Lennard-Jones potential determination via the Schrodinger ... kouetcha… · Lennard-Jones...

Lennard-Jones potential determination via theSchrodinger equation

Daniella Nguemalieu Kouetcha.1, Hamidreza Ramezani∗,2 and Nathalie Cohaut11 Universite d’Orleans, ICMN, UMR CNRS 7374, Interfaces, Confinement, Materiaux et Nanostructures,1b rue de la Ferollerie, CS 40059, 45071 Orleans cedex, France,

2Ecole Polytechnique de l’Universite d’Orleans, Universite d’Orleans, ICMN, UMR CNRS 7374, Interfaces,Confinement, Materiaux et Nanostructures, 8 rue Leonrad de Vinci, 45072 Orleans, France.∗Corresponding author: 8 rue Leonard de Vinci, 45072 Orleans, France, [email protected]

Abstract: The accurate atomic potentialdetermination is an essential task in themolecular simulations, e.g. Grand Canoni-cal Monte Carlo (GCMC). The so-called ab-initio simulations using the quantum me-chanics would be of great interest in thecomputational physics. The numerical sim-ulation of the adsorption phenomenon re-quires knowing the interactions parametersbetween the atoms that make up the sys-tems and it sustains the potential that itwould be exploited either. In the presentcommunication, the determination of theinteraction parameters for the molecularsimulation purposes have been extracted bymeans of solving the Schrodinger’s equa-tion using the finite element method. ThePartial Differential Equation (PDE) imple-mentation of the Schrodinger has been suc-cessfully achieved via COMSOL MP.

Keywords: Schrodinger equation; Wavefunction; Custom PDE; FEA; Hydrogenatom

1 Introduction

There is a bunch of studies pertaining tothe purification of water by means of thephysically- or chemically-motivated ways.The physical adsorption can be addressedas a prominent method of purification.The adsorption is a surface phenomenonin which the particles or molecules (ad-sorbate) bind to the top layer of a ma-terial (adsorbent) [1]. However, the ph-ysisorption deals with the Van der Waalsinteractions between adsorbent and adsor-bate, whereas, the chemisorption handlesthe chemical bonds that are involved be-tween the adsorbate and adsorbent.To achieve the physisorption adsorption,

it is necessary to know the values of theparameters of molecules interactions. Thedetermination of these parameters can bedone via the quantum mechanics involv-ing the Schrodinger equation. The Lennard-Jones’ molecular parameters, i.e. ε and σare usually determined from physical prop-erties such as the critical constants, sec-ond virial coefficient, viscosity and crys-tal properties [2, 3]. The empirical po-tentials are customarily applied to performthe molecular simulations, e.g. GrandCanonical Monte Carlo (GCMC). The mostcommonly-used pairwise additive poten-tial has been proposed by Lennard Jones(Equation 1). This potential is increasinglyused to determine the intermolecular pa-rameters for simple molecules (H2, O2, N2,CO, CH4) [3], in which we add the inter-action terms in some cases potential iodide(KI) molecule [4, 5].

ULJij (r) = 4εij

[(σijrij

)12

−(σijrij

)6]

(1)

However, there are some additionalpotentials such as the Morse’s potential(Equation 2) [6, 7]. The aforementioned po-tential takes advantage of four parametersand it provides the smaller separation dis-tances between the atoms as well.

UM(rij) =

D [exp(−2β(rij − rc))− 2 exp(−β(rij − rc))]

+ E∞(2)

where, D, rij , rc, β and E∞ are the dissocia-tion energy, internuclear separation, meanbond length and constants extracted fromthe spectroscopic data, respectively. Onecan also address some further potentials,e.g. Rydberg’s potential, Buckingham’spotential, Murrel-Mottram’s potential and

Excerpt from the Proceedings of the 2015 COMSOL Conference in Grenoble

Biswas-Hamann’s potential (See appendixA for more details). The COMSOL MP isused to solve the Schrodinger’s equation inthe case of the hydrogen molecule herein[8, 9]. The paper is organized as following:The second section deals with the meth-ods and theory about Schrodinger’s equa-tion. The use of Comsol Multiphysics hasbeen brought in the third section. The Re-sults and discussion in fourth section. Atlast, some conclusions and outlooks are dis-cussed.

