Quantum-Assisted Metrology of Neutral Vitamins in the Gas Phase

Abstract

It has recently been shown that matter-wave interferometry can be used to imprint a periodic nanostructure onto a molecular beam, which provides a highly sensitive tool for beam displacement measurements. Here, we use this feature to measure three electronic properties of provitamin A, vitamin E and vitamin K1 in the gas phase, for the first time. The shift of the matter-wave fringes in a static electric field encodes the molecular susceptibility and the time-averaged dynamical electric dipole moment. The dependence of the fringe pattern on the intensity of the central standing light-wave diffraction grating is used to determine the molecular optical polarizability. The comparison of our experimental findings with molecular dynamics simulations and density functional theory provides a rich picture of the electronic structure and dynamics of these biomolecules in the gas-phase with β-carotene as a particularly interesting example.

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Angewandte
International Edition
A Journal of the Gesellschaft Deutscher Chemiker
www.angewandte.org
Chemie
Accepted Article
Title: Quantum-assisted metrology of neutral vitamins in the gas-phase
Authors: Lukas Mairhofer, Sandra Eibenberger, Joseph P. Cotter,
Marion Romirer, Armin Shayeghi, and Markus Arndt
This manuscript has been accepted after peer review and appears as an
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of the final Version of Record (VoR). This work is currently citable by
using the Digital Object Identifier (DOI) given below. The VoR will be
published online in Early View as soon as possible and may be different
to this Accepted Article as a result of editing. Readers should obtain
the VoR from the journal website shown below when it is published
to ensure accuracy of information. The authors are responsible for the
content of this Accepted Article.
To be cited as: Angew. Chem. Int. Ed. 10.1002/anie.201704916
Angew. Chem. 10.1002/ange.201704916
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Quantum-assisted metrology of neutral vitamins in the gas-phase
L. Mairhofer,[a] S. Eibenberger,[a,b] J. P. Cotter, [a,c] M. Romirer, [a] A. Shayeghi,[a] and M. Arndt*[a]
Abstract:
It has recently been shown that matter-wave interferometry can be
used to imprint a periodic nanostructure onto a molecular beam, which
provides a highly sensitive tool for beam displacement measurements.
Here, we use this feature to measure three electronic properties of
provitamin A, vitamin E and vitamin K1 in the gas phase, for the first
time. The shift of the matter-wave fringes in a static electric field
encodes the molecular susceptibility and the time-averaged
dynamical electric dipole moment. The dependence of the fringe
pattern on the intensity of the central standing light-wave diffraction
grating is used to determine the molecular optical polarizability. The
comparison of our experimental findings with molecular dynamics
simulations and density functional theory provides a rich picture of the
electronic structure and dynamics of these biomolecules in the gas-
phase with β-carotene as a particularly interesting example.
Experimental studies with neutral biomolecules in the gas-phase
are important because they allow their intrinsic electronic
properties to be assessed without perturbation by matrix effects.[1]
In particular, vitamins in the gas-phase have recently received
renewed theoretical interests[2] and analytical experiments using
these ubiquitous but thermally sensitive particles in the gas-phase
have been carried out using mass spectrometry[3] and microwave
spectroscopy.[4] Here, we utilize the benefits of near-field quantum
interference to measure optical polarizabilities and electric
susceptibilities of molecules in the same setup. We compare
experimental data with molecular dynamics (MD) simulations
combined with density functional theory (DFT) for -tocopherol
(vitamin E, C29H50O2), phylloquinone (vitamin K1, C31H46O2) and
β-carotene (provitamin A, C40H56). They are similar in complexity
and mass, but differ in their symmetry, polarity, and thermal
folding dynamics. This influences their static and optical
polarizability, as well as their permanent and vibration-induced
electric dipole moment.[5]
Following Louis de Broglie,[6] quantum mechanics assigns a
wave-nature to matter, for instance to the center-of-mass of entire
molecules as well as to the electrons inside.[7] While electron
delocalization is the basis of covalent chemical bonding,[8] the
quantum nature of the center-of-mass motion of molecules is less
commonly observed, since it requires the dedicated preparation
of delocalization on the micrometer scale. We do this in our
Kapitza-Dirac-Talbot-Lau interferometer (KDTLI),[9] illustrated in
Figure 1. All vitamins are evaporated from a ceramic oven at
temperatures between 400 K and 460 K to form a molecular beam
in high vacuum. They pass through the interferometer and are
detected using electron impact ionization quadrupole mass
spectrometry, about two meters downstream from the source.
Thermal fragments occur but are rejected by the quadrupole mass
filter. We select a velocity class of the molecular beam by defining
a free-flight parabola in the gravitational field of the Earth using
three slits. The transmitted velocity distribution is determined by
chopping the molecular beam in a pseudo-random sequence and
measuring the time-of-flight to the detector.[10] For all vitamins
described here the velocity distribution has a mean of v ≃200 m/s
with a spread of about 45% (FWHM). This corresponds to de
Broglie wavelengths  =ℎ/v of 3-6 pm, where ℎ is Planck’s
constant and  the mass of the individual molecules. The
wavelength  is almost three orders of magnitude smaller than
the size of each molecule. However, at the position of the second
grating the center-of-mass wave function is delocalized across
the molecular beam over one million times .
The setup is as follows: The three gratings G1, G2 and G3 have
equal period of d = 266 nm and are positioned at equal distances
along the molecular beam. G1 and G3 are machined into 190 nm
thick silicon nitride. The first grating acts as an array of point-like
Figure 1. Nea r-field matter-wave interferometer with the addition of a high
voltage deflection electrode. The effusive source emits a beam of vit amins,
which diffract at the gr atings G1, G2 and G3. G1 and G3 are nanomechani cal
gratings (black), G2 is an optical grating (green). The deflection electrode
(brown) displaces the observed interference fringes. The quadrupole mass
spectrometer (QMS) ionizes and mass selects the molecules.
[a] Dr. Lukas Mairhof er, Dr. Sandra Eibenberger, Dr. Joseph P. Cotter,
Marion Romirer, Dr. A rmin Shayeghi, Prof. Dr. Markus Arndt*
Faculty of Physics, VCQ, University of Vienna
Boltzmanngasse 5, A -1090 Vienna
E-mail: markus.arndt@univie.ac.at
[b] Dr. Sandra Eibe nberger
Lyman Laboratory
Harvard Universit y, Department of Ph ysics,
17 Oxford Street, Cambridge, MA 02138, USA
[c] Dr. Joseph P. Cotter
Centre for cold matter, Blackett Laborat ory, Imperial College, Prince
Consort Road, Lond on SW7 2BW, UK
Supporting information for this article is given via a link at the end of the
document.
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sources of width s = 110 nm. It follows from Heisenberg’s
uncertainty relation, that each molecule passing through any of
the slits of G1 acquires a transverse momentum uncertainty of
Δ ≥ ℎ/Δx = h/s. This delocalization increases linearly with the
distance behind G1. When molecules arrive at the second grating
L=10.5 cm downstream from G1, the indeterminacy of the
molecular position – their transverse coherence[11] – has grown
to 2/ ≃ 1 µ such that it covers several periods of G2. The
diffraction grating G2 is an optical standing wave with a period of
/2 that is obtained by retro-reflecting a laser beam with a
wavelength = 532 nm from a mirror. Each molecule, with
optical polarizability (), passing through the electric field 
of G2 experiences an electric dipole potential 
 =
−() 2()/4 which modulates the de Broglie phase of the
molecular center-of-mass wave function. Free evolution of the
matter wave behind G2 leads to the formation of a periodic
molecular density distribution. Close to multiples of the Talbot
length =2/ this pattern is an image of the grating of
period d.
This interferometer concept is common in optics,[12] and has been
demonstrated in atom optics[13] as well as medical x-ray
imaging[14]. Recently, it has allowed revealing the wave nature of
molecules as massive as 10000 amu.[15] Here, we use this method
to study electronic properties of neutral vitamins in the gas-phase.
The interference pattern is detected by scanning the
nanomechanical mask G3 over the molecular beam, another 10.5
cm behind G2. If the molecular fringes and the grating G3 are in
phase, the number of molecules arriving at the spectrometer ()
is maximized. The fringe visibility of the sine fit to the data is
defined as =( − )/( +). It depends on the
optical polarizability  () and absorption cross
section () of the molecules as well as the laser intensity in
the center of the Gaussian light beam = 2/  with the
horizontal and vertical waist  and .
High visibillity interference patterns are observed for all three
vitamins. In Figure 2, we show a typical high-contrast
interferogram of β-carotene, which is a clear evidence for the
quantum nature of its motional state.[16] The dependence of the
fringe visibility on the laser power P allows us to determine their
optical polarizability[17]. We plot a typical V(P)-curve for the cases
of -tocopherol and phylloquinone in Figure 3. A molecule may
also absorb a photon in transit through the diffracting laser field.
This is an additional matter-wave beam splitting mechanism,
which also modulates the fringe visibility.[10] A correct
interpretation of the V(P) curve (Figure 3) thus hinges on good
knowledge of the laser intensity in G2 which we have calibrated
in situ using C60 molecules.11
The nanoscale period of the molecular density pattern enables
high-resolution molecule deflectometry: Any transverse force
acting on the molecular beam will displace the fringes by a
distance Δ . Using a tailored pair of electrodes we create a
homogeneous force close to the second grating that shifts the
interference fringes[18] by Δ = ⋅ (/)⋅()/v2. It grows
with the electric susceptibility-to-mass ratio (/), the gradient of
the square of the deflection field  and inversely with the square
of the molecular velocity v. The geometrical factor K contains
information about the electrode geometry and position, here
calibrated using C60 molecules. The deflection Δ follows the
same law found in classical deflectometry[19] but quantum
interference adds the nanometric fine structure which is valuable
for achieving a spatial resolution of the shift of the order of 10 nm.
In Figure 4 we demonstrate the molecular fringe deflection for
α
-
tocopherol and phylloquinone in an electric field of varying
strength. For rigid non-polar molecules, the static polarizability
can be directly extracted from such deflection measurements. In
most vitamins, however, molecular vibrations induce fluctuations
of the squared electric dipole moment µ2 whose thermal average
〈µ2⟩ contributes to the total electric susceptibility[20] through the
van Vleck relation = +〈µ2〉/3, where  is
Boltzmann’s constant.
To extract the electronic properties from our experiments we
compare the data to theory. The center-of-mass motion is
described by a well-established quantum formalism in phase
space[16]. In the absence of radiation, collisions and other
interactions with the environment, the internal degrees of freedom
are decoupled from the molecular center-of-mass motion and are
described by a combined MD and DFT approach. A full quantum
treatment of the internal states is conceivable[21].
Figure 2:
Molecular interference pattern of
β
-
carotene. Data points (red
circles) show the number of detected molecules as a function of the transverse
position of grating G3. We observe a sinusoidal variation in the number of
counts, the high amplitude of which is a cl ear evidence of quantum
interfe
rence. The solid line is a sinusoidal fit to the data
, from which we extract
a fringe visibility o f
V = 32 ± 2 %. The grey area hig hlights the dark counts.
Figure 3. Experimental interference fringe visibilities of phylloquinone (blue
diamonds)
and α-tocopherol (green squares)
as a function of the diffracting
laser power in G2
. In the interaction region, the hori zontal waist 
was 20
µm
and the vertical waist 920
µm
. The error bars sh ow the uncertainties of the
visibility resulting from an error propagation of the amplitude and offset of the
sine curve using
68% confidence intervals for the sine fit.
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Here, we sample the molecular configurational space by MD
simulation to account for the floppiness of the vitamins and
compute their static and optical polarizabilities (at 532 nm) using
the Coupled-Perturbed Kohn-Sham method, as well as the
electric dipole moments for subsequent time steps using DFT.
The conformational space was scanned by a MD simulation using
the LAMMPS package[22] with CHARMM[23] force field parameters
obtained from the multipurpose atom-typer for CHARMM[24].
