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City University of New York (CUNY) City University of New York (CUNY)
CUNY Academic Works CUNY Academic Works
Dissertations and Theses City College of New York
2021
Comparison of Calculated Normal Mode Molecular Vibrations Comparison of Calculated Normal Mode Molecular Vibrations
with Experimental Gas-Phase Infrared Spectroscopy with Experimental Gas-Phase Infrared Spectroscopy
Anila Renis Sutar CUNY City College
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Comparison of Calculated Normal Mode Molecular Vibrations
with Experimental Gas-Phase Infrared Spectroscopy
By
Anila R. Sutar
Submitted in partial fulfillment of the requirements for the degree of Master of
Science in Chemistry in the Department of Chemistry and Biochemistry of the City
College of New York
December 2021
Advisor:
Prof. Glen Kowach
Thesis Committee:
Prof. Urs Jans
Prof. Barbara Zajc
2
Table of Contents
Acknowledgements
Abstract
1. Introduction 13
1.1 Vibrational motion of molecules 15
1.1.1 Diatomic molecular vibrations 17
1.1.1.1 Harmonic oscillator 18
1.1.1.2 Anharmonic oscillator 21
1.1.2 Polyatomic molecular vibrations 23
1.1.2.1 Computationally smaller molecule 25
1.1.2.2 Computationally larger molecule 26
1.2 Interaction of light with matter 27
1.2.1 Transitions between different energy states 28
1.2.2 Study of matter using light 30
1.2.3 Transition dipole moment 31
1.2.4 Vibrational selection rules 33
1.2.5 Infrared Spectroscopy 34
1.2.5.1 FTIR principle and analysis process 36
2. Computational Chemistry 39
2.1 Correct approach for calculations 40
2.2 Introduction to Psi4 42
3
2.3 Best Computational methods 42
2.4 Type of methods 43
2.4.1 Hartree-Fock (HF) Theory 43
2.4.1.1 Restricted Hartree-Fock (RHF) [Default] 45
2.4.1.2 Unrestricted Hartree-Fock (UHF) 45
2.4.1.3 Restricted Open-Shell Hartree-Fock (ROHF) 45
2.4.2 Coupled Cluster Theory 46
2.4.3 Density Functional Theory 47
2.4.4 Møller-Plesset Perturbation Theory 48
2.5 Basis sets 49
2.6 Example of input files and output files 50
2.6.1 Input files 50
2.6.2 Output files 51
2.7 Performed calculations on several molecules 54
3. Experimental Methods 55
3.1 FTIR 55
3.2 Avogadro 57
3.3 Psi4 57
4. Results and Discussion 59
4.1 Scaling factor 59
4.2 Diatomic molecules 60
4.2.1 H2 - Dihydrogen 61
4.2.2 C2 - Dicarbon 62
4
4.2.3 HCl - Hydrogen Chloride 64
4.2.4 CO - Carbon Monoxide 66
4.3 Triatomic molecule 69
4.3.1 H2O - Water 69
4.4 Trigonal planar molecule 73
4.4.1 CH2O - Formaldehyde 73
4.5 Tetrahedral molecule 76
4.5.1 CH4 - Methane 76
4.6 Other polyatomic molecules 79
4.6.1 CH2O2 - Formic acid 79
4.6.2 CH3COOH - Acetic acid 82
4.6.3 C3H6O - Acetone 85
4.6.4 C6H12 - Cyclohexane 88
5. Conclusion 92
6. References 93
5
List of Figures
Figure 1: Visualization of all possible molecular vibrations[11] [Reprinted from “M. Mutsaers,
Zika vector control: Near infrared spectroscopy predicting Wolbachia infection in post-mortem
Aedes aegypti, 4365/328726218, June 19, (2018).”]
Figure 2: It implies that when a force is exerted to a spring, the spring's extension is proportional
to the force applied[25][Reprinted from “J. P. Carter, SP - 1, EP - 27, Who needs Constitutive
Models? Australian Geomechanics Vol 41 No 2 June (2006).”]
Figure 3: A particle's potential energy that can be mapped using simple harmonic oscillations [26]
[Reprinted from “S. J. Ling, J. Sanny and B. Moebs, The Quantum Harmonic Oscillator.,
November 5, OpenStax CNX, https://phys.libretexts.org/@go/page/4531, (2020).”]
Figure 4: Morse potential curve and vibrational energy levels [20] [Reprinted from “A. sanli, M.
Lyyra, Transition Dipole Moment and Lifetime Study of Na2 And Li2 Electronic States Via Autler-
Townes And Resolved Fluorescence Spectroscopy, May 1, (2017).”]
Figure 5: Potential energy surface of water molecule [23] [Reprinted from “Lewars, G. Errol
Computational Chemistry Volume 98 || The Concept of the Potential Energy
Surface.,10.1007/978-90-481-3862-3(Chap. 2), 9–43. doi:10.1007/978-90-481-3862-3_2,
(2011).”]
Figure 6: An illustration of the electromagnetic spectrum, emphasizing the narrow window of
visible light that can be detected by the human eye [24] [Reprinted from “J. Zwinkels, Light,
Electromagnetic Spectrum. Encyclopedia of Color Science and Technology, 1–
8. doi:10.1007/978-3-642-27851-8_204-1, (2015).”]
6
Figure 7: Transitions between different energy states of molecules after interaction with a photon
are illustrated in Jablonski diagram, here S0 is ground state and S is excited state [27]
[Reprinted from “U. Gehlsen, A. Oetke, M. Szaszák, N. Koop, F. Paulsen, A. Gebert, G.
Huettmann, P. Steven, Two-photon fluorescence lifetime imaging monitors metabolic changes
during wound healing of corneal epithelial cells in vitro. , 250(9), 1293–
1302. doi:10.1007/s00417-012-2051-3, (2012).”]
Figure 8: The fundamental elements of a Fourier transforms infrared spectrometer[36] [Reprinted
from “M. A. Mohamed, Membrane Characterization || Fourier Transform Infrared (FTIR)
Spectroscopy, 3–29. doi:10.1016/B978-0-444-63776-5.00001-2, (2017).”]
Figure 9: Internal features of the Fourier transform infrared spectrometer[37] [Reprinted from
“F. Zadeh, F. Anaya, Nelson M., Schiffman, Laura A., O. Craver, Vinka. Fourier transform
infrared spectroscopy to assess molecular-level changes in microorganisms exposed to
nanoparticles. Nanotechnology for Environmental Engineering, 1(2), 1–. doi:10.1007/s41204-
016-0001-8, (2016).”]
Figure 10: Typical Fourier transform infrared spectrum of cellulose membranes with the different
types of bonds those are common [36] [Reprinted from “M. A. Mohamed, Membrane
Characterization || Fourier Transform Infrared (FTIR) Spectroscopy, 3–29. doi:10.1016/B978-0-
444-63776-5.00001-2, (2017).”]
Figure 11: FTIR vibrational spectra of HCl molecule [Reprinted from “Data from NIST Standard
Reference Database 69: NIST Chemistry Webbook”]
7
Figure 12: FTIR vibrational spectra of carbon monoxide molecule [Reprinted from “Data from
NIST Standard Reference Database 69: NIST Chemistry Webbook”]
Figure 13: FTIR vibrational spectra of water molecule [Reprinted from “Data from NIST Standard
Reference Database 69: NIST Chemistry Webbook”]
Figure 14: FTIR vibrational spectra of CH2O molecule [Reprinted from “Data from NIST
Standard Reference Database 69: NIST Chemistry Webbook”]
Figure 15: FTIR vibrational spectra of CH4 molecule [Reprinted from “Data from NIST Standard
Reference Database 69: NIST Chemistry Webbook”]
Figure 16: Collected high-resolution FTIR spectra of formic acid
Figure 17: Collected high-resolution FTIR spectra of acetic acid
Figure 18: Collected high-resolution FTIR spectra of acetone
Figure 19: FTIR vibrational spectra of Cyclohexane molecule [Reprinted from “Data from NIST
Standard Reference Database 69: NIST Chemistry Webbook”]
8
List of Tables
Table 1: Vibrational and/or electronic energy levels water molecule
Table 2: Vibrational and/or electronic energy levels of methylamine
Table 3: Performed computational calculations on several molecules
Table 4: List of Samples characterized using FTIR
Table 5: Listed molecules to analyze the cost effectiveness of computational methods and basis
sets
Table 6: Scaling factor value varies on different theory and basis set
Table 7: Psi4 computational calculations of H2 molecule
Table 8: NIST computational calculations of H2 molecule
Table 9: Psi4 computational calculations of C2 molecule
Table 10: NIST computational calculations of C2 molecule
Table 11: Psi4 computational calculations of HCl molecule
Table 12: NIST computational calculations of HCl molecule
Table 13: Comparison of experimental with calculated frequency of HCl molecule
Table 14: Psi4 Computational calculations of CO molecule
Table 15: NIST computational calculations of CO molecule
Table 16: Comparison of experimental with calculated frequency of CO molecule
Table 17: Psi4 computational calculations of H2O molecule
Table 18: NIST computational calculations of H2O molecule
Table 19: Comparison of calculated and experimental vibrational frequencies of water
Table 20: Psi4 computational calculations of CH2O molecule
9
Table 21: NIST computational calculations of CH2O molecule
Table 22: Comparison of experimental and calculated frequencies of formaldehyde
Table 23: Psi4 computational calculations of CH4 molecule
Table 24: NIST computational calculations of CH4 molecule
Table 25: Comparison of experimental and calculated frequencies of methane
Table 26: Comparison of experimental and calculated frequencies of formic acid
Table 27: Comparison of experimental and calculated frequencies of acetic acid
Table 28: Comparison of experimental and calculated frequencies of acetone
Table 29: Comparison of fundamental and scaled frequencies of cyclohexane
10
Acknowledgements
It cannot be argued with that the most influential person for my thesis work, Dr. Glen
Kowach my thesis advisor, who give me an opportunity to carry out my research work in his lab.
My thesis would not have been possible without his kind support and encouragement. I would like
to thank him for his invaluable support has guided me through these past two years.
I feel greatly honored to acknowledge the contribution and continuous guidance of Dr. Urs
Jans as my thesis committee member.
I owe my most sincere gratitude to Dr. Barbara Zajc as my graduate advisor as well as a
thesis committee member. I am so grateful for her advice and untiring help.
A special thank you goes out to my lab partner Estefania Herrera, for being incredible all
the time.
I would like to express my deep and sincere thanks to my best friends for their continuous
encouragement and understanding, starting from my research work till its end.
I express my gratitude to my family whose selfless love, constant encouragement, sincere
prayers, expectations, and blessings have always been vital source of inspiration in my life.
11
Abstract
Computational vibrational spectroscopy serves as an important tool in the interpretation of
experimental infrared (IR) spectra. Analysis of computational results provides a perspective over
broader wavelength ranges and at higher precision. Although there are issues regarding accuracy,
this can be approximated by using a scaling factor. High-resolution gas-phase FTIR spectroscopy
at a resolution of 0.125 cm-1 can partially resolve rovibrational transitions in the P, Q, and R bands
and therefore identify fundamental frequencies with approximately 1 cm-1 precision.
This research has compared high-resolution gas-phase FTIR absorption peaks to calculated
vibrational frequencies. In the calculation of normal mode frequencies, reliability, feasibility, and
ease of interpretation are still a matter of concern due to the interplay of several factors (time,
accuracy, and memory requirement of computer hardware). Calculations were performed using
different methods (theories and basis sets), such as, B3LYP-D/cc-pVDZ, MP2-cc-pVDZ,
CCSD/cc-pVDZ, CCSD(T)/aug-cc-pVDZ, CCSD(T)/aug-cc-pVTZ, CCSD/cc-pVDZ and
CCSD/cc-pVTZ on different diatomic and polyatomic molecules using the ab initio computational
software package, Psi4. H2, C2, HCl and CO required 15-20 minutes to get the output file with ± 0
to 8 cm-1 variations compared to experimental values. Couple cluster theory (CC) yields quite
accurate calculations for diatomic molecules. Whereas, with polyatomic H2O, CH4 and CH2O
required a time of 45 minutes to 1 hour with variation of ± 0 to 20 cm-1. For H2O couple cluster
theory is best. For CH4 and CH2O density functional theory (DFT) provides the nearly similar
number to experimental frequencies. The molecules, CH2O2, CH3COOH and C3H6O, required up
to 48 hours due to the number of electrons (N) in each molecule and the number of calculations
needed for a particular theory, for example MP2 scales as O(N5) and CCSD(T) scales as
O(N7). Vibrational frequencies were calculated using only B3LYP-D/cc-pVDZ as a
12
computational method and basis set for polyatomic molecules, since it requires less computational
power as compared to other methods and basis sets. In addition, a slightly larger molecule,
cyclohexane (C6H12), was used to evaluate the effectiveness of B3LYP-D/cc-pVDZ. Larger
molecular systems have the conflict between accuracy and computational cost in terms of time,
accuracy, and memory requirement of the hardware. In contrast, the smaller molecules consume
less time to calculate and make it possible to use a high level of theory that gives accurate results.
The initial goal of this research was to find the best theoretical method and basis set for
diatomic molecules such as HCl. The HCl calculated vibrational frequency is 2999 cm-1, while
experimental frequency is 2991 cm-1 with an 8 cm-1 difference in comparing experimental values
with the CCSD(T)/cc-pVTZ method. For C3H6O using B3LYP-D/cc-pVDZ, the observed Psi4
frequency of the CH3 degenerate stretching (d-str) vibrational mode is 1425 cm-1, while the
experimental value is 1426 cm-1 and demonstrates very small difference of 1 cm-1. Density
functional theory proved accurate for polyatomic molecules, while for diatomic molecules couple
cluster theory performed well. CCSD(T)/cc-pVTZ required less than 500 megabytes for diatomic
molecules. To calculate energy, Psi4 needed 8 s while optimization was 1 m 59 s and frequency
was 2 m 35 s for HCl. To calculate energy Psi4 needed 1 h 5 m, while optimization 54 m 19 s and
frequency took 1 h 54 m for calculation for acetone. High-resolution FTIR gas-phase spectra of
molecules were collected and compared to calculated molecular vibrational frequencies. This
study revealed that couple cluster theory is the best approach for diatomic molecules, and density
functional theory is more accurate for polyatomic molecules.
13
1. Introduction
Infrared radiation is electromagnetic radiation that is slightly less energetic than visible
light. Infrared radiation can be absorbed by molecules, which causes them to vibrate at different
frequencies. For example, in the infrared spectrum of water, the frequencies at which water absorbs
radiation and thus vibrates are observed. We used the ab initio computational software package,
Psi4, to calculate the normal modes of vibration that would be observed in the infrared spectrum
of a molecule.
Analysis of the interaction of compound with electromagnetic waves is known as
spectroscopy. With the help of Infrared (IR) Spectroscopy, scientists can analyze the interaction
of light and matter. It has been a valuable tool in the study and characterization of a wide variety
of molecular systems in chemistry. As a result, several modes of interaction exist for radiation:
absorption, emission, diffraction, impedance, resonance, and inelastic scattering. As a result, it is
used to characterize/detect matter (atoms, molecules, and nuclei) by observing the spectra
produced and their interaction with radiation. [1-2]
Infrared spectroscopy is extensively used in both industry and research. Chemists use it to
evaluate functional groups in substances and to elucidate the structures of organic compounds. IR
spectroscopy is a technique to determine the functional groups of atoms by evaluating their
vibrations. The structure of a complex molecule is instructive to study. [3]. It can evaluate the
symmetry of an organic and inorganic molecule. Additionally, it provides precise values for
pertinent spectroscopic data, including band positions and absorption cross sections. [4] It is a
straightforward and dependable technique being used in the industries for unit of measure, quality
inspection with improvement, and robust measurement. It aids in the study of reaction progression.
14
IR spectroscopy is a technique to evaluate the presence of particulates in a compound.
Additionally, the presence of water in a sample can be determined.
The ability of state-of-the-art quantum mechanical (QM) methods to predict molecular
properties has been demonstrated to be critical for the analysis of molecular systems. In recent
years, theoretical computations have developed into powerful and widely used tools for assigning
and predicting experimental spectra, as well as for gaining a better understanding of the various
effects that contribute to the observed spectroscopic properties. [5-6] In the case of infrared
spectroscopy, QM calculations at an appropriate level of theory enable the forecasting of reliable
vibrational spectra for small to medium-sized molecules. [7]
This thesis is motivated by the desire to obtain computational normal vibrational modes of
frequency using by different theories and basis sets on the Psi4 platform. To obtain accurate
calculations a few combinations of theories and basis sets were evaluated. To obtain high-quality
IR data, it is necessary to collect the spectra in the gas-phase using high-resolution FTIR. We
compared low-resolution IR data from the literature to our high-resolution data. Since high-
resolution data is more effective so, we compared our experimental high-resolution gas-phase
FTIR data to our calculated IR data. The purpose of this research is to encourage future chemists
to use high-resolution gas-phase FTIR as their preferred spectroscopic technique as well as use
that data to compare with calculated vibrational frequency and find the effective computational
method and basis set to gain the accurate computational vibrational data.
15
1.1 Vibrational motion of molecules
There are three basic types of motions that demonstrate themselves such as translations
which is external motion, rotations which is internal motion, and vibrations which is internal
motion. In contrast to diatomic molecules, polyatomic molecules exhibit more complex vibrations,
known as normal modes, which are more diverse. Due to the lack of an experimental method for
directly observing how atoms move at a particular frequency, the explanation of the experimental
molecular vibrational spectrum is highly dependent on the theoretical analysis of vibrational
modes. [8,9]
While a molecule has translational and rotational motion, each atom has its own motion relative
to the other atoms. An IR or Raman spectrum represents the vibrational modes. To observe a
mode in the infrared spectrum, changes must take place in the permanent dipole (i.e., not
diatomic molecules). Raman spectroscopy detects diatomic molecules but not infrared
spectroscopy. Since homonuclear diatomic molecules where the two atoms are the same, only
have one band but no permanent dipole, so they only have one vibration. For instance, H2,C2,
N2. [10] Unsymmetric diatomic molecules, on the other hand (e.g., CO), do absorb in the
infrared spectrum. There are more complex vibrations in polyatomic molecules that can be
summed up or resolved into the normal modes of vibration of the molecules. When more than
two atoms interact, a variety of vibrations such as stretching, scissoring, rocking, wagging, and
twisting can occur. Visualization of all possible molecular vibrations; arrows indicate
movements caused by electronegativity-induced repulsion. Specific molecules vibrate at a
particular frequency, which absorbs energy when it equals that of light shown in figure 1.
16
Figure 1: Visualization of all possible molecular vibrations[11] [Reprinted from “M. Mutsaers,
Zika vector control: Near infrared spectroscopy predicting Wolbachia infection in post-mortem
Aedes aegypti, 4365/328726218, June 19, (2018).”]
The degree of freedom refers to the number of different factors required to completely
describe the motion of a particle. Three coordinates are sufficient for an atom moving in three-
dimensional space, indicating that its degree of freedom is three. It moves in a purely translational
fashion. When a molecule is composed of N atoms (or ions), the degree of freedom increases to
3N, as each atom has three degrees of freedom.
The degree of freedom is equal to the sum of the vibrational mode and the rotational mode
plus the translational mode.
Additionally, because these atoms are bonded, not all motions are translational; some
become rotational, while others become vibrational. For non-linear molecules, all rotational
motion can be described in terms of rotations around three axes; the rotational degree of freedom
is three, while the remaining 3N-6 degrees of freedom are associated with vibrational motion.
However, rotation around one's own axis is not rotation for a linear molecule because it does not
17
change the molecule. Thus, any linear molecule has only two degrees of freedom of rotation,
leaving 3N-5 degrees of freedom for vibration. [12,13]
The total number of Degrees of Freedom (DoF) for linear molecules is (3N - 5) whereas
for non-linear molecule is (3N - 6). There is a difference in the type of radiation produced by IR
and visible and ultraviolet radiation because the IR radiation has less energy. For molecules with
low energy differences between rotation and vibration, IR radiation absorption is a common
phenomenon. A molecule's dipole moment changes when it vibrates or rotates, which is a criterion
for IR absorption. For example, the charge distribution between hydrogen and bromine in HBr is
not evenly distributed, because bromine is more electronegative and has a greater electron density
than hydrogen. Thus, HBr has a high dipole moment and is polar. Charge difference and distance
between the two centers of charge determine dipole moment. To understand how radiation
interacts with matter, it is necessary to understand how the dipole moment of the molecule
fluctuates as it vibrates. Absorption occurs when the radiation's frequency matches the molecular
vibration, and this alters the molecular vibration's magnitude. This also occurs when asymmetric
molecules rotate around their centers, resulting in a change in their dipole moment, allowing them
to interact with the radiation field. (Note that because molecules such as O2, N2, I2, and Br2 do not
exhibit a changing dipole moment during vibrational and rotational motions, they cannot absorb
infrared radiation.) [14]
1.1.1 Diatomic molecular vibrations
There are two types of diatomic molecules exist. One is homonuclear and another one is
heteronuclear. H2,N2,C2, I2 consider as homonuclear and it has no dipole moment between atoms
able to absorb IR radiation. While with heteronuclear molecules for instance HBr, HCl, CO have
18
high dipole moment and is polar. Heteronuclear diatomic molecules can absorb IR radiation
because of electronegativity of atoms while homonuclear are not. A precise illustration of a
molecule's potential energy as a function of interatomic distance is required for determining its
molecular structure. The analytical potential energy function provides the most comprehensive and
compact information about the molecule's structure.
1.1.1.1 Harmonic oscillator
The harmonic oscillator is a fundamental model in both classical and quantum mechanics.
It serves as a model for the mathematical treatment of a wide variety of phenomena, including
elasticity, acoustics, alternating current circuits, molecular and crystal vibrations, electromagnetic
fields, and matter's optical properties.
In classical mechanics, a simple realization of the harmonic oscillator is a particle that is
subjected to a restoring force proportional to its displacement from its equilibrium position. In
terms of motion in a single dimension, this means
𝐹 = −𝑘𝑥 Eq. 1.1
The force constant k represents the spring's stiffness. At equilibrium, the variable x is set
to zero, positive for stretching, and negative for compression. Equation 1.1's negative sign reflects
the fact that F is a restoring force that always acts in the opposite direction of the displacement x.
19
Figure 2: It implies that when a force is exerted to a spring, the spring's extension is
proportional to the force applied[25][Reprinted from “J. P. Carter, SP - 1, EP - 27, Who needs
Constitutive Models? Australian Geomechanics Vol 41 No 2 June (2006).”]
As illustrated in Figure 2, such a force could originate from a spring that obeys Hooke's
law. The restoring force is determined by Hooke's law, which can be applied to a real spring with
sufficiently small deflections. This means that the restoring force is proportional to the deflection.
Figure 3: A particle's potential energy that can be mapped using simple harmonic oscillations
[26] [Reprinted from “S. J. Ling, J. Sanny and B. Moebs, The Quantum Harmonic Oscillator.,
November 5, OpenStax CNX, https://phys.libretexts.org/@go/page/4531, (2020).”]
20
The energy splitting is ħω equal to hv0. The energy divisions are identical. There are no
asymptotic lines on the y axis, which means that the particles can theoretically pass through each
other. When a system is perturbed from equilibrium, it responds with a force proportional to the
magnitude of the perturbation. A harmonic oscillator is an idealized expression of Hooke's Law.
To calculate vibrational frequencies in harmonic oscillator the duration r is,
𝑟 = 2𝜋√𝜇
𝑘 Eq. 1.2
where k is the constant of force and μ is reduced mass. As long as the photon's frequency
matches one of its normal vibrational modes, it can be absorbed by a single molecule. When a
photon of the same frequency hits a molecule in its ground state, the molecule is able to absorb the
photon because the molecule is vibrating at the same frequency. The frequency of vibration is
usually expressed in terms of how many vibrations would occur in the time it takes for light to
travel one centimeter in chemistry. However, this is not always the case. An expression for the
vibrational frequency of simple harmonic motion can be written as:
�̅� = 1
2𝜋𝑐√𝑘
𝜇 Eq. 1.3
For v equals the frequency in centimeters per second, c equals the speed of light in
centimeters per second, k equals the force constant in erg/cm2, and μ equals the reduced mass in
grams. Reduce mass is equal to m1.m2/m1+m2 where m is atomic masses.
In nature, idealized situations degenerate and are unable to accurately describe linear
equations of motion. Anharmonic oscillation occurs when the restoring force no longer equals the
displacement.
21
1.1.1.2 Anharmonic oscillator
This concept of anharmonic oscillation refers to an oscillator which does not follow a
simple harmonic motion. For diatomic molecules, the Morse Potential [15] is the most widely
accepted function for describing the vibrational interactions between atoms in the molecules.
Atoms oscillate back and forth because of the interaction of the forces acting on them. They repel
each other with a powerful force when they are close to one another. Both positively charged nuclei
and electrons in their inner orbits are attracted to each other by electrostatic forces. As the distance
between them rises, the attractive force takes over as the dominant force. When the distance
between the atoms becomes too great, the atoms lose their electric field interaction. At this
distance, the bond's dissociation energy is equivalent to the energy present. Molecular
spectroscopy has used the Morse function to calculate the frequencies and intensity of overtones
of stretching in diatomic molecules and in molecular dynamics [15,17,18].
Diatomic molecules can be modelled using the Philip M. Morse potential. Due to the
explicit inclusion of the impacts of bond breaking, including the existence of unbound states, it is
considered a better approach for the vibrational structure of a molecule than simple harmonic
oscillator. [19] Anharmonicity and transition probabilities are adaptive and combination bands are
also taken into consideration. The Morse potential is a simplistic function that is used to design
the potential energy of a diatomic molecule as a function of internuclear distance. A Morse
potential energy diagram is depicted in Figure 4.
22
Figure 4: Morse potential curve and vibrational energy levels [20] [Reprinted from “A. sanli,
M. Lyyra, Transition Dipole Moment and Lifetime Study of Na2 And Li2 Electronic States Via
Autler-Townes And Resolved Fluorescence Spectroscopy, May 1, (2017).”]
Morse [15] pioneered the use of a convenient parameter analytical function to approximate
the shape of the anharmonic potential energy curve for a diatomic molecular oscillator in 1929.
The atomic potential model is a critical component of a molecular dynamic’s numerical method.
This investigation is focused on atoms that can vibrate freely within molecules. When two atoms
that are capable of bonding do so, they form a diatomic molecule. The Morse potential is a
convenient representation of a diatomic molecule's potential energy. [21]
𝑉(𝑟) = 𝐷(1 − 𝑒−𝛽(𝑟− 𝑟𝑒))2 Eq. 1.4
There are several variables in this equation: r represents the distance between atoms, re
represents the equilibrium bond distance, De represents the well depth (defined in terms of the
dissociated atoms) and controls the "width" of the potential (the ‘β’ is, the greater the well). As the
two atoms are moved closer together or away from one another, the potential energy, V(r), reflects
23
the potential energy of the system. To compute the dissociation energy of the bond, the zero-point
energy E(0) is subtracted from the depth of the well. By considering the partial derivatives of the
potential energy expression, it can be demonstrated that the parameter, β, is the force constant of
a bond. Where force constant is ke. [22].
𝛽 = √𝑘𝑒
2𝐷𝑒 Eq. 1.5
Based on vibrational energy the principal components of stationary states on the Morse
potential are defined as follows:
𝐺(𝑣) = 𝜔𝑒 (𝑣 + 1
2) − 𝜔𝑒𝑥𝑒 (𝑣 +
1
2)2
Eq. 1.6
𝜔𝑒 = 𝛽 (𝐷ℎ × 102
2𝜋2𝑐𝜇)1 2⁄
Eq. 1.7
𝜔𝑒𝑥𝑒 = ℎ𝛽2 × 102
8𝜋2𝜇𝑐 Eq. 1.8
A morse oscillator has precisely two terms, and G(v)(Eq. 1.6) is the standard symbol for
vibrational energy levels.[21] Morse potential factors can be closely attributed to the constants
ωe (Eq. 1.7) and ωeχe (Eq. 1.8) in this approach. G(v) is a good estimation for the genuine
vibrational structure of diatomic molecules that do not rotate. Indeed, actual molecule spectra are
frequently fitted to the equation 1.4.
1.1.2 Polyatomic molecular vibrations
One of the most common ways to describe the system's potential energy (PES) is to look
at how the position of each of the elements in the system influences that potential energy. One or
more coordinates can be used to define a potential energy curve, or an energy profile if the surface
has only one coordinate. The analogy of a landscape is instructive: for a system with two degrees
24
of freedom (e.g., two bond lengths), the energy value, (equivalency: the height of the land) is a
function of two bond lengths, equivalency: the coordinates of the position on the ground. For each
atom or molecular structure, there exists a unique potential energy associated with it. The Potential
Energy Surface represents this concept. This results in a smooth energy "landscape," and chemistry
can be viewed from a topological standpoint (of particles evolving over "valleys" and
“passes"). [23]
Atom positions can be represented by the elements in the r-vector, which describes the
geometry of the set of atoms. It's possible that the vector r represents the atoms' Cartesian
coordinates, but it could also represent the distances and angles between the atoms themselves.
Based on r, we can calculate the value of the energy function V(r) for all possible values of this
parameter. Using the introduction's landscape analogy, V(r) denotes the height on the "energy
landscape," introducing the concept of a potential energy surface. [23]
Figure 5: Potential energy surface of water molecule [23] [Reprinted from “Lewars, G.
Errol Computational Chemistry Volume 98 || The Concept of the Potential Energy
Surface.,10.1007/978-90-481-3862-3(Chap. 2), 9–43. doi:10.1007/978-90-481-3862-3_2,
(2011).”]
25
The PES for water (Figure 5) illustrates the energy minimum associated with an optimized
molecular structure with a water-O-H bond length of 0.0958 nm and an H-O-H bond angle of
104.5°.
1.1.2.1 Computationally smaller molecule
i. H2O-Water
Table 1: Vibrational and/or electronic energy levels water molecule (Source: NIST)
Vibrational and/or electronic energy levels
Water (H2O)
Vibrational modes Vibration
No.
Recommended
Freq.
Infrared
spectra
Raman
Spectra
Value|Phase Value|Phase
Symmetric stretch 1 3657 3656.65 gas 3654 gas
Bending 2 1595 1594.59
gas
Asymmetric stretch 3 3756 3755.79 gas
Table 1 illustrates three different vibrational modes of water as well as frequency with the
help of spectroscopy. A equal to 0~1 cm-1 uncertainty IR spectra is more accurate compared to
selected frequency value and shown all of three vibrational modes while Raman only shows
symmetric stretch. IR spectroscopy measures the relative frequencies when sample absorbs
radiation, in contrast, Raman spectroscopy quantifies frequencies where a sample scatters
radiation. Raman rely on polarizability of a molecule when IR spectroscopy totally depend on
dipole moment. For frequency measurement of molecule, IR spectrum is more effective than
Raman spectrum.
26
1.1.2.2 Computationally bigger molecule
i. CH3NH2-Methylamine
Table 2: Vibrational and/or electronic energy levels of methylamine (Source: NIST)
Vibrational and/or electronic energy levels
Methylamine (CH3NH2) gas-phase
Vibrational modes
Vibration No.
Recommended Freq.
Infrared spectra
Raman Spectra
CH3 d-str 1 3361 3361 3360
CH3 s-str 2 2961 2961 2960
NH2 scis 3 2820 2820 2820
CH3 d-deform 4 1623 1623
CH3 s-deform 5 1473 1473 1460
CH3 rock 6 1430 1430
CN str 7 1130 1130
NH2 wag 8 1044 1044 1044
NH2 a-str 9 780 780 781
CH3 d-str 10 3427 3427 3470
CH3 d-deform 11 2985 2985
NH2 twist 12 1485 1485
CH3 rock 13 1419
Torsion 14 1195 1195
CH3 d-str 15 268 268
Table 2 explains about different vibrational modes of methylamine as well as respective
frequency of IR and Raman spectrum. Methylamine contains seven atoms, so it has total number
of degrees of freedom is 3N-6 which is 15 total number of vibrational modes are possible. IR
spectra is more accurate and shown all fifteen vibrational modes while Raman shows few. Also,
presenting this spectrum all collected in gas-phase. However, Raman is not able gives us all data
as IR spectrum do. IR spectroscopy is better than Raman spectroscopy when it comes to measuring
frequencies of all vibrational modes.
27
1.2 Interaction of light with matter
Light, or Visible Light, is a term that is frequently used to refer to electromagnetic radiation
that the human eye can detect. With wavelengths ranging from meters to less than 1x10-11 meters,
the electromagnetic spectrum is extremely wide, covering everything from low-energy radio
waves to gamma radiation. Fluctuations in electric and magnetic fields are referred to as
electromagnetic radiation, which travels at the speed of light about 300,000 kilometers per second
through a vacuum. Massless energy packets, known as photons, can also be used to describe the
flow of light. Same photons close to the speed of light and have wavelike properties. A photon is
the smallest unit of energy that can be transmitted, and it was the discovery that light travels in
discrete quanta that gave rise to Quantum Theory.
Figure 6: An illustration of the electromagnetic spectrum, emphasizing the narrow window of
visible light that can be detected by the human eye [24] [Reprinted from “J. Zwinkels, Light,
Electromagnetic Spectrum. Encyclopedia of Color Science and Technology, 1–
8. doi:10.1007/978-3-642-27851-8_204-1, (2015).”]
28
Visible light is not fundamentally distinct from the rest of the electromagnetic spectrum,
except that the human eye can detect visible waves. Only a small portion of the electromagnetic
spectrum is covered by these wavelengths, which range from 400 nm for violet light to 700 nm for
red light. Ultraviolet (UV) radiation is that which is shorter than 400 nm in wavelength and Infra-
Red (IR) radiation is that which is longer than 700 nm in wavelength, neither of which can be
observed by the human eye. In contrast, advanced scientific detectors like those manufactured by
Andor, can detect and measuring photons over a much wider range of the electromagnetic
spectrum and at much lower quantities of photons (i.e., much weaker light levels) than the human
eye can even see.[24]
1.2.1 Transitions between different energy states
Humans' ability to 'see' light is not accidental. We perceive the world around us primarily
through light. To be sure, when it comes to studying the universe, light detection is an extremely
powerful tool. When light interacts with matter, it is altered, and many of the characteristics of that
object can be defined by studying light that emerged or contacted matter.
Matter is made up of atoms, ions, and molecules, and it is their interactions with light that
generate the various phenomena that can aid in our understanding of matter's nature. Atoms, ions,
and molecules all have defined energy levels, which are typically associated with the energy levels
that matter's electrons can hold. There are many ways in which light can interact with the energy
levels of matter, and it is possible for the matter to generate light.
29
Figure 7: Transitions between different energy states of molecules after interaction with a
photon are illustrated in Jablonski diagram, here S0 is ground state and S is excited state [27]
[Reprinted from “U. Gehlsen, A. Oetke, M. Szaszák, N. Koop, F. Paulsen, A. Gebert, G.
Huettmann, P. Steven, Two-photon fluorescence lifetime imaging monitors metabolic changes
during wound healing of corneal epithelial cells in vitro. , 250(9), 1293–
1302. doi:10.1007/s00417-012-2051-3, (2012).”]
Figure 7 depicts a Jablonski diagram, a way to represent the energy levels of matter. A
photon can be absorbed by an atom or molecule in its lowest energy state, the ground state, and
the atom or molecule will be elevated to a higher energy level, the excited state. As a result, matter
can absorb light with specific wavelengths. Typically, an atom or molecule remains in an excited
state for a very brief period before reverting to the ground state via a variety of mechanisms. The
excited atom or molecule loses energy not by emitting a photon, but by internal processes that
typically result in the matter heating up. As a result, the intermediate energy level relaxes to the
ground state by emitting a lower-energy photon (longer wavelength) in comparison of absorbed
photon in the initial stage.
