thesis of doctorate degree
Infrared Spectroscopy
of Small Biomolecules
in the Gas Phase
Åke Andersson
Department of Physics
University of Gothenburg
Gothenburg, Sweden 2025
Doctoral Dissertation in Physics
Department of Physics
University of Gothenburg
412 96 Gothenburg, Sweden
Main supervisor:
Prof. Vitali Zhaunerchyk, Department of Physics, University of Gothenburg
Examiner:
Prof. Dag Hanstorp, Department of Physics, University of Gothenburg
Opponent:
Prof. Timothy S. Zwier, Department of Chemistry, Purdue University
© Åke Andersson, 2025
ISBN: 978-91-8115-306-4 (printed)
ISBN: 978-91-8115-307-1 (online)
URL: http://hdl.handle.net/2077/87128
Cover:
The dominant conformer AAFA-1 of the neutral capped tetrapeptide
Ace-Ala-Ala-Phe-Ala-NH2, with H-bonds drawn as dotted cyan lines.
Printed by Stema Specialtryck AB, Borås
Typeset in X LEATEX with Times Ten and Akzidenz-Grotesk
ii
Abstract
Proteins are macromolecules essential to all forms of life. Their biological function
is related to how they fold, which is determined by their primary sequence, but the
exact relationship is not understood. Physical modeling offers a bottom-up answer
to this question, but the models need calibration on experimental data of similar
biomolecules.
Infrared (IR) spectroscopy is a powerful tool for determining molecular structure,
and in turn validating physical models. The infrared range corresponds to molecu-
lar vibrations, which are intimately related to the geometric structure. Thus, by
performing IR spectroscopy on biomolecules, it is in the aggregate possible to
judge physical models.
Simulations of molecules are best done in an isolated environment. To match
this, the spectroscopic experiments should be performed in the gas phase, where
molecules are isolated, and the intrinsic properties can be studied. The low density
of the gas phase requires particular action spectroscopy techniques to be used.
This thesis focuses on action spectroscopy techniques for biomolecules in the gas
phase, and all theory required to describe it. Four such techniques are applied to
small biomolecules: IR multiple photon dissociation (IRMPD) of ions, IRMPD
of neutrals combined with ionizing vacuum ultraviolet radiation (IRMPD–VUV),
ultraviolet (UV) resonance-enhanced multiple photon ionization (REMPI), and
conformer-specific IR–UV ion dip spectroscopy derived from REMPI. In addition,
an experimental setup for the last two techniques has been constructed.
The established IRMPD technique is applied to proton-bound dimers of asparagine.
Using partial isotope labeling, the IR resonances are assigned vibrational modes.
Corresponding quantum chemical calculations are undertaken to predict the spec-
trum and infer the structure.
Also, the recently developed IRMPD–VUV technique is applied to the pentaala-
nine peptide, in belief of it forming a miniature helix. Both experiment and theory
indicate multiple populated conformers. Although their structures are not able to
be determined, the chemical environment of the carboxyl (COOH) group can be
inferred.
Finally, the conformer-specific IR–UV ion dip spectroscopy technique is applied
to phenylated polyalanine peptides. The phenyl group allows for REMPI and ion
dip spectroscopy. In contrast to pentaalanine, the experiment shows only one con-
former, which is supported by theory.
iii
Sammandrag
Proteiner är makromolekyler som är nödvändiga för alla livformer. Deras biolo-
giska funktion är kopplad till hur de veckas, vilket bestäms av deras primärsekvens,
men det exakta sambandet är inte förstått. Fysikalisk modellering erbjuder en lös-
ning från grunden i denna fråga, men modellerna behöver kalibreras med experi-
mentella data från liknande biomolekyler.
Infraröd (IR) spektroskopi är ett kraftfullt verktyg för att bestämma molekylär
struktur och därmed validera fysikaliska modeller. Det infraröda området mot-
svarar molekylära vibrationer, som är nära relaterade till den geometriska struk-
turen. Genom att utföra IR-spektroskopi på biomolekyler är det sammantaget
möjligt att bedöma fysikaliska modeller.
Simuleringar av molekyler görs bäst i en isolerad miljö. För att matcha detta bör
spektroskopiska experiment utföras i gasfas, där molekyler är isolerade och de
inneboende egenskaperna kan studeras. Den låga densiteten i gasfasen kräver att
särskilda tekniker för verkansspektroskopi används.
Denna avhandling fokuserar på verkansspektroskopitekniker för biomolekyler i
gasfas och all teori som krävs för att beskriva dessa. Fyra sådana tekniker tilläm-
pas på små biomolekyler: IR flerfotondissociation (IRMPD) av joner, IRMPD
av neutrala molekyler kombinerad med joniserande vakuumultraviolett strålning
(IRMPD–VUV), ultraviolett (UV) resonansförstärkt flerfotonjonisering (REMPI),
och konformerspecifik IR–UV-jon-dip-spektroskopi baserad på REMPI. Dessu-
tom har en experimentell uppställning för de två sista teknikerna konstruerats.
Den etablerade IRMPD-tekniken tillämpas på protonbundna dimerer av aspara-
gin. Genom partiell isotopmärkning tilldelas IR-resonanserna vibrationsmoder.
Motsvarande kvantkemiska beräkningar utförs för att förutsäga spektrumet och
dra slutsatser om strukturen.
Vidare tillämpas den nyligen utvecklade IRMPD–VUV-tekniken på peptiden pen-
taalanin, i tro om att den ska bilda en miniatyrhelix. Både experiment och teori
indikerar flera förekommande konformer. Även om deras strukturer inte kan
bestämmas, kan den kemiska miljön kring karboxylgruppen (COOH) avläsas.
Slutligen tillämpas den konformerspecifika IR–UV-jon-dip-spektroskopitekniken
på fenylerade polyalaninpeptider. Fenylgruppen möjliggör REMPI och jon-dip-
spektroskopi. I motsats till pentaalanin visar experimentet endast en konformer,
vilket stöds av teorin.
iv
Preface
This thesis is a compilation of papers written during my time as a PhD student.
Chapter 1 provides a foundational overview accessible to university students, and
may be skipped by readers familiar with spectroscopy. The remaining chapters
follow the IMRaD standard for scientific writing.
Abbreviations
ADMP Atom-centered Density Matrix Propagation
B3LYP Becke’s 3-parameter Lee–Yang–Parr (functional)
BOMD Born–Oppenheimer Molecular Dynamics
DFM Differential Frequency Mixing
DFT Density Functional Theory
FEL Free Electron Laser
FELIX Free Electron Laser for Infrared eXperiments
FWHM Full Width at Half Maximum
GU University of Gothenburg
HF Hartree–Fock
IR InfraRed
IRMPD InfraRed Multi-Photon Dissociation
IVR Intramolecular Vibrational energy Redistribution
MALDI Matrix-Assisted Laser Desorption/Ionization
MCP Micro-Channel Plate (detector)
MD Molecular Dynamics
Nd:YAG Neodymium-doped Yttrium Aluminum Garnet
OPA Optical Parametric Amplifier
OPO Optical Parametric Oscillation
PES Potential Energy Surface
RDG Reduced Density Gradient
REMPI Resonance-Enhanced Multi-Photon Ionization
RMSD Root Mean Square Distance
SHG Second Harmonic Generation
TOF Time-Of-Flight
UV UltraViolet
VPT2 2nd-order Vibrational Perturbation Theory
VUV Vacuum UltraViolet
v
List of papers
This thesis consists of an introductory text and the following papers:
I IRMPDSpectroscopy of Homo- andHeterochiral Asparagine Proton-Bound
Dimers in the Gas Phase
Åke Andersson, Mathias Poline, Kas J. Houthuijs, Rianne E. van Outersterp,
Giel Berden, Jos Oomens, and Vitali Zhaunerchyk
Published in: The Journal of Physical Chemistry A, 125, pp. 7449–7456, 2021,
doi: https://doi.org/10.1021/acs.jpca.1c05667.
My contributions: Partaking the in experiment, experimental data analysis, some
theoretical analysis, and writing the whole manuscript.
II Indication of 310-Helix Structure in Gas-Phase Neutral Pentaalanine
Åke Andersson, Vasyl Yatsyna, Mathieu Linares, Anouk Rijs, and
Vitali Zhaunerchyk
Published in: The Journal of Physical Chemistry A, 127, pp. 938–945, 2023,
doi: https://doi.org/10.1021/acs.jpca.2c07863.
My contributions: Experimental analysis, theoretical analysis, and writing the
whole manuscript.
III IR Spectroscopic Studies of Gas-Phase Peptides
Åke Andersson, and Vitali Zhaunerchyk
Submitted to publisher in 2024 as a review chapter.
My contributions: Literature search and writing the majority of the manuscript.
IV IR-UV Ion-Dip Spectroscopy of Capped Phenylated Polyalanines in the
Gas Phase
Åke Andersson, Piero Ferrari, Imre Bakó, and Vitali Zhaunerchyk
In manuscript.
My contributions: Project planning, partaking in the experiment, experimental
analysis, theoretical analysis, and writing the whole manuscript.
vi
Additional papers
Additionally, I have contributed to the following papers:
V Structure of Proton-Bound Methionine and Tryptophan Dimers in the Gas
Phase Investigatedwith IRMPDSpectroscopy andQuantumChemical Cal-
culations
Åke Andersson, Mathias Poline, Meena Kodambattil, Oleksii Rebrov, Estelle Loire,
Philippe Maître, and Vitali Zhaunerchyk
Published in: The Journal of Physical Chemistry A, 124, pp. 2408–2415, 2020,
doi: https://doi.org/10.1021/acs.jpca.9b11811.
My contributions: Partaking in the experiment, experimental data analysis, and
writing the whole manuscript.
VI Single-Photon Hot-Electron Ionization of C70
Åke Andersson, Luca Schio, Robert Richter, Michele Alagia, Stefano Stranges,
Piero Ferrari, Klavs Hansen, and Vitali Zhaunerchyk
Published in: Physical Review A, 107, p. 013103, 2023,
doi: https://doi.org/10.1103/PhysRevA.107.013103.
My contributions: Partaking in the experiment and experimental data analysis.
VII Comment on “CumulantMapping As the Basis ofMulti-Dimensional Spec-
trometry” by Leszek J. Frasinski, Phys. Chem. Chem. Phys., 2022, 24,
20776–20787
Åke Andersson
Published in: Phys. Chem. Chem. Phys., 25, pp. 32723–32725, 2023,
doi: https://doi.org/10.1039/D3CP02525J.
My contributions: All of it.
VIII Parametric Cumulant Mapping: A Multidimensional Correlation Method
Intended for Experiments with Fluctuating Event Rates
Anna Brandt, Åke Andersson, and Vitali Zhaunerchyk
Submitted to Journal of Physics B.
My contributions: Development of the theory.
vii
Contents
1 Foundation 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Biomolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Introduction 9
2.1 The Protein Folding Problem . . . . . . . . . . . . . . . . . . . . . . 9
2.2 IR Spectroscopy in the Gas Phase . . . . . . . . . . . . . . . . . . . 10
2.3 Action Spectroscopy Techniques . . . . . . . . . . . . . . . . . . . . 11
2.4 Prediction of Vibrational Spectra . . . . . . . . . . . . . . . . . . . 13
3 Experimental Methods 15
3.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Spectroscopy Techniques . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 IRMPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.2 REMPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.3 IR–UV Ion dip . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Setup Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.1 Molecular Sources . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.2 Tunable Light Sources . . . . . . . . . . . . . . . . . . . . . 22
3.3.3 Mass Spectrometers . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Local GU Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.1 Source Chamber . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.2 Interaction Cross . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.3 Mass Spectrometer . . . . . . . . . . . . . . . . . . . . . . . 31
viii
4 Computational Methods 33
4.1 Goal and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Molecular Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Conformer Searching . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.1 Conformer Distance . . . . . . . . . . . . . . . . . . . . . . 35
4.3.2 Energy-Blind Sampling . . . . . . . . . . . . . . . . . . . . . 35
4.3.3 Energy-Guided Sampling . . . . . . . . . . . . . . . . . . . 36
4.3.4 Dynamical Sampling . . . . . . . . . . . . . . . . . . . . . . 37
4.4 Energy Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.4.1 Wavefunction Approximations . . . . . . . . . . . . . . . . 38
4.4.2 Density Functionals . . . . . . . . . . . . . . . . . . . . . . . 40
4.4.3 Empirical Methods . . . . . . . . . . . . . . . . . . . . . . . 41
4.5 Frequency Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5.1 Harmonic Approximations . . . . . . . . . . . . . . . . . . . 42
4.5.2 Anharmonic Perturbation . . . . . . . . . . . . . . . . . . . 43
4.5.3 Semiclassical Dynamics . . . . . . . . . . . . . . . . . . . . . 45
4.6 Hydrogen Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5 Results and Discussion 49
5.1 IRMPD Spectroscopy of Proton-Bound Asparagine Dimers . . . . 49
5.2 IRMPD–VUV Spectroscopy of Neutral Pentaalanine . . . . . . . . 54
5.3 IR–UV Ion Dip Spectroscopy of Phenylated Polyalanines . . . . . 60
6 Conclusions and Outlook 67
Acknowledgements 71
Bibliography 73
ix

1
Foundation
This chapter is meant to give an accessible explanation of the field and its basic
concepts. Terms written in italics are defined in that sentence.
1.1 Motivation
Proteins are versatile macromolecules, essential to all known forms of life. They
are structured like long chains, consisting of linked amino acids. A protein is char-
acterized by its sequence of constituent amino acids, which remarkably determines
its folded shape and biological function. Understanding these relationships is a
great unsolved problem in computational biology.
Physical simulations of proteins could solve the folding problem. Although an
adequate physical theory at the atomic level have been known for a century, its
colossal complexity renders a proper simulation of protein folding computation-
ally intractable. Consequently, physicists have needed to come up with simpler
theories, sacrificing accuracy while chipping off complexity. At the extreme of
simplicity lie parameterized models which do not model electrons directly, and
must therefore be calibrated to account for their effects.
Parameterized models are typically calibrated to large sets of results of complex
computations or experiments on isolated molecules. The final calibrated model is
simple, but only valid on molecules similar to those in the calibration set, and lim-
ited in accuracy by the original results. Thus the desire to simulate proteins implies
a demand for accurate experiments on isolated molecules similar to protein.
What kind of experiment should be performed to satisfy this demand? The title
of this work, infrared spectroscopy of small biomolecules in the gas phase, is my
answer to this. I will explain why by analyzing its parts.
1
Chapter 1. Foundation
Small biomolecules refers to the objects of study in this work: proton-bound or
peptide-linked oligomers of amino acids. They are subsets of proteins and there-
fore representative in every aspect except for size. Being considerably smaller,
they are much easier to study, both experimentally and theoretically.
Spectroscopy is the study of interaction between matter and light. In this work,
the matter is a molecule and the light is infrared or ultraviolet. Infrared light in
particular is used to reveal the vibrational frequencies of the molecule, which re-
lates to its structure and energy landscape, which is useful in model calibration.
Ultraviolet light is also used, to ionize the molecule when desired.
Gas-phase spectroscopy studies the molecule in the gas phase, in the absence
of environmental effects. This makes the experiment directly comparable with
theoretical predictions, which almost always neglect the environment for reasons
of complexity. Delivering large molecules to the gas phase is an art that requires
both the low-pressure environment of a vacuum chamber and some trick.
To summarize, gas-phase spectroscopy of small biomolecules gives us accurate
experimental results of molecules in the absence of environmental effects. These
results can then be used to calibrate parameterized models for similar molecules.
And those models can then be applied to simulate the folding of proteins and other
biological processes at the atomic level.
1.2 Biomolecules
Amino acids are the building blocks of the biomolecules
considered in this work. As the name hints at, they are car-
boxylic acids—molecules with a carboxyl (COOH) group—
that also contain an amino (NH2) group. Additionally, ev-
ery amino acid species has a characteristic side chain (R)
which determines its properties. A total of 22 species occur
naturally. In each of those the three groups meet at a single
point, the 𝛼-carbon, as depicted in Fig. 1.1. This makes the Figure 1.1: A natural
molecule chiral (unless R = H), meaning that it is distinct amino acid. The pink
ball R is its side chain.
from its mirror image. The two possible configurations are
called enantiomers l and d. Only l occurs naturally.
The carboxyl group of one amino acid can react with the amino group of another
to form a peptide link, binding the two together. Thus every amino acid can bond
with two neighbors, forming a peptide-linked chain. If such a chain contains more
than a few dozen amino acids, it is called a protein, otherwise it is only a peptide. A
peptide or protein is characterized by its sequence of linked amino acids, known as
the primary structure. Figure 1.2 shows how a peptide is formed from amino acids.
2
1.2. Biomolecules
+ + → +
Figure 1.2: Reaction scheme of peptide formation. Three amino acids condense into one
tripeptide by forming peptide links between the carboxyl and amino groups.
propanol (gauche) propanol (trans) isopropanol methoxyethane
Figure 1.3: Related propanol molecules. All molecules above have the same formula
C3H8O, and are therefore structural isomers. Furthermore, the two molecules on the left
differ by only rotations along single bonds, and are thus conformers.
The three-dimensional structures of a molecule, called conformations, are gen-
erally not stable over time. At room temperature, single-bonds often are free to
rotate, changing the structure. This is called conformational isomerism, and stable
structures that differ only by rotations about single bonds are called conformers.
More distantly related are structural isomers, often just called isomers, which have
the same chemical formula but district bond structures. Figure 1.3 shows conform-
ers and isomers of propanol.
The central -[N-C-C]- sequence of a protein is referred to as its backbone. De-
spite containing only single bonds, the backbone structure is relatively stable. This
stability arises from intramolecular interactions, which constrain the molecules
and prevent the rotations about single bonds. Examples of intramolecular inter-
actions include hydrogen bonding, steric repulsion, and hydrophobic attraction.
These interactions stabilize the protein into rigid patterns known as secondary
structures. Figure 1.4 shows the two most common secondary structures: the α-
helix and the β-sheet.
