1. Central Department of Physics, Tribhuvan University, Kathmandu, Nepal; lekhnath.regm@gmail.com

*             Correspondence: lekhnath.regm@gmail.com

Keywords: hydrogen cyanide, Hartree-Fock, MØller-Plesset Second Order

1. 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 almond-like 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]. Many-body systems require many mathematical operations, which take time. The computational method is ideal for studying many-body 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 many-electron 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 semi-empirical. 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 semi-empirical method, one uses one-electron 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: Hartree-Fock (HF) self-consistent-field, density functional theory (DFT), and quantum Monte-Carlo (QMC). A one-particle schrödinger equation governs electron motion in the Hartree-Fock self-consistent-field approximation, a rigorous one-electron 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 electron-electron interaction. In HF approximation, instantaneous electron motion correlation is ignored. Many methods, including Møller-Plesset Perturbation approximation (MP), configuration interaction (CI), coupled-cluster method, multi-configurational self-consistent-field method, and multi-reference configuration interaction [8], account for instantaneous electron motion. We use the Møller-Plesset Perturbation approximation and configuration interaction to study correlation effects caused by instantaneous electron motion. In Møller-Plesset 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 many-electron system is described in terms of density of electrons which gives the physical properties of the system just like as wave function in Hartree-Fock self-consistent-field approximation. The degrees of freedom decreases from 3N-6 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.

2. 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];

1. Computational model should give a unique energy for a system defined.
2. It should be size consistent. The margin of the error should be proportional to the molecular size.
3. Computational model should be practical. It should be able to define problems ranging from simple to complex and those problems which are experimentally impossible.
4. 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 first-principles 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, semi-empirical 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 (Moller-Plesset 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].

Semi-empirical 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 Kohn-Sham 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, Noble-Prize (1998 for development of computational methods in quantum chemistry) laureate John People and his research group at Carnegie-Mellon 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;

1. Creating GAUSSIAN input by defining the computational model and defining the
2. Specifying the location of the various scratch files. The very useful scratch files
3. used in GAUSSIAN are formatted checkpoint files (.fch) and checkpoint files (.chk).
4. Initiating program execution.
5. 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 case-insensitive. 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, STO-nG. People-based 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 split-valence basis set. 3-21G is a split-valence basis set. The 3-21G 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 asterisk-denoted polarization function increases wave function flexibility. Single asterisk means “d” primitives are added to non-Hydrogen 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 function-like. Polarized basis set is 3-21G* and diffuse function is 6-31+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 co-ordinates for each atom in the system or by defining Z-matrix [28]. Z-matrix includes bond distance, angles and dihedral angles. The molecular specification in the Z-matrix fashion is shown in the Appendix I. The Z-matrix molecular specification can be explained by the scheme in figure 03.

Figure 03: Molecular specification using Z-matrix 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 co-ordinate we have to list all the atoms with their (x, y , z) co-ordinate in the molecular specification section as shown in the figure 04.

Figure 04: Molecular specification using Cartesian co-ordinate 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, full-featured graphical user interface for Gaussian. Gauss View supports all Gaussian features, and it includes graphical facilities for analyzing the results of the calculations through state-of-the-art 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), GAMESS-UK, 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 platform-independent open-source 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 Levenberg-Marquardt 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 built-in 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 three-dimensions
• 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/UV-Vis/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.

3. RESULTS AND DISCUSSION

The study have performed the first-principles (ab initio) calculation to study about Hydrogen Cyanide (HCN) molecule. We have carried out the Hartree-Fock (HF), Møller-Plesset second order perturbation (MP2), Quadratic Configuration Interaction Single (double) method (QCISD) and Becke-style 3-Parameter Lee-Yang-Parr 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 body-effects 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 d-type 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
	EHF(a.u.)	EMP2(a.u.)	EQCISD(a.u.)	EDFT(a.u.)
3-21G	-92.35408	-92.56503	-92.56677	-92.90763
3-21G*	-92.35408	-92.56503	-92.56677	-92.90769
6-31G	-92.82832	-93.04343	-93.04408	-93.39249
6-31G*	-92.87520	-93.15894	-93.16730	-93.42261
6-31G(3d, p)	-92.88363	-93.19414	-93.20095	-93.43051
6-311G	-92.84956	-93.07639	-93.07680	-93.41889
6-311G*	-92.89698	-93.19268	-93.19972	-93.44914
6-311G(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 many-body 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: Many-body contribution to the total energy of HCN molecule