30
1.2.2 Study of matter using light
When light is split into its constituent wavelengths after it has interacted with matter, the
resulting spectral signature tells us an enormous amount about the matter itself because it is
composed of photons that have been absorbed or expelled. The broad field of spectroscopy
encompasses a variety of spectroscopic techniques, including Raman spectroscopy,
absorption/transmission/reflection infrared spectroscopy, atomic spectroscopy, and laser induced
breakdown spectroscopy. These techniques provide a wealth of useful information about the
scientific properties of atoms and molecules, as well as the ability to precisely identify and quantify
their presence in a sample. [28]
In time-dependent quantum mechanics, spectroscopy is one of the most important topics
for chemist. This refers to the study of matter through light fields (electromagnetic radiation). A
resonant interaction between charged particles and an oscillating electromagnetic field is the
classical mechanism for light-matter interactions. In order to begin solving this problem, we must
first construct a Hamiltonian for the light-matter interaction, which takes the form
𝐻 = 𝐻𝑀 + 𝐻𝐿 + 𝐻𝐿𝑀 Eq. 1.9
While the Hamiltonian of subject HM is typically (but not always) time independent, the
electromagnetic field HL and its engagement with subject HLM are time dependent. For various
modes of electromagnetic radiation, a quantum mechanical procedure of light would describe it in
terms of photons. [28]
It is needed to solve the coupled motion equations of quantum electrodynamics and
quantum dynamics such as time-dependent Schrodinger equation for the light field and the material
system. In this case even for easy systems, this is a difficult task. So, the electromagnetic field
31
represented by Maxwell's equations will be applied to the light. It disrupts the fundamental
Hamiltonian in some way. As a result of the interaction between light and matter represent as,
𝐻 ≈ 𝐻𝑀 + 𝐻𝐿𝑀 Eq. 1.10
We are only concerned in resolving the material electromagnetic field at this time; the
interaction’s effects on light aren’t important. While the material system is influence by light, the
light itself has no influence of the material system.[29]
1.2.3 Transition dipole moment
Transition is a colloquial term for one of the various types of dipoles. As with other
dipole moments, the transition dipole describes the charge difference between two regions of
a molecule. Excited states of an electron produce a transition dipole as they move from ground
states. The ground state has a different charge distribution than the excited state, and it is this
difference in electron density between the two states that results in the transition dipole.
A molecular orbital transition in which an electron moves from π the to a π* orbital called
electron transition from bonding to anti bonding transition. Diatomic molecule shows a possible
transition from a π molecular orbital in the orbital to the π* orbital. For example, when electron
gets excited from ground state π to excited state π*, there are more electrons in orbitals π* than
there are in ground state π, and this is where the dipole points.[30]
Additionally, we will explore photons and molecular excitation of the transition dipole moment.
The position of the transition dipole is critical in determining the presence or absence of molecular
excitation. To excite an electron from its ground state to an excited state, it must not only be
provided with the appropriate amount of energy by a photon, however the photon's electric field
32
should also correspond with the transition dipole. When a photon of light excites an electron, all
its energy is absorbed by the event. This is referred to as absorption.
The electron density of the excited state π* orbitals is different from the system. The dipole
arrow is aligned with the lobe of the excited state π* orbital with the maximum electron density,
because of our changes to the electron density. It is not necessary to have precise alignment
between both the electromagnetic field and the transition dipole. However, when the angle between
the transition dipole and the electric field grows, the likelihood of absorbing a photon
diminishes.[30]
It generates further transition dipoles and eventually obtain the perfectly aligned photon. It
is the electric dipole moment that is associated with the transition between two states, and it is
commonly represented by the symbol dnm for a transition between two states with initial state m
and final state n, respectively. Since the transition dipole moment includes the phase differences
between the two states, it is typically a complex vector quantity. Transition dipole moment is
measured in Coulombmeters (Cm), although a more practical unit is the Debye-meter (Dm) (D).
[31]
In general, the transition dipole moment is important for determining whether the
electric dipole interaction allows for transitions. As an example, the transition from a bonding
π orbital to an antibonding π* orbital is permitted since the integral describing the transition
dipole moment is greater than zero in value. As the dipole operator of the r-function is an odd
function of r, the integrand is an even function, and so this transition takes place between even
and odd orbitals. It is not always the case that the integral of an odd function over the symmetric
limits produces a result of zero. [21][31]
The parity selection criteria for electric dipole transitions reflects this fact.
33
(𝜇𝓏)12 = ∫𝜓2∗𝜇2𝜓1𝑑𝜏 Eq. 1.11
As a result of the triple integral yielding an ungerade (odd) product, it is disallowed to
compute the integral of the transition moment of an electronic transition among similar atomic
orbitals, such as s-s or p-p. For example, electrons can only be redistributed between orbitals in
a single transition. The transition is permitted if the triple integral yields a gerade (even)
product. The term (µz) 12 denotes the z component of the dipole moment of transition between
states 1 and 2. This means there would be no transitions between states 1 and 2, because
(µz,)12=0. The dipole transition moment underpins the previously postulated selection
principles. Transitions take place only between states with a non-zero transition moment. [32]
1.2.4 Vibrational Selection Rules
The ground state of the majority of molecules is completely symmetrical at the zero-point
vibrational level. As a result, completely symmetric modes of the ionic state are observed. If the
quantum numbers of the vibrational levels are n, then the selection rule is n = 0, ±1, ±2, etc. There
will be no restrictions on how you can transition between the ground electronic state and excited
electronic states for any totally symmetric vibration. For large molecules, it is possible to
interdigitate the structures of several symmetric modes.
When it comes to vibrational transitions, a molecule's dipole moment must change during
the motion. For example, Homonuclear diatomic are infrared inactive – stretching the bond does
not change the molecule's dipole moment, which remains zero. Bond stretching increases the
distance between the molecule's positive and negative ends, increasing the dipole moment of
heteronuclear diatomic, which may make them infrared-active. Selection rule does not necessitate
34
molecules with permanent electric dipole moments; the change may be from one state to another
without dipoles. [33]
In terms of the quantum number v, the specific vibrational selection rule is as follows: The
gaps among vibrational energy levels are greater than the gaps between rotational energy levels,
to the point where molecules are almost always in their vibrational ground state, ν = 0 at room
temperature. Thus, a vibrational spectrum is frequently composed of a single line representing the
transition from ν = 0 to ν = 1. The term "fundamental transition" describes this process. Other
transitions are only faintly observed, owing to the minuscule populations of the lower states
involved. [33]
A parabola estimation of the potential energy curve leads to transitions at slightly different
frequencies, which is why this approximation is only an approximation. So, it is possible to see
several lines in the spectrum, especially if the molecule is formed in some vibrationally excited
states (ν > 0), because transitions from these states will have higher intensities than would
otherwise be the case. [33]
1.2.5 Infrared spectroscopy
Infrared radiation stimulates molecular vibrations when absorbed by molecules. A
molecule dipole moment changes during vibrations in the mid- and far-infrared spectral ranges
when the frequencies of both light and vibration are equal. Due to the intermolecular and
intramolecular influences on vibrational frequency, the chance of absorption is also affected. The
vibrating masses and the kind of bond (single, double, triple) determine the estimated position of
an infrared absorption band; the actual position is dictated by the intra- and intermolecular
35
environment's electron withdrawing or donating effects, as well as coupling with other vibrations.
Absorption strength increases as the polarity of the vibrating connections increases. [34]
Infrared (IR) absorption spectroscopy is a well-established bulk characterization technique
that provides chemical information about a material's molecular structure by identifying
characteristic vibrational bands associated with bonded units. Fourier methods have been used to
improve the signal-to-noise ratios and resolution for generally weak IR vibrational signals
associated with biomedically relevant species, which has resulted in a significant improvement in
IR interrogation of surfaces. Nonetheless, FTIR analysis of surfaces is frequently signal-limited
and, in many cases, non-specific, as many characteristic IR absorption bands are shared by many
species of interest. Due to the large amount of IR sampling depth used to obtain a sufficient surface
signal, it is difficult to distinguish and resolve the specific surface analytical zone from the bulk-
derived IR signals once a sufficient surface signal has been obtained (up to a micron). Only when
these "surface analytical" configurations are configured to remove the bulk signal do they return
information from thousands of molecular layers beneath their actual interface. These surface-
sensitive modes of investigation have yielded a wide range of results, from the most accurate to
the most inaccurate. [35]
A Michelson interferometer modulates the light, and a computer-based fast Fourier
transform (FT) analysis is used in today's instruments for IR absorption spectroscopy, known as
the FT type. Colorful light sources (such as a ceramic element that glows in the dark) produce a
nearly continuous spectrum in the range of molecular vibrations. An interferogram can be
generated from the modulated light passing through the sample compartment and then converted
to an IR spectrum using an inverse Fourier transform (FT). In conventional IR-absorption
spectroscopy, the work is carried out in a transmission mode, with light being transmitted through
36
the sample. This setup frequently necessitates more complex sample preparation, such as the
creation of slurries or solutions that can be expanded across two optical windows. [35]
1.2.5.1 FTIR principle and analysis process
FTIR spectrometer has the power to handle the most advanced research-level experiments,
routine analyses are performed just as conveniently. In comparison to the traditional
spectrophotometer, the Fourier transform spectrophotometer rapidly produces the infrared
spectrum. In the diagram shown in Fig. 8, the basic components of an FTIR spectrophotometer are
depicted. [36]
Figure 8: The fundamental elements of a Fourier transforms infrared spectrometer[36]
[Reprinted from “M. A. Mohamed, Membrane Characterization || Fourier Transform Infrared
(FTIR) Spectroscopy, 3–29. doi:10.1016/B978-0-444-63776-5.00001-2, (2017).”]
Working function of the energy strikes at the beam splitter and produces two beams of
roughly the same intensity. One beam strikes the stationary mirror and returns to the beam splitter.
37
The other beam goes to the moving mirror. When these two beams meet up again at the beam
splitter, they recombine. The recombined beam passes through the sample and the repeat the same
process shown in figure 9.
Figure 9: Internal features of the Fourier transform infrared spectrometer[37] [Reprinted from
“F. Zadeh, F. Anaya, Nelson M., Schiffman, Laura A., O. Craver, Vinka. Fourier transform
infrared spectroscopy to assess molecular-level changes in microorganisms exposed to
nanoparticles. Nanotechnology for Environmental Engineering, 1(2), 1–. doi:10.1007/s41204-
016-0001-8, (2016).”]
The FTIR spectrometer range is primarily in the mid-IR region 2.5 to 25 μm between 4000
and 400 cm-1. Functional groups can be identified by the appearance of an absorption band that
occurs in the mid-IR region (4000 to 400 cm-1), where it is possible to find transition energies
corresponding to changes in the vibrational energy state of numerous functional groups. FTIR
spectra can typically be used to analyze four distinct regions of bond types. Single bonds (O-H, C-
H, and N-H) are detectable at higher wavenumbers, as illustrated in Fig. 10. (2500-4000 cm-1).
Additionally, the triple and double bonds can be detected in the middle wavenumber ranges of
2000-2500 cm-1 and 1500-2000 cm-1, respectively. Additional to this, the vibration of the molecule
in its entirety gives rise to a complex pattern of vibrations in the low wavenumber region between
38
650 and 1500 cm-1, which can be used for the identification of the molecule itself. That region
called fingerprint region.[36]
Figure 10: Typical Fourier transform infrared spectrum of cellulose membranes with the
different types of bonds those are common [36] [Reprinted from “M. A. Mohamed, Membrane
Characterization || Fourier Transform Infrared (FTIR) Spectroscopy, 3–29. doi:10.1016/B978-
0-444-63776-5.00001-2, (2017).”]
39
2. Computational Chemistry
Computational chemistry can characterize the structure of a molecule using computational
method. Additionally, it aids in simulating the IR spectrum of a molecule and obtaining high-
quality IR data, making it a valuable tool for drug discovery in both academic research and
industry. The complexity of normal IR spectra, on the other hand, puts the accuracy and efficiency
of commonly used empirical methods to the test. Quantum mechanics (QM)-based approaches are
increasingly being investigated to improve the accuracy of IR spectrum description. In principle,
QM calculations account for all contributions to energy, accounting for terms that are typically
omitted from empirical force fields, and thus provide a higher degree of transferability. This
research presented applications of QM methods to the study of high-quality infrared data.
Numerical techniques and algorithms used in Psi4 are the most advanced available in the field of
quantum chemistry. Some of the code uses shared-memory parallelization to run on multi-core
machines more efficiently. Using a Python-based parser, a user can input data in a very
straightforward manner for routine computations, but it also could perform highly complicated
tasks.
The goal of using computational chemistry in vibrational spectroscopy is to build a more
realistic tool. The primary disadvantage of vibrational spectroscopy from a practical standpoint is
the absence of a direct spectrum–structure relationship. As a result, analyzing a molecule's
vibrational spectrum to determine its structure is difficult or impossible. Nuclear Magnetic
Resonance (NMR) is an important spectroscopic technique but has some drawbacks. The majority
of these are related to the inherent sensitivity of vibrational spectroscopy, which enables the
detection of extremely small amounts. Additionally, there are advantages to vibrational
spectroscopy's broader scope, such as its applicability to solids, liquids, and gases, as well as
40
adsorbed layers. IR spectroscopy also has a lower instrument cost than other spectroscopic
techniques.
To summarize, it's clear that vibrational spectroscopy has many advantages that could be
enhanced with the development of a reliable method for predicting vibrational spectra. An
approach like this could be used to estimate the spectra that new structures will detect.[38] A
comparison to the observed spectrum would establish the identity of a product, even if it was an
entirely new molecule. The restricted Hartree–Fock model (RHF), the restricted open-shell
Hartree–Fock (ROHF) quantum theory, and the unrestricted Hartree–Fock (UHF) theory are the
most appropriate for this purpose. This research is based on using computational method to get the
accurate frequency of normal vibrational modes and compare that with the IR experimental
frequency of normal modes and find the right approach for molecular calculations.
2.1 Correct approach for calculations
Numerical techniques and algorithms used in Psi4 are the most advanced available in the
field of quantum chemistry. There are three basic types of calculations available in Psi4; single-
point energy predicts stability and reaction mechanisms when optimization of geometry predicts
shape and frequency predict spectra. The molecular structure and vibrational spectra of substances
can be analyzed and understood using quantum chemistry approaches. The FTIR spectroscopic
data in the bibliography will be used to assess the molecular structure, which has been derived
theoretically by quantum chemical methods and empirically.
Theories such as HF, DFT, DCT, CC, TDSCF, MP2, CI and many other quantum chemical
methods are accessible in Psi4. There are theories such as HF, DFT,MP2,CCSD are ab initio
calculations uses Schrödinger's equation, but with approximations. CCSD, CCSD(T), and DFT are
41
the best approaches for obtaining close data. As far as medium-sized molecules are concerned, the
DFT and CCSD(T) approaches are extremely useful. For wavenumber computations, they appear
to be more accurate than Hartree-Fock (HF) and MP2 approaches, as well as more efficient.
Computational Methods are system of equations or computations used to determine the energetics
of a molecule. Different methods use different approximations (or levels of theory) to produce
results of varying levels of accuracy. There is a tradeoff between accuracy and computational time.
The basis set functions used in quantum chemical computations in current computational
chemistry are limited. Using a finite basis set that expands toward an infinite complete set of
functions is known as approaching the complete basis set limit. To model molecular orbitals, a
basis set is a linear combination of functions. It is possible to consider the atoms' atomic orbitals
as basis set functions in quantum chemical computations since the equations that define molecular
orbitals are otherwise extremely difficult to solve.
Over the years, several standard basis sets have been thoroughly evaluated and optimized.
To model molecular orbitals as accurately as feasible, a user should, in general, use the largest
basis set available. Because calculation costs increase exponentially as the basis set grows, a
tradeoff must be made between precision and cost. As a result, the "theoretical model chemistry"
is achieved, which is the combination of a well-defined energy technique (e.g., Hartree-Fock) with
a well-defined basis set. [39] The CCSD(T) method is used with the aug-cc-pVTZ basis set. The
prefix "aug" denotes the addition of one set of diffuse functions for each angular momentum in the
basis set. Again, the accuracy of a calculation is dependent on both the method and the type of
basis set applied to it. There is a tradeoff between accuracy and time. Larger basis sets will describe
the orbitals more accurately but take longer to solve.
42
2.2 Introduction to Psi4
Psi4 offers a wide range of quantum chemistry approaches, all of which are implemented
utilizing the most advanced numerical methods and algorithms available. Some of the code uses
shared-memory parallelization to run on multi-core machines more effectively.
Compared to Psi3, Psi4 is a completely different product. Code has been rewritten from
scratch using a strong new set of C++ libraries, although some existing libraries and modules have
been preserved. Much of the tedious work of executing the same computation on all molecules in
a test set can now be done automatically thanks to a new Python front-end for Psi4. There are
numerous functionals available in Psi4 that were not present in Psi3. Psi4 makes extensive use of
density fitting, resulting in some of the most effective DFT, MP2,CCSD and CCSD(T) code
around. For non-covalent interactions, Psi4 introduces a wide range of powerful characteristics for
energy component analysis via symmetry-adaptive perturbation theory. Analytic gradients for
perturbation theory and coupled-cluster methods have been added. It could utilize open-source
libraries for implicit solvent (PCM), density-matrix renormalization group CI, Grimme dispersion
corrections, effective fragment potentials and linked cluster theory. [40]
To get output file from Psi4, needed to create input file based on Psi4 coding format. Any
conventional text modifier can be used to create a text file that Psi4 can read. The default input
and output file names are input.dat and output.dat, respectively. The command line can be used to
override these defaults by identifying the input and output file names.
2.3 Best computational methods
Couple cluster approach such as CCSD, CCSD(T) is currently the best methods for
analyzing diatomic molecules. For polyatomic molecules B3LYP-D method is appropriate method
43
for calculation and it comes under Density functional theory approach. Compared to previous
ways, we can acquire far more accurate data in a shorter period using these methods.
2.4 Type of methods
Many electronic structure approaches, including Hartree–Fock molecular orbital theory,
coupled cluster theory, and full configuration interaction are accessible in the PSI4 package.
