Atomic orbitals probability densities

The magnitude and shape of such a mean-field potential is shown below [21] in figure B3.1.4 for the two 1 s electrons of a beryllium atom. The Be nucleus is at the origin, and one electron is held fixed 0.13 A from the nucleus, the maximum of the Is orbital s radial probability density. The Coulomb potential experienced by the second electron is then a function of the second electron s position along the v-axis (coimecting the Be nucleus and the first electron) and its distance perpendicular to the v-axis. For simplicity, this second electron... 

Its charge density distribution is like that of the cation (with sign reversal) because the added electron goes into the nonbonded orbital with a node at the central carbon atom. The probability of finding that electron precisely at the central carbon atom is zero. 

The magnitude and "shape" of sueh a mean-field potential is shown below for the Beryllium atom. In this figure, the nueleus is at the origin, and one eleetron is plaeed at a distanee from the nueleus equal to the maximum of the Is orbital s radial probability density (near 0.13 A). The radial eoordinate of the seeond is plotted as the abseissa this seeond eleetron is arbitrarily eonstrained to lie on the line eonneeting the nueleus and the first eleetron (along this direetion, the inter-eleetronie interaetions are largest). On the ordinate, there are two quantities plotted (i) the Self-Consistent Field (SCF) mean-field... 

The spacial distribution of electron density in an atom is described by means of atomic orbitals Vr(r, 6, (p) such that for a given orbital xp the function xj/ dv gives the probability of finding the electron in an element of volume dv at a point having the polar coordinates r, 6, 0. Each orbital can be expressed as a product of two functions, i e. 0, [Pg.1285]

All s-orbitals are independent of the angles 0 and c[>, so we say that they are spherically symmetrical (Fig. 1.31). The probability density of an electron at the point (r,0,ct>) when it is in a ls-orbital is found from the wavefunction for the ground state of the hydrogen atom. When we square the wavefunction (which was given earlier, but can also be constructed as RY from the entries for R and V in Tables 1.2a and 1.2b) we find that... 

Tj FIGURE 1.33 The three s-orbitals of 5 lowest energy. The simplest way of drawing an atomic orbital is as a g boundary surface, a surface within which there is a high probability (typically 90%) of finding the electron. We shall use blue to denote s-orbitals, but that color is only an aid to their identification. The shading Jp within the boundary surfaces is an 9 approximate indication of the electron density at each point. 

FIGURE 1.34 The radial wavefunctions of the first three s-orbitals of a hydrogen atom. Note that the number of radial nodes increases (as n 1), as does the average distance of the electron from the nucleus (compare with Fig. 1.32). Because the probability density is given by ip3, all s-orbitals correspond to a nonzero probability density at the nucleus. 

Atomic orbitals are actually graphical representations for mathematical solutions to the Schrodinger wave equation. The equation provides not one, but a series of solutions termed wave functions t[ . The square of the wave function, is proportional to the electron density and thus provides us with the probability of finding an electron within a given space. Calculations have allowed us to appreciate the shape of atomic orbitals for the simplest atom, i.e. hydrogen, and we make the assumption that these shapes also apply for the heavier atoms, like carbon. 

Figure 2.1 Radial probability density plots for Is and 2s orbitals of hydrogen atom...

The wavefunction of an electron associated with an atomic nucleus. The orbital is typically depicted as a three-dimensional electron density cloud. If an electron s azimuthal quantum number (/) is zero, then the atomic orbital is called an s orbital and the electron density graph is spherically symmetric. If I is one, there are three spatially distinct orbitals, all referred to as p orbitals, having a dumb-bell shape with a node in the center where the probability of finding the electron is extremely small. (Note For relativistic considerations, the probability of an electron residing at the node cannot be zero.) Electrons having a quantum number I equal to two are associated with d orbitals. 

The hydrogen atom orbitals are functions of three variables the coordinates of the electron. Their physical interpretation is that the square of the amplitude of the wave function at any point is proportional to the probability of finding a particle at that point. Mathematically, the electron density distribution is equal to the square of the absolute value of the wave function ... 

The further development of the hgand field concept takes place in Molecular Orbitals (MO) Theory. As an atomic orbital is a wave fimction describing the spatial probability density for a single electron bound to the nucleus of an atom, a molecular orbital is a wave function, which describes the spatial probabihty density for a single electron bond to the set of nuclei, which constitute the framework of a molecule. 

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