The Adiabatic and Born-Oppenheimer Approximations

Let us first review the Bom-Oppenheimer approximation in a bit more detail. The total Hamilton operator can be written as the kinetic and potential energies of the nuclei and electrons. 

The Hamilton operator is first transformed to the centre of mass system, where it may be 

Here Hg is the electronic Hamilton operator and H p is called the mass-polarization (Mtot is the total mass of all the nuclei and the sum is over all electrons). We note that He depends only on the nuclear positions (via Vne and Vnn, see eq. (3.23)) and not on their momenta. 

Assume for the moment that the full set of solutions to the electronic Schrddinger equation is available, where R denotes nuclear positions and r electronic coordinates. 

Since the Hamilton operator is hermitic (J H jdr = J yH, dr), the solutions can be chosen to be orthogonal and normalized orthonormal). 

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THE ADIABATIC AND BORN-OPPENHEIMER APPROXIMATIONS symbol. Expanding (3.7) gives... 

Suppose we wish to know the dipole moment of, say, the HCl molecule, the quantity that tells us important information about the charge distribution. We look up the output and we do not find anything about dipole moment. The reason is that all molecules have the same dipole moment in any of their stationary state y, and this dipole moment equals to zero, see, e.g., Piela (2007) p. 630. Indeed, the dipole moment is calculated as the mean value of the dipole moment operator i.e., ft = (T l/i l ) = ( F (2, q/r,) T), index i runs over all electrons and nuclei. This integral can be calculated very easily the integrand is antisymmetric with respect to inversion and therefore ft = 0. Let us stress that our conclusion pertains to the total wave function, which has to reflect the space isotropy leading to the zero dipole moment, because all orientations in space are equally probable. If one applied the transformation r -r only to some particles in the molecule (e.g., electrons), and not to the other ones (e.g., the nuclei), then the wave function will show no parity (it would be neither symmetric nor antisymmetric). We do this in the adiabatic or Born-Oppenheimer approximation, where the electronic wave function depends on the electronic coordinates only. This explains why the integral ft = ( F F) (the integration is over electronic coordinates only) does not equal zero for some molecules (which we call polar). Thus, to calculate the dipole moment we have to use the adiabatic or the Born-Oppenheimer approximation. 

The well-known Born-Oppenheimer approximation (BOA) assumes all couplings Kpa between the PES are identically zero. In this case, the dynamics is described simply as nuclear motion on a single adiabatic PES and is the fundamental basis for most traditional descriptions of chemistry, e.g., transition state theory (TST). Because the nuclear system remains on a single adiabatic PES, this is also often referred to as the adiabatic approximation. 

In this thesis work. Dr. Ren also studied the non-adiabatic effect in the F -I- D2 reaction, where the F ( Pi/2) is expected to be non-reactive according to the Bom-Oppenheimer approximation. He measured accurately the population ratio of F( P3/2) and F ( Pi/2) in the beam using synchrotron radiation single photon autoionization, then determined the relative reactivity of F and F with D2. For the first time, he found that F ( Pi/2) is more reactive than F( P3/2) at low collision energy, providing a clear case of the breakdown of Born-Oppenheimer approximation. This is the first accurate experimental measurement of the non-adiabatic effects of this important system. 

Pack R T and Hirschfelder J O 1970 Energy corrections to the Born-Oppenheimer approximation. The best adiabatic approximation J. Chem. Phys. 52 521-34... 

We have derived time-reversible, symplectic, and second-order multiple-time-stepping methods for the finite-dimensional QCMD model. Theoretical results for general symplectic methods imply that the methods conserve energy over exponentially long periods of time up to small fluctuations. Furthermore, in the limit m —> 0, the adiabatic invariants corresponding to the underlying Born-Oppenheimer approximation will be preserved as well. Finally, the phase shift observed for symmetric methods with a single update of the classical momenta p per macro-time-step At should be avoided by... 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

When the Drude particles are treated adiabatically, a SCF method must be used to solve for the displacements of the Drude particle, d, similarly to the dipoles Jtj in the induced dipole model. The implementation of the SCF condition corresponding to the Born-Oppenheimer approximation is straightforward and the real forces acting on the nuclei must be determined after the Drude particles have attained the energy minimum for a particular nuclear configuration. In the case of N polarizable atoms with positions r, the relaxed Drude particle positions r + d5CF are found by solving... 

Fig. 3. Vibrational population distributions of N2 formed in associative desorption of N-atoms from ruthenium, (a) Predictions of a classical trajectory based theory adhering to the Born-Oppenheimer approximation, (b) Predictions of a molecular dynamics with electron friction theory taking into account interactions of the reacting molecule with the electron bath, (c) Born—Oppenheimer potential energy surface, (d) Experimentally-observed distribution. The qualitative failure of the electronically adiabatic approach provides some of the best available evidence that chemical reactions at metal surfaces are subject to strong electronically nonadiabatic influences. (See Refs. 44 and 45.)...

The important fact that must be remembered is that in the Born-Oppenheimer approximation, Equation 2.8, the potential energy for vibrational motion is Eeiec(S) which is independent of isotopic mass of the atoms. In the adiabatic approximation, the potential energy function is Eeiec(S)+C and this potential will depend on nuclear mass if C depends on nuclear mass. 

Fig. 2.1 The adiabatic correction to the Born-Oppenheimer approximation for H2 and HD schematic, not to scale AC = C(H2)-C(HD). In each case the uncorrected potential lies to the left, the corrected to the right...

The electronic contributions to the g factors arise in second-order perturbation theory from the perturbation of the electronic motion by the vibrational or rotational motion of the nuclei [19,26]. This non-adiabatic coupling of nuclear and electronic motion, which exemplifies a breakdown of the Born-Oppenheimer approximation, leads to a mixing of the electronic ground state with excited electronic states of appropriate symmetry. The electronic contribution to the vibrational g factor of a diatomic molecule is then given as a sum-over-excited-states expression... 

Born-Oppenheimer approximation (physchem) The approximation, used in the Born-Oppenheimer method, that the electronic wave functions and energy levels at any instant depend only on the positions of the nuclei at that instant and not on the motions of the nuclei. Also known as adiabatic approximation. born ap an.hT-mar 3,prak s3,ma shan J... 

One of the necessary conditions for a many-body description is the validity of the decomposition of the system under consideration on separate subsystems. In the case of very large collective effects we cannot separate the individual parts of the system and only the total energy of the system can be defined. However, in atomic systems the inner-shell electrons are to a great extent localized. Therefore, even in metals with strong collective valence-electron interactions, atoms (or ions) can be identified as individuals and we can define many-body interactions. The important role in this separation plays the validity for atom- molecular systems the adiabatic or the Born-Oppenheimer approximations which allow to describe the potential energy of an N-atom systeni as a functional of the positions of atomic nuclei. 

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