# Solution of the Rovibrational Schrödinger Equation of a Molecule Using the Volterra Integral Equation

Authors: Mahmoud Korek; Nayla El-Kork
Journal: Advances in Physical Chemistry
 (2018)

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## Abstract

By using the Rayleigh-Schrödinger perturbation theory the rovibrational wave function is expanded in terms of the series of functionsϕ0,ϕ1,ϕ2,…ϕn, where ϕ0 is the pure vibrational wave function and ϕι are the rotational harmonics. By replacing the Schrödinger differential equation by the Volterra integral equation the two canonical functions α0 and β0 are well defined for a given potential function. These functions allow the determination of (i) the values of the functions ϕι at any points; (ii) the eigenvalues of the eigenvalue equations of the functions ϕ0,ϕ1,ϕ2,…ϕn which are, respectively, the vibrational energy Ev, the rotational constant Bv, and the large order centrifugal distortion constants Dv,Hv,Lv….. Based on these canonical functions and in the Born-Oppenheimer approximation these constants can be obtained with accurate estimates for the low and high excited electronic state and for any values of the vibrational and rotational quantum numbers v and J even near dissociation. As application, the calculations have been done for the potential energy curves: Morse, Lenard Jones, Reidberg-Klein-Rees (RKR), ab initio, Simon-Parr-Finlin, Kratzer, and Dunhum with a variable step for the empirical potentials. A program is available for these calculations free of charge with the corresponding author.


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## Body

## 1. Introduction

For the gas-phase of molecules, the rovibrational spectroscopy remains the most accurate and reliable source for understanding the molecular structure and the development of quantum mechanics. The theoretical determination of the rovibrational constants enables the prediction of line positions, which can be used in guiding experimental investigations and greatly facilitating the detection of unknown molecules. The fundamental atomic and molecular data are very important for interpreting observations and modelling the atmospheres of planetary and stellar objects. In the last two decades hundreds of extrasolar planets, cool stars, the atmospheres of exoplanets, and brown dwarfs have been detected where their other proposal uses molecular states.Since diatomic molecules exhibit very long decoherence times, the molecular electronic states (electronic, vibrational, and/or rotational) are chosen recently to act as the quantum bits (qubits). A femtosecond laser pulse can be shaped to produce high fidelity binary shaped laser pulses to act as a quantum logic gate on the chosen qubits. By using the vibrational states of a linear molecule of N atoms as the qubit basis the number of vibrational degrees of freedom is 3N – 6. A huge quantity of spectroscopic data is required to implement experiments using shaped laser pulses for molecular quantum computing. However, these data are still not available in literature.The quantum mechanical canonical function method is a powerful technique to compute the very large rotation–vibration constants. The vibration rotation energy of an electronic state of a diatomic molecule is commonly represented byEvJ=Ev+λBv+λ2Dv+λ3Hv+λ4Lv+λ5Mv…, where λ=J(J+1), v and J are, respectively, the vibrational and rotational quantum numbers, Ev is the pure vibrational energy, Bv the rotational constant, and Dv,Hv,Lv+… are the centrifugal distortion constants (CDC) related to a potential energy curve U(r) and v. In the conventional approach of the Raleigh-Schrödinger perturbation theory (RSPT) [1] Albritton et al. [2] derived the first analytical expressions of the centrifugal distortion constants (CDC). Because of the complexity and the tedious computation of these expressions, Hutson derived, by using the Numerov difference equation, an algorithm [3] that allows the determination of the constants Dv, Hv, Lv, and Mv only in terms of the vibrational wave function ψv. However, difficulties appeared in this algorithm for some potentials (like the Lennard-Jones potential) in treating high vibrational level (near dissociation for example) [4]. Because of the development in spectroscopic techniques, higher orders are needed [5]. Tellinghuisen [4] introduced an improvement to Huston algorithm, but it was not sufficient to accede these levels and reach large order of the centrifugal distortion constants.In this work, we present the canonical function method (superposition of difference equation) which is an advantageous technique with respect to other difference equation methods for solving the vibration-rotation eigenvalue problem [6]. Moreover, with this canonical approach with variable step [7], the large order of CDC and the calculation of the vibration-rotation energy levels, even near dissociation [8], for any potential energy curves (either RKR [9], ab initio [10–12], or empirical [13]) and for any electronic state can be done by one single and simple routine. A program is available for these calculations free of charge with the corresponding author, with applications on the six potential energy curves: Morse potential, Lenard Jones potential, RKR potential, ab initio potential, Simon-Parr-Finlin (Kratzer) potential, and Dunhum potential. Since the canonical function method is proven to be valid for any type of potential function, the program is structured in different subroutines for the different cited potentials in order to give the possibility to use the same program for other types of potentials that can be added.


## 2. The Theory

### 2.1. Vibration-Rotation Canonical Functions

In the Born-Oppenheimer approximation, the vibration-rotation motion of a diatomic molecule is described by the wave functionΨvJ and the energy EvJ that are, respectively, the eigenfunction and the eigenvalue of the radial Schrödinger equation [18](1)d2ΨvJdr2+kEvJ-Vr-JJ+1r2ΨvJr=0where k=2μ/ħ2, μ is the reduced mass, ħ is the Planck constant, v and J are the vibrational and rotational quantum numbers, respectively, and r is the internuclear distance. This equation can be simply represented by(2)Ψ′′vJx=fvJxΨvJx,where x=r-re(re) is the value of r at the equilibrium, and(3)fvJx=-kE-Ux+λx+re2with λ=JJ+1. Equation (1) is equivalent to the Voltera integral equation [19](4)ΨvJx=ΨvJ0+x′Ψλ′0+∫0xx-tfvJtΨvJtdtin the sense that any solution of (4) is solution of (1). By inserting ΨvJ(x) by its expression many times in the integral, one can find:(5)ΨvJx=ΨvJ0αx+Ψ′vJ0βxwhere αx and βx are two canonical functions [20, 21] defined by6-1αx=∑i=0∞AixwithAi=∫0xx-tfvJtAi-1tdtandA0x=16-2βx=∑i=0∞BixwithBix=∫0xx-tfvJtBi-1tdtandB0x=xwith the well determined initial values7α0=1,α′0=0β0=0,β′0=1The initial value Ψ′vJ0 for the unnormalized wave function ΨvJx can be deduced from αx and βx by using (5), on one hand, and on the other hand the boundary conditions [5]8Ψx→∞x→-revJx→0We find9Ψ′vJ0ΨvJ0=limx→∞⁡-αxβxΨ′vJ0ΨvJ0=limx→-re⁡-αxβx


### 2.2. The Rotational Schrödinger Equations

In the Rayleigh-Schrödinger perturbation theory (RSPT), the eigenvalue and the eigenfunction of (1) are given, respectively, by10Eλ=∑n=o∞enλn11ΨvJx=∑n=0∞Φnλnwhere e0=Ev is the pure vibrational energy, e1=Bv is the rotational constant, e2=-Dv, e3=Hv,… are the centrifugal distortion constants, φ0 is the pure vibrational wave function, and φi is the rotational correction.Therefore, the energy factorfvJx can be written as12fvλx=∑n=0∞εixλiwhere13-1ε0=-ke0-Ux13-2ε1=-ke1+1r0+x2⋯13-nεn=kenfori≥2From this expression of fvJx, the functions α(r) and β(r) are given by 14αr=∑n=0∞anr-rin15βr=∑n=0∞bnr-rinBy replacing αvJx, βvJx by their expressions 14 and 15 in (5) we find16ΨvJx=Ψv+∑n=1∞φnxλnA set of Schrödinger equations is obtained by replacing the wave function ΨvJx by its expression 16 in (2) to obtain a set of Schrödinger equations17-0φ′′0x-ε0xφ0x=017-1φ′′1x-ε0xφ1x=ε1xφ0x17-2φ′′2x-ε0xφ2x=ε1xφ1x+ε2xφ0x⋯17-nφ′′nx-ε0xφnx=∑m=1nεmxφn-mxwhere the first of these equations is the radial Schrödinger equation of pure vibration. All the other equations are called the rotational Schrödinger equations.


### 2.3. Analytic Expressions of the Rotation Harmonics

#### 2.3.1. Pure VibrationΨvx

For one electronic state and for a given potential function, the solution of the vibrational Schrödinger equation17-0 is given by [21]18Ψvx=Ψv0αvx+Ψ′v0βvxwhere αvx and βvx are the pure vibration canonical functions defined by (5) and 7 in which we replace fvJx by fvx (i.e., we take J=0).


#### 2.3.2. Calculation of the Rotational HarmonicsΦnx

A rotational Schrödinger equation17-0, 17-1, 17-2, 17-n is given by19φ′′nx-ε0xφnx=∑m=1nεmxφn-mxIf we multiply this equation by x-t and integrate it between zero and x, it will become20φnx=φn0+φ′n0x+∑m=1n∫0xx-tεmtφn-mtdtBy replacing εmt by its expression 13-1, 13-2, 13-n in 17-0, 17-1, 17-2, 17-n we obtain21φnx=φn0αvx+φ′0βvx+σnxwhere22σnx=∑m=0∞kmnx23-0k0nx=∫0xx-tεntφ0tdt23-1k1nx=∫0xx-t∫0tt-t1εntφ0tdtdt23-2kmn=∫0xx-tεntkm-1tdtfor x = 024kon0=k1n0=⋯=kmn0=⋯=0k0n′0=k1n′0=⋯=kmn′0=⋯=0Therefore25σn0=σ′n0=0for the derivation of order i26d′kmn0dx′=0fori≥2and27d′σn0dx′≠0For the unnormalized wave function, we chose Ψv0=φ00=1, and by using 16 we obtain28φn0=0By replacing φn0 by its value in 21 we obtain29φnx=φ′n0βvx+σnxOn the other hand, the rotation harmonics must be vanished at the boundaries 8; thus 29 gives30-σnxβvx=lnand the rotation harmonic φnx is given by [7]31φnx=σnx+lnβvxThis expression is valid without any restriction for the form of a given potential function.


### 2.4. Numerical Method

#### 2.4.1. Calculation of the Vibrational Wave FunctionΦ0r

For one electronic state and for a given potential function, the vibrational wave function is given by32φ0x=φ00αvx+φ′00βvxThe determination of φ0x requires the calculation of αvx, βvx, and φ′0x.(1) Calculation of α v x and β v x. For a given potential, and on one interval, Ii=ri,ri+1 has the polynomial form33Ur=∑n=0Nγnir-rinThe canonical functions αx and βx are particular solutions of 17-0, because Ux is expanded in polynomial [8]; αx and βx also can be expanded as34αr=∑n=0∞anr-rin35βr=∑n=0∞bnr-rinBy representing αr and βr by the same function y(r) for a given potential U(r) and energy E, the function y(r) is given by36yr=∑n=0∞Cnir-rinBy using 17-0, we obtain the following recursion relation:37n+2n+1Cn+2i=-kECni+k∑m=0nCmiγn-miwith38C0i=yri,C1i=yriThe initial values yr1 and y′ri are given by39yri=∑n=0∞Cni-1ri-ri-1n40y′ri=∑n=0∞nCni-1ri-ri-1n-1where y(0) = 1, y′(0)=0 for α function and y(0) = 0, y′(0)=1 for β functionTherefore, the canonical functionsαr and βr are well determined at any point r.(2) Calculation of Φ ′ 0. From equation (5), the wave function φ0x is given by41φ0r=αr+φ′00βrBy using the boundary conditions 8 one can find42φ′00φ00=limr→∞r→-re⁡-αrβr=limr→∞r→-re⁡α′rβ′rFor the unnormalized wave function φ00=1, the vibration function φ0r is determined at any point r.


### 2.5. Diatomic Centrifugal Distortion Constants (CDC)

The rotational equations17-0, 17-1, 17-2, and 17-n are all of the form43z′′+ke0-vz=SrWe multiply this equation by φ0 and integrate between r0 and ∞. By making use of 17-0, 17-1, 17-2, and 17-n, we obtain44∫r0∞z′′φ0-zφ′′0dr=∫r0∞srφ0rdrThen we make use of the boundary conditions for φ0 and z (at r≈∞) on one hand, and of 8 on the other hand, and we find45-1z′r0=-∫r0∞srφ0rdrand similarly, for the other boundary condition (at r≈0),45-2z′r0=-∫r00srφ0rdrThe continuity condition for s(r) implies the equality of z′r0 given by 44; therefore46-∫0∞srφ0rdr=-∫r00srφ0rdr,∫0∞srφ0rdr=s∣φ0=0This equation gives for the successive value of s 47-0φ0∣e1-R∣φ0=047-1φ0∣e1-R∣φ1+e2φ0∣φ2=0⋯47-nφ0∣e1-R∣φn-1+∑m=2nemφ0∣φn-m=0.These equations give simple expression of e1,e2,…,en in terms of φ0,φ1,…,φn-148-1I0e1=R048-2I0e2=R1-e1I148-3I0e3=R2-e1I2-e2I1⋯48-nI0en=Rn-1-∑m=1n-1emIn-mwhere49In=φ0∣φn,Rn=φ0∣R∣φnOnce e0=Ev is given, the determination of e1,e2,… is reduced to that of simple definite integrals I0,I1,I2,… and R0,R1,R2,… depending on φ0,φ1,φ2,…. [19].


## 2.1. Vibration-Rotation Canonical Functions

In the Born-Oppenheimer approximation, the vibration-rotation motion of a diatomic molecule is described by the wave functionΨvJ and the energy EvJ that are, respectively, the eigenfunction and the eigenvalue of the radial Schrödinger equation [18](1)d2ΨvJdr2+kEvJ-Vr-JJ+1r2ΨvJr=0where k=2μ/ħ2, μ is the reduced mass, ħ is the Planck constant, v and J are the vibrational and rotational quantum numbers, respectively, and r is the internuclear distance. This equation can be simply represented by(2)Ψ′′vJx=fvJxΨvJx,where x=r-re(re) is the value of r at the equilibrium, and(3)fvJx=-kE-Ux+λx+re2with λ=JJ+1. Equation (1) is equivalent to the Voltera integral equation [19](4)ΨvJx=ΨvJ0+x′Ψλ′0+∫0xx-tfvJtΨvJtdtin the sense that any solution of (4) is solution of (1). By inserting ΨvJ(x) by its expression many times in the integral, one can find:(5)ΨvJx=ΨvJ0αx+Ψ′vJ0βxwhere αx and βx are two canonical functions [20, 21] defined by6-1αx=∑i=0∞AixwithAi=∫0xx-tfvJtAi-1tdtandA0x=16-2βx=∑i=0∞BixwithBix=∫0xx-tfvJtBi-1tdtandB0x=xwith the well determined initial values7α0=1,α′0=0β0=0,β′0=1The initial value Ψ′vJ0 for the unnormalized wave function ΨvJx can be deduced from αx and βx by using (5), on one hand, and on the other hand the boundary conditions [5]8Ψx→∞x→-revJx→0We find9Ψ′vJ0ΨvJ0=limx→∞⁡-αxβxΨ′vJ0ΨvJ0=limx→-re⁡-αxβx


## 2.2. The Rotational Schrödinger Equations

In the Rayleigh-Schrödinger perturbation theory (RSPT), the eigenvalue and the eigenfunction of (1) are given, respectively, by10Eλ=∑n=o∞enλn11ΨvJx=∑n=0∞Φnλnwhere e0=Ev is the pure vibrational energy, e1=Bv is the rotational constant, e2=-Dv, e3=Hv,… are the centrifugal distortion constants, φ0 is the pure vibrational wave function, and φi is the rotational correction.Therefore, the energy factorfvJx can be written as12fvλx=∑n=0∞εixλiwhere13-1ε0=-ke0-Ux13-2ε1=-ke1+1r0+x2⋯13-nεn=kenfori≥2From this expression of fvJx, the functions α(r) and β(r) are given by 14αr=∑n=0∞anr-rin15βr=∑n=0∞bnr-rinBy replacing αvJx, βvJx by their expressions 14 and 15 in (5) we find16ΨvJx=Ψv+∑n=1∞φnxλnA set of Schrödinger equations is obtained by replacing the wave function ΨvJx by its expression 16 in (2) to obtain a set of Schrödinger equations17-0φ′′0x-ε0xφ0x=017-1φ′′1x-ε0xφ1x=ε1xφ0x17-2φ′′2x-ε0xφ2x=ε1xφ1x+ε2xφ0x⋯17-nφ′′nx-ε0xφnx=∑m=1nεmxφn-mxwhere the first of these equations is the radial Schrödinger equation of pure vibration. All the other equations are called the rotational Schrödinger equations.


## 2.3. Analytic Expressions of the Rotation Harmonics

### 2.3.1. Pure VibrationΨvx

For one electronic state and for a given potential function, the solution of the vibrational Schrödinger equation17-0 is given by [21]18Ψvx=Ψv0αvx+Ψ′v0βvxwhere αvx and βvx are the pure vibration canonical functions defined by (5) and 7 in which we replace fvJx by fvx (i.e., we take J=0).


### 2.3.2. Calculation of the Rotational HarmonicsΦnx

A rotational Schrödinger equation17-0, 17-1, 17-2, 17-n is given by19φ′′nx-ε0xφnx=∑m=1nεmxφn-mxIf we multiply this equation by x-t and integrate it between zero and x, it will become20φnx=φn0+φ′n0x+∑m=1n∫0xx-tεmtφn-mtdtBy replacing εmt by its expression 13-1, 13-2, 13-n in 17-0, 17-1, 17-2, 17-n we obtain21φnx=φn0αvx+φ′0βvx+σnxwhere22σnx=∑m=0∞kmnx23-0k0nx=∫0xx-tεntφ0tdt23-1k1nx=∫0xx-t∫0tt-t1εntφ0tdtdt23-2kmn=∫0xx-tεntkm-1tdtfor x = 024kon0=k1n0=⋯=kmn0=⋯=0k0n′0=k1n′0=⋯=kmn′0=⋯=0Therefore25σn0=σ′n0=0for the derivation of order i26d′kmn0dx′=0fori≥2and27d′σn0dx′≠0For the unnormalized wave function, we chose Ψv0=φ00=1, and by using 16 we obtain28φn0=0By replacing φn0 by its value in 21 we obtain29φnx=φ′n0βvx+σnxOn the other hand, the rotation harmonics must be vanished at the boundaries 8; thus 29 gives30-σnxβvx=lnand the rotation harmonic φnx is given by [7]31φnx=σnx+lnβvxThis expression is valid without any restriction for the form of a given potential function.


## 2.3.1. Pure VibrationΨvx

For one electronic state and for a given potential function, the solution of the vibrational Schrödinger equation17-0 is given by [21]18Ψvx=Ψv0αvx+Ψ′v0βvxwhere αvx and βvx are the pure vibration canonical functions defined by (5) and 7 in which we replace fvJx by fvx (i.e., we take J=0).


## 2.3.2. Calculation of the Rotational HarmonicsΦnx

A rotational Schrödinger equation17-0, 17-1, 17-2, 17-n is given by19φ′′nx-ε0xφnx=∑m=1nεmxφn-mxIf we multiply this equation by x-t and integrate it between zero and x, it will become20φnx=φn0+φ′n0x+∑m=1n∫0xx-tεmtφn-mtdtBy replacing εmt by its expression 13-1, 13-2, 13-n in 17-0, 17-1, 17-2, 17-n we obtain21φnx=φn0αvx+φ′0βvx+σnxwhere22σnx=∑m=0∞kmnx23-0k0nx=∫0xx-tεntφ0tdt23-1k1nx=∫0xx-t∫0tt-t1εntφ0tdtdt23-2kmn=∫0xx-tεntkm-1tdtfor x = 024kon0=k1n0=⋯=kmn0=⋯=0k0n′0=k1n′0=⋯=kmn′0=⋯=0Therefore25σn0=σ′n0=0for the derivation of order i26d′kmn0dx′=0fori≥2and27d′σn0dx′≠0For the unnormalized wave function, we chose Ψv0=φ00=1, and by using 16 we obtain28φn0=0By replacing φn0 by its value in 21 we obtain29φnx=φ′n0βvx+σnxOn the other hand, the rotation harmonics must be vanished at the boundaries 8; thus 29 gives30-σnxβvx=lnand the rotation harmonic φnx is given by [7]31φnx=σnx+lnβvxThis expression is valid without any restriction for the form of a given potential function.


## 2.4. Numerical Method

### 2.4.1. Calculation of the Vibrational Wave FunctionΦ0r

For one electronic state and for a given potential function, the vibrational wave function is given by32φ0x=φ00αvx+φ′00βvxThe determination of φ0x requires the calculation of αvx, βvx, and φ′0x.(1) Calculation of α v x and β v x. For a given potential, and on one interval, Ii=ri,ri+1 has the polynomial form33Ur=∑n=0Nγnir-rinThe canonical functions αx and βx are particular solutions of 17-0, because Ux is expanded in polynomial [8]; αx and βx also can be expanded as34αr=∑n=0∞anr-rin35βr=∑n=0∞bnr-rinBy representing αr and βr by the same function y(r) for a given potential U(r) and energy E, the function y(r) is given by36yr=∑n=0∞Cnir-rinBy using 17-0, we obtain the following recursion relation:37n+2n+1Cn+2i=-kECni+k∑m=0nCmiγn-miwith38C0i=yri,C1i=yriThe initial values yr1 and y′ri are given by39yri=∑n=0∞Cni-1ri-ri-1n40y′ri=∑n=0∞nCni-1ri-ri-1n-1where y(0) = 1, y′(0)=0 for α function and y(0) = 0, y′(0)=1 for β functionTherefore, the canonical functionsαr and βr are well determined at any point r.(2) Calculation of Φ ′ 0. From equation (5), the wave function φ0x is given by41φ0r=αr+φ′00βrBy using the boundary conditions 8 one can find42φ′00φ00=limr→∞r→-re⁡-αrβr=limr→∞r→-re⁡α′rβ′rFor the unnormalized wave function φ00=1, the vibration function φ0r is determined at any point r.


