Nuclear configuration space

The values of this (approximate) fi (qx) calculated from this equation are smaller than 0.08 kcal/mol over the entire nuclear configuration space involved, and to a very good approximation can be neglected. 

Let S be any simply connected surface in nuclear configuration space, bounded by a closed-loop L. Then, if 4>(r,R) changes sign when transported adiabatically round L, there must be at least one point on S at which (r, R) is discontinuous, implying that its potential energy surface intersects that of another electronic state. 

The presence of this term can also introduce numerical inefficiency problems in the solution of Eq. (31). Since the ADT matrix U(qx) is arbitrary, it can be chosen to make Eq. (31) have desirable properties that Eq. (15) does not possess. The parameter U(q> ) can, for example, be chosen so as to automatically minimize W (Rx) relative to W l ad(R/ ) everywhere in internal nuclear configuration space and incorporate the effect of the geometric phase. Next, we will consider the structure of this ADT matrix for an ra-electronic-state problem and a general evaluation scheme that minimizes the magnitude of W Rx). 

Hoffmann s review.2) The number of specific examples mentioned in the text is severely limited in order to save space they can be easily found elsewhere.2) Instead, space is devoted to detailed discussion of topics likely to be less familiar to the organic chemist, such as some of the properties of potential energy hypersurfaces in multidimensional nuclear configuration space, etc. When in doubt, the author erred on the side of sounding too explicit and trivial at the risk of offending the reader with good physical background. 

Even in highly symmetrical molecules, at most points in the nuclear configuration space the molecule has very low or no symmetry. 

The inconvenience introduced by the labelling system used here is that the physical nature of any given state, say Si, will often change, possibly rather abruptly, as the nuclei wander throughout the nuclear configuration space (Fig. 1). It may be nn at some geometries, nn at others, singly excited at some, doubly excited at others. 

Once this equation is solved for all relevant regions of the nuclear configuration space, in the BO framework, the nuclear motion can be treated either via a classical mechanical analysis with the help of computer simulations [6], or it can be treated quantum mechanically for simple models [54], In the latter scheme, the nuclear Schrodinger equation must be solved ... 

It is common practice to treat the nuclear configurational space X as a classical object. In this approach the forces on the nuclei are needed. The Hellman-Feynman formula for the force acting on the k-th nuclei (position coordinate Xk) is given by ... 

It is clear that arbitrary one-particle densities of a molecular system need not have the same topology. In fact, only those belonging to the same structural region will share this property. To make these concepts clearer, consider two nuclear configurations X and Y belonging to the nuclear configuration space in the context of the Born-Oppenheimer approximation. The corresponding one-electrons densities are p r X) and p(r T), respectively. Consider the... 

In rigorous quantum mechanics, something like an electronic base function parametrically dependent on nuclear configuration space cannot be. Such dependence would imply that the electronic quantum number of the base function depends upon the particular selected region of nuclear configuration space. 

The difficulty eq. (14) may put to a chemist is that the amplitude s time dependence is given in a closed form integration over the whole electronic and nuclear configuration space must be carried out [10-12]. 

Confining and asymptotic GED states A hint at molecular structure. The change from a configuration space where the nuclear wave function is defined to a special role as the positive source (nuclear) configuration space Q in eq. (7), is achieved by an isometry mapping the distances are invariant but the ideology is changed. 

A rigorous electro-nuclear separability scheme has been examined. Therein, an equivalent positive charge background replaces the nuclear configuration space the coordinates of which form, in real space, the -space. Diabatic potential energy hypersurfaces for isomers of ethylene in -space were calculated by adapting standard quantum chemical packages. 

If the electron density is regarded as a function p(r,K) of the nuclear arrangement K, then the MIDCOs G(a,K) and the enclosed density domains DD(a,K) are also functions of the nuclear configuration K. How these functions vary with K is one of the main questions of chemistry. It is worthwhile to study this question using the nuclear configuration space approach [2,40],... 

Replace the set of second-order coupled equations in the nuclear configuration space by a set of first-order equations appropriate to the time-dependent Schrbdinger equation. 

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