By- Prof. R.S. Gupta GOVERNMENT (AUTONOMOUS) P.G. COLLEGE Department of Physics SATNA (M.P.) ATOMIC AND MOLECULAR PHYSICS By- Prof. R.S. Gupta

BORN-OPPENHEIMER-APPROXIMATION   Molecules being a rigid structure posses translational motion in which the intrinsic structure of molecule is not affected because this part of motion has no properties of molecular structure and consequently no quantum feature. The rotation of atomic arrangement can also take place about same axis within the molecule. In addition to this, there may be internal vibrations of the molecule. Further, the electrons within the molecule make transitions between the electronic energy levels. Therefore, the molecular energy levels can be divided into electronic, vibrational and rotational energy levels. The order of magnitudes is Ee > Ev > Er . Where suffixes e, v, r, stands for electronic, vibration and rotation respectively.

It should be possible to devise a method of approximate solution of the wave equation involving its separation into three equation. One dealing with the motion of the electrons, other with the vibrational motion of the nuclei and third with the rotational motion of the nuclei. Considering the fact that the mass of every atomic nucleus is several thousand times as great as the mass of an electron. Born and Oppenheimer showed that an approximate solution of the complete wave equation for a molecule can be obtained by solving the wave equation for the electrons alone with the nuclei in a fixed configuration; and then solving a wave equation for the nuclei alone, in which characteristic energy value of the electronic wave equation, regarded as a function of the internuclear distance occurs as a potential function. The complete wave equation for molecule consisting of n electrons and N nuclei is [-ħ2/2m i2 -2/2Mj j2 + V]= EΨ ……….(1) Where Mj is the mass of the jth nucleus and m the mass of each electron, V is the potential energy of the system. i.e. sum of electrostatic interaction between all pairs of electrons and nuclei.

Let represents the 3N co-ordinates of N nuclei, relative to axis fixed in space and n co-ordinates of n electrons, relative to axes determined by the co-ordinates of the nuclei. Again let v represent the quantum numbers associated with the motion of nuclei, and represents the quantum numbers associated with the motion of electrons. According to Born and Oppenheimer’s treatment, an approximate solution , V(η, ξ) of equation (1) may be obtained in the form , V (η, ξ) = (η, ξ) , V(ξ) …………… (2) The function (η, ξ) which may be called as electronic wave functions. These functions can be obtained by solving the wave equation for the electrons alone, the nuclei being restricted to fixed configuration.

The wave equation is [-ħ2/2m i2 + V (η, ξ)] (η, ξ) = U(ξ) (η, ξ) …………(3) The equation (3) is obtained with the help of equation (1) by substituting U(ξ) for E and (η, ξ) for The term involving i2 has been omitted. It is observed that the characteristic values U(ξ) of the electronic energy are continuous function of nuclear co-ordinates ξ. Knowing U(ξ) as a function of ξ for given set of values of the electronic quantum number , we next obtain the expressions for the nuclear wave functions , V(ξ) . The nuclear wave equation is [2/2mj j2 + U()] Ψ, V(ξ) = E,V Ψ V(ξ) ……….(4)   The values of E, V are the characteristic energy values for the entire molecule.

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