Hamiltonian, clamped nuclei

Here, t is the nuclear kinetic energy operator, and so all terms describing the electronic kinetic energy, electron-electron and electron-nuclear interactions, as well as the nuclear-nuclear interaction potential function, are collected together. This sum of terms is often called the clamped nuclei Hamiltonian as it describes the electrons moving around the nuclei at a particular configrrration R. 

If the last term in (28) is ignored, then what is left seems to be the right sort of nuclear motion Hamiltonian for present purposes. The function Ep(q) + F"(q) is clearly a potential in the nuclear variables and the kinetic energy operator depends only on the nuclear variables too. It is not quite true that (25) is actually the clamped nuclei Hamiltonian but it is shown in (7) that the correspondence is very close. 

The irreducible representation index of the symmetry group of the clamped nuclei Hamiltonian (see Appendix C available at booksite.elsevier.com/978-0-444-59436-5)... 

The Ho does not contain the kinetic energy operator of the nuclei, but it does contain all the other terms (this is why it is called the electronic or clamped nuclei Hamiltonian) the first term stands for the kinetic energy operator of the electrons, and V means the potential energy corresponding to the Coulombic interaction of aU particles. The first term in the operator H (i.e., — Ajj), denotes the kinetic energy operator of the nuclei, while the operator H" couples the motions of the nuclei and electrons. ... 

The detailed form of Vi,n,r R) is obtained from Uk R) of Eq. (6.30) and therefore from the solution of the Schrddinger Eq. (6.24) with the clamped nuclei Hamiltonian. In principle, there... 

We are, therefore, beyond the adiabatic approximation (which requires a single vibronic state, a product function) and the very notion of the single potential eneigy hypersurface for the motion of the nuclei becomes irrelevant. In the adiabatic approximation, the electronic wave function is computed from Eq. (6.8) with the clamped nuclei Hamiltonian i.e., the electronic wave function does not depend on what the nuclei are doing, but only where they are. In other words, the electronic structure is determined [by finding a suitable R) through solution of the... 

The function Vr(f R) represents an eigenfunction of the electronic Hamiltoiiian Ho(R) i.e., the Hamiltonian H, in which the kinetic energy operator for the nuclei is assumed to be zero (the clamped nuclei Hamiltonian)... 

The eigenvalue of the clamped nuclei Hamiltonian depends on positions of the nuclei and in the Born-Oppenheimer approximation, it is mass-independent This eneigy as a function of the configuration of the nuclei lejaesents the potential energy for the motion of the nuclei (Potential Energy Surface, or PES). 

Born-Oppenheimer approximation (p. 272) branching plane (p. 312) branching space (p. 311) clamped nuclei Hamiltonian (p. 264) conical intersection (p. 312) continuum states (p. 297)... 

Adiabatic case. Suppose we have a Hamiltonian H(r R) that depends on the electronic coordinates r and parametrically depends on the configuration of the nuclei R. In practical applications, most often 7Y(r R) = Ho(r R), the electronic clamped nuclei Hamiltonian corresponding to eq. (6.8) (generalized to polyatomic molecules). The eigenfunctions (r R) and the eigenvalues Ei R) of the Hamiltonian ii(r R) are called adiabatic. Fig. 6.11. If we take 7Y =Ho(r, R), then in the adiabatic approximation (p. 227) the total wave function is represented by the... 

To solve this problem in detail let us limit ourselves to the simplest situation the two-state model (Appends D). Let us consider a diatomic molecule and such an intemuclear distance Rq that the two electronic adiabatic states r/rj (r Rq) and 4>2(r Rq)) correspond to the non-degenerate (but close on the energy scale) eigenvalues of the clamped nuclei Hamiltonian Ho(Ro)-... 

Abstract Arguments are advanced to support the view that at present it is not possible to derive molecular structure from the full quantum mechanical Coulomb Hamiltonian associated with a given molecular formula that is customarily regarded as representing the molecule in terms of its constituent electrons and nuclei. However molecular structure may be identified provided that some additional chemically motivated assumptions that lead to the clamped nuclei Hamiltonian are added to the quantum mechanical account. 

We concentrate on two broad themes. It is obvious that the whole collection of isomers supported by a given molecular formula share the same Coulomb Hamiltonian. The first part of the chapter is concerned with how this fundamental fact has been treated in quantiun chemistry through the introduction of the clamped nuclei Hamiltonian. This involves two crucial assumptions (1) the nuclei can be treated as fixed ( clamped ) classical particles that merely provide a classical external potential for the electrons and (2) formally identical nuclei can be treated as distinguishable. The second part of the chapter discusses in a general way the basic quantum mechanical theory of the clamped nuclei Hamiltonian, concentrating particularly on its symmetry properties. 

It is customary to incorporate the nuclear repulsion energy into O Eq. 2.2 the nuclear repulsion term is merely an additive constant and so does not affect the form of the electronic wavefunctions. Its inclusion modifies the spectrum of the clamped nuclei Hamiltonian only trivially by changing the origin of the energy. The eigenvalue equation for the clamped nuclei Hamiltonian is then... 

It is sometimes asserted that the clamped nuclei Hamiltonian can be obtained from the Coulomb Hamiltonian by letting the nuclear masses increase without limit. The Hamiltonian that would result if this were done would be... 

It is thus not at all clear to precisely which question the clamped nuclei Hamiltonian provides the answer and a further discussion of the properties of the Coulomb Hamiltonian is required before the clamped nuclei problem can be put into an appropriate form for yielding a potential, There are two main ways in which such a discussion can be attempted. If it is desired to stay with the Coulomb Hamiltonian in its lab oratory-fixed form then the solutions must be expressed in coherent state (wave-packet) form to allow for their continuum nature. If the solutions are required as L -normalizable wavefunctions, then the translational motion must be separated from the Coulomb Hamiltonian and the solutions of the remaining translationally invariant part must be sought It is in this second approach that it is easiest to make contact with the standard arguments and this will be considered in the following section. 

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