Probability density of electron

For low density electron ensembles such as electrons in semiconductors, where electrons are usually allowed to occupy energy bands much higher and much lower than the Fermi level, the probability density of electron energy distribution may be approximated by the Boltzmann fimction of Eqn. 1-3, as shown in Fig. 1-3. The total concentration, n.,of electrons that occupy the allowed electron... 

Fig. 1-3. Probability density of electron energy distribution, fli), state density, D(t), and occupied electron density. Die) fit), in an allowed energy band much higher than the Fermi level in solid semiconductors, where the Boltzmann function is applicable.

Fig. 6-4. Electron energy levels of an adsorbate particle broadened by interaction with adsorbent metal crystal M adsorbent metal R = atomic adsorbate particle = adsorbed particle W= probability density of electron energy states x = distance to adsorbate particle, xq = distance to adsorbed particle.

Fig. 8-33. Energy diagram showing a shift of redox electron level due to complexation of reductant and oxidant particles (1) afSnity for complexation is greater with oxidants than with reductants, (2) affinity for complexation is greater with reductants than with oxidants. COMPLEX z ligand-coordinated complex redox particles HYDRATE = simply hydrated redox particles W = probability density of electron states e., ) - standard Fermi level of hydrated redox particles - standard Fermi level of ligand-coordinated...

Since DFT calculations are in principle only applicable for the electronic ground state, they cannot be used in order to describe electronic excitations. Still it is possible to treat electronic exciations from first principles by either using quantum chemistry methods [114] or time-dependent density-functional theory (TDDFT) [115,116], First attempts have been done in order to calculate the chemicurrent created by an atom incident on a metal surface based on time-dependent density functional theory [117, 118]. In this approach, three independent steps are preformed. First, a conventional Kohn-Sham DFT calculation is performed in order to evaluate the ground state potential energy surface. Then, the resulting Kohn-Sham states are used in the framework of time-dependent DFT in order to obtain a position dependent friction coefficient. Finally, this friction coefficient is used in a forced oscillator model in which the probability density of electron-hole pair excitations caused by the classical motion of the incident atom is estimated. 

Finally, we define the total electron density in ra as the sum of the probability densities of electrons 1 and 2 ... 

The probability density of electrons p x) in a quantum-mechanical system is given by the diagonal element of the first-order reduced-density matrix, (the superscript s indicates... 

In Equation 1.86, we found that if we try to put two electrons with the same spin at the same point, the wave function is equal to zero. It is quite easy to see in Equation 1.87 that if the two electrons approach each other, the determinant wave function tends to zero and is proportional to the distance between them, 6. (Set 1 = fitt and 2 = 2a = (5 + 5)a in Equation 1.87 and use the Taylor expansion to get Vji( 2) = Vji(h + 5). The result is a sum of two Slater determinants where one has two columns equal and the other one is proportional to 5.) This means that the density of electrons with the same spin, that is, the absolute square of the wave function, tends to zero as 5. If the position of an electron is assumed fixed, the probability density of electrons with the same spin tends to zero near to the fixed electron. The excluded probability density amounts to a full electron, as will be proven for a Slater determinant in Chapter 2. This hole is called the exchange hole. Electrons with the same spin are thus correlated in a Slater determinant. The correlation problem is the problem of accounting for a correlated motion between the electrons. 

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... 

The nature of the intemuclear distance, r, is the object of interest in this chapter. In Eq. (5.1) it has the meaning of an instantaneