[1]Solmar Varela

[4]Vladimiro Mujica

[1]Institute for Materials Science and Max Bergmann Center of Biomaterials, TU
Dresden, 01062 Dresden, Germany

2]Dresden Center for Computational Materials Science (DCMS), TU Dresden, 01062
Dresden, Germany

3]Departamento de Física, Colegio de Ciencias e Ingeniería, Universidad San
Francisco de Quito, 170901 Quito, Ecuador

[4]School of Molecular Sciences, Arizona State University, Tempe, AZ 85287 USA

# Electron Spin Polarization as a Predictor of Chiroptical Activity in Helical
Molecules

*<EMAIL_ADDRESS>Rafael Gutierrez  Gianaurelio Cuniberti  Ernesto Medina<EMAIL_ADDRESS>*  [  [  *

###### Abstract

Chiral structures, breaking spatial inversion symmetry, exhibit non-zero
chiroptical activity (COA) due to the interaction between their electric and
magnetic responses under external electromagnetic fields, an effect that is
otherwise absent in achiral systems. Non-magnetic chiral structures also
exhibit Chiral-Induced Spin Selectivity (CISS), where spin-polarization (SP)
emerges without external magnetic influence. We have obtained a COA-SP
connection for a model system of an electron constrained to a helix including
spin-orbit coupling (SOC), and in the presence of an external electromagnetic
field. Despite its simplicity, this model captures the relevant physics
required to address the problem. In particular, our results reveal that the
norm of the SP vector can be used as a predictor of COA. In addition to SOC
and the breaking of space inversion, a non-vanishing SP requires the breaking
of time-reversal symmetry (TRS), as demanded by Onsager’s reciprocity. Beyond
the relationship between SP and COA, we obtain the novel result that TRS
breaking is also necessary to yield a non-vanishing contribution of the SOC to
the COA.

###### keywords:

CISS, chiroptical activity, spin-orbit coupling, perturbation theory

Spin polarization (SP) in molecules and solids is usually understood as a
response to an external magnetic field. The discovery that for chiral
materials, molecules, solids, and interfaces, this magnetic response can be
obtained even in the absence of external magnetic fields in processes
involving electron transfer, electron transport, and bond polarization through
chiral centers, has had profound consequences in fundamental physics,
chemistry, and biology, as well as in important applications in spintronics,
NMR, and Quantum Information Sciences (QIS)[1, 2, 3, 4, 7, 8, 11, 9, 5, 6,
10]. The theoretical description of this phenomenon, known as the Chiral-
Induced Spin Selectivity (CISS) effect, requires first, the inclusion of spin-
orbit coupling (SOC), and second, in addition to space-inversion symmetry
breaking associated with chirality, the breaking of time-reversal symmetry
(TRS) [14, 15, 12, 13]. These basic symmetry requirements do not exclude the
fact, recently established in a number of theoretical studies, that a true
comprehension of the physics of the CISS effect demands the inclusion of
electron-phonon, spin-phonon, and electron-electron interactions in addition
to non-adiabatic effects[18, 17, 19, 16, 20]. The CISS effect has been
extensively investigated in experiments involving photoemission[21, 23, 22,
24], electron transport[25, 26, 27, 28], electron transfer[29, 30], and more
recently, in electrochemistry and photoluminescense [34, 32, 35, 31, 33, 36,
37].

The interpretation of the CISS effect as a magnetic response linked to
electron SP in chiral systems strongly suggests that there must exist a
connection with the optical activity response observed in these systems when
exposed to circularly polarized light [31, 38, 39, 40, 41, 42].This connection
arises from the interplay between the electric and magnetic induced dipole
moments in chiral systems. For optically active molecules, the optical
response is normally measured either by the rotational power, which expresses
the angle of rotation of the polarization plane of the light, or through the
Chiroptical Activity (COA), related to the difference in the intensity of
absorption of polarized photons. On the other hand, SP is defined as a three-
dimensional vector containing the expectation value of the three Pauli spin
matrices, which in the CISS mechanism is associated with electron transport,
electron transfer, or bond polarization in chiral systems. Both SP and COA can
be interpreted as responses of a chiral system to perturbations, but the
underlying physical mechanisms for these two phenomena are fundamentally
different because COA requires photon absorption, whereas CISS-SP does not
necessarily involve photoexcitation. The latter might explain why the explicit
connection between them has so far remained elusive [43, 44, 45].

