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In continuation of our previous paper of the anharmonic potentials analysis through the Floquet representation, we performed in this work a systematic calculation of the diatomic vibrational energy levels as well as the corresponding wave functions. The solution of Schr
ödinger equation according to Morse potential, which is a suitable model to describe the diatomic vibrational spectra, has been introduced; thus the explicit formulas to the second order have been established. As an illustration, the dissociation energies of some molecules species (
*i*.
*e*. ScN, LiH, Cl
_{2} and NO) have been computed, as well as the wave functions and the corresponding probability densities, relating to the (ScN) molecule have been represented. Comparisons of our results with those of literature have been made.

In view of the importance that has the harmonic, pseudo-harmonic and anharmonic oscillators in various fields of physics, they have been extensively used for solving most of physical problems. To this end, significant efforts are devoted to determining the general formulas of vibration-rotation energy spectra and wave functions for the molecular systems. The simple harmonic oscillator is a typical model, and a major tool in understanding many physical phenomena. In particular, the introduction of this model in quantum mechanics theory plays a fundamental role in the mathematical formalism of this theory, and it is one of the exactly solvable potentials of the Schrödinger equation [

Our motivation is to continue exploring the principal formulas given in our previous paper of the analysis of anharmonic potentials through the Floquet representation [_{2} and NO), to find the finite number of bound states and the corresponding dissociation energy as well as the wave functions. Besides, to clarify and confirm our results, we have also made the calculations, according to the graphical Birge-Sponer plot [

The organization of the present paper is as follows: In Section 2 we give the outline of the theoretical context and the basic equations. In Section 3 we apply the analytic expressions to obtain numerical results and we give comparison. Section 4 presents the discussion. Section 5 contains the conclusions.

The anharmonic Morse potential is frequently used to describe the vibrational spectra of diatomic molecules. Its analytical expression is such as [

V ( q ) = D e ( 1 − e − ρ q ) 2 (1)

where D e is a parameter that controls the depth of the potential well, ρ is a parameter that measures the curvature of the potential well (width of attraction), and q is the variation of interatomic distance with respect to the equilibrium distance.

Let us do the Taylor development of the exponential term in V(q) to fourth order around q = 0 . Collecting the coefficients of the obtained series expansion gives,

V ( q ) = D e ρ 2 q 2 − D e ρ 3 q 3 + 7 12 D e ρ 4 q 4 (2)

The Morse oscillator Hamiltonian of a diatomic molecule of reduced mass m is given by,

H M = p 2 2 m + V ( q ) = p 2 2 m + D e ρ 2 q 2 − D e ρ 3 q 3 + 7 12 D e ρ 4 q 4 (3)

Let us make the changes of the constant factors in the following notations,

ω 0 = ρ 2 D e m (4a)

μ 1 = − D e ρ 3 m ω 0 2 ℏ m ω 0 (4b)

μ 2 = 7 12 D e ρ 4 ℏ m 2 ω 0 3 (4c)

where ω 0 is the vibrational constant of the diatomic molecule.

Then we find the anharmonic oscillator which Hamiltonian is such that,

H ( q ^ ) = p 2 2 m + 1 2 m ω 0 2 q 2 + μ 1 ℏ ω 0 q ^ 3 + μ 2 ℏ ω 0 q ^ 4 (5)

where q ^ = m ω 0 ℏ q and where μ 1 ℏ ω 0 q ^ 3 and μ 2 ℏ ω 0 q ^ 4 are the cubic and quadric anharmonic perturbations with amplitudes μ 1 and μ 2 respectively.

According to Equations (3) and (4), we have the similar situation given by the cubic and quadric anharmonic oscillators, for which the Hamiltonian is also given by Equation (5).

Using Equations (24) and (35) of our paper [

E ( 2 a ) n = ℏ ω 0 ( n + 1 2 ) + 7 16 D e ρ 4 ℏ 2 m 2 ω 0 2 ( 2 n 2 + 2 n + 1 ) − 15 4 D e 2 ρ 6 ℏ 2 m 3 ω 0 4 ( n 2 + n + 11 30 ) − 49 1152 D e 2 ρ 8 ℏ 3 m 4 ω 0 5 ( 34 n 3 + 51 n 2 + 59 n + 21 ) (6)

We emphasize that the Morse potential anharmonicity effect appears clearly to the second order calculation and that it affects the high vibrational energy levels.

