Quantum-mechanical model atoms with orbitals

The Quantum-Mechanical Model Atoms with Orbitals ... 

In the Bohr model, an orbit is a circular path—analogous to a baseball s path—that shows the electron s motion around an atomic nucleus. In the quantum-mechanical model, an orbital is a probability map, analogous to the probability map drawn by our catcher. It shows the relative likelihood of the electron being found at various locations when the atom is probed. Just as the Bohr model has different orbits witir different radii, the quantum-mechanical model has different orbitals with different shapes. 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

In this section, you saw how the ideas of quantum mechanics led to a new, revolutionary atomic model—the quantum mechanical model of the atom. According to this model, electrons have both matter-like and wave-like properties. Their position and momentum cannot both be determined with certainty, so they must be described in terms of probabilities. An orbital represents a mathematical description of the volume of space in which an electron has a high probability of being found. You learned the first three quantum numbers that describe the size, energy, shape, and orientation of an orbital. In the next section, you will use quantum numbers to describe the total number of electrons in an atom and the energy levels in which they are most likely to be found in their ground state. You will also discover how the ideas of quantum mechanics explain the structure and organization of the periodic table. 

You know that a covalent bond involves the sharing of a pair of electrons between two atoms each atom contributes one electron to the shared pair. In some cases, such as the hydronium ion, HsO", one atom contributes both of the electrons to the shared pair. The bond in these cases is called a co-ordinate covalent bond. In terms of the quantum mechanical model, a co-ordinate covalent bond forms when a filled atomic orbital overlaps with an empty atomic orbital. Once a co-ordinate bond is formed, it behaves in the same way as any other single covalent bond. The next Sample Problem involves a polyatomic ion with a co-ordinate covalent bond. 

As far as qualitative quantum-mechanical models are concerned, the first attempts applied the principles of valence-bond theory. For example, the bent geometry of the water molecule was initially interpreted in terms of the overlap of the Is atomic orbitals of the H atoms with two 2p atomic orbitals (with mutually perpendicular axes) of the O atom, each one contributing one electron ... 

The periodic table provides us with an excellent way to organize information about the elements. You should be familiar with the basic layout of the table as well as the names for sped lie groups of elements. The Quantum mechanical model of atomic structure is far too difficult to be explained in detail in an AP Chemistry course. However, some aspects of the theory are appropriate, and you should know them. These include the predicted number and shapes of orbitals in each energy level the number of electrons found in each orbital, sublevel, and energy level and the meaning of the four quantum numbers. 

The idea of electrons existing in definite energy states was fine, but another way had to be devised to describe the location of the electron about the nucleus. The solution to this problem produced the modern model of the atom, often called the quantum mechanical model. In this new model of the hydrogen atom, electrons do not travel in circular orbits but exist in orbitals with three-dimensional shapes that are inconsistent with circular paths. The modern model of the atom treats the electron not as a particle with a definite mass and velocity, but as a wave with the properties of waves. The mathematics of the quantum mechanical model are much more complex, but the results are a great improvement over the Bohr model and are in better agreement with what we know about nature. In the quantum mechanical model of the atom, the location of an electron about the nucleus is described in terms of probability, not paths, and these volumes where the probability of finding the electron is high are called orbitals. 

An orbital is a volume of space about the nucleus where the probability of finding an electron is high. Unlike orbits that are easy to visualize, orbitals have shapes that do not resemble the circular paths of orbits. In the quantum mechanical model of the hydrogen atom, the energy of the electron is accurately known but its location about the nucleus is not known with certainty at any instant. The three-dimensional volumes that represent the orbitals indicate where an electron will likely be at any instant. This uncertainty in location is a necessity of physics. 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

According to the quantum-mechanical model, each energy state of an atom is associated with an atomic orbital, a mathematical function describing an electron s motion in three dimensions. We can know the probability that the electron is within a particular tiny volume of space, but not its exact location. This probability decreases quickly with distance from the nucleus. 

Each solution to the equation (that is, each energy state of the atom) is associated with a given wave function, also called an atomic orbital. It s important to keep in mind that an orbital in the quantum-mechanical model bears no resemblance to an orbit in the Bohr model an orbit was, supposedly, an electron s path around the nucleus, whereas an orbital is a mathematical function with no direct physical meaning. 