We have used a model Hamiltonian of an electron confined to a helical box in
the presence of SOC induced by the confining field of the inversion asymmetric
helix [46]. In the absence of SOC, the model was initially solved exactly by
Tinoco et al. [47], and we extend it further by including an external
electromagnetic field. Using a perturbative approach to solve the Schrödinger
equation, we approximately obtain the two components of the $1/2-$spinors for
this problem. We then proceed to calculate both the COA and the three
components of the CISS-induced magnetic response, where the $z$-component
corresponds to the average value of $\sigma_{z}$ Pauli matrix, assuming
electron motion along the $z$-axis. Despite the apparent simplicity of our
model, it is the first explicit analytical solution to a long-standing problem
in the field of CISS-induced molecular SP and how it can be connected to COA.
Our model is also related to the description of the phenomenon of field-
mediated chirality transfer ref.[41]. The conclusion is that the interaction
with circularly polarized radiation carries information both about SP and COA,
an important result that we are only beginning to understand and that might
have important consequences in the description of SP using chiral photons in
molecules with emerging topological features in the electronic structure as,
for example, spin textures[48].

Our model also affords an important digression into the inclusion of
controlled schemes to take into account TRS breaking. This is a fundamental
aspect of any formalism used to describe SP because it cannot occur in a
system where TRS is preserved, as implied by Onsager relations and the onset
of Kramers’ degeneracy [49, 50]. We have used a simple realization of a
Büttiker’s probe to show that TRS, and hence spin polarization, is extremely
sensitive to decoherence, a fact that permits folding within a single concept
the qualitative effect of electron correlation, electron-phonon interaction,
spin textures, and non-adiabatic effects, that have been invoked to explain
the anomalous high value of the spin-orbit coupling that is apparently
required to reproduce the experimental values of spin polarization[16, 51,
17]. In fact, depending on how TRS is broken, either through a decoherence
probe or by changing the boundary conditions of the helix, and hence the
relative spin populations, we can use this result to reproduce important
asymmetries in the enantio-specific photon absorption that have been observed
in complex chiral solids and interfaces[40, 39]. The inclusion of TRS breaking
is also essential in understanding the dependence of both SP and COA on the
geometrical factors of the helix, e.g. its length, which has been analyzed
experimentally [52, 44, 53, 45, 54].

##  1 Results and discussion

For optically active molecules, the chiroptical properties are usually
quantified in terms of the anisotropic dissymmetry factor $g_{CD}$, which
characterizes the COA. It can be written in terms of the extinction
coefficients $\varepsilon_{-}(\varepsilon_{+})$ for left-(right-) polarized
incident light, corresponding to the optical transition between the $n$-th and
$m$-th electronic states, as:

$g_{CD}=2\frac{(\varepsilon_{-}-\varepsilon_{+})}{(\varepsilon_{-}+\varepsilon_{+})}=\frac{4R_{nm}}{|\boldsymbol{\mu}_{nm}|^{2}+|\mathbf{m}_{nm}|^{2}}.$
(1)

Here, $R_{nm}$ is the rotatory strength, and $\boldsymbol{\mu}_{nm}$ and
$\mathbf{m}_{nm}$ are the transition electric dipole moment, and the
transition magnetic dipole moment, respectively. The rotatory strength
$R_{nm}$, also known as Rosenfeld’s tensor, is defined as the imaginary part
of the scalar product of the two transition moments:

$\displaystyle R_{nm}$ $\displaystyle=$
$\displaystyle\imaginary(\boldsymbol{\mu}_{nm}\cdot\boldsymbol{m}_{mn}).$ (2)

This definition is the usual one in the dipole approximation, which excludes
quadrupole and higher-order contributions[55]. For an isotropic system,
$R_{nm}$ can be written as
$R_{nm}=(1/3)\left(R_{11}+R_{22}+R_{33}\right)_{nm}$, with $(R_{ii})_{nm}$
being the tensor components relative to the direction of propagation of light,
e.g., $(R_{33})_{nm}$ refers to light incident along the $z$-direction [47].

