Publication History
Submitted: November 25, 2023
Accepted: December 14, 2023
Published: January 31, 2024
Identification
D0209
Citation
Lekh Nath Regmi (2024). First Principles Study of the Stability and Bonding Analysis of Hydrogen Cyanide and its Dimer. Dinkum Journal of Natural & Scientific Innovations, 3(01):5880.
Copyright
© 2024 DJNSI. All rights reserved
5880
First Principles Study of the Stability and Bonding Analysis of Hydrogen Cyanide and its DimerOriginal Article
Lekh Nath Regmi ^{1*}
 Central Department of Physics, Tribhuvan University, Kathmandu, Nepal; lekhnath.regm@gmail.com
* Correspondence: lekhnath.regm@gmail.com
Abstract: In the present work, first principles calculations are performed to study the stability and topology analysis of Hydrogen Cyanide (HCN) molecule and its dimer. We have carried out the HartreeFock (HF), MØllerPlesset Second Order (MP2), Configuration Interaction (CI) approximation with quadratic, single and double states (QCISD) and BeckeStyle 3Parameter LeeYangParr correlation functional theory (B3LYP) calculations to estimate the equilibrium configuration, ground state energy, Binding Energy and Dipole Moment. All these calculations have been carried out using the Gaussian 03 set of programs. We have performed the MP2 calculation choosing basis set 6−31G (3d,p) to estimate total energy in different ionized states, Electron Affinity (E.A.), Ionization Energy (I.E.) and Proton Affinity (P.A.). The many body effects have been incorporated by considering the MP2 and QCISD levels of approximation. The value of bond length are 1.07 Å and 1.17 Å of HC and C≡N bond respectively which are in agreement in experimental values within the 1%. We have also calculated the binding energy and its positive value means the molecule is stable. Similarly, Ionization Energy has been calculated to be 1319.26 KJ/mol, which is in agreement with experimental value within 1%. We have also calculated Electron Affinity and Proton Affinity. In the electronic properties, HOMOLUMO energy gap, molecular hardness, softness, electronegativity, chemical potential and global electrophilic index are analyzed. The high value of HUMOLUMO energy (18.06 eV) gap shows excitability of the HCN is very low. We have also studied the total density of states, bond order analysis and thermochemistry. The topology analysis performed to obtain the density of all electrons and laplacian of electron density at bond critical points to study the nature of bonding shows the ve values of laplacian of electron density for HCN means the existence of strong type of bonding. We have also plot the contour map of electrostatic potential distributed on molecular Vander Waals surface of which shows nitrogen atom in the studied molecule is more electron negative than carbon and hydrogen atoms. We also calculated the Dimerization energy of Hydrogen Cyanide dimer which was found to be 0.163 eV.
Keywords: hydrogen cyanide, HartreeFock, MØllerPlesset Second Order
 INTRODUCTION
The term cyanide refers to any compound that contains the cyanide ion (CN), consisting of a carbon atom triple bonded to a nitrogen atom. Cyanides have been being used for mining operations from 1890s. It can take many forms such as HCN, NaCN, KCN etc. One of the forms is Hydrogen Cyanide (HCN), which is a simple linear organic molecule, colorless or pale blue liquid or gas with a faint bitter almondlike odor. As being toxic gas, HCN had been used as a weapon in World War II [1]. It is an important precursor for amino acid, nucleic acid, polymer to pharmaceuticals manufacturing. HCN has been detected in the interstellar medium [2]. Since then, extensive studies have probed formation and destruction pathways of HCN in various environments and examined its use as a tracer for a variety of astronomical species and processes. Computer use is rising across all sciences. Computational physics uses computers to solve theoretical physics problems that analytical theory cannot. In this new field of physics, computers are used for experiments. Computer performance improves equation solving time [3]. Manybody systems require many mathematical operations, which take time. The computational method is ideal for studying manybody physics. Quantum chemistry solves chemistry problems using quantum mechanics. All branches of chemistry are influenced by quantum chemistry. To solve the hydrogen atom problem in quantum mechanics, the simplest problem can be solved analytically to obtain accurate wave functions [4]. If one has to solve the problem of manyelectron system, one must rely on approximations to simplify problems. In quantum mechanics, a system is described in terms of wave functions. For a molecular system, the wave functions is designed by the combinations of wave functions of atoms. In quantum mechanical approach, the concept of distribution of electrons in a molecule is given in terms of molecular orbitals, is formed by the linear combinations of atomic orbitals (LCAO) [5]. In LCAO approximations, when two electrons are brought together, their atomic orbitals overlap to form molecular orbitals and their energy levels get splitted [6]. The study of molecular orbitals in quantum mechanics is classified into two main categories, ab initio and semiempirical. The meaning of the word ab initio is ‘from the beginning’. In ab initio calculations, there are no experimental biases expect for fundamental constants like electronic charge(e), electronic mass(m), Planck’s constant(h), Boltzmann constant(kB) etc. In semiempirical method, one uses oneelectron Hamiltonian and takes the bond integral as adjustable parameter to give some of the results that are already known from experiment[3]. The ab initio calculations have been much prominent in studying the electronic structure of a wide range of molecules and their properties. The method enables us to unearth various physical properties such as ground state energy, electric dipole moment, electric field gradient parameters, nuclear quardrupole moment, ionization potential, polarizability, vibration energy and vibration frequency of many electron systems [7] etc. Recent supercomputer advances have made ab initio calculations popular in biophysics and nanotechnology. This method examines DNA and RNA’s physical and chemical properties [4]. There are three main ab initio approaches: HartreeFock (HF) selfconsistentfield, density functional theory (DFT), and quantum MonteCarlo (QMC). A oneparticle schrödinger equation governs electron motion in the HartreeFock selfconsistentfield approximation, a rigorous oneelectron approximation. Each electron is assumed to move in the field created by the nuclei and remaining electrons. An antisymmetric electron is described. HF method is variational and uses interaction to calculate system energy and wave function. Coulomb and exchange interactions explain electronelectron interaction. In HF approximation, instantaneous electron motion correlation is ignored. Many methods, including MøllerPlesset Perturbation approximation (MP), configuration interaction (CI), coupledcluster method, multiconfigurational selfconsistentfield method, and multireference configuration interaction [8], account for instantaneous electron motion. We use the MøllerPlesset Perturbation approximation and configuration interaction to study correlation effects caused by instantaneous electron motion. In MøllerPlesset Perturbation theory, we take HF Hamiltonian as unperturbed Hamiltonian and the correlation is given by small perturbation. It is not a variational method. The configuration interaction method is based on variational principle like HF method, only the difference between HF and CI is that the CI method takes multi determinant trial wave functions and gives correlation also [9]. The method of configuration interaction approximates the tool wave functions, by a linear combination of all possible electronic configuration functions. In DFT, a manyelectron system is described in terms of density of electrons which gives the physical properties of the system just like as wave function in HartreeFock selfconsistentfield approximation. The degrees of freedom decreases from 3N6 to 6 for N electron system. In DFT, the contributions due to correlation and exchange terms are not separated. To take the exchange and correlation effects, a functional is defined in terms of density of electrons only in local density approximation effects and is defined in terms of density and its gradient in case of generalized gradient approximation.
 MATERIALS AND METHODS
2.1 Computational methods
Computational methods are based on a sufficiently well developed mathematical method [22] with good technical skills. The computational model can be automated for the implementation on a computer. Using the computational simulation, we can understand different problems, natural phenomena and moreover carry out Physics in a different way than the scientific methods. In a scientific method, based on a logical conclusion from the known facts a hypothesis is formulated. The formulated hypothesis is compared to see its consistency with the available data. The hypothesis becomes a theory if hypothesis is found to be in agreement with the known facts. This theory is able to explain the observed phenomena and predict experimental result and if the theory remains correct for many years, it is stated as the scientific law [27]. In the case of the computational model, a computational model is constructed such that it can describe and predict the scientific results without performing the complex mathematical calculations based on a rigorous theory. The computational model simplifies the scientific method by construction of the computational model which can efficiently solve a difficult and time consuming problem. The computational method provides a bridge between the experimental and theoretical calculations. The computational model for the accurate solution should fulfill different properties. Some of the main properties that is needed for a successful computational model are [22];
 Computational model should give a unique energy for a system defined.
 It should be size consistent. The margin of the error should be proportional to the molecular size.
 Computational model should be practical. It should be able to define problems ranging from simple to complex and those problems which are experimentally impossible.
 Highly desirable but not necessarily a computational model should be a variational model.
As mentioned in the theoretical background chapter, the many body problem consisting of more than two particles is a difficult problem to solve analytically. Computational methods can be implemented to solve many body problems. There are different software packages like GAUSSIAN, QUANTUM ESPRESSO, GAMESS etc. that incorporates the mathematics of the many body problem to define a computational model and solve the many body problem. In the present work, we have carried out the calculation of the firstprinciples of silicon hydrides using GAUSSIAN 03 sets of software packages.
2.2 Molecular modeling method
GAUSSIAN is capable of running all of the major methods in molecular modeling including molecular mechanics, ab initio, semiempirical and density functional theory (DFT) [23]. Figure 01 shows that the GAUSSIAN software packages involves method of calculations ranging from classical mechanics to solve Newton’s equation of motion to Schrodinger equation.
Figure 01: Methods implemented in GAUSSIAN
The different types of methods of calculations implemented in GAUSSIAN are;
Molecular Mechanics
Molecular mechanics energy expression consist of simple algebraic equation for energy of compound. It does not use wave function or total electron density. Constants in the energy expression equation are obtained from spectroscopic data or ab initio calculation. A set of equation with their associated constants is called a force field [28]. Molecular mechanics implemented in GAUSSIAN packages are Assisted model building with energy confinement (AMBER), universal force field (UFF) etc.
ab initio
In ab initio (latin meaning from beginning) approximations made are mathematical only and no experimental data are included . Some of the ab initio calculations involved are HF, MPn (MollerPlesset nth order perturbation; n > 2, coupled cluster (CC), configuration interaction (CI) etc. HF model does not include correlation where as other models define the correlation [28]. In all of the mention ab initio model, the calculation is done using the wave function method. For multi determinant methods like CI, CC natural orbitals are used to increase the computational efficiency. Natural orbitals are the eigen function of the first reduced density matrix [28]. MPn is not a variational model. The energy obtained after computer simulation using MPn model should be checked with variational model [28].
Semiempirical Mechanics
In this method certain information are omitted; usually the core electrons are not included. To correct the errors introduced by exclusion of the information parameters are defined by fitting results to ab initio method or experimental data [21]. Some of the semi empirical methods included in GAUSSIAN are Intermediate neglect of differential overlap (INDO), common neglect of differential overlap (CNDO) etc.
Density functional method
In density functional method, we have to calculate the electron density that gives the minimum energy. The electron density is expressed in terms of the KohnSham orbitals. The dependence on electron density make this method more time efficient.
2.3 GAUSSIAN
GAUSSIAN is a computational chemistry software program capable of running all of the major methods ranging from laws of classical to quantum mechanics in molecular modeling [29]. It can be used by chemists, chemical engineers, biochemists, physicists and others scientist for their research and it will decrease their time as well as cost [27,29]. GAUSSIAN was developed in 1970 by theoretical chemist, NoblePrize (1998 for development of computational methods in quantum chemistry) laureate John People and his research group at CarnegieMellon University as GAUSSIAN 70 [27,31]. The GAUSSIAN name originates from the People’s bass set which uses Gaussian type of orbitals (GTOs) instead of the Slater type of orbitals (STOs) to speed up the computed calculation [28]. The software is being updated time to time and recently it is in new version GAUSSIAN 03 [29]. Running GAUSSIAN involves the following activities;
 Creating GAUSSIAN input by defining the computational model and defining the
 Specifying the location of the various scratch files. The very useful scratch files
 used in GAUSSIAN are formatted checkpoint files (.fch) and checkpoint files (.chk).
 Initiating program execution.
 Analyzing the output file.
Figure 02: Scheme of the GAUSSIAN input file
Figure 02 shows the scheme of the GAUSSIAN input file. Input is a free format and caseinsensitive. The GAUSSIAN input file includes several different sections. They are described below;
Link 0 and Title section
The link 0 section is also called % lines as this line starts with the %. In this line, we have to define the number of processors used. If more than one using (% nproc= number of processor), location of the scratch files using (% chk=file location), memory specification using (% mem= memory) etc. While in the title section the programmer have to give brief description of the calculation but it is not necessary [28].
Route section
The route section is the important part of the input file in GAUSSIAN. This section includes the description of the job type and theoretical model. The theoretical model is defined by defining the method of the calculations and the basis sets. The route section starts with the # sign. Wide range of the molecular properties can be computed using frequency analysis, population analysis, NMR chemical shifts etc. Different jobs can be carried out in the GAUSSIAN by giving different commands. Some of the job type with their command in the bracket are Optimization (opt), frequency calculation (freq), single point energy calculations (sp), potential energy surface scan (scan) etc. According to the molecule taken, the basis set and the molecular modeling methods can be defined to give the best possible result.
Basis set
Basis sets describe the shape of atom orbitals. Linear combination of basis function and angular function creates molecular orbitals and entire wave function [29,32]. The GAUSSIAN has pseudo potentials and basis sets. The single contraction of n GTO orbitals forms the smallest basis sets, STOnG. Peoplebased sets exist. This basis set uses one basis function for each core atomic orbital and a larger basis for the valence orbitals [31]. So it’s called splitvalence basis set. 321G is a splitvalence basis set. The 321G basis set describes each core orbital by a single contraction of three GTO primitives, and each valence shell by two and one GTO primitives, respectively [31]. The asteriskdenoted polarization function increases wave function flexibility. Single asterisk means “d” primitives are added to nonHydrogen atoms, while double asterisk means “p” primitives are added to hydrogen. Diffuse functions are added to define wave functions far from the nucleus, especially for anions with larger electron distribution. Plus signs indicate diffuse function [28]. Single plus and double plus are polarization basis functionlike. Polarized basis set is 321G* and diffuse function is 631+G.
Molecular specification
The last section in the input of the GAUSSIAN is the molecular specification. In this section, we have to define the charge, multiplicity, molecules geometry and extra information. Computing the geometry of a molecule is one of the most basic function of a computational chemistry program. Molecule geometry can be specified either by using the cartesian coordinates for each atom in the system or by defining Zmatrix [28]. Zmatrix includes bond distance, angles and dihedral angles. The molecular specification in the Zmatrix fashion is shown in the Appendix I. The Zmatrix molecular specification can be explained by the scheme in figure 03.
Figure 03: Molecular specification using Zmatrix for a hypothetical molecule consisting of atoms A, B, C and D.
In the figure 03, the first column defines the atoms in the molecule and second column defines the atom to which the length (BA, CA and DA in ) refers in the third column. The fourth column defines the atom to which the angle (CAB, DAB in degree ( ° )) are refered in the fifth column. While the sixth column consist of the atom to which the dihedral angle (DABC in degree ( ° )) is referred in the seventh column. Similarly, for the molecular specification using cartesian coordinate we have to list all the atoms with their (x, y , z) coordinate in the molecular specification section as shown in the figure 04.
Figure 04: Molecular specification using Cartesian coordinate for a hypothetical molecule consisting of atoms A, B, C and D.
By defining all the sections of the input panel of the GAUSSIAN as shown in the figure 01, the GAUSSIAN software is run and the output is studied using the stream output file created after completion of the program. The input file for the GAUSSIAN is also made by using the GAUSS VIEW software.
2.4 GAUSS View, GAUSSUM and ORIGIN
GAUSS View is an affordable, fullfeatured graphical user interface for Gaussian. Gauss View supports all Gaussian features, and it includes graphical facilities for analyzing the results of the calculations through stateoftheart visualization features. Currently, it stands as a part of the Gaussian Package. By subscribing to Gaussian, direct access to run Gauss View is possible. GAUSSUM is a GUI application that can analyze the output of ADF, GAMESS (US), GAMESSUK, GAUSSIAN, Jaguar and PC GAMESS to extract and calculate useful information. This includes the progress of the SCF cycles, geometry optimization, UVV is/ IR/Raman spectra, MO levels, MO contributions and more. It is written by Noel O’Boyle and is available for free under the GNU Public License [21]. Origin is a proprietary computer program for interactive scientific graphing and data analysis. It is produced by Origin Lab Corporation, and runs on Microsoft windows. It has inspired several platformindependent opensource clones like QtiPlot or SciDAVis. Graphing support in Origin includes various 2D/3D plot types. Data analyses in Origin include statistics, signal processing, curve fitting and peak analysis. Origin’s curve fitting is performed by the nonlinear least squares better which is based on the LevenbergMarquardt algorithm. Origin imports data files in various formats such as ASCII text, Excel, NI TDM, DIADem, NetCDF, SPC, etc. It also exports the graph to various image file formats such as JPEG, GIF, EPS, TIFF, etc. There is also a builtin query tool for accessing database data via ADO, and an enhanced Digitizer tool to capture data from images of existing graphs.
2.5 Multiwfn
Multiwfn is a powerful wavefunction analysis program, supporting almost all of the most important wavefunction analysis methods. Primary functions of Multiwfn are
 Showing molecular structure and viewing orbitals.
 Calculating real space function in one, two and threedimensions
 Topology analysis for electron density, Laplacian etc.
 Critical points and gradient paths can be searched and visualized in terms of 3D or plane graph.
 Population analysis, Orbital composition analysis and Bond order analysis are supported.
 Plotting of TDOS/PDOS,/OPDOS is conveniently defined.
 Plotting of IR/Raman/UVVis/ECD/VCD spectrum.
 Quantitative analysis of molecular surface (surface area, enclosed volume, average value and standard of mapped functions can be computed).
 Basin analysis, electron excitation analysis and Charge decomposition analysis.
Other useful function of utilities involved in Qunatum Chemistry analysis such as weak interaction analysis, plotting scatter map etc.
 RESULTS AND DISCUSSION
The study have performed the firstprinciples (ab initio) calculation to study about Hydrogen Cyanide (HCN) molecule. We have carried out the HartreeFock (HF), MøllerPlesset second order perturbation (MP2), Quadratic Configuration Interaction Single (double) method (QCISD) and Beckestyle 3Parameter LeeYangParr correlation functional theory (B3LYP) calculation to study the equilibrium of hydrogen cyanide molecule. We have also carried out MP2 calculation to study the equilibrium geometry of HCN dimer. Calculations have also been performed to estimate the electron affinity, proton affinity, ionization energy and topology analysis. The many bodyeffects have been incorporated in the HF calculations by considering the MP2 perturbation approximation. In the present work, these calculations have been carried out using the Gaussian 03 set of programs.
3.1 The Ground State Energy, Equilibrium Configuration and Binding Energy of Hydrogen Cyanide (HCN)
We have carried out the HF, MP2, QCISD and DFT(B3LYP) levels of calculation to obtain the total energy of the HCN molecule using the basis sets 3−21G, 3−21G*, 6−31G, 6−31G*, 6−31G(3d,p), 6−311G, 6−311G* and 6−311G(3d,p). The symbol (*) and (3d,p) in the basis sets represent the inclusion of dtype and (3d,p)type Gaussian polarization functions respectively. The calculation of the total energy of the molecule is presented in table 01 and the plot of the basis set dependence of the HF, MP2, QCISD and DFT energy values is shown in figure 05
Table 01: Total energy of HCN molecule obtained in the HF, MP2, QCISD and DFT levels of approximation with different basis sets
Basis Sets  Levels of Approximations  
E_{HF}(a.u.)  E_{MP2}(a.u.)  E_{QCISD}(a.u.)  E_{DFT}(a.u.)  
321G  92.35408  92.56503  92.56677  92.90763 
321G*  92.35408  92.56503  92.56677  92.90769 
631G  92.82832  93.04343  93.04408  93.39249 
631G*  92.87520  93.15894  93.16730  93.42261 
631G(3d, p)  92.88363  93.19414  93.20095  93.43051 
6311G  92.84956  93.07639  93.07680  93.41889 
6311G*  92.89698  93.19268  93.19972  93.44914 
6311G(3d, p)  92.90399  93.22220  93.22795  93.45463 
Figure 05: Total energy of HCN molecule versus increasing size and flexibility of basis sets obtained in the HF, MP2, QCISD and DFT levels of approximation.
Table 4.1 clearly shows that the results of the HF, MP2, QCISD and DFT calculations for the total energy of HCN molecule are basis set convergent. Also, the energy values of HCN molecule obtained with the MP2, QCISD and DFT calculations are lower than the corresponding HF energy values. Moreover, DFT energy values are lower than the QCISD values, which in turn are lower than the corresponding MP2 values. The manybody contributions to the total energy of HCN molecule in the MP2 and QCISD levels of approximation can be obtained with the help of following relations
( E_{MB})_{MP2 }= E_{MP2} – E_{HF} (4.1)
( E_{MB})_{QCISD }= E_{QCISD} – E_{HF} (4.2)
Where (δE_{MB})_{MP2} and (δE_{MB})_{QCISD} are many body contributions to the total energy of HCN molecule in the MP2 and QCISD levels of approximation respectively and E_{HF}, E_{MP2} and E_{QCISD} are the total energy of HCN molecule in the HF, MP2 and QCISD levels of approximation respectively. Table 02 shows the values of (δE_{MB})_{MP2} and (δE_{MB})_{QCISD} for HCN molecule obtained with the choice of basis sets mentioned above.
Table 02: Manybody contribution to the total energy of HCN molecule
Basis Sets  ManyBody Contributions (eV)  
( E_{MB})_{MP2}  ( E_{MB})_{QCISD}  
321G  5.74  5.79 
321G*  5.74  5.79 
631G  5.85  5.87 
631G*  7.72  7.95 
631G(3d, p)  8.45  8.63 
6311G  6.18  6.27 
6311G*  8.05  8.24 
6311G(3d, p)  8.66  8.82 
A close look at the table 02 shows that the values of (δE_{MB}) for HCN molecule obtained in the QCISD level of approximation are lower than the corresponding values obtained in the MP2 level of approximation by around 2%5%. Assuming that the basis sets with higher flexibility would provide the better result, we have estimated the many body contributions to the ground state energy of the HCN molecule to be 8.66 eV and 8.82 eV with MP2 and QCISD level of approximation respectively. Knowing the total energy of the nitrogen atom, carbon atom, hydrogen atom and HCN molecule, it is straight forward to calculate the binding energy with the aid of following relation:
B.E. = E(H) + E(C) + E(N) – E(HCN) (4.3)
The total energy of the nitrogen atom, carbon atom and hydrogen atom with different basis sets at different levels of approximation is presented in the Appendix.
Table 03: Binding energy and Dipole moment of HCN at different level of approximations with different basis sets
Methods
Basis Sets 
HF  MP2  QCISD  DFT(B3LYP)  
Binding energy (eV)  Dipole Moment (Debye)  Binding energy (eV)  Dipole Moment (Debye)  Binding energy (eV)  Dipole Moment (Debye)  Binding energy (eV)  Dipole Moment (Debye)  
321G  13.43293  3.04  17.36044  3.08  16.11359  2.77  17.73760  2.72 
321G*  13.43293  3.04  17.36044  3.08  16.11359  2.77  17.73923  2.72 
631G  13.59131  3.24  17.27499  3.31  15.96365  3.01  17.47010  2.93 
631G*  14.84660  3.21  18.43423  3.26  16.84831  2.98  18.15966  2.91 
631G(3d, p)  14.86700  3.24  18.78853  3.27  17.07880  2.98  18.30089  2.99 
6311G  13.47593  3.25  17.19499  3.31  15.89616  3.03  17.26901  2.98 
6311G*  14.68360  3.22  18.37790  3.26  16.78790  2.98  18.04346  2.96 
6311G(3d, p)  14.84714  3.18  18.67614  3.22  17.06492  2.93  18.18524  2.92 
Table 03 shows that the binding energy for the HCN molecule is greater for
the basis set 631G(3d,p) than the basis sets 321G, 321G*, 631G, 631G*, 6311G, 6311G*, 6311G(3d,p) in all levels of approximation. Hence, we have considered the basis sets 631G(3d,p) for the further calculation. The equilibrium configuration of HCN molecule obtained in the MP2 level of
approximation using the basis set 631G(3d,p) is presented in Figure 06.
Figure 06: Equilibrium configuration of HCN molecule with the basis set 631G(3d,p) at the MP2 level of approximation. The bond length value is in Angstrom.
Here H, C, and N refer the position of the corresponding atoms to find the bond length and bond angle as given in table 4.4. We have compared the bond length and bond angle of HCN molecule with choice of basis sets 631G(3d,p) at MP2 level of approximation with experimental values.
Table 04: Calculated values of bond length and bond angle of HCN molecule with choice of basis set 631G(3d,p) at MP2 level of approximation and experimental values.
R(HC)  R(CN)  A(RCN)  
631G(3d,p)  1.07  1.17  180 
Experiment  1.06  1.16  180 
 The bond length is in angstrom (A) and bond angles.
We have also studied the variation of total energy of HCN molecule with the variation of bond length with the choice of basis set 631G(3d,p) at MP2 level of approximation.
Figure 07: Total Energy of HCN molecule versus bond length(R) CH and CN with choice of basis set 631G(3d,p) at MP2 level of approximation.
It is clearly seen from figure 07 that total energy for the HCN molecule decreases with increase in bond length R(CH) from 1.02 Å to 1.07 Å and the minimum energy (Emin) of 93.19414 a.u. occurs at a bond length 1.07 Å. Beyond 1.07 Å, the value of total energy increases with increase in bond length in the range (1.07Å to 1.12 Å). Similarly from the figure, the total energy deceases with increase in bond length R(CN) from 1.12 Å to 1.17 Å and the minimum energy (E_{min}) of – 93.19414 a.u. occurs at a bond length 1.17 Å. Beyond 1.17 Å, the value of total energy increases with increase in bond length in the range (1.17 Å to 1.24 Å) studied.
3.2 The Total Energy of Hydrogen Cyanide Ionized State
We have calculated the total energy of HCN molecule in different ionized state with basis set 631G(3d,p) at HF, MP2, QCISD and DFT(B3LYP) levels of approximation.
Table 05: Total Energy of HCN molecule in different ionized state with basis set 631G(3d,p) at HF, MP2, QCISD and DFT(B3LYP) levels of approximation.
Ionized States  Total Energy (a.u.)  
HF  MP2  QCISD  DFT  
(HCN)^{—}  92.26071  92.60971  92.62328  92.60979 
(HCN)^{‑}  92.73714  93.05428  93.06645  93.31360 
(HCN)˚  92.8836  93.1941  93.20100  93.43050 
(HCN)^{+}  92.44754  92.69204  92.71806  92.94031 
(HCN)^{++}  91.5806  91.79528  91.83693  92.03517 
(H_{2}CN)^{+}  93.41280  93.69256  93.72534  93.97756 
From these calculations, we have estimated the E.A., I.E. and P.A. with choice of basis set 631G(3d,p) at HF, MP2, QCISD and DFT(B3LYP) levels of approximation.
Table 06: Electron Affinity, Ionization Energy and Proton Affinity of HCN molecule with basis set 631G(3d,p) at HF, MP2, QCISD and DFT(B3LYP) levels of approximation
Methods
Parameters 
HF  MP2  QCISD  DFT(B3LYP)  Experiment 
Electron Affinity (KJ/mol)  384.902  367.482  353.399  307.181  – 
Ionization Energy (KJ/mol)  1145.826  1319.268  1268.793  1288.001  1312.196 
Proton Affinity (KJ/mol)  1389.415  1308.707  1376.304  1436.306  – 
It is clearly seen from the table 4.6 that the HF, MP2, QCISD and DFT values of the Electron Affinity, Ionization Energy and Proton Affinity of the HCN molecule obtained with the basis set 631G(3d,p) are different for different level of approximation. Here we have found following results:
 The Electron Affinity (in negative value) is greater for HF level than MP2 level where as MP2 level has greater value than the QCISD level and lowest for the DFT level.
 Ionization Energy value is the lowest for the HF level and highest for the MP2 level. The Ionization energy for DFT level has greater value than the QCISD level.
 The experimental value of Ionization Energy is in within the accuracy of 1% for MP2 level with choice of basis set 631G (3d,p)
 The Proton Affinity value is the lowest for the MP2 level and the greatest for the HF level. The Proton Affinity for QCISD level has greater value than the DFT level with basis set 631G (3d,p).
3.3 HOMOLUMO Energy gap in HCN molecule
Molecular orbitals are very useful in qualitative descriptions of bonding and reactivity. Orbitals are actually mathematical conveniences and not physical quantities. The atomic orbital contributions for each atom in the molecule are given for each molecular orbital, numbered in order of increasing energy. For HCN there are 59 molecular orbitals. Out of them, 7 are occupied and remaining 52 are unoccupied (virtual) orbitals. Molecular orbital number 7 is the highest occupied molecular orbital (HOMO) and molecular orbial number 8 is the lowest unoccupied molecular orbital (LUMO). With the help of Gauss view 5.0.8 set of programs under NBO populations analysis, we have also performed the visualization of occupied and unoccupied molecular orbitals for HCN molecule. Figure 08 shows energy levels diagram representing the occupied and unoccupied molecular orbitals and HOMOLUMO of HCN molecule with the basis set 631G(3d,p) at the MP2 level of approximation. HOMO and LUMO are distinguished by shading on the energy level diagram. From the energy values for HOMO (0.48378 a.u.) and LUMO (0.18007 a.u.), we have obtained the HOMOLUMO gap for HCN molecules and corresponding value of HOMOLUMO gap is 0.66385 a.u. (18.06469eV). The difference of the energies of the HOMO and LUMO is called the band gap which sometimes serves as a measure of the excitability of the molecules i.e. the smaller the energy gap, more easily it will be excited. Figure 08 shows that the excitability for HCN molecule is very low since there is high band gap.
Figure 08: HOMO and LUMO energies and HOMO and LUMO of HCN with the basis set 631G (3d,p) at the MP2 level of approximation.
Table 07: Calculation of different parameters on the basis of HOMOLUMO energy with the basis set 631G(3d,p) at the MP2 levels of approximation.
Parameters  HCN 
Homo Energy (E_{H})  13.1646 Ev 
Lumo Energy (E_{L})  4.9009 eV 
HomoLumo gap [E_{G}= E_{L} – E_{H} ]  18.0655eV 
Hardness (H) =  9.0327eV 
Softness (S) =  0.1107eV^{1} 
Chemical Potential( ) =  4.1318eV 
Electronegativity ( ) = –  4.1318eV 
Global electrophilic index ( ) =  0.9450eV 
We have estimated the value of Global Hardness, Global Softness, Chemical potential, Electronegativity and Global electrophilicity of HCN molecule. The value of hardness was found to be 9.0327eV. The higher the value of hardness, the lesser is its reactivity and hence greater stability. The global softness is the inverse concept of the hardness and is useful for the straightforward prediction of the chemical reactivity. The value of softness was found to be 0.1107eV^{1}. A molecule having low S value is less reactive than a molecule having higher S value. That means increase in softness changes electron density more easily than hard molecule and hence increases the chemical reactivity. The chemical potential is the tendency of electrons to flow from region of higher potential areas to areas with lower potential until becomes uniform throughout. Therefore, it is the middle point of the HOMO and LUMO and is estimated it to be 4.1318eV. In mulliken sense, electronegativity is the negative of the chemical potential and it was found to be 4.1318eV. The global electrophilic index express the strength of electrophilicity of the species and it was estimated to be 0.9450eV.
3.4 Total Density of State (TDOS) of HCN Molecule
The detailed theory of Total Density of State (TDOS) have been described, Figure the Total Density of State (TDOS) spectrum of HCN molecule with basis set 631G(3d,p) at MP2 levels of approximation. The line originated from 0.0 in the figure represents the DOS spectrum of HCN molecule. Figure 09 shows that DOS spectrum of the HCN molecule is dense in the region of energy greater than of 4 eV. Also in the energy range below the 13 eV, there are few spectra of the HCN molecule. The lines left to the HUMOLUMO gap in the figure represents the occupied orbitals and the lines right to HOMOLUMO gap represent the unoccupied orbitals. The energy difference between highest occupied orbital (13.18070 eV) and lowest unoccupied orbital (4.88480 eV) gives the HOMOLUMO energy gap of 18.0655 eV, which exactly match with the values calculated from figure 08
Figure 09: Total Density of States spectrum of HCN molecule with basis set 631G(3d,p) at the MP2 levels of approximation
3.5 Study of Mulliken and Natural Population Analysis of charge distribution of HCN molecule
Here we have tabulated the charges computed by Mulliken population analysis and the Natural population analysis (NPA) for HCN molecule with the basis set 631G(3d,p) at MP2 level of pproximation.
Table 08: Mulliken and Natural net atomic charges of HCN with basis set 631G(3d,p) inHF+MP2 level of approximation.
Atoms  Charges by  
Mulliken population analysis  Natural Population analysis  
C  0.260  0.115 
H  0.329  0.239 
N  0.069  0.354 
The Mulliken analysis places the more negative charge on the carbon atom while less negative charge on the nitrogen atom than NPA. This comparison of charge distributions can be better depicted by using histogram as in figure 10
Figure 10: Mulliken and Natural net atomic charge distributions of HCN with basis set 631G(3d,P) at MP2 level of approximation.
3.6 Topology Analysis
In chemical topology analysis we are just concerned with the properties of the scalar fields like the density of all electron [ρ(r)] and their corresponding second derivative ^{2} , the Laplacian of electron density. Here we have studied the topology analysis of molecules by taking bond critical points (BCP) only.
3.7 Bond Critical Points (BCPs), Density of all electron at BCPs (ρ), Laplacian of electron density at BCPs ( 2 ) of HCN molecule
We have performed the MP2 levels of approximation with basis set 631G(3d,p) to study the topology of the HCN molecule. The different bond critical points of the HCN molecule at MP2 levels of approximation with the basis set 631G(3d,p) is shown in figure 11. It is seen from the figure that there is a single BCP exists at each the bonds. The density of all electron and Laplacian of electron density at different BCPs are presented in the table 4.9
Figure 11: Bond Critical Points (BCP) of HCN with the basis sets 6 31G (3d,p) at the MP2 level of approximation.
Table 09: Density of all electrons and Laplacian of electron density of HCN molecule at BCP along different bonds with basis set 631G(3d,p) at MP2 levels of approximation.
Bonds  Density of all electrons
(ρ) (a.u.) 
Laplacian of electron density
( ^{2}ρ) (a.u.) 
CH  0.28720  0.11725 
CN  0.45872  0.22536 
Table 10: Variation of density of all electrons and laplacian of electron density with respect to bond length of HCN with the basis set 631G(3d,p) in the MP2 level of approximation.
Bond length (Å)  Density of all electrons ρ (a.u)  Laplacian of electron density ρ(a.u) 
1.10  0.52808  0.45855 
1.11  0.51720  0.34987 
1.12  0.50667  0.24536 
1.13  0.49647  0.14468 
1.14  0.48659  0.04754 
1.15  0.47701  0.04636 
1.16  0.46773  0.13726 
1.17  0.45872  0.22537 
1.18  0.44997  0.31086 
1.19  0.43319  0.47446 
1.20  0.43319  0.47446 
It is seen from the table that the values of ρ at BCP along CH bond and BCP along CN bonds are 0.28720 and 0.48751 a.u. respectively. Also the value of at BCP along CN bond is higher in negative value i.e. 0.22536 a.u. than the laplacian at BCP along CH bond having value of 0.11725 a.u. This indicates the potential energy is dominant at these critical points and hence negative charges are concentrating and bonds are strong type. Figure 12 shows that the variation of density of all electrons of HCN with decreasing the bond length between carbon and nitrogen atom [C N] using 631G(3d,p) basis set at the MP2 levels of approximation. From figure 12, it is seen that on decreasing the bond length between carbon and nitrogen atom from 1.20 Å to 1.10 Å, the density of all electrons increases continuously from 0.43319 a.u. to 0.52808 a.u., which is characterized by linear fit equation y = mx + c . On decreasing 0.1 Å bond length between carbon and nitrogen atom we have found that increase in density of all electrons is 0.00949 a.u. (0.25821 eV). In the above figure, the linear fit equation is obtained as y = 0.97736x +1.60169 with error in ‘m’ is 0.02372 and error in ‘c’ is 0.02729.
Figure 12:Variation of bond length with [a] density of all electrons and [b] laplacian of electron density of HCN with the basis set 631G(3d,p) in the MP2 levels of approximation.
Figure 4.8(b) shows that the variation of laplacian of electron density of HCN with decreasing the bond length between carbon and nitrogen [CN] using 631G(3d,p) basis set at the MP2 levels of approximation. From figure 4.8(b), it is seen that on decreasing the bond length between carbon and nitrogen atom from 1.20 Å to 1.10 Å, the laplacian of electron density increases continuously from 0.47446 a.u. to 0.45855 a.u., which is characterized by linear fit equation y = mx + c . On decreasing 0.1 Å bond length between carbon and nitrogne atom we have found that increase in laplacian of electron density is 0.09330 (2.53890 eV). In the above figure, the linear fit equation is obtained as y = 9.59682x + 10.99789 with error in ‘m’ is 0.23532 and error in ‘c’ is 0.27072.
3.8 Quantitative Analysis of Molecular Surface of HCN molecule
Using the Multiwfn set of programs version 3.3.4 [23], we have studied the global maximum and global minimum of HCN molecule. Here we analyze electrostatic potential (ESP) of studied molecules on vdW surface.
Figure 13: Contour map of electrostatic potential distributed on molecular Van der Waals surface of HCN molecule.
In figure 12, the bold line corresponds to vdW surface which is the isosurface of electron density 0.001 a.u.[defined by R. F. W. Bader].From this graph it is clear that H atom is more positively charged and N atom is more negatively charged than C because the local vdW surface closed to H atom largely intersected solid contour lines and N atom largely intersected dotted contour lines . Because of this reason, we can see that the hydrogen atom is more positively and nitrogen atom is more negatively charged.
Figure 14: Global Maxima and Global Minima of HCN with basis set 631G(3d,p) in MP2 level of approximation.
Figure 14 shows the maximum and minimum points of HCN molecule on the vdW surface with choice of 631G(3d,p) basis set at MP2 level of approximation. We can see that there are one maximum point and one minimum point. Maximum point is near hydrogen atom where as the minimum point is near nitrogen atom. The global maximum point has the value of 2.19790 eV. The global minimum point has the value of 1.41925 eV and lies near the nitrogen atom.
3.9 Study of Thermochemistry parameters and Isotopic Shift on HCN molecule
Zeropoint vibrational energy (E0), total energy (E), specific heat capacity (Cv) and entropy (S) of HCN at temperature 273.15 K and 1 atm pressure and frequency scale factor 1=1/1.12 was performed. We have estimated the total thermal energy, zero point energy, specific heat capacity and entropy of the HCN molecule and its isotopologs with the choice of basis set 631G(3d,p) at the MP2 level of approximation.
Table 11: Thermochemistry of HCN with basis set 631G(3d,p) in the MP2 level of approximation where N = N^{14} and C = C^{12}.
Parameters
Isotops 
Parameters


Total Thermal energy ( KJ/mol)  Zero point vibrational energy ( KJ/mol)  Specific heat capacity (J mol^{1 }K^{1})  Entropy (J mol^{1 }K^{1})  
HCN  43.14  36.88  28.11  199.70 
HCN^{15}  42.96  36.69  28.14  200.41 
HC^{13}N  42.82  36.54  28.22  200.44 
HC^{14}N  42.53  36.25  28.31  201.14 
DCN  37.03  30.41  30.58  203.77 
From the table 4.11, it is seen that the total thermal energy and zero point vibrational energy of the HCN molecule decrease where as specific heat capacity and entropy increase for the N^{15} isotope than N^{14} in HCN molecule. Similarly, the values of total thermal energy and zero point vibrational energy are greater for C^{12 }than C^{13 }and C^{14}. However, values of specific heat capacity and entropy are greater for C^{14} than C^{13} and lowest for C^{12}. The value of total thermal energy and zero point vibrational energy is the lowest where as specific heat capacity and entropy is the highest when hydrogen atom is replaced by deuteron atom.
3.10 Equilibrium geometry and Bond critical points in HCN dimer
Figure 15: Equilibrium configuration of HCN dimer with the basis set 631G(3d,p) at the MP2 level of approximation
The equilibrium configuration of HCN dimer obtained in the MP2 level of approximation using the basis set 631G(3d,p) is presented in figure 4.11. The equilibrium geometry of HCN molecule shows the values of equilibrium bond length R(1H−4N) and R(3N−6H) are 3.04 Å and 3.32 Å respectively. Also the values of equilibrium bond length of R(2C−5C) is 3.21Å.. The dimerization energy of the HCN dimer is estimated as
Dimerization Energy = 2E(HCN) − E(HCN) 2 (4.4)
and was found to be 0.163eV.
 CONCLUSION
The linear organic molecule with carbonnitrogen triple bond HCN is known variously as Prussic acid, Formic anammonide, Methanenitrile. The highly valuable precursor to many chemical compounds ranging from polymers to pharmaceuticals, HCN, was first isolated from a blue pigment (Prussian blue). In the present work we have used the GaussView to build the initial structure and the structure was then optimized using Gaussian03 software to get the equilibrium configuration. We have carried out the HartreeFock (HF), MØllerPlesset second order (MP2), truncated configuration interaction (CI) approximation with quadratic, single and double states (QCISD) and Beckestyle 3Parameter LeeYangParr correlation functional theory (B3LYP) calculations to estimate the equilibrium configuration, ground state energy, Binding Energy and Dipole Moment of HCN molecule. The ground state energy, equilibrium configurations and binding energy of the HCN Molecule is estimated with the basis sets 631G(3d,P) in the MP2 level of approximation. The calculated values of bond length of HCN in the MP2 levels of approximations agree with experimental value with in around 1% respectively. The calculated binding energy of HCN molecules in the MP2 level of approximation in the basis sets 631G(3d,P) and found to be 18.78853 eV. The positive value of the binding energy shows that the system is stable. The ground state energy of the neutral, positively charged and negativelycharged HCN has been calculated. This shows that the neutral hydrogen cyanide has less value of ground state energy indicating it is more stable than its charged(ionized) states. The calculation also shows that the negatively single charged state is more stable than positively single charged state.
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Publication History
Submitted: November 25, 2023
Accepted: December 14, 2023
Published: January 31, 2024
Identification
D0209
Citation
Lekh Nath Regmi (2024). First Principles Study of the Stability and Bonding Analysis of Hydrogen Cyanide and its Dimer. Dinkum Journal of Natural & Scientific Innovations, 3(01):5880.
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