Beyond secondary structures there are also tertiary and quaternary, although
they are out of scope for this thesis. The tertiary structure comprises the full three-
dimensional structure of the protein, stabilized by hydrophobic interactions and
hydrogen bonds of topologically distant groups. Multiple proteins may combine
and stabilize into more complex enzymes. Such a combination is called a quater-
nary structure.
3
Chapter 1. Foundation
Figure 1.4: Example of secondary structures. Left: an α-helix. Right: an antiparallel β-
sheet. Shaded green strips have been added to show the curving of the backbone.
+ + → +
Figure 1.5: Protolysis scheme of dimer formation. The hydronium cation transfers its sur-
plus proton to an amino acid, enabling it to form a hydrogen bond to the other amino acid.
Amino acids and peptides can be protonated, meaning they contain one extra
proton, which usually sits on the amino group. Such protonated amino groups
tend to participate in strong hydrogen bonds because of the surplus charge. In
the gas phase, protonated and neutral amino acids can form clusters stabilized by
such bonds. The simplest example is the proton-bound dimer, shown in Fig. 1.5,
which consists of one protonated and one neutral amino acid. While not naturally
occurring, it is interesting as a model system for hydrogen bonding.
Some spectroscopy schemes that will be introduced later require the molecule
to resonantly absorb light in a particular wavelength range (ultraviolet, more on
this later) at which biomolecules seldom do. The solution is then to add a chro-
mophore, a functional group which narrowly absorbs the desired wavelength. The
most popular choice in this field is the phenyl group, which naturally occurs in the
amino acid phenylalanine. With such an absorption feature, it will be possible to
perform spectroscopy on and deduce the three-dimensional structure of a specific
conformer.
1.3 Spectroscopy
Spectroscopy is broadly defined as the study of interaction between matter and
light. There are three major types of it: emission, scattering and absorption spec-
troscopy; corresponding to the creation, reflection and destruction of photons,
which carry energy proportional to their frequency.
4
1.3. Spectroscopy
Light Una ected matter
Abs. A ected matter
Matter Transmitted light
Figure 1.6: Abstract diagram of general absorption spectroscopy. Light and matter inter-
act, resulting in some matter absorbing light and becoming affected. In an experiment, at
least three of the five quantities above are measured to deduce the absorption rate.
This work is confined within absorption spectroscopy, where experiments typi-
cally study the absorption rate as a function of wavelength by colliding molecules
with photons, see Fig. 1.6. In such experiments, three signal amounts are observed:
relaxed matter, excited matter, and transmitted light. In the context of gas-phase
spectroscopy of biomolecules, the rate of absorption is relatively low, which makes
the amount of transmitted light an unreliable variable. Instead, the amounts of re-
laxed and excited matter is measured. This is called action spectroscopy because it
measures the action of light unto matter.
We are used to thinking of light frequency (or wavelength) as a scale from red
to violet, but that is just the visible part. Light is really electromagnetic radiation
which can have any frequency. Examples include (Fig. 1.7) infrared (IR) light at
60THz (2000 cm−1), visible orange light at 500THz (600nm), and ultraviolet (UV)
light at 1200THz (250nm). In this thesis, a wide array of light frequencies will be
used, from 10.5THz (350 cm−1) to 2540THz (118nm).
The way light interacts with matter depends on its frequency. Infrared light
is resonant with the vibrational modes in molecules. In particular, the mid-IR
range of 24–120THz (800–4000 cm−1) contains resonances with local vibrations
  800 cm -1   4000 cm -1   700 nm  400 nm   200 nm
Far-IR Mid-IR Near-IR Vis. UV VUV
10 THz 30 THz 100 THz 300 THz 1000 THz 3000 THz
Figure 1.7: Light frequency regions, from far-IR to vacuum UV.
5
Chapter 1. Foundation
involving a few nuclei, while the far-IR range of 0.3–24THz (10–800 cm−1) con-
tains the frequencies of molecule-wide vibrational modes. The set of resonances in
the range of 100–1400 cm−1 are virtually unique for a given conformer, and there-
fore referred to as its molecular fingerprint. By performing spectroscopy in these
IR frequency ranges and cross-validating with theoretical analyses, the molecular
geometry can be inferred.
In contrast to IR, UV photons carry enough energy to excite the electronic state
of the molecule, or even ionize it by knocking off an electron. For example, the
previously mentioned phenyl group can be electronically excited by an UV photon
of frequency 1122THz (267.3nm), and then ionized by another. Higher energy
UV photons (above 1500THz or equivalently below 200nm) can directly ionize
molecules by themselves, and must therefore be transported in vacuum. Hence,
such UV photons are also called vacuum UV (VUV) photons.
By combining IR and UV photons, even more information can be extracted.
Most notably, conformer-specific spectroscopy becomes possible when a chromo-
phore is used. The end result is then not just a single molecular geometry, but
rather one molecular geometry per conformer. Descriptions of different spec-
troscopy schemes including this one are given in Section 3.2.
1.4 Computation
In order to infer the structure from a vibrational spectrum, theoretical predictions
for each plausible structure must be computed. Such plausible structures are called
stable conformers, and finding them is a difficult task. When they are found, their
predicted stability and vibrational spectrum can be calculated. This is also a hard
task, but it has established solutions. Finally, the experimental spectrum can be
compared to predictions, in order to confirm a structure.
Finding stable conformers is difficult, because the space of candidate conformers
has a high dimension and cannot be exhaustively covered. Instead, strategies for
conformer searching rely on randomness and adaptation. Even then, assessing the
completeness of a search is as hard as the searching problem itself. Some strategies
for finding conformer are described in Section 4.3, along with a discussion of their
strengths and weaknesses.
The probability 𝑝 of a molecule adopting a certain conformer at a given tempera-
ture and pressure is directly related to its Gibbs energy 𝐺, through the Boltzmann
equation 𝑝 ∝ exp(−𝐺/𝑘B𝑇 ). Accurately determining the energy (Gibbs or any
other) of a molecule is a hard but well-studied problem, that requires a solution
based on quantum mechanics. Section 4.4 discusses this in detail, but a softer in-
troduction is given below:
6
1.4. Computation
Quantum mechanics provides a fundamental framework for understanding the
behavior of atoms and molecules. It discards the classical mechanics concepts
of precise position and velocity, replacing them with a wavefunction that assigns
a probability to each possible state of the physical system. This wavefunction
depends on the coordinates of all particles, resulting in an exponential compu-
tational complexity scaling with system size. Therefore exact solutions are in-
tractable for practical purposes, and approximate methods are necessary to study
actual molecules.
To reduce the complexity of a molecule, the Born–Oppenheimer approximation
is always used. It relies on the fact that electrons have a much smaller mass than
the nuclei, and thus “see” them as fixed when solving the Schrödinger equation.
Together with the adiabatic approximation, which assumes that electronic states
adjust instantaneously to nuclear motion, the Schrödinger equation for the nu-
clei can be solved separately. In some contexts the semi-classical approximation is
made, meaning that the nuclei are treated classically.
Even with the above approximations made, the complexity of the wavefunction
is an exponential function of the number of electrons. Density functional theory
(DFT) handles this problem by replacing the wavefunction with the electron den-
sity, whose complexity does not inherently scale with the number of electrons.
This is possible because the ground state properties of a system depends in princi-
ple only on the electron density, as guaranteed by the Hohenberg–Kohn theorem.
In practice, determining properties from only the electron density is difficult and
requires approximations.
Practical implementations of DFT are quite technical and beyond the scope of
this thesis. What’s worth knowing is that DFT functionals, methods for determin-
ing the energy from the electron density, can be placed on a ladder of increasing
complexity and accuracy. At the bottom is the local density approximation, in
which the energy of each small volume element is obtained by comparison with
a homogeneous electron gas of the same density. Above this level lie generalized
gradient approximations, which consider not only the local density, but also its gra-
dient. Higher up still are hybrid methods, which additionally considers nonlocal
effects.
7

2
Introduction
This chapter introduces the field of gas-phase IR spectroscopy, starting from its
connection to protein folding.
2.1 The Protein Folding Problem
Proteins are essential molecules present in all forms of life. They are long polymers
of amino acids, folded into particular shapes determined by the primary struc-
ture. The shape of a protein then determines its biological function. For example,
the protein in Fig. 2.1 has folded into a restriction enzyme that cleaves DNA at a
specific site. Understanding how primary structure, protein shape, and biological
function relate is known as the protein folding problem, and is still an open area
of research. [9]
Attempts to solve the protein folding problem by accurately simulating the dy-
namics of folding are hindered by complexity. The laws of quantum physics are not
in question, but simply intractable for molecules as large as proteins. It has there-
Figure 2.1: The restriction endonuclease enzyme
EcoRI, which consists of two identical folded proteins.
Its function is to cleave DNA (red) at specific sites, and
it can do this because of its particular shape, which is
stabilized by the relatively stiff secondary structures:
α-helices (magenta) and β-strands (cyan).
9
Chapter 2. Introduction
fore been necessary develop simpler molecular force field models such as AM1,
[10, 11] MM3, [12] and more recently PM6. [13, 14] Over time, computational costs
have decreased exponentially, enabling simulation of folding of proteins. [15, 16]
Simplified models of molecular forces have many free parameters which need
to be set, typically by comparison with experimental data. It is also possible to fit
the parameters to computed properties of a trusted method, but such a method
is trusted because of its good agreement with experiments. In the end, accurate
empirical data on similar molecules are required to develop the methods.
Separate from physical simulations of proteins, there are machine learning so-
lutions to the protein folding problem, notably AlphaFold. [17–19] Such methods
apply statistical reasoning to determine the final protein shape, and thus also re-
quire empirical data.
2.2 IR Spectroscopy in the Gas Phase
Spectroscopy methods study matter through its interaction with electromagnetic
radiation. IR spectroscopy in particular uses frequency-tunable IR light, which
covers the frequency range of molecular vibrations. The vibrational frequencies
of a molecule are sensitive to the molecular structure, meaning that the latter can
be inferred from the former when combined with theoretical predictions. Thus,
IR spectroscopy is applicable to satisfy the need for empirical data on protein-like
molecules. [20]
The IR spectrum of a biomolecule consists of resonances corresponding to vibra-
tional modes, which lie in the frequency range of 10–4000 cm−1. Figure 2.2 shows
this frequency range and the amide modes, which are local vibrational modes,
Figure 2.2: IR frequency region with the locations of amide vibrational modes.
10
2.3. Action Spectroscopy Techniques
common to all peptides. As a general rule, modes involving few nuclei have high
frequencies, such as the amide A stretching mode. Conversely, modes at low fre-
quencies (less than say, 1400 cm−1) involve many nuclei. This makes them sensitive
functions of the molecular structure, and they are therefore called molecular fin-
gerprints. [21] Going further, the far-IR range corresponds to completely delocal-
ized vibrational modes, which are sensitive enough to tell the specific conformer.
[22, 23]
Proteins are naturally found in aqueous solution, where they interact with the
environment. While there are IR spectroscopy techniques that study proteins un-
der natural conditions, [24, 25] the properties of such protein depend on the envi-
ronment is a way that requires advanced modeling. [26, 27]
In contrast to the above, gas-phase IR spectroscopy studies molecules in a iso-
lated environment, and therefore give insights into intrinsic molecular properties.
[21, 28] This is much closer to electronic structure simulations, which are done
in vacuum by default. However, most biomolecules are too fragile to simply be
boiled into the gas phase. Instead, specific soft evaporation techniques such as
electrospray ionization [29] or laser desorption [30] are used. The result is that the
biomolecules are delivered to the low-pressure gas phase.
An arguable downside to gas-phase spectroscopy is that it does not include ef-
fects from the environment. However, this is not completely true. To understand
the effects of the solvent, it is possible to progressively add solvent molecules to
the object of study, a practice known as microsolvation. [31, 32]
2.3 Action Spectroscopy Techniques
Because biomolecules are delivered to the gas phase at a low pressure and density,
some spectroscopy techniques are not available. The low density makes the total
absorption cross-section quite small, meaning most photons will not interact with
any molecule. This makes transmission spectroscopy unfeasible, as it depends on
the ratio of transmitted photons, which cannot be measured with good signal-to-
noise. Instead, action spectroscopy must be applied, which measures the ratio of
molecules that experience some action caused by photon absorption. This thesis
will demonstrate four different action spectroscopy techniques, and discuss their
strengths and weaknesses.
The first technique is IR multiple-photon dissociation (IRMPD) spectroscopy
of trapped ions, which has been widely employed to study biomolecules. [1, 5,
33–38] In IRMPD, the detectable action is the dissociation of trapped parent ions
into fragments, triggered by the absorption of multiple IR photons. Due to charge
conservation, only one fragment inherits the charge of the parent ion. Thus the
action is detectable in the ion mass spectrum, both as an decrease in the parent
11
Chapter 2. Introduction
count, and increase in total fragment count. Combining the two gives an accurate
measure of the fragmentation rate, which is closely related to the IR absorption
rate. [39] However, it is known that the IRMPD spectrum can be slightly shifted
compared to single-photon IR absorption. [40]
A limitation of the IRMPD studies mentioned so far is the requirement of an ion
trap, and in turn the charged state of the studied molecule. It was recently demon-
strated that IRMPD can be performed in a molecular beam of neutral molecules
by using the intense and long-lasting IR pulses of a free electron laser. [41] To
distinguish and count the parent and fragment neutrals, they are ionized with a
VUV laser pulse and accelerated by an electric field into a mass spectrometer.
The method is therefore known as the IRMPD–VUV method, and has since been
applied to neutral peptides. [42, 43]
Because the IRMPD–VUV method does not use a trap, and therefore fragments
molecules with a single pulse, it should in theory be applicable only to molecules
up to a maximal size. One of the papers in this thesis applies the method to the
largest molecule yet: the pentaalanine peptide. [2]
Conformer-Sensitive Techniques Some biomolecules adopt several conform-
ers, each with a distinct IR spectrum. An experiment using the IRMPD techniques
above can only obtain a weighted sum of those IR spectra, which complicates the
comparison with predictions. One way to perform conformer-specific IR spec-
troscopy builds on resonance-enhanced multi-photon ionization (REMPI) using
UV photons. [21]
REMPI spectroscopy counts the molecules ionized by a tunable UV laser. It
is not possible with all biomolecules, but rather requires the presence of a UV-
absorbent functional group. The group, called a chromophore, has an excited
electronic state with a long lifetime. such electronic states are conformer-sensitive,
meaning that the REMPI spectrum a biomolecule consists of resolved resonances
belonging to its populated conformers.
Potential options for the chromophore include aromatic rings, with the simplest
example being the phenyl group, a simple six-carbon ring structure. Three of the
amino acids naturally found in proteins contain an aromatic ring: phenylalanine,
[44] tyrosine, [45] and tryptophan. [46] It is also possible to put a phenyl-based cap
on the end of a peptide. [47]
Conformer-specific IR spectroscopy can be achieved by adding an IR laser to
the REMPI experiment. The absorption of an IR photon will then reduce the
effectiveness of REMPI, causing the ion signal to dip. This technique is therefore
called double-resonance IR–UV ion dip spectroscopy, [48] but also resonant ion-
dip IR (RIDIR) spectroscopy. [49] Such studies are popular because they result
in one IR spectrum per conformer, which can then be compared with predictions
12
2.4. Prediction of Vibrational Spectra
and validate not only the spectra, but also the relative abundances. [44–47, 50, 51]
2.4 Prediction of Vibrational Spectra
The obtained IR spectrum from an experiment is not by itself reversible to a molec-
ular structure (conformer), although it does indicate the presence or absence of
important functional groups. Instead, the experimental spectrum must be com-
pared to predicted spectra of likely conformers. Generating the conformers and
predicting their spectra are two separate tasks.
Generating the conformers amounts to choosing values for the torsional angles
in a molecule whose bond structure is already known. The space of conformers
is therefore high-dimensional and impossible to fully explore. Instead, a sparse
sampling strategy must be employed. There are a few algorithms for this pur-
pose, based on independently randomized angles, [52] randomized internucleic
distances, [53] mutation of known conformers, [54] and molecular dynamics simu-
lations. [55] The output of such an algorithm is a long list of conformers, of which
a subset the vibrational spectra are computed. It is hard to gauge whether suffi-
ciently many conformers have been generated, but agreement between a predicted
spectrum and experiment is considered proof of correctness.
Given a specific conformer, its energy can be computed using density func-
tional theory (DFT), which is well established. [56] This is used to rule out high-
energy conformers, but also relates to the vibrational spectrum. By considering
the derivatives of energy with respect to the positions of the nuclei, it is possible to
predict the vibrational spectrum. [57] Such predictions are decently accurate, and
better if higher order derivatives are considered. [58, 59]
An alternative method for vibrational spectra prediction is the Born–Oppen-
heimer molecular dynamics (BOMD) method. [60] Rather than considering a fix
geometry, it is based on molecular dynamics simulations, thereby exploring the
surrounding part of the conformational space. Although it has correctly predicted
the spectra of some peptides, [61] the BOMD method has some fundamental flaws
that will be discussed. In particular, the anharmonic effects and temperature de-
pendence will be problematized.
13

3
Experimental Methods
This chapter is meant to explain how action spectroscopy is performed in practice,
down to every significant component.
3.1 Objective
The objective of every experiment in this thesis is to obtain some spectrum of a
molecular species in the gas phase. At the very minimum, an experimental setup
for action spectroscopy must contain three components: a source that produces
gas-phase molecules, a wavelength-tunable laser that produces photons to collide
with said molecules, and a detector that counts the molecules afterwards in order
to determine the absorption rate. All of these are placed in a vacuum chamber
which require pumps for operation. The exact realization of these components
depends on the spectroscopy technique.
The relevant spectroscopy techniques will be listed in the next section, and the
realization of these components will be described in the subsections of Section 3.3.
3.2 Spectroscopy Techniques
There are a few spectroscopy techniques used in this thesis, differentiated by what
kind of photons the target molecules absorb.
3.2.1 IRMPD
Infrared multi-photon dissociation (IRMPD) is an action spectroscopy technique
in which the molecule sequentially absorbs IR photons until it dissociates. [21] It
15
Chapter 3. Experimental Methods
Figure 3.1: Schematic representation of IRMPD. Upon absorbing a photon, the resonant
vibrational mode becomes excited and unable to absorb from the monochromatic laser
beam. After some energy redistribution, the molecule can absorb again. Eventually, the
molecule dissociates.
is accomplished by exposing the molecules to a sufficiently strong IR laser beam.
The ratio of fragmented to parent molecules can then be used to estimate the ab-
sorption rate.
To understand the rate of IRMPD, one must take anharmonic effects into ac-
count. In a harmonic oscillator the vibrational energy levels would be equidistant,
but in reality the spacing decreases as the mode energy increases. The consequence
for spectroscopy is that after a vibrational mode absorbs a photon, its resonance
frequency decreases and no longer aligns with the laser frequency, making fur-
ther absorption unlikely. However, another anharmonic effect allows the excited
mode to relax by transferring its energy to other modes, a phenomenon known
as intramolecular vibrational energy redistribution (IVR). [62] After IVR has oc-
curred, which takes about 100 fs, [63] the molecule can absorb once again. After
a few (typically 5–10) cycles of absorption and IVR, the molecule dissociates be-
cause the added energy is sufficient to break a bond. Figure 3.1 gives a schematic
representation of the entire IRMPD process.
Because IRMPD involves relatively slow IVR cycles, the molecules must be ex-
posed to IR radiation for a sufficiently long time. This can be accomplished in two
ways: with a long pulse or with an ion trap. Free electron laser macropulses are
about 10µs long, which is sufficient. Electromagnetic traps such as Penning traps
can be used to hold ions in place, allowing multiple pulses with shorter duration
to cause IRMPD.
A common experimental setup for IRMPD spectroscopy is to have an electro-
spray ionization source pointing into a Penning trap. [39] The trapped ions can
then be exposed to a laser beam long enough to cause IRMPD, causing some ions
to fragment. The ratio of fragmented ions can then be inferred from the radiation.
More details are given in the relevant subsections of Section 3.3.
16
3.2. Spectroscopy Techniques
As a rule of thumb, spectroscopic events involving 𝑛 photons scale with the laser
pulse energy 𝐸 like 𝐸𝑛. This has been shown not to be the case for IRMPD; the
fragment fluence is zero when the pulse energy is below a threshold and increases
nonlinearly beyond it. [39] An explanation for this behavior is that the IRMPD
rate is limited by the first absorption. The nonlinear scaling makes normalization
with respect to pulse energy a challenge.
For a fixed pulse energy, the IRMPD rate 𝐴 at some frequency 𝜈 can be calcu-
lated from the amounts of fragmented and parent ions. If there are 𝑁0 parent ions
initially, and during the pulse each remaining parent ion has a constant chance of
fragmentation, then 𝑁pa = 𝑁0 exp(−𝐴𝑡) parent ions will remain after time 𝑡, and
𝑁fr = 𝑁0(1−exp(−𝐴𝑡)) fragment ions will have been created. Solving for 𝐴 gives
the IRMPD rate
𝐴(𝜈) = 1 (1 + 𝑁ln fr(𝜈)𝑡 𝑁 ). (3.1)pa(𝜈)
This formula becomes unstable when 𝑁pa is small, which physically means that the
laser pulse is strong enough to fragment most ions. The solution is then to lower
the pulse energy.
Figure 3.2 shows how the pulse energy affects the IRMPD spectrum of the pro-
tonated methionine dimer Met2H
+. The highly IR-active modes around 1400 and
1750 cm−1 involve bending and stretching of the carboxyl functional group, respec-
tively. Low pulse energy (about 5mJ) is required to see the features; a higher pulse
energy would saturate and broaden the signal. On the other hand, the less IR-
active modes below 1000 cm−1 are best seen with high pulse energy (about 20mJ).
At 1070 cm−1 both effects are seen: the shoulder peak is the most visible at mid
pulse energy (about 12mJ) because at higher energies it becomes subsumed by the
larger peak at 1150 cm−1.
Met2H
+
High
Mid
Low
600 800 1000 1200 1400 1600 1800
Frequency (cm!1)
Figure 3.2: IRMPD spectrum of the protonated methionine dimer, taken from Ref. [5].
17
IRMPD rate
Chapter 3. Experimental Methods
IRMPD-VUV Neutral molecules are not as easily trappable as ions because they
do not respond to electromagnetic fields, but it is still possible to perform IRMPD
spectroscopy on a neutral molecular beam. [41–43] The lack of a trap places a
requirement on pulse duration, which is satisfied by free electron lasers. In order
to separately count the fragments and parent molecules, they are ionized by high
energy so called vacuum ultraviolet (VUV) photons and accelerated into a time-
of-flight mass spectrometer. This setup is called IRMPD-VUV spectroscopy for
that reason.
Although the intended purpose of the VUV pulse is to ionize the fragments and
parent molecules, there is a side effect of additional fragmentation. This is proven
by the fact that fragments are detected when only the VUV pulse is applied. One
way to compensate for this effect is to subtract the VUV-only signal from the main.
Using subscripts to denote whether the IR laser is on or off, the IRMPD rate can
be estimated as
𝐴(𝜈) = (1 + 𝑁fr,on(𝜈) 𝑁ln fr,off𝑁pa,on(𝜈)
) − ln(1 + 𝑁 ). (3.2)pa,off
The derivation of this formula is given in Ref. [2]. It relies on some assumptions
whose validity needs further exploration, notably that the ionization probability of
the parent molecule is equal to the sum of the ionization probabilities of its frag-
ments, averaged over fragmentation patterns. It is tempting to think that because
the right term in Eq. (3.2) does not depend on the IR frequency 𝜈, it only needs
to be measured once. However, due to the fact that the amount of initial parent
molecules is difficult to control, the IR pulses are fired with every one out of two
VUV pulses.
3.2.2 REMPI
Resonance-enhanced multi-photon ionization (REMPI) is an action spectroscopy
technique which selectively ionizes molecules using UV photons. In the context
of this thesis, two photons of the same frequency are used to ionize the molecule.
The rate of ionization is greatly increased when the photon energy is resonant with
an intermediate electronic state. Thus the electronic structure of a molecule can
be probed by a UV laser with tunable frequency.
Molecules containing chromophores such as a phenyl group have an intermedi-
ate state with long lifetime, corresponding to a quality factor of more than 10 000.
[4] This makes REMPI very efficient and selective; it is possible to distinguish
not only isomers but also conformers from their REMPI spectra. [64] This useful
property, shown in Fig. 3.3, enables conformer-specific spectroscopy, something
that will be the focus of the next subsection.
18
3.2. Spectroscopy Techniques
Conf. A Conf. B
+
0 - 0 0 - 0
* Conf. A
1 - 0
0
UV Frequency
Figure 3.3: Schematic representation of REMPI spectroscopy. REMPI occurs when the
photon energy is resonant with an intermediate electronic state. Such states are sharp
enough to resolve conformers and vibrational quanta.
The REMPI rate as a function of pulse energy is often approximated as linear
function, despite being a two-photon process. Consider a three-state model where
at time 𝑡 there are many molecules in the ground state, 𝑒(𝑡) in the excited interme-
diate state, and 𝑖(𝑡) in the ionized state. Molecules in the ground state are exited
with rate 𝐴, and excited molecules are ionized with rate 𝐵𝑒 and relax with rate 𝐶𝑒.
This gives the equations
d
𝑡𝑒 = 𝐴 − 𝐵𝑒 − 𝐶𝑒;
d
𝑡𝑖 = 𝐵𝑒 and 𝑒(0) = 𝑖(0) = 0. (3.3)d d
Assuming 𝐴, 𝐵, and 𝐶 are proportional to the pulse power, then the ion yield is a
function of the pulse energy 𝐸:
𝑎𝑏
IonYield(𝐸) = −(𝑏+𝑐)𝐸(𝑏 + 𝑐)2 (𝑒 + (𝑏 + 𝑐)𝐸 − 1), (3.4)
where 𝑎, 𝑏, and 𝑐 are proportionality constants. This function is quadratic at low
energies ((𝑏 + 𝑐)𝐸 ≪ 1) but becomes asymptotically linear at high energies ((𝑏 +
𝑐)𝐸 ≫ 1). Figure 3.4 shows empirical proof of this nonlinear behavior.
AAFA data
Model fit
0 1 2 3 4 5 6 7 8 9 10
Pulse energy (mJ)
Figure 3.4: REMPI signal as a function of pulse energy, measured on phenylated
capped tetraalanine (AAFA). The relation is approximately linear but better described
by Eq. (3.4).
19
Ion signal
Ion signal
Chapter 3. Experimental Methods
One flaw with Eq. (3.4) is that it does not handle saturation correctly. It shows
linear behavior in the high-energy limit, but this is absurd as no more than 100%
of molecules can be ionized. A solution is to modify the derivation of the model by
adding another variable 𝑔 for the number of ground state molecules, and replace
𝐴 with 𝐴𝑔.
3.2.3 IR–UV Ion dip
IR–UV ion dip is an action spectroscopy technique based on REMPI that allows
the IR spectra of specific conformers to be measured. It exploits the fact that
REMPI is selective enough to only ionize a specific conformer from its ground
state. By exposing the molecule to IR photons, the ground state is depopulated,
causing the ion signal to dip. By measuring and comparing the ion signal with
and without IR light, the IR absorption can be calculated. Figure 3.5 shows an
overview of the technique.
A common setup for ion dip experiments involves a neutral molecular beam,
tunable IR and UV lasers, and some ion detector. Before the main experiment,
the UV range of the chromophore is explored. For the phenyl group, this range
is typically within 37 400–37 700 cm−1. [4, 65] This reveals REMPI resonances of
the molecules, each belonging to a specific conformer. In the actual ion dip exper-
iment, the UV frequency is kept fixed at a REMPI resonance while the IR laser
is scanned in a wide range. The result is an single-photon IR absorption spectrum
for each resonance.
A simple quantitative model for IR–UV ion dip spectroscopy exists, and its
derivation will be given here. Initially there are 𝑁0 molecules in the ground state.
If a molecule is exposed to an IR pulse of energy 𝐸, it will remain in the ground
state with probability exp(−𝑎(𝜈)𝐸), where 𝑎 is the absorbance that depends on
the IR frequency 𝜈. Then, REMPI will ionize a fraction 𝑟 of the ground state
+
*
IR on
IR off
0
IR Frequency IR Frequency
Figure 3.5: Schematic and data representation of IR–UV ion dip spectroscopy. REMPI
lets two UV photons resonantly ionize a specific conformer in the ground state. Any IR
absorption depopulates the ground state, causing the ion signal to dip. The absorption
spectrum can then be calculated from the dip.
20
Ion signal
Absorption
3.3. Setup Components
molecules. The two measured quantities are the ion signal with IR 𝑁on(𝜈) =
𝑟𝑁0 exp(−𝑎(𝜈)𝐸), and without 𝑁off = 𝑟𝑁0. The absorbance 𝑎 can then be found
as
𝑎(𝜈) = 1 𝑁𝐸 ln
off
𝑁 (𝜈). (3.5)on
Just like with the IRMPD-VUV method, 𝑁off does not depend on the IR fre-
quency 𝜈 and could in principle be measured once. However, due to the reality
that 𝑁0 is difficult to control, 𝑁on and 𝑁off are measured equally often by firing IR
pulses with every one out of two UV pulses.
3.3 Setup Components
This section describes the crucial components used in experimental setups in this
thesis, categorized as molecular sources, tunable light sources, and mass spec-
trometers. It also describes the setup in the local laboratory in Fysikforskarhuset,
Gothenburg.
3.3.1 Molecular Sources
The role of a molecular source is to produce gas-phase molecules for the spectro-
scopic experiment. Biomolecules in particular are fragile and require a soft deliv-
ery to the gas phase. There are a few ways to do this depending on the type of
molecule and the desired charge state.
Electrospray Ionization Electrospray Ionization (ESI) is a soft ionization tech-
nique used to transfer molecules from a liquid solution into the gas phase as pos-
itive (or negative) ions. [29] First, the sample to be analyzed is dissolved in an
acidic (or basic) solution. This solution is then passed through a capillary needle
held at a positive (or negative) electric potential on the order of kV. The nee-
dle points toward a grounded electrode, causing a strong electric field, which pulls
the protonated (or deprotonated) ions to the tip of the needle. As the solution is
pushed through the needle, it ejects charged droplets that quickly evaporate and
fission due to Coulomb repulsion, until only singular molecules and small clusters
remain. The gentle nature of ESI makes it ideal for analyzing biomolecules like
peptides, which are otherwise prone to denaturation.
21
Chapter 3. Experimental Methods
Laser Desorption Laser desorption, meaning firing a brief laser pulse onto the
sample, is a relatively soft evaporation technique that produces mostly neutral but
also ionized molecules. [66] The rapid and localized heating of the pulse vaporizes
the sample, delivering molecules to the gas phase without significant fragmentation
due to the short duration of the pulse. The laser frequency can be in the UV range
[67] or in the IR range. [30] For the experiments in this thesis, laser pulses with
frequency 1064nm and energy on the order of 1mJ are used.
To improve the yield, the sample is often mixed with a compound that absorbs
well at the laser frequency, the technique is then called matrix-assisted laser des-
orption/ionization (MALDI). In the experimental setups in this thesis, carbon
power was used as a matrix compound because of its efficient absorption of 1064nm
light.
Supersonic Jets Although not a source, supersonic jet expansion can be used
to cool gas-phase molecules. The basic setup is a high-pressure reservoir with a
controllable nozzle into a vacuum chamber. [68] When the nozzle opens for a few
100µs, the gas sprays into the chamber at supersonic speeds in the shape of a jet,
causing it to cool.
The modeling of supersonic flows is quite complex and given elsewhere, [69] but
a simple argument for the cooling exists. Because the expansion is rapid, it can be
assumed to be adiabatic, implying that 𝑝𝑉 𝛾 is constant, where 𝛾 is the heat capacity
ratio, equal to 5/3 for monoatomic and 7/5 for diatomic gasses. Combined with
the ideal gas law 𝑝𝑉 = 𝑁𝑘B𝑇 , one finds 𝑇 ∝ 𝑉 1−𝛾, implying that the temperature
decreases as the volume increases.
Gas-phase molecules of study can be cooled using supersonic jet expansion in
at least two ways. The first is direct mixing with the carrier gas in the reservoir,
before the expansion happens. This is only possible if the molecule is naturally in
the gas phase. The second is a combination with MALDI: By timing the nozzle
opening just after the laser pulse, the expansion will blow onto the MALDI fumes
and mix the two gases during expansion.
A supersonic jet expansion is typically combined with a skimmer, which is an
inverse funnel used to select the central part of the jet, which becomes a molecu-
lar beam. Because location and velocity are strongly correlated in a cold jet, the
molecular beam is collimated.
3.3.2 Tunable Light Sources
This subsection will describe the tunable light sources used in this thesis. Free
electron lasers have a flexible frequency down to the far-IR range, and a relatively
large pulse energy. Table top laser systems have superior frequency resolution, but
cannot cover a wide range without reconfiguration.
22
3.3. Setup Components
100 ms ~7 s
1 ns ~1 ps
Figure 3.6: Anatomy of FELIX pulsing. Every 100ms, FELIX fires a train of micropulses.
Infrared Free Electron Lasers The free electron laser (FEL) FELIX at Rad-
boud University, Nijmegen was used for three experiments in this thesis. The de-
sign and theory of FELs are described elsewhere, [70, 71] and only a brief descrip-
tion will be given here, with a focus on the produced radiation.
An FEL generates light from bunches of electrons that oscillate when passing
through a static but periodically alternating magnetic field. The frequency of the
light is a function of the magnetic field strength, and can thus be varied continu-
ously. The linewidth (frequency resolution) is inversely proportional to the num-
ber of periods.
FELIX has a frequency range of 66–3600 cm−1 (until 2022 the upper limit was
2000 cm−1 [72]), and a typical linewidth just under 1%. Its IR pulses are emitted
every 100ms, see Fig. 3.6. Those pulses last for about 7 µm and consist of mi-
cropulses separated by 1ns intervals. The fact that the pulses are relatively long
has positive implications for IRMPD spectroscopy; between micropulses, the vi-
brational energy has time to redistribute itself.
Table Top Lasers Systems The first part of any laser system is the pump laser,
which in all experiments of this thesis has been some Nd:YAG laser, which outputs
pulses of light with wavelength 1064nm (𝜈0 = 9400 cm−1) at a rate of 10 or 20Hz.
In a linear optical component, the frequency of light is preserved. But there are
also nonlinear crystals such as KD2PO4 that enable processes which change the
frequency of light. The simplest example is probably second harmonic generation
(SHG), which implies that two photons of frequency 𝜈 are consumed to produce a
photon of frequency 2𝜈. This process requires a phase-matching condition which
in practice means that the crystal must be precisely angled relative to the laser
beam. A SHG KD2PO4 crystal is often installed in Nd:YAG lasers to allow for
generation of 532nm (2𝜈0 = 18 800 cm−1) light.
Another nonlinear process is sum and differential frequency mixing (SFM and
DFM), in which two incoming frequencies 𝜈1 and 𝜈2 are converted into either
23
Chapter 3. Experimental Methods
Nd:YAG OPA DFM
SHG OPO
1064 nm OPO idler Mid-IR
532 nm (OPO signal) OPA idler
Figure 3.7: Schematic setup for the generation of IR light from an Nd:YAG laser. Colors
are not to scale, but in the correct order. A part of the 1064nm fundamental is doubled
into 532nm and fed to an OPO crystal. The output OPO idler becomes the input signal in
the OPA, pumped by the fundamental. The resulting OPA idler can cover the vibrational
stretching range 2000–4000 cm−1. With other crystals, the signal and idler from the OPA
can be combined in a DFM crystal to generate light in the wide range of 560–2500 cm−1.
