center and the chemical environment. These
topics are addressed separately in the Subsections below.

###  IV.1 Correlation and relativistic effects

In this Section we investigate the effect of relativity and electron
correlation in the calculations of $M_{U,iso}^{PV}$ and $\sigma_{U,iso}^{PV}$.
To analyze both effects, we consider the differences between the 4c and the NR
values, and between the DFT/PBE0-based and the DHF-based calculations,
respectively.

Table 3:  Isotropic shifts of PV-NSR and “normalized” PV-NMR-shielding constants ($2\,|M^{PV}_{U,iso}|$, in $\mu$Hz, and $2\,|g_{U}\,\sigma^{PV}_{U,iso}|$, in $\mu$ppm, respectively) for the 235U nucleus in the NUHF$X$ ($X=$ Cl, Br, I) set of molecules, using different methods. Hamiltonian | Method | $2\,|M^{PV}_{U,iso}|$ | $2\,|g_{U}\,\sigma^{PV}_{U,iso}|$
---|---|---|---
$X=$ Cl |  $X=$ Br |  $X=$ I |  $X=$ Cl |  $X=$ Br |  $X=$ I
LL (NR) | DHF-RPA | 28.4 | 35.0 | 41.8 | – | – | –
DFT/PBE0 | 23.4 | 21.6 | 26.4 | – | – | –
DC (4c) | DHF-RPA | 604.0 | 652.0 | 542.0 | 67.4 | 80.2 | 65.7
DFT/LDA | 200.0 | 240.0 | 220.0 | 0.8 | 0.7 | 1.8
DFT/PBE | 196.6 | 238.0 | 214.0 | 0.7 | 0.0 | 0.4
DFT/CAMB3LYP | 99.8 | 117.4 | 119.2 | 4.4 | 4.9 | 5.8
DFT/PBE0 | 121.4 | 142.6 | 141.0 | 2.8 | 3.5 | 5.4

In Table 3 and Fig. 2(a), the inclusion of relativity and electronic
correlation are seen to have an opposite effect. At the DHF-RPA level of
theory (4c-RPA vs NR-RPA), relativistic effects increase the values of
$|\Delta M_{X,iso}^{PV}|$ significantly (with a 4c/NR ratio between 13 and
22). However, the calculated relativistic and NR values of $|\Delta
M_{X,iso}^{PV}|$ are closer in magnitude when electron correlation is included
using the DFT methodology (4c-PBE0 vs NR-PBE0), with a 4c/NR ratio of just
between 5 and 7. The above observations demonstrate the importance of using a
relativistic framework for meaningful investigations of these properties.
Correlation effects are more pronounced in the relativistic than in the NR
regime. For $|g_{U}\,\Delta\sigma_{U,iso}^{PV}|$ (Fig. 2(b)), the effect of
electron correlation is also to decrease the calculated values significantly.
Therefore, reliable calculations of isotropic PV-NSR and PV-NMR shielding
constants require simultaneous inclusion of both relativistic and correlation
effects. Instabilities of the Kramer’s restricted DHF and DKS wave functions
appear in the LL calculations of $\sigma_{U,iso}^{PV}$ and that is why they
are not reported in this work.

(a)

(b)

Figure 2: Calculated values of (a) $|\Delta M_{U,iso}|=2\,|M^{PV}_{U,iso}|$
(in $\mu$Hz) and (b)
$|g_{U}\,\Delta\sigma_{U,iso}|=2\,|g_{U}\sigma^{PV}_{U,iso}|$ (in $\mu$ppm)
for 235U in NUHF$X$ molecular systems (with $X=$ Cl, Br, I) employing the LL
and DC Hamiltonians, at the DHF, DFT-LDA, DFT-PBE, DFT-CAMB3LYP, and DFT-PBE0
levels of approach using the dyall.cv3z basis set for all elements.

We also evaluate the dependence of the calculated properties on the chosen
functional at the 4c-DFT level of theory. While
$|g_{U}\,\Delta\sigma_{U,iso}^{PV}|$ is stable under the change of the DFT
functional, $|\Delta M_{U,iso}^{PV}|$ varies more significantly (see Figs.
2(a) and 2(b)).

Figs. 3(a) and 3(b) show the relativistic effects at the DFT/PBE0 level of
theory in the calculations of $M_{iso}^{PV}$ for the tungsten and uranium
nuclei in the NW$XYZ$ and NU$XYZ$ systems (with $X,Y,Z=$ H, F, Cl, Br, I),
respectively. The H- and F-containing systems exhibit the highest isotropic
PV-NSR constants in both the NR and 4c regimes.

(a)

(b)

Figure 3: (Top) Calculated values of $M^{PV}_{iso}$ for (a) 183W and (b) 235U
in the NW$XYZ$ and NU$XYZ$ systems ($X,Y,Z=$ H, F, Cl, Br, I), respectively,
employing the LL and DC Hamiltonians at the DFT-PBE0 level of theory and using
the dyall.cv3z basis set for all elements. (Bottom) Relativistic effects taken
as the difference between the 4c and NR values of $M^{PV}_{iso}$. All values
are in $\mu$Hz.

In Fig. 4 a similar behavior can be observed for $g_{W}\,\sigma^{PV}_{W,iso}$
in the NW$XYZ$ set of molecules (with $X,Y,Z=$ H, F, Cl, Br, I).

Figure 4: (Top) Calculated values of $g_{W}\,\sigma^{PV}_{W,iso}$ for the 183W
nucleus in the NW$XYZ$ systems ($X,Y,Z=$ H, F, Cl, Br, I), employing the LL
and DC Hamiltonians at the DFT-PBE0 level of theory and using the dyall.cv3z
basis set for all elements. (Bottom) Relativistic effects taken as the
difference between the 4c and NR values of $g_{W}\,\sigma^{PV}_{W,iso}$. All
values are given in $\mu$ppm.

###  IV.2 Fixing chiral centers. Analysis of the environment

Initially, we focus our analysis on the effects of the ligands on the size of
the calculated $|\Delta M^{PV}_{K,iso}|$ and
$|g_{K}\,\Delta\sigma^{PV}_{K,iso}|$ ($K=$ 183W and 235U) on the NW$XYZ$ and
NU$XYZ$ series of molecules ($X,Y,Z=$ H, F, Cl, Br, I), as seen in Table 4.