2 Methods and theory

The adsorption of different molecules, e.g.H2, Pb and Cd, on different forms ofcarbon-based materials are of great inter-est in the adsorption society (See Figure 1,Figure 2 and Figure 3).

Figure 1: Typical illustration of nanotube.

Figure 2: Typical illustration of graphite(slit-shaped).

Figure 3: Typical illustration of conical shapes[10].

The above-mentioned adsorptions may becarried out by means of the chemical orphysical methods. For physical method,the real problem is the definition of theinteraction potential of the molecules that

they should be adsorbed. Thus, differ-ent methods are presented in the liter-ature to determine the potential that isbest for a particular molecule includingDensity Functional Theory (DFT) meth-ods. The finite element method can be uti-lized to solve the Schrodinger’s equationas well. In the quantum mechanics, theSchrodinger’s equation can be described asa PDE and it highlights how the quantumstate of a physical system changes withintime. It was formulated in late 1925 bythe Australian physicist Erwin Schrodinger[11]. This equation has different formsfor a specific situation. Thus, it may betime-independent, time-dependent, 1-D, 3-D, 1-particle and N-particles. The gen-eral Schrodinger equation is displayed asEquation 3.

Hψ = Eψ (3)

Where H is Hamiltonian operator of sys-tem, E represent energy, and ψ is quantummechanical wave function. However, thedefinition of the expression of Hamiltoniandepends on the number of nuclei and elec-tron system studied (monoatomic or poly-atomic). If we consider the case of an atom(N nuclei and n electron) expression of theHamiltonian is given by Equation 4.

H = −N∑k=1

~2

2Mk∇2k︸︷︷︸

Nuclear kinetic energy

−n∑i=1

~2

2me∇2i︸︷︷︸

Electron kinetic energy

− 1

4πε0

N∑k=1

n∑i=1

zke

Rik︸︷︷︸Electron/nuclear attraction

+1

4πε0

n∑i=1

n∑j 6=1

e2

rij︸︷︷︸Electron/electron repulsion

+1

4πε0

∑k<l

zkzlRkl︸︷︷︸

Nuclear/nuclear repulsion

(4)

Where the equation parameters are:

~ = h2π , h is Plank’s constant.

Mk is the mass of kth nuclei and me is themass of an electron.

Rik is distance between the electron andnuclei.

Rkl is distance between nuclei.

rij is distance between electron.


e is charge of the electron.

zk and zl are charge of kth and lth nuclei.

By applying the condition of Born-Oppenheimer [12] ”when we consider thenuclei set”, the Hamiltonian is decoupledpart in nuclear and electronics. The reso-lution of the electronic part allows for thewave function and the energy of the sys-tem. In addition, this is the square of thewave function that has a physical meaningand represents the probability of findingan electron.Thus, the validation of the 3Dmodel of the hydrogen was performed bysolving the Schrodinger’s equation fromCOMSOL software, the electrostatic po-tential in Cartesian coordinates is given byEquation 6.

3 Results

3.1 Hydrogen atom

To make the adsorption of differentmolecules by numerical simulations, it isuseful to determine the parameters of inter-action between different atoms that makeup the system, this will define the adequatepotential to explain the phenomenon.

This article shows how to calculatemolecule hydrogen ion the interactionparameters by solving the Schrodinger’sequation and using the finite elementsmethod.

The 2D model of atom hydrogen istreated by exploiting the cylindrical coor-dinates (ρ, φ, z). Thus, we must define theLaplacian in cylindrical coordinates and theelectronic wave function ψ as the productof two function, one with a radial and z-coordinate dependence , and the other anangular dependence (ψ = ψ1(ρ, z)ψ2(φ))[13], the results of the model in 2D are pre-sented in Figure 1.

