Molecules According to Quantum Mechanics

ON THE STRUCTURE OF THE SPECTRA OF TWO-ATOMIC MOLECULES ACCORDING TO QUANTUM MECHANICS... 

On the Structure of the Spectra of Two-Atomic Molecules According to Quantum Mechanics E. Wigner and E. E. Witmer Z. Phys. 51, 859 (1928)... 

In this form the Pauli principle cannot be understood by students who have not studied quantum mechanics and its consequences for the distribution of electrons in a molecule is not apparent. Even before they take a course in quantum mechanics beginning university students are, however, introduced to the idea that the electrons in a molecule are in constant motion and that according to quantum mechanics we cannot determine the path of any one electron but only the probability of finding an electron in an infinitesimal volume surrounding any particular point in space. It can be shown that a consequence of the Pauli principle is that... 

Secondly, information is obtained on the nature of the nuclei in the molecule from the cusp condition [11]. Thirdly, the Hohenberg-Kohn theorem points out that, besides determining the number of electrons, the density also determines the external potential that is present in the molecular Hamiltonian [15]. Once the number of electrons is known from Equation 16.1 and the external potential is determined by the electron density, the Hamiltonian is completely determined. Once the electronic Hamiltonian is determined, one can solve Schrodinger s equation for the wave function, subsequently determining all observable properties of the system. In fact, one can replace the whole set of molecular descriptors by the electron density, because, according to quantum mechanics, all information offered by these descriptors is also available from the electron density. 

Molecular entropies For a perfect monoatomic gas, there is only translational motion. According to quantum mechanics, the translational energy of molecules in a box is quantized and the size of the quantum is proportional to the reciprocal of the atomic weight. Heavier gases have smaller gaps and the number of states available and degeneracies are greater. 

According to classical physics, a solid body like a ball can rotate with any energy. According to quantum mechanics, however, rotational energy is quantized, and a body can rotate only with certain energies—thar is, only at certain speeds. Let s see what that means for a diatomic molecule AB, with atomic masses mA and mn and bond length R. The molecule... 

The reason that chemical laws are not simply reduced to electrostatics is that the electrons behave under the influence of their own or applied electric fields, not according to classical mechanics, but according to quantum mechanics obeying the singular Pauli principle. In fact, electrostatics and dielectric constants are simpler applications of the electrical structure of molecules and use outside macroscopic homogeneous electric fields interacting with microscopic inhomogeneous fields. 

According to quantum mechanics, the vibration is Raman-active if one of these six components of the polarizability changes during the vibration. Thus, it is obvious that the vibration of a homopolar diatomic molecule is Raman-active but not IR-active, whereas the vibration of a heteropolar diatomic molecule is both IR- and Raman-active. 

A diatomic molecule has two axes around which it can physically rotate see Figure 2. These axes are equivalent, and correspond to a single moment of inertia L I determines the spacing of rotational energy levels of the molecule, and is used to define the rotational constant B, where B = h/(8ai I) and h is Planck s constant. According to quantum mechanics, rotational energy levels can only take on certain discrete values, i.e. they are quantized, and we label them with a quantum number called J. The energies of the rotational levels are ... 

According to quantum mechanics, isolated molecules do not have a finite boundary, but rather fade away into the regions of low electron density. It has been well established, however, from properties of condensed matter and molecular interactions, that individual molecules occupy a finite and measurable volume. This notion is at the core of the concept of molecular structure. 33 A number of physical methods yield estimations of molecular dimensions. These methods include measurements of molar volumes in condensed phases, critical parameters (lattice spacings and bond distances), and collision diameters in the gas phase. 34 From these results, one derives values of atomic radii from which a number of empirical molecular surfaces can be built. Note that the values of the atomic radii depend on the physical measurement chosen. 35-i37... 

What would happen if one prepared the system in a given state which does not represent a stationary state For instance, one may deform a molecule by using an electric field and then switch the field off. The molecule will turn out to be suddenly in state x/r i.e., not in its stationary state. Then, according to quantum mechanics, the state of the molecule will start to change according to the time evolution equation (time-dependent Schrddinger equation) ... 

Then, take the electronic charge— physically (in macroscopic or observable sense) indivisible below the elementary charge carry— however, there is rarely integer amounts of elementary charges are attributed to certain atoms in a molecule, according with quantum mechanical calculation at any level of approximation for a given many-electronic many-nuclear bonded structure. 

According to quantum mechanics the two nuclei in a diatomic molecule cannot remain at a constant distance from each other the molecule must vibrate. The vibrational energy levels are determined by the masses of the vibrating nuclei and by the shape of the potential energy curve. 

According to quantum mechanics the allowed vibrational energies of such a molecule is given by... 

According to quantum mechanics, the stationary states of a molecule are described by the eigenfunctions of the Hamiltonian, Ti, corresponding to a quantized set of eigenvalues. 

According to quantum mechanics, any system as an atom or a molecule is described by a function of both the coordinates of the particles constituting the system and the time. They are normally referred to as wave functions, ij/, because of their mathematical expressions which correspond to those traditionally used for describing undulatory processes. itself does not have physical meaning, it is only a mathematical function that can also have negative value or be an... 

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