Atomic nucleus Born-Oppenheimer

CSFs into the wavefunction expansion. Although unattainable in molecular calculations, the second limiting case, corresponding to full Cl for a complete orbital set, is called the complete Cl expansion s. The eigenvalues of the complete Cl expansion are the exact energies within the clamped-atomic-nucleus Born-Oppenheimer approximation. A correspondence may then be established with the bracketing theorem between the lowest eigenvalues of a limited CSF expansion and those of the exact complete Cl expansion. This is illustrated schematically in Fig. 2. 

In the clamped-nucleus Born-Oppenheimer approximation, with neglect of relativistic effects, the molecular Hamiltonian operator in atomic units takes the form... 

Below we will use Eq. (16), which, in certain models in the Born-Oppenheimer approximation, enables us to take into account both the dependence of the proton tunneling between fixed vibrational states on the coordinates of other nuclei and the contribution to the transition probability arising from the excited vibrational states of the proton. Taking into account that the proton is the easiest nucleus and that proton transfer reactions occur often between heavy donor and acceptor molecules we will not consider here the effects of the inertia, nonadiabaticity, and mixing of the normal coordinates. These effects will be considered in Section V in the discussion of the processes of the transfer of heavier atoms. 

It is mentioned in passing that the proper masses mA and mB to be used in Equation 3.3 are the atomic masses (nucleus + electrons) rather than the respective nuclear masses as might be expected from a strict Born-Oppenheimer approximation. For further discussion of this point, reference should be made to the reading lists at the end of this chapter and of Chapter 2. The combination of Equations 3.1 and 3.2 corresponds to a classical harmonic oscillator with force constant f and mass p. The harmonic oscillator frequency v is given by the well-known formula... 

Schrodinger equations for atoms and molecules use the the sum of the potential and kinetic energies of the electrons and nuclei in a structure as the basis of a description of the three dimensional arangements of electrons about the nucleus. Equations are normally obtained using the Born-Oppenheimer approximation, which considers the nucleus to be stationary with respect to the electrons. This approximation means that one need not consider the kinetic energy of the nuclei in a molecule, which considerably simplifies the calculations. Furthermore, the... 

For the N-electron atom, we have seen (Section 3.7) several terms in the Hamiltonian operator. We collect here some more terms, to come to a "final list," within the Born-Oppenheimer approximation of a fixed nucleus ... 

The geometrical and electronic structure for molecular systems in general will depend on the balance between the different terms in the Hamiltonian i.e. electron-nucleus, electron-electron and nucleus-nucleus interaction including the valence as well as the core electrons of the constituent atoms. The full Hamiltonian for the molecular system is normally separated into a Hamiltonian Hn for the nuclei and another one Hgi for the electrons with fixed positions for the nuclei according to Born Oppenheimer approximation [31]. 

In Equation 9.1, which is expressed in atomic units, is the charge on nucleus A located at R the nuclei are treated as stationary (Born-Oppenheimer approximation). The quantity p(r) is the electronic density, which is the average number of electrons in each volume element dr. Since the average (static) electronic density is being used, V(r) is labeled the electrostatic potential. 

The Born-Oppenheimer approximation takes us a long way in that direction by assuming that the rapid motion of the electrons is separable from the nuclear motion. The Coulomb forces acting on the electrons and nuclei in a molecule are of comparable magnitude, but force is mass times acceleration, and each nucleus has at least 10 times more mass than each electron. Therefore, the forces accelerate the electrons to much higher speeds than the nuclei. The Born-Oppenheimer approximation extends this difference in speed to the limit in which we treat the electrons as traveling around stationary nuclei. This means we don t have to solve for the motions of the electrons and nuclei at the same time the electronic and nuclear coordinates are separable, in a way similar to the separation of angular and radial coordinates for the one-electron atom (Section 3.1). In the one-electron atom, we solve the... 

The most common initial approach to molecular quantum mechanics is the Hartree-Fock (HF) calculation, introduced in Section 4.2. Applying the Born-Oppenheimer approximation, we permit ourselves to solve only the electronic wavefunction initially, for some fixed geometry of the nuclei. Therefore, the approach is no diflferent for molecules than for atoms, except that the potential energy includes contributions from more than one nucleus. For simplicity here, we will confine ourselves to the Hartree equation, rather than the HF equation that is applied to Slater determinants. Our Eq. 4.26 for the atom becomes... 

For a single atom the nucleus is fixed at the origin of coordinates. For molecules, a big simplification results from the fact that electrons rearrange so much faster than nuclei that the positional coordinates of the nuclei can be kept fixed in the calculation of electronic energies, and the molecular wavefiinction depends only on the coordinates of electrons. This is called the Born-Oppenheimer assumption (there are no potential energy terms for nuclei in the hamiltonian 3.39). The total electronic energy is the expectation value of the hamiltonian operator, equation 3.8 ... 

First, from a physical perspective, it is not feasible to account for absolutely all degrees of freedom of a system in calculation of a partition function. Taking into account all subatomic degrees of freedom is impractical. Approximations such as the Born-Oppenheimer approximation that the nucleus of atoms is at the ground state are then indispensable, but the constant contribution to the total partition function is arbitrary. 

The large mass of a nucleus compared to that of an electron permits an approximate separation of electronic and nuclear motion, which is the basis for the treatments discussed in the preceding sections. Historically, there were several attempts to expand the solutions of the Schrodinger equation, i.e. the energies and the wavefunctions, in powers of a small quantity related to the ratio of electronic and nuclear masses. The breakthrough has been achieved by Born and Oppenheimer in their classic paper in which they have chosen k = (1/M) / as the small quantity.done in the preceding sections, M is some average nuclear mass and we work in atomic units where the electronic mass is 1. 

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