𝜈1+𝜈2 or |𝜈1−𝜈2|. For example, Nd:YAG lasers often also contain a SFM KD2PO4
crystal that outputs light of wavelength 355nm (3𝜈0 = 28 200 cm−1). DFM ArGaSe2
crystals can be used to mix long-wavelength light up to 18µm (555 cm−1).
Yet another nonlinear process is optical parametric oscillation (OPO), which
can be thought of as SFM in reverse, inside an optical resonator. Starting from a
single pump frequency, two lower frequencies are produced. By convention the
greater output frequency is called signal, and the smaller idler. The frequency of
the signal (and hence the idler) depends on the crystal-to-beam angle. An OPO
crystal can also be used outside the resonator as a single-pass optical parametric
amplifier (OPA). The input is then the pump and the signal, and the output is an
amplified signal and some idler.
Figure 3.7 shows how all of the above can be used to cover a large part of the
vibrational spectrum, including stretching and bending modes. [73, 74] Starting
with an Nd:YAG, both the fundamental with frequency 9400 cm−1 (1064nm) and
its second harmonic are produced. The latter is fed into an OPO crystal, which
produces an idler tunable in the range of 5400–7400 cm−1 (1850–1350nm). An-
other crystal is used as an OPA to subtract this frequency from the fundamental,
resulting in an OPA idler of 2000–4000 cm−1 (5000–2500nm). Already, this range
covers the OH and NH stretching modes that exist in peptides. Finally, the two
outputs of the OPA are mixed using a ArGaSe2 DFM to yield a frequency of 1530–
4190 cm−1. This additionally covers the C=O stretching mode of peptides. Using
a similar setup, it is possible to achieve a range of 18000–600nm. [74]
In the system shown in Fig. 3.7, the OPO unit can be replaced with a dye laser
and a DFM. Similar to OPO, a dye laser needs a pump as input and gives a signal
(but no idler) as output. The OPO idler would then be replaced by the output of
the DFM when fed the dye laser output and the second harmonic.
A simpler setup can be used to generate UV light using a dye laser, see Fig. 3.8.
24
3.3. Setup Components
Nd:YAG
SHG SFM Dye SHG
1064 nm 355 nm UV
532 nm C153 out
Figure 3.8: Schematic setup for the generation of UV light from an Nd:YAG laser. Colors
are not to scale, but in the correct order. The 1064nm fundamental is doubled and then
summed with itself to yield its 355nm third harmonic, which is fed to a dye laser filled
with Coumarin 153. The dye laser emits tunable light in the range of 518–574nm, which is
doubled to 259–287 nm.
Again, starting with an Nd:YAG, its internal KD2PO4 optics produce the third
harmonic of frequency 28 200 cm−1 (355nm). This is then used to pump a dye laser
filled with Coumarin 153, whose output is tunable in the 17400–19 300 cm−1 (574–
518nm) range. A SHG BaB2O4 crystal is used to double this range into 34800–
38 600 cm−1 (287–259nm). This interval includes several chromophores, notably
phenylalanine at approximately 267 nm.
To summarize, table top lasers can be used to generate mid-IR light in the range
of 1530–4190 cm−1 and UV light in the range of 34 800–38 600 cm−1. These two
ranges in combination enable ion dip spectroscopy.
Laser Scanning Algorithms For nonlinear optical crystals to function, they
must be angled such that a phase-matching condition occurs. [75] The tolerance
of this angle is on the order of 100µrad, and depends on the light frequency. Thus,
scanning the laser requires all crystals to be precisely rotated. This is accomplished
by using stepper motors and specialized scanning algorithms.
One approach to scanning is to first construct a table where each row contains a
laser frequency and the appropriate crystal angles. The laser can then be set to a
specific frequency by looking up and interpolating between the closest rows in the
table. In practice such a table only works for scanning in one direction, because
of backlash in the stepper motors. Construction of these tables requires another
scanning algorithm.
A table-less approach is to use a fraction of the beam to infer the angle error.
Because the beam profile has finite width, it has some variance in Fourier space.
When the crystal is angled with a small error, it will function better for some non-
central wavevectors, and therefore stray to the side after a distance. By employing
a differential two-pixel photodiode, it is therefore possible to infer the angle error,
and correct it using linear feedback.
In some contexts, the two-pixel photodiode cannot be used and must be replaced
25
Chapter 3. Experimental Methods
with a one-pixel photodiode. Linear theory then fails, because the photodiode
signal is then an even function of angle error, resembling a Gaussian. For scanning
in this case, an algorithm developed during this thesis called Zigzag can be used.
A simplified presentation of Zigzag in Python is
for index in range(patience):
direction = index%2
top_energy = 0
while True:
motor[index].increment_angle()
energy = diode.get_energy()
if energy > top_energy:
top_energy = energy
elif energy < 0.9*top_energy:
break
It works by changing direction when the energy falls below a threshold, which is
continuously updated.
3.3.3 Mass Spectrometers
In spectroscopic experiments there is not only the molecule of study present; there
are also fragments of it and the background. For this reason, the detector must be
sensitive to mass and produce a mass spectrum rather than a total count. Two types
of such spectrometers are employed in the experiment of this thesis.
Ion Cyclotron For spectroscopy of ions, a cylindrical Penning trap is used. It
confines the ions in the radial direction with a strong homogeneous magnetic field,
and in the axial direction with a linear electric field. The strong magnetic field is
usually accomplished with a superconductive electromagnet, which requires liquid
helium cooling.
The motion of an ion in the Penning trap is a combination of cyclotron oscillation
with frequency
𝜔 = 𝑞𝐵𝑚 (3.6)
and a negligible oscillation due to the electric field. Hence mass is related to fre-
quency, and by measuring the outgoing near-field radiation of the trap and taking
the Fourier transform, it is possible to obtain a mass spectrum. This principle of
mass measurement is known as Fourier-transform ion cyclotron resonance (FT-
ICR, see Fig. 3.9), and can reach a relative accuracy better than 1 in 1 000 000. [76]
26
3.3. Setup Components
+V 0 0 +V'
B
Ions
Detector
Figure 3.9: Schematic presentations of FT-ICR and TOF mass spectrometers. Left (FT-
ICR): ions are trapped by a strong homogeneous magnetic field. The resulting cyclotron ra-
diation is measured and Fourier transformed into an inverse mass spectrum. Right (TOF):
molecules are ionized and immediately accelerated by a static electric field, and drift to-
wards a detector. The optional midway reflectron extends the drift time by creating an-
other point where the flight time is independent of ionization position.
Time-of-flight On the other hand, neutral atoms cannot be directly guided by
electromagnetic fields. Therefore, spectroscopy techniques on neutrals such as
IRMPD–VUV and IR–UV ion dip produce ions after the IR absorption has oc-
curred. In time-of-flight (TOF) mass spectrometers, these ions are accelerated
upon ionization using a static electric field, giving them a velocity of
𝑣 = √2𝑞𝑈𝑚 , (3.7)
where 𝑈 is the electric potential at the point of ionization. The ions then drift
through a grounded tube for a time proportional to inverse velocity, or equiva-
lently to mass squared. At the end of the drift path, singular ions are detected
with precise time resolution using a micro-channel plate (MCP) detector.
The accuracy of TOF mass spectrometers is limited not by time resolution (which
could be coped with by extending the drift path), but the spread in acceleration po-
tential due to the width of the molecular beam. Ions created further away from
the drift region, where the potential is higher, initially lag behind but eventually
catch up at a point and surpass those created at the center of the beam. This point
is a time focus and the best place for the detector.
By introducing a reflectron as part of the drift path, it is possible to delay the ions
with high kinetic energy, and thereby creating a time focus further away. This in-
creases the relative accuracy of the instrument, because the drift time is increased
without affecting the time resolution. A typical relative mass accuracy of 1 in 2000
is reachable with a reflectron added. Figure 3.9 (right) shows a TOF mass spec-
trometer with a reflectron.
27
Chapter 3. Experimental Methods
3
G
L E
2
1
Figure 3.10: The local experimental setup, affectionately known as the Giraffe. Some
major components. 1: The source chamber containing the nozzle for supersonic jet expan-
sion, the skimmer, and the MALDI setup. On its opposite side (not visible), the chamber
connects to the sluice and the sample delivery system. 2: Interaction cross where the ion-
ization happens. Two CaF2 windows allow UV and IR light to enter the cross. 3: Tube for
time-of-flight mass spectrometry. On the top (not visible) sits the reflectron. L: Nd:YAG
laser used for laser desorption. G: Gate valve for isolating the high-vacuum part of the
chamber. E: Electric field acceleration plates for the mass spectrometer. The same piece
holds the extraction plates, which reach down into the cross.
28
3.4. Local GU Laboratory
3.4 Local GU Laboratory
One of the goals set at the start of this PhD project was to build an experimental
setup capable of spectroscopic experiments in our laboratory in Fysikforskarhuset,
Gothenburg. The chosen design was inspired by a vacuum chamber setup for
REMPI and IR–UV ion dip spectroscopy at Radboud University, Nijmegen. It
consists of three major parts (Fig. 3.10): a source chamber, an interaction cross,
and a time-of-flight mass spectrometer.
3.4.1 Source Chamber
The source chamber, shaped like a rectangular block, is designed to generate a
neutral molecular beam directed into the interaction region. The two most impor-
tant components for this purpose are the skimmer and the source. The skimmer
is fixed and mounted to the side of the source chamber that faces the interaction
region, see Fig. 3.11. The source is mounted on a XY translational stage on the
opposite side. By design it is detachable, and at present there are two sources to
choose from: oven and pulsed valve. The oven is used to evaporate molecules that
allow it, such as fullerene. The pulsed valve is used in tandem with MALDI to
achieve supersonic jet cooling of fragile molecules such as peptides.
N S
M
V
H
Figure 3.11: Virtual cross-cut of the source chamber. Important components are labeled.
N: Pulsed valve and nozzle through which a carrier gas expands in a supersonic jet. It is
mounted to a XY translational staged on the left chamber wall. S: Skimmer that selects
the central part of the jet, and leads into the interaction cross. M: MALDI setup, the blue
and yellow part is the quickly replaceable sample holder. The grey linear feedthrough is
seen in the background. V: Vertical translational stage, used to adjust where the MALDI
fumes enter the jet expansion. H: Horizontal translational stage used to move the sample
bar. Rests on a blue shelf which is mounted to the chamber wall.
29
Chapter 3. Experimental Methods
F
h
S s
G a
M f
Figure 3.12: Virtual and real image of the sample delivery system. M: MALDI setup,
with the sample holder colored blue. G: Gate valve. S: Sluice chamber for fast venting.
F: Linear feedthrough that fits in the sluice in its shortened state, and reaches the MALDI
setup in its elongated state. h: MALDI sample bar holder. s: Leaf spring that gently fixes
the sample bar holder. a: Adapter than the linear feedthrough can dock to using a M6
screw. f: Linear feedthrough arm in extended state, docked to the sample bar holder.
In the source chamber, below the pulsed valve, is a MALDI setup. Its purpose
is to deliver fragile molecules into the gas phase. It sits on a shelf mounted to the
wall of the source, with two linear translation stages in between. The vertical stage
is needed because the efficiency of cooling is sensitive to the distance between the
desorption point and the gas expansion. The horizontal stage moves the setup
such that the desorption point contains new molecules to desorb. A small 1064nm
Nd:YAG laser sits on top of the source chamber, and emits the necessary pulses to
the desorption point.
The MALDI sample bar lasts for just less than an hour, and must then be re-
placed. To minimize time spent pumping down, a sluice with a linear feedthrough
is installed, see Fig. 3.12. The feedthrough connects to the sample bar holder and
can be used to pull it out into the sluice. The sluice gate can then be closed and
the sluice opened on the other side, allowing the sample bar to be replaced. A
dedicated pump clears the sluice within minutes, and the sample bar holder can
then be reinstalled.
3.4.2 Interaction Cross
The interaction cross is, as the name suggests, shaped like a six-directional cross.
The neutral molecular beam enters it from one direction and intersects the laser
beams that enter through windows on the sides perpendicular to the molecular
beam. Any ions created are accelerated upwards into the mass spectrometer tube.
The turbo pump connects to the bottom side.
30
3.4. Local GU Laboratory
FWHM = 0.043 Da
92 92.5 93 93.5
0 10 20 30 40 50 60 70 80 90 100
Mass (Da)
Figure 3.13: Mass spectrum of toluene, obtained from the local time-of-flight mass spec-
trometer. The pink inset shows the peaks of toluene and its first C13 isotopologue. The
full width half maximum of peaks is measured to be 0.043Da.
3.4.3 Mass Spectrometer
The mass spectrometer used in our laboratory is a slightly adapted commercial
solution from Jordan TOF. Specifically it is a reflectron time-of-flight mass spec-
trometer. It consists of four parts: the extraction plates, the time-of-flight tube,
the reflectron, and the microchannel plate (MCP) detector. The extraction plates
reach down into the interaction cross because their purpose is to accelerate ions
into the tube.
Figure 3.13 shows the REMPI mass spectrum of toluene measured by the mass
spectrometer in the local laboratory. At a mass of 92–93Da, the full width at half
maximum is 0.043Da, implying a relative accuracy of more than 1 in 2000. Fig-
ure 3.14 shows the corresponding REMPI spectrum.
37200 37250 37300 37350 37400 37450 37500 37550
UV Frequency (cm!1)
Figure 3.14: The REMPI spectrum of toluene, as defined by the pink region in Fig. 3.13.
31
Ion yield (arb.) Count (arb.)

4
Computational Methods
This chapter is meant to give an overview of the methods to simulate spectroscopic
results employed in this thesis.
4.1 Goal and Overview
A spectroscopic experiment typically results in a spectrum. To interpret this spec-
trum and extract information about the underlying molecular properties, candi-
date hypotheses are simulated and compared to the experimental result. If a sim-
ulation matches the experiment, its hypothesis is deemed more likely.
Candidate hypotheses for IR spectra typically include three parts: one or more
conformer that are believed to be present in the experiment, an energy surface
model believed to describe those conformers, and a model for describing the vi-
brational frequencies. While the latter models can easily be found “off-the-shelf”,
finding the relevant conformers to a molecule in practical time is an hard problem.
Section 4.3 will categorize the available methods, with a focus on those used in my
work.
Searching for conformers in the vast conformational space is indeed a hard prob-
lem. The result of a conformer search is a large count of conformers, potentially
more than 100 000. These are then further optimized with a hierarchy of meth-
ods to obtain accurate three-dimensional (3D) structures, and their energies. Sec-
tion 4.4 will describe those and rank them on the hierarchy of accuracy and com-
plexity. Finally, Section 4.5 will describe some ways the vibrational frequency
spectrum can be computed, including the recent Born–Oppenheimer molecular
dynamics method.
33
Chapter 4. Computational Methods
SMILES XMol XYZ Gaussian Z-matrix
FC(F)F 5 0 1
fluoroform
C 0.00 0.00 0.00 C
H 1.09 0.00 0.00 H 1 1.09
F -0.48 1.26 0.00 F 1 1.35 2 111
F -0.48 -0.63 1.09 F 1 1.35 2 111 3 120
F -0.48 -0.63 -1.09 F 1 1.35 2 111 4 120
Figure 4.1: Data representations of fluoroform (CHF3). The SMILES format is brief and
omits hydrogens. The XMol XYZ format includes the Cartesian coordinates. The Gaus-
sian Z-matrix format specifies each nuclei relative to up to three previous ones.
4.2 Molecular Formats
Before diving into the various computational methods and softwares, it is worth
describing how molecules are represented on computers. There are at least three
important classes of formats: string, Cartesian, and Z-matrix. Conversion between
formats is possible with the wonderful Open Babel tool. [77]
String formats like SMILES (Simplified Molecular Input Line Entry System) are
very compact descriptions of molecules, but they do not specify exact positions
of the nuclei, and therefore contain no conformational data. They are useful as
a starting point before the conformational search. SMILES in particular has an
intuitive tree-like syntax, exemplified with fluoroform in Fig. 4.1.
Cartesian formats like XMol XYZ are text files that fully specify the positions
of all nuclei. They are comprised of a small header followed by one row for each
nuclei. The row representation includes at least element and position, but some
formats also include neighborhood information and chemical properties.
Z-matrix formats like Gaussian Z-matrix also use one row per nuclei to spec-
ify their positions. They differ from Cartesian formats in that they use internal
coordinates, meaning bond lengths, bond angles, and dihedral angles. This rep-
resentation is more practical when handling conformers, because they are defined
by their dihedral angles. Another advantage of Z-matrix over Cartesian formats is
that 𝑁 nuclei require only 3𝑁 − 6 coordinates rather than 3𝑁 , because the origin
and orientation of the basis is never chosen.
4.3 Conformer Searching
When an experiment is performed on molecular species, its general bond structure
but not its preferred conformer is known. The first step of a simulation is therefore
34
4.3. Conformer Searching
to find a large set of plausible conformers, which live in the conformational space
generated by dihedral angle rotations around single bonds. This space is exponen-
tially large and thus infeasible to exhaust for any peptide. Consequently, sparse
sampling strategies are required to explore the conformational space.
4.3.1 Conformer Distance
When searching the conformational space, it is possible to find the same conformer
twice, represented in two different ways. In order to mitigate this inefficiency, the
algorithm needs to define and check for duplicates. A good definition for duplicity
is that the conformer distance, to be defined in the following paragraph, is less than
some threshold value on the order of 1Å.
