Wave function quantum species

There will be many occasions when we shall need to multiply symmetry species or, in the language of group theory, to obtain their direct product. For example, if H2O is vibrationally excited simultaneously with one quantum each of Vj and V3, the symmetry species of the wave function for this vibrational combination state is... 

In a recent paper. Mo and Gao [5] used a sophisticated computational method [block-localized wave function energy decomposition (BLW-ED)] to decompose the total interaction energy between two prototypical ionic systems, acetate and meth-ylammonium ions, and water into permanent electrostatic (including Pauli exclusion), electronic polarization and charge-transfer contributions. Furthermore, the use of quantum mechanics also enabled them to account for the charge flow between the species involved in the interaction. Their calculations (Table 12.2) demonstrated that the permanent electrostatic interaction energy dominates solute-solvent interactions, as expected in the presence of ion species (76.1 and 84.6% for acetate and methylammonium ions, respectively) and showed the active involvement of solvent molecules in the interaction, even with a small but evident flow of electrons (Eig. 12.3). Evidently, by changing the solvent, different results could be obtained. 

Vibrations may be decomposed into three orthogonal components Ta (a = x, y, z) in three directions. These displacements have the same symmetry properties as cartesian coordinates. Likewise, any rotation may be decomposed into components Ra. The i.r. spanned by translations and rotations must clearly follow the appropriate symmetry type of the point-group character table. In quantum formalism, a transition will be allowed only if the symmetry product of the initial and final-state wave functions contains the symmetry species of the operator appropriate to the transition process. Definition of the symmetry product will be explained in terms of a simple example. 

The hydrogen nucleus is classified as a Eermi particle with nuclear spin I = 1/2. Because of Pauli exclusion principle, hydrogen molecule is classified into two species, ortho and para. Erom the symmetry analysis of the wave functions, para-hydrogen is defined to have even rotational quantum number J with a singlet nuclear spin function, and ortho-hydrogen is defined to have odd J with a triplet nuclear spin function. The interconversion between para and ortho species is extremely slow without the existence of external magnetic perturbation. 

A quantum mechanical separation scheme is introduced between electronic and nuclear degrees of freedom. The electronic stationary wave function determines the stationary geometry ofthe external Coulomb source completing the characterization ofthe molecular species. The hypothesis reflected by the formal equality... 

The physics of condensed phases is commonly formulated as of infinite extent. However, solid and liquid objects in the laboratory are of finite size and terminate discontinuously in a surface (in vacuum) or an interface, under all other conditions. Atoms or molecules at the surface or interface of the condensed object find themselves in a completely different environment, compared to those in the interior of the body. They are less confined in at least one direction, which means that the wave function looks different in this direction - it is less classical. It is implied that surface or interfacial species show more quantum-mechanical behaviour, compared to the bulk. This is the basic reason for the special properties of surfaces and the origin of all interfacial phenomena. Surface chemistry should therefore be formulated strictly in terms of quantum theory, but this has never been attempted. In its present state of development it still is an empirical science, although many physico-chemical concepts are introduced to rationalize the behaviour of interfaces. 

In summary, the rigged Born-Oppenheimer framework permits a general description of chemical reactions. By retaining the stationary geometry structures determined with modern electronic wave function methods the relationship between quantum electronic state and molecular species is established. The relaxation processes involve serial changes of quantum states. 

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