Due to the fact that organic molecules are primarily composed of low atomic
number elements, atomic spin-orbit coupling (ASOC) is considered to be very
weak, hence the usual definition of the rotatory strength in eq.(2) omits the
contribution of the electronic spin and any influence of ASOC on the circular
dichroism (CD) spectrum. Nonetheless, the discovery of the CISS effect
presents an opportunity to investigate the role of an effective SOC that can
be substantially different from the bare ASOC involved in spectroscopy. The
inclusion of spin effects requires a redefinition of the $R_{nm}$ to consider
the total magnetic moment $\mathbf{J}$, adding the spin contribution
$\mathbf{m}_{s}$ to the orbital one, i.e.,
$\mathbf{J}=\mathbf{m}+\mathbf{m}_{s}$.

Figure 1: Helical system, energy spectrum, and the Bloch space. a, Structure
of the helix in the molecular coordinate system $(\hat{x},\hat{y},\hat{z})$ in
the presence of an external time-dependent electromagnetic field with
frequency $\omega$. The structural parameters are given by the radius $a$, the
pitch $2\pi b$, and the $\varphi$ angle such that $0\leq\varphi\leq 2\pi K$,
with $K$ the number of turns. An effective SO magnetic field $\mathbf{B}_{SO}$
is considered perpendicular to the helix axis. b, Energies of electrons in
units of $\hbar\omega_{0}$ on an counter-clockwise helix ($\zeta=+1$) as a
function of the SOC strength, with $K=2$. Each energy represents the two
Kramers’ pairs, i.e. $E_{\tilde{n},+}^{-,+}=E_{\tilde{n},-}^{+,+}$, and
$E_{\tilde{n}+1,+}^{+,+}=E_{\tilde{n}+1,-}^{-,+}$. For convenience, we have
defined frequencies $\omega_{SO}=2\alpha a/\hbar(a^{2}+(b/2\pi)^{2})$ and
$\omega_{0}=\hbar/m(a^{2}+(b/2\pi)^{2})$[46]. c-f Schematic representation of
the vector $\Psi$ and the corresponding states $\psi_{s}^{\nu}$ for the
enantiomer $\zeta=+1$ on Bloch sphere, showing the role of the SOC, and the
dephasing (changes in $\delta$ phase) and voltage (changes in $z$ coefficient)
probes.

We then consider the Hamiltonian for an electron with charge $e$ and rest mass
$m_{e}$ as $\mathcal{H}=H^{0}+H^{\prime}$, where $H^{0}$ and $H^{\prime}$
refer to the non-perturbed system and the perturbation, respectively. The non-
perturbed Hamiltonian includes the kinetic energy and the SOC via a Rashba-
like term,
$H_{SO}=\boldsymbol{\sigma}\cdot(\mathbf{p}\times\boldsymbol{\alpha})$, where
$\sigma_{i}$ are Pauli’s matrices, $\mathbf{p}$ is the linear momentum
operator, and $\boldsymbol{\alpha}$ is a parameter of the model that includes
the electric field and a coupling constant that controls the magnitude of the
SOC. The SOC in a tight-binding model of a real molecule is connected to the
ASOC ($\backsim$6 meV for carbon atoms) and can be enhanced by the orbital
overlap between neighboring $\pi$-orbitals, resulting in an effective
intrinsic SOC of the order of meV (like in carbon nanotubes [56]), which
depends explicitly on the chirality and geometry of the molecule [57, 58, 59,
60]. An effective SOC in a helical model can also be related to the confining
molecular electrostatic field $\textbf{E}_{helix}$, which generates a SOC
whose magnitude is given by
$\boldsymbol{\alpha}=(e\hbar/4m_{e}^{2}c^{2})\mathbf{E}_{helix}$, and where
the electric field has helical symmetry[46, 61]. In both scenarios, the
electron is subject to an effective momentum-dependent, spin-orbit magnetic
field $\mathbf{B_{SO}}$, which interacts with its spin. We assume here that
$\mathbf{B_{SO}}$ is oriented perpendicular to the longitudinal axis of the
helix (see Fig.1a). The perturbation $H^{\prime}$ includes an external
electromagnetic radiation of frequency $\omega$.