Let us remark that,

2 n 2 + 2 n + 1 = 2 ( n + 1 2 ) + 1 2

( n 2 + n + 11 30 ) = ( n + 1 2 ) 2 + 7 60

and ( 34 n 3 + 51 n 2 + 59 n + 21 ) = 34 ( n + 1 2 ) 3 + 67 2 ( n + 1 2 )

Then Equation (6) can be written in term of ( n + 1 2 ) such as,

E ( 2 a ) n = ℏ ρ 2 D e m [ 1 − 3283 18432 ℏ 2 ρ 2 m D e ] ( n + 1 2 ) − ℏ 2 ρ 2 2 m ( n + 1 2 ) 2 − 833 2304 ℏ 3 ρ 3 m 2 m 2 D e ( n + 1 2 ) 3 (7)

Since the experiments data neglected the ( n + 1 2 ) 3 term, we can read Equation (7) as,

E ( 2 a ) n = ℏ ρ 2 D e m [ 1 − 3283 18432 ℏ 2 ρ 2 m D e ] ( n + 1 2 ) − ℏ 2 ρ 2 2 m ( n + 1 2 ) 2 (8)

Taking the derivative with respect to n of Equation (8) gives,

d E ( 2 a ) n d n = ℏ ρ 2 D e m [ 1 − 3283 18432 ℏ 2 ρ 2 m D e ] − ℏ 2 ρ 2 m ( n + 1 2 ) (9)

Equations (8) and (9) play the basic role in our analysis of the diatomic vibrational energy.

Let us note that the energies levels spacing will decrease with increasing values of n, therefore when d E ( 2 a ) n d n = 0 , the vibrational quantum number takes the

maximum value n max , i.e. the value for the vibrational quantum number where dissociation occurs, which allows us to determine the dissociation energy of the diatomic molecule.

Let us also note that the function d E ( 2 a ) n d n versus ( n + 1 2 ) decreases as a linear function of the variable ( n + 1 2 ) .

Using the expressions of the evolution operator, to the second order given by Equations (3) and (4b) [

ψ ( 2 a ) n ( q ^ ) = ( m ω 0 ℏ π ) 1 4 e − q ^ 2 2 2 n n ! ∑ j = − 8 n + 8 K n + j n H n + j ( q ^ ) (10)

where H n ( q ^ ) are the Hermite polynomials.

And where the different coefficients K n + j n are given by,

K n − 8 n = μ 2 2 32 n ! ( n − 8 ) !

K n − 7 n = 2 192 μ 1 μ 2 n ! ( n − 7 ) !

K n − 6 n = 1 2 [ μ 1 2 9 + ( n − 11 6 ) μ 2 2 2 ] n ! ( n − 6 ) !

K n − 5 n = 17 24 μ 1 μ 2 ( n − 104 85 ) n ! ( n − 5 ) !

K n − 4 n = 1 4 [ μ 2 + ( 2 n − 3 2 ) μ 1 2 + ( n − 1 ) ( 2 n − 7 ) μ 2 2 ] n ! ( n − 4 ) !

K n − 3 n = [ μ 1 3 + ( 21 n 2 − 74 n + 29 ) μ 1 μ 2 16 ] n ! ( n − 3 ) !

K n − 2 n = 1 2 [ ( 2 n − 1 ) μ 2 + ( 7 n 2 − 19 n + 1 ) μ 1 2 4 − ( 4 n 3 + 60 n 2 − 16 n + 15 ) μ 2 2 8 ] n ( n − 1 )

K n − 1 n = 1 2 [ 3 n μ 1 − ( 81 n 3 + 834 n 2 + 203 n + 274 ) μ 1 μ 2 48 ] n

K n n = 1 + μ 1 2 144 ( 2 n + 1 ) ( 82 n 2 + 82 n + 87 ) + 1 256 μ 2 2 ( 65 n 4 + 130 n 3 + 487 n 2 422 n + 156 )

K n + 1 n = 1 4 [ − 3 ( n + 1 ) μ 1 + ( − 81 n 3 + 591 n 2 + 1222 n + 816 ) μ 1 μ 2 48 ]

K n + 2 n = 1 8 [ − ( 2 n + 3 ) μ 2 + ( 7 n 2 + 33 n + 27 ) μ 1 2 4 + ( 4 n 3 + 48 n 2 + 142 n + 87 ) μ 2 2 8 ]

K n + 3 n = 1 8 [ − μ 1 3 + ( 21 n 2 + 116 n + 124 ) μ 1 μ 2 16 ]

K n + 4 n = 1 64 [ − μ 2 + ( 2 n + 7 2 ) μ 1 2 + ( 2 n 2 + 13 n + 13 ) μ 2 2 ]

K n + 5 n = 17 768 ( n + 108 85 ) μ 1 μ 2

K n + 6 n = 1 128 [ μ 1 2 9 + ( n + 17 6 ) μ 2 2 2 ]

K n + 7 n = μ 1 μ 2 1536

K n + 8 n = μ 2 2 8192

As illustration the first three states are given below.

