Nuclear geometry

Consequently, an accurate description of molecular shape should include a smearing effect resulting from nuclear motions and uncertainties. A number of solutions are proposed in the literature, including the use of nuclear wavefunctions, 2 o open sets of nuclear configurations,and a fuzzy set approach to molecular shape. We shall use the term dynamical shape for a description that takes into account nuclear flexibility. Such a characterization is performed with dynamic shape descriptors. The term static shape descriptors is reserved to those defined at a frozen nuclear geometry. 

It is possible to design shape descriptors that incorporate nuclear vibrations. However, the simpler approach to shape dynamics is to apply static descriptors to entire domains of nuclear configurations. We then observe the interrelation between molecular shape and molecular motions (e.g., vibrations, reaction paths).32,33 in this case, we find a range of values in static parameters. The span of such a range (e.g., the fluctuations in a static descriptor) can characterize shape dynamics.3  

The next sections focus mostly on the properties of absolute shape descriptors. Special attention is devoted to those that are used to study static conformations and dynamics. Among the myriad of shape descriptors in the literature, we deal with a subset of those that are conceptually distinct and serve as examples for the construction of many others. 

In addition to parameters commonly used in many applications, this chapter includes some dD descriptors and molecular pD models that are not found in the standard computational chemistry literature. This is the case, for instance, for the topological and geometrical analyses of elastic chains and elastic surfaces. Their inclusion aims at giving the reader a broader perspective on the tools used in other related fields. 

The selected descriptors are defined, and their known properties, practical implementation, and some applications are discussed. The analysis of large molecules is emphasized. 

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The adiabatic picture is the standard one in quantum chemistry for the reason that, not only is it mathematically well defined, but it is also that used in ab initio calculations, which solve the electronic Hamiltonian at a particular nuclear geometry. To see the effects of vibronic coupling on the potential energy surfaces one must move to what is called a diabatic representation [1,65,180, 181]. 

If more than one electronic state is involved, then the electronic wave function is free to contain components from all states. For example, for non-adiabatic systems the elecbonic wave function can be expanded in the adiabatic basis set at the nuclear geometry R t)... 

H(I) and H(II). This fact does not provide any information on the nuclear sti ucture of this species at the energy minimum. By symmetry, it is clear that the system has three equivalent minima on the ground-state surface, which were designated as the three diatomic pairs. The nuclear geometry of each of these minima is quite different from that of the other two. 

In our treatment of molecular systems we first show how to determine the energy for a given iva efunction, and then demonstrate how to calculate the wavefunction for a specific nuclear geometry. In the most popular kind of quantum mechanical calculations performed on molecules each molecular spin orbital is expressed as a linear combination of atomic orhilals (the LCAO approach ). Thus each molecular orbital can be written as a summation of the following form ... 

Once the electronic Schrodinger equation has been solved for a large number of nuclear geometries (and possibly also for several electronic states), the PES is known. This can then be used for solving the nuclear part of the Schrodinger equation. If there are N nuclei, there are 3N coordinates that define the geometry. Of these coordinates, three describe the overall translation of the molecule, and three describe the overall rotation of the molecule with respect to three axes. Eor a linear molecule, only two coordinates are necessary for describing the rotation. This leaves 3N-6(5) coordinates to describe the internal movement of the nuclei, the vibrations, often chosen to be... 

Consider now the Hamilton operator. The nuclear-nuclear repulsion does not depend on electron coordinates and is a constant for a given nuclear geometry. The nuclear-electron attraction is a sum of terms, each depending only on one electron coordinate. The same holds for the electron kinetic energy. The electron-electron repulsion, however, depends on two electron coordinates. 

Another aspect of wave function instability concerns symmetry breaking, i.e. the wave function has a lower symmetry than the nuclear framework. It occurs for example for the allyl radical with an ROHF type wave function. The nuclear geometry has C21, symmetry, but the Cay symmetric wave function corresponds to a (first-order) saddle point. The lowest energy ROHF solution has only Cj symmetry, and corresponds to a localized double bond and a localized electron (radical). Relaxing the double occupancy constraint, and allowing the wave function to become UHF, re-establish the correct Cay symmetry. Such symmetry breaking phenomena usually indicate that the type of wave function used is not flexible enough for even a qualitatively correct description. 

The adiabatic electronic potential energy surfaces (a function of both nuclear geometry and electric field) are obtained by solving the following electronic eigenvalue equation... 

Equation (28) is the set of exact coupled differential equations that must be solved for the nuclear wave functions in the presence of the time-varying electric field. In the spirit of the Born-Oppenheimer approximation, the ENBO approximation assumes that the electronic wave functions can respond immediately to changes in the nuclear geometry and to changes in the electric field and that we can consequently ignore the coupling terms containing... 

Both current methods share the limitation of being conducted for fixed nuclear geometry—typically the ground-state equilibrium geometry. However,... 

This new, approximate macromolecular density matrix (q K ), K [A]) for the new, slightly distorted nuclear geometry K1 is also idempotent with respect to multiplication involving the actual new overlap matrix S(K... 

For the new, slightly distorted macromolecular nuclear geometry K, the electronic density can be expressed as the improved approximation... 

If the original macromolecular density matrix is already available, then such approximate macromolecular electron densities for slightly distorted nuclear geometries are simpler to calculate than the full recalculation of an ADMA macromolecular density matrix that involves a new fragmentation procedure. 

Although FEP is mostly useful for binding type of simulations rather than chemical reactions, it can be valuable for reduction potential and pKa calculations, which are of interest from many perspectives. For example, prediction of reliable pKa values of key groups can be used as a criterion for establishing a reliable microscopic model for complex systems. Technically, FEP calculation with QM/MM potentials is complicated by the fact that QM potentials are non-seperable [78], When the species subject to perturbation (A B) differ mainly in electronic structure but similar in nuclear connectivity (e.g., an oxidation-reduction pair), we find it is beneficial to use the same set of nuclear geometry for the two states [78], i.e., the coupling potential function has the form,... 

In Eq. (2.39), AE(R) is the minimum (over all pairs of electronic states considered) of the absolute value of the energy gap at the given nuclear geometry, threshold is a numerical parameter taken small enough (fa 10 5) that it has no effect on the dynamics, and AEc and AEw are the center and range of the switching function, respectively. 

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