The establish conformer distance, often simply called root-mean-square dis-
tance (RMSD), is the least square distance between point sets, taken over the
group generated by rotations and translations. [78] To be explicit, if two conform-
ers have position matrices 𝐴 and 𝐵 of size 𝑁 × 3 the distance is
RMSD(𝐴,𝐵) = √1 min ||𝐴𝑄 + 1𝑣 − 𝐵|| , (4.1)
𝑁 𝑄,𝑣 𝐹
where 𝑄 is a 3×3 rotation matrix, 1 is an 𝑁×1 vector of ones, 𝑣 is a 1×3 translation
vector, and || ⋅ ||𝐹 is the Frobenius norm. This has an explicit solution
RMSD(𝐴,𝐵) = √1 ||𝐴𝑈̃ − ?̃?𝑉 ||
𝑁 𝐹
(4.2)
where 𝑈𝑆𝑉 𝑇 = 𝐴𝑇̃ ?̃? (4.3)
𝐴̃ = 𝐴 − 1 𝐴 ?̃? = 𝐵 − 1𝑁 1 and 𝑁 1𝐵. (4.4)
Equation (4.3) means that 𝑈 and 𝑉 are given by the singular value decomposition
of the right hand side.
The RMSD is directly applicable to Cartesian formats, which Z-matrix can be
converted into. It is zero between identical conformers, and naturally invariant
with respect to the translations and orthogonal rotations, but not row permuta-
tions. Therefore, conformers that are separated by only translation and orthogo-
nal rotation can still have nonzero RMSD, for example if they are related by a 60°
methyl rotation. This issue can be resolved by taking the smallest RMSD across
methyl rotations or the like.
4.3.2 Energy-Blind Sampling
The most straightforward approach to generating conformations is to randomly
select dihedral angles from a distribution on the domain [−180°, 180°]. This can
35
Chapter 4. Computational Methods
be an efficient way to sample small molecules such as dipeptides. [42, 43] Not all
angles need to be varied, for example the peptide link H-N-C=O dihedral angle
is almost always 180°, because of the nearby double bond. Random sampling is
easily implemented in high-level languages such as Matlab or Python, but there
is also an open source implementation called Confab [52] than is integrated into
Open Babel.
Independently randomizing angles can lead to unphysical self-intersecting con-
formations, especially when applied to long molecules such as peptides. Distance
geometry embedding addresses this by first establishing upper and lower bounds
for all atom-pair distances using bond information. A candidate distance matrix
is then randomly generated within these bounds, and the conformation is fitted to
match this distance matrix. An implementation of distance geometry embedding
is freely available in the software package RDKit. [53]
4.3.3 Energy-Guided Sampling
Because low-energy conformer are more interesting than high-energy ones, it is
natural to employ coarse energy calculations in order to steer the search into lower
energies. There are some simple and fast energy models for this purpose which are
detailed in Section 4.4. Even if they have very poor correlation with DFT for stable
conformers, they help avoiding unphysical behaviors such as self-intersection.
The most famous example of energy-guided sampling is the Metropolis–Hastings
algorithm, [79, 80] in which the system is iteratively mutated such that the trajec-
tory obeys Boltzmann statistics. Specifically the logic loop can be written as
state = initiate_state()
for _ in range(patience):
new_state = mutate_state(state)
accept_prob = exp(-beta*(energy(new_state)-energy(state)))
if uniform() < accept_prob:
state = new_state
save_state(state)
This loop is known to generate states following a Boltzmann distribution if the
function mutate_state is symmetric in the sense that for all X and Y, the probabili-
ties of Y = mutate_state(X) and X = mutate_state(Y) are equal. On mutations
of dihedral angles, which is the natural choice for conformer generation, this is sat-
isfied by adding a random variable with symmetric distribution.
The beta parameter is inversely proportional to temperature and must be cho-
sen carefully. A too high value of beta makes the program find the closest energy
minimum and stay there, while a too low value yields unphysical high-energy con-
formers. There is a Goldilocks zone of decent values of beta, but it is also possible
36
4.3. Conformer Searching
to start with a low value and increase it over time. This results in a different dis-
tribution of conformers than running in the Goldilocks zone, analogous to how
annealed metals differ from quenched. [81]
A downside of pure Metropolis–Hastings sampling is that low-energy conform-
ers are frequently revisited. This redundancy can be mitigated by so-called pol-
ing, meaning adding repulsive potential terms centered on past conformers. [82]
Another way of mitigating redundancy is to exclude entire regions based on pre-
viously generated conformers. [83]
Another algorithm entirely, which avoids redundancy by design rather than as
an afterthought, is the basin-hopping algorithm. Its main idea is to sequentially
explore the neighborhoods of every optimized state once. A simplified version
can be written
states = [initiate_state()]
for state in states:
for _ in range(num_attempts):
new_state = optimize_state(mutate_state(state))
if energy(state) < max_energy and state not in states:
states.append(state)
save_state(state)
Rather than an explicit patience constant, the theoretical runtime is related to the
values of num_attempts and max_energy. Roughly speaking, the former affects the
density of conformers and the latter the volume of the feasible space, both which
increase runtime. In practice the runtime is difficult to estimate, and the algorithm
is terminated externally.
An implementation of the basin-hopping algorithm is included in the software
package Tinker. [54] The implementation is named scan and makes some context-
specific alterations to the code above. For example, rather than independently
mutating the molecule num_attempts times, scan chooses num_attempts distinct
vibrational modes and stimulates the molecules by adding momentum to them.
The mutated molecule is then obtained by simulating the system with Newtonian
dynamics for a while. Energies and forces are calculated from cheap molecular
force field methods that can be externally supplied.
4.3.4 Dynamical Sampling
A physics-inspired approach to generate conformers is through classical molecular
dynamics (MD) simulations. The equations of motions are often solved with the
Verlet integrator. [84] In theory, the molecule should eventually find a low-energy
conformer, just as in reality. Unfortunately this is inefficient in practice because
the integration time step (about 1 fs) is much lower than the time between distinct
37
Chapter 4. Computational Methods
conformers (about 10ps), meaning most computation is wasted on details within
conformer basins.
There are some unphysical tricks that makes MD better at generating conform-
ers. Replica-exchange MD [85, 86] starts by running multiple MD simulations in
parallel, but at different temperatures or with different Hamiltonians. Periodi-
cally, coordinates of parallel runs are exchanged according to Metropolis logic.
Such exchanges make conformer generation more efficient by overcoming energy
barriers. The CREST software designed for conformer searching uses a replica-
exchange MD algorithm that additionally includes poling, meaning the addition of
repulsive bias in the MD potential for each previously identified conformer. [55]
4.4 Energy Methods
Arguably, the most fundamental problem in molecular physics is to determine the
energy of a given system. For this task there are a plethora of methods, which can
be classified on a scale from cheap (short runtime) to accurate. These methods syn-
ergize because they have different use-cases; an energy-guided conformer search
would use a cheap force field method like MM3, but the final energy evaluation of
a conformer would use a composite method like G4 if possible.
4.4.1 Wavefunction Approximations
The main difference between cheap and accurate methods is what approximations
they make. Most methods used in this thesis make some common approximations
that will be described here.
In principle, the energy of any quantum-physical system can be found by solving
the time-independent Schrödinger equation
?̂? |𝜓⟩ = 𝐸 |𝜓⟩ , (4.5)
which is an eigenvalue problem for the energy scalar 𝐸, given an Hamiltonian op-
erator ?̂? . In the case of a molecule consisting of 𝑛e electrons and 𝑛n nuclei with co-
ordinates 𝑟𝑖 and𝑅𝑖, the wavefunction depends on all,𝜓 = 𝜓(𝑟1,… , 𝑟𝑛 , 𝑅1,… ,𝑅e 𝑛 )n
and the Hamiltonian is
?̂? = 𝑇̂ + 𝑇̂ + 𝑉 ̂e n ee + 𝑉̂ ̂en + 𝑉nn, (4.6)
where 𝑇̂e and 𝑇̂n are the kinetic operators of the electrons and nuclei, and 𝑉⋅̂ ⋅ are
Coulomb potentials.
In practice, the complexity and size of the wavefunction 𝜓 is too great to be
solved analytically or numerically, and must be broken down successively into sim-
pler forms.
38
4.4. Energy Methods
An implicit approximation for light nuclei such as those in biomolecules, is the
neglect of relativistic effects, implying that 𝜓 is scalar complex valued, and that the
kinetic operators can be written
𝑛e ℏ2 𝑛e ℏ2𝑇̂ 2e = −∑ 2𝑚 ∇𝑟 and 𝑇
̂ 2
n = −∑ 2𝑀 ∇𝑅 . (4.7)𝑖 𝑖𝑖=1 e 𝑖=1 𝑖
The most important approximation is the Born–Oppenheimer one, which sep-
arates the solution of the nucleic and electronic part. [87] First, the nuclei are
considered fixed at positions ?⃗? and the electronic Schrödinger equation
(𝑇̂ + 𝑉 ̂e ee + 𝑉̂ ̂en(?⃗?) + 𝑉nn(?⃗?)) ∣𝜓e(?⃗?)⟩ = 𝐸e(?⃗?) ∣𝜓e(?⃗?)⟩ (4.8)
is solved, yielding the electronic energy as a function of nucleic positions 𝐸e(?⃗?),
often called the potential energy surface (PES). Then, the nucleic terms are rein-
troduced and solved for in the nucleic Schrödinger equation
(𝑇̂n +𝐸e) |𝜓n⟩ = 𝐸 |𝜓n⟩ (4.9)
The Born–Oppenheimer approximation thus reduces the Schrödinger equation
into two eigenvalue problems with much smaller wavefunctions. The electronic is
the more challenging one, because there are always more electrons than nuclei in
a molecule.
One important approximation that significantly reduces the complexity of the
wavefunction is the single Slater determinant approximation, which is famously
used in the Hartree–Fock (HF) method. The idea is that the electronic many-
electron wavefunction is a product of one-electron wavefunctions, up to anti-sym-
metrization. Explicitly, the wavefunction is written
𝜙1(𝑟1) 𝜙2(𝑟1) ⋯ 𝜙𝑛e(𝑟1)
𝜓(𝑟 11,… , 𝑟𝑛 ) = √𝑛 ∣ ⋮ ⋮ ⋱ ⋮ ∣. (4.10)e e 𝜙1(𝑟𝑛 ) 𝜙2(𝑟𝑛 ) ⋯ 𝜙𝑛 (𝑟e e e 𝑛 )e
To appreciate how greatly this form limits 𝜓, count the dimension of its space.
Assuming the one-electron wavefunctions belong to some vector space 𝑉 of large
dimension 𝑑, the space of many-electron wavefunctions is the 𝑛eth exterior power
of 𝑉 , which has dimension ( 𝑑 ) ≈ 1 𝑛𝑛 𝑛 !𝑑 e . On the other hand, the subspace one e
the form of Eq. (4.10) is parameterized by 𝑉 𝑛e , which has dimension 𝑛e𝑑. This
counting does not take normalization into account, but the point remains that the
number of degrees of freedom in 𝜓 becomes a linear function of 𝑑.
The downside of such a strong approximation is that correlation information is
lost, because the Slater determinant is essentially a product approximation, which
39
Chapter 4. Computational Methods
in terms of probabilities means independence. Despite this, the HF method allows
accounting for around 99% of the total electronic energy. [88] The remainder, due
to correlation effects requires relaxing this approximation, which is what so-called
post-HF methods do.
The one-electron wavefunctions are in principle normalized functions from real
3D space to the complex numbers. In practice, they are restricted to finite-dimen-
sional subsets with some useful property. Slater-type orbitals, inspired by the hy-
drogen system, have basis functions with a radial factor 𝑅(𝑟) = 𝑟𝑛𝑒−𝑎𝑟. On the
2
other hand, Gaussian-type orbitals use 𝑅(𝑟) = 𝑟𝑛𝑒−𝑎𝑟 , which makes inner prod-
ucts between them computable analytically. It is common to combine the strength
of both types, by using as basis functions fixed linear combinations of Gaussian-
type orbitals that approximate Slater-type orbitals. Two such implementations are
the Pople [89] and Dunning [90, 91] basis sets.
4.4.2 Density Functionals
Another approach to determining the energy system is density functional theory
(DFT), which replaces the wavefunction 𝜓 with the electron probability density
𝜌(𝑟) = ∫ d3𝑟 ⋯∫ d3𝑟 |𝜓(𝑟, 𝑟 ,… , 𝑟 )|22 𝑛 2 𝑛 . (4.11)e e
Although this seems like a massive loss of information, the ground state wavefunc-
tion and energy can in principle be determined from the corresponding electron
density. Formally, the second Hohenberg–Kohn theorem [92] states that there
exists an universal functional 𝐹 such that the functional
𝐸[𝜌] = 𝐹 [𝜌] +∫𝑉 (𝑟)𝜌(𝑟) d3𝑟 (4.12)
has a global minimum (𝜌0, 𝐸[𝜌0]) consistent with the ground state solution to the
electronic Schrödinger equation with external (nucleic-electronic) potential𝑉 . This
implies that any evaluation of 𝐸 gives an upper bound on the true energy 𝐸[𝜌0].
Unfortunately, 𝐹 is not explicitly defined in the proof of the theorem, but must
rather be approximated.
A popular way to practically use DFT is with the Kohn–Sham formalism, [56]
which splits the universal functional into
𝐹[𝜌] = 𝑇s[𝜌] + 𝐽[𝜌] + 𝐸xc[𝜌], (4.13)
where 𝑇s[𝜌] is the Kohn–Sham kinetic energy that can be found by solving a HF-
like problem, 𝐽[𝜌] is the Coulomb energy
2
𝐽[𝜌] = 1 𝑒 ∫d3𝑟 ∫d3𝑟 𝜌(𝑟1)𝜌(𝑟2)2 4𝜋𝜖 1 20 |𝑟1 − 𝑟 |
, (4.14)
2
40
4.4. Energy Methods
and 𝐸xc[𝜌] is the exchange-correlation functional. There are many competing ap-
proximations for 𝐸xc[𝜌], including separate parts for the exchange and the cor-
relation part. For example, one could use Becke’s 1988 functional [93] for the
exchange, and Lee, Yang, and Parr’s functional [94] for the correlation. This com-
bined method is known as B-LYP, and this naming convention is generally upheld.
The DFT functionals used in the papers of this thesis include B3LYP, [95] M06-
2X, [96] andωB97X-D. [97, 98] Although not purely DFT, composite methods such
as CBS-4M [99, 100] and G4(MP2) [101] have also been used.
Many DFT functionals do not correctly model London dispersion forces. While
two distant atoms should experience a weak attractive force corresponding to a
𝐶𝑟−6 potential, the DFT energy decreases exponentially. A popular solution is
then to add an empirical dispersion term. The papers of this thesis consistently
use Grimme’s dispersion [102] with Becke–Johnson damping, [103] which for a
diatomic molecules simplifies to
𝐸 (𝑟) ∝ − 𝐶6 𝐶8disp 1 + (𝑟/𝑟 −6)6 1 + (𝑟/𝑟 )8
, (4.15)
8
where 𝐶𝑛 and 𝑟𝑛 depend on the atom number of the nuclei. For larger molecules,
the equation above is applied on all pairs of nuclei, and there is also a three-body
term. [102]
4.4.3 Empirical Methods
The simplest methods for calculating energies are linear combinations of analytical
expressions of bond lengths, bond angles, and dihedral angles. Because they can
easily be differentiated to give forces, they are known as molecular force fields.
Examples include AMBER, [104] CHARMM, [105] and MM3. [12] The main
difference is the expressions used and the data used to fit the coefficients. Force
fields have poor accuracy, being almost uncorrelated with DFT. [106] Thus, they are
mostly used in energy-guided sampling, where the absolute energy is not crucial.
Semiempirical methods are more complex than force fields. They start from
wavefunctions and Hamiltonians, but neglect some integrals or replace the with
empirical expressions. The recently developed PM7 method [107] implemented in
the freely available MOPAC software by Stewart [108] is one such method. In the
papers of this thesis, PM7 is often used as a middle step between the conforma-
tional search and DFT, because it correlates decently (20–40%) with DFT.
41
Chapter 4. Computational Methods
4.5 Frequency Methods
The vibrational frequency spectrum of a molecule is intimately related to spectro-
scopic experiments, and is a sensitive function of molecular structure. Therefore,
much effort has gone into developing accurate methods for computing the fre-
quency spectrum from the 3D structure.
Conventional frequency calculations are based on energy derivatives of a spe-
cific 3D structure. These are used to fit a simpler model to the potential energy
surface, such that the Schrödinger equation for nucleic motion can be solved on
this model.
More recently, frequency methods based on molecular dynamics have risen in
usage. They only need first-order derivatives of the potential energy surface, and
instead explore many points around the equilibrium. This strategy along with its
(dis)advantages will be discussed in Subsection 4.5.3.
4.5.1 Harmonic Approximations
One of the simplest quantum systems for which the Schrödinger equation can be
analytically solved is the harmonic oscillator. Explicitly, the one-dimensional (1D)
equation is
𝑝̂2(2𝑚 +
1
2𝑘𝑥
2̂ ) |𝜓⟩ = 𝐸 |𝜓⟩ , (4.16)
and has the eigenvalues 𝐸𝑛 = ℏ𝜔(𝑛 + 12) for non-negative 𝑛, where 𝜔 = √𝑘/𝑚.
The corresponding eigenstates |𝑛⟩ are Gaussian function multiplied by Hermite
polynomials. An important property of the eigenstates are that the expectation
value
∞
⟨𝑛′|𝑥|̂ 𝑛⟩ = ∫ 𝜓𝑛′(𝑥)𝑥𝜓𝑛(𝑥) d𝑥 (4.17)
−∞
is nonzero precisely when 𝑛′ = 𝑛 ± 1. This value corresponds to the probability
amplitude of an operator proportional to 𝑥̂ changing the vibrational quanta from
𝑛 to 𝑛′. In the context of absorption spectroscopy, this operator is the electric
dipole, and the integral statement implies that a harmonic system can only absorb
photons increasing the quanta by one. Thus the absorption energy is given by
𝐸𝑛+1 −𝐸𝑛 = ℏ𝜔.
The 𝑑-dimensional quantum harmonic oscillator is intuitively similar to the 1D
oscillator. In fact, it can be split into 𝑑 independent 1D oscillators by a linear
change of coordinates. The new basis vectors are called vibrational modes and
can be found by solving an eigenvalue problem involving the mass and stiffness
42
4.5. Frequency Methods
matrices. The absorption energies of the whole system is {ℏ𝜔𝑚}, where the 𝑑 an-
gular frequencies 𝜔𝑚 come from the eigenvalue problem.