Table 4:  Calculations of $2\,|M^{PV}_{K,iso}|$ (in $\mu$Hz) and $2\,|g_{K}\,\sigma^{PV}_{K,iso}|$ (in $\mu$ppm) for the $K=$ 183W and 235U nuclei in the N$KXYZ$ ($X,Y,Z=$ H, F, Cl, Br, I) molecules. The DC Hamiltonian was used at the DFT/PBE0 level of approach. |  $2\,|M^{PV}_{K,iso}|$ ($\mu$Hz) |  $2\,|g_{K}\,\sigma^{PV}_{K,iso}|$ ($\mu$ppm)
---|---|---
$XYZ$ |  $K=$ W |  $K=$ U |  $K=$ W |  $K=$ U
HFCl | 50.40 | 121.40 | 0.93 | 2.77
HFBr | 62.40 | 142.60 | 1.66 | 3.48
HFI | 69.80 | 141.00 | 3.26 | 5.40
HClBr | 7.70 | 25.80 | 0.77 | 0.84
HClI | 15.74 | 42.20 | 2.50 | 3.66
HBrI | 0.90 | 12.56 | 1.73 | 2.54
FClBr | 4.10 | 4.36 | 0.08 | 0.38
FClI | 4.24 | 1.79 | 0.19 | 0.81
FBrI | 0.67 | 6.90 | 0.10 | 0.31
ClBrI | 1.08 | 3.52 | 0.00 | 0.06

In Fig. 5 we have classified all the systems in three different groups:
systems containing (i) both H and F atoms, (ii) only H atoms without fluorine,
and (iii) systems containing no H atoms. In Fig. 5(a) it is seen that the size
of $|\Delta M^{PV}_{X,iso}|$ follows (i) $>$ (ii) $>$ (iii).

As stated in Sec. IV.1, the H- and F-containing systems exhibit the highest
isotropic PV-NSR constants. This can be due to the fact that an increasingly
asymmetric electronic distribution appears in the molecular chiral center when
the difference of electronegativities between two of the ligands also
increases. The electronegativities of the atoms considered in this work follow
the tendency
$\chi_{\text{F}}>\chi_{\text{Cl}}>\chi_{\text{Br}}>\chi_{\text{I}}>\chi_{\text{H}}$.
It implies that F and H have the largest and smallest electronegativities,
respectively, and hence the largest PV effects are found for the NWHF$X$ and
NUHF$X$ molecules (with $X=$ Cl, Br, I). On the other hand, as Cl and Br have
similar electronegativities, the presence of these ligands will not generate a
strong asymmetry in the electronic cloud at the chiral center position, and so
the resulting PV effects will be between the lowest for the studied series of
molecules, in spite of them being heavier than the HF-containing molecules.

On the other hand, $\sigma^{PV}_{X,iso}$ does not follow the same tendency, as
seen in Fig. 5(b). For $|g_{X}\,\Delta\sigma^{PV}_{X,iso}|$, groups (i) and
(ii) show a more similar behavior, while the systems that do not contain
hydrogen atoms (i.e., those of group (iii)) still yield the lowest values. The
effect of the environment is remarkably large for these properties, and more
significant than the atomic number of the chiral center, in many cases. For
example, $|\Delta M^{PV}_{X,iso}|$ is much larger in NWHFI than in the heavier
NUClBrI. Therefore, we can identify the H- and F-containing systems as the
most promising candidates for measurements in the investigated series. Similar
conclusions were reached in an earlier paper that investigated the shieldings
in the NW$XYZ$ compounds[42].

(a)

(b)

Figure 5: Calculated values of (a) $|\Delta
M_{X,iso}|=|M^{S}_{X,iso}-M^{R}_{X,iso}|=2\,|M^{PV}_{X,iso}|$ (in $\mu$Hz) (b)
$|g_{X}\,\Delta\sigma_{X,iso}|=2\,|g_{X}\,\sigma^{PV}_{X,iso}|$ (in $\mu$ppm)
for the $X$ nuclei in N$XABC$ systems (with $X=$ 183W, 235U; and $A,B,C=$ H,
F, Cl, Br, I), employing the DC Hamiltonian at the DFT-PBE0 level of theory
and using the dyall.cv3z basis set for all the elements.

###  IV.3 Fixing environment. Analysis of chiral centers

In general, U-containing systems have higher PV-NSR and PV-NMR shielding
constants than W-containing systems (for the same ligand environment).
Furthermore, in both cases there is not a significant effect of the ligand
when it is different to H and F. Therefore, we proceeded to study the effect
of the chiral center while keeping the H and F ligands in the systems under
study.

To analyze the effects produced by the PV interactions as a function of the
chiral center, we included in the study further molecules containing elements
with 6 valence electrons, and classified our sample into two groups according
to their metal valence open-shell orbitals: $p$-group elements: Se, Te, Po
($s^{2}p^{4}$) and transition metals: Cr, Mo ($d^{5}s^{1}$), W ($d^{4}s^{2}$),
and U ($f^{3}d^{1}s^{2}$). The obtained results are displayed in Table 5.

Table 5:  Calculations of $2\,|M^{PV}_{K,iso}|$ (in $\mu$Hz) and
$2\,|g_{K}\,\sigma^{PV}_{K,iso}|$ (in $\mu$ppm) for the $K$ nuclei (with $K=$
53Cr, 77Se, 95Mo, 125Te, 183W, 209Po, and 235U) in the N$K$HF$X$ ($X$= Cl, Br,
I) series of molecules. The DC Hamiltonian was used in the calculations at the
DFT/PBE0 level of approach. DFT/PBE was used the for Po-containing molecules
to avoid instabilities of the Kramer’s restricted DKS wave functions appearing
in the DFT/PBE0 calculations of these systems.

|  $2\,|M^{PV}_{K,iso}|$ ($\mu$Hz) |  $2\,|g_{K}\,\sigma^{PV}_{K,iso}|$ ($\mu$ppm)
---|---|---
$K$ | N$K$HFCl | N$K$HFBr | N$K$HFI | N$K$HFCl | N$K$HFBr | N$K$HFI
Cr | 0.52 | 0.61 | 0.59 | 0.02 | 0.02 | 0.10
Se | 6.34 | 9.76 | 10.40 | 0.08 | 0.08 | 0.04
Mo | 6.18 | 7.96 | 8.86 | 0.09 | 0.20 | 0.47
Te | 14.72 | 25.80 | 28.20 | 1.07 | 1.41 | 1.26
W | 50.40 | 62.40 | 69.80 | 0.93 | 1.66 | 3.26
Po | 286.00pbe | 478.00pbe | 600.00pbe | 3.94pbe | 3.26pbe | 10.34pbe
U | 121.40 | 142.60 | 141.00 | 2.77 | 3.48 | 5.40

  * a

Calculated using the DFT/PBE functional.