An important ingredient of our model is that the stationary wavefunctions
$\Psi_{\tilde{n}}^{\zeta}$, necessary to describe the system to first order in
the perturbation[55], can be constructed as linear combinations of eigenstates
of $H_{0}$, $\psi_{\tilde{n},s}^{\nu,\zeta}$. These eigenfunctions
$\psi_{\tilde{n},s}^{\nu,\zeta}$ have been derived in a previous work [46] and
the different quantum numbers correspond to the direction of electron
propagation $\nu=+1(-1)$, the label for spin components $s=\pm 1$, and the
helicity of the state $s\nu$, i.e., the projection of the spin angular
momentum in the direction of propagation. The label $\zeta=\pm 1$ corresponds
to the two enantiomers, and $n$ labels the energy channels
$\tilde{n}=(n-K)/2K$, $n=1,2,3...$. Explicitly, the wavefunctions for the two
enantiomers are given by:

$\Psi_{\tilde{n}}^{+}=ze^{i\delta}(\psi_{\tilde{n}+1,+}^{+,+}+\psi_{\tilde{n},-}^{+,+})-qe^{i\eta}(\psi_{\tilde{n}+1,-}^{-,+}+\psi_{\tilde{n},+}^{-,+}),$
(3)
$\Psi_{\tilde{n}}^{-}=ze^{i\delta}(\psi_{\tilde{n}+1,+}^{-,-}+\psi_{\tilde{n},-}^{-,-})-qe^{i\eta}(\psi_{\tilde{n}+1,-}^{+,-}+\psi_{\tilde{n},+}^{+,-}),$
(4)

where the coefficients $ze^{i\delta}$ and $qe^{i\eta}$ have been added to
explore the influence of decoherence on the model by changing the phases and
amplitudes of the wavefunctions. These wavefunctions satisfy the same boundary
conditions used in Tinoco’s landmark article [47], i.e., for a helix of length
$2\pi K$ we have
$\Psi^{\zeta}_{\tilde{n}}|_{\varphi=0}=\Psi^{\zeta}_{\tilde{n}}|_{\varphi=2\pi
K}=0$. The eigenfunctions also obey Kramers’ degeneracy, which implies that
states with equal helicity are degenerate. The inclusion of SOC opens a gap
$\Delta$ separating states with $s\nu=+1$ from those with $s\nu=-1$, but
Kramers’ degeneracy is preserved unless TRS is broken (see Fig.1(b) for
eigenvalues energy spectrum).