## 2.4.1. Calculation of the Vibrational Wave FunctionΦ0r

For one electronic state and for a given potential function, the vibrational wave function is given by32φ0x=φ00αvx+φ′00βvxThe determination of φ0x requires the calculation of αvx, βvx, and φ′0x.(1) Calculation of α v x and β v x. For a given potential, and on one interval, Ii=ri,ri+1 has the polynomial form33Ur=∑n=0Nγnir-rinThe canonical functions αx and βx are particular solutions of 17-0, because Ux is expanded in polynomial [8]; αx and βx also can be expanded as34αr=∑n=0∞anr-rin35βr=∑n=0∞bnr-rinBy representing αr and βr by the same function y(r) for a given potential U(r) and energy E, the function y(r) is given by36yr=∑n=0∞Cnir-rinBy using 17-0, we obtain the following recursion relation:37n+2n+1Cn+2i=-kECni+k∑m=0nCmiγn-miwith38C0i=yri,C1i=yriThe initial values yr1 and y′ri are given by39yri=∑n=0∞Cni-1ri-ri-1n40y′ri=∑n=0∞nCni-1ri-ri-1n-1where y(0) = 1, y′(0)=0 for α function and y(0) = 0, y′(0)=1 for β functionTherefore, the canonical functionsαr and βr are well determined at any point r.(2) Calculation of Φ ′ 0. From equation (5), the wave function φ0x is given by41φ0r=αr+φ′00βrBy using the boundary conditions 8 one can find42φ′00φ00=limr→∞r→-re⁡-αrβr=limr→∞r→-re⁡α′rβ′rFor the unnormalized wave function φ00=1, the vibration function φ0r is determined at any point r.


## 2.5. Diatomic Centrifugal Distortion Constants (CDC)

The rotational equations17-0, 17-1, 17-2, and 17-n are all of the form43z′′+ke0-vz=SrWe multiply this equation by φ0 and integrate between r0 and ∞. By making use of 17-0, 17-1, 17-2, and 17-n, we obtain44∫r0∞z′′φ0-zφ′′0dr=∫r0∞srφ0rdrThen we make use of the boundary conditions for φ0 and z (at r≈∞) on one hand, and of 8 on the other hand, and we find45-1z′r0=-∫r0∞srφ0rdrand similarly, for the other boundary condition (at r≈0),45-2z′r0=-∫r00srφ0rdrThe continuity condition for s(r) implies the equality of z′r0 given by 44; therefore46-∫0∞srφ0rdr=-∫r00srφ0rdr,∫0∞srφ0rdr=s∣φ0=0This equation gives for the successive value of s 47-0φ0∣e1-R∣φ0=047-1φ0∣e1-R∣φ1+e2φ0∣φ2=0⋯47-nφ0∣e1-R∣φn-1+∑m=2nemφ0∣φn-m=0.These equations give simple expression of e1,e2,…,en in terms of φ0,φ1,…,φn-148-1I0e1=R048-2I0e2=R1-e1I148-3I0e3=R2-e1I2-e2I1⋯48-nI0en=Rn-1-∑m=1n-1emIn-mwhere49In=φ0∣φn,Rn=φ0∣R∣φnOnce e0=Ev is given, the determination of e1,e2,… is reduced to that of simple definite integrals I0,I1,I2,… and R0,R1,R2,… depending on φ0,φ1,φ2,…. [19].