Figs. 6 and 7 present the values of $|\Delta M_{X,iso}|$ and $|g_{X}\
\Delta\sigma_{X,iso}|$ for the $X$ nuclei in N$X$HF$Y$ systems with $X=$ 53Cr,
77Se, 95Mo, 125Te, 183W, 209Po, and 235U, and $Y=$ Cl, Br, I as a function of
the atomic number of the chiral center, respectively. As expected, the PV-NSR
and PV-NMR-shielding constants increase with the atomic number; however, it is
interesting to remark that this dependency is different for the two groups of
molecules. Generally, the PV-NSR and PV-NMR-shieldings increase faster with
the atomic number in the systems containing $p$ open shell orbitals than in
those with $d$ open shells. We found that the excitations from the valence
shells provide the highest contributions to these properties in the linear
responses of Eqs. (5) and (6). We can thus explain the difference in the
trends by the fact that the valence electrons in the $p$ orbitals experience
higher relativistic effects and contract more than those in $d$ and $f$
shells, bringing the electrons closer to the nucleus This leads us to find the
largest PV effects in Po-containing systems. Notice that we have included
systems containing U (heavier than Po) in our study.

(a)

(b)

(c)

Figure 6: Calculated values of $|\Delta M_{X,iso}|=2\,|M^{PV}_{X,iso}|$ (in
$\mu$Hz) for the $X$ nuclei in N$X$HF$Y$ systems with $X=$ 53Cr, 77Se, 95Mo,
125Te, 183W, 209Po, and 235U, for $Y=$ (a) Cl, (b) Br, and (c) I, employing
the DC Hamiltonian at the DFT-PBE0 level of theory (except for NPoHF$X$, where
the DFT-PBE functional was used instead) and using the dyall.cv3z basis set
for all the elements.

(a)

(b)

(c)

Figure 7: Calculated values of
$|g_{X}\,\Delta\sigma_{X,iso}|=2\,|g_{X}\,\sigma^{PV}_{X,iso}|$ (in $\mu$ppm)
for the $X$ nuclei in N$X$HF$Y$ systems with $X=$ 53Cr, 77Se, 95Mo, 125Te,
183W, 209Po, and 235U, for $Y=$ (a) Cl, (b) Br, and (c) I, employing the DC
Hamiltonian at the DFT-PBE0 level of theory (except for NPoHF$X$, where the
DFT-PBE functional was used instead) and using the dyall.cv3z basis set for
all the elements.

On the other hand, Figs. 6(a), 6(b), 6(c), and also Figs. 7(a), 7(b), and 7(c)
show the same dependency of both properties on the metal atomic number when
$Y$ is varied in N$X$HF$Y$ molecules. There is only one exception to this
tendency, for $|g_{\text{Se}}\ \Delta\sigma_{\text{Se},iso}|$ in NSeHFI.

##  V Conclusions

NSD-PV contributions to the NSR and NMR shielding tensors were obtained in
this work using the relativistic expression proposed in Ref. 35 for a series
of light- and heavy-element-containing tetrahedral molecules. We have shown
that relativity plays a crucial role in describing the PV effects in these
parameters for the chiral centers of the N$AXYZ$ series of molecules (with
$A=$ Cr, Mo, W, Se, Te, Po, U; and $X,Y,Z=$ H, F, Cl, Br, I). Electron
correlation effects are as important as relativity for describing the PV-NSR
and PV-NMR-shielding constants in these molecules, and are more pronounced in
the relativistic than in the NR calculations. Therefore, reliable calculations
of $M_{iso}^{PV}$ and $\sigma_{iso}^{PV}$ require simultaneous inclusion of
both relativistic and correlation effects.

We have shown that molecules containing both H and F atoms exhibit the highest
isotropic PV-NSR and PV-NMR shielding constants, making this choice of ligand
the most promising for such investigations.

We also provide a study of the effect of the chiral center (keeping the H and
F ligands). Here, we included further tetrahedral molecules containing
elements with 6 valence electrons as chiral centers. The isotropic PV-NSR and
PV-NMR-shielding constants increase with the atomic number of the chiral
center. However, depending on the electronic structure of the metal atom,
different tendencies emerge; that is the PV-NSR and PV-NMR-shieldings increase
faster with the atomic number in systems containing $p$ open-shell orbitals
than in those with $d$ open shells. We found the highest value of PV-NSR
constant shift for the 209Po nucleus in NPoHFI, on the order of 0.6 mHz.

While the PV-NSR constants given in this work do not reach the experimental
sensitivity limit for NSR constants, on the order of 1 to 10 Hz[103, 104],
they give crucial insights for the experimental search of PV effects in
molecules. A natural extension of this work is to search for promising
realistic molecules for measurements that are both experimentally accessible
and benefit from larger PV-NSR contributions. We will use the insights from
this and earlier works on PV-NMR-shieldings[105, 106, 107, 108, 109, 42, 32]
as a starting point to identify new molecular candidates for further
computational investigations.

## Supplementary Material

See supplementary material for the optimized structural parameters of the
chiral molecules studied in this work.

###### Acknowledgements.

This work has been performed under the Project HPC-EUROPA3
(INFRAIA-2016-1-730897), with the support of the EC Research Innovation Action
under the H2020 Programme; in particular, IAA gratefully acknowledges the
support of the University of Groningen and the computer resources and
technical support provided by SURFsara. We would like to thank as well the
Center for Information Technology of the University of Groningen in The
Netherlands, and the Institute for Modeling and Innovation on Technologies
(IMIT) of Argentina for their support and for providing access to the
Peregrine and IMIT high performance computing clusters. This work also made
use of the Dutch national e-infrastructure with the support of the SURF
Cooperative using grant no. EINF-3247. IAA acknowledges partial support from
FONCYT by grants PICT-2016-2936 and PICT-2020-SerieA-00052. This publication
is part of the project High Sector Fock space coupled cluster method:
benchmark accuracy across the periodic table (with project number
VI.Vidi.192.088 of the research programme Vidi which is financed by the Dutch
Research Council (NWO)).

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# OSC-MC: Online Secure Communication Model for Cloud Environment

Deepika Saxena and Ashutosh Kumar Singh D. Saxena and A. K. Singh are with the
Department of Computer Applications, National Institute of Technology,
Kurukshetra, India. E-mail<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
###### Abstract

A malicious cloud user may exploit outsourced data involved in online
communication, co-residency, and hypervisor vulnerabilities to breach and
hamper sensitive information, and inject malicious traffic based congestion,
rendering services to other benign users. To address this critical and
challenging problem, this letter proposes an Online Secure Communication Model
for Cloud (OSC-MC) by identifying and terminating malicious VMs and inter-VM
links prior to occurrence of security threats. The anomalous network traffic,
bandwidth usage, and unauthorised inter-VM links are security breach
indicators which guides secure cloud communication and resource allocation.
The simulation and comparison of the proposed model with existing approaches
reveal that it significantly improves authorised inter-communication links up
to 34.5% with reduction of network hogs, and power consumption by 66.46% and
39.31%, respectively.