During the MD simulation, a single molecule is propagated over
100 ns (after an equilibration run of 10 ns) in time steps of 1 fs at
the respective experimental oven temperatures of around 450 K
controlled by a Nosé-Hoover thermostat[25] with a relaxation time
of 0.1 ps. Assuming that the MD time-evolution of a single
molecule in vacuum covers a sufficiently large conformational
phase space the single time sequence samples a statistically
representative ensemble of conformations in the hot molecular
beam. Short ab initio molecular dynamics (AIMD) simulations
using NWChem[26] at the PBE0/3-21G level of theory[27] over 50
ps indicate that the CHARMM force field is a reasonable
approximation for our high temperature simulations.
The molecular structure was extracted from the MD simulation
every 2 ns and fed into DFT calculations at the CAM-
B3LYP/Def2TZVP level of theory[28] using the Gaussian program
package.[29] The range-separated hybrid exchange-correlation
functional CAM-B3LYP has been shown to perform well for
calculations of electronic (hyper)polarizabilities of organic
compounds.[30]
Figure 5 displays the electronic parameters for α-tocopherol.
Simulation data for the other two vitamins are compiled in the
Supplementary Information. Conformational changes occur on
the picosecond scale. Even under vigorous fluctuations the
optical and the static polarizability stay constant within a few
percent, while the dipole moment fluctuates by up to 400% peak-
to-peak when sampled at the nanosecond scale. Such
calculations allow us to determine the van Vleck susceptibility
χ
which we compare with our measurements in Table 1. We show
the DFT values for the electric dipole moment, extracted from a
thermal average over the MD time steps. Molecular dynamics
suggests that all-trans β-carotene is more rigid than the other two
vitamins, non-polar in its ground state. However, modelling shows
that β carotene can develop a non-zero average dipole moment
and may undergo thermally induced cis-trans isomerization in the
gas phase. In solution[31], this transition has been observed at a
temperature as low as 350 K. Cis-geometries are therefore
expected to contribute to the experimental results. Furthermore,
spectra in solution indicate that β-carotene exhibits strong
wavelength shifts even in moderate electric fields[32] , and its
optical polarizability should therefore depend on the field.
It surprises that the MD averaged optical polarizability of all-trans
β-carotene exhibits a negative sign, even though the diffraction
laser is red-detuned to the nearest expected dipole allowed
transition around 440 nm. Although the experiment is insensitive
to the sign of opt (532 ) we have cross-checked its value
using the global hybrid PBE0 functional. While for phylloquinone
and α-tocopherol PBE0 yields very similar predictions for all
electronic properties, β-carotene is again a special case as the
optical polarizability maintains the negative sign but its value
changes from −152 Å3 (CAM-B3LYP) to − 107 Å3 (PBE0). This
is not surprising as the polarizability spectra are sensitive to the
percentage of Hartree-Fock exchange[33]. This lower value is also
closer to the experimental result (83 ±10 Å3).
To further elucidate the origin of the sign of ⟨opt⟩ at the
wavelength of the grating laser (532 nm), we have calculated
optical polarizabilities and spectra from time-dependent DFT both
for the vibrational ground state with inversion symmetry and
distorted carotene geometries. While opt (532 ) is positive for
the inversion symmetric geometry it exhibits a negative value for
distorted structures. This correlates with the calculated optical
response: We find an intense dipole transition in the range 600-
650 nm for a distorted carotene but not for its ground state. W ith
respect to this transition, the grating laser is blue-detuned,
explaining the negative sign of ⟨opt (532 )⟩. Our
computation is consistent with recent experiments, showing that
this transition is accessible in two-photon processes[34] and near-
edge x-ray absorption combined with UV photoelectron
spectroscopy[35].
Figure 4: Molecular fringe displ acement of phylloquinone (blue diamonds, left
scale)
and α-tocopherol (g reen squares, right scale)
as a function of the
deflection
voltage. As expected it
depends quadratically on the voltage. The
axes of the
two curves are offset vertically for clarity.
The velocities were 180
m/s for
-tocopherol and 195 m/s for phylloquinone.
The inset sketches the
molecular fringe position for two deflection voltages (r ed
= 6 kV, black = 1 kV ).
The error bars
are 68% confidence intervals for
the relative phase of the two
sinusoidal interfere nce curves
. At high voltage the uncertainty increases
,
because the fring e shift and visibility are sensitive to the finite v elocity spread o f
the molecular beam
. A shift of  corresponds to a beam
displacement of half a
grating period.