2.4.1 Hartree-Fock (HF) theory
Hartree–Fock theory brings out the optimized molecular orbitals (MOs) {𝜓𝑖},
𝜓𝑖(�⃗�1) = 𝐶𝜇𝑖∅𝜇(�⃗�1) Eq. 2.1
In Psi4, {∅𝜇} the basis functions, referred to as atomic orbitals, are contracted Cartesian
Gaussian functions (AOs). Hartree–Fock constrained variational parameters, the MO coefficients,
are contained in the matrix 𝐶𝜇𝑖 . For the smallest conceivable antisymmetric wave function, a single
Slater determinant, molecular orbitals are employed.
|ψ0⟩ = 1
√𝑁!|
𝜓1(�⃗�1) 𝜓2(�⃗�1)
𝜓1(�⃗�2) 𝜓2(�⃗�2)
⋯ 𝜓𝑁(�⃗�1)
⋯ 𝜓𝑁(�⃗�2)⋮ ⋮
𝜓1(�⃗�𝑁) 𝜓2(�⃗�𝑁)⋱ ⋮⋯ 𝜓𝑁(�⃗�𝑁)
| Eq. 2.2
This form of the Hartree–Fock wavefunction is like considering the electron correlation as
a mean field repulsion instead of a more sophisticated effect.
In view of electronic Hamiltonian,
�̂� =∑−1
2𝑖
∇𝑖2 + ∑∑−
𝑍𝐴𝑟𝑖𝐴
𝐴𝑖
+ ∑1
𝑟𝑖𝑗𝑖>𝑗
𝐸𝑞. 2.3
According to Slater's rules, the Hartree–Fock energy is
44
𝐸HF = ⟨Ψ0|�̂�|Ψ0⟩ = ∑⟨𝑖|ℎ̂|𝑖⟩
𝑖
+1
2∑[𝑖𝑖|𝑗𝑗]
𝑖,𝑗
− [𝑖𝑗|𝑗𝑖]
= 𝐷𝜇𝜈𝛼 (𝐻𝜇𝜈 + 𝐹𝜇𝜈
𝛼 ) + 𝐷𝜇𝜈𝛽(𝐻𝜇𝜈 + 𝐹𝜇𝜈
𝛽) Eq. 2.4
Here, H denotes the one-electron potential in the AO-basis, which contains both electron-nuclear
attraction and kinetic energy.
𝐻𝜇𝑣 = (𝜇|−1
2∇2 + ∑ −
𝑍𝐴
𝑟1𝐴𝐴 | 𝑣) Eq. 2.5
The orbital coefficients construct the AO-basis density matrix to get D.
𝐷𝜇𝜈𝛼 = 𝐶𝜇𝑖
𝛼𝐶𝜈𝑖𝛼 Eq. 2.6
as well as an effective one-body potential known as the Fock matrix(F), which is the
current density value
𝐹𝜇𝜈𝛼 = 𝐻𝜇𝜈 + (𝐷𝜆𝜎
𝛼 + 𝐷𝜆𝜎𝛽)(𝜇𝜈|𝜆𝜎)⏟
𝐽
+ 𝐷𝜆𝜎𝛼 (𝜇𝜆|𝜎𝜈)⏟
𝐾𝛼
Eq. 2.7
In chemists' terminology, the tensor (μv|λσ) is an AO electron-repulsion integral (ERI).
(𝜇𝜈|𝜆𝜎) = ∬ 𝜙𝜇ℝ6(𝑟→
1)𝜙𝜈(𝑟→
1)1
𝑟12𝜙𝜆(𝑟
→
2)𝜙𝜎(𝑟→
2)𝑑3𝑟1𝑑
3𝑟2 Eq. 2.8
The MO coefficients are discovered as the Fock matrix's generalized eigenvectors.
𝐹𝛼𝐶𝛼 = 𝑆𝐶𝛼𝜖𝛼 Eq. 2.9
The orbital energies are represented by the eigenvalues ϵ, and the metric matrix S would be the
AO-basis overlap matrix.
𝑆𝜇𝜈 = (𝜇|𝜈) Eq. 2.10
There are a few things to keep in mind about the Fock matrix, which is based on both alpha
and beta density (orbitals). Non-linearity is a feature of SCF, which ceases as soon as the producing
orbitals and the Fock matrix they form are self-consistent.
45
The computing work of the SCF technique is dominated by the creation of Coulomb matrix
J and exchange matrix Kα. Diagonalization of the Fock matrix can be a significant problem for
very large systems. [40][41]
2.4.1.1 Restricted Hartree-Fock (RHF) [Default]
Slater-determinant-based RHF can be used to calculate the electronic energy and
configuration of closed-shell molecule systems in its fundamental singlet state, as it is built up
with doubly occupied orbitals that minimize total energy.[42]
2.4.1.2 Unrestricted Hartree-Fock (UHF)
It is suitable for most open-shell systems and is quite simple to converge. It is important to
note that the spatial sections of the alpha and beta orbitals are completely independent of one
another, which provides for a great deal of freedom in the wavefunction. This flexibility, on the
other hand, comes at the expense of spin symmetry; UHF wavefunctions do not have to be
eigenfunctions of the S2> operator to be valid. The output file contains a line indicating how far
this operator has deviated from its expected value. To avoid the problem of "spin contamination,"
it is recommended that you switch to a ROHF if the deviation is higher than a few hundredths of
a millimeter.
2.4.1.3 Restricted Open-Shell Hartree-Fock (ROHF)
Suitable for open-shell systems in which spin contamination is a concern. Consistent
positive spin polarization is required to achieve convergence (the alpha and beta doubly occupied
orbitals are identical).
46
2.4.2 Coupled Cluster theory
Many-body systems can be described numerically using a technique called coupled cluster
(CC). Its most common application is in the field of computational chemistry, but it is also
employed in nuclear physics and atomic physics. This method is similar to Hartree-Fock molecular
orbital theory, but it uses the exponential cluster operator to build multi-electron wavefunctions.
This approach is used in some of the most precise computations for molecules of tiny to medium
size. [43]
As a quantum chemistry technique, linked clusters are one of the most precise and
trustworthy. An exponential expansion rather than a linear expansion of the wavefunction is used
in the connected cluster instead of configuration interaction
|ψ⟩ = 𝑒�̂�|ϕ0⟩
= (1 + �̂� + 1
2�̂�2 +
1
3!�̂�3 + ⋯) |∅0⟩ Eq. 2.11
�̂� is written as the sum of operators that generate single, double, and multiple excited
determinants:
�̂� = �̂�1 + �̂�2 + �̂�3 +⋯+ �̂�𝑁 Eq. 2.12
with
�̂�1|∅0⟩ = ∑∑𝑡𝑖𝑎
𝑣𝑖𝑟
𝑎
|∅𝑖𝑎
𝑜𝑐𝑐
𝑖
⟩
Eq. 2.13
�̂�2|∅0⟩ = ∑∑𝑡𝑖𝑗𝑎𝑏
𝑣𝑖𝑟
𝑎<𝑏
|∅𝑖𝑗𝑎𝑏
𝑜𝑐𝑐
𝑖<𝑗
⟩
This is since the linked cluster singles and doubles (CCSD) model reaches its limit at this
point. T̂ = T̂1 + T̂2
47
The incorporation of these products makes coupled-cluster methods size-extensive, which
means that the performance of computation must not diminish for complex molecules. According
to the CCSD algorithm, the computing cost of CCSD scales as O(o2v4), where o represents
occupied orbitals and v represents virtual orbitals, where o is the number of occupied and v is
virtual orbitals.
The CCSD(T) technique [44] is an improvement on the CCSD method in that it
incorporates a perturbative estimate of the energy contributed by the �̂�3 operator. For molecules
containing more than a dozen heavy atoms or so, the computational cost of this new term scales
as O(o3v4) making it prohibitively expensive to compute. However, in some circumstances, when
this procedure is inexpensive, it produces extremely high-quality outcomes.[45]
2.4.3 Density Functional Theory
A computational quantum mechanical concept used in chemistry to study the electronic
structure, particularly the ground state of several systems, in specific atoms and molecules is
known as density-functional theory (DFT). By employing functionals, it is possible to determine
the properties of a many-electron system. DFT is most widely use method among
all computational chemistry methods. Instead of focusing on wavefunctions and orbitals, DFT
focuses on the electron density. Simplified Kohn–Sham density functional theory is one of the
most important tools in modern computational chemistry because it's cost effectiveness and
accuracy.
DFT is established on the Hohenberg–Kohn theorems, which says that energy is a universal
functional of the one-particle electronic density and there appears a set of noninteracting atomic
orbitals with the same density as the accurate set of electrons, with the quasiparticle states defined
48
as eigenvectors of an effective one-body potential encompassing the true N-body quantum
phenomena. The former concept allows for the consideration of the electronic density rather than
the much more complicated wavefunction, whereas the later concept enables the treatment of the
perplexing kinetic energy factor via implicit one-body Kohn–Sham orbitals. As can be seen from
the energy expression, KS-DFT inherits most of the machinery from Hartree–Fock.
𝐸KS = ∑⟨𝑖|ℎ̂|𝑖⟩ + 1
2∑[𝑖𝑖|𝑗𝑗] + 𝐸𝑋𝐶[𝜌𝛼 , 𝜌𝛽]
𝑖,𝑗𝑖
= 𝐷𝜇𝑣𝑇 (𝑇𝜇𝑣 + 𝑉𝜇𝑣) +
1
2𝐷𝜇𝑣𝑇 𝐷𝜆𝜎(𝜇𝑣|𝜆𝜎)
𝑇 + 𝐸𝑋𝐶 [𝜌𝛼, 𝜌𝛽] Eq. 2.14
In this case, T is the noninteracting quantum mechanical kinetic energy operator, V denotes
the nucleus-electron attraction potential, DT denotes the total electron density matrix, and
𝐸𝑋𝐶[𝜌𝛼 , 𝜌𝛽] denotes the exchange, coupling, and residual kinetic energy functionals. Since the
residual kinetic energy term is typically fairly modest and is frequently neglected, 𝐸𝑋𝐶 is frequently
referred to as the exchange-correlation functional.[46]
2.4.4 Møller-Plesset Perturbation Theory
Møller-Plesset perturbation probably the most frequently used technique for approximating a
molecule's correlation energy. Beyond the Hartree-Fock approximation, second order Møller-
Plesset perturbation theory (MP2) is one of the easiest and most usable levels of theory. MP2
theory says that the Hartree-fock energy lies above the exact HF energy so the first correction
should be negative.
Through second order, Møller–Plesset theory or Many-Body Perturbation Theory (MBPT) has the
spin-orbital formula:
49
𝐸total(2)
= 𝐸Reference −𝑓𝑖𝑎𝑓𝑖𝑎
𝜖𝑎 − 𝜖𝑖−1
4
⟨𝑖𝑗||𝑎𝑏⟩2
𝜖𝑎 + 𝜖𝑏 − 𝜖𝑖 − 𝜖𝑗 Eq. 2.15
Here, i and j denote occupied spin orbitals, a and b denote virtual spin orbitals, fia and ov denote
Fock Matrix elements, ϵ denotes orbital eigenvalues, and ij||ab denotes anti-symmetrized
physicists' ERIs. The singles correction is used to synchronized RHF and UHF sources.
𝐸MBPT(1)
= −𝑓𝑖𝑎𝑓𝑖𝑎
𝜖𝑎 − 𝜖𝑖 Eq. 2.16
is zero according to the Brillioun Condition, as well as the perturbation series's first contribution
is of second order:
𝐸MBPT(2)
= −1
4
⟨𝑖𝑗|𝑎𝑏⟩2
𝜖𝑎 + 𝜖𝑏 − 𝜖𝑖 − 𝜖𝑗 Eq. 2.17
The MP2 module always evaluates the first-order contribution, or "single energy."[47]
2.5 Basis sets
Basis set is collection of mathematical functions use to help the Schrodinger equation. It
describes the electronic states of molecules; utilizing molecular orbitals, it generates
wavefunctions for the electronic states. These wavefunctions are approximations to Schrödinger
formula solutions. A mathematical function for a molecular orbital is generated, ψi as a linear
combination of other functions, φj, referred to as basis functions since they serve as the foundation
for expressing the molecular orbital, Cij are the linear combination coefficients.
𝜓𝑖 = ∑ 𝑐𝑖𝑗𝜑𝑗𝑗 Eq. 2.18
The Hartree-Fock approach or density-functional theory both use basis sets to describe
the electronic wave function to convert the partial differential equations in the model into
algebraic equations that can be efficiently implemented on computers.
50
When it comes to solid state physics, plane waves and real-space techniques, or even
atomic orbitals, can be employed as the basis set (resulting in the linear combination of atomic
orbitals approach) as is common in quantum chemistry. It is possible to employ several different
types of atomic orbitals: numerical atomic orbitals, Slater-type orbitals, as well as Gaussian-type
orbitals. [48] Since Gaussian-type orbitals are the most employed of the three, the post Hartree-
Fock approach can be efficiently implemented.
There are different basis set such as aug-cc-pVTZ, aug-cc-pVDZ, cc-pVDZ, cc-pVTZ to
calculate energy, frequency, and optimization of geometry. The prefix "aug" denotes the addition
of one set of diffuse functions for each angular momentum in the basis set. Again, the accuracy of
a calculation is dependent on both the method and the type of basis set applied to it. There is a
tradeoff between accuracy and time. Larger basis sets will describe the orbitals more accurately
but take longer to solve.
2.6 Example of input files and output files
Presented diatomic molecule such as H2 (singlet state) to get the basic understanding of
content inside the input and output files.
2.6.1 Input files
The input files of energy, optimize, and frequency using by Psi4 for the H2 molecule are
shown below. All other molecules follow a similar logic in terms of building input files for these
various calculations.
i Hydrogen input file of energy:
51
ii. Hydrogen input file of optimize:
iii. Hydrogen input file of frequency:
After optimizing the geometry of the molecule, the atomic coordinates from the “optimize” output
file must me placed into the input file for the “frequencies” calculation. These atomic coordinates
can be used to determine the equilibrium bond distance based on the level of theory and basis set
that was used for the calculation in “optimize”.
2.6.2 Output files
Output files of energy, frequency, and optimization Psi4 are shown below as example only.
i. Hydrogen output file of energy:
set basis aug-cc-pVTZ
molecule {
0 1
H 0.000000000000 0.000000000000 -0.370500000000
H 0.000000000000 0.000000000000 0.370500000000
}
energy('CCSD(T)')
set basis aug-cc-pVTZ
molecule {
0 1
H 0.000000000000 0.000000000000 -0.370500000000
H 0.000000000000 0.000000000000 0.370500000000
}
optimize('CCSD(T)')
set basis aug-cc-pVTZ
molecule {
0 1
H 0.000000000000 0.000000000000 -0.371443708316
H 0.000000000000 0.000000000000 0.371443708316
}
frequency('CCSD(T)')
52
==> Geometry <==
Molecular point group: d2h
Full point group: D_inf_h
Geometry (in Angstrom), charge = 0, multiplicity = 1:
Center X Y Z Mass
------------ ----------------- ----------------- ----------------- ---------------------------
H 0.000000000000 0.000000000000 -0.370500000000 1.007825032230
H 0.000000000000 0.000000000000 0.370500000000 1.007825032230
Running in d2h symmetry.
Rotational constants: A = ************ B = 60.92632 C = 60.92632 [cm^-1]
Rotational constants: A = ************ B = 1826525.07181 C = 1826525.07181 [MHz]
Nuclear repulsion = 0.714139285654521
Charge = 0
Multiplicity = 1
Electrons = 2
Nalpha = 1
Nbeta = 1
==> Post-Iterations <==
Orbital Energies [Eh]
---------------------
Doubly Occupied:
1Ag -0.594355
Virtual:
1B1u 0.052561 2Ag 0.053107 2B1u 0.190348
1B2u 0.208884 1B3u 0.208884 3Ag 0.284713
1B2g 0.297439 1B3g 0.297439 4Ag 0.417419
3B1u 0.420473 4B1u 0.737393 5Ag 0.791345
1B1g 0.791345 2B2g 0.883931 2B3g 0.883931
6Ag 0.903172 2B2u 0.931436 2B3u 0.931436
5B1u 1.047236 1Au 1.048153 6B1u 1.048153
3B2u 1.066995 3B3u 1.066995 7Ag 1.523326
3B2g 1.938548 3B3g 1.938548 7B1u 2.125321
53
ii. Hydrogen output file of optimize:
iii. Hydrogen output file of frequency:
The output files are lengthy, and only the numerical values that are presented are those
used to compare to experimental values.
8B1u 2.596272 8Ag 2.935935 9Ag 3.615630
2B1g 3.615630 4B2u 3.633791 4B3u 3.633791
10Ag 4.156435 2Au 4.406151 9B1u 4.406151
5B2u 4.430945 5B3u 4.430945 4B2g 4.488261
4B3g 4.488261 10B1u 5.190245 5B2g 5.784832
5B3g 5.784832 11Ag 5.991099 11B1u 7.121462
Final Occupation by Irrep:
Ag B1g B2g B3g Au B1u B2u B3u
DOCC [ 1, 0, 0, 0, 0, 0, 0, 0 ]
@RHF Final Energy: -1.13302546567342
=> Energetics <=
Nuclear Repulsion Energy = 0.7141392856545209
One-Electron Energy = -2.5056204781932943
Two-Electron Energy = 0.6584557268653561
Total Energy = -1.1330254656734173
Computation Completed
Final optimized geometry and variables:
Molecular point group: d2h
Full point group: D_inf_h
Geometry (in Angstrom), charge = 0, multiplicity = 1:
H 0.000000000000 0.000000000000 -0.371443708316
H 0.000000000000 0.000000000000 0.371443708316
==> Harmonic Vibrational Analysis <==
Vibration 6
Freq [cm^-1] 4429.9786
54
2.7 Performed calculations on several molecules
Computational calculation of diatomic and polyatomic molecules are performed using
different computational methods and basis set including B3LYP-D/cc-pVDZ, MP2-cc-pVDZ,
CCSD/cc-pVDZ, CCSD(T)/aug-cc-pVDZ, CCSD(T)/aug-cc-pVTZ, CCSD/cc-pVDZ and
CCSD/cc-pVTZ respectively. In table 3 shown all molecules are calculated to observe the
effectiveness of method and basis set.