In the case of the resolution of theSchrodinger equation and 3D Cartesian co-ordinate of the hydrogen atom, the Hamil-tonian should be formulated as:

H = − ~2

2me(∂2

∂x2+

∂2

∂y2+

∂2

∂z2) + V (x, y, z)

(5)

where V (x, y, z) = − e2

4πε0√x2 + y2 + z2

.

−∇.(c∇u+ αu) + β.∇u+ au = daλu (6)

where the coefficient values of PDE equa-

tion (Equation 6) are : c =~2

2me, α = 0,

a = V , da = 1.However, the solution is more accu-

rate when more mesh density is used,this means that more RAM memory is re-quired during the computations (Figure 1,Figure 2).

Figure 1: Mesh density of the hydrogen atommodel in 2D.

Figure 2: Mesh (6804786 Degree Of Freedoms(DOFs)) density of 3D the hydrogen atom

model.

However, to validate the model, a com-parison has been made with the 2D modelthat is located in the model library COM-SOL. To solve the problem in 3D, it is veryimportant to define the boundary condi-tions in COMSOL Multiphysics properly,


and also define the terms of the various co-efficients of eigenvalue PDE (Equation 6).

The resolution of the equation for thehydrogen atom shows different forms ofatomic orbitals.

The results on the Schrodinger’s equa-tion solution for a hydrogen atom havebeen made to an eigenvalue of −2 ×10−18[J ]. The solution gives the mini-mum value of the energy near to −2.180 ×10−18[J ] (0.50 [Hartree]) (≈ −2.176 ×10−18[J ] (0.499 [Hartree]) which is the en-ergy of the hydrogen atom in the groundstate) Figure 3 and Figure 4 .

Figure 3: Atomic orbital shape, eigenvalue andwave function for the ground state of the

hydrogen atom in 2D.

Figure 4: Atomic orbital shape, eigenvalue andwave function for the ground state of the

hydrogen atom in 3D.

The following result gives the wave

function and eigenvalue −5.450 × 10−19[J ](0.125 [Hartree])(≈ −5.44 × 10−19[J ] (0.125[Hartree]) energy levels of hydrogen atomfor n=2) for quantum number (n=2) for hy-drogen (Figure 5 and Figure 6).

Figure 5: Atomic orbital shape, eigenvalue andwave function of the hydrogen atom in 2D.


The following outcomes provide thewave function and eigenvalue 2.422 ×10−19[J ] (0.056 [Hartree]) (≈ 2.417 ×10−19[J ] (0.055 [Hartree]) energy levels ofhydrogen atom for n=3) for the quantumnumber (n=3) for hydrogen (Figure 7 andFigure 8).




3.2 The hydrogen molecularion H+

2

The hydrogen molecule ion is the simplestmolecule and the only molecule for whichwe can solve the electronic Schrodingerequation exactly [14, 15, 16, 17].

e

r1 r2

H1 r H2

Figure 9: Geometry of the H+2 .

The two protons are labeledH1 andH2, andthe distance from each proton to the elec-

tron e are r1 and r2, respectively. r is the dis-tance between H1 and H2. So, the Hamilto-nian for the hydrogen molecular ion with-out the Born-Oppenheimer approximationis given by:

H = − ~2

2M(∇2

1 +∇22)−

~2

2me∇2e

− e2

4πε0r1− e2

4πε0r2+

e2

4πε0r(7)

The Born-Oppenheimer shows that theHamiltonian for electronic motion given byEquation 8, because the mass of nuclei areso much greater than that of electrons, thatthe velocity of nuclei will be much smallerthan that of the electron.

H = − ~2

2me∇2e −

e2

4πε0r1− e2

4πε0r2+

e2

4πε0r(8)

The first term is the electronic kineticenergy, the two other terms are the elec-trostatic attractions of the electrons to eachnuclear, and the last term is the nu-clear electrostatic repulsion. Solving thisSchrodinger’s equation in the case of hy-drogen ion shows the influence of the dis-tance between the nuclei.

