Electrons probabilities of finding

From a quantum mechanical perspective, an atom or molecule would be considered to have no permanent dipole moment if the probability of finding electrons is symmetric about the nucleus. For example the probability of finding the electron in the ground state of hydrogen is constant with respect to its solid... 

The traditional view of molecular bonds is that they are due to an increased probability of finding electrons between two nuclei, as compared to a sum of the contributions of the pure atomic orbitals. The canonical MOs are delocalized over the whole molecule and do not readily reflect this. There is, furthermore, little similarity between MOs for systems which by chemical measures should be similar, such as a series of alkanes. The canonical MOs therefore do not reflect the concept of functional groups. 

This expression indicates that there is a decreased probability (indicated by the term —2(/>A(/>B) of finding the electrons in the region between the two nuclei. In fact, there is a nodal plane between the positive and negative (with respect to algebraic sign) of the two regions of the molecular orbital. As a simple definition, we can describe a covalent bond as the increased probability of finding electrons between two nuclei or an increase in electron density between the two nuclei when compared to the probability or density that would exist simply because of the presence of two atoms. 

Each orbital has its own associated energy, and each represents information about where, inside the atom, the electrons would spend most of their time. Scientists cannot determine the actual paths of the moving electrons. However, orbitals indicate where there is a high probability of finding electrons. 

The wave function P contains all information of the joint probability distribution of the electrons. For example, the two-electron density is obtained from the wave function by integration over the spin and space coordinates of all but two electrons. It describes the joint probability of finding electron 1 at r, and electron 2 at r2. The two-electron density cannot be obtained from elastic Bragg scattering. 

Because we are principally interested in the probability of finding electrons at various points in space, we shall be more concerned with tlie squares of the radial functions than with the functions themselves, it is the square of the wave function... 

Modern atomic theory explains the structures and behaviors of the atom much better than the earlier atomic theories. This theory explains the probability of finding electrons around the nucleus by virtue of quantum numbers and orbitals. The quantum numbers are the integer numbers designating the energy levels of the electrons in an atom, and the orbitals are the probable regions in which the electrons might be found around the nucleus. 

Figure 5.8 The stick picture of pyrrole on which is superimposed the probability of finding electrons at different points in the molecule obtained using quantum mechanics...

Quantum mechanics is useful for calculating the values of ionization potentials, electron affinities, heats of formation and dipole moments and other physical properties of atoms and molecules. It can also be used to calculate the relative probabilities of finding electrons (the electron density) in a structure (Figure 5.8). This makes it possible to determine the most likely points at which... 

The correct answer is (A). This statement is pretty much a paraphrase of Heisenberg s principle, which eventually led to the quantum model of the atom, based on probabilities of finding electrons in certain regions. 

The wave mechanical treatment of the hydrogen atom does not provide more accurate values than the Bohr model did for the energy states of the hydrogen atom. It does, however, provide the basis for describing the probability of finding electrons in certain regions, which is more compatible with the Heisenberg uncertainty principle. Note that the solution of this three-dimensional wave equation resulted in the introduction of three quantum numbers (n, /, and mi). A principle of quantum mechanics predicts that there will be one quantum number for... 

The term a2 0X2 represents the probability of finding electrons from atom (1) and a2 (j)22 is the probability from atom (2). A covalent bond can be defined as the increased probability of finding electrons between two atoms resulting from electron sharing. As shown in Eq. (2.21), the term 2ci a2(j) (j)2 is proportional to the increased probability of finding electrons between the atoms caused by the bond between them. 

In Eq. (2.22), the term -2aia2(j)i(j)2 leads to a decreased probability of finding electrons between the two atoms. In fact, there is a nodal plane between them where the probability goes to zero. 

Second, in some cases, as shown in Figure 2.11, the orbitals overlap so that there is favorable overlap in one region that is canceled in another. The result is that S = 0 and there is no overall increased probability of finding electrons shared between the two atoms. That is, the wave functions are said to be orthogonal, and these cases are referred to as nonbonding. 

Region or volume in space in which the probability of finding electrons is highest. Atomic Radius... 

The traditional view of molecular bonds is that they are due to an increased probability of finding electrons between two nuclei, as ompared to a sum of the -contributions of the pure atomic orbitals. The canonical MOs arc-dclocaUzcd over the... 

To express the wavefunction as a product of functions, one for each ctron, is in agreement with the fact that, in the absence of interaction ctween the electrons, the probability of finding electron 1 in point jJa> a) and electron 2 in point (xt,yb,z is the product of the individual sbabilities < fs(xa,yaj a) and lJ Xb,yh,Zc)- This is the result of the joint abability for two independent events being simply the product of the indi-idual probabilities ... 

We can illustrate this basic principle by considering the combination of the Ir atomic orbitals on two different atoms (Figure 9-2). When these orbitals are occupied by electrons, the shapes of the orbitals are plots of electron density. These plots show the regions in molecules where the probabilities of finding electrons are the greatest. 

By 1935, the current model of the atom had evolved. This model explains electron behavior by interpreting the emission spectra of aU the elements. It pictures energy levels as regions of space where there is a high probability of finding electrons. Before going on to the modern atomic theory, take another look at what you already know about atoms and electrons. 

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