## 3. Results and Discussion

### 3.1. Calculated Data

In order to test the accuracy and the validity of the present method for the calculation of the different rovibrational parameters we used as examples for the empirical potentials Morse potential [14, 15], Lenard Jones potential [4, 8, 22], Dunhum Potential [23], and Kratzer potential [24]. For the investigation of the ab initio potential [17], we employed the state averaged Complete Active Space Self Consistent Field (CASSCF) followed by Multireference Configuration Interaction (MRCI) method with Davidson correction (+Q), single and double excitations. In these entire calculations, we used the computational chemistry program MOLPRO [25] with the advantage of the graphical user interface GABEDIT [26]. The CASSCF configuration space was used as the reference in the MRCI calculations. The potential energy U(r) is determined in terms of the internuclear distance r with a step equal to 0.02Å. The Schrödinger equation is solved by using the cubic spline interpolation between every two consecutive points. A Rydberg-Klein-Rees (RKR) potential is obtained experimentally and defined by their turning points [27–29]. The calculation has been done by an interpolation between every two consecutive points also by using the cubic spline interpolation. If the number of the turning points is limited we extrapolate the potential function curve to the right by [30] U(x) = a/r6 +b/r8 +c/r10 and to the left by U(x) = d/r15 where the constants a, b, c can be obtained from literature (if available) and d is calculated using the used potential energy curve.The comparison of our calculated values of the vibrational energy levelsEv, by using the empirical Morse potential for the molecule CO, with those given in [14, 15] shows, respectively, excellent agreements with the relative differences 0.00%<ΔEv/Ev<1.25% and 0.00%<ΔEv/Ev<0.03% (Table 1). Similar excellent agreement ΔEv/Ev = 0.00% is obtained by comparing our calculated values of the different vibrational levels Ev with those given by Martin et al. [16] of the RKR potential of the molecule I2 (Table 2). Our method is also tested in case of anab initio potential for the molecule ZnBr [17] (Table 3). Recently [31], we calculated, by using the present method, the rovibrational parameters Ev, Bv, Dv of the molecule LiSr by using anab initio potential. The comparison of the calculated values of Ev and Bv with those available in literature shows the good agreement with the relative differences 0.1%<ΔEv/Ev<7.5% for Ev and acceptable agreement 10.9%<ΔBv/Bv<12.9% for Bv. However, the comparison of our calculated values of Bv, by the present method, with those given in literature shows a very good agreement with the relative difference 1.1%<ΔBv/Bv<2.8% for the molecule NaSr [31].Table 1
The energy eigenvaluesEv, the rotational constants Bv, and the centrifugal distortion constants Dv, Hv, Lv of the pure vibrational energy levels of the Morse potential of CO molecule.
V E v cm−1 Δ E v / E v B v cm−1 D v  × 106 cm−1 H v  × 1012 cm−1 L v  × 1017 cm−1 0 1081.78a  1.9225 6.11954 5.8008 3.6445 1081.61b 0.02 1081.5857c 0.02  1 3225.05  1.9050 6.11885 5.6556 3.7160 3225.05 0.00 3224.8568 0.03  2 5341.84  1.8875 6.11849 5.5100 3.7848 5342.11 0.00 5341.6478 0.00  3 7432.22  1.8700 6.11844 5.3659 3.8409 7432.79 0.01 7432.0247 0.00  4 9496.25  1.8525 6.11863 5.2272 3.8678 9497.09 0.18 9496.0550 0.00  5 11534.03  1.8350 6.11892 5.1007 3.8397 11535.01 0.01 11533.80787 0.00  6 13545.61  1.8175 6.11910 4.9970 3.7184 13546.56 0.00 13545.3543 0.00  7 15531.12  1.8001 6.11884 4.9323 3.4491 15531.72 0.00 15530.767 0.00  8 17490.66  1.7826 6.11761 4.9298 2.9556 17490.1202 0.00  9 19424.39  1.7652 6.11467 5.0224 2.2224 19423.4901 0.00  10 21332.48  1.7478 6.10895 5.2545 0.8581 21330.9539 0.01  11 23195.48  1.7353 6.3063 5.1176 7.4807 23212.5907 0.07  12 25046.37  1.7179 6.3265 4.8608 7.7559 25068.4807 0.09  13 26870.69  1.7004 6.3476 4.5905 8.0499 26898.7056 0.10  14 28668.43  1.6829 6.3695 4.3061 8.3641 28703.3478 0.00  15 30439.59  1.6653 6.3922 4.0066 8.7004 30482.4908 0.14  16 32184.18  1.6476 6.4159 3.6911 9.0606 32236.2189 0.16  17 33902.19  1.6299 6.4405 3.3586 9.4470 33964.6167 0.18  18 35593.63  1.6121 6.4661 3.0081 9.8617 35667.7691 0.21  19 37258.49  1.5943 6.4927 2.6384 10.307 37345.7608 0.23  20 38896.77  1.5764 6.5204 2.2483 10.787 38998.6766 0.26  21 40508.47 0.29 1.5584 6.5493 1.8363 11.303 40626.6001  22 42093.6  1.5403 6.5794 1.4011 11.859 42229.6144 0.32  23 43652.15  1.5221 6.6108 9.4096 12.460 43807.8009 0.36  24 45184.13  1.5039 6.6435 4.5424 13.109 45361.2396 0.39  25 46689.52  1.4856 6.6775 6.0978 13.812 46890.0081 0.43  26 48168.35  1.4672 6.7131 6.0676 14.573 48394.1814 0.47  27 49620.59  1.4488 6.7502 1.1853 15.398 49873.8318 0.51  28 51046.26  1.4302 6.7890 1.7992 16.295 51329.0276 0.55  29 52445.35  1.4116 6.8295 2.4510 17.270 52759.8334 0.60  30 53817.87  1.3929 6.8718 3.1438 18.332 54166.3090 0.65  31 55163.8  1.3741 6.9160 3.8807 19.496 55548.5091 0.70  32 56483.17  1.3552 6.9623 4.6654 20.757 56906.4825 0.75  33 57775.95  1.3362 7.0107 5.5017 22.142 58240.2716 1.11  34 59042.17  1.3171 7.0614 6.3942 23.660 59711.7 1.12  35 60281.79  1.2979 7.1145 7.3475 25.327 60835.4299 0.92  36 61494.85  1.2785 7.1702 8.3670 27.299 62096.8449 0.98  37 62681.33  1.2591 7.2286 9.4589 29.181 63334.1657 1.04  38 63841.23  1.2396 7.2899 10.630 31.412 63547.399 0.46  39 64974.55  1.2200 7.3543 11.887 33.881 65736.5076 1.17  40 66081.3  1.2002 7.4220 13.238 36.619 66901.4905 1.24 aThe first entry is for the present work. bThe second and the cthird entries are, respectively, for [14, 15].Table 2
The energy eigenvaluesEv, the rotational constants Bv, and the centrifugal distortion constants Dv, Hv, Lv of the pure vibrational energy levels of the RKR potential of the electronic state XO+g of the molecule I2.
v E v cm−1 B v  × 102 cm−1 D v  × 109 cm−1 -Hv  × 1016 cm−1 -Lv  × 1022 cm−1 0 107.10a 3.731 4.547 5.115 2.858 107.0981b  1 320.40 3.719 4.572 5.380 3.115 320.3986  2 532.47 3.708 4.597 5.554 3.345 532.4712  3 743.31 3.696 4.624 5.750 3.462 743.3082  4 952.90 3.684 4.651 5.959 3.589 952.9017  5 1161.24 3.673 4.679 6.282 4.258 1161.2438  6 1368.33 3.661 4.708 6.504 3.546 1368.3260  7 1574.14 3.649 4.739 6.857 5.389 1374.1399  8 1778.68 3.637 4.770 7.131 4.302 1778.6764 3.637 4.768 5.57 3.03  9 1981.93 3.625 4.803 7.365 5.531 1981.9263 3.625 4.794 7.18 3.23  10 2183.88 3.613 4.837 7.742 6.076 2183.8800 3.613 4.825 7.50 3.46  11 2384.53 3.601 4.872 7.949 5.614 2384.5275 3.601 4.864 7.85 3.73  12 2583.86 3.589 4.909 8.316 6.790 2583.8536 3.589 4.901 8.32 4.02  13 2781.86 3.576 4.947 8.679 6.859 2781.8628 3.577 4.939 8.64 4.35  14 2978.53 3.564 4.987 9.049 6.970 2978.5290 3.564 4.980 9.08 4.71  15 3173.84 3.551 5.029 9.510 7.467 3173.8461 3.551 5.019 9.54 5.11  16 3367.80 3.539 5.072 10.01 8.106 3367.8023 3.540 5.509 10.04 5.56  17 3560.38 3.526 5.117 10.52 8.463 3560.3875 3.527 5.114 10.57 6.05  18 3751.58 3.513 5.163 11.10 3751.5835 3.513 5.166 11.13  19 3941.38 3.121 3941.3830 aThe first entry is for the present work. bThe second entry is for [16].Table 3
The eigenvaluesEv, the rotational constants Bv, and the centrifugal distortion constants Dv, Hv, Lv of the pure vibrational levels of the ab initio potential of the electronic state 22Σ+ of ZnBr molecule.
v E v B v  × 102 cm−1 D v  × 108 cm−1 H v  × 1014 cm−1 L v  × 1019 cm−1 0 73.93a 4.45103 1.61218 0.91241 0.14521 1 221.79 4.44744 1.63363 0.90472 0.19051 2 368.99 4.44350 1.66127 0.92463 0.05234 3 515.45 4.43935 1.67943 0.87615 0.27340 4 661.24 4.43481 1.70614 1.00984 0.15840 5 806.32 4.43028 1.71652 0.84228 0.29044 6 950.77 4.42537 1.73609 0.94056 0.11801 7 1094.61 4.42030 1.75034 0.11996 0.47912 8 1237.82 4.41370 1.78434 0.05349 0.84506 9 1380.27 4.40576 1.79032 0.59008 1.70130 10 1522.02 4.39775 1.73959 2.38350 2.47151 11 1663.35 4.39179 1.74384 0.88805 5.98681 12 1804.32 4.38529 1.88228 0.62346 2.15342 13 1944.46 4.37762 1.95563 3.05568 8.56892 14 2083.65 4.37184 1.86765 0.65588 4.32335 15 2222.28 4.36596 1.88941 2.90845 0.21492 16 2360.26 4.35694 1.90747 1.64760 0.79699 17 2497.54 4.34855 1.87664 320136 8.65452 18 2634.29 4.34149 1.96854 0.20066 4.88072 19 2770.33 4.33340 2.00230 1.64988 6.52090 20 2905.64 4.32574 1.95889 0.24895 5.04669 21 3040.34 4.31742 1.98347 0.26314 1.24147 22 3174.38 4.30858 2.00303 2.70697 1.67678 23 3307.78 4.30041 2.03418 0.91178 1.17857 24 3440.52 4.29194 2.06118 1.10390 3.42636 25 3572.59 4.28340 2.05726 0.54873 1.82649 26 3704.00 4.27461 2.07284 0.96785 1.25401 27 3834.77 4.26566 2.09089 1.58438 0.12407 28 3964.89 4.25677 2.10901 0.80808 1.31828 29 4094.36 4.24767 2.11876 1.04840 0.30910 30 4223.18 4.23849 2.12936 0.86572 0.05289 31 4351.37 4.22914 2.13889 1.08629 0.85910 32 4478.92 4.21971 2.15187 1.10438 0.34952 33 4605.84 4.21019 2.16476 0.78563 0.70987 34 4732.14 4.20049 2.17260 0.99447 1.02375 35 4857.80 4.19069 2.18507 1.11534 0.64514 36 4982.85 4.18081 2.19293 0.94266 0.49012 37 5107.27 4.17082 2.20322 1.05993 0.17041 38 5231.08 4.16074 2.21393 0.94721 1.19382 39 5354.28 4.1505 2.22180 0.97556 1.04020 40 5476.87 4.14026 2.23134 1.12858 0.13659 41 5598.85 4.12991 2.24135 0.94846 0.81640 42 5720.22 4.11945 2.24801 1.01250 0.74932 43 5840.99 4.10891 2.25680 1.06147 0.53996 44 5961.17 4.09830 2.26532 0.88668 0.28173 45 6080.74 4.08758 2.27180 1.09059 0.75800 46 6199.73 4.07680 2.28052 1.12210 0.61602 47 6318.12 4.06596 2.28691 0.91408 0.02149 48 6435.92 4.05503 2.29364 1.02652 0.71959 49 6553.13 4.04403 2.30155 1.04383 .25215 50 6669.76 4.03296 2.30781 0.95909 0.10453 51 6785.81 4.02183 2.31237 1.14847 0.87011 52 6901.28 4.01063 2.32080 1.05833 0.81187 53 7016.17 3.99938 2.32662 0.95641 0.09161 54 7130.49 3.98806 2.33122 1.13254 0.51320 55 7244.23 3.97669 2.33750 1.03295 0.81658 56 7357.41 3.96527 2.34305 0.93553 0.06755 57 7470.02 3.95378 2.34775 1.09995 0.66094 58 7582.06 3.94225 2.35311 1.13554 0.84554 59 7693.54 3.93068 2.35746 0.95176 0.06318 60 7804.46 3.91905 2.36103 1.09731 0.78457 61 7914.82 3.90737 2.36686 1.11190 0.93709 62 8024.62 3.89566 2.37097 0.90113 0.83185 63 8133.87 3.88389 2.37344 0.99341 0.85369 64 8242.57 3.87209 2.37792 1.14385 0.17720 65 8350.73 3.86026 2.38109 0.98947 0.70085 aData published in [17] calculated by the present software.