###  1.1 Bloch sphere and Expectation Values

In general, the superposition of states can be represented as Bloch vectors
residing on the Bloch sphere, where the state $\Psi_{\tilde{n}}^{\zeta}$ can
be written as
$\ket{\Psi}=\cos{(\theta^{\prime}/2)}\ket{0}+e^{i\varphi^{\prime}}\sin{(\theta^{\prime}/2)}\ket{1}$.
A similar representation can be made for the states $\psi_{s}^{\nu}$, where,
for simplicity, we omit the labels $\tilde{n}$ and $\zeta$. The influence of
SOC and TRS on the states on the Bloch sphere for $\tilde{n}=1$ and $\zeta=+1$
is illustrated in Figure 1c-f. In the simplest case, when SOC is zero and TRS
is preserved (Fig. 1c), the angle of inclination $\theta$ of the spinors
$\psi_{s}^{\nu}$ with respect to the vertical $z$-axis is equal to zero. Each
of these states represents a pure state in the direction of $\ket{0}$,
resulting in $\Psi$ being in the direction of $(\ket{0}+\ket{1})/\sqrt{2}$.
The degree of polarization can be defined as
$P=\sqrt{P_{x}^{2}+P_{y}^{2}+P_{z}^{2}}$, where $P_{\imath}$ represent the
expectation values of $\ket{\Psi}$[62]. Hence, in this case, $P=1$, since
$P_{x}=1$, and $P_{y}=P_{z}=0$. The effect of SOC is depicted in Figure 1d.
The SO-induced magnetic field $\mathbf{B}_{SO}$ causes all the states
$\psi_{\tilde{n},s}^{\nu,\zeta}$ to rotate by an angle $\theta$, and the
spinor $\Psi_{\tilde{n}}^{\zeta}$ to tilt by an angle $\theta^{\prime}$ with
respect to the molecular axis, allowing us to interpret the impact of SOC as a
relaxation of the state vector along the direction of the SO field. This
rotation results in a decrease in $P_{x}$ and an increase in $P_{z}$, causing
$P$ to be less than 1.

The inclination angle $\theta^{\prime}$ of the spinor $\Psi$ can be modulated
by decoherence in the presence of SOC, as shown in Figures 1e-f. While a
change in the $\delta$ phase (or $\eta$) does not affect the orientation of
the states $\psi$, a change in the $z$ (or $q$) coefficient breaks the
degeneracy of states with the same helicity (Kramers’ pairs), resulting in an
additional rotation of the states that adds to the $\theta$ angle, and
therefore is also reflected in $\theta^{\prime}$. In other words, breaking TRS
by a change of the amplitudes in the wavefunction acts as an additional
contribution to the magnitude of the SO interaction, causing the angle
$\theta$ of certain states to increase (or decrease) due to this effect.

###  1.2 Time Reversal Symmetry, Voltage Probes and Decoherence Effects

TRS plays an important role in this work because Onsager’s reciprocity
relation precludes the possibility of having non-zero SP in the linear regime
unless TRS is explicitly broken. The fact that in the original experiments on
electron SP in gas phase molecules, the measured polarization was very low[62]
can be understood as a natural consequence of the fact that in gas phase TRS
is difficult to break unless multi-photon effects are included [63]. The
coefficients $ze^{i\delta}$ and $qe^{i\eta}$ in Eqs.(3,4) have been introduced
to simulate spin-insensitive Büttiker’s probes, allowing for the exploration
of decoherence effects on the model, which introduces a disruption of symmetry
in both $\nu$ directions and breaks TRS[12]. These Büttiker’s probes can serve
as dephasing probes, which preserve unitarity, where only complex phases are
changed[64], or as voltage probes, where coefficients are modified leading to
non-unitary evolution[65, 67, 66]. It can be seen that when
$ze^{i\delta}=qe^{i\eta}=1$, TRS is preserved.

In what follows, we will consider the combined influence of SOC and the
breaking of TRS in our model on both the chiroptical response characterized by
the anisotropic dissymmetry factor $g_{CD}$ and the spin polarization vector
$\mathbf{P}=(P_{x},P_{y},P_{z})$. We will show one of our central results
that, under certain conditions, the norm of the polarization vector $P$ is a
predictor of chiroptical activity.

###  1.3 The Effect of SOC and TRS Breaking on the Rotatory Strength and COA

A central element in establishing the connection between COA and the SP is to
generalize the definition of the Rosenfeld tensor to include the spin
contributions. The rotatory strength $R_{nm}$ (and its components) can then be
expressed in the general form:

$R_{nm}=\frac{3e^{2}}{4m_{e}c}\left[\left(\frac{a^{2}b}{a^{2}+b^{2}}\right)r_{nm}+2\hbar
S_{nm}\right],$ (5)

where $r_{nm}$ is a dimensionless function that contains the contribution of
the orbital magnetic moment and depends on the quantum numbers $n$ and $m$ and
the number of turns $K$. The spin contribution to the total orbital magnetic
moment is included in the coefficient $S_{nm}$, and SOC is hidden in the
wavefunctions with which the transition matrix elements are built (see
Supplementary Material for more details).