For a molecule with 𝑁 nuclei, there are 3𝑁 degrees of freedom. Define 3𝑁
coordinates 𝑥⃗ = {𝑥𝑖} such that 𝑥⃗ = 0⃗ corresponds to the equilibrium geometry,
assuming that it is known. Now, the potential energy 𝑉 is approximated by its
second-order Maclaurin expansion around this point:
3𝑁 3𝑁
𝑉 (𝑥)⃗ ≈ 𝑉 (0)⃗ +∑𝑉 ′𝑖 (0)⃗ 𝑥𝑖 +
1 ″
2 ∑ 𝑉𝑖𝑗(0)
⃗ 𝑥𝑖𝑥𝑗. (4.18)
𝑖=1 𝑖,𝑗=1
Because the expansion was made around the equilibrium geometry, the first-order
derivatives of the potential vanish. The Schrödinger equation on this harmonic
potential can now be solved as previously described, with the set of second-order
derivatives 𝑉 ″ as the stiffness matrix.
Although there are 3𝑁 coordinates, 6 (5 for linear molecules) of these describe
translations and rotations. For this reason, only 3𝑁−6 computed vibrational mode
frequencies are nonzero, within numerical accuracy. Each of these modes con-
tribute to the vibrational spectrum with an intensity proportional to the squared
derivative of the electric dipole with respect to the mode coordinate.
In summary, the harmonic frequency method computes the vibrational spec-
trum by comparison with a quantum harmonic oscillator. Two harmonic approx-
imations are made: that the restoring force on nuclei is a linear function of their
coordinates, and that the electric dipole of the molecule is also a linear function of
the same coordinates.
The computed harmonic frequency spectra of molecules, when compared to ex-
perimental data, consistently underestimate vibrational frequencies by a few per-
cent. By multiplying the computed result with an empirical scaling factor, agree-
ment is improved and the typical error is around 30 cm−1. [58] However, the need
for this scaling factor suggests that the harmonic approximation (of force) is too
simplistic.
4.5.2 Anharmonic Perturbation
When a Maclaurin approximation is inaccurate, the natural solution is to include
more terms. Adding only third order terms to the potential would make it un-
bounded in a pathological way, so the next logical step is to add third and fourth
order terms. This family of potentials is not analytically solvable. What can be
done is to start from the harmonic theory, and add higher order terms considered
as small perturbations. This is the rationale for second-order vibrational pertur-
bation theory (VPT2). [57]
43
Chapter 4. Computational Methods
VPT2 splits the Hamiltonian like ?̂? = ?̂?HO + ?̂?extra, where ?̂?HO corresponds
to the understood harmonic oscillator, and ?̂?extra to higher order terms but also a
Coriolis term. The perturbed energy for a given state with quantum number vector
?⃗? is then obtained as
⟨?⃗?∣?̂?extra∣?⃗?⟩ ⟨?⃗?∣?̂?𝐸 = ⟨?⃗?∣?̂? ∣?⃗?⟩ + ⟨?⃗?∣?̂? ∣?⃗?⟩ + ∑ extra
∣?⃗?⟩
?⃗? HO extra , (4.19)
?⃗?∶?⃗?≠?⃗? ⟨?⃗?∣?̂?HO∣?⃗?⟩ − ⟨?⃗?∣?̂?HO∣?⃗?⟩
which with some VPT2-specific assumptions simplifies to [57]
3𝑁−6 3𝑁−6 𝑟
𝐸 1 1 1?⃗? = 𝐺0 + ∑ ℏ𝜔𝑟(𝑛𝑟 + )+ ∑ ∑ℏ𝜒𝑟𝑠(𝑛𝑟 + )(𝑛𝑠 + ). (4.20)
𝑟=1 2 𝑟=1 𝑠=1 2 2
In the expression above 𝐺0, 𝜔𝑟, and 𝜒𝑟𝑠 are constants that can be computed from
geometry and the potential derivatives of order up to four. In fact, only the fourth-
order derivatives of the form 𝑉 ⁗𝑟𝑟𝑠𝑡 need to computed, which saves complexity.
The ground state absorption spectrum can then be obtained by subtracting the
ground state energy 𝐸0⃗ from every energy. In theory any transition is allowed, but
in practice only excitations of up to two quanta are computed. The one-quantum
excitations are called fundamental frequencies and have (angular) frequencies
3 1 3𝑁−6𝜈𝑟 = ℏ−1(𝐸𝑒𝑟⃗ −𝐸0⃗) = 𝜔𝑟 + 2𝜒𝑟𝑟 + 2 ∑ 𝜒𝑟𝑠, (4.21)𝑠=1
where 𝑒𝑟⃗ is a unit vector in coordinate 𝑟. The two-quanta excitations are called first
overtones if both quanta lie in the same mode, and combination bands otherwise.
Their frequencies are
𝜈𝑟𝑟 = ℏ−1(𝐸2𝑒⃗ −𝐸0⃗) = 2𝜈𝑟 + 2𝜒𝑟𝑟 and (4.22)𝑟
𝜈 −1 𝑟≠𝑠𝑟𝑠 = ℏ (𝐸𝑒𝑟⃗ +𝑒⃗ −𝐸0⃗) = 𝜈𝑟 + 𝜈𝑠 + 𝜒𝑟𝑠. (4.23)𝑠
The corresponding intensities involve derivatives of the electric dipole and are de-
scribed in Ref. [57].
A known problem with VPT2 is that the expression 𝜔𝑟+𝜔𝑠−𝜔𝑡 appears in some
denominators of the 𝜒𝑟𝑠. Such ratios become excessively large when 𝜔𝑟+𝜔𝑠 ≈ 𝜔𝑡,
a condition known as Fermi resonance, leading to absurdly large contributions to
the vibrational spectrum. There are some different strategies for removing Fermi
resonances. [57, 109, 110]
The computational complexity of VPT2 is naturally higher than that of a har-
monic frequency analysis, which only requires second-order derivatives of the po-
tential at a point. The bottleneck of VPT2 is the computation of all potential
44
4.5. Frequency Methods
derivatives of the form 𝑉 ⁗𝑟𝑟𝑠𝑡. Most functionals allow up to second-order deriva-
tives to be computed at a point. Further derivatives must be numerically com-
puted with a finite displacement, implying that all second-order derivatives must
be computed at 1 + 2(3𝑁 − 6) points. There are economical versions of VPT2
that reduces the cost by computing only a subset of the fourth-order derivatives,
making the method applicable to larger systems. [111, 112]
VPT2 with many options is implemented in the Gaussian software. [113] Bench-
marking studies [59, 114] show a typical error of around 15 cm−1, an improvement
over harmonic theory. However the presence of Fermi resonances sometimes
makes VPT2 inapplicable. [5]
4.5.3 Semiclassical Dynamics
A recently popularized idea for computing vibrational spectra is the Born-Oppen-
heimer molecular dynamics (BOMD) method. Unlike harmonic frequency anal-
ysis and VPT2 which analyzes the potential from a point, the BOMD method cal-
culates a spectrum based on the trajectory of a BOMD simulation. [115, 116]
The semiclassical BOMD simulation treats the nuclei as classical particles af-
fected by a potential found by solving the Schrödinger equation with the Born–
Oppenheimer approximation. This is possible thanks to the Hellman–Feynman
theorem, which states that [88, 117]
?̂?𝜆 |𝜓𝜆⟩ = 𝐸𝜆 |𝜓𝜆⟩
d𝐸 d?̂?
implies 𝜆𝜆 = ⟨𝜓 ∣
𝜆
𝜆 𝜆 ∣𝜓𝜆⟩. (4.24)d d
Setting 𝜆 to a nucleic coordinate gives an expression for the corresponding force.
The motion of the nuclei is then computed with an integration scheme for New-
tons equations, such as the Verlet integrator. [84] The result is a trajectory of the
molecule, notably including the electric dipole.
A common time-saving trick is to estimate the initial wavefunction or electron
density based on values from the previous time step. In particular, the atom-
centered density matrix propagation (ADMP) method [118–120] implemented in
Gaussian [113] gives the electron density fictitious inertia such that its dynamics
can be included in the simulation. ADMP is thus an performance improvement to
plain BOMD.
Given a trajectory of the electric dipole 𝜇,⃗ the BOMD method proceeds to com-
pute the absorption spectrum as the inverse Fourier transform of the autocorrela-
tion of the centered electric dipole 𝜇̃ = 𝜇⃗ − ⟨𝜇⟩⃗ :
∞ ∞
𝜎(𝜔) ∝ 𝜔2∫ 𝑑𝜏 𝑒𝑖𝜔𝜏 ∫ 𝑑𝑡𝜇(̃ 𝑡) ⋅ 𝜇(̃ 𝑡 + 𝜏). (4.25)
−∞ −∞
45
Chapter 4. Computational Methods
Verlet Solver on Anharmonic Oscillator
Verlet HO Verlet AO
Simulation
Corrected !cor"t = 2sin(
1
2!sim"t)
correction
correction 1 2 1 4
prediction V (x) = 2x + 50x
Exact HO Exact AO
0 0.2 0.4 0.6 0.8 1
Time step !0"t
Figure 4.2: Principle and example of frequency shift compensation in the BOMD method.
Left: by simulating a harmonic oscillator (HO) with finite time step, a correction formula
can be derived and then applied to an anharmonic oscillator (AO). Right: AO frequency
as a function of simulation time step, raw and with correction.
This formula can be derived from linear response theory. [121, 122] Another way
to think about it is through the Wiener–Khintchine theorem, which states that the
(inverse) Fourier transform of the autocorrelation equals the squared absolute
value of the (inverse) Fourier transform. Thus Eq. (4.25) relates the absorption
spectrum to the Fourier spectrum.
When running BOMD simulations in practice, some parameters must be cho-
sen. The temperature or average mode energy is the most important, because
anharmonic effects scale linearly with simulation temperature. [123] The energy
distribution across modes can also be chosen in a few ways. [124] The duration of
the simulation relates to the spectral resolution of the Fourier transform. Finally,
the time step much be small enough to accurately describe the dynamics.
If the time step is far too large, energy conservation breaks down. However, if
the time step is only slightly too large, the consequence is a systematic frequency
shift. [123] This frequency shift depends only on the time step and the integration
scheme, but not on the molecule. For example, using Verlet integration with time
step Δ𝑡 to simulate a harmonic oscillator with angular frequency 𝜔0 results in a
trajectory with angular frequency 𝜔sim such that 𝜔simΔ𝑡 = 2 arcsin(12𝜔0Δ𝑡). [123]
This knowledge can be used to run simulations with a relatively high time step,
and the resulting frequency shift can be compensated for afterwards. Figure 4.2
shows this trick applied to a 1D anharmonic oscillator simulated at various large
time steps.
In spite of popular belief, the BOMD method does not adequately model an-
harmonic frequency shifts. This is easily seen by considering a toy system of a 1D
harmonic oscillator with a small quartic perturbation. The trajectory will then be
periodic with a fundamental frequency that depends on the energy. To match the
46
Frequency !=!0
1.03 1.04 1.05 1.06 1.07 1.08
4.6. Hydrogen Bonds
exact fundamental frequency of this quantum system, the temperature must sat-
isfy 𝑘B𝑇 ≈ ℏ𝜔, which is 1–2 orders of magnitude too high for diatomic molecules.
[125] Furthermore, the BOMD method predict that the first overtone is exactly
twice the fundamental, contrary to VPT2 (see Eq. (4.22)) and experiment. [126]
The strength of the BOMD method is rather to handle floppy system which do
not have a unique equilibrium geometry. An example of this is microsolvated clus-
ters where the water molecules can sit at many sites. [127, 128] Another notable
example where the BOMD method performed better than harmonic analysis was
on a capped dipeptide. [61]
4.6 Hydrogen Bonds
Hydrogen bonds (H-bonds) are attractive non-covalent interactions, most often
between hydrogen and either nitrogen, oxygen, or fluoride. The formal definition
of a H-bond is not agreed upon, but there is a suggestion. Johnson and coauthors
[129] suggest a classification of non-covalent interactions based on the electron
density and its derivatives. Specifically, three values are computed at every point:
the density 𝜌, the reduced density gradient (RDG) ∝ 𝜌−4/3|∇𝜌|, and the median
eigenvalue 𝜆2 of ∇∇⊺𝜌. H-bonds are then said to occur where the density is large,
the RDG is small, and where the median eigenvalue is negative.
Figure 4.3 shows an example analysis of triglycine. The threshold value of 0.4
is chosen for the RDG, selecting a subset of 3D space. The subset is colored ac-
cording to its sign(𝜆2)𝜌 value, and consists of four components, each marked with
a colored boundary in both halves of the figure. Two of them (gold and red) a
high density and are considered H-bond, while the other two (purple and blue)
are consider van der Waals interactions.
Figure 4.3: Non-covalent interaction analysis of a triglycine conformer. Left: scatterplot
of electron density and reduced density gradient (RDG). Some regions are highlighted.
Right: 3D structure of the molecule, together with the highlighted regions.
47

5
Results and Discussion
This chapter summarizes the original results found in the papers.
5.1 IRMPD Spectroscopy of Proton-Bound
Asparagine Dimers
This section summarizes the findings of Ref. [1], which used IRMPD to study
proton-bound dimers of asparagine (Asn). Specifically, the homo- and heterochi-
ral dimers were analyzed with the intent of detecting minute chiral differences.
Asparagine has four functional groups that can form hydrogen bonds (two amino,
one carbonyl, and one carboxyl), and therefore it was believed that the two di-
astereomers (the homo- and the heterodimer) would adopt significantly different
structures (see Fig. 5.1) stabilized by chiral-specific intermolecular interactions,
and that this would be visible in the IR spectrum. Because the laser intensity is
somewhat fluctuating, the two diastereomers were analyzed simultaneously to rule
out false differences.
Theoretical The theoretical investigation started with a conformer search, which
consisted of many parallel MD simulations in the DFTB+ software using the the
density functional-based tight binding method. Many different initial geometries
were obtained by randomly generating dihedral angles without concern to energy,
and then optimizing the structure. The MD simulations were then started with
velocities drawn from a 298K Maxwell–Boltzmann distribution. Frames of inter-
est of the resulting trajectories were structurally optimized into conformers by the
B3LYP functional with the 6-311++G** Pople basis set, with which their harmonic
spectra were also calculated. Anharmonic VPT2 spectra were also calculated, but
49
Chapter 5. Results and Discussion
Conf. DD-B1 Conf. DL-A1
Figure 5.1: The lowest Gibbs energy conformer of dd-Asn +2H and dl-Asn H
+
2 , according
to the G4MP2 method on a structure optimized with B3LYP-GB3BJ/6-311++G**. Inter-
and intramolecular interactions are shown as dashed lines annotated with their length in
Å. The conformers names contain A and B in reference to which functional group the
protonated amino group binds to.
are elided because of problems with Fermi resonances. Finally, their energies were
computed with the Gaussian G4MP2 method. The most stable (lowest Gibbs en-
ergy) conformer of each diastereomer is seen in Fig. 5.1.
Three different types of conformers were found, differing in which functional
group the protonated alpha amino group formed H-bonds with. [130] Type A im-
plies an H-bond with the alpha amino group of the other moiety, and type B with
the carboxyl group of the other moiety. Type Z is similar to type B, but also im-
plies that the neutral moiety is an zwitterion, meaning the proton on the carboxyl
has moved to the alpha amino group. Significantly, the most stable conformer of
the homodimer was type B, while the heterodimer was predicted to adopt type A,
a clear chiral effect seen in Fig. 5.1.
Experimental To make the results more understandable, a brief summary of the
experiment will be given. The experiment was conducted at Radboud University,
using the free electron laser FELIX in the frequency range of 500–1875 cm−1 but
also a table-top OPO laser in the frequency range of 3000–3600 cm−1. A solution
of both enantiomers (d-Asn and 15N-labeled l-Asn) was prepared and delivered to
the gas-phase using electrospray ionization. This produced dimers of three kinds:
dd, dl, and ll. Because the l-enantiomers were isotopically labeled, the three
kinds of dimers were mass resolved at 𝑚 = 265, 267, and 269Da. A quadrupole
filter was used to direct two of the three kinds into a Penning trap. By means of
Fourier transform ion cyclotron resonance, the mass spectrum of ions inside the
trap were continuously measured. The absorption spectra could of both diastere-
omers could then be determined by irradiating the trapped ions and measuring the
amounts of whole and fragment molecules.
50
5.1. IRMPD Spectroscopy of Proton-Bound Asparagine Dimers
500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 3400 3500 3600
Figure 5.2: Experimental IRMPD spectra of the three kinds of asparagine dimers: (top)
dd-Asn H+, (middle) dl-Asn H+2 2 , and (bottom) ll-Asn H
+
2 . The intensity scale is in
arbitrary units, but the horizontal grid lines have constant spacing in all three frequency
regions. The IRMPD rate is averaged over several scans to increase precision. Features
of interest are highlighted with black triangles. The frequency range of 3000–3350 cm−1 is
omitted because it contained no sharp features, only a very broad NH stretching.
Figure 5.2 shows the minutely different IRMPD spectra of all three types of as-
paragine dimers. The most noticeable effect is that some peak frequencies are non-
constant affine functions of mass. For example, the frequency of NH-stretching is
3415, 3411, and 3407 cm−1 for dd, dl, and ll respectively. This frequency shift
is due to the 15N-labeling and thus a proof of the presence of nitrogen in the ob-
served vibrational mode. Inversely, the frequency of OH-stretching is 3540 cm−1
for all three kinds, proving that nitrogen does not participate in the corresponding
vibrational mode. Following such logic, it is possible to assign vibrational modes
to the observed peaks.
Analysis Figure 5.3 shows a comparison of the experimental spectra and the pre-
dicted spectra of the most stable conformer of each type. Theoretical frequencies
are scaled with appropriate fixed constants. [58] The conformer type (A, B, or
Z) has direct implications for the spectrum, for example the carboxyl stretching
mode in the range of 1750–1850 cm−1. In this range, type A conformers have two
close bands, because the two carboxyl group have similar chemical environments.
