Quantum Mechanical Computation By Sasha Payne N. Diaz CHEMISTRY 47-L 1 January Introduction Backgrounds for computational chemistry Understanding the behavior of materials at the atomic scale is fundamental to modern science and technology. As many properties and phenomena are ultimately controlled by the details of the atomic interactions, simulations of atomic systems provide useful information. A wide variety of models have been developed to describe atomic interactions. Quantum Mechanics ultimately provides the best description of matter. In quantum mechanics, a particle is distributed through space like a wave. This wave function contains information about all the properties of the system. By solving the Schrödinger equationh Ψ= EΨ (where H is the Hamiltonian operator and E is the value of energy) We can find the wave function of any given system. If we let the Hamiltonian operator operate on this wave function, we can get the expectation value of the sum energy of a given system. The same can apply to the diatomic system which will be investigated in depth in this lab report. However, often the time, these wave functions and expectation values of energy are not easily accessible by experiments or by analytical method. Therefore, the solution of the Schrödinger equation apart from a few very simple examples has to be performed numerically using computers. In this experiment, the program Gaussian will be used to calculate out the potential energy. This program will generate a linear combination of a series of Gaussian functions as the solution to the Schrödinger equation.
Background information about vibrational motion An important feature of a diatomic system is the vibrational motion of the two atoms. Atoms in a diatomic molecules and solids vibrate around their mean positions as bonds stretch and compress. The detection and interpretation of vibrational frequencies is the basis of infrared spectroscopy. A particle undergoes harmonic motion if it experiences a restoring force proportional to its displacement: F = kx (the Hooke's law), where k is the force constant. Because force is related to potential energy by F = dv/d x, it corresponds to the particle having a potential energy: 1 V ( x) = kx The energy of a molecule with respect to bond length is often approximated using a harmonic oscillator potential and is a parabola when put into graph: E p = k( r re ) As can be seen from the equation, the potential energy of a diatomic molecule relates to the equilibrium bond distance r e and the distance between the two atoms. It can be predicted that the potential energy will reach its lowest value when distance between the two atoms equals to the equilibrium bond distance r e. Both compressed and stretched molecular bond result in the increase of the potential energy. However, in the real diatomic molecule, this curve is different from that of harmonic oscillator. In the region near the energy minimum, the PE vs. distance curve will fit to a third-order polynomial with an offset x - value : PE ( d) = a + a1( d dequilibrum ) + a ( d dequilibrum ) + a ( d dequilibrum ) Note that a is equivalent to the force constant k. a, the anharmonisity value, needs to be as small as possible to get accurate a. Fundamental vibrational frequency can be calculated by the following equation: ν = 1 π k where k is the force constant and µ is the reduced mass of the two µ atoms.
This fundamental frequency describes the energy needed to vibrate the molecular bond. Objectives: 1. Using Gaussian to compute potential energy curve of diatomic molecule H, LiF, LiCl and LiBr.. Fit the curve with a third order polynomial to obtain the force constant of diatomic bond.. Compare the literature value of fundamental vibrational frequency and bond length with the computational value and discuss the applicability of harmonic oscillator model and the computing ability of Gaussian. Experimental Experiment1. Computation of different potential energy of H molecules with ten different interatomic distances. Experiment 1 is carried out by creating an input file using a text editor first. This enables the Gaussian to know what it needs to calculate. Create hh.inp as shown below. Save the text as inp format to be ready for the Gaussian to read. In this file, RHF stands for Restricted Hartree-Fock method and 6-11G refers to the basis set Here means unpaired and 1 means multiplicity (.,.,.) is the position of hydrogen atom in a D coordinate system. Next step is to run Gaussian through terminal using hh.inp as input data. After a short while of calculation, Gaussian will emit a file called hh.log. Open this file and find for the calculation result begin with SCF Done. The energy is in the unit of hartress:
Record the energy and the distance. Repeat the process with different interatomic distances to obtain 1 energy data. Use Igor to visualize the general trend. Experiment. Computation of different potential energy of H molecule of 11 different interatomic distances starting from.å. Noticing that 1 data are far from enough to obtain any information accurately and this method require lots of time and energy, another input text is used instead. In this scan test, R starts from. and increase with a step size of.. It will calculate for 1 times and generate 1 energy values. The emitted log file is shown below. Copy these columns and create another text.