Figure 10: Atomic orbital shape, eigenvalueE = 124.788662 [Hartree] and wave function of

the hydrogen ion in 3D, r = 0.05 [A].

The Figure 10 show that, when r −→ 0the two nuclei close each other, thereforeshare the same electronic orbital.


Figure 11: Atomic orbital shape, eigenvalueE = 27.732068 [Hartree] and wave function of

the hydrogen ion in 3D, r = 0.1[A].

The above results (Figure 10 and Figure 11)show that when the distance betweenthe nuclei is very low the electron istrapped between the two cores, and inter-acts equiprobable manner.

Figure 12: Atomic orbital shape, eigenvalueE = −0.609516 [Hartree] and wave function of

the hydrogen ion in 3D, r = 1 [A].

The above result (Figure 12) shows thatthere is a distance where the interaction be-tween the electron and the nucleus becomesvery strong, at this distance the energy ofsystem is the lowest. In this case the elec-tron is in orbit around the two nuclei.


the hydrogen ion in 3D, r = 8 [A].


the hydrogen ion in 3D , r = 10 [A].

However, when the distance between thecores becomes very large, the electron or-bit is in between one of the two nuclei(Figure 13 and Figure 14).

3.2.1 Lennard-Jones Potential of H+2

The Lennard-Jones potential is function ofdistance between the centers of two parti-cles, and for the molecule hydrogen it isrepresent by Figure 15 and Figure 16. Theseresults have found numerically by PDE


method, the bonding energy has a mini-mum at Re = 1 [A] (Figure 16) and the ex-perimental value is 1.06 [A] [18, 16]. Whenthe internuclear separation r = Re, theenergy of system is lowest, and dissocia-tion energy ε = 3.02 [eV] and experimentalvalue is 2.78 [eV].

−500

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

Energy

in[eV]

0 1 2 3 4 5 6 7 8 9 10

Distance r in [A]

Energy curve versus r

Figure 15: Molecule hydrogen ion energy versusr (separation of two protons).

−20

−15

−10

−5

0

5

10

15

20

25

30

35

Energy

in[eV]

0 1 2 3 4 5 6 7 8

Distance r in [A]

σ

ǫ

Energy curve versus r

Figure 16: Lennard-Jones parameters ofhydrogen ion, σ = 0.44 [A], ε = 3.023 [eV].

4 Conclusions and outlooks

We have validate the 3D model of the hy-drogen atom by solving the Schrodinger’sequation and used this model to determinethe parameters of Lennard Jones in the caseof the molecule of hydrogen ion by finiteelements methods. Moreover, the effect ofthe distance between the protons are welldemonstrated that the mesh are not madevery well. The next problem to solve isto determine by exploiting the finite ele-ment method and COMSOL software, set-tings interactions potential Lennard-Jonesamong two molecules of hydrogen, and us-ing the approach used in the case Hydrogenmolecule ion.

Acknowledgements

The authors gratefully thank CaSciModOT(Calcul Scientifique et ModelisationOrleans Tours) facility for some numericalexperiments in the present paper.

References

[1] M.P. Allen and D.J. Tildesley.Computer simulation of liquids. Ox-ford science publications. ClarendonPress, 1989.

[2] Nota. Intermolecular parameters forsimple molecules. Port. Quim, 13:254–257, 1971.

[3] M. Oobatake and T. Ooi. Determina-tion of energy parameters in lennard-jones potential from second virial co-efficients. Progress of theoreticalphysics, 1972.

[4] R. A. Serway and J. W. Jemett. Physicsfor Scientists and Engineers withmodern physics. 2012.

[5] M. Elstner and T. Kubar. Biomolecularmodeling. Theoretical chemistry, 2011.

[6] Teik-Cheng Lim. Connection amongclassical interatomic potential func-tions. Journal of MathematicalChemistry, 36(3):261–269, 2004.