Ground state (n = 0)

ψ ( 2 a ) 0 ( q ^ ) = 1 23040 ( m ω 0 ℏ π ) 1 4 e − q ^ 2 2 ∑ j = 0 8 C j 0 q ^ j (11)

C 0 0 = 45 ( 512 + 288 μ 2 − 272 μ 1 2 − 407 μ 2 2 ) , C 1 0 = − 720 ( 32 + 27 μ 2 ) μ 1

C 2 0 = 360 ( − 48 μ 2 + 88 μ 1 2 + 129 μ 2 2 ) , C 3 0 = 480 ( 261 μ 2 − 16 ) μ 1

C 4 0 = 120 ( − 48 μ 2 + 88 μ 1 2 − 81 μ 2 2 ) , C 5 0 = 576 μ 1 μ 2

C 6 0 = 160 ( 8 μ 1 2 + 39 μ 2 2 ) , C 7 0 = 1920 μ 1 μ 2

C 8 0 = 720 μ 2 2

First excited state (n = 1)

ψ ( 2 a ) 1 ( q ^ ) = 2 23040 ( m ω 0 ℏ π ) 1 4 e − q ^ 2 2 ∑ j = 0 9 C j 1 q ^ j (12)

C 0 1 = − 120 ( 2065 μ 2 − 384 ) μ 1 , C 1 1 = − 45 ( 1488 μ 1 2 + 3023 μ 2 2 − 1440 μ 2 − 512 )

C 2 1 = − 240 ( 833 μ 2 + 192 ) μ 1 , C 3 1 = 120 ( 568 μ 1 2 + 537 μ 2 2 − 240 μ 2 )

C 4 1 = 960 ( 207 μ 2 − 8 ) μ 1 , C 5 1 = 120 ( 152 μ 1 2 − 21 μ 2 2 − 48 μ 2 )

C 6 1 = 10176 μ 1 μ 2 , C 7 1 = 160 ( 8 μ 1 2 + 57 μ 2 2 )

C 8 1 = 1920 μ 1 μ 2 , C 9 1 = 720 μ 2 2

Second excited state (n = 2)

ψ ( 2 a ) 2 ( q ^ ) = 1 46080 ( m ω 0 ℏ π ) 1 4 e − q ^ 2 2 ∑ j = 0 10 C j 2 q ^ j (13)

C 0 2 = − 45 ( 2480 μ 1 2 + 5041 μ 2 2 + 1440 μ 2 + 512 ) , C 1 2 = − 240 ( 9349 μ 2 − 1632 ) μ 1

C 2 2 = − 90 ( 3632 μ 1 2 + 7139 μ 2 2 − 3936 μ 2 − 512 ) , C 3 2 = − 960 ( 1823 μ 2 + 136 ) μ 1

C 4 2 = 120 ( 1528 μ 1 2 + 75 μ 2 2 − 624 μ 2 ) , C 5 2 = 768 ( 663 μ 2 − 20 ) μ 1

C 6 2 = 80 ( 632 μ 1 2 − 141 μ 2 2 − 144 μ 2 ) , C 7 2 = 37632 μ 1 μ 2

C 8 2 = 80 ( 32 μ 1 2 + 29 μ 2 2 ) , C 9 2 = 3840 μ 1 μ 2

C 10 2 = 1440 μ 2 2

In attempt to illustrate the established Equations (8) and (9), we performed the numerical computation using the Maple software to obtain the values of the parameters, corresponding to the previously mentioned diatomic molecules.

Solving the following equation: d E ( 2 a ) n d n = 0 , round down to the nearest integer, leads to find the maximum vibrational quantum number n max associated with the highest bound state energy level such as,

n max = 2 D e m ℏ ρ − 3283 2 18432 ℏ ρ D e m − 1 2 (14)

Therefore, substituting the obtaiend value n max in Equation (8), gives the theoretical dissociation energy D e t h connected to the equilibrium position (i.e.the depth of the Morse potential well) of the molecule.