###### Index Terms:

online communication, inter-VM relation, network cascading, malicious traffic.

##  I Introduction

The cloud platforms enabled with easy provisioning, rapid deployment, resource
scalability, hardware consolidation etc., have become critical avenues to many
commercial, academic, and research organizations by providing them access to
millions of customers at moderate cost [1]. However, the cloud communications
suffer from high security breaches due to discrepancies and vulnerabilities
associated with the networking devices, hypervisor and susceptibilities of
side channels [2], [3]. A malicious user may inject network congestion, launch
multiple VMs and exploit sequential or parallel VM placement to hamper real
time communication and compromise benign user’s VMs [4]-[7]. Therefore,
securing online data execution and communication among different cloud users’
applications, is a crucial and bottleneck problem. To the best of the authors’
knowledge, there is no existing model which enables proactive protection
against security threats via congestion, co-location, network cascading
effect, and vulnerability.

This letter proposes a novel Online Secure Communication Model for Cloud
Environment (OSC-MC) to provide secure and network efficient data execution
and transmission in cloud data centre networks. In this model, the inter-VM
relations on a server are exclusively monitored, formulated, and analyzed as
suspicious or non-suspicious by comparing with the authorized inter-
communication links specified for a VM in a log maintained by a resource
manager. OSC-MC investigates the inter-VM behaviour, identifies and suspends
malicious VMs, and terminates all the malicious links before the completion of
sufficient time required to accomplish successful attack. Furthermore, the
incoming traffic is predicted online and distributed into clusters on the
basis of bandwidth consumption of future applications to combat the network
congestion based security attacks. The consecutive prediction and grouping of
incoming traffic into clusters help in ($i$) determining the network
hogs/congestion and mitigating them prior to occurrence, ($ii$) autoscaling of
sufficient number of VMs and their scheduling on selected network-efficient
servers with minimized bandwidth wastage.

##  II OSC-MC

Consider $m$ users {$U_{1}$, $U_{2}$, …, $U_{m}$}$\in\mathds{U}$ have
requested execution of their application (i.e., Bag of Tasks {$t_{1}$,
$t_{2}$, …, $t_{n}$}) to Resource Manager (RM) on a set of $q$ VMs {$V_{1}$,
$V_{2}$, …, $V_{q}$} $\in\mathds{V}$ hosted on $p$ servers {$S_{1}$, $S_{2}$,
…, $S_{p}$} $\in\mathds{S}$. For instance, VMs: {$V_{1}$, $V_{2}$, …, $V_{s}$}
are deployed on $S_{1}$; {$V_{1^{\prime}}$, $V_{2^{\prime}}$, …, $V_{r}$} and
{$V_{1^{\prime\prime}}$, $V_{2^{\prime\prime}}$, …, $V_{q}$} on $S_{2}$ and
$S_{p}$, respectively as shown in Fig. 1.

Figure 1: Online Secure Cloud Communication Model

RM distributes the users’ tasks among the VMs purchased by them and allocates
these VMs to the selected network-efficient (i.e., optimized bandwidth)
servers. The periodic estimation of future resources (viz., CPU, memory,
bandwidth etc.) utilization information of the VMs and predicted bandwidth
usage based clustering of incoming traffic into groups assist the RM in
deciding an efficient workload distribution while avoiding the future network
congestion. Concurrently, RM generates and maintains Inter-VM Communication
Log (IVCL) which consists of the information of user specified authorized
access links among the inter-dependent VMs based on the scheduling of their
respective tasks. Security Manager (SM) utilizes IVCL for investigation of
possibility of any security threat by distinguishing unauthorized links among
all the current inter-VM links on a server and terminating them beforehand.
Each server consists of a hypervisor and resource pool to deploy different
users’ VMs and process tasks, respectively. An exclusive Monitor layer is
employed to surveil inter-communication links among VMs on the respective
server.

##  III Secure Inter-VM Communication

Let the applications of users {$U_{1}$, $U_{2}$, … , $U_{A}$, …, $U_{m}$} are
executed on different VMs {${V}_{i}^{k}$: $i\in[1,q]$, $k\in[1,m]$} as
illustrated in Fig. 2. Each server deploys a Monitor that maintains a VM Link
Access Matrix (VLAM) for the surveillance of the current inter-communication
behaviour among the VMs hosted on it. Taking the assumptions: (i) All of the
VMs {$V^{A}_{1}$, $V^{A}_{2}$, …, $V^{A}_{22}$} $\in V^{Mal}$ belonging to
user $U_{A}$ are malicious VMs and inter-links among them are malicious links,
(ii) RM is unaware about malicious user or attacker ($U_{A}$), and (iii) All
the links established between a user VM and other known VM with access
permission assigned by the respective user have authorized inter-communication
links ($\mathds{AL}$) which are recorded in IVCL. SM investigates the
probability of threat by comparing the current information of VLAM with the
information available in IVCL, periodically. Any mismatch between the observed
inter-VM links and set of authorized inter-communication links is an indicator
of security threat.

Figure 2: Secure inter-VM communication

Let $V_{i,z}$ and $V_{i^{\ast},z^{\ast}}$ specify $i^{th}$ VM of $z^{th}$
benign user and ${i^{\ast}}^{th}$ VM of ${z^{\ast}}^{th}$ malicious user,
respectively and their mapping ($\omega$) on $k^{th}$ server is denoted as
$\omega^{z}_{ik}$ and $\omega^{z^{\ast}}_{{i^{\ast}}k}$, respectively. The
term $\omega_{ik}$=1, if $i^{th}$ VM is hosted on $k^{th}$ server, else,
$\omega_{ik}$=0. The unauthorized inter-VM access
$\Theta^{col}_{ik,z\rightarrow{i^{\ast}}k,z^{\ast}}$ from $V_{i,z}$ to
$V_{i^{\ast},z^{\ast}}$ co-located on $k^{th}$ server over time-interval
{$t_{1}$, $t_{2}$} is stated in Eq 1; where
$\uplus^{V}_{ik,z\rightarrow{i^{\ast}}k,z^{\ast}}$ specifies inter-VM relation
between $V_{i,z}$ and $V_{i^{\ast},z^{\ast}}$.

