The breakdown of the Born-Oppenheimer approximation for a diatomic molecule: The dipole moment and nuclear quadrupole coupling constants

Abstract

In the Born-Oppenheimer approximation the dipole moment of the vibrational levels of a 1Σ electronic state of a heteronuclear diatomic molecule can be expressed as a power series in [ ( B e ω e )(v + 1 2 ) ], where v is the vibrational quantum number, and to order ( B e ω e ) 2 this expression is μ v=[μ e+( B e ω e ) 2μ c]+μ 1[( B e ω e )(v+ 1 2 )]+μ 2( B e ω e )(v+ 1 2 )] 2 Similarly the nuclear quadrupole coupling constant eQq of each nucleus in the molecule can be expressed as eQq=[eQq e+( B e ω e ) 2eQq c]+eQq 1[( B e ω e )(v+ 1 2 )]+eQq 2( B e ω e )(v+ 1 2 )] 2 In this paper the effect of the breakdown of the Born-Oppenheimer approximation on the expressions for μ v and eQq for an isolated 1Σ ground electronic state of a heteronuclear diatomic molecule is determined. The effect is to change only μ c and eQq c and, therefore, to alter the relationship between the μ v or eQq values of two isotopes of a molecule. The intensities of the lines in the rotation and rotation-vibration spectrum are also slightly modified by this effect. For the HCl molecule we find that μ v=[1.0908+( B e ω e )(164)]+8.6[( B e ω e )(v+ 1 2 )]−9.5[( B e ω e )(v+ 1 2 )] 2D where the second term (+164 D) would have the value −4 D in the Born-Oppenheimer approximation. Similarly for the 35Cl nucleus of the HCl molecule we have eQq=[−66.806+( B e ω e )(2460)]−472.23[( B e ω e )(v+ 1 2 )]+750[( B e ω e )(v+ 1 2 )] 2MHz where the second term (+2460 MHz) would be −110 MHz in the Born-Oppenheimer approximation.