Figure 5: Calculated electronic properties of
α
-tocopherol. Evoluti on of the
static ( 

,
red squares) and optic al (


(532 )) , blue diamonds)
polarizability as well as its electric dipole moment ( µ, black circles) d uring the
MD simulation displayed in 2 ns steps. Th e DFT calculations p erformed along
the MD trajectory ar e o
btained at the CAM-B3LYP/ Def2TZVP level of the ory.
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COMMUNICATION
Table 1: Molecula r electronic properties from experiments and theory. The
theoretical values represent a thermal average at the temperature of the
experiment. The co mputational result is the mean over all consid ered MD steps.
We estimated the experimental errors by testi ng the robustness of th e result
against the combined standard deviations of the contributi ng factors (see Suppl.
Info).
a) 1st row: CAM-B3LYP/Def2T ZVP; 2nd row: PBE0/Def2TZVP; 3rd row:
Experiment
b) Convert to SI units by multiplying wit h 40
In summary, we have shown that quantum-interference assisted
metrology opens a window to measuring electrical, optical and
dynamical information of biomolecules in a single comprehensive
setting. We have shown this here for the three pro/vitamins α-
tocopherol, phylloquinone and β-carotene. We find good
agreement with computational chemistry and also see that the
fully conjugated electronic structure of β-carotene opens a
number of interesting questions. Future studies in molecule
interference shall address them also using highly sensitive single-
photon recoil spectroscopy around 640 nm[36]. Sources of
internally cold molecules[37] will allow to further elucidate the role
of conformations, which can be supported by more elaborate
AIMD simulations. Matter-wave assisted metrology thus proves to
be an interesting link between quantum optics and chemistry. It
can be readily extended to magnetic, optical and collisional
properties, and thus help benchmarking computational models of
complex biomolecules.
Acknowledgements
We acknowledge financial support through the European Research
Council ERC AdvG 320694, as well as through the FWF within
W1210-25. JC acknowledges a VCQ fellowship. The computational
results presented have been achieved using the Vienna Scientific
Cluster (VSC). We are thankful to M.C. Böhm, E. Voyiatzisis and G.
G. Rondina for useful discussions.
Keywords: Vitamins • Matter-waves • Interferometry • Electronic
structure • Deflectometry • DFT • MD
The authors declare no financial interest in this work
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Vitamin E
α
-Tocopherol
K1
Phylloquinone
Pro-A
β
-Carotene
Sum formula C29H50O2 C31H46O2 C40H56
T / K 400 ± 5 450 ± 5 460 ± 5
Mass / amu 430.7 450.7 536.9
〈
µ
〉a
/ D 1.8 ± 0.1
1.8 ± 0.1
1.1 ± 0.1
1.1 ± 0.1
1.3 ± 0.1
1.3 ± 0.1
〈
αstat
〉a,,b
/
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54 ± 1
54 ± 1
58 ± 1
59 ± 1
211 ± 4
229 ± 3
〈
αopt
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Å
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56 ± 1
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62 ± 1
61 ± 1
52 ± 7
- 152 ± 11
- 107 ± 4
(-) 83 ± 10
χa,b /
Å
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78 ± 3
80 ± 8
65 ± 1
66 ± 1
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229 ± 4
240 ± 3
n.a.
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COMMUNICATION
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10.1002/anie.201704916
Angewandte Chemie International Edition
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COMMUNICATION
Entry for the Table of Contents
COMMUNICATION
Molecule interferometry exploits the
quantum wave-nature of molecules
to prepare a nanoscale density
pattern in high vacuum. Here this is
used to determine various electric
properties of vitamins isolated in the
gas-phase, including β-Carotene
(Provitamin A),
α
-Tocopherol
(Vitamin E), and
Phylloquinone (Vitamin K1).
Lukas Mairhofer, Sandra Eibenberger,
Joseph Cotter, Marion Romirer, Armin
Shayeghi and Markus Arndt*
Page 1 – 
Quantum-assisted metrology of
neutral vitamins in the gas-phase
10.1002/anie.201704916
Angewandte Chemie International Edition
This article is protected by copyright. All rights reserved.