$\int_{t_{1}}^{t_{2}}\Theta^{col}_{ik,z\rightarrow{i^{\ast}}k,z^{\ast}}dt={\int_{t_{1}}^{t_{2}}(\omega_{ik}^{z}\times\omega_{{i^{\ast}}k}^{z^{\ast}}\times\uplus^{V}_{ik,z\rightarrow{i^{\ast}}k,z^{\ast}}})dt$
(1)

The sets of authorized links ($\mathds{AL}_{legal}^{V_{i,z}}$) and current
inter-VM links ($\mathds{L}^{V_{i,z}}$) specified for VM $V_{i,z}$ in access
log (IVCL) and VLAM are stated in Eqs. (2) and (3), respectively.

$\\{\mathds{AL}_{1}^{V_{i,z}},\mathds{AL}_{2}^{V_{i,z}},...,\mathds{AL}_{n}^{V_{i,z}}\\}\in\mathds{AL}_{legal}^{V_{i,z}}$
(2)
$\\{\mathds{L}_{1}^{V_{i,z}},\mathds{L}_{2}^{V_{i,z}},...,\mathds{L}_{n}^{V_{i,z}}\\}\in\mathds{L}^{V_{i,z}}$
(3)

An inter-VM link between $V_{i,z}$ and $V_{i^{\ast},z^{\ast}}$ placed on
$k^{th}$ server is denoted as:
$\mathds{L}_{ik,z\rightarrow{i^{\ast}}k,z^{\ast}}$; where
$\forall\\{i,i^{\ast}\\}\in[1,q]$, $i\neq i^{\ast}$, $k\in[1,p]$, $z\in[1,m]$
and the relationship ($\uplus^{V}_{ik,z\rightarrow{i^{\ast}}k,z^{\ast}}$)
between them is revealed via comparison of VLAM and IVCL, is evaluated in Eq.
(4). The value of $\uplus_{ik,z\rightarrow{i^{\ast}}k,z^{\ast}}$ is $1$ for
suspicious or unauthorized relation, otherwise, it is $0$.

$\uplus^{V}_{ik,z\rightarrow{i^{\ast}}k,z^{\ast}}=\begin{cases}$1$,&{If\quad{\mathds{L}_{ik,z\rightarrow{i^{\ast}}k,z^{\ast}}}}\nsubseteq\mathds{AL}_{legal}^{V_{i,z}}\\\
$0$,&{{\textit{otherwise}}}\end{cases}$ (4)

Eq. (5) evaluates network cascading attack
($\Theta^{cas}_{({i}k,z\rightarrow{i^{\ast\ast}}k^{\ast},z^{\ast})}$) between
VMs $V_{i,z}$ and $V_{i^{\ast\ast},z^{\ast}}$ deployed on $k^{th}$ and
${k^{\ast}}^{th}$ servers, respectively via $V_{i^{\ast},z^{\ast}}$ on
$k^{th}$ server, where $\forall\\{i,i^{\ast},i^{\ast\ast}\\}\in[1,q]$,
$k\in[1,p]$,
$\uplus^{V}_{({{i}k},{z}\rightarrow{i^{\ast\ast}}k^{\ast},z^{\ast})}$ is
defined using Eq. (6)

$\begin{aligned}
\int_{t_{1}}^{t_{2}}\Theta^{cas}_{({i}k,z\rightarrow{i^{\ast\ast}}k^{\ast},z^{\ast})}dt=\int_{t_{1}}^{t_{2}}{\omega_{ik}^{z}\times\omega_{{i^{\ast}}k}^{z^{\ast}}\times\omega_{{i^{\ast\ast}}k}^{z}}\\\
\times\uplus^{V}_{({{i}k},{z}\rightarrow{i^{\ast\ast}}k^{\ast},z^{\ast})}dt\end{aligned}$

(5)

$\displaystyle\uplus^{V}_{({{i}k},{z}\rightarrow{i^{\ast\ast}}k^{\ast},z^{\ast})}=\uplus^{V}_{{{i}k,z}\rightarrow{i^{\ast}}k,z^{\ast}}\times\uplus^{V}_{{i^{\ast}}k,z^{\ast}\rightarrow{i^{\ast\ast}}k^{\ast},z^{\ast}}$
(6)

Eq. (7) specifies server vulnerability based unauthorised access
($\Theta^{vul}_{{ik,z}\rightarrow S_{k}}$) from $V_{ik,z}$ to server $S_{k}$
during time $\\{t_{1},t_{2}\\}$, where $\uplus^{S}_{ik,z}$ is evaluated using
Eq. (8). The relation between $V_{i,z}$ and $k^{th}$ server ($S_{k}$) is
‘suspicious’ if available throughput (${TP}_{V_{ik,z}}^{avl}$) and bandwidth
(${BW}_{V_{ik,z}}^{avl}$) are lesser than guaranteed threshold
($\mathds{TH}_{v_{i,z}}^{gtd}$), otherwise, ‘non-suspicious’.

$\int_{t_{1}}^{t_{2}}\Theta^{vul}_{{ik,z}\rightarrow
S_{k}}dt={\int_{t_{1}}^{t_{2}}\omega_{ik,z}\times\uplus^{S}_{{ik,z}\rightarrow
S_{k}}}dt\quad\forall\\{i\\}\in q,k\in p$ (7) $\uplus^{S}_{{ik,z}\rightarrow
S_{k}}=\begin{cases}$1$,&{If({{{TP}}_{V_{ik,z}}^{avl}\wedge{BW}_{V_{ik,z}}^{avl})\times\delta_{t}\leq{\mathds{TH}_{V_{ik,z}}^{gtd}}}{}}\\\
$ 0$,&{{\textit{otherwise}}}\end{cases}$ (8)

The guaranteed threshold (i.e., $\mathds{TH}_{v_{i,z}}^{gtd}$) of performance
parameters: throughput ($TP$) and bandwidth ($BW$) of $i^{th}$ VM of $z^{th}$
user is defined as
$\mathds{TH}_{V_{ik,z}}^{gtd}=\\{\mathds{TH}^{{TP}}_{V_{ik,z}}\wedge\mathds{TH}^{{BW}}_{V_{ik,z}}\\}$.
The total security breaches information ($\Theta^{DC}_{z}$) of $z^{th}$ user
can be compiled by applying Eq. (9):

$\begin{aligned}
\int_{t_{1}}^{t_{2}}\Theta^{DC}_{z}dt=\int_{t_{1}}^{t_{2}}\Theta^{col}_{ik,z\rightarrow{i^{\ast}}k,z^{\ast}}dt+\int_{t_{1}}^{t_{2}}\Theta^{cas}_{({i^{\ast}}k,z^{\ast}\rightarrow{i^{\ast\ast}}k^{\ast},z)}dt+\int_{t_{1}}^{t_{2}}\Theta^{vul}_{ik,z}dt\end{aligned}$