We have chosen the molecules (ScN, LiH, Cl_{2} and NO) whose parameters are given in

The calculated values of the maximum quantum numbers, the theoretical dissociation energy and the dissociation energy with respect to the zero point level corresponding to (ScN, LiH, Cl_{2} and NO) molecules are given in

Next, from Equation (9) we can compute the differences between any two successive energies levels: Δ E = E ( 2 a ) n − E ( 2 a ) n − 1 , which can be written as follows,

Δ E = d E ( 2 a ) n d n + ℏ 2 ρ 2 2 m = ℏ ρ 2 D e m [ 1 − 3283 18432 ℏ 2 ρ 2 m D e ] + ℏ 2 ρ 2 2 m − ℏ 2 ρ 2 m ( n + 1 2 ) (15)

Thence the graphics of the Birge-Sponer plots, corresponding to ScN, LiH, and Cl_{2} and NO molecules are given respectively in

In

and the probability densities of the wave-packet ( | ψ ( 2 a ) n ( q ^ ) | 2 ) in the cases where the number states are n = 0 and n = 5 respectively, for the (ScN) molecule.

Molecule | ρ ( 10 10 m − 1 ) | m(amu) | D e ( eV ) |
---|---|---|---|

ScN [ | 1.50680 | 10.682771 | 4.56 |

LiH [ | 1.128 | 0.8801221 | 2.515287 |

Cl_{2} [ | 2.0087 | 17.608328 | 2.513926 |

NO [ | 2.7534 | 7.521478 | 6.613502 |

Molecule | This work | Literature | |||
---|---|---|---|---|---|

n max | D e t h ( eV ) | D 0 t h ( eV ) | n max | D e ( eV ) | |

ScN | 101 | 4.5596 | 4.5147 | --- | 4.56 [ |

LiH | 28 | 2.5128 | 2.426 | 29 [ | 2.515287 [ |

2.515267 [ | |||||

Cl_{2} | 72 | 2.5136 | 2.4790 | 72 [ | 2.513926 [ |

NO | 55 | 6.6115 | 6.4940 | 55 [ | 6.613502 [ |

The main goal of our paper is to give an alternative way, to solve the quantum anharmonic oscillator problem [

The second order calculation of vibrational energy levels according to the Morse oscillator where performed in Equations (6) and (7). Therefore, Equation (7) is used for the numerical calculation, based on the parameter values of the molecules selected from

Note that the graphs of Δ E versus ( n + 1 2 ) of Equation (15) are lines with negative slopes. Therefore the spacing Δ E between adjacent vibrational energy levels is a decreasing linear function of the variable ( n + 1 2 ) . Thus the plots of

The value of ( n max + 1 2 ) is the intersection of the curve with the abscissa axis, as well as integral of Δ E over ( n + 1 2 ) from ( n = 0 to n = n max ) to obtain the area under the curve, gives the dissociation energy D 0 . Then we can compute D e t h = D 0 + E ( 2 a ) 0 .

In this work, we performed the calculations of the vibrational energy levels as well as the wave functions of the diatomic molecules, according to a systematic approach. Our computations were carried out using the parameters of the Morse potential of some molecules, namely (ScN, LiH, Cl_{2} and NO). Then, with the help of the Maple software, we have determined the value of the maximum vibrational quantum number ( n max ) and the theoretical dissociation energy D e t h corresponding to the previous mentioned molecules.

Moreover, we also represented in

function of the variable ( n + 1 2 ) . This method allowed us as well to find again

the values: n max and D e t h of these molecules. From

We project in futures works to improve the calculations, where V(q) will be developed with more higher power terms, and we will study the ro-vibrational energy of diatomic molecules, according to the Deng-Fan potential.

Finally, we would like to point out that our study could be a useful tool for future research students to understand the quantum dynamics of anharmonic systems.

The authors declare no conflicts of interest regarding the publication of this paper.

Idrissi, M.J., Fedoul, A. and Sayouri, S. (2020) Systematic Approach to Compute the Vibrational Energy Levels of Diatomic Molecules. Journal of Applied Mathematics and Physics, 8, 2463-2474. https://doi.org/10.4236/jamp.2020.811182^{ }

V ( q ) : Morse potential

D e : Dissociation energy

ρ : Parameter that controls the width of attraction

H M : Morse oscillator Hamiltonian

m: Reduced mass of the diatomic system

ω 0 : Unperturbed oscillator frequency

p: Impulsion operator

q: Position operator

ℏ : Reduced Planck's constant

E ( 2 a ) n : Quasi-energy to second order

n: Quantum number

n max : Maximum quantum number

ψ ( 2 a ) n ( q ^ ) : Wave-functions to second order

H n ( q ^ ) : Hermite polynomials