following Born Oppenheimer approximation deduce the complete wave equation of a molecule
Answers
{\displaystyle \Psi _{\mathrm {total} }=\psi _{\mathrm {electronic} }\otimes \psi _{\mathrm {nuclear} }
Computation of the energy and thewavefunction of an average-size molecule is simplified by the approximation. For example, the benzene molecule consists of 12 nuclei and 42 electrons. The time independent Schrödinger equation, which must be solved to obtain the energy and wavefunction of this molecule, is a partial differential eigenvalue equation in 162 variables—the spatial coordinates of the electrons and the nuclei. The BO approximation makes it possible to compute the wavefunction in two less complicated consecutive steps. This approximation was proposed in 1927, in the early period of quantum mechanics, by Born and Oppenheimer and is still indispensable in quantum chemistry.
In the first step of the BO approximation theelectronic Schrödinger equation is solved, yielding the wavefunction {\displaystyle \psi _{\text{electronic}}}depending on electrons only. For benzene this wavefunction depends on 126 electronic coordinates. During this solution the nuclei are fixed in a certain configuration, very often the equilibrium configuration. If the effects of the quantum mechanical nuclear motion are to be studied, for instance because avibrational spectrum is required, this electronic computation must be in nuclear coordinates. In the second step of the BO approximation this function serves as a potential in a Schrödinger equation containing only the nuclei—for benzene an equation in 36 variables.
The success of the BO approximation is due to the difference between nuclear and electronic masses. The approximation is an important tool of quantum chemistry: all computations of molecular wavefunctions for large molecules make use of it, and without it only the lightest molecule, H2, can be handled. Even in the cases where the BO approximation breaks down, it is used as a point of departure for the computations.
The electronic energies consist of kinetic energies, interelectronic repulsions, internuclear repulsions, and electron–nuclear attractions. In accord with the Hellmann-Feynman theorem, the nuclear potential is taken to be an average over electron configurations of the sum of the electron–nuclear and internuclear electric potentials.
In molecular spectroscopy, because the ratios of the periods of the electronic, vibrational and rotational energies are each related to each other on scales in the order of a thousand, the Born–Oppenheimer name has also been attached to the approximation where the energy components are treated separately.
{\displaystyle E_{\mathrm {total} }=E_{\mathrm {electronic} }+E_{\mathrm {vibrational} }+E_{\mathrm {rotational} }+E_{\mathrm {nuclear} }
The nuclear spin energy is so small that it is normally omitted.
Answer:
One of the fundamental ideas that underlies the description of the quantum states of molecules is the Born-Oppenheimer approximation.
Explanation:
The motion of the nuclei and the motion of the electrons can be distinguished using this approximation. For us, this is hardly a fresh concept. When explaining the electronic absorption spectra of cyanine dyes using the particle-in-a-box model without taking into account the nuclei's mobility, we previously used this approximation. Then, without mentioning the motion of the electrons, we know about the translational, rotational, and vibrational motion of the nuclei. We will look more extensively at the importance and ramifications of this crucial approximation in this chapter. Note that in this context, the term "nuclear" refers to atomic nuclei as components of molecules rather than the nucleus' fundamental structure. The nuclei move significantly more slowly than the electrons as a result of this distinction. Additionally, a mutually attracting force of opposite charges exists between the two - .
Thus, according to the Born-Oppenheimer approximation, the full Hamiltonian contains the nuclear kinetic energy terms.