(9)

The attack coverage ($\mathds{A}_{coverage}^{z^{\ast}}$) by ${z^{\ast}}^{th}$
malicious user is defined as the ratio of the number of malicious links
($\mathds{L}^{Mal}_{i^{\ast},z^{\ast}}$) between malicious VM
($V^{Mal}_{{z^{\ast}}}$) and benign VMs ($V_{z}$) i.e.,
$|\sum_{k=0}^{p}\sum_{i^{\ast}=0}^{m^{\ast}}\mathds{L}^{Mal}_{i^{\ast}k,z^{\ast}}|$
and total number of malicious VMs ($Tot\\_V^{z^{\ast}}$) during time interval
{$t_{1}$, $t_{2}$}. $\mathds{A}_{coverage}^{z^{\ast}}$ is computed using Eq.
(10) and $\mathds{L}^{Mal}_{i^{\ast},z^{\ast}}$ are determined using Eq. (11).

$\int_{t_{1}}^{t_{2}}\mathds{A}_{coverage}^{z^{\ast}}dt=\int_{t_{1}}^{t_{2}}\frac{|\sum_{k=0}^{p}\sum_{i^{\ast}=0}^{m^{\ast}}\mathds{L}^{Mal}_{i^{\ast}k,z^{\ast}}|}{Tot\\_V^{z^{\ast}}}dt$
(10)

$\begin{aligned}
\int_{t_{1}}^{t_{2}}\mathds{L}^{Mal}_{i^{\ast},z^{\ast}}dt=\int_{t_{1}}^{t_{2}}\mathds{L}^{V_{i^{\ast},z^{\ast}}}-(\mathds{L}^{V_{i^{\ast},z^{\ast}}}\cap\mathds{AL}^{V_{i^{\ast},z^{\ast}}}_{legal})dt\end{aligned}$

(11)

Monitor utilizes VLAM, IVCL, and Eqs. (4)-(9) to identify malicious VM
($V^{Mal}$) and transmits the knowledge of $V^{Mal}$ among all the
neighbouring servers to terminate all the malicious links before the
propagation of the network security threat at extreme level. Further, all the
malicious VMs are penalized by suspending them while notifying $U_{A}$.

##  IV Congestion Avoidance and VM Allocation

A machine learning based predictor is deployed to analyse resource
requirements of VMs for execution of future applications and estimate the
bandwidth hog in real time. The future applications are grouped into $N$
clusters depending on their predicted bandwidth usage which guides the
allocation of VMs to the selected network-efficient servers while mitigating
the congestion proactively. Let $\mathds{T}_{\sigma}$ be the deviation of
network traffic from the estimated traffic ($\mathds{T}$) over duration
$\delta_{t}\in$ {$t_{1}$, $t_{2}$}, $\mathds{T}_{\sigma}^{thr}$ and
$\delta_{t}^{thr}$ are traffic deviation and time-period thresholds
respectively. Eq. (12) investigates real time security threat in data centre
($\Theta^{con}_{dc}$) due to network congestion.

$\int_{t_{1}}^{t_{2}}\Theta^{con}dt=\begin{cases}1,&{If(\mathds{Tr}_{\sigma}\times\delta_{t}>\mathds{T}_{\sigma}^{thr}\times\delta_{t}^{thr})}\\\
-1,&{If(\mathds{Tr}_{\sigma}\times\delta_{t}<0)}\\\
0,&{\text{otherwise}}\end{cases}$ (12)

If congestion is anticipated  (i.e., $\Theta^{con}_{dc}$=$1$), the network
traffic is diverted across multiple paths and assigned to servers reserved for
handling network hogs. Similarly, if underload ($\Theta^{con}_{dc}$=$-1$) is
detected, the load is shifted from underloaded to network-efficient servers;
otherwise, the traffic is normal. A neural network based predictor is
periodically trained with historical and live resource utilization of VMs
executing different applications which allows extraction of useful patterns
and learning of correlations and helps to predict resources (viz. CPU, memory
and bandwidth) utilization in real time. The predicted applications are
grouped into $N$ clusters to filter bandwidth hogs (which may generate
congestion based security attacks) and schedule them on selected network-
efficient servers. K-means clustering is applied to keep the clusters as
different as possible while grouping predicted applications of similar
bandwidth usage so that the sum of the squared distance ($\mathds{G}$) between
their bandwidth requirement and centroid of the cluster is minimum by applying
Eq. (13), where ${w}_{ic}$ defines mapping between bandwidth of $i^{th}$
application ($BW_{i}$) and centroid ($\mu_{c}$) of $c^{th}$ cluster.

$\mathds{G}=\sum_{i=1}^{m}\sum_{c=1}^{N}{{w}_{ic}{|{BW}_{i}-\mu_{c}|}^{2}}$
(13)

The deployment of $i^{th}$ VM ($V_{i}$) on $j^{th}$ server ($S_{j}$) must
satisfy the resource (CPU ($C$), Memory ($M$), Bandwidth (${BW}$)) capacity
constraints stated in Eq. (14), where $V_{i}^{\mathds{R}}$ specifies resource
requirement of $V_{i}$ and $S_{k}^{\mathds{R}}$ is available resource capacity
of server $S_{k}$.

$\sum_{i=1}^{q}{V_{i}^{\mathds{R}}\times\omega_{ik}\leq
S_{k}^{\mathds{R}}};\quad\forall_{k}\in\\{1,p\\},{\mathds{R}}\in\\{C,M,BW\\}$
(14)

The resource utilization ($RU$) of data centre is evaluated by using Eqs. (15)
and (16), where $\mathds{N}$ is the number of monitored resources, $RU^{C}$,
$RU^{M}$ and $RU^{BW}$ are CPU, memory and bandwidth utilization of server. If
$k^{th}$ server $S_{k}$ is active i.e., it hosts VM ($\eta_{k}$ = 1),
otherwise, it is inactive ($\eta_{k}$ = 0).

