TY - JOUR

T1 - Solution of the Rovibrational Schrödinger Equation of a Molecule Using the Volterra Integral Equation

AU - Korek, Mahmoud

AU - El-Kork, Nayla

N1 - Publisher Copyright:
© 2018 Mahmoud Korek and Nayla El-Kork.

PY - 2018

Y1 - 2018

N2 - By using the Rayleigh-Schrödinger perturbation theory the rovibrational wave function is expanded in terms of the series of functions φ0,φ1,φ2,.φn, where φ0 is the pure vibrational wave function and φi 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 φi 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.

AB - By using the Rayleigh-Schrödinger perturbation theory the rovibrational wave function is expanded in terms of the series of functions φ0,φ1,φ2,.φn, where φ0 is the pure vibrational wave function and φi 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 φi 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.

UR - http://www.scopus.com/inward/record.url?scp=85055590084&partnerID=8YFLogxK

U2 - 10.1155/2018/1487982

DO - 10.1155/2018/1487982

M3 - Article

AN - SCOPUS:85055590084

SN - 1687-7985

VL - 2018

JO - Advances in Physical Chemistry

JF - Advances in Physical Chemistry

M1 - 1487982

ER -