|
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the assumption that the motion of atomic nuclei and electrons in a molecule can be separated. The approach is named after Max Born and J. Robert Oppenheimer. In mathematical terms, it allows the wavefunction of a molecule to be broken into its electronic and nuclear (vibrational, rotational) components. : Computation of the energy and the wavefunction 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 the ''electronic'' Schrödinger equation is solved, yielding the wavefunction The nuclear spin energy is so small that it is normally omitted. ==Short description== The Born–Oppenheimer (BO) approximation is ubiquitous in quantum chemical calculations of molecular wavefunctions. It consists of two steps. In the first step the nuclear kinetic energy is neglected,〔This step is often justified by stating that "the heavy nuclei move more slowly than the light electrons." Classically this statement makes sense only if the momentum ''p'' of electrons and nuclei is of the same order of magnitude. In that case ''m''nuc >> ''m''elec implies ''p''2/(2''m''nuc) << ''p''2/(2''m''elec). It is easy to show that for two bodies in circular orbits around their center of mass (regardless of individual masses), the momentum of the two bodies is equal and opposite, and that for any collection of particles in the center of mass frame, the net momentum is zero. Given that the center of mass frame is the lab frame (where the molecule is stationary), the momentum of the nuclei must be equal and opposite to that of the electrons. A hand waving justification can be derived from quantum mechanics as well. Recall that the corresponding operators do not contain mass and think of the molecule as a box containing the electrons and nuclei and see particle in a box. Since the kinetic energy is ''p''2/(2''m''), it follows that, indeed, the kinetic energy of the nuclei in a molecule is usually much smaller than the kinetic energy of the electrons, the mass ratio being on the order of 104).〕 that is, the corresponding operator ''T''n is subtracted from the total molecular Hamiltonian. In the remaining electronic Hamiltonian ''H''e the nuclear positions enter as parameters. The electron–nucleus interactions are ''not'' removed and the electrons still "feel" the Coulomb potential of the nuclei clamped at certain positions in space. (This first step of the BO approximation is therefore often referred to as the ''clamped nuclei'' approximation.) The electronic Schrödinger equation :: is solved (out of necessity, approximately). The quantity r stands for all electronic coordinates and R for all nuclear coordinates. The electronic energy eigenvalue ''E''e depends on the chosen positions R of the nuclei. Varying these positions R in small steps and repeatedly solving the electronic Schrödinger equation, one obtains ''E''e as a function of R. This is the potential energy surface (PES): ''E''e(R) . Because this procedure of recomputing the electronic wave functions as a function of an infinitesimally changing nuclear geometry is reminiscent of the conditions for the adiabatic theorem, this manner of obtaining a PES is often referred to as the ''adiabatic approximation'' and the PES itself is called an ''adiabatic surface''.〔It is assumed, in accordance with the adiabatic theorem, that the same electronic state (for instance the electronic ground state) is obtained upon small changes of the nuclear geometry. The method would give a discontinuity (jump) in the PES if electronic state-switching would occur.〕 In the second step of the BO approximation the nuclear kinetic energy ''T''n (containing partial derivatives with respect to the components of R) is reintroduced and the Schrödinger equation for the nuclear motion〔This equation is time-independent and stationary wavefunctions for the nuclei are obtained, nevertheless it is traditional to use the word "motion" in this context, although classically motion implies time-dependence. 〕 : is solved. This second step of the BO approximation involves separation of vibrational, translational, and rotational motions. This can be achieved by application of the Eckart conditions. The eigenvalue ''E'' is the total energy of the molecule, including contributions from electrons, nuclear vibrations, and overall rotation and translation of the molecule. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Born–Oppenheimer approximation」の詳細全文を読む スポンサード リンク
|