$RU^{DC}=\frac{\sum_{k=1}^{p}{RU_{k}^{C}}+\sum_{k=1}^{p}{RU_{k}^{M}}+\sum_{k=1}^{p}{RU_{k}^{BW}}}{|\mathds{N}|\times\sum_{k=1}^{p}{\eta_{k}}}$
(15)

$RU_{k}^{\mathds{R}}=\frac{\sum_{i=1}^{q}{\omega_{ik}}\times
V_{i}^{\mathds{R}}}{S_{k}^{\mathds{R}}}\quad\forall_{k}\in\\{1,p\\},\mathds{R}\in\\{C,M,BW\\}$
(16)

The total power consumption ($PW^{DC}$) during time-interval {$t_{1}$,
$t_{2}$} is computed by applying Eq. (17), where ${PW_{i}}^{max}$,
${PW_{i}}^{min}$, and ${PW_{i}}^{idle}$ are maximum, minimum, and idle state
power consumption, respectively of $i^{th}$ server.

$PW^{DC}=\sum_{i=1}^{P}{[{PW_{i}}^{max}-{PW_{i}}^{min}]\times{RU}+{PW_{i}}^{idle}}$
(17)

##  V  Operational Design and Illustration

OSC-MC utilizes historical and current bandwidth usage of different
applications to anticipate the demand of bandwidth for the VM executing the
application in real-time, and accordingly, balances the load on different
servers to mitigate any probability of network congestion based security
attacks. A module named Monitor keeps track of inter-VM links among VMs on a
server in a matrix and analyses them to determine probability of co-residency
and network cascading attacks.

###  V-A Complexity Computation

Algorithm 1 presents the detailed operational summary of OSC-MC, where the
complexity of steps 1 and 2 are $O(1)$ and $O(mpq)$, respectively. VMs are
allocated using First-Fit Decreasing algorithm based on the decreasing order
of predicted bandwidth so as to reduce bandwidth wastage. The steps 3-15
repeat for $t$ intervals. Steps 4-6 execute $q$ times to predict resources
requirement of $q$ VMs. Step 7 calls K-Means clustering with time complexity
of $O(m^{2})$. Again, the step 8 maps $q$ VMs to $p$ servers consumes
complexity of $O(pq)$. Steps 9-14 execute for $q^{2}$ times. Hence, the total
complexity comes out to be $O(m^{2}q^{2}pt)$.

1 Initialize: $List_{\mathds{U}}$, $List_{\mathds{V}}$, $List_{\mathds{S}}$;

2 Allocate $V_{1}$, $V_{2}$, …, $V_{q}$ to $S_{1}$, $S_{2}$, …, $S_{p}$ by
defining a mapping $\mathds{U}\times\mathds{V}\mapsto\mathds{S}$ and generate
IVCL ;

3 for  _each time-interval $\\{t_{1},t_{2}\\}$_ do

4 for  _each VM $V_{i}$_ do

5 [${V_{t_{2}}}^{Pred(C_{i},M_{i},BW_{i})}$] $\Leftarrow$ Resource
prediction(${V}_{t_{1}}$) $\forall i\in[1,q]$ ;

6

7 end for

8 Clustering ($C_{1}$, $C_{2}$, …, $C_{N}$) $\leftarrow$
K-Means($\mathds{V}^{BW}_{t_{2}}$);

9 Schedule $V_{1}$, $V_{2}$, …, $V_{q}$ on servers and generate VLAM for each
server;

10 if  _$\mathds{AL}\nsubseteq\mathds{AL}_{legal}$_ then

11 Identify and terminate malicious VM ($V^{Mal}$) by comparing IVCL and VLAM;

12

13 else

14 Keep ${V}_{i}$ at same server until user terminates it;

15 end if

16

17 end for

Algorithm 1 OSC-MC: Operational Summary

###  V-B Illustration

Consider four sets of inter-dependent VMs such that {$V_{1}$, $V_{2}$,
$V_{3}$, $V_{4}$}, {$V_{5}$, $V_{6}$, $V_{7}$ }, {$V_{8}$, $V_{9}$, $V_{10}$,
$V_{11}$}, {$V_{12}$, $V_{13}$, $V_{14}$, $V_{15}$} belong to $U_{1}$,
$U_{2}$, $U_{3}$, and $U_{4}$, respectively. Taking the assumption that
{$V_{1}$, $V_{3}$, $V_{11}$} are hosted on server $S_{1}$, {$V_{2}$, $V_{7}$,
$V_{12}$} are deployed on $S_{2}$, {$V_{5}$, $V_{6}$, $V_{8}$} on $S_{3}$,
{$V_{4}$, $V_{9}$, $V_{14}$} on $S_{4}$, and {$V_{13}$, $V_{10}$, $V_{15}$} on
$S_{5}$. Let a set of VMs {$V_{8}$, $V_{9}$, $V_{10}$, $V_{11}$} of user
$U_{3}$ be malicious. The inter-dependent VMs on same or different servers
(viz. $V_{1}$ and $V_{3}$ hosted on $S_{1}$, $V_{5}$ and $V_{6}$ hosted on
$S_{3}$ etc.) exchange data among them which is sensed by the malicious VMs
$V_{8}$, $V_{9}$, $V_{10}$, $V_{11}$ to access sensitive data by establishing
unauthorised links, side-channels, exploiting hypervisor vulnerability or
capturing traces of valuable data from cache of server hosting them. Based on
given allocation, colocation attacks occur on $S_{1}$, $S_{3}$ and $S_{5}$
(i.e., $\Theta^{col}=60\%$) and cascading attack can propagate via multiple
connections (i.e., $\Theta^{cas}=100\%$). To eliminate such attacks, OSC-MC
employs an exclusive Monitor on each server to surveil all the current inter-
VM links established on a server in VLAM, compare them with the authorised
link information in IVCL and terminate all malicious links and suspend
malicious VMs $V_{8}$, $V_{9}$, $V_{10}$, $V_{11}$.

##  VI Performance Evaluation and Discussion

The simulation experiments are executed on a server machine assembled with two
Intel® Xeon® Silver 4114 CPU with 40 core processor and 2.20GHz clock speed,
deployed with 64-bit Ubuntu 16.04 LTS having main memory of 128 GB in Python
3.1. The data centre environment is set up with quad and dual core processor
types of server having CPU: 2000 MIPS, RAM: 2048 MB, BW: 10,000. Two types of
VMs having (500, 512, 1000) and (1000, 1024, 1000) as CPU, RAM and BW
utilization respectively are used. Three-layered feed-forward neural network
is used for resource prediction and the hypervisor vulnerability score
associated to each server is generated in the range [0, 10] randomly where a
value $\geq 7$ indicates high probability of attack. We experimented with
benchmark Bitbrains dataset containing resources (CPU, Memory and Bandwidth)
usage of more than 1400 VMs with a periodic interval of 5 minutes, the number
of users equals to one-third of total number of VMs, and inter-VM links are
randomly generated during run-time.