[7] A. L. Kantor, L. N. Long, and M. M.Micci. Molecular dynamics simu-lation of dissociation kinetics. ThePennsylvania State University, 2000.

[8] COMSOL AB.Comsol MultiPhysics: User’s guide,November 2008. COMSOL 3.5a.

[9] COMSOL AB.Comsol MultiPhysics: User’s guide,November 2010. COMSOL 4.1.

[10] A. Gotzias, H. Heiberg-Andersen,M. Kainourgiakis, and Th. Steriotis.Grand canonical monte carlo simula-tions of hydrogen adsorption in car-bon cones. Applied Surface Science,256(17):5226 – 5231, 2010.

[11] N. Zettili. Quantum Mechanics:Concepts and Applications. John Wi-ley & Sons, 2009.


[12] P. Le Floch. Introduction a la chimiemoleculaire par la theorie des orbitalesfrontieres. Polytechnique, 2006.

[13] Comsol MultiPhysics: Model Library.2008.

[14] A. Ishikawa, H. Nakashima, andH. Nakatsuji. Solving the schrodingerand dirac equations of hydrogenmolecular ion accurately by the freeiterative complement interactionmethod. Jounal of chemecal physics,128:,124103, 2008.

[15] E. Tuckerman. The hydrogen moleculeion. 2011.

[16] A. R. Rossi. Molecular structure. 2013.

[17] P. Manninen. 554017 AdvancedComputational Chemistry. 2009.

[18] M. D. Fayer. Molecular bonding, 2007.

[19] T. C. Lim. Relationship betweenmorse and murrell-mottram potentialsat long range. Journal of mathematicalchemistry, 2004.

Appendices

A Potential equations

A.1 Buckingham potential

The Buckingham (Equation 9) potential isa formula which describes the Pauli repul-sion energy and Van der Waals energy forthe interaction of two atoms that are di-rectly bonded as a function of the inter-atomic distance r.

UBK(rij) = A exp(−Brij)−C

r6ij(9)

Where A, B, C are constants. The fist term ofthis equation is a repulsion and the secondis an attraction. This equation in theoreticalstudy of equation of state for gaseous he-lium, neon and argon.

A.2 Rydberg potential

The Rydberg potential was proposed by Ry-dberg to evaluate the vibration spectrum of

diatomic molecules, the same properties areobtained with the Morse potential.

URyd(rij) = −γ[1 + a

(rij −RR

)]×

exp

(−a(rij −RR

))(10)

A.3 Biswas-Hamann potential

The Biswas-Hamann potential (Equation 11)have two distinct parts a repulsive term andan attractive terms.

UBH(rij) = A exp(−αrij) +B exp(−βrij)(11)

A.4 Murrel-Mottram

The Murrel-Mottram potential function ex-ist as a summation of 2-body and a 3-bodyterm [19]. For 2-body term the parame-ters γMM and a are determined by fittingphonon and elastic constant data, and thelattice energies and lattice constant of crys-talline phase, this equation displayed asEquation 12.

UMM(rij) = −γMM

[1 + a

(rij −RR

)]×

exp

(−a(rij −RR

))(12)

Nomenclature

Constants

ε01

µ0C2 Permittivity of free space8.8542× 10−12 in [C2/N.m2]

εij Parameter of Lennard-Jones in [J]

~ h2π 1.054571628× 10−34 in [J.s]

σij Distance at which the two particles areat equilibrium in [m]

e Charge on an electron 1.60218 × 10−19

in [C]

h Plank’s constant 6.62606896× 10−34 in[J.s]

M Mass of the nuclei in [Kg]

me Mass of electron 9.1094×10−31in [Kg]


Scalar quantities

ψ Wave function in [−]

E Energy eigenvalue in [J]

m Magnetic quantum number in [−]

rij Distance in [m]

ULJij (r) Lennard-Jones potential in [J]

V Potential energy [J]


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