Quantum mechanical model wave functions

Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

It was particularly the efforts to understand the spectrum of the hydrogen atom that led to the discovery of quantum mechanics. The wave functions of the hydrogen atom function as model orbitals in other atoms and molecules. 

How are the electrons distributed in an atom You might recall from your general chemistry course that, according to the quantum mechanical model, the behavior of a specific electron in an atom can be described by a mathematical expression called a wave equation—the same sort of expression used to describe the motion of waves in a fluid. The solution to a wave equation is called a wave function, or orbital, and is denoted by the Greek letter psi, i/y. 

Dixon et al. [75] use a simple quantum mechanical model to predict the rotational quantum state distribution of OH. As discussed by Clary [78], the component of the molecular wave function that describes dissociation to a particular OH rotational state N is approximated as... 

A new and accurate quantum mechanical model for charge densities obtained from X-ray experiments has been proposed. This model yields an approximate experimental single determinant wave function. The orbitals for this wave function are best described as HF orbitals constrained to give the experimental density to a prescribed accuracy, and they are closely related to the Kohn-Sham orbitals of density functional theory. The model has been demonstrated with calculations on the beryllium crystal. 

As we have seen, the nucleons reside in well-defined orbitals in the nucleus that can be understood in a relatively simple quantum mechanical model, the shell model. In this model, the properties of the nucleus are dominated by the wave functions of the one or two unpaired nucleons. Notice that the bulk of the nucleons, which may even number in the hundreds, only contribute to the overall central potential. These core nucleons cannot be ignored in reality and they give rise to large-scale, macroscopic behavior of the nucleus that is very different from the behavior of single particles. There are two important collective motions of the nucleus that we have already mentioned that we should address collective or overall rotation of deformed nuclei and vibrations of the nuclear shape about a spherical ground-state shape. 

Schrodinger s quantum mechanical model of atomic structure is framed in the form of a wave equation, a mathematical equation similar in form to that used to describe the motion of ordinary waves in fluids. The solutions (there are many) to the wave equation are called wave functions, or orbitals, and are represented by... 

The quantum mechanical model proposed in 1926 by Erwin Schrodinger describes an atom by a mathematical equation similar to that used to describe wave motion. The behavior of each electron in an atom is characterized by a wave function, or orbital, the square of which defines the probability of finding the electron in a given volume of space. Each wave function has a set of three variables, called quantum numbers. The principal quantum number n defines the size of the orbital the angular-momentum quantum number l defines the shape of the orbital and the magnetic quantum number mj defines the spatial orientation of the orbital. In a hydrogen atom, which contains only one electron, the... 

Fig. 4.10. Electron momentum distributions for neon ( 75oi = 0.79 a.u. and /. 02 = 1.51 a.u.) subject to a linearly polarized monochromatic field with frequency ui = 0.057 a.u. and intensity I = 3.0 x 1014W/cm2, as functions of the electron momentum components parallel to the laser-field polarization. The left and the right panels correspond to the classical and to the quantum-mechanical model, respectively. The upper and lower panels have been computed for a contact and Coulomb-type interaction Vi2, respectively. In panels (a) and (d), and (h) and (e), the second electron is taken to be initially in a Is, and in a 2p state, respectively, whereas in panels (c) and (/) the spatial extension of the bound-state wave function has been neglected. The transverse momenta have been integrated over...

In 1926, Erwin Schrodinger used de Broglie s idea that matter has wavelike properties. Schrodinger proposed what is now known as the quantum mechanical model of the atom. In this new model, he abandoned the notion of the electron as a small particle orbiting the nucleus. Instead, he took into account the particle s wavelike properties, and described the behaviour of electrons in terms of wave functions. 

The different contributions to the HF, BLYP and B3LYP interaction energy of CO on the Pt 3(7,3,3) cluster model representation of the Pt(lll) surface as obtained at the various steps of the CSOV method are reported on Figure 2. The most important result of this comparison is that the qualitative picture of the chemisorption bond arising from ab initio HF and DFT quantum chemical approaches is essentially the same the relative importance of the different mechanisms remaining unchanged. This is an important conclusion because it validates many previous analysis of the chemisorption bond carried out in the framework of Hartree-Fock cluster model wave functions. " ... 

In the quantum mechanical model the electron is described as a wave. This representation leads to a series of wave functions (orbitals) that describe the possible energies and spatial distributions available to the electron. 

We adopt a simplified microscopic quantum-mechanical model of a 2D Wannier-Mott exciton, in which the polarization (eqn C.3) can be taken to vanish for L > Lw/2 and inside the well to be given by the product of the Is-wave function of the relative motion of the electron and hole at the origin, with the lowest subband envelope functions for the electron and hole in the approximation of... 

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. 

List the most important ideas of the quantum mechanical model of the atom. Include in your discussion the terms or names wave function, orbital, Heisenberg uncertainty principle, de Broglie, Schrodinger, and probability distribution. 

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 0. 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 probabWty density, 0. The electron density distribution is a map of the probability of finding the electron at all points in space. 

When n is infinite, the shape of the wave function approaches that of a straight line. The peaks and valleys of the standing wave all blur together into a single continuum. Hence, the quantum mechanical model approximates the classical one at very large values of n, a property which is known as the correspondence princpie. In order for any new theory of the atom to be valid, it must not only explain and predict new behavior but it must also incorporate all the experimental evidence that preceded it. 

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