### 3.2. Advantage of the Present Method

The first explicit analytical expressions for the calculation of the distortion constants were derived by Albritton et al. [2]. The conventional Rayleigh-Schrödinger perturbation approach makes use of the wavefunction expansion given in 11 that leads to the expressions for el,e2,e3,… as first- and higher-order perturbation energies 10. These expressions of the distortion constants were considered fairly good for low-lying vibrational levels only; for higher levels they failed mainly because of the omission of the contribution from continuum levels.A modification of the perturbation method enabled Hutson [3] to derive simple expressions for el,e2,e3,… in terms of the internuclear distance r, ϕo, ϕ1, ϕ2 and to elaborate an efficient routine to solve the diatomic distortion constants problem by using the Numerov method described by Cooley [32]. By using this method, Tellinghuisen [4] pointed out a “pesky problem of instability” in the computation of ϕ1, ϕ2 in the Hutson approach. Therefore, he described a modification of the Hutson method in order to improve it by eliminating “nuisance instabilities in the nonclassical regions.”The advantages of the method given in the present work are as follows:(i) By using the variable step method, for an empirical potential, the number of mesh steps is largely reduced. Forab initio and RKR potentials, the step is the distance between two consecutive points. In the present method we step out to the left and to right by starting from an arbitrary origin. The calculation is stopped when the wave function tends to zero (see 42) which reduces the range of integration.(ii) The instability problem and its solution presented by Tellinghuisen [4] are both implied by the perturbation formulation as used by Hutson [3] (which indirectly implies an initial value problem for ϕ1,ϕ2…). The present work eliminates this instability problem by giving exact analytical initial values for ϕ1,ϕ2… (see 32 and 41) at an arbitrary origin.(iii) The present method is free from any conventional problems related to those of Hutson and Tellinghuisen; it does not imply any trial value or any “matching” problem (or initial values assumptions) on any iteration; it is equally easy for low or high levels.(iv) The computing “effort” to calculateel is roughly the same as that to obtain e2,e3,…. For each constant two new integrals, In and Rn (see 49), are required. The effort necessary to calculate Bv,Dv,Hv,Lv,Mv… is roughly equivalent to the average required to solve the vibrational eigenvalue.(v) Calculation of the energy eigenvalueEv near the dissociation energy is another advantage. The spectroscopic constants used in the present work for Morse potential of the molecule CO [14, 15] are ωe = 2169.81358 cm−1 and ωexe = 13.28831 cm−1. The dissociation energy for this potential is given by De=ωe2/4ωexe = 88575.80 cm−1. We present in Table SF7 (Supplementary Material file) the vibrational energy levels Ev, the rotational constants Bv, and the centrifugal distortion constants Dv, Hv, Lv of the of the Morse potential for the molecule CO near dissociation (from v = 41 to v = 81). Our calculated value for Ev = 88575.55 cm−1 for v = 81 is in excellent agreement with that calculated using the formula De=ωe2/4ωexe.We are now considering the application of this method as a verification procedure, for additional future and previously characterized molecules. The calculated values of the rovibrational energy levelsEvJ of the potential energy curves Morse, RKR, and ab initio for different values of v and J are given in Tables (SF1, SF2, SF3) in Supplementary Material file. The accuracy of our calculated values of the rotational constants Bv and the centrifugal distortion constants Dv, Hv, Lv for the six considered potentials by using the present software is presented in Tables (SF1, SF4, SF5-SF6) in Supplementary Material file. In these tables, we compare the calculated values of the rovibrational energy levels EvJ by using (1) and 11. This comparison shows the high accuracy of the present technique.