Figure 2 shows the influence of the SOC strength as well as of decoherence on
the rotatory strength $R$ for the transition from the fundamental state $n=1$
to the first excited state $m=2$. Figure 2a illustrates first Tinoco’s case
when TRS is preserved. Breaking TRS via a dephasing probe (change in relative
phase $\delta-\eta$) in the presence of SOC shifts the spectrum, but by an
amount which is independent of the SOC magnitude. On the other hand, the
introduction of a voltage probe, implying a change in the amplitude $z$ in the
wavefunction, results in a boundary condition determining a non-zero SP,
analogous to a magnetic tip in a junction. This manifests in varying
rotational strengths (see Fig.2b). The tensor components $R_{11}$ and $R_{22}$
for the first transition, corresponding to incident radiation along the
$\mathbf{B}_{SO}$ direction (perpendicular to the molecular axis), are the
only affected by the voltage probe inducing decoherence. This change affects
the components in a way that leaves the average $R$ components almost
unchanged.

Figure 2: Rotatory strength with SO coupling and decoherence. Rotatory
strength $R$ and its components $R_{\imath\imath}$ for the transition from
$n=0$ to $m=1$ as a function of the magnitude of the SO coupling. a shows
Tinoco’s case where TRS is preserved, and b shows the effect of decoherence
induced with a voltage probe. The characteristic values of the helix used are
$a=0.075$ nm, $2\pi b=3.54$ nm, and $K=1$, for positive chirality $\zeta=+1$.

The CD spectrum of the helix for an isotropic system, for the first
transitions from the fundamental state, is shown in Fig.3 when TRS is
preserved. This spectrum remains the same with and without SO coupling (see
3a), indicating that the electronic spin effect is negligible when TRS is
preserved. This result can be interpreted as a manifestation of Bardarson’s
theorem[50]. Both the dephasing and voltage probe effects are observable in
the spectrum as a change in the magnitude of the CD intensity in graphs b and
c, respectively. But, while dephasing is evidenced as an asymmetry in the CD
spectra of the two enantiomers, the effect of the voltage probe generates the
same change in the magnitude of the CD intensity for both enantiomers, in
addition to a shift in the maximum absorption peaks, so that the spectra for
the two enantiomers are mirror images, even in the presence of SOC.

Figure 3: Circular dichroism spectrum with SOC and decoherence. The CD
spectrum for a helix is shown as a function of the incident radiation
wavelength $\lambda$ for a helix with preserved TRS (a) with SO coupling; (b)
with a dephasing probe; and (c) with a voltage probe, for both enantiomers
$\zeta$. The CD spectrum of the helix for an isotropic system, for the
transitions from the fundamental state $n=1$ to $m$, was defined in terms of
the rotatory strength in the form CD$=\frac{A}{\Gamma\sqrt{\pi}}\sum_{m\neq
1}R_{1m}Exp{\left[-\left(\frac{\lambda-\lambda_{m1}}{\Gamma}\right)^{2}\right]}$,
with $\Gamma$ the half-width of the Gaussian, $\lambda$ the incident radiation
wavelength, and $\lambda_{m1}=\lambda_{m}-\lambda_{1}$[68].

Figure 4: Circular dichroism components for oriented helix with SOC and
decoherence. Figures a-c illustrate the effect of the voltage probe on the
CDıı components without SO coupling for incident radiation in the $x$, $y$,
and $z$ directions, with $\imath=1,2,3$ respectively. d-f, Effect of
decoherence coupled with SO interaction on the CD components, revealing an
asymmetry in CD11 and CD22 components for both enantiomers (solid lines). We
have used $\delta=\eta=0$, and $q=1$.