The model that we have developed for the structure of atoms has been further refined. This more sophisticated model, known as the quantum mechanical model, retains most of the general features that we have deduced for atomic structure. Within this model, the electrons in atoms occupy specific regions of space known as orbitals, with a maximum of two electrons occupying each orbital. There are three orbitals in a/ subshell and one orbital in each s subshell. The idea that the two electrons in a given orbital must have opposite spins was first proposed by Wolfgang Pauli in 1925, and is known as the Pauli Exclusion Principle. Most general chemistry texts have some discussion of these ideas. An interesting introduction to the ideas of quantum mechanics can be found in Sections 3.13 and 3.15 of Chemistry Structure Dynamics, by J. N. Spencer, G. M. Bodner, and L. H. Rickard (Fourth Edition). You should read the appropriate sections of your text to become familiar with the terms and basic ideas of this model. 

To begin with, we recall that atomic orbitals are mathematical functions that come from the quantum mechanical model for atomic structure. (Section 6.5) To explain molecular geometries, we can assume that the atomic orbitals on an atom (usually the central atom) mix to form new orbitals called hybrid orbitals. The shape of any hybrid orbital is different from the shapes of the original atomic orbitals. The process of mixing atomic orbitals is a mathematical operation called hybridization. The total number of atomic orbitals on an atom remains constant, so the number of hybrid orbitals on an atom equals the number of atomic orbitals that are mixed. 

QUANTUM MECHANICS AND ORBITALS (SECTION 6.5) In the quantum mechanical model of the hydrogen atom, the behavior of the electron is described by mathematical functions called wave functions, denoted with the Greek letter ip. Each allowed wave function has a precisely known energy, but the location of the electron cannot be determined exactly rather, the probability of it being at a particular point in space is given by the probabilify densify, The electron density distribution is a map of the probability of Ending the electron at all points in space. 

How did something as seemingly abstract as the quantum mechanical model of I the atom help the development of modern materials and light sources The concept of orbital shapes and energies ultimately provided ways to predict chemical reactivities and optimize conditions in which reactions occur. Halogen lamps are predicated on the chemical reaction of halogen molecules with the mngsten wire of... 

Figure 3-5 is a blank energy level diagram you can use to depict electrons for any particular atom. Not all the known orbitals and subshells are shown. But with this diagram, you should be able to do most anything you need to. (If you don t have a clue what orbitals, subshells, or all those numbers and letters in the figure have to do with the price of beans, check out the Quantum mechanical model section, earlier in this chapter. Fun read, lemme tell ya.)... 

This is the case, for example, for Mg. For Ca, the 4s and 4p orbitals overlap. There is a continuous energy band, but a band that is fiUed by only two electrons per atom. Mg and Ca are, therefore, metals. This case corresponds closely to the classical Drude model when electrons may take on any energy value. To make the simplest possible quantum mechanical model, we only need to treat the electrons as in the box model. We have to observe the Pauli principle and occupy the orbitals with two electrons until the actual total number has been reached. This is essentially the FEM for metals. 

An example of such an approximation may be found in the applied mathematical field of quantum mechanics, by which the behavior of electrons in molecules is modeled. The classic quantum mechanical model of the behavior of an electron bound to an atomic nucleus is the so-called particle-in-a-box model. In this model, the particle (the electron) can exist only within the confines of the box (the atomic orbital), and because the electron has the properties of an electromagnetic wave as well as those of a physical particle, there are certain restrictions placed on the behavior of the particle. For example, the value of the wave function describing the motion of the electron must be zero at the boundaries of the box. This requires that the motion of the particle can be described only by certain wave functions that, in turn, depend on the dimensions of the box. The problem can be solved mathematically with precision only for the case involving a single electron and a single nuclear proton that defines the box in which the electron is found. The calculated results agree extremely well with observed measurements of electron energy. 

The quantum-mechanical model of the atom replaced the Bohr model in the early twentieth century. In the quantum-mechanical model Bohr orbits are replaced with quantum-mechanical orbitals. Orbitals are different from orbits in that they represent, not specific paths that electrons follow, but probability maps that show a statistical distribution of where the electron is likely to be found. The idea of an orbital is not easy to visualize. Quantum mechanics revolutionized physics and chemistry because in the quantums-mechanical model, electrons do not behave like particles flying through space. We cannot, in general, describe their exact paths. An orbital is a probability map that shows where the electron is likely to be found when the atom is probed it does not represent the exact path that an electron takes as it travels through space. 

The Quantum-Mechanical Model The quantum-mechanical model for the atom describes electron orbitals, which are electron probability maps that show the relative probability of finding an electron in various places surrounding the atomic nucleus. Orbitals are specified with a number (n), called the principal quantum number, and a letter. The principal quantum number (n = 1,2,3. . . ) specifies the principal shell, and the letter (s, p, d, or f) specifies the subshell of the orbital. In the hydrogen atom, the energy of orbitals depends only on n. In multi-electron atoms, the energy ordering is Is 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s. 

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