Quantum mechanical descriptions of atoms

In the quantum-mechanical description of atoms and molecules, electrons have characteristics of waves as well as particles. In the familiar case of the hydrogen atom, the orbitals Is, 2s, 2p,... describe the different possible standing wave patterns of electron distribution, for a single electron moving in the potential field of a proton. The motion of the electrons in any atom or molecule is described as fully as possibly by a set of wave functions associated with the ground and excited states. 

The Schrodinger wave equation, Hip = Eip, lies at the heart of the quantum mechanical description of atoms. Recall from the preceding discussion that H represents an operator (the Hamiltonian) that extracts the total energy E (the sum of the potential and kinetic energies) from the wave function. The wave function ip depends on the x, y, and z coordinates of the electron s position in space. 

The Lewis theory of chemical bonding provides a relatively simple way for us to visualize the arrangement of electrons in molecules. It is insufficient, however, to explain the differences beuveen the covalent bonds in compounds such as Hi, Fi, and HF. Although Lewis theory describes the bonds in these three molecules in exactly the same way, they really are quite different from one another, as evidenced by their bond lengths and bond enthalpies listed in Table 9.3. Understanding these differences and why covalent bonds form in the first place requires a bonding model that combines Lewis s notion of atoms sharing election pairs and the quantum mechanical descriptions of atomic orbitals. 

This discussion may well leave one wondering what role reality plays in computation chemistry. Only some things are known exactly. For example, the quantum mechanical description of the hydrogen atom matches the observed spectrum as accurately as any experiment ever done. If an approximation is used, one must ask how accurate an answer should be. Computations of the energetics of molecules and reactions often attempt to attain what is called chemical accuracy, meaning an error of less than about 1 kcal/mol. This is suf-hcient to describe van der Waals interactions, the weakest interaction considered to affect most chemistry. Most chemists have no use for answers more accurate than this. 

According to the quantum mechanical description of the 1 s orbital of the hydrogen atom, what relation exists between the surface of a sphere centered about the nucleus and the location of an electron ... 

Unfortunately, the Schrodinger equation for multi-electron atoms and, for that matter, all molecules cannot be solved exactly and does not lead to an analogous expression to Equation 4.5 for the quantised energy levels. Even for simple atoms such as sodium the number of interactions between the particles increases rapidly. Sodium contains 11 electrons and so the correct quantum mechanical description of the atom has to include 11 nucleus-electron interactions, 55 electron-electron repulsion interactions and the correct description of the kinetic energy of the nucleus and the electrons - a further 12 terms in the Hamiltonian. The analysis of many-electron atomic spectra is complicated and beyond the scope of this book, but it was one such analysis performed by Sir Norman Lockyer that led to the discovery of helium on the Sun before it was discovered on the Earth. 

Wavefunction The quantum mechanical description of a system such as an atom or molecule. Information about the system is derived by operating on the wavefunction with the appropriate operator. 

Solution of (12) gives the complete non-relativistic quantum-mechanical description of the hydrogen atom in its stationary states. The wave function is interpreted in terms of... 

An understanding of the structure of molecules requires a proper quantum mechanical description of the covalent bond that cannot be captured by the use of central pair potentials. We therefore extend our linear combination of atomic orbitals (LCAO) treatment of the s-valent dimer to three-, four-, five-, and six-atom molecules respectively. Following eqs (3.46) and (4.17), we write the binding energy per atom for an. -atom molecule as... 

If a solution, being in contact with an electrode, contains photosensitive atoms or molecules, irradiation of such a system may lead to photoelectro-chemical reactions or, to be more exact, electrochemical reactions with excited particles involved. In such reactions the electrons pass either from an excited particle to the electrode (the anodic process) or from the electrode to an excited particle (the cathodic process). In this case, an elementary act of charge transfer has much in common with ordinary (dark) electrochemical redox reactions, which opens a possibility of interpreting certain aspects of photochemical processes under consideration with the use of concepts developed for general quantum mechanical description of electrode processes. 

Quantum-mechanical Description of the Quadrivalent Carbon Atom. —The description that is given above of the quadrivalent carbon atom as forming four sp8 bonds is somewhat idealized. In a later section (Sec. 4-5) it is pointed out that the bond orbitals have some d and / character. Moreover, the four valence electrons are not closely described by the electron configuration sp8, even aside from the contribution of configurations involving d and / orbitals. v... 

In ihe quantum-mechanical description of a hydrogen atom. Ihe radial portion of the probability densily distribution is the same in all directions from Ihc nucleus, hul only for the case I = 0 is Ihe magnitude of the dislrihulion the same in all radial directions. For all other values of /. the magnitude of the distribution is a function of the angular direction, defined... 

A quantum mechanical theory is in principle needed to describe molecular phenomena in both few-atom and many-atom systems. In some cases a single electronic state is involved, and it is possible to gain valuable insight using only classical molecular dynamics, which can be relatively easy to apply even for a system of many atoms. A quantum mechanical description of molecular phenomena is however clearly needed for electronic states, insofar these have pronounced wavemechanical properties. The need for a quantum description of nuclear motions in molecular dynamics is less apparent, but it is required in some important situations. If we consider a generic interaction between two species A(a) and B(j3) leading to formation of two others, C(7) and D(6), all of them in the specified quantum states, so that... 

In the quantum mechanical description of molecules (atoms and clusters) one problem has been the identification and validity of adiabatic separations of electronic ip) md nuclear (R) coordinates [30] This problem has been with us ever since the Bom-Oppenheimer (BO) theory was published in 1927 [1,2]. But this approach, as implemented in quantum chemistry, has serious deceiving aspects. 

The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64], To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72], They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability aeff to demonstrate that the effect of the image potential on aeff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal-molecule system in a molecularly shaped cavity [62,73-78], In particular, the molecule was treated at the Hartree-Fock, DFT or ZINDO level, while for the metal different models have been explored for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. 

The quantum-mechanical description of the hydrogen atom is of central importance as the only example from which to derive many generalizations... 

The previous chapters have focused on different methods for obtaining more or less accurate solutions to the Schrodinger equation. The natural by-product of determining the electronic wave function is the energy however, there are many other properties that may be derived. Although the quantum mechanical description of a molecule is in terms of positive nuclei surrounded by a cloud ofmegative electrons, chemistry is still formulated as atoms held together by bonds . This raises questions... 

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