Type B conformers, on the other hand, have one band redshifted, corresponding
to the carboxyl group that participates in the intermolecular H-bond. Finally, type
Z has only one band in this range, because the deprotonated carboxyl group has
51
555 556 557
565 565 565
591 591 591
759 762
782 782
835 835
854 849
1145 1147 1149
1296 1296 1296
1394 1398 1402
1413 1413 1413
1443 1446
1456 1456 1456
1590 1592 1594
1592 1597 1602
1692 1696 1702
1757 1761 1757
1784 1790 1786
3407 3411 3415
3493 3498
3503
3519 3519
3540 3540 3540
Chapter 5. Results and Discussion
600 800 1200 1400 1600 1800 3500 600 800 1200 1400 1600 1800 3500
Figure 5.3: Predicted spectra of stable asparagine conformers compared with experiment.
The B3LYP functional with the N07D basis set is used for calculating the vibrational fre-
quencies, which are then scaled and broadened. Two scaling constants are used: one below
2000 cm−1 and one above.
shifted all the way to 1660 cm−1. In experiment, two peaks were seen in the 1800–
1900 cm−1 range, ruling out type Z as a dominant conformer.
By combining the vibrational spectra of predicted conformers with the afore-
mentioned isotope shifts, it was possible to assign vibrational modes to all fre-
quencies greater than 1500 cm−1, and many lesser. The full assignment with proof
is given in the discussion of the original paper. [1] The colors of the bands in Fig. 5.3
refers to this assignment. Red colors are used to indicate movement of an oxygen,
such as the alcohol stretching at 3540 cm−1. Green indicates movement of the
nitrogen in the side chain amino group, which occurs in the symmetric and anti-
symmetric stretching at 3407 and 3519 cm−1. Blue is used in the modes where the
nitrogen of the alpha amino group moves, such as the umbrella motion of the pro-
tonated amino group at 1500 cm−1. Colors are combined when multiple apply, for
example the carboxyl group at 1790 cm−1 is colored both red (oxygen) and blue
(alpha nitrogen).
The assignment of vibrational modes can be used to infer the conformer type in
a more systematic fashion. By contrasting the experimental frequencies of a set of
modes with the theoretical modes of a conformer, it is possible to give a grade to
the combination of theoretical method and conformer. Figure 5.4 shows precisely
this for some five popular theoretical methods and the most stable conformer of
each type, using two measures. The first measure is the root-mean-square-error,
52
5.1. IRMPD Spectroscopy of Proton-Bound Asparagine Dimers
30
20
10
0
0.2
0.4
0.6
B3LYP/ B3LYP/ B3LYP/ M062X/ B97XD/ B3LYP/ B3LYP/ B3LYP/ M062X/ B97XD/
N07D aug-cc-pVDZ 6-311++G** 6-311++G** 6-311++G** N07D aug-cc-pVDZ 6-311++G** 6-311++G** 6-311++G**
Figure 5.4: Quantified difference between experimental and predicted spectra.√The top
half shows the root-mean-square-error of predictions, and the bottom half shows 1 − 𝑆2,
where 𝑆 is the Pearson correlation coefficient.
√
and the second is 1 − 𝑆2, where 𝑆 is the Pearson correlation coefficient. The
theoretical frequencies are scaled with appropriate fixed constants. [58]
Figure 5.4 shows that the B3LYP functional with the relatively small N07D ba-
sis set is marginally better than other methods, and that type B conformers best
predict the experimental spectra of both diastereomers. For the homodimer this
is expected, but the most stable conformer of the heterodimer is DL-A1. How-
ever, there one more conformer of the heterodimer, DL-B3, which is within 𝑘B𝑇
of DL-A1. The presence of both would explain the mixed indications from the
experiment: that type A is better around 1900 cm−1 but type B is better around
1400 cm−1.
Summary The existence of a chiral effect was not able to be proven experimen-
tally. While the initial belief was that the spectra would look different due to the
homodimer adopting a type B conformer and the heterodimer type A, in reality
the experiment showed very similar spectra. The biggest observed difference was
in the shoulder feature at 1450 cm−1. However, this difference is quite small and
is arguable an effect of the 15N labeling. Future experiments at cryogenic tem-
peratures, using techniques like cryogenic traps or storage rings, would increase
resolution of features and enhance capacity for chiral differentiation.
In summary, the homo- and heterochiral proton-bound dimers of asparagine
were studied with simultaneous IRMPD spectroscopy in the frequency ranges of
500–1875 cm−1 and 3000–3600 cm−1. By using 15N labeling and theoretical meth-
ods, it was possible to assign vibrational modes to most frequencies. No chiral
effect was proven to exist.
53
Chapter 5. Results and Discussion
5.2 IRMPD–VUV Spectroscopy of Neutral
Pentaalanine
This section summarizes the results of Ref. [2], which investigated neutral pen-
taalanine (Ala5) with IRMPD–VUV spectroscopy. In proteins, long sequences
of alanine residues are known to fold into alpha helices, and so it was curious to
see if gas-phase Ala5, b would do the same. This experiment was also interesting
because it was the largest molecule yet for IRMPD–VUV spectroscopy to be ap-
plied to. It was not obvious at the time that fragmentation would occur, because in
IRMPD–VUV the molecular beam is irradiated only for a short time, and larger
molecule naturally take longer to fragment.
Experimental The experiment took place at Radboud University, and used FE-
LIX in the frequency range of 340–1820 cm−1. The sample of Ala5 in dehydrated
form was delivered to the gas phase using MALDI. A supersonic jet expansion of
argon gas cooled down the molecules and swept them towards a skimmer. Only
the central, coolest part (approximately 10K) passed through the skimmer and
formed a molecular beam. That beam was then exposed to two laser beam pulses:
first an IR pulse from FELIX of duration 7 µs and energy 30–80mJ1 (depending on
frequency), and then a VUV pulse of duration 3ns and energy 1µJ. The resulting
ions were then counted by a reflectron-type time-of-flight mass spectrometer.
Because FELIX pulses are subdivided into micropulses separated by 1ns peri-
ods, the 7 µs FELIX pulse comprises 7000 micropulses, between which IVR oc-
curs. IVR has a timescale of 100 fs, so it is safe to assume that the absorbing mode
is relaxed after each 1ns period of darkness.
Figure 5.5 show the experimental IRMPD–VUV spectrum of Ala5 in the fre-
quency range of 340–1820 cm−1. The spectrum will be discussed after the theory.
1Ref. [2] incorrectly writes the unit as mW.
Ala5 IRMPD-VUV experiment.
400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800
Frequency (cm!1)
Figure 5.5: Experimental IRMPD–VUV spectrum of neutral Ala5, obtained with FELIX.
54
Intensity (arb.)
5.2. IRMPD–VUV Spectroscopy of Neutral Pentaalanine
C
G+ 0 G–
χi+1 A+ 180 A–
ψi ω Ti
φ ψCN φi+1
ωC
χ1 i=1–4
Figure 5.6: Definition of dihedral angles of Ala5. The part is brackets should be read as
repeated four times. There are four types of dihedral angles: C-N-C-C (𝜑), N-C-C-N (𝜓),
C-C-N-C (𝜔), and N-H-H-C (𝜒). Some exceptions occur at the endpoints, for example the
𝜑N angle is defined by H̄NCC, where H̄ is the center between the amino hydrogens. The
inset shows the rule for classifying dihedreal angles as cis (C), gauche (G), anticlinal (A),
or trans (T). Because the 𝜔 angles are locked into trans configuration, and the 𝜒 angles
only affect non-interacting methyl groups, conformers can be defined from only the 𝜑 and
𝜓 angles.
Theoretical Ala5 has a simple linear structure; neglecting methyl rotations, the
conformer is determined by 10 dihedral angles of the backbone. Figure 5.6 shows
the definition of all dihedral angles in the molecule.
Because Ala5 is a relatively large molecule, it is not feasible to sample its con-
formational space with energy-blind methods. Instead, the basin-hopping algo-
rithm scan from the Tinker software was employed to generate conformers. A
low-energy subset of these were further optimized with the B3LYP functional and
the Jun-cc-pVTZ basis set, and their harmonic frequencies were calculated with
the same method. The CBS-4M method was used to compute the Gibbs ener-
gies at 400K, an estimate of the temperature after laser desorption. To quanti-
tatively determine the existence and location of hydrogen bonds (H-bonds), the
NCI method [129] was employed.
Figure 5.7 shows the structure of the 8 most stable conformers, which have a
combined abundance of 95%, and are logically named A1–A8 after abundance
rank at 400K. Additionally, Fig. 5.7 shows the H-bonds of each conformer. Most
of the listed conformers form a closed loop by connecting the carboxyl group to
the amino or carbonyl group of the other end. This is not realistic for a protein,
and detracts from Ala5 as a model system for helices.
55
Chapter 5. Results and Discussion
H-bond strength (a!30 )
0.043 0.038 0.033 0.028 0.023 0.018
OO OO
A1 (0.0) 25% HH HHNN NN OO HH
TCG−A+G+G+G+CG−A+ H
H NN NN NN
HH HH HH
OO OO OO
OO OO
A2 (0.7 20% HH HH) NN NN OO HH
A−CA−A+G+G−G−G+G+A− H
H NN NN NN
HH HH HH
OO OO OO
OO OO
H H
A3 (1.6) 15% H HNN NN OO HH
H N N N
A−CG−G+G+G−G−G+G+G+ H N N NHH HH HH
OO OO OO
OO OO
H H
A4 (2.9) 10% H HNN NN OO HH
TCA−CG−G+G+G− −
H N N N
G A+ H N N NHH HH HH
OO OO OO
OO OO
A5 (3.0) 10% HH HHNN NN OO HH
A−CA−
H N N N
CG−G+G+G−G−A+ H N N NHH HH HH
OO OO OO
OO OO
A6 (4.4) 6% HH HHNN NN OO HH
A+CG−G+G+A−
H N N N
G−CA−G+ H N N NHH HH HH
OO OO OO
OO OO
A7 (5.2) 5% HH HHNN NN OO HH
A−CG−CA−G+
H N N N
G+G−G−T H N N NHH HH HH
OO OO OO
OO OO
H H
A8 (5.6) 5% H HNN NN OO HH
− − − + + − − − + HH NN NN NA G G G G A G CA G NHH HH HH
OO OO OO
Figure 5.7: Structure and hydrogen bonds (H-bonds) of the 8 most stable conformers of
Ala5. The left column shows the name, Gibbs energy at 400Kin kJ/mol, abundance, and
backbone dihedral angles of each conformer. The right column shows the locations of the
H-bonds of the corresponding structures. Theirs strengths are indicated by the color, and
non-covalent interactions just short of the H-bond definition (𝜌 ∈ [15, 18]× 10−3a −30 ) are
dashed.
56
5.2. IRMPD–VUV Spectroscopy of Neutral Pentaalanine
Experiment Prediction
CH3-wa COH-be am-III CH3-um am-II am-I
conformer-speci-c NH-wa NH2-tw/wa NH-ro CH-ro CH3-sc NH2-sc COOH
a A1 25%
b A2 20%
c A3 15%
d A4 10%
e A5 10%
f A6 6%
g A7 5%
h A8 5%
400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800
Frequency (cm!1)
Figure 5.8: Harmonic spectra of the 8 most abundant conformers as colored lines, com-
pared with the experimental IRMPD–VUV spectrum as black lines. The top of the figure
lists important vibrational modes at their frequency. The abbreviations used are: wagging
(wa), twisting (tw), bending (be), rocking (ro), amide (am), umbrella (um), and scissoring
(sc).
57
Intensity (arb.)
Chapter 5. Results and Discussion
Analysis The harmonic spectra of these conformers can be seen in Fig. 5.8, to-
gether with the experimental spectrum for comparison. Of particular interest is
the carboxyl C=O stretching mode at 1760 cm−1, clearly resolved in experiment
but predicted by only A1, A3, and A7. In A1 and A7 the carboxyl group is free
to move, and in A3 it is only weakly bond by an H-bond. In all other conformers,
the carboxyl group is connected by a strong H-bond to some other group, which
causes its stretching frequency to decrease. Another feature of interest is the car-
boxyl COH bending mode at 1135 cm−1, which is correctly predicted by A1 and
A7, but shifted to 1190 cm−1 by A3. Thus there is strong evidence for the presence
of A1 and/or A7, with the former being more likely due to the computed energies.
A1 and A7 cannot, however, explain the wide peak at 630 cm−1, likely a combi-
nation of several modes due to its width. A3 has many modes in the vicinity, but
also A2, A6, A7, and A8 are options. Another feature not explained by any of the
mentioned conformers is the lack of absorption at 820 cm−1, where only A4 (and
to a lesser degree A8) has a band. It is therefore likely that the experiment contains
many conformers, a conclusion consistent with the computed abundances.
Dynamical Spectra In addition to its harmonic analysis, the spectrum of A1
was also determined with the BOMD method using the same functional, though
with a smaller basis set: N07D. The BOMD simulations consisted of 30 runs using
a time step of 0.5 fs, each lasting for 2–3ps. The runs were made with Gaussian
using the ADMP keyword.
A consequence of the relatively√short run duration 𝜏 is that the spectral fre-
quency resolution is limited by 1/( 3𝜏). In the worst case of 𝜏 = 2ps, this res-
olution becomes 10 cm−1. Below 1000 cm−1, this value is greater than the 1%
linewidth assumed for FELIX, effectively broadening the far-IR BOMD spectra
more than experiment.
At temperatures 50K (used in Ref. [61]) and 100K, the results were very similar
to the harmonic spectrum, other than the aforementioned BOMD broadening.
This can be understood by the fact that the typical deviation from equilibrium is
too small to explore anharmonic effects because 𝑘B𝑇 ≪ ℏ𝜔.
On the other hand, at 500K and 1000K, the molecule eventually or immediately
fragmented in simulation. A quick calculation shows that at 400K the total energy
in all 153 vibrational modes of Ala5 would sum to 624kJ/mol, exceeding the lit-
erature [131] C-C bond energy of 346kJ/mol. This means that fragmentation is
permitted but not necessarily probable.
To evaluate whether the time step of 0.5 fs had shifted the simulated BOMD
spectra, the effect of time step on simulated frequencies was systematically inves-
tigated. Actually, as one finds from dimensional analysis, the relative shift depends
only on the product 𝜈Δ𝑡 of the frequency and the time step. It was therefore suf-
ficient to simulate a few small molecules in order to find this relation.
58
5.2. IRMPD–VUV Spectroscopy of Neutral Pentaalanine
#10-3
15 BOMD frequencies
Fitted power law
10
5
0
0 0.05 0.1 0.15
Harmonic frequency#"t
Figure 5.9: Dimensionless BOMD frequency shift of nine small biomolecules. The BOMD
frequencies were obtained by fitting Gaussian functions to the BOMD spectra of 1K sim-
ulations. The error bar size is equal to the full width half maximum of the corresponding
peak in its BOMD spectrum. A power law is fitted to the the frequency shifts.
Figure 5.9 shows the BOMD frequencies of a few small biomolecules simulated
at 1K with a rather large time step. The dimensionless frequency shift is well de-
scribed by a power law:
(𝜈D − 𝜈H)Δ𝑡 = 2.95(𝜈HΔ𝑡)3.25, (5.1)
where 𝜈H is the harmonic and 𝜈D is the BOMD frequency. Thus when Ala5 was
simulated using a time step of 0.5 fs, the frequency shift of its modes (below 1900 cm−1)
was at most 2 cm−1. Had the time step been 1 fs, the frequency shift would have
been up to 9 cm−1, a noticeable amount. In principle Eq. (5.1) could be used to
“unshift” the BOMD spectra to normal, but this was not done.
Summary The IRMPD–VUV spectrum of Ala5, the largest molecule studied
with this method, was determined in the frequency range of 340–1820 cm−1. A
theoretical investigation of the molecule predicted many populated conformers,
and a comparison of spectra confirmed this. Specifically, the carboxyl group was
very informative when inferring the structure, but also caused a loop-like rather
than a helix-like structure. Future experiments could seal this functional group to
improve the chances of seeing a helix. On a side note, the BOMD method was used
to predict spectra, but did not offer better results than harmonic theory, in spite
of its larger computational cost. The effect of time step on BOMD frequencies
was investigated and found to be systematic, implying that frequency errors are
correctable.
59
(BOMD!Harm.)#"t
Chapter 5. Results and Discussion
5.3 IR–UV Ion Dip Spectroscopy of Phenylated
Polyalanines
This section summarizes the findings of Ref. [4], which intended to overcome dif-
ficulties encountered in the study of Ala5. One of the conclusions of that study
was that Ala5 adopted a mixture of conformers. Because that experiment was not
conformer selective, it was difficult to say exactly which conformers were present.
The present paper solves this by replacing one alanine residue with phenylalanine.
Another problem was that the carboxyl group had a tendency to form strong H-
bonds with the opposite end, and in doing so prevented a helical structure. Hence,
caps have been added to both termini.
The present paper studied the two capped phenylated polyalanine peptides Ac-
Ala-Ala-Phe-Ala-NH2 (AAFA) and Ac-Ala-Ala-Phe-Ala-Ala-NH2 (AAFAA).
Utilizing the added phenyl group, conformer-specific IR–UV ion dip spectroscopy
was performed in the frequency range of 300–1900 cm−1 using FELIX and also
3200–3600 cm−1 using a table top OPO laser.
Theoretical An extensive conformational search was carried out for both AAFA
and AAFAA, the results of which is seen in Fig. 5.10. To begin with, four par-
allel conformer searches were executed for each molecule. The purpose of this
redundancy was to estimate the risk of missing conformers in the searches. Each
search consisted of the basin-hopping Tinker scan generating 25 000 conformers.