## 3.1. Calculated Data

In order to test the accuracy and the validity of the present method for the calculation of the different rovibrational parameters we used as examples for the empirical potentials Morse potential [14, 15], Lenard Jones potential [4, 8, 22], Dunhum Potential [23], and Kratzer potential [24]. For the investigation of the ab initio potential [17], we employed the state averaged Complete Active Space Self Consistent Field (CASSCF) followed by Multireference Configuration Interaction (MRCI) method with Davidson correction (+Q), single and double excitations. In these entire calculations, we used the computational chemistry program MOLPRO [25] with the advantage of the graphical user interface GABEDIT [26]. The CASSCF configuration space was used as the reference in the MRCI calculations. The potential energy U(r) is determined in terms of the internuclear distance r with a step equal to 0.02Å. The Schrödinger equation is solved by using the cubic spline interpolation between every two consecutive points. A Rydberg-Klein-Rees (RKR) potential is obtained experimentally and defined by their turning points [27–29]. The calculation has been done by an interpolation between every two consecutive points also by using the cubic spline interpolation. If the number of the turning points is limited we extrapolate the potential function curve to the right by [30] U(x) = a/r6 +b/r8 +c/r10 and to the left by U(x) = d/r15 where the constants a, b, c can be obtained from literature (if available) and d is calculated using the used potential energy curve.The comparison of our calculated values of the vibrational energy levelsEv, by using the empirical Morse potential for the molecule CO, with those given in [14, 15] shows, respectively, excellent agreements with the relative differences 0.00%<ΔEv/Ev<1.25% and 0.00%<ΔEv/Ev<0.03% (Table 1). Similar excellent agreement ΔEv/Ev = 0.00% is obtained by comparing our calculated values of the different vibrational levels Ev with those given by Martin et al. [16] of the RKR potential of the molecule I2 (Table 2). Our method is also tested in case of anab initio potential for the molecule ZnBr [17] (Table 3). Recently [31], we calculated, by using the present method, the rovibrational parameters Ev, Bv, Dv of the molecule LiSr by using anab initio potential. The comparison of the calculated values of Ev and Bv with those available in literature shows the good agreement with the relative differences 0.1%<ΔEv/Ev<7.5% for Ev and acceptable agreement 10.9%<ΔBv/Bv<12.9% for Bv. However, the comparison of our calculated values of Bv, by the present method, with those given in literature shows a very good agreement with the relative difference 1.1%<ΔBv/Bv<2.8% for the molecule NaSr [31].Table 1
The energy eigenvaluesEv, the rotational constants Bv, and the centrifugal distortion constants Dv, Hv, Lv of the pure vibrational energy levels of the Morse potential of CO molecule.
V E v cm−1 Δ E v / E v B v cm−1 D v  × 106 cm−1 H v  × 1012 cm−1 L v  × 1017 cm−1 0 1081.78a  1.9225 6.11954 5.8008 3.6445 1081.61b 0.02 1081.5857c 0.02  1 3225.05  1.9050 6.11885 5.6556 3.7160 3225.05 0.00 3224.8568 0.03  2 5341.84  1.8875 6.11849 5.5100 3.7848 5342.11 0.00 5341.6478 0.00  3 7432.22  1.8700 6.11844 5.3659 3.8409 7432.79 0.01 7432.0247 0.00  4 9496.25  1.8525 6.11863 5.2272 3.8678 9497.09 0.18 9496.0550 0.00  5 11534.03  1.8350 6.11892 5.1007 3.8397 11535.01 0.01 11533.80787 0.00  6 13545.61  1.8175 6.11910 4.9970 3.7184 13546.56 0.00 13545.3543 0.00  7 15531.12  1.8001 6.11884 4.9323 3.4491 15531.72 0.00 15530.767 0.00  8 17490.66  1.7826 6.11761 4.9298 2.9556 17490.1202 0.00  9 19424.39  1.7652 6.11467 5.0224 2.2224 19423.4901 0.00  10 21332.48  1.7478 6.10895 5.2545 0.8581 21330.9539 0.01  11 23195.48  1.7353 6.3063 5.1176 7.4807 23212.5907 0.07  12 25046.37  1.7179 6.3265 4.8608 7.7559 25068.4807 0.09  13 26870.69  1.7004 6.3476 4.5905 8.0499 26898.7056 0.10  14 28668.43  1.6829 6.3695 4.3061 8.3641 28703.3478 0.00  15 30439.59  1.6653 6.3922 4.0066 8.7004 30482.4908 0.14  16 32184.18  1.6476 6.4159 3.6911 9.0606 32236.2189 0.16  17 33902.19  1.6299 6.4405 3.3586 9.4470 33964.6167 0.18  18 35593.63  1.6121 6.4661 3.0081 9.8617 35667.7691 0.21  19 37258.49  1.5943 6.4927 2.6384 10.307 37345.7608 0.23  20 38896.77  1.5764 6.5204 2.2483 10.787 38998.6766 0.26  21 40508.47 0.29 1.5584 6.5493 1.8363 11.303 40626.6001  22 42093.6  1.5403 6.5794 1.4011 11.859 42229.6144 0.32  23 43652.15  1.5221 6.6108 9.4096 12.460 43807.8009 0.36  24 45184.13  1.5039 6.6435 4.5424 13.109 45361.2396 0.39  25 46689.52  1.4856 6.6775 6.0978 13.812 46890.0081 0.43  26 48168.35  1.4672 6.7131 6.0676 14.573 48394.1814 0.47  27 49620.59  1.4488 6.7502 1.1853 15.398 49873.8318 0.51  28 51046.26  1.4302 6.7890 1.7992 16.295 51329.0276 0.55  29 52445.35  1.4116 6.8295 2.4510 17.270 52759.8334 0.60  30 53817.87  1.3929 6.8718 3.1438 18.332 54166.3090 0.65  31 55163.8  1.3741 6.9160 3.8807 19.496 55548.5091 0.70  32 56483.17  1.3552 6.9623 4.6654 20.757 56906.4825 0.75  33 57775.95  1.3362 7.0107 5.5017 22.142 58240.2716 1.11  34 59042.17  1.3171 7.0614 6.3942 23.660 59711.7 1.12  35 60281.79  1.2979 7.1145 7.3475 25.327 60835.4299 0.92  36 61494.85  1.2785 7.1702 8.3670 27.299 62096.8449 0.98  37 62681.33  1.2591 7.2286 9.4589 29.181 63334.1657 1.04  38 63841.23  1.2396 7.2899 10.630 31.412 63547.399 0.46  39 64974.55  1.2200 7.3543 11.887 33.881 65736.5076 1.17  40 66081.3  1.2002 7.4220 13.238 36.619 66901.4905 1.24 aThe first entry is for the present work. bThe second and the cthird entries are, respectively, for [14, 15].Table 2
The energy eigenvaluesEv, the rotational constants Bv, and the centrifugal distortion constants Dv, Hv, Lv of the pure vibrational energy levels of the RKR potential of the electronic state XO+g of the molecule I2.
v E v cm−1 B v  × 102 cm−1 D v  × 109 cm−1 -Hv  × 1016 cm−1 -Lv  × 1022 cm−1 0 107.10a 3.731 4.547 5.115 2.858 107.0981b  1 320.40 3.719 4.572 5.380 3.115 320.3986  2 532.47 3.708 4.597 5.554 3.345 532.4712  3 743.31 3.696 4.624 5.750 3.462 743.3082  4 952.90 3.684 4.651 5.959 3.589 952.9017  5 1161.24 3.673 4.679 6.282 4.258 1161.2438  6 1368.33 3.661 4.708 6.504 3.546 1368.3260  7 1574.14 3.649 4.739 6.857 5.389 1374.1399  8 1778.68 3.637 4.770 7.131 4.302 1778.6764 3.637 4.768 5.57 3.03  9 1981.93 3.625 4.803 7.365 5.531 1981.9263 3.625 4.794 7.18 3.23  10 2183.88 3.613 4.837 7.742 6.076 2183.8800 3.613 4.825 7.50 3.46  11 2384.53 3.601 4.872 7.949 5.614 2384.5275 3.601 4.864 7.85 3.73  12 2583.86 3.589 4.909 8.316 6.790 2583.8536 3.589 4.901 8.32 4.02  13 2781.86 3.576 4.947 8.679 6.859 2781.8628 3.577 4.939 8.64 4.35  14 2978.53 3.564 4.987 9.049 6.970 2978.5290 3.564 4.980 9.08 4.71  15 3173.84 3.551 5.029 9.510 7.467 3173.8461 3.551 5.019 9.54 5.11  16 3367.80 3.539 5.072 10.01 8.106 3367.8023 3.540 5.509 10.04 5.56  17 3560.38 3.526 5.117 10.52 8.463 3560.3875 3.527 5.114 10.57 6.05  18 3751.58 3.513 5.163 11.10 3751.5835 3.513 5.166 11.13  19 3941.38 3.121 3941.3830 aThe first entry is for the present work. bThe second entry is for [16].Table 3
The eigenvaluesEv, the rotational constants Bv, and the centrifugal distortion constants Dv, Hv, Lv of the pure vibrational levels of the ab initio potential of the electronic state 22Σ+ of ZnBr molecule.
v E v B v  × 102 cm−1 D v  × 108 cm−1 H v  × 1014 cm−1 L v  × 1019 cm−1 0 73.93a 4.45103 1.61218 0.91241 0.14521 1 221.79 4.44744 1.63363 0.90472 0.19051 2 368.99 4.44350 1.66127 0.92463 0.05234 3 515.45 4.43935 1.67943 0.87615 0.27340 4 661.24 4.43481 1.70614 1.00984 0.15840 5 806.32 4.43028 1.71652 0.84228 0.29044 6 950.77 4.42537 1.73609 0.94056 0.11801 7 1094.61 4.42030 1.75034 0.11996 0.47912 8 1237.82 4.41370 1.78434 0.05349 0.84506 9 1380.27 4.40576 1.79032 0.59008 1.70130 10 1522.02 4.39775 1.73959 2.38350 2.47151 11 1663.35 4.39179 1.74384 0.88805 5.98681 12 1804.32 4.38529 1.88228 0.62346 2.15342 13 1944.46 4.37762 1.95563 3.05568 8.56892 14 2083.65 4.37184 1.86765 0.65588 4.32335 15 2222.28 4.36596 1.88941 2.90845 0.21492 16 2360.26 4.35694 1.90747 1.64760 0.79699 17 2497.54 4.34855 1.87664 320136 8.65452 18 2634.29 4.34149 1.96854 0.20066 4.88072 19 2770.33 4.33340 2.00230 1.64988 6.52090 20 2905.64 4.32574 1.95889 0.24895 5.04669 21 3040.34 4.31742 1.98347 0.26314 1.24147 22 3174.38 4.30858 2.00303 2.70697 1.67678 23 3307.78 4.30041 2.03418 0.91178 1.17857 24 3440.52 4.29194 2.06118 1.10390 3.42636 25 3572.59 4.28340 2.05726 0.54873 1.82649 26 3704.00 4.27461 2.07284 0.96785 1.25401 27 3834.77 4.26566 2.09089 1.58438 0.12407 28 3964.89 4.25677 2.10901 0.80808 1.31828 29 4094.36 4.24767 2.11876 1.04840 0.30910 30 4223.18 4.23849 2.12936 0.86572 0.05289 31 4351.37 4.22914 2.13889 1.08629 0.85910 32 4478.92 4.21971 2.15187 1.10438 0.34952 33 4605.84 4.21019 2.16476 0.78563 0.70987 34 4732.14 4.20049 2.17260 0.99447 1.02375 35 4857.80 4.19069 2.18507 1.11534 0.64514 36 4982.85 4.18081 2.19293 0.94266 0.49012 37 5107.27 4.17082 2.20322 1.05993 0.17041 38 5231.08 4.16074 2.21393 0.94721 1.19382 39 5354.28 4.1505 2.22180 0.97556 1.04020 40 5476.87 4.14026 2.23134 1.12858 0.13659 41 5598.85 4.12991 2.24135 0.94846 0.81640 42 5720.22 4.11945 2.24801 1.01250 0.74932 43 5840.99 4.10891 2.25680 1.06147 0.53996 44 5961.17 4.09830 2.26532 0.88668 0.28173 45 6080.74 4.08758 2.27180 1.09059 0.75800 46 6199.73 4.07680 2.28052 1.12210 0.61602 47 6318.12 4.06596 2.28691 0.91408 0.02149 48 6435.92 4.05503 2.29364 1.02652 0.71959 49 6553.13 4.04403 2.30155 1.04383 .25215 50 6669.76 4.03296 2.30781 0.95909 0.10453 51 6785.81 4.02183 2.31237 1.14847 0.87011 52 6901.28 4.01063 2.32080 1.05833 0.81187 53 7016.17 3.99938 2.32662 0.95641 0.09161 54 7130.49 3.98806 2.33122 1.13254 0.51320 55 7244.23 3.97669 2.33750 1.03295 0.81658 56 7357.41 3.96527 2.34305 0.93553 0.06755 57 7470.02 3.95378 2.34775 1.09995 0.66094 58 7582.06 3.94225 2.35311 1.13554 0.84554 59 7693.54 3.93068 2.35746 0.95176 0.06318 60 7804.46 3.91905 2.36103 1.09731 0.78457 61 7914.82 3.90737 2.36686 1.11190 0.93709 62 8024.62 3.89566 2.37097 0.90113 0.83185 63 8133.87 3.88389 2.37344 0.99341 0.85369 64 8242.57 3.87209 2.37792 1.14385 0.17720 65 8350.73 3.86026 2.38109 0.98947 0.70085 aData published in [17] calculated by the present software.