OSC-MC is compared to Security Embedded Dynamic Resource Allocation (SEDRA)
[8], Previously Selected Server First (PSSF) [4] and Secure and Energy-Aware
Load Balancing (SEA-LB) [9]. PSSF maintains list of cloud users and allocates
their requested new VMs on the same server which have previously hosted their
VMs. SEDRA considers performance of network traffic and inter-VM links to
detect and mitigate VM threats by utilizing a random tree classifier. SEA-LB
assigns VMs to energy-efficient servers subject to maximum resource
utilization, minimum power consumption and side-channel attacks by using a
modified multi-objective genetic algorithm approach. The number of shared
servers among users are minimized to provide security.

###  VI-A Results

Table I shows achieved results of different performance metrics for OSC-MC
with various size of data centre and varying number of benign and malicious
users. The average number of malicious links ($\mathds{L}^{Mal}$),
unauthorized inter-VM access ($\Theta$), and VMs (${V}^{Mal}$) increases
unevenly with the number of malicious users and growing size of the data
centre. Also, the average number of $\mathds{NW}^{Hogs}$ varies with dynamic
network traffic and size of the data centre. The mean of $RU$ (%) ranges from
61% to 64% while $PW$ ($KW$) scales up with rising size of the data centre.

TABLE I: Performance metrics for OSC-MC

VMs |  $U$# ($U_{A}$%) | $\mathds{L}^{Mal}$ | $V^{Mal}$ |  $\Theta$ (#) | $\mathds{NW}^{Hogs}$ |  $RU$ (%) |  $PW$($KW$)
---|---|---|---|---|---|---|---
200 | 66 (5) | 43 | 6 | 43 | 13 | 63.45 | 9.58
66 (20) | 58 | 17 | 58 | 12 | 62.06 | 9.96
66 (90) | 306 | 131 | 306 | 16 | 62.48 | 9.29
500 | 166 (5) | 12 | 9 | 12 | 22 | 61.73 | 22.73
166 (20) | 109 | 64 | 109 | 23 | 60.42 | 22.58
166 (90) | 1053 | 319 | 1053 | 21 | 61.48 | 23.02
800 | 266 (5) | 77 | 34 | 77 | 42 | 62.23 | 44.13
266 (20) | 456 | 116 | 456 | 39 | 62.41 | 43.47
266 (90) | 1297 | 518 | 1297 | 41 | 62.07 | 43.18
1100 | 366 (5) | 99 | 43 | 99 | 46 | 63.20 | 78.34
366 (20) | 387 | 161 | 387 | 41 | 64.05 | 78.47
366 (90) | 1918 | 706 | 1918 | 47 | 63.88 | 78.18

###  VI-B Comparison

The number of network hogs obtained with OSC-MC, SEDRA and without-OSC-MC
(W-OSC) for data centre of sizes 200 VMs ($\xi_{200}$), 500 VMs ($\xi_{500}$),
800 VMs ($\xi_{800}$), and 1100 VMs ($\xi_{1100}$) are shown in Fig. 3. The
number of hogs in case of OSC-MC, is below 70 for every size of data centre
which scales down with the time period because of periodic learning of neural-
network predictor in real-time. OSC-MC reduces congestion up to 66.46% and
89.94% over SEDRA and W-OSC, respectively for $\xi_{1100}$. Fig. 4 compares
inter-VM communication links over period {$t_{1}$, $t_{2}$} observed for OSC-
MC with that of SEDRA, SEA-LB, and PSSF for varying number of $U_{A}$. The
authorised link percentage $\mathds{AL}$ (%) has reached up to $100\%$ in case
of OSC-MC while it is observed to be 77%-97%, 57%-97%, and 57%-96% for SEDRA,
SEA-LB, and PSSF, respectively.

Figure 3: Network hogs Figure 4: Authorised Inter-VM Links over time

Fig. 5 compares $\mathds{AL}$ ($\%$) with respect to various sizes of data
centre where, OSC-MC detects and mitigates almost every $\mathds{L}^{Mal}$ and
$V^{Mal}$ before the completion of enough time required for occurrence of
actual security threat. OSC-MC scales down active $V^{Mal}$ and
$\mathds{L}^{Mal}$ among VMs by monitoring mutual inter-relations among them
and raises the average legal access percentage by 32.1%, 34.5% and 21.5%
against PSSF, SEA-LB, and SEDRA, respectively for 500 VMs.

Figure 5: Average authorised access percent for different size of data centre

Table II compares power consumption ($PW$) and resource utilization ($RU$) by
OSC-MC with other three schemes; where $PW$ is reduced up to 39.31%, 31.95%
and 17.65% against PSSF [4], SEA-LB [9], and SEDRA [8], respectively for 1100
VMs. $RU$ is evaluated using Eqs. (15) and (16) which increases as: PSSF $<$
SEDRA $<$ SEA-LB $<$OSC-MC. $RU$ is independent of the size of data centre and
varies in the constant ranges of 55%-56%, 59%-60%, and 61.5%-62.5% for PSSF,
SEDRA, and OSC-MC, respectively. The reason behind above shown results is the
online prediction of required resources beforehand that assisted in
consolidation of VMs on least number of active servers.

TABLE II: Power consumption and Resource utilization

VMs | Metrics | SEA-LB | PSSF | SEDRA | OSC-MC
---|---|---|---|---|---
200 |  $PW$ ($KW$) | 30.92 | 32.13 | 14.23 | 9.17
$RU$ (%) | 60.12 | 58.12 | 59.12 | 62.06
500 |  $PW$ ($KW$) | 65.66 | 74.29 | 51.24 | 22.02
$RU$ (%) | 60.92 | 58.62 | 59.92 | 61.73
800 |  $PW$ ($KW$) | 84.36 | 93.39 | 69.09 | 43.35
$RU$ (%) | 60.81 | 58.94 | 59.94 | 62.06
1100 |  $PW$ ($KW$) | 129.67 | 143.58 | 107.28 | 88.35
$RU$ (%) | 60.36 | 58.31 | 59.46 | 61.28

##  VII Conclusion

A novel online security ingrained cloud communications model: OSC-MC is
proposed that monitors inter-VM relations and detects malicious VMs to
mitigate security breaches. OSC-MC incorporates machine learning based
congestion prediction and maintains IVCL and VLAM to determine and terminate
suspicious inter-VM links. It avoids security breaches due to known and
unknown VMs and substantially decreases malicious congestion, network
cascading threats, and co-residency threats. The achieved results support
influential performance of OSC-MC against the compared approaches.

## Acknowledgment

This work is financially supported by National Institute of Technology,
Kurukshetra, India.

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