In the Igor window, select data load wave chose load general text. Skip the first column. Go to window new graph and select corresponding column for X axis and Y axis. Because the energy and distance units are non-si unit, adjustment must be made to convert Angstrom to nanometer and Hartree to kj/mol (1 Hartree=6. kj/mol). The energy corresponding to a distance where the atoms are non-bonded must be computed to zero so that the depth of the energy can be easily read from the diagram. After these adjustment, fit the curve with a third order polynomial. Select poly_xffset function with four polynomial terms. Adjust the range in the data options tab to best fit the curve so that the red fitting line come through the center of each circle, the standard deviation is less than 1% of the original value and k being as small as possible. After obtaining the k value, convert the unit of k (kj/molonm ) into the unit of force constant k(n/m). Calculate the Fundamental vibrational frequency using the equation ν = 1 π k. Obtain the equilibrium bond distance by finding the point with µ lowest potential energy. Experiment. Investigating LiF, LiCl and LiBr Repeat this process for three additional molecules LiF, LiCl and LiBr. Substitute H for Li and the halogen in the input file. Combine three curve into one graph for comparison. OResults, calculation and discussion Experiment 1 Table 1: Potential Energy of selected interatomic distance Distance (Å) Energy (hartree). -1.6164717.7-1.178188
.8-1.11467 1. -1.968694. -.76766477491 1. -.787814179. -.718999 4. -.718971841 6. -.716779116 1. -.71496887 Data were taken directly from hh.log. As can be seen from the table, the energy experiences a decrease following by an increase as the distance change from. Å to 1. Å. Little changes as the distance goes bigger than 4. Å, which indicates a gentle slope of the curve. The energy reaches the lowest at.7 Å. It can be postulated that the equilibrium bond distance is somewhere near.7 Å. Experiment Figure 1: Potential energy curve of a H molecule. The red line indicates the curve fitting from the forth to the thirteenth data point. Coefficient values and standard deviation in the box are the outcome of
curve fitting using poly_xffset function. Table : Coefficient values and standard deviation for fig.1 Coefficient Coefficient values Standard deviation k ( kj / mol nm ).181 1 8.96 1 6 k ( kj / mol nm ) -1.868 1 1.1 1 µ =1/m H 1/ 1.794 g/mol k 1 k= 9 (1 ) N A =.98 N/m k literature =7 N/m ν = 1 π k = µ π k 1 N 1 1 A -9 NA 1 m H 1 =7.8 1 Hz According to Igor: E(minloc)= E(min)= -961.694 kj/mol Equilibrium bond distance r e =.7 Å literature r e =74 pm As can be seen from the outcome, Gaussian successfully predicts the equilibrium bond distance. However using a harmonic oscillator to model the hydrogen atom is less satisfying. There is a quite large difference between the literature value of the force constant and the computation. This maybe partly due to the imprecise curve fit. Experiment -1 Potential Energy (kj/mol) - - -4 - Coefficient values ± one standard deviation K =-4.44 ±.98 K1 =-4. ± 81.1 K =1.486e+ ± 4.e+ K =-1.181e+6 ± 6.e+4 Constant: X =.14.1.. Interatomic Distance (nm) LiF.4.