Table 3: Performed computational calculations on several molecules
Performed calculations on diatomic and polyatomic
Chemical Structure
Molecule # Of Atoms
Ground State
Chemical Structure
Molecule # Of Atoms
Ground State
H2 Dihydrogen 2 singlet CH2O2 Formic acid 5 singlet
Li2 Dilithium 2 singlet CH4 Methane 5 singlet
C2 Dicarbon 2 singlet CH3OH Methanol 6 singlet
N2 Dinitrogen 2 singlet CH5N Methylamine 7 singlet
F2 Difluorine 2 singlet C2H4O2 Acetic acid 8 singlet
Na2 Disodium 2 singlet CH5N3 Guanidine 9 singlet
HCl Hydrogen Chloride
2 singlet C3H6O Acetone 10 singlet
CO Carbon monoxide
2 singlet C4H8O Tetrahydrofuran (THF)
13 singlet
H2O Water 3 singlet C6H12 Cyclohexane 18 singlet
CH2O Formaldehyde 4 singlet
55
3. Experimental Methods
3.1 Fourier-Transform Infrared Spectroscopy (FTIR)
Fourier Transform Infrared (FTIR) spectroscopy is a form of vibrational spectroscopy that
provides excellent selectivity for material identification. A Thermo Scientific Nicolet 6700 FTIR
spectrometer equipped with the OMNIC FTIR software was used to capture the IR spectrums of
the sample. To start collecting the spectra of the isolated solvents, the experimental parameters are
set in the FTIR instrument, starting with a high-resolution of 0.125 cm−1 to 5 cm−1 collected for
every 4 scans. The x- axis of the spectra were set to wavenumber cm-1 while, the y-axis of the
spectra was set to %Transmittance. %𝑇 =𝐼
𝐼0× 100, where 𝐼 determine the intensity of the light
evaluated with a sample in the beam and the 𝐼0 is the intensity of the light evaluated with no sample
in. By identifying the functional groups present in the samples that absorb infrared radiation at
certain wavenumbers, the molecules in the solvents could be identified. This enables the structure
of unidentified molecules to be determined, making the FTIR spectra a valuable analytical tool.
The set-up of the FTIR spectrometer was modified to allow vacuuming and continuous
nitrogen purging during the run of the experiment to help reduce volume of H2O and CO2 in the
FTIR chamber. After 15-20 minutes into the nitrogen purging, the first background scan was taken
with an empty sample glass cell set in the FTIR chamber. The empty sample glass cell was then
removed. Using the 50 𝜇𝐿 glass syringe 10 𝜇𝐿 of the solvent were placed into the glass cell. The
filled sample glass cell was then mounted in the FTIR chamber. The glass cell had a path length
of 10 𝑐𝑚 with windows of 2 𝑐𝑚 (width) made of NaCl (𝑉 = 31.415 𝑐𝑚3). Volume of the glass
cell is V=πr2h. Based on that determined the volume of the sample. Volume of the sample was
10𝜇𝐿 to collect the FTIR gas-phase spectra. FTIR scans were then taken immediately and every
second after, until a smooth IR spectrum was obtained. To attain the 4 smoothest scans, the internal
56
environment of the FTIR chamber was manually adjusted until water vapor peaks were minimized
and the spectrums of the sample collected. There is a list of molecules in table 4 were characterized
by FTIR high-resolution gas-phase spectroscopy.
Table 4: List of Samples characterized using high-resolution gas-phase FTIR
Chemical Structure Molecule # of Atoms
CH2O2 Formic Acid 5
C2H4O2 Acetic Acid 8
C3H6O Acetone 10
Vapor pressure analysis of solvent is important factor in the FTIR gas-phase spectroscopy.
One or both energy levels involved in a molecular collision are displaced if the molecule's
rotational energy changes due to radiation adsorption, as the overall pressure of the gas phase
sample increases. High-resolution spectra were studied using acetone, formic acid, and acetic acid
as solvents. Various molecular configurations, molecular weights, and vapor pressures of each of
these solvents affect their infrared radiation absorption. The smaller the molecular weight, the
lower the vapor pressure of the solvent. Vapor pressure has solved by the Antoine equation:
log10(𝑃) = 𝐴 − (𝐵/(𝐶 + 𝑇)), in bar units at the room temperature in Kelvin (K). Molecular
weight of formic acid is 46.03 g/mol and it is smaller than acetone and acetic acid. So, formic acid
has low vapor pressure than acetone and acetic acid. Acetone has a molecular weight of 58.08
g/mol, which is quite comparable to that of acetic acid, which is 60.09 g/mol. Both solvents have
a C=O functional group, while acetic acid contains an OH functional group, which might also
illustrate the reason that acetone has a higher vapor pressure in comparison of acetic acid. This
difference in functional groups between the two solvents will be useful to spot in the IR spectrum.
57
3.2 Avogadro
The computational journey began with the creation of a molecule using the Avogadro
software. It started with the choice of elements and how they are bonded together in terms of bond
order. Once completed the designing of a molecule with the help of various tools, chose Extension
from the tool bar to improve the molecule's geometry. Created Psi4 input file with the help
of extension bar, clicking on "Psi4". In the Psi4 input window, first entered the title and
then selected the calculation type, such as energy, optimize, or frequency. Furthermore, selected
the specific method and basis set and saved as an input file. That file was opened in Notepad
(Windows). Accessed through the terminal, gave a command to use certain input file to create an
output file with the use of Psi4. Made sure the input files were named correctly since those input
files were used to create the output files by the computer.
3.3 Psi4
The Schrödinger equation is the basis of quantum mechanics and gives a complete
description of the electronic structure of a molecule. If the equation could be fully solved all
information pertaining to a molecule could be determined. DFT, Couple Cluster Theory, MP2
Theory calculations were performed with the Psi4 programs. DFT, Couple Cluster Theory, MP2
Theory calculations were performed with the Psi4 programs. These theories are numerical
technique used for describing many-body systems. These are post-Hartree–Fock ab initio quantum
chemistry methods in the field of computational chemistry. Tried above theories with different
basis set such as cc-pVDZ, cc-pVTZ, aug-cc-pVDZ and aug-cc-pVTZ. Obtained Output file of
energy which gives us HOMO, LUMO energies and bond distance. Optimized energy to get
optimized bond distance as well as gives us Total energy. Obtained frequency output file to get all
58
vibrational frequency of the molecule. Molecular output file contains various numerical data;
however, we are using those data which are needed such as vibrational frequencies to compare
with experimental IR vibrational spectrum. Table 3 illustrates a list of molecules calculated by
computational method in Psi4. After observing the effectiveness and accuracy of different methods
with same size of molecules; eliminated some molecule observed with same tendency and chosen
selected ones with different number of atoms to demonstrate the frequency of computational
vibrational normal modes and compare that with high-resolution gas-phase IR spectra. Table 5
shows the listed molecules to analyze the cost effectiveness of computational methods and basis
set.
Table 5: List of molecules to analyze the cost effectiveness of computational methods and
basis sets
Chemical Structure
Molecule # Of
Atoms Chemical Structure
Molecule # Of Atoms
H2 Dihydrogen 2 CH2O2 Formic acid 5
C2 Dicarbon 2 CH4 Methane 5
HCl Hydrogen Chloride
2 CH3OH Methanol 6
CO Carbon
monoxide 2 C2H4O2 Acetic acid 8
H2O Water 3 C3H6O Acetone 10
CH2O Formaldehyde 4 C6H12 Cyclohexane 18
59
4. Results and Discussion
In the following, evaluate the molecular computational frequency of vibrational normal
modes to compare with experimental gas-phase IR spectra. Numerical data being trustworthy and
reliable are rely on the type of calculation in term of method and basis set to be used. Based on
accuracy and time selected the theory and basis set for accurate calculation.
Different methods use different approximations (or levels of theory) to produce results of
varying levels of accuracy. There is a tradeoff between accuracy and computational time. In
chemistry a basis set is a group of mathematical functions used to describe the shape of the orbitals
in a molecule, each basis set is a different group of constants used in the wavefunction of the
Schrödinger equation. Once again there is a tradeoff between accuracy and time. Larger basis sets
will describe the orbitals more accurately but take longer to solve and it also applicable with
different theory too.
The accuracy of a calculation is dependent on both the model and the type of basis set
applied to it. Obtained energy, optimization, and frequencies of molecules with the help of
computational methods. Here comparing Psi4 calculations with NIST computational calculation
with IR frequency of vibrational modes and comparing HOMO energy and ionization energy.
Using seven different method and basis set to collected Psi4 calculated frequencies and compare
that with NIST calculated frequencies as well as experimental IR frequencies to determine the
effective method and basis set among of all.
4.1 Scaling factor
The scaling factor is an essential factor in vibrational computational calculations.
Computational vibrational frequencies obtained by ab initio programs must be multiplied by a
60
scaling factor between 0.8 to 1.0 cm-1 to match with the experimental vibrational frequencies.
Scaling factor can be determined by averaging known experimental frequencies using multiple
molecules and the calculated frequencies. There is a different scaling factor for each combination
of theory and basis set. The scaling factor (c) is equal to experimentally observed vibrational
frequencies (νi) divided by the calculated vibrational frequencies (ωi) and standard deviation is
± 0.0213. Table 6 shows the scaling factor value varies depending on computational method and
basis set. When the scaling factor is 1, then the experimental values match the calculated values.
Table 6: Scaling factor value varies on different theory and basis set
Theory Basis Set Scaling Factor
MP2 cc-pVDZ 0.9525
B3LYP cc-pVDZ 0.9700
CCSD(T) cc-pVDZ 0.9788
CCSD(T) cc-pVTZ 0.9748
CCSD cc-pVDZ 0.9473
CCSD(T) aug-cc-pVTZ 0.9610
CCSD(T) aug-cc-pVDZ 0.9650
4.2 Diatomic molecules
Calculated vibrational frequency of several HOMO nuclear and heteronuclear diatomic
using computational method and basis set in Psi4. HOMO nuclear diatomic molecules are
observed in the Raman spectra but not in the IR spectra. This is since diatomic molecules have one
band and no permanent dipole, and therefore one single vibration.
Compared calculated Psi4 frequency with NIST calculated frequency found out the
effective computational method and basis set for molecules. Purpose of using diatomic is to get
the far visualization for polyatomic. Performed Psi4 calculation on several molecules to examine
the tendency of theory and basis set for different diatomic such as HOMO nuclear and
61
heteronuclear. Analyzed same rhythm of theory and basis set for different diatomic molecules. So,
displayed popular molecule such as H2, C2 as HOMO nuclear and HCl, CO as heteronuclear
diatomic molecules.
4.2.1 H2 - Dihydrogen
Presented Psi4 calculated as well as NIST literature calculated numerical values in table 7
and 8.
Table 7: Psi4 computational calculations of H2 molecule
Method Basis Set HOMO Energy (Ha)
HOMO-LUMO Gap Energy
(Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-pVDZ -0.592487 0.653968 -1.163402 4638.6367
CCSD(T) cc-pVTZ -0.594381 0.761498 -1.172336 4432.9723
CCSD(T) aug-cc-pVTZ
-0.594355 0.646916 -1.172635 4429.9786
B3LYP-D cc-pVDZ -0.42988 0.505092 -1.173604 4659.5389
MP2 cc-pVDZ -0.592109 0.789305 -1.155207 4692.3967
CCSD(T) aug-cc-pVDZ
-0.592487 0.653968 -1.164899 4635.6359
CCSD cc-pVDZ -0.592109 0.789357 -1.163672 4667.1267
Table 8: NIST computational calculations of H2 molecule
Method Basis Set HOMO Energy
(Ha)
HOMO-LUMO Gap Energy
(Ha)
Full Energy
(Ha)
Frequency (cm-1)
CCSD(T) cc-pVDZ -0.583 0.775 -1.163669 4382
CCSD(T) cc-pVTZ -0.591 0.756 -1.172333 4411
CCSD(T) aug-cc-pVTZ
-0.591 0.643 -1.172632 4402
B3LYP cc-pVDZ -0.426 0.496 -1.173602 4368
MP2 cc-pVDZ -0.588 0.783 -1.155218 4501
CCSD(T) aug-cc-pVDZ
0.583 0.644 -1.164896 4344
CCSD cc-pVDZ -0.586 0.779 -1.163673 4383
62
Ionization energy of H2 identified by NIST is 0.566891 ha, whereas the value of HOMO
energy obtained using Psi4 CCSD/cc-pVDZ is 0.592109 ha and using NIST CCSD/cc-pVDZ is -
0.586 ha. This is given in tables 7 and 8. Among all the methods, the closet one is CCSD/cc-pVDZ.
The HOMO-LUMO gap energy using Psi4 CCSD/cc-pVDZ is 0.789357 ha, while using NIST, it
is 0.779 ha. The full energy using Psi4 CCSD/cc-pVDZ is 1.163672 ha, while using NIST, it is -
1.163673 ha. It is the total potential energy of the molecule. HOMO nuclear diatomic molecules
such as H2 no dipole moment and are IR inactive. With the help of computational methods, we can
get the vibrational frequency, however it isn’t possible to collect with IR spectroscopy. The
frequency obtained using Psi4 CCSD/cc-pVDZ is 4667.1267 cm-1, while using NIST, it is
4383 cm-1. The couple cluster theory is proved to be the most efficient amongst all the seven
method and basis sets.
4.2.2 C2 - Dicarbon
Table 9 and 10 clearly shows Psi4 calculations and NIST literature calculation respectively.
Table 9: Psi4 computational calculations of C2 molecule
Method Basis
Set HOMO Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-
pVDZ -0.454941 0.348602 -75.731108 1820.6994
CCSD(T) cc-
pVTZ -0.457095 0.34556 -75.814713 1863.4012
CCSD(T) aug-cc-pVTZ
-0.457961 0.342054 -75.815021 1865.4120
B3LYP-D cc-
pVDZ -0.327402 0.067416 -75.891846 1874.2962
MP2 cc-
pVDZ -0.454922 0.348588 -75.703127 1876.018
CCSD(T) aug-cc-pVDZ
-0.457653 0.341082 -75.738391 1987.2745
CCSD cc-
pVDZ -0.454941 0.348602 -75.703073 1864.4752
63
Table 10: NIST computational calculations of C2 molecule
Method Basis Set HOMO
Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-pVDZ -0.44644 0.33978 -75.727818 1828
CCSD(T) cc-pVTZ -0.45336 0.54779 -75.783164 1846
CCSD(T) aug-cc-pVTZ
-0.4538 0.33785 -75.786379 1841
B3LYP cc-pVDZ -0.32464 0.0643 -75.889248 1875
MP2 cc-pVDZ -0.44732 0.34067 -75.699437 1873
CCSD(T) aug-cc-pVDZ
-0.44888 0.33212 -75.734751 1814
CCSD cc-pVDZ -0.44937 0.34277 -75.699956 1861
Ionization energy of C2 identified by NIST is 0.418942 ha, whereas the value of HOMO
Energy obtained using Psi4 MP2/cc-pVDZ is 0.454922 ha and using NIST MP2/cc-pVDZ is
0.44732 ha. This is given in tables 9 and 10. Among all the methods, the closet one is MP2/cc-
pVDZ. The HOMO-LUMO gap energy using Psi4 MP2/cc-pVDZ is 0.348588 ha, while using
NIST, it is 0.34067 ha. The full energy using Psi4 MP2/cc-pVDZ is 75.703127 ha, while using
NIST, it is 75.699437 ha. It is the total potential energy of the molecule. HOMOnuclear diatomic
molecules such as C2 no dipole moment and are IR inactive. With the help of computational
methods, we can get the vibrational frequency, however it isn’t possible to collect with IR
spectroscopy. The frequency obtained using Psi4 MP2/cc-pVDZ is 1876.018 cm-1, while using
NIST, it is 1873 cm-1. The Møller–Plesset perturbation theory is proved to be the most efficient
amongst all the seven methods and basis sets.
64
4.2.3 HCl - Hydrogen Chloride
HCl diatomic molecule has one normal mode of vibration. HCl is heteronuclear diatomic
and do have dipole moments hence have IR active vibrations and easily observe in IR spectrum.
Table 11 illustrate the computational calculations of Psi4, where table 12 shows the study of NIST
literature computational calculations.
Table 11: Psi4 computational calculations of HCl molecule
Method Basis Set HOMO Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-pVDZ -0.470718 0.611726 -460.260262 3024.4608
CCSD(T) cc-pVTZ -0.474617 0.593444 -460.369653 2999.1172
CCSD(T) aug-cc-pVTZ -0.476457 0.505435 -460.377976 3002.9553
B3LYP-D cc-pVDZ -0.326788 0.329473 -460.822184 2930.6587
MP2 cc-pVDZ -0.470709 0.611764 -460.242035 3081.0261
CCSD(T) aug-cc-pVDZ -0.476619 0.511125 -460.278878 2976.55
CCSD cc-pVDZ -0.470718 0.611726 -460.257716 3038.0015
Table 12: NIST computational calculations of HCl molecule
Method Basis Set HOMO Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-pVDZ -0.47085 0.6133 -460.254636 3015
CCSD(T) cc-pVTZ -0.47509 0.59749 -460.337216 3001
CCSD(T) aug-cc-pVTZ
Not available
Not available -460.343241 2992
B3LYP cc-pVDZ -0.32703 0.33252 -460.822164 2921
MP2 cc-pVDZ -0.47121 0.61753 -460.235826 3076
CCSD(T) aug-cc-pVDZ
Not available
Not available -460.272302 2969
CCSD cc-pVDZ -0.47087 0.61361 -460.25227 3032
65
Ionization energy of HCl identified by NIST is 0.468333 ha, whereas the value of HOMO
energy obtained using Psi4 MP2/cc-pVDZ is 0.470709 ha and using NIST MP2/cc-pVDZ is
0.47121 ha. This is given in tables 11 and 12. Among all the methods, the closet one is MP2/cc-
pVDZ. The HOMO-LUMO gap energy using Psi4 MP2/cc-pVDZ is 0.611764 ha, while using
NIST, is 0.61753 ha. The full energy using Psi4 MP2/cc-pVDZ is 460.242035 ha, while using
NIST, is 460.235826 ha. Heteronuclear diatomic molecules such as HCl have dipole moment and
are IR active. With the help of computational methods, we can get the vibrational frequency, and
compare that with experimental IR frequency. The frequency obtained using Psi4 MP2/cc-pVDZ
is 3081.0261 cm-1, while using NIST, it is 3076 cm-1. The Møller–Plesset perturbation theory is
proved to be the most efficient amongst all the seven method and basis sets for energy.