In contrast, the impact of the voltage probe on the CD components CDıı,
relative to the direction of incidence of the radiation field, is shown in
figure 4. Inducing decoherence (without SOC) affects the spectrum in such a
way that the three CD components show mirror images (4, a\- c) However, the
coexistence of the SOC with decoherence generates an asymmetry in CD11 and
CD22 for the two enantiomers (see figures 4, d-e), in contrast to CD33 which
is symmetrical. This asymmetry is related to the orientation of the helix with
respect to the direction of the incident radiation. In this way, the voltage
probe simulates a magnetic probe as in the experiments, which moves the local
field of the helix, changing the momentum and modifying the spin population
injected into the molecule[39, 40, 54].

###  1.4 Spin Polarization as a Predictor of Chiroptical Activity

As mentioned in the introductory part, one of the main motivations for this
work is to establish an explicit connection between the SP induced by the CISS
effect and the COA. Before entering into the details of this connection, one
should emphasize a fundamental difference: while COA requires photon
absorption, SP can occur as a ground-state magnetic response associated with
electron transport, electron transfer, and bond polarization that occurs in
the absence of external magnetic fields and also in photo-induced ET
reactions. The relationship presented here is related to a very interesting
interplay between SP (connected to the $P_{z}$ component), and spin
coherence[69]. Here, we will concentrate on the simpler case of the comparison
between ground state SP and COA, because it offers important insights into the
connection between the two phenomena and the interpretation of recent
experiments by Waldeck et al. [31] about a correlation between the length
dependence of both magnitudes and the possible use of SP as a predictor of
COA.

Since the COA is a scalar and the SP is a vector quantity, we have considered
as a plausible scalar predictor the norm of the polarization vector $P$, which
carries information about the global SP and is independent of the coordinate
system. Our conjecture is also inspired by the fact that the polarization
vector is simply related to the spin component of the total magnetic moment by
the equation $\mathbf{m}_{s}=\it{C}\bf{P}$, where $\it{C}$ is proportional to
the gyromagnetic ratio. This connection of the CISS-related response to the
magnetic moment is very significant, because it emphasizes the nature of this
effect as a magnetic response to electron transport. It should be stressed
that in our model, transport is mimicked by changing the boundary conditions
with a parameter $\epsilon$, defining the stationary states in eqs.(3,4). In
this case, we consider
$\Psi^{\zeta}_{\tilde{n}}|_{\varphi=0+\epsilon}\neq\Psi^{\zeta}_{\tilde{n}}|_{\varphi=2\pi
K}$. From this point of view, our system is equivalent to an electron in a box
with adjustable boundary conditions.

Figure 5: Length dependence of SP and COA. Norm of the polarization vector $P$
as a function of $K$ for the fundamental state ($n=1$) (a) when TRS is
preserved, and (c) when TRS is broken by boundary-induced decoherence, in
presence of SOC. Dissymmetry factor $g_{CD}$ as a function of $K$ for the
transition from the ground state with $n=1$ to the first excited state $m=2$
(b) when TRS is preserved, and (d) when TRS is broken using a voltage probe
with SOC. Total average magnetic moment for the ground state as a function of
the helix length for (e) the Tinoco’s case, and (f) when TRS is breaking by a
voltage probe in the presence of SOC. We have used $\delta=\eta=0$, and $q=1$.

Figures 5 displays the graphical relationship between $P$, calculated for the
ground state $n=1$, and the dissymmetry factor $g_{CD}$, corresponding to the
transition $n=1\rightarrow m=2$ as a function of the number of turns in the
helix $K$, in presence of SOC when: i) TRS is preserved (panels a and b); ii)
TRS is broken by decoherence and changing the boundary conditions ( panels c
and d). Despite the fact that both phenomena have very different physical
origins, there is a clear correspondence that originates from the fact that
both quantities are sensitive to the CISS-induced magnetic response. That the
SP component is the key predictor is also apparent from the analysis of the
plot of the total average magnetic moment for the ground state, fig.5e as a
function of the length of the helix. Although this magnitude also increases
with length in the presence of SOC and TRS breaking, it does not show the
saturation behavior of both $P$ and $g_{CD}$, which indicates that it is
dominated by the orbital angular momentum contribution. It is important to
notice that the predictor, the SP, exhibits a rather small variation as a
function of length, mostly because of the constraints imposed by TRS in gas
phase. The numerical proportionality between the two magnitudes is simply
calculated to be $P\approx c10^{-6}g_{CD}$. This type of relationship, one of
our central results, should survive the transition to more realistic models
and also to chiral interface, where the breaking of TRS arises from either
non-linear conditions or because the system under consideration has open
boundaries.