All found conformers were then optimized with the semi-empirical PM7 method,
AAFA-1 (0.00) AAFA-2 (23.85) AAFA-3 (26.29) AAFA-4 (26.49)
AAFAA-1 (0.00) AAFAA-2 (3.36) AAFAA-3 (3.99) AAFAA-18 (23.82)
Figure 5.10: Selected conformers of AAFA and AAFAA. Their corresponding Gibbs
energies are given in kJ/mol. AAFAA-18 is included because of its predicted spectrum.
60
5.3. IR–UV Ion Dip Spectroscopy of Phenylated Polyalanines
O O
H H H
N N NH
N N
H H
O O O
AAFA-1 Ph
Figure 5.11: The strongly dominant conformer AAFA-1, as a 3D structure and as a skele-
ton formula. The bright cyan lines show the H-bonds that stabilizes its β-hairpin structure.
which has a better correlation with DFT. The most stable conformers, meaning
those with energy within 21kJ/mol = 𝑘B ⋅ 2500K of the minimum, were further
optimized with B3LYP/Jun-cc-pVTZ and finally evaluated with CBS-4M.
The results for AAFA was remarkable: within 1000 iterations, each search found
the same extremely stable conformer, named AAFA-1 and seen in Fig. 5.11. This
conformer has a Gibbs energy 24kJ/mol lower than any other, because of its four
strong H-bonds, a tetramer configuration known as β-hairpin. [28, 132]
On the other hand, the results for AAFAA were not as clear. The parallel con-
former searches did not produce a common most stable (lowest Gibbs energy)
conformer, suggesting that 25 000 iterations were insufficient to reliably sample
the conformational space of AAFAA. For this reason, a new generation of four
more searches of 25 000 iterations each were performed. This new generation pro-
duced a new most stable conformer, but simultaneously failed to find the top three
most stable conformers from the previous generation. It was clear that this type
of search was not reliable for AAFAA, and other strategies were considered.
Another conformer search was then performed in the CREST software, which
uses a sophisticated replica-exchange MD algorithm. Similarly to before, this con-
former search found a new most stable conformer, but failed to find other low-
energy conformers. In fact, the top three most stable conformers had only been
found once — in three different searches. At this point, the non-reproducible na-
ture of the searches became an interesting message in and of itself.
To better understand why the conformational searches of AAFAA were failing,
another generation of four searches of 25 000 iterations each was carried out for
the capped pentaalanine peptide (AAAAA). These were similarly inefficient at
finding a most stable conformer, likely due to the large size of the molecule.
In a final act of desperation for AAFAA conformers, phenyl groups were sub-
stituted onto the central residue of the nine most stable AAAAA conformers, and
the molecule was optimized using B3LYP/Jun-cc-pVTZ. This resulted in AAFAA
conformers with a very similar backbone structure. Unfortunately, none of the
conformers produced were significant. Moreover, as seen in Fig. 5.12, the addi-
tion of the phenyl group did not preserve conformer rank.
61
Chapter 5. Results and Discussion
20 AAAAA
AAFAA
15
10
5
0
1 2 3 4 5 6 7 8 9
Rank of initial AAAAA conformer
Figure 5.12: Computed energies of AAAAA and the corresponding AAFAA conformers.
B3LYP was used to compute the electronic energy with added zero-point correction. The
computation phenylated version of AAAAA-4 did not converge in time.
Experimental The experimental setup was the same as for Ala5, except that the
VUV laser setup was substituted for a tunable UV laser setup, covering the fre-
quency range of 268.3–262.4nm = 37 280–38 120 cm−1. To begin with, a REMPI
scan was performed in this interval for both molecules, revealing resonances (3 for
AAFA, 1 for AAFAA) in a subset of the range, shown in Fig 5.13.
The UV laser was then parked on each REMPI resonance while the IR laser
scanned its entire range. Figure 5.14 shows the resulting three IR–UV ion dip
spectra of AAFA, one for each resonance in its REMPI spectrum. The three spec-
tra are extremely similar, proving that they all three resonances come from the
same conformer. This is consistent with the prediction of a dominant conformer.
Looking back at Fig. 5.13. the leftmost peak at 𝜈0 = 37 498 cm−1 is believed
to be the fundamental transition, and the other two are most likely vibrational
overtones 37 515 cm−1 = 𝜈0+17 cm−1 and 37 530 cm−1 = 𝜈0+32 cm−1. Indeed, the
ground state vibrational spectrum calculated with B3LYP/Jun-cc-pVTZ contains
two small frequencies: 17.0 and 32.3 cm−1.
AAFA 37498 AAFAA 37562
37515
37530
37475 37500 37525 37550 37500 37525 37550 37575
UV frequency (cm!1)
Figure 5.13: Experimental REMPI spectra of AAFA and AAFAA. The curves show the
ion yields of the molecules, averaged over multiple scans. Each of the labeled peaks in the
AAFA spectra was used for ion dip spectroscopy.
62
Ion yield (arb.)
B3LYP EE+ZPE (kJ/mol)
5.3. IR–UV Ion Dip Spectroscopy of Phenylated Polyalanines
UV freq. (cm!1)
37498
37515
37530
400 500 600 700 800 900 1000 1100 1200 1300 1400
1450 1500 1550 1600 1650 1700 1750 3250 3300 3350 3400 3450 3500 3550
IR frequency (cm!1)
Figure 5.14: Experimental IR–UV ion dip spectra of AAFA. Red, yellow, and green colors
used to draw the absorption signal signify the UV frequency used, as seen in the legend.
Above 3000 cm−1, the spectra are vertically offset to reveal otherwise identical features.
Below 800 cm−1, there is a difference in saturation, possibly because the IR intensity was
not correctly recorded. Overall the spectra are very similar and most likely originate from
the same conformer.
Analysis Figure 5.15 shows the predicted harmonic spectra compared to the ion
dip experiment of both molecules.
Starting with AAFA, where the conformation search was unambiguous, the
agreement between experiment and theory is quite good. In the stretching re-
gion above 3000 cm−1, four out of five predicted bands align with experiment. The
erring band is a NH stretching mode at 3295 cm−1, corresponding to N terminus
NH group. This group participates in the strongest H-bond in the molecule, pos-
sibly making its potential anharmonic and causing the harmonic prediction to fail.
Continuing with AAFA, the agreement is a bit difficult to assess below 2000 cm−1
because of the sheer number of features. Most major features are accounted for by
the predicted harmonic spectra of AAFA-1, and this conformer certainly fits the
spectrum better than the alternatives. Overall, AAFA-1 was correctly predicted
to be the dominant conformer and matches the experimental ion dip spectrum.
The dominant conformer AAFA-1 adopts a β-hairpin structure, characterized
by a β-turn in the middle. Phenylated alanine tripeptides also include this β-turn,
[50] but longer alanine peptides are expected to form a α-helices. [133]
Looking at AAFAA, the agreement is worse. The experimental spectrum con-
tains a band at 3510 cm−1, believed to be free or weakly interacting NH2 stretching.
The most stable conformer with this feature is AAFAA-18. Even if its high energy
is overlooked, AAFAA-18 is not a great fit for the rest of the spectrum. Therefore,
63
Absorption (arb.)
Chapter 5. Results and Discussion
AAFA-1 Exp.
400 500 600 700 800 A90A0 F10A00-21100 1200 1300 1400 1500 1600 1700 3300 3400 3500
400 500 600 700 800 A90A0 F10A00-31100 1200 1300 1400 1500 1600 1700 3300 3400 3500
400 500 600 700 800 A90A0 F10A00-41100 1200 1300 1400 1500 1600 1700 3300 3400 3500
400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 3300 3400 3500
IR frequency (cm!1)
AAFAA-1 Exp.
400 500 600 700 800 A90A0 F10A00A1-1200 1200 1300 1400 1500 1600 1700 3300 3400 3500
400 500 600 700 800 A90A0 F10A00A1-1300 1200 1300 1400 1500 1600 1700 3300 3400 3500
400 500 600 700 800 A90A0 F10A00A1-110081200 1300 1400 1500 1600 1700 3300 3400 3500
400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 3300 3400 3500
IR frequency (cm!1)
Figure 5.15: Harmonic spectra of conformers of AAFA and AAFAA as colored lines
versus experimental IR–UV ion dip spectra in black. The harmonic spectra are calculated
with B3LYP/Jun-cc-pVTZ and scaled by 0.980 below 2000 cm−1 and 0.955 above.
64
Absorption (arb.) Absorption (arb.)
5.3. IR–UV Ion Dip Spectroscopy of Phenylated Polyalanines
it is likely that the conformer searches of AAFAA failed to find the true conformer
seen in experiment.
The failure of a large conformer searches to fully explore AAFAA raises ques-
tions about how many iterations are needed to explore a given molecule, or more
generally when a conformer search can be said to have converged. The present
paper used a naive strategy of running multiple searches in parallel, and requiring
their results to be similar to each other. This is computationally inefficient because
the same structures are optimized and explored multiple times. Instead, the search
algorithm could be rewritten to remember how many times each conformer have
been found, and terminate only if all the most stable conformers have been found
multiple times.
Summary The IR–UV ion dip spectra of AAFA and AAFAA were successfully
recorded in the frequency ranges of 300–1900 and 3200–3600 cm−1. Ironically, both
molecules adopted a singe dominant conformer, so the conformer-specific nature
of the experiment was unnecessary. The dominant conformer of AAFA was found
to be a β-hairpin structure common to multiple tetramers, and was validated on
both Gibbs energy and vibrational spectrum.
The slightly larger AAFAA molecule proved a challenge to analyze. Multiple
large (𝑁 > 200 000) conformer searches were carried out, but the results were
not conclusive. From comparison with experiment, it seems likely that the true
conformer was not found. This is a sobering result about the validity of conformer
searches done on large molecules, which frequently employ far smaller conformer
searches.
Corresponding AAFAA and AAAAA conformers were compared and found
to have differing energy ranks, implying that the addition of a phenyl group changes
which conformers are the most stable in a molecule. This is interesting because
phenyl groups are often added onto molecules to enable IR–UV ion dip spec-
troscopy, but the conclusions about such phenylated molecules should not imme-
diately be transferred to their original counterparts.
65

6
Conclusions and Outlook
This chapter briefly presents the conclusions of the experimental papers and the
theoretical methods. The implications of the latter on the field is discussed. Also,
the present state of the local laboratory is summarized.
Experimental Four spectroscopy techniques were applied in the experiments of
this thesis. IRMPD and IRMPD–VUV spectroscopy can be applied to arbitrary
charged and neutral molecules respectively, in contrast to REMPI and IR–UV ion
dip spectroscopy, which both require a chromophore. The three IR techniques are
in tandem with theoretical predictions sensitive to the molecular structure. In par-
ticular, positions of IR bands in the mid-IR range tell the chemical environments
of functional groups, which allows for many conformers to be excluded. Further-
more, the spectrum in the far-IR range is a fingerprint that allows a specific con-
former to be confirmed.
The IRMPD study of labeled and unlabeled asparagine dimers [1] demonstrated
how the vibrational modes of a molecule can be unambiguously assigned to exper-
imental bands. By tracing how the frequency minutely changed as a function of
mass labeling, it was possible to tell how much the labeled atom species partici-
pated in a given band. The study also demonstrated how the IR spectra of two
similar molecules could be recorded simultaneously, eliminating laser fluctuations
as a source of false differences.
The experimental investigation of pentaalanine [2] stressed a known weakness
of the IRMPD–VUV method: that it is not conformer-sensitive. Because of this
and due to the many populated conformers, it was not possible to precisely deter-
mine which conformers were actually seen in experiment. The vibrational modes
of the unique carboxyl group gave information about its chemical environment,
67
Chapter 6. Conclusions and Outlook
which while helpful for excluding conformers, is not realistic as a model system
for proteins.
Finally, the IR–UV ion dip study of capped phenylated polyalanines [4] gave
mixed results. Focusing on the tetramer for the moment, the experiment and the-
ory both found a strongly dominant conformer containing a central β-turn. Its
structure is common to other tetramers, and is known as β-hairpin. [132] This is
distinct from the α-helix structure that longer polyalanines are predicted to adopt,
meaning the future attempts to create gas-phase helices should use longer pep-
tides.
Theoretical The conformer search of AAFAA revealed that more than 200 000
iterations were not enough to sufficiently sample the conformational space. This
finding is motivated by two facts: that the most stable conformers were found only
once across several parallel runs, and that not one of their spectra matches the
experiment. It is a troubling message, as many studies have been made with con-
siderably fewer iterations, and may be similarly incomplete.
With the knowledge of this pathology in conformer searching, future similar
searches should include a test of method convergence. The study of AAFAA used
a naive test: comparing the result of parallel runs. This is computationally ineffi-
cient because it causes redundant calculations, but it does work. A better strategy
would be to extend the basin-hopping method to track not only which conformers
have been found, but also how many times they have been found. The latter could
then be used as a measure of convergence of the search.
The harmonic predictions of spectra using B3LYP or similar functionals are gen-
erally quite good, and capture the differences between conformers, as seen in the
experimental papers. However, harmonic theory does fail to describe some vibra-
tional modes involving groups that participate in a strong H-bond, such as the pro-
tonated amino group in the proton-bound dimer of asparagine. Most notably, the
stretching mode of this group was several 100 cm−1 broad, clearly not describable
by a single band. Also, its umbrella inversion mode was consistently mispredicted
by harmonic theory.
The anharmonic VPT2 method does generally increase the accuracy of individ-
ual mode frequencies, and does not require an arbitrary scaling factor. Unfor-
tunately, it occasionally produces some unphysically strong IR intensities, which
worsens the overall agreement with experiment. Because of the unpredictable na-
ture of this method, it could not be used to compare conformers.
The BOMD method for spectrum generation was employed for pentaalanine,
where it resulted in spectra similar to harmonic ones, and subsequently critically
analyzed. One conclusion from this analysis is that requiring BOMD to accurately
describe anharmonic shifts of fundamental frequencies results in multiple contra-
68
dictory constraints on the temperature. Specifically, each vibrational mode should
have a distinct temperature of ℏ𝜔/𝑘B, which is hot enough to make the molecule
dissociate. Another negative finding is that the BOMD method is incorrect for
overtone already in diatomic systems, where it incorrectly predicts the first over-
tone to be exactly twice of the fundamental.
On a more positive note, the analysis of the BOMD method found that the fre-
quency shifts due to the choice of time step are predictable and easily corrected,
allowing accelerated simulations. In fact, the shifts are described by a universal re-
lation dependent only on the integration scheme. Such a relation can be obtained
by simulating a plain harmonic oscillator with the given integration scheme, and
then be applied in post-processing of BOMD spectra to reduce the error. The uni-
versal relations of the Verlet integrator and the integration scheme of the Gaussian
ADMP method were provided in this work.
Laboratory Part of this PhD project has been about building a chamber setup
for experiments in the local GU laboratory, now known as the Giraffe. When I
started out, there were design plans inspired by a setup at FELIX, but nothing
was built. Since then, its design plans have been extended, the Giraffe has been
assembled, and REMPI experiments have performed with it.
The Giraffe features a switchable source system consisting of a gas source, a
oven source, and a MALDI source. Molecules produced from the selected source
are seeded into a supersonic jet expansion, and form a collimated molecular beam
after passing through a skimmer. Optical windows allow the molecular beam to be
irradiated, and any consequent ionization will accelerate the ions into a reflectron
type TOF mass spectrometer.
A REMPI experiment of toluene was performed using the gas source, with the
purpose of testing the mass spectrometer. The resulting mass spectrum showed
the toluene peak at 92Da with an FWHM of 0.043Da. This implies a resolution
greater than 1 in 2000, sufficient for the study of peptides with less than 20 residues.
In May 2025, a new IR laser (LaserVision) was delivered to the laboratory, and
will soon be installed at the time of writing. When this laser is operational, the cur-
rent REMPI setup will be able to perform IR–UV ion dip spectroscopy, enabling
conformer-specific studies of small biomolecules. I hope that this development
will be used by future students to advance the research of our group.
69

Acknowledgements
My five years as a PhD student have been a transformative experience filled with
learning, possible only thanks to some inspiring people I have met in Gothenburg
and while traveling. I will attempt to list the most prominent.
Although we only worked together for a brief amount of time, Vasyl Yatsyna
taught me much about spectroscopy experiments and supported me in designing
the Giraffe. His PhD defense was the first I attended, and it provided me with
valuable insights. On the theoretical side, Mathias Poline from Uppsala helped
me understand conformer searching and optimization, during the two papers we
collaborated on.
In the local laboratory, the specific expertise of Ruslan Chulkov was invaluable
for understanding and optimizing our laser systems. Di Lu was always very helpful
in answering questions about and finding design solutions for vacuum chambers.
Manufacturing parts for the Giraffe was made possible thanks to Jan-Åke Wiman
and his workshop. Installing high-voltage electrodes and water-cooling in turbo
pumps was safely and reliably managed by Mats Rostedt. Development of the
switchable sources were greatly aided by Viola D’mello.
Many of the experiments were performed in labs outside of Sweden. At the
ELETTRA synchrotron facility, I learned about coincidence spectroscopy data
analysis from Robert Richter. My understanding of statistical physics of molecules
and nanoparticles came from Klavs Hansen in the time we worked together. At
the FELIX facility, the hard work of Kas Houthuijs and later Piero Ferrari made
our experiments possible. The knowledge of Jos Oomens was also very useful
In parallel with my research, I have taught in courses at Chalmers and GU.
Thanks to Thomas Wernstål I was able to obtain teaching roles in mathematics,
notably a course by Dennis Eriksson, whom I returned to work with many times.
Elisabeth Eriksson was always helpful in finding substitute teaching opportunities
and lending course literature.
For ten years I have been active in the International Physicists’ Tournament, a
physics competition to which Andreas Isacsson initially introduced me. Thanks
to him, Jana Madjarova, and many others, this competition has expanded from a
student activity to a supervised course.
Finally, I owe the most to Vitali Zhaunerchyk, without whom I would not have
met half of the listed people. He has been instrumental in all parts of my education
as a researcher, and I am deeply appreciative for all of his contributions to my
education. Thank you, Vitali!
71

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