Figure 11: FTIR vibrational spectra of HCl molecule [Reprinted from “Data from NIST
Standard Reference Database 69: NIST Chemistry Webbook”]
In figure 11 shows the FTIR spectra of HCl and observed peak absolutely match with
calculated vibrational frequency.
66
Table 13: Comparison of experimental with calculated frequency of HCl molecule
Mode Experimental Fundamental freq. (cm-1)
Experimental Harmonic freq. (cm-1)
Calculated Scaled freq. (cm-1)
CCSD(T)/ cc-pVTZ Psi4 freq. (cm-1)
Stretching 2886 2991 2925 2999
As illustrated in Table 13, the value obtained for experimental harmonic frequency using
NIST is 2991 cm-1, while using Psi4, it is 2999 cm-1. These 2 values are close and accurate. Scaling
factor also affects the calculated frequency. After getting the value of scaling factor of component
seems like this is very close calculated frequency to experimental harmonic frequency. The couple
cluster theory has been the most effective for frequency, in comparison to all the other methods.
4.2.4 CO - Carbon Monoxide
Below table 14 explains the computational calculations of Psi4, where table 15 illustrate
the study of NIST literature computational calculations.
Table 14: Psi4 Computational calculations of CO Molecule
Method Basis Set HOMO
Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-pVDZ -0.549029 0.702924 -113.058635 2145.5994
CCSD(T) cc-pVTZ -0.553803 0.691742 -113.180495 2190.6234
CCSD(T) aug-cc-pVTZ
-0.555139 0.624423 -113.191836 2169.9305
B3LYP-D cc-pVDZ -0.373142 0.350606 -113.321350 2201.1502
MP2 cc-pVDZ -0.554888 0.633784 -113.040681 577.8662
CCSD(T) aug-cc-pVDZ
-0.554888 0.633784 -113.078360 2262.6183
CCSD cc-pVDZ -0.549029 0.702924 -113.047578 2212.0959
67
Table 15: NIST computational calculations of CO molecule
Method Basis Set HOMO Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-pVDZ -0.35569 0.47509 -113.054976 2144
CCSD(T) cc-pVTZ -0.35764 0.46399 -113.155578 2154
CCSD(T) aug-cc-pVTZ
-0.35764 0.46399 -113.162193 2144
B3LYP cc-pVDZ -0.37357 0.34926 -113.321369 2201
MP2 cc-pVDZ -0.55098 0.69784 -113.036807 2114
CCSD(T) aug-cc-pVDZ
-0.3627 0.43449 -113.074052 2104
CCSD cc-pVDZ -0.54989 0.69993 -113.043969 2209
Ionization energy of CO identified by NIST is 0.515004 ha, whereas the value of HOMO
energy obtained using Psi4 CCSD/cc-pVDZ is 0.549029 ha and using NIST CCSD/cc-pVDZ is
0.54989 ha. This is given in tables 14 and 15. Among all the methods, the closet one is CCSD/cc-
pVDZ. The HOMO-LUMO gap energy using Psi4 CCSD/cc-pVDZ is 0.702924 ha, while using
NIST, it is 0.69993 ha. The full energy using Psi4 CCSD/cc-pVDZ is -113.047578 ha, while using
NIST, it is -113.043969 ha. It is the total potential energy of the molecule. Heteronuclear diatomic
molecules such as CO have dipole moment and are IR active. With the help of computational
methods, we can get the vibrational frequency, and compare that with experimental IR frequency.
The frequency obtained using Psi4 CCSD/cc-pVDZ is 2212.0959 cm-1, while using NIST, it is
2209 cm-1. The couple cluster theory is proved to be the most efficient amongst all the seven
method and basis sets.
Figure 12 shows the high-resolution gas-phase FTIR spectra of CO molecule can partially
resolve rovibrational transitions in the P, Q, and R bands.
68
Figure 12: FTIR vibrational spectra of carbon monoxide molecule [Reprinted from “Data from
NIST Standard Reference Database 69: NIST Chemistry Webbook”]
Table 16: Comparison of experimental with calculated frequency of CO molecule
Mode Experimental Fundamental freq. (cm-1)
Experimental Harmonic freq. (cm-1)
Calculated Scaled freq. (cm-1)
CCSD(T)/ aug-cc-pVTZ Psi4 freq. (cm-1)
Stretching 2143 2169 2080 2169
As illustrated in Table 16, the value obtained for experimental harmonic frequency using
NIST is 2169 cm-1, while using Psi4, it is 2169 cm-1. The Psi4 value is precise as the value
completely matches the NIST experimental value. The couple cluster theory-CCSD(T) and aug-
cc-pVTZ have been the most effective method and basis set respectively, for frequency, in
comparison to all the other methods and basis sets.
69
4.3 Triatomic molecule
4.3.1 H2O - Water
The non-linear triatomic H2O molecule forms an angle with oxygen at its center. Since this
molecule is nonlinear, it has oscillating dipole moments, making it an IR active molecule. and will
exhibit absorption bands in the range (3N - 6). Where n denotes the atomic number. Thus, the
water molecule (N = 3 atoms) will have 3N - 6 = 3. Three degrees of freedom in terms of vibration.
This indicates that the three vibrations in the molecule have a dipole moment. During the vibration,
the electric dipole moment changes on a periodic basis. The dipole moment can occur parallel to
or across the symmetry axis. Each of these vibrational modes is associated with a dipole moment
and is IR active. These vibrational modes are as follows: stretching in symmetry, asymmetrical
stretching, and symmetrical bending (scissoring).
Table 17 Illustrate the computational calculations of Psi4, where table 18 explains the study
of NIST literature computational calculations.
Table 17: Psi4 computational calculations of H2O molecule
Method Basis Set HOMO Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-pVDZ -0.492521 0.675986 -76.2432 1677.512 3769.842 3894.546
CCSD(T) cc-pVTZ -0.503717 0.644644 -76.3457 1667.822 3857.18 3962.082
CCSD(T) aug-cc-pVTZ
-0.509561 0.538821 -76.3575 1666.87 3704.911 3804.972
B3LYP-D cc-pVDZ -0.287343 0.33673 -76.4206 1656.339 3749.073 3852.621
MP2 cc-pVDZ -0.492515 0.675985 -76.0259 1668.732 3781.355 3918.96
CCSD(T) aug-cc-pVDZ
-0.508636 0.543824 -76.2761 1645.185 3748.703 3867.298
CCSD cc-pVDZ -0.492521 0.675986 -76.2401 1689.125 3776.989 3897.542
70
Table 18: NIST computational calculations of H2O molecule
Method Basis Set HOMO Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-pVDZ -0.49299 0.67464 -76.2413 3820 1690 3926
CCSD(T) cc-pVTZ -0.50399 0.64467 -76.3322 3840 1668 3945
CCSD(T) aug-cc-pVTZ
-0.50946 0.53866 -76.3423 3812 1645 3921
B3LYP-D cc-pVDZ -0.28775 0.33698 -76.4206 3752 1660 3853
MP2 cc-pVDZ -0.49357 0.67731 -76.2286 3852 1678 3971
CCSD(T) aug-cc-pVDZ
Not available Not available -76.2739 3788 1637 3907
CCSD cc-pVDZ -0.49298 0.67493 -76.2382 3846 1697 3950
Ionization energy of H2O identified by NIST is 0.463813 ha, whereas the value of HOMO
energy obtained using Psi4 CCSD(T)/cc-pVDZ is 0.492521ha and using NIST CCSD(T)/cc-
pVDZ is 0.49299 ha. This is given in tables 17 and 18. Among all the methods, the closet one is
CCSD(T)/cc-pVDZ. The HOMO-LUMO gap energy using Psi4 CCSD(T)/cc-pVDZ is 0.675986
ha, while using NIST, it is 0.67464 ha. The full energy using Psi4 CCSD(T)/cc-pVDZ is -76.2432
ha, while using NIST, it is -76.2413 ha. It is the total potential energy of the molecule. With the
help of computational methods, we can get the vibrational frequency, and compare that with
experimental IR frequency. The frequency obtained by Psi4 CCSD(T)/cc-pVDZ is 1677 cm-1,
3769 cm-1, 3894 cm-1, while using NIST, it is 3820 cm-1, 1690 cm-1, 3926 cm-1, The couple cluster
theory CCSD(T) and basis set cc-pVDZ are provided to be the most efficient amongst all the seven
methods and basis sets for energy.
71
Figure 13: FTIR vibrational spectra of water molecule [Reprinted from “Data from NIST
Standard Reference Database 69: NIST Chemistry Webbook”]
Above, figure 13, shows the FTIR spectra of H2O molecule. look at the infrared spectrum
of the H2O molecule, the frequencies 3832 cm-1 and 1649 cm-1 correspond to symmetric vibrations,
while the highest frequency 3943 cm-1corresponds to an asymmetric mode. The low frequency
could be attributed to vibration caused by angle bending. However, because the two O-H bonds in
H2O are identical, their stretching vibrations cannot produce absorptions at different frequencies
independently. Each absorption should not be assigned to a distinct bond's vibration. The
vibrations of the two bonds are coupled (overlapped) to produce two distinct vibrations, one with
symmetric stretching and one with asymmetric stretching. Normal vibrations refer to these three
vibrational motions because all three atoms in the molecule vibrate at the same frequency and in
phase with one another.
72
Table 19: Comparison of calculated and experimental vibrational frequencies of water
Mode Experimental Fundamental freq. (cm-1)
Experimental Harmonic freq. (cm-1)
Calculated Scaled freq. (cm-1)
CCSD(T)/ cc-pVTZ Psi4 freq. (cm-1)
Symmetric Stretching
3657 3832 3744 3857
Symmetric Bending Mode
1595 1649 1626 1667
Asymmetric Stretching
3756 3943 3846 3962
Table 19 shows the experimental infrared frequencies of the H2O molecule, the frequency
observed at 3832 cm-1 corresponds to a symmetric stretching mode and 1649 cm-1 correspond to a
symmetric bending mode, while the highest frequency, 3756 cm-1 corresponds to an asymmetric
stretching mode. However, with calculated data (CCSD(T)/cc-pVTZ) observed a symmetric
stretching mode at 3857 cm-1, a symmetric bending mode at 1667 cm-1 and an asymmetric
stretching mode at 3962 cm-1. Couple cluster theory and specially CCSD(T) method with cc-pVTZ
gives us the nearest frequency to IR spectral frequency. Also, there is scaling factor affect the
calculated frequency. After solving the scaling factor, these frequency values are approximately
the same as the experimental harmonic frequencies. NIST has calculated frequency data available.
NIST calculated data of H2O molecule, the frequency observed at 3945 cm-1 corresponds to a
symmetric stretching mode and 1668 cm-1 correspond to a symmetric bending mode, while the
highest frequency, 3840 cm-1 corresponds to an asymmetric stretching mode. The values computed
by Psi4 are more accurate as compared to NIST calculated values, and also matches with the
experimental values.
73
4.4 Trigonal planar molecule
4.4.1 CH2O-Formaldehyde
Formaldehyde is nonlinear, it has oscillating dipole moments, making it an IR active
molecule. and will exhibit absorption bands in the range (3N - 6). Where N denotes the atomic
number. Thus, the water molecule (N = 4 atoms) will have 3N - 6 = 6. Six degrees of freedom in
terms of vibration. This indicates that the six vibrations in the molecule have a dipole moment.
Table 20, 21 and 22 represent the numerical data of Psi4 calculations and NIST literature
experimental and calculated data.
Table 20: Psi4 computational calculations of CH2O molecule
Method Basis Set
HOMO Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-
pVDZ -0.4383 0.568101 -114.2232 1177 1267 1536 1788 2929 2991
CCSD(T) cc-
pVTZ -0.444 0.557648 -114.3624 N/A
CCSD(T) aug-cc-pVTZ
-0.4463 0.472163 -114.3753 N/A
B3LYP-D cc-
pVDZ -0.2675 0.21809 -114.5081 1187 1249 1512 1830 2848 2902
MP2 cc-
pVDZ -0.4383 0.56811 -114.1975 1199 1276 1549 1785 2971 3041
CCSD(T) aug-cc-pVDZ
-0.445 0.477525 -114.2501 1146 1257 1543 1886 3135 3210
CCSD cc-
pVDZ -0.4383 0.568101 -114.213 1197 1280 1551 1831 2957 3021
Ionization energy of CH2O identified by NIST is 0.399832 ha, whereas the value of HOMO
energy obtained using Psi4 CCSD(T)/cc-pVDZ is 0.4383 ha and using NIST CCSD(T)/cc-pVDZ
is 0.43362 ha. This is given in tables 20 and 21. Among all the methods, the closet one is
CCSD(T)/cc-pVDZ. The HOMO-LUMO gap energy using Psi4 CCSD(T)/cc-pVDZ is 0.568101
74
ha, while using NIST, it is 0.56269 ha. The full energy using Psi4 CCSD(T)/cc-pVDZ is -114.2232
ha, while using NIST, it is -114.219 ha. It is the total potential energy of the molecule. With the
help of computational methods, we can get the vibrational frequency, and compare that with
experimental IR frequency. In formaldehyde, vibrational CH2 wagging mode for Psi4 is 1177 cm-
1 while for NIST, it is 1176 cm-1, vibrational CH2 rocking mode for Psi4 is 1267 cm-1 while for
NIST, it is 1265 cm-1, vibrational CH2 scissoring mode for Psi4 is 1536 cm-1 while for NIST, it is
1535 cm-1, vibrational CO stretching mode for Psi4 is 1788 cm-1 while for NIST, it is 1787 cm-1,
vibrational CH2 symmetric stretching mode for Psi4 is 2929 cm-1 while for NIST, it is 2925 cm-1,
vibrational CH2 asymmetric stretching mode for Psi4 is 2991 cm-1 while for NIST, it is 2987 cm-
1. The frequency calculations for psi4 and NIST are almost same for CCSD(T)/cc-pVDZ.
Table 21: NIST computational calculations of CH2O molecule
Method Basis Set
HOMO Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-
pVDZ -0.43362 0.56269 -114.219 2925 1787 1535 1176 2987 1265
CCSD(T) cc-
pVTZ -0.43972 -0.43972 -114.3338 2929 1781 1543 1192 2996 1274
CCSD(T) aug-cc-pVTZ
-0.44222 0.46821 -114.3429 2933 1766 1530 1181 3001 1261
B3LYP-D
cc-pVDZ
-0.26706 0.22291 -114.5076 2866 1832 1515 1186 2919 1253
MP2 cc-
pVDZ -0.43609 0.5682 -114.1932 2965 1783 1547 1199 3034 1275
CCSD(T) aug-cc-pVDZ
-0.44159 0.47399 -114.2452 2932 1737 1515 1167 3010 1245
CCSD cc-
pVDZ -0.43502 0.56856 -114.2089 2952 1829 1549 1195 3016 1279
75
Figure 14: FTIR vibrational spectra of CH2O molecule [Reprinted from “Data from NIST
Standard Reference Database 69: NIST Chemistry Webbook”]
Figure 14 shows the absorption peak in the infrared region with experimental spectroscopy
which observed closer to the calculated frequency.
Table 22: Comparison of experimental and calculated frequencies of formaldehyde
Mode Experimental Fundamental freq. (cm-1)
Calculated Scaled freq. (cm-1)
B3LYP-D/ cc-pVDZ Psi4 freq. (cm-1)
CH2 s-str 2782 2780 2848
CO stretch 1746 1777 1830
CH2 scissors 1500 1470 1512
CH2 wag 1167 1151 1187
CH2 a-str 2843 2831 2902
CH2 rock 1249 1215 1249
In formaldehyde, vibrational CH2 wagging mode for Psi4 is 1187 cm-1 while for
experimental fundamental frequency, it is 1167 cm-1, vibrational CH2 rocking mode for Psi4 is
1249 cm-1 while for experimental fundamental frequency, it is 1249 cm-1, vibrational CH2
scissoring mode for Psi4 is 1512 cm-1 while for experimental fundamental frequency, it is 1500
76
cm-1, vibrational CO stretching mode for Psi4 is 1830 cm-1 while for experimental fundamental
frequency, it is 1746 cm-1, vibrational CH2 symmetric stretching mode for Psi4 is 2848 cm-1 while
for experimental fundamental frequency, it is 2782 cm-1, vibrational CH2 asymmetric stretching
mode for Psi4 is 2902 cm-1 while for experimental fundamental frequency, it is 2843 cm-1.
Observed CH2 rocking mode and CH2 scissoring mode in Psi4, and experimental fundamental
frequency are same. After calculating the scaling factors, other frequencies compute to close values
in B3LYP-D/ cc-pVDZ. The frequency calculations for psi4 and NIST are almost identical for
B3LYP-D/cc-pVDZ.
4.5 Tetrahedral molecule
4.5.1 CH4 - Methane
CH4 (Methane) is a tetrahedral molecule, the simplest hydrocarbon. Methane is a
symmetric five atomic molecule characterized by nine vibrational degrees of freedom with a
vanishing permanent dipole moment. It is a very high-symmetry molecule of the Td symmetry
group containing several degenerate modes. In methane there are multiple vibrations have the same
energy. In below table 23 and 24 there is comparison between Psi4 computational calculations and
NIST literature calculations.
Ionization energy of CH4 identified by NIST is 0.463408 ha, whereas the value of HOMO
energy obtained using Psi4 CCSD(T)/cc-pVDZ is -0.542 ha and using NIST CCSD(T)/cc-pVDZ
is -0.535 ha. This is given in tables 23 and 24. Among all the methods, the closet one is
CCSD(T)/cc-pVDZ. The HOMO-LUMO gap energy using Psi4 CCSD(T)/cc-pVDZ is 0.7347 ha,
while using NIST, it is 0.726 ha. The full energy using Psi4 CCSD(T)/cc-pVDZ is -40.39 ha, while
using NIST, it is -40.38 ha. It is the total potential energy of the molecule. With the help of
77
computational methods, we can get the vibrational frequency, and compare that with experimental
IR frequency.
Table 23: Psi4 computational calculations of CH4 molecule
Method Basis Set
HOMO Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-
pVDZ -0.542 0.7347 -40.39 1333 1333 1333 1552 1552 3055 3191 3191 3191
CCSD(T) cc-
pVTZ -0.543 0.6872 -40.45 1353 1353 1353 1585 1586 3054 3158 3159 3159
B3LYP-D cc-
pVDZ -0.389 0.4639 -40.51 1305 1305 1305 1518 1518 2987 3121 3121 3121
MP2 cc-
pVDZ -0.542 0.7346 -40.36 1340 1340 1340 1568 1568 3088 3241 3241 3241
CCSD(T) aug-cc-
pVDZ -0.543 0.58 -40.39 1321 1321 1321 1538 1538 3018 3147 3147 3147
CCSD cc-
pVDZ -0.542 0.7347 -40.38 1343 1343 1343 1560 1560 3059 3193 3193 3193
Table 24: NIST computational calculations of CH4 molecule
Method Basis Set
HOMO Energy (Ha)
HOMO-LUMO Gap Energy (Ha)
Full Energy (Ha)
Frequency (cm-1)
CCSD(T) cc-
pVDZ -0.535 0.726
-40.38
3041 1550 1550 3176 3176 3176 1333 1333 1333
CCSD(T) cc-
pVTZ -0.541 0.684
-40.43
3036 1570 1570 3155 3155 3155 1343 1343 1343
B3LYP-D cc-
pVDZ -0.388 0.461
-40.51
3027 1531 1531 3149 3149 3149 1310 1310 1310
MP2 cc-
pVDZ -0.54 0.731
-40.36
3082 1565 1565 3234 3234 3234 1338 1338 1338
CCSD(T) aug-cc-
pVDZ N/A N/A
-40.39
3017 1535 1535 3145 3145 3145 1319 1319 1319
CCSD cc-
pVDZ -0.535 0.726
-40.38
2893 1476 1476 3018 3018 3018 1271 1271 1271
In methane, vibrational Symmetric stretch for Psi4 is 3055 cm-1 while for NIST, it is 3041
cm-1, vibrational degenerated stretching mode for Psi4 is 3191 cm-1 while for NIST, it is 3176 cm-
1, vibrational mode of degenerated deformation for Psi4 is 1333 cm-1 while for NIST, it is 1333
78
cm-1, one more degenerated deformation observed of vibrational mode for Psi4 is 1552 cm-1 while
for NIST, it is 1550 cm-1 for CCSD(T)/cc-pVDZ. The couple cluster theory CCSD(T) and basis
set cc-pVDZ are effective amongst all the seven methods and basis sets for energy. Cost
effectiveness of CCSD(T) and aug-cc-pVTZ are high so cannot obtained the data from it. However,
with growing world future chemist can obtain data from it with high speed and large memory
component of the bigger computer.
Figure 15 experimental vibrational frequency matches with the computational vibrational
frequencies
Figure 15: FTIR vibrational spectra of CH4 molecule [Reprinted from “Data from NIST
Standard Reference Database 69: NIST Chemistry Webbook”]
Table 25: Comparison of experimental and calculated frequencies of methane
Mode Experimental Fundamental freq. (cm-1)
Calculated Scaled freq. (cm-1)
B3LYP-D/cc-pVDZ Psi4 freq. (cm-1)
B3LYP-D/cc-pVDZ NIST freq. (cm-1)
sym stretch 2917 2936 2987 3027
deg. deform
1534 1485 1518 1531
deg. stretch 3019 3054 3121 3149
deg. deform
1306 1271 1305 1310
79
In methane, vibrational Symmetric stretch for Psi4 is 2987 cm-1 while experimental
frequency, it is 2917 cm-1, vibrational degenerated stretching mode for Psi4 is 3121 cm-1 while for
experimental frequency, it is 3019 cm-1, vibrational mode of degenerated deformation for Psi4 is
1305 cm-1 while for experimental frequency, it is 1306 cm-1, one more degenerated deformation
observed of vibrational mode for Psi4 is 1518 cm-1 while for experimental frequency, it is 1534
cm-1 for B3LYP-D/cc-pVDZ. After calculating the scaling factors, experimental fundamental
frequencies compute to close values in B3LYP-D/ cc-pVDZ. Density functional theory B3LYP-
D/cc-pVDZ is effective amongst all the seven methods and basis sets to calculate vibrational
frequency. B3LYP-D/cc-pVDZ calculated vibrational frequency seems closer to the experimental
IR frequency.
4.6 Other polyatomic molecules
4.6.1 CH2O2 - Formic acid
Formic acid has the chemical formula of HCOOH or CH2O2 where a hydrogen atom is
attached to the -COOH group to form the simplest carboxylic acid. It is also known as methanoic
acid. HCOOH is polar in nature so it can absorb IR. exhibit absorption bands in the range (3N -
6). Where N denotes the atomic number. Thus, formic acid molecule (N = 5 atoms) will have 3N
- 6 = 9. Nine degrees of freedom in terms of vibration. This indicates that the nine vibrations in
the molecule have a dipole moment. In table 26 is a comparison between experimental and
calculated frequencies from the NIST and calculated frequencies from Psi4 calculations.
80
Table 26: Comparison of experimental and calculated frequencies of formic acid
Mode Experimental Fundamental freq. (cm-1)
Calculated Scaled freq. (cm-1)
B3LYP-D/cc-pVDZ Psi4 freq. (cm-1)
B3LYP-D/cc-pVDZ NIST freq. (cm-1)
OH str 3570 3568 3677 3678
CH str 2943 2943 3022 3034
C=O str 1770 1789 1842 1845
CH bend 1387 1353 1391 1395
OH bend 1229 1273 1313 1313
C-O str 1105 1105 1136 1139
OCO deform
625 609 634 628
CH bend 1033 1015 1046 1047
Torsion 638 680 696 701
Ionization energy of CH2O2 identified by NIST is 0.416369 ha, whereas the value of
HOMO energy obtained using Psi4 CCSD(T)/cc-pVDZ is 0.4650 ha and using NIST doesn’t have
calculated energy available for CCSD(T)/cc-pVDZ. Among all the methods, the closet one is
CCSD(T)/cc-pVDZ. The HOMO-LUMO gap energy using Psi4 CCSD(T)/cc-pVDZ is 0.6341 ha,
while NIST doesn’t have calculated energy of HOMO-LUMO gap. The Full Energy using Psi4
CCSD(T)/cc-pVDZ is -189.3152 ha, where using NIST, it is -189.3006 ha. The energy calculation
obtained from CCSD(T)/cc-pVDZ are accurate than MP2 and B3LYP-d method.
In the table 26 there is a comparison between experimental and calculated frequency by
DFT theory and in below figure 16 there is a high-resolution FTIR spectra of Formic acid. It seems
getting calculated frequency numbers are nearly match with the spectra received from FTIR.
81
Figure 16: Collected high-resolution FTIR spectra of formic acid
With the help of computational methods, we can get the vibrational frequency, and
compare that with experimental IR frequency in figure 18 and table 26. In Formic acid, vibrational
OCO deform mode for Psi4 is 634 cm-1, For NIST, it is 701 cm-1 while for experimental
fundamental frequency, it is 625 cm-1, vibrational CH bend mode for Psi4 is 1046 cm-1, For NIST,
it is 1047 cm-1 while for experimental fundamental frequency, it is 1033 cm-1, vibrational CH
bending mode for Psi4 is 1391 cm-1, For NIST, it is 1395 cm-1 while for experimental fundamental
frequency, it is 1387 cm-1, vibrational C-O stretching mode for Psi4 is 1136 cm-1, For NIST, it is
1139 cm-1 while for experimental fundamental frequency, it is 1105 cm-1. Calculated vibrational
frequency of OCO deform mode, CH bend modes, and C-O stretching mode has similarities in
values in comparison of experimental, other modes such as C=O str, OH bend and Torsion has
variation in the values as shown in table 26, however, after calculating scaling factor frequencies
seems closer. Able to achieve Competitive frequency numbers from density functional theory
(B3LYP-D) is accurate among of all other methods.
0
20
40
60
80
100
120
500 1000 1500 2000 2500 3000 3500 4000
Tran
smit
tan
ce (%
)
Wavenumber (cm-1)
82
Since formic acid have five atoms, it is challenging to run couple cluster theory with cc-
pVTZ, aug-cc-pVTZ basis set. Seven different method and basis set combinations were attempted,
the coupled cluster calculations didn’t work, such as CCSD(T)/cc-pVTZ, CCSD(T)/aug-cc-pVTZ,
CCSD(T)/aug-cc-pVDZ. These theory and basis set are so accurate; however, it consumes longer
time to run as well as another factor affecting is such as speed and memory of the device.
CCSD(T)/cc-pVDZ, CCSD/cc-pVDZ, MP2/cc-pVDZ and B3LYP-D/cc-pVDZ are worked.
4.6.2 CH3COOH - Acetic acid
This molecule does have total number of eight atoms. It is Polar molecule and have dipole
moment to absorb IR. It has 18 degrees of freedom in terms of vibration. Geometry of the molecule
is carbon 1-tetrahedral, carbon-2-trigonal planar and oxygen-3-bent. It contains the following
bonds: carbon-oxygen double C=O, carbon-oxygen single C-O, oxygen-hydrogen O-H, carbon-
hydrogen C-H, carbon-carbon single, C-C.
Since acetic acid contains eight atoms, it is computationally bigger molecule to run
calculations. Able to calculate only three computational calculations among all seven theory and
basis set. Those methods worked are B3LYP-D/cc-pVDZ, MP2/cc-pVDZ and CCSD/cc-pVDZ
and among all these three B3LYP-D performed well. It didn’t provide sharp accurate frequency;
however, its frequencies are better than MP2 and CCSD.
This is complex molecule to calculate frequency with the help of heavy computational
method (CCSD(T)) in Psi4. In table 27 have illustrate the different modes of vibrational frequency
collected by Density Functional theory with cc-pVDZ basis set on Psi4 platform as well as NIST
literature calculated frequency data.
83
Ionization energy of CH2O2 identified by NIST is 0.391380 ha, whereas the value of
HOMO energy obtained using Psi4 MP2/cc-pVDZ is 0.447292 ha and using NIST is 0.44894 ha.
Among all the methods we were able to calculate MP2 and B3LYP-D/cc-pVDZ, the closet one is
MP2/cc-pVDZ for energy calculations. The HOMO-LUMO gap energy using Psi4 MP2/cc-pVDZ
is 0.622045 ha, while NIST calculated energy of HOMO-LUMO gap is 0.62591 ha . The Full
Energy using Psi4 MP2/cc-pVDZ is -229.1001 ha, where using NIST, is -228.4811 ha. The energy
calculation obtained from MP2/cc-pVDZ are accurate than B3LYP-D only for energies.
Table 27: Comparison of experimental and calculated frequencies of acetic acid
Mode Experimental Fundamental freq. (cm-1)
B3LYP-D/cc-pVDZ Psi4 freq. (cm-1)
B3LYP-D/cc-pVDZ NIST freq. (cm-1)
OH str 3045 3703 3701
CH3 d-str 2969 3104 3123
CH3 s-str 2943 3025 3054
C=O str 1754 1845 1845
CH3 d-deform
1454 1436 1446
CH3 s-deform
1445 1430 1441
OH bend 1371 1330 1336
C-O str 1207 1211 1215
CH3 rock 1166 985 988
CC str 925 673 682
OCO deform 767 584 682
CCO deform 318 418 417
CH3 d-str 3012 3156 3175
CH3 d-deform
1443 1396 1400
CH3 rock 1168 1050 1054
C=O op-bend 1032 863 867
C-O torsion 332 542 584
CH3 torsion 130 88 79
84
Figure 17: Collected high-resolution FTIR spectra of acetic acid
With the help of computational methods, we can get the vibrational frequency, and
compare that with experimental IR frequency in figure 17 and table 27. In Acetic acid, vibrational
CH3 d-deform mode for Psi4 is 1436 cm-1, For NIST, it is 1446 cm-1 while for experimental
fundamental frequency, it is 1454 cm-1, vibrational CH3 s-deform mode for Psi4 is 1430 cm-1, For
NIST, it is 1441 cm-1 while for experimental fundamental frequency, it is 1430 cm-1, vibrational
OH bending mode for Psi4 is 1330 cm-1, For NIST, it is 1336 cm-1 while for experimental
fundamental frequency, it is 1371 cm-1, vibrational CH3 torsion mode for Psi4 is 88 cm-1, For
NIST, it is 79 cm-1 while for experimental fundamental frequency, it is 130 cm-1. Calculated
vibrational frequency of CH3 d-deform mode, CH3 s-deform modes, CH3 torsion and OH bending
mode has similarities in values in comparison of experimental, other modes such as OH str, CH3
rock, CC str, OCO deform, C-O torsion and CCO deform has variation in the values as shown in
table 27. However, after calculating scaling factor frequencies seems a bit closer. Able to achieve
Competitive frequency numbers from density functional theory(B3LYP-D) than MP2 theory.
0
20
40
60
80
100
120
500 1000 1500 2000 2500 3000 3500 4000
Tran
smit
tan
ce (%
)
Wavenumber (cm-1)
85
Since acetic acid have eight atoms, it is challenging to run couple cluster theory with cc-
pVTZ, aug-cc-pVTZ basis set. Tried seven different method and basis set, didn’t work such as
CCSD(T)/cc-pVTZ, CCSD(T)/cc-pVDZ, CCSD(T)/aug-cc-pVTZ, CCSD(T)/aug-cc-pVDZ and
CCSD/cc-pVDZ. These theory and basis set are accurate; however, it consumes longer time to run
as well as another factor affecting is such as speed and memory of the device. MP2/cc-pVDZ and
B3LYP-D/cc-pVDZ are less cost effective so these were able to compute the frequency.
4.6.3 C3H6O - Acetone
Acetone is a computationally large molecule, and it contains ten atoms. This molecule is
nonlinear, it has oscillating dipole moments, making it an IR active molecule. and will exhibit
absorption bands in the range (3N - 6). Where n denotes the atomic number. Thus, the water
molecule (N = 10 atoms) will have 3N - 6 = 24. Twenty-six degrees of freedom in terms of
vibration. This indicates that the twenty-six vibrations in the molecule have a dipole moment. It is
complex molecule to run bigger theory and basis set to collect computational vibrational spectra.
Only three methods and basis set were able to do calculation. One of the best is B3LYP-Dwith
cc-pVDZ which gives us better comparison then other methods. Frequencies obtained from
Density Functional theory is much better than NIST literature calculated frequencies. Psi4
calculated frequencies are much closer to the experimental high-resolution FTIR frequencies.
Below tables represent the numerical data of Psi4 calculations and NIST literature experimental
and calculated frequencies.
86
Table 28: Comparison of experimental and calculated frequencies of acetone
Mode Experimental Fundamental freq. (cm-1)
B3LYP-D/cc-pVDZ Psi4 freq. (cm-1)
B3LYP-D/cc-pVDZ NIST freq. (cm-1)
CH3 d-str 3019 3134 3153
CH3 s-str 2937 3004 3033
CO str 1731 1814 1812
CH3 d-deform 1435 1429 1458
CH3 s-deform 1364 1365 1370
CH3 rock 1066 1068 1104
CC str 777 785 869
CCC deform 385 372 489
CH3 d-str 2963 3072 3091
CH3 d-deform 1426 1425 1427
CH3 rock 877 862 880
Torsion 77 56 43
CH3 d-str 2972 3079 3098
CH3 s-str 1454 1446 1438
CH3 d-deform 1091 1091 1471
CH3 s-deform 484 481 531
CC str 125 148 377
CH3 rock 3019 3133 3152
CO ip-bend 2937 2998 3026
CH3 d-str 1410 1417 1435
CH3 d-deform 1364 1354 1361
CH3 rock 1216 1222 1229
CO op-bend 891 874 1071
Torsion 530 533 789
Ionization energy of acetone identified by NIST is 0.356578 ha, whereas the value of
HOMO energy obtained using Psi4 CCSD/cc-pVDZ is 0.408044 ha and using NIST doesn’t have
calculated energy available for CCSD/cc-pVDZ. Among all the methods we were able to calculate
MP2, CCSD and B3LYP-D/cc-pVDZ, the closet one is CCSD/cc-pVDZ for energy. The HOMO-
LUMO gap energy using Psi4 CCSD/cc-pVDZ is 0.550858 ha, while NIST doesn’t have
87
calculated energy of HOMO-LUMO gap. The full energy using Psi4 CCSD/cc-pVDZ is -192.641
ha, , while NIST doesn’t have calculated full energy. The energy calculation obtained from
CCSD/cc-pVDZ are accurate than MP2 and B3LYP-D for energy calculations.
Below figure 18, FTIR experimental vibrational frequencies seems nearly simiar to the
calculated vibrational frequency.
Figure 18: Collected high-resolution FTIR spectra of acetone
With the help of computational methods, we can get the vibrational frequency, and
compare that with experimental IR frequency in figure 18 and table 28. In acetone, vibrational
CH3 s-deform mode for Psi4 is 1365 cm-1, For NIST, it is 1370 cm-1 while for experimental
fundamental frequency, it is 1364 cm-1, vibrational CH3 d-deform mode for Psi4 is 1429 cm-1, For
NIST, it is 1458 cm-1 while for experimental fundamental frequency, it is 1435 cm-1, vibrational
CC str mode for Psi4 is 785 cm-1, For NIST, it is 869 cm-1 while for experimental fundamental
frequency, it is 777 cm-1, vibrational CH3 rock mode for Psi4 is 1068 cm-1, For NIST, it is 1104
0
20
40
60
80
100
120
500 1000 1500 2000 2500 3000 3500 4000
Tran
smit
tan
ce (%
)
Wavenumber (cm-1)
88
cm-1 while for experimental fundamental frequency, it is 1066 cm-1. Calculated vibrational
frequency of (3)CH3 d-deform mode, (3)CH3 s-deform modes, (3)torsion and (3)CH3 rock mode
has nearly similar values in comparison of experimental, other modes such as CO str, CC str, CO
ip-bend and CO-op-bend has a little variation in the values as shown in table 28. However, after
calculating scaling factor frequencies seems a bit closer. Able to achieve Competitive frequency
numbers from density functional theory(B3LYP-D) than MP2 and CCSD method. Compare of
experimental vibrational frequencies to Psi4 calculated vibrational frequencies are highly accurate
than NIST calculated vibrational frequencies as seen in table 28.
Since acetone have ten atoms, it is challenging to run high level theory such as couple
cluster theory(CCSD(T)) with cc-pVTZ, aug-cc-pVTZ basis set. Tried seven different method and
basis set, didn’t work such as CCSD(T)/cc-pVTZ, CCSD(T)/cc-pVDZ, CCSD(T)/aug-cc-pVTZ
and CCSD(T)/aug-cc-pVDZ. These theory and basis set are accurate; however, it consumes longer
time to run as well as another factor affecting is such as speed and memory of the device. MP2/cc-
pVDZ and B3LYP-D/cc-pVDZ are less cost effective so these were able to compute the frequency.
4.6.4 C6H12 - Cyclohexane
Cyclohexane is a large molecule contains 18 atoms. It has 48 degrees of freedom in terms
of vibration. The forty-eight normal modes of vibration of cyclohexane be divided into the
following groups: 12 C-H stretching vibrations, 6H-C-H bending vibrations, 6 C-C stretching
vibrations, 6 C-C-C bending vibrations and 18 CH2 rocking and twisting vibrations. Observed
range of frequency for cyclohexane is shown in table 29.
Ionization energy of cyclohexane identified by NIST is 0.363083 ha, whereas the value of
HOMO energy obtained using Psi4 MP2/cc-pVDZ is 0.422794 ha and using NIST, it is 0.42346
89
ha. Among all seven methods we were only able to calculate MP2 energy. The HOMO-LUMO
gap energy using Psi4 MP2/cc-pVDZ is 0.601618 ha, while NIST, it is 0.60291 ha. The full energy
using Psi4 MP2/cc-pVDZ is -235.10 ha, while NIST, it is -235.08. The energy calculation is only
possible with MP2/cc-pVDZ, since cyclohexane is computationally larger molecule.
Table 29: Comparison of fundamental and scaled frequencies of cyclohexane
Mode Experimental Fundamental freq. (cm-1)
Scaled Frequency (cm-1)
B3LYP-D/cc-pVDZ Psi4 freq. (cm-1)
B3LYP-D/cc-pVDZ NIST freq. (cm-1)
CH2 a-str 2930 2976 3131 3124
CH2 s-str 2852 2920 1485 1482
CH2 scis 1465 1429 1481 1479
CH2 rock 1157 1127 1184 1184
CC str 802 793 833 832
CCC deform + CC torsion
383 375 394 393
CH2 twist 1383 1321 1385 1383
CH2 wag 1157 1097 1155 1152
CC str + CC torsion
1057 1070 1071 1069
CH2 wag 1437 1273 1389 1387
CH2 twist 1090 1018 1126 1124
CH2 a-str 2915 2987 3072 3066
CH2 s-str 2860 2918 1503 1501
CH2 scis 1437 1411 1389 1387
CH2 rock 1030 997 1048 1047
CCC deform
523 495 520 520
CH2 a-str 2930 2980 3136 3129
CH2 s-str 2897 2918 3070 3064
CH2 scis 1443 1400 1390 1388
CH2 wag 1347 1322 1339 1337
CH2 twist 1266 1239 1286 1284
90
CC str 1027 1014 1066 1065
CH2 rock 785 761 799 799
CCC deform + CC torsion
426 405 425 425
CH2 a-str 2933 2975 3143 3136
CH2 s-str 2863 2918 3070 3063
CH2 scis 1457 1409 1473 1470
CH2 wag 1355 1318 1303 1301
CH2 twist 1261 1223 1286 1284
CH2 rock 907 876 921 920
CC str 863 851 895 894
CCC deform + CC torsion
248 228 239 239
With the help of computational methods, we can get the vibrational frequency, and
compare that with experimental IR frequency in figure 19 and table 29. In cyclohexane, vibrational
CCC deform torsion mode for Psi4 is 520 cm-1, For NIST, it is 520 cm-1 while for experimental
fundamental frequency, it is 523 cm-1, vibrational CH2 rock mode for Psi4 is 799 cm-1, For NIST,
it is 785 cm-1 while for experimental fundamental frequency, it is 799 cm-1, vibrational CCC
deform + CC torsion mode for Psi4 is 425 cm-1, For NIST, it is 425 cm-1 while for experimental
fundamental frequency, it is 426 cm-1, vibrational CCC deform + CC torsion mode for Psi4 is 239
cm-1, For NIST, it is 239 cm-1 while for experimental fundamental frequency, it is 248 cm-1.
Calculated vibrational frequency of CCC deform torsion mode, CH2 rock modes, CCC deform +
CC torsion and other CCC deform + CC torsion mode has similar values in comparison of
experimental, other modes such as CH2 a-str, CH2 s-str, CH2 scissoring and CC str has a quite
variation in the values as shown in table 29. However, after calculating scaling factor frequencies
seems a bit closer.
91
Figure 19: FTIR vibrational spectra of Cyclohexane molecule [Reprinted from “Data from NIST
Standard Reference Database 69: NIST Chemistry Webbook”]
Based on figure 19, comparison of NIST experimental Fundamental frequency with FTIR
frequencies are quite similar when MP2/cc-pVDZ Psi4 calculated frequencies are vary specially
in functional group region while fingerprint regions absorbed frequencies are quite similar with
experimental fundamental frequencies.
Vibrational modes observe in fingerprint region frequency values are closer in both
experimental and computational. With higher frequency consumption of vibrational modes values
are far different than each other in terms of comparison between computational and experimental.
Since it is a large molecule, we were able to calculate frequency only with MP2/cc-pVDZ. Other
theory and basis set didn’t work such as CCSD(T)/cc-pVTZ, CCSD(T)/cc-pVDZ, CCSD(T)/aug-
cc-pVTZ,CCSD/cc-pVDZ, B3LYP-D/cc-pVDZ and CCSD(T)/aug-cc-pVDZ. These theory and
basis set are accurate; however, it consumes longer time to run as well as another factor affecting
is such as speed and memory of the device. MP2/cc-pVDZ and B3LYP-D/cc-pVDZ are less cost
effective. Able to calculate the frequency with MP2/cc-pVDZ only.
92
5. Conclusion
Based on the results of this study, computational chemistry of larger molecular systems
have the conflict between accuracy and computational cost in terms of time, accuracy, and memory
requirement of hardware. In contrast, the smaller molecules consume less time to calculate and
able to use high level of theory and provides accurate results.
Few researchers use gas-phase FTIR spectroscopy as the preferred spectroscopic method.
This study shows the effectiveness of high-resolution gas-phase FTIR spectroscopy (resolution at
0.125 cm-1) while comparing with computational vibrational frequencies. High-resolution FTIR
spectra tend to give very narrow and sharp peaks compared to other spectroscopic methods. With
the use of highly accurate theories and basis sets the precise vibrational frequencies and high-
resolution experimental vibrational frequencies can be compared. With the use of couple cluster
theory, diatomic molecules take about 15-20 minutes to get the output file with ± 0 to 8 cm-1
variation compared to experimental wavenumbers. While polyatomic molecules took up to 48
hours (depending on size of molecule) to give the results.
This study emphasizes that future chemists may use high-resolution gas-phase FTIR
spectroscopy as the preferred method among the other vibrational spectroscopic methods. For
computational spectroscopy to choose couple cluster theory for computationally smaller molecules
and density functional theory for computationally larger molecules to perceive accurate normal
vibrational modes. The hope is that future chemists will be able to use couple cluster theory for
polyatomic molecules as computer performance improves in memory and speed. Ultimately, a
high-level theory and a sophisticated basis set will obtain accurate vibrational frequencies for
complex molecules and lead to structural determination.
93
6. References
1. J. M. Hollas. Modern Spectroscopy. Fourth Edition, John Wiley & Sons, Ltd., ISBN 0 470
84416 7, (2004).
2. Daniel C. Harris, Quantitative Chemical Analysis, Michelson Laboratory, Eighth Edition,
ISBN-13: 978-1-4292-1815-3, (2007).
3. T. Theophanides, Introduction to Infrared Spectroscopy, Infrared Spectroscopy – Materials
Science, Engineering and Technology, ISBN: 978-953-51-0537-4, (2012).
4. I. Carnimeo, C. Puzzarini, N. Tasinato, P. Stoppa, A. P. Charmet, M. Biczysko, C. Cappelli,
and V. Barone, Anharmonic theoretical simulations of infrared spectra of halogenated organic
compounds. The Journal of Chemical Physics, 139 (7), 074310–. doi: 10.1063 / 1.4817401,
(2013).
5. V. Barone, A. Baiardi, M. Biczysko, J. Bloino, C. Cappelliad and F. Lipparini,
Implementation and validation of a multi-purpose virtual spectrometer for large systems in
complex environments. Physical Chemistry Chemical Physics, 14(36), 12404–
. doi:10.1039/c2cp41006k, (2012).
6. I. Carnimeo, C. Puzzarini, N. Tasinato, P. Stoppa, A. P. Charmet, M. Biczysko, C. Cappelli
and V. Barone, Anharmonic theoretical simulations of infrared spectra of halogenated organic
compounds. The Journal of Chemical Physics, 139(7), 074310–. doi:10.1063/1.4817401, (2013).
7. R. A. Friesner, Ab initio quantum chemistry: Methodology and applications. Proceedings of
the National Academy of Sciences, 102(19), 6648–6653. doi:10.1073/pnas.0408036102. (2005).
8. Inakollu, V. S. Sandeep, Yu, Haibo. A systematic benchmarking of computational vibrational
spectroscopy with DFTB3: Normal mode analysis and fast Fourier transform dipole
94
autocorrelation function. Journal of Computational Chemistry, 39(25), 2067–
2078. doi:10.1002/jcc.25390, (2018).
9. S. Wang, Efficiently Calculating Anharmonic Frequencies of Molecular Vibration by
Molecular Dynamics Trajectory Analysis, ACS Omega 4 (5), 9271-9283, DOI:
10.1021/acsomega.8b03364, (2019).
11. M. Mutsaers, Zika vector control: Near infrared spectroscopy predicting Wolbachia infection
in post-mortem Aedes aegypti, 4365/328726218, June 19, (2018).
12. P. L. Polavarapu, Vibrational Spectra: Principles and Applications with emphasis on optical
activity Volume 85 || Chapter 5 Vibrational analysis, (1998).
13. P. L. Polavarapu, Vibrational Spectra: Principles and Applications with emphasis on optical
activity Volume 85 || Chapter 4 Polyatomic molecules, 61–79. doi:10.1016/S0167-
6881(98)80040-2, (1998).
14. F. Siebert and P. Hildebrandt, Vibrational Spectroscopy in Life Science, WILEY-VCH
Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40506-0, (2008).
15. P. M. Morse, Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels.
Physical Review, 34(1), 57–64. doi:10.1103/PhysRev.34.57, (1929).
16. Z. Rong , H. G. Kjaergaaed and M. L. Sage, Comparison of the Morse and Deng-Fan
potentials for X-H bonds in small molecules, Molecular Physics: An International Journal at the
Interface Between Chemistry and Physics, DOI:10.1080/0026897031000137706, 101:14, 2285-
2294, April 22, (2003).
95
17. J. M. Hu, J. P. Zhai, F. M. Wu,§ and Z. K. Tang, Molecular Dynamics Study of the
Structures and Dynamics of the Iodine Molecules Confined in AlPO4-11 Crystals, J. Phys.
Chem. B 2010, 114, 16481–16486, doi:10.1021/jp1076615, October 25, (2010).
18. L. Lessinger, Morse Oscillators, Birge-Sponer Extrapolation, and the Electronic Absorption
Spectrum of I2. Journal of Chemical Education, Volume 71, Number 5, 388-
doi:10.1021/ed071p388, May (1994).
19. E. Fidiani, Modeling of diatomic molecule using the Morse potential and the Verlet
algorithm, 1719, 030001–. doi:10.1063/1.4943696, March 11, (2016).
20. A. sanli, M. Lyyra, Transition Dipole Moment and Lifetime Study of Na2 And Li2
Electronic States Via Autler-Townes And Resolved Fluorescence Spectroscopy, May 1, (2017).
21. P. F. Bernath. Spectra of atoms and molecules, New York oxford, Oxford University Press,
ISBN 0-19-507598-6, QC454.A8B47, (1995).
22. D. R. Lide, Handbook of chemistry and physics, October 27, Editor-in-chief, 84th Edition,
(2003 - 2004).
23. Lewars, G. Errol Computational Chemistry Volume 98 || The Concept of the Potential
Energy Surface.,10.1007/978-90-481-3862-3(Chap. 2), 9–43. doi:10.1007/978-90-481-3862-3_2,
(2011).
24. J. Zwinkels, Light, Electromagnetic Spectrum. Encyclopedia of Color Science and
Technology, 1–8. doi:10.1007/978-3-642-27851-8_204-1, (2015).
25. J. P. Carter, SP - 1, EP - 27, Who needs Constitutive Models? Australian Geomechanics Vol
41 No 2 June (2006).
26. S. J. Ling, J. Sanny and B. Moebs, The Quantum Harmonic Oscillator., November 5,
OpenStax CNX, https://phys.libretexts.org/@go/page/4531, (2020).
96
27. U. Gehlsen, A. Oetke, M. Szaszák, N. Koop, F. Paulsen, A. Gebert, G. Huettmann, P. Steven,
Two-photon fluorescence lifetime imaging monitors metabolic changes during wound healing of
corneal epithelial cells in vitro, 250(9), 1293–1302. doi:10.1007/s00417-012-2051-3, (2012).
28. A. Tokmakoff, MIT Department of Chemistry, March 8, Interaction of light with matter,
(2007).
29. J. Keeling, Light-Matter Interactions and Quantum Optics, August 28, ISBN-13: 978-
1500976347, (2014).
30. V. Shahani, J. Jenkinson, U. Krull, N. Woolridge, The efficacy of interactive analogical
models in the instruction of bond energy curves, Chemversity – Transition Dipole,
10.1039/C5RP00194C, (2016).
31. P. J. Walla, Modern Biophysical Chemistry, Detection and Analysis of Biomolecules,
Second Edition by Wiley-VCH Verlag GmbH & Co. ISBN-13: 978-3527337736, (2014).
32. D. A. McQuarrie, University of California, Davis, J. D. Simon, B. George, Geller Professor
of Chemistry, Duke University/ Selection Rules Are Derived from Time-Dependent Perturbation
Theory/ physical chemistry, molecular approach. ISBN 0-935702-99-7, QD453.2.M394 (1997).
33. Stephanos, Joseph J., Electrons, Atoms, and Molecules in Inorganic Chemistry || Vibrational
Rotational Spectroscopy, 505–584. doi:10.1016/B978-0-12-811048-5.00009-2, (2017).
34. A. Barth, Infrared spectroscopy of proteins, A. Barth / Biochimica et Biophysica Acta
(1767), 1073–1101, doi:10.1016/j.bbabio.2007.06.004, (2007).
35. Grainger, D.W. Comprehensive Biomaterials || Surface Analysis and Biointerfaces: Vacuum
and Ambient In Situ Techniques, 1–22. doi:10.1016/B978-0-08-055294-1.00082-9, (2011).
36. M. A. Mohamed, Membrane Characterization || Fourier Transform Infrared (FTIR)
Spectroscopy, 3–29. doi:10.1016/B978-0-444-63776-5.00001-2, (2017).
97
37. F. Zadeh, F. Anaya, Nelson M., Schiffman, Laura A., O. Craver, Vinka. Fourier transform
infrared spectroscopy to assess molecular-level changes in microorganisms exposed to
nanoparticles. Nanotechnology for Environmental Engineering, 1(2), 1–. doi:10.1007/s41204-
016-0001-8, (2016).
38. Palafox, M. Alcolea. Computational chemistry applied to vibrational spectroscopy: A tool for
characterization of nucleic acid bases and some of their 5-substituted derivatives. Physical
Sciences Reviews, 2(8), –. doi:10.1515/psr-2016-0132, (2017).
39. Dr. Andrew Gilbert, Prof. John Herbert , Q-CHEM 5.3 User’s Manual handbook Version 5.3,
August 28, Mol. Phys. 113, 184–215, (2020).
40. D. G. Smith, L. A. Burns, A. C. Simmonett, R. M. Parrish, M. C. Schieber, R. Remigio, E. G.
Hohenstein, B. P. Pritchard, B. R. Brooks, H. F. Schaefer III, J. M. Turney, T. D. Crawford, and
C. D. Sherrill , "PSI4 1.4: Open-source software for high-throughput quantum chemistry", J.
Chem. Phys. 152, 184108, https://doi.org/10.1063/5.0006002, (2020).
41. J. M. Turney, A. C. Simmonett, R. M. Parrish, E. G. Hohenstein, F. Evangelista, J. T.
Fermann, B. J. Mintz, L. A. Burns, J. J. Wilke, M. L. Abrams, N. J. Russ, C. D. Sherrill, and T.
D. Crawford, “Psi4: An open-source ab initio electronic structure program,” HF : Har t r ee –
Fock T heor y , WIREs Comput. Mol. Sci. 2, 556, doi: 10.1002/wcms.93, (2012)
42. Y. G. Smeyers , The half projected Hartree-Fock model for determining singlet excited
states. Advance in quantum chemistry, Vol 36, 0065-327NW, (2002)
43. Cramer, Christopher J., Essentials of Computational Chemistry, Theories and Models,
Second Edition, Chichester: John Wiley & Sons, Ltd. (224-227) ISBN 0-470-09182-7 (2004).
44. T. Daniel. Crawford and Henry. F. Schaefer III, Rev. An introduction to couple cluster theory
for computational chemists. Vol 14, 33 (2000).
98
45. C. D. Sherrill and H. F. Schaefer III, The Configuration Interaction Method: Advances in
highly correlated approaches, Vol 34, Academic Press, CA, (1999).
46. R. M. Parrish, J. M. Turney, Daniel G, A. Smith, R. M. Parrish, DFT: Density Functional
Theory, psicode.org/psi4manual, (1-4) (2017).
47. J. E Thirman, M. P. Head-Gordon, W. H. Miller, B. Smit, E. Joel, Moore A second-order
Møller_Plesset perturbation theory energy decomposition analysis for intermolecular
interactions: design, implementation, and application, University of California, Berkeley, (9-11)
Spring (2017).
48. Lehtola, Susi, A review on non‐relativistic, fully numerical electronic structure calculations
on atoms and diatomic molecules. International Journal of Quantum Chemistry, (23-25),
e25968–. doi:10.1002/qua.25968, (2019).