Figures 6 $\bf a$ and $\bf b$ show the behavior of the magnetic moment
expectation values for the ground state $|\mathbf{m}_{11}|$ due to the SOC,
compared to spinless electrons, and the magnitude of the transition magnetic
dipole moment $|\mathbf{m}_{12}|$ for the first transition, respectively.
While both magnitudes have a monotonic behavior when TRS is preserved with
SOC, interference effects that arise from the inclusion of decoherence can
make for the non-monotone behavior of the transition moment. In any event, it
is clear that these magnitudes do not have the same predicting behavior on COA
as the SP, despite the fact that they depend on length.

Figure 6: Length dependence of magnetic moment. a, Magnetic moment expectation
value for the ground state as a function of the length $K$. b, Transition
magnetic dipole moment as a function of $K$ for the transition from $n=1$ to
$m=2$. We have used $\delta=\eta=0$, and $q=1$.

##  2 Conclusions and Final Remarks

Using a very simple model of an electron in a helix, including the spin degree
of freedom and spin-orbit coupling, we have explored a fundamental connection
between optical activity and the CISS effect in chiral systems. We have also
investigated the fundamental role of time-reversal symmetry in strongly
modulating both the spin polarization and the chiroptical response, as well as
the influence of spin-orbit coupling on the optical activity that involves a
strongly asymmetric enantiomeric response, not observed in simple chiral
molecules in gas phases or in solutions, but that has been found in chiral
solids and interfaces.

All our findings seem to point to a deeper connection that necessarily
involves the CISS-modulated magnetic response and the fact that the
chiroptical activity involves a dephasing of the electric and magnetic fields
as they propagate through the chiral structure. This dephasing should be
connected to the fact that the onset of the CISS effect triggers an
intramolecular magnetic response that needs to be explicitly included in the
spin polarization of the system. We are currently exploring this fundamental
connection and also the extension of our model to describe the photon-induced
spin polarization in electron transfer processes.

Supplementary information

Acknowledgments The authors acknowledge fruitful discussion with Prof. Jesus
M. Ugalde. S.V. acknowledges the support given by the Eleonore-Trefftz-
Programm and the Dresden Junior Fellowship Programme by the Chair of Materials
Science and Nanotechnology at the Dresden University of Technology, and the
W.M. Keck Foundation through the grant “Chirality, spin coherence and
entanglement in quantum biology.” R.G. and G.C. acknowledge the support of the
German Research Foundation (DFG) within the project Theoretical Studies on
Chirality-Induced Spin Selectivity (CU 44/55-1), and by the transCampus
Research Award Disentangling the Design Principles of Chiral-Induced Spin
Selectivity (CISS) at the Molecule-Electrode Interface for Practical
Spintronic Applications (Grant No. tCRA 2020-01), and Programme trans-Campus
Interplay between vibrations and spin polarization in the CISS effect of
helical molecules (Grant No. tC2023-03). E.M. acknowledges funding from
project POLI17945 of USFQ and the Dresden Fellowship Programme. V.M
acknowledges the support of Ikerbasque, the Basque Foundation for Science, the
German Research Foundation for a Mercator Fellowship within the project
Theoretical Studies on Chirality-Induced Spin Selectivity (CU 44/55-1), and
the W.M. Keck Foundation through the grant “Chirality, spin coherence and
entanglement in quantum biology.”

## Author contributions

…

## Competing interests

The authors declare no competing interests.

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