## 3.2. Advantage of the Present Method

The first explicit analytical expressions for the calculation of the distortion constants were derived by Albritton et al. [2]. The conventional Rayleigh-Schrödinger perturbation approach makes use of the wavefunction expansion given in 11 that leads to the expressions for el,e2,e3,… as first- and higher-order perturbation energies 10. These expressions of the distortion constants were considered fairly good for low-lying vibrational levels only; for higher levels they failed mainly because of the omission of the contribution from continuum levels.A modification of the perturbation method enabled Hutson [3] to derive simple expressions for el,e2,e3,… in terms of the internuclear distance r, ϕo, ϕ1, ϕ2 and to elaborate an efficient routine to solve the diatomic distortion constants problem by using the Numerov method described by Cooley [32]. By using this method, Tellinghuisen [4] pointed out a “pesky problem of instability” in the computation of ϕ1, ϕ2 in the Hutson approach. Therefore, he described a modification of the Hutson method in order to improve it by eliminating “nuisance instabilities in the nonclassical regions.”The advantages of the method given in the present work are as follows:(i) By using the variable step method, for an empirical potential, the number of mesh steps is largely reduced. Forab initio and RKR potentials, the step is the distance between two consecutive points. In the present method we step out to the left and to right by starting from an arbitrary origin. The calculation is stopped when the wave function tends to zero (see 42) which reduces the range of integration.(ii) The instability problem and its solution presented by Tellinghuisen [4] are both implied by the perturbation formulation as used by Hutson [3] (which indirectly implies an initial value problem for ϕ1,ϕ2…). The present work eliminates this instability problem by giving exact analytical initial values for ϕ1,ϕ2… (see 32 and 41) at an arbitrary origin.(iii) The present method is free from any conventional problems related to those of Hutson and Tellinghuisen; it does not imply any trial value or any “matching” problem (or initial values assumptions) on any iteration; it is equally easy for low or high levels.(iv) The computing “effort” to calculateel is roughly the same as that to obtain e2,e3,…. For each constant two new integrals, In and Rn (see 49), are required. The effort necessary to calculate Bv,Dv,Hv,Lv,Mv… is roughly equivalent to the average required to solve the vibrational eigenvalue.(v) Calculation of the energy eigenvalueEv near the dissociation energy is another advantage. The spectroscopic constants used in the present work for Morse potential of the molecule CO [14, 15] are ωe = 2169.81358 cm−1 and ωexe = 13.28831 cm−1. The dissociation energy for this potential is given by De=ωe2/4ωexe = 88575.80 cm−1. We present in Table SF7 (Supplementary Material file) the vibrational energy levels Ev, the rotational constants Bv, and the centrifugal distortion constants Dv, Hv, Lv of the of the Morse potential for the molecule CO near dissociation (from v = 41 to v = 81). Our calculated value for Ev = 88575.55 cm−1 for v = 81 is in excellent agreement with that calculated using the formula De=ωe2/4ωexe.We are now considering the application of this method as a verification procedure, for additional future and previously characterized molecules. The calculated values of the rovibrational energy levelsEvJ of the potential energy curves Morse, RKR, and ab initio for different values of v and J are given in Tables (SF1, SF2, SF3) in Supplementary Material file. The accuracy of our calculated values of the rotational constants Bv and the centrifugal distortion constants Dv, Hv, Lv for the six considered potentials by using the present software is presented in Tables (SF1, SF4, SF5-SF6) in Supplementary Material file. In these tables, we compare the calculated values of the rovibrational energy levels EvJ by using (1) and 11. This comparison shows the high accuracy of the present technique.


## 4. Conclusion

In the present work, the Volterra integral equation is used to solve the rovibrational Schrödinger equation of a diatomic molecule. This method showed that a high order precision is obtained for large order centrifugal distortion constant asDv,Hv,…Qv…. By using the present canonical function approach one can obtain these constants with the high values of vibrational levels even near dissociation by one single and simple routine. This method provides strong evidence for our assumption that the higher order of Dv,Hv,Lv… and the higher vibrational v and rotational level J for any electronic state and any type of potential energy curves (either experimental, empirical, or theoretical) are as accurate as the low-order values.



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