Figure : Potential energy curve of a LiF molecule. Method used is the same as that of the H molecule. Potential Energy (kj/mol) -1 - - -4 - 'energy LiF' 'energy LiCl' 'energy LiBr'.1. Figure : Potential energy curve of molecule LiF, LiCl and LiBr... Interatomic Distance (nm).4. An obvious trend of lower absolute potential energy and larger equilibrium bond distance can be observed from Fig. as the halogen increases in atomic mass and radius. Table : Coefficient values and standard deviation for Fig. Molecule k (kj/molonm ) k σ k (kj/molonm ) σ k LiF 1486 6 4 4. 1-1.181 1 6. 1 LiCl 996 4.66 1 -.1 1. 1 LiBr 6766 4 1.67e 1 -.441 1 1.6 1 Note that only the curve fitting of LiF suffices all the three curve fitting criteria. LiCl and LiBr, however, get the k value with a standard deviation that exceed 1% of the original value. If a smaller standard deviation of k is to get, the k (anharmonicity) value has to be larger than the one shown in the table. Therefore the harmonic oscillator model failed to work ideally for LiCl and LiBr.
Table 4: Atomic weight and reduced mass of LiF, LiCl and LiBr. Calculated value and literature value for force constant. Molecule m 1 (g/mol) m (g/mol) µ (g/mol) k (N/m) k literature (N/m) [4] LiF 6.941 18.998.84 46.7 LiCl 6.941.4.8 14. 14 LiBr 6.941 79.94 6.86 1.886 1 Table : Calculated fundamental vibrational frequency, wave number, wave length, energy difference and the literature value of wave number of LiF, LiCl and LiBr. Molecule ν (Hz) ν (m -1 ) λ (m) E(J ) ν (cm -1 ) literature LiF 1.71879 1 LiCl 1.178 1 LiBr 1.98 1 1 966.6 671.84 1.9 1.14 1 1.1191 1 1 ν =9 [] 1.48974 1 8.891 1 ν =64.7(7) [6] 1.8864 1 6.6119 1 ν =4 [7] The computation value and literature value of force constant and wave number match correspondingly with minor discrepancy. Overall, the harmonic oscillator model is applicable to diatomic molecules and makes good approximation. As the molecule change from LiF to LiBr, the force constant k and the fundamental frequency ν decreases. As ν describes the energy needed to vibrate the molecular bond, it also indicates that the molecular bond gets weaker. This trend is in accordance with the expectation that halogen with larger electronegativity forms ionic bond with Lithium at a higher force constant. Table 6: The equilibrium bond distance (re) of LiF, LiCl and LiBr read from the curve where potential energy (PE) reaches the lowest. Minimum location is the sequence number of data that corresponds to re and the lowest potential energy. Molecule r e (Å) Min location PEmin (kj/mol) r e literature (Å) [4] LiF 1. -84.988 1.69 LiCl. -4.7.7
LiBr. -.67.174 The computation value and literature value of equilibrium bond distance match correspondingly with minor discrepancy. We can conclude that Guassian is a qualified computation grogram to compute potential energy curve of diatomic molecules. References [1] Albert Bartók-Pártay, The Gaussian Approximation Potential, SpringerLink: Heidelberg; Springer-Verlag, 1; 1, [] Peter Atkins; Julio de Paula; Ron Friedman, Quanta, Matter, and Change, W. H. Freeman and Company, United States and Canada, 9 [] Bernhard Schrader, Infrared and Raman Spectroscopy; VCH Verlagsgesellschaft mbh, Germany, 199 [4] W.M. Haynes; David R. Lide, CRC Handbook of Chemistry and Physics, 91st edition; CRC Press, 1-11; Section 9 Molecular structure and Spectroscopy 8,. [] Guido L. Vidale, THE INFRARED SPECTRUM OF THE GASEOUS LITHIUM FLUORIDE (LiF) MOLECULE1; J. Phys. Chem., 196, 64 (), 14 18 [6]Thompson, G. A.;Maki, A. G.; Olson, Wm. B.; Weber, A,High-resolution infrared spectrum of the fundamental band of LiCl at a temperature of 8 C; Journal of Molecular Spectroscopy, Volume 14, Issue 1, p. 1-18 [7] Bruce S. Ault, George C. Pimentel, Matrix isolation infrared studies of lithium bonding; J. Phys. Chem., 197, 79 (6), 61 66 Acknowledgements The author thanks Prof. Joseph A. DiVerdi for helpful talk and revision of the lab report outline and provision of insightful lab roadmap.