Probability of finding an electron

In theories of bonding the term is often used to indicate the probability of finding an electron at a particular point. 

Another example of the difficulty is offered in figure B3.1.5. Flere we display on the ordinate, for helium s (Is ) state, the probability of finding an electron whose distance from the Fie nucleus is 0.13 A (tlie peak of the Is orbital s density) and whose angular coordinate relative to that of the other electron is plotted on the abscissa. The Fie nucleus is at the origin and the second electron also has a radial coordinate of 0.13 A. As the relative angular coordinate varies away from 0°, the electrons move apart near 0°, the electrons approach one another. Since both electrons have opposite spin in this state, their mutual Coulomb repulsion alone acts to keep them apart. 

Note that the sums of the squares of the coefficients in a given MO must equal 1 (e.g., 0.3717 + 0.6015 + 0.3717 + 0.6015 = 1.0 for Pi) because each of the AOs represents a probability distribution of finding the electron at a given point in space. The total probability of finding an electron in all space for an MO must be unity, exactly as for its constituent AOs. We now can see that the LCAO approximation is only one of many possibilities to describe the electron density (= probability) for MOs. We do not have to express the electron density as a linear combination of the electron densities of AOs centered at the atoms. We could also... 

Electron density represents the probability of finding an electron at a poin t in space. It is calcii lated from th e elements of th e den sity matrix. The total electron density is the sum of the densities for alpha and beta electrons. In a closed-shell RUE calculation, electron densities are the same for alpha and beta electrons. 

The Is orbital /i = is correct but not normalized. The normalized function governing the probability of finding an electron at some distance r along a fixed axis measured from the nucleus in units of the Bohr radius oq = 5.292 x 10 " m is... 

What is the probability of finding an electron between 0.6 and 1.2 Bohr radii of the nucleus. Assume the electron to be in the Is orbital of hydrogen. 

Sketch the probability of finding an electron in the 2s orbital of hydrogen at distance r from a hydrogen nucleus as a function of r as a contour map with heavy lines at high probability and light lines at low probability. How does this distribution differ from the Is orbital ... 

One of the limitations of HF calculations is that they do not include electron correlation. This means that HF takes into account the average affect of electron repulsion, but not the explicit electron-electron interaction. Within HF theory the probability of finding an electron at some location around an atom is determined by the distance from the nucleus but not the distance to the other electrons as shown in Figure 3.1. This is not physically true, but it is the consequence of the central field approximation, which defines the HF method. 

Instead of probability distributions it is more common to represent orbitals by then-boundary surfaces, as shown m Figure 1 2 for the Is and 2s orbitals The boundary sur face encloses the region where the probability of finding an electron is high—on the order of 90-95% Like the probability distribution plot from which it is derived a pic ture of a boundary surface is usually described as a drawing of an orbital... 

FIGURE 1 2 Boundary surfaces of a Is orbital and a 2s orbital The boundary surfaces enclose the volume where there is a 90-95% probability of finding an electron... 

The characteristic feature of valence bond theory is that it pictures a covalent bond between two atoms in terms of an m phase overlap of a half filled orbital of one atom with a half filled orbital of the other illustrated for the case of H2 m Figure 2 3 Two hydrogen atoms each containing an electron m a Is orbital combine so that their orbitals overlap to give a new orbital associated with both of them In phase orbital overlap (con structive interference) increases the probability of finding an electron m the region between the two nuclei where it feels the attractive force of both of them... 

Optically pure (Section 7 4) Descnbing a chiral substance in which only a single enantiomer is present Orbital (Section 1 1) Strictly speaking a wave function i i It is convenient however to think of an orbital in terms of the probability i i of finding an electron at some point relative to the nucleus as the volume inside the boundary surface of an atom or the region in space where the probability of finding an electron is high... 

Boundary surface (Section 1.1) The surface that encloses the region where the probability of finding an electron is high (90-95%). 

The electron density (probability of finding an electron) at a certain position r from a single molecular orbital containing one electron is given as the square of the MO. 

Electron density in H The depth of color is proportional to the probability of finding an electron in a particular region. 

Orbital Region of space in which there is a high probability of finding an electron within an atom, 141 band theory and, 654—655 hybrid, 186,187 relation to ligand, 418... 

Quantum mechanics provides a mathematical framework that leads to expression (4). In addition, for the hydrogen atom it tells us a great deal about how the electron moves about the nucleus. It does not, however, tell us an exact path along which the electron moves. All that can be done is to predict the probability of finding an electron at a given point in space. This probability, considered over a period of time, gives an averaged picture of how an electron behaves. This description of the electron motion is what we have called an orbital. 

The F lends itself to the usual probabilistic interpretation, that is, F P di is equal to the probability of finding an electron in the volume element dv, thus, a plot of this quantity as a... 

Cjl.115 Wavefunctions are normalized to 1. This term means that the total probability of finding an electron in the system is 1. Verify this statement for a particle-in-the-box wavefunction (Eq. 10). 

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