Basis Sets	Many-Body Contributions (eV)
	( EMB)MP2	( EMB)QCISD
3-21G	-5.74	-5.79
3-21G*	-5.74	-5.79
6-31G	-5.85	-5.87
6-31G*	-7.72	-7.95
6-31G(3d, p)	-8.45	-8.63
6-311G	-6.18	-6.27
6-311G*	-8.05	-8.24
6-311G(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)
3-21G	13.43293	3.04	17.36044	3.08	16.11359	2.77	17.73760	2.72
3-21G*	13.43293	3.04	17.36044	3.08	16.11359	2.77	17.73923	2.72
6-31G	13.59131	3.24	17.27499	3.31	15.96365	3.01	17.47010	2.93
6-31G*	14.84660	3.21	18.43423	3.26	16.84831	2.98	18.15966	2.91
6-31G(3d, p)	14.86700	3.24	18.78853	3.27	17.07880	2.98	18.30089	2.99
6-311G	13.47593	3.25	17.19499	3.31	15.89616	3.03	17.26901	2.98
6-311G*	14.68360	3.22	18.37790	3.26	16.78790	2.98	18.04346	2.96
6-311G(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 6-31G(3d,p) than the basis sets 3-21G, 3-21G*, 6-31G, 6-31G*, 6-311G, 6-311G*,  6-311G(3d,p) in all levels of approximation. Hence, we have considered the basis sets 6-31G(3d,p) for the further calculation. The equilibrium configuration of HCN molecule obtained in the MP2 level of
approximation using the basis set 6-31G(3d,p) is presented in Figure 06.

Figure 06: Equilibrium configuration of HCN molecule with the basis set 6-31G(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  6-31G(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 6-31G(3d,p) at MP2 level of approximation and experimental values.

	R(HC)	R(CN)	A(RCN)
6-31G(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 6-31G(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 6-31G(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 6-31G(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 6-31G(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
(H2CN)+	-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 6-31G(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 6-31G(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 6-31G(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 6-31G (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 6-31G (3d,p).

3.3 HOMO-LUMO 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 HOMO-LUMO of HCN molecule with the basis set 6-31G(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 HOMO-LUMO gap for HCN molecules and corresponding value of HOMO-LUMO 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 6-31G (3d,p) at the MP2 level of approximation.

Table 07: Calculation of different parameters on the basis of HOMO-LUMO energy with the basis set 6-31G(3d,p) at the MP2 levels of approximation.

Parameters	HCN
Homo Energy (EH)	-13.1646 Ev
Lumo Energy (EL)	4.9009 eV
Homo-Lumo gap [EG=| EL – EH| ]	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 6-31G(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 HUMO-LUMO gap in the figure represents the occupied orbitals and the lines right to HOMO-LUMO gap represent the unoccupied orbitals. The energy difference between highest occupied orbital (-13.18070 eV) and lowest unoccupied orbital (4.88480 eV) gives the HOMO-LUMO 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 6-31G(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 6-31G(3d,p) at MP2 level of  pproximation.

Table 08: Mulliken and Natural net atomic charges of HCN with basis set 6-31G(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 6-31G(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 6-31G(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 6-31G(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 6-31G(3d,p) at MP2 levels of approximation.

Bonds	Density of all electrons (ρ) (a.u.)	Laplacian of electron density ( 2ρ) (a.u.)
C-H	0.28720	-0.11725
C-N	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 6-31G(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 C-H bond and BCP along C-N bonds are 0.28720 and 0.48751 a.u. respectively. Also the value of  at BCP along C-N bond is higher in negative value i.e. -0.22536 a.u. than the laplacian at BCP along C-H 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 6-31G(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 6-31G(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 [C-N] using 6-31G(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 6-31G(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 6-31G(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

Zero-point 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 6-31G(3d,p) at the MP2 level of approximation.

Table 11: Thermochemistry of HCN with basis set 6-31G(3d,p) in the MP2 level of approximation where N = N¹⁴ and C = C¹².

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
HCN15	42.96	36.69	28.14	200.41
HC13N	42.82	36.54	28.22	200.44
HC14N	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 N¹⁵ isotope than N¹⁴ in HCN molecule. Similarly, the values of total thermal energy and zero point vibrational energy are greater for C¹² than C¹³ and C¹⁴. However, values of specific heat capacity and entropy are greater for C¹⁴ than C¹³ and lowest for C¹². 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 6-31G(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 6-31G(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.

3. CONCLUSION

The linear organic molecule with carbon-nitrogen 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 Gaussian-03 software to get the equilibrium configuration. We have carried out the Hartree-Fock (HF), MØller-Plesset second order (MP2), truncated configuration interaction (CI) approximation with quadratic, single and double states (QCISD) and Becke-style 3-Parameter Lee-Yang-Parr 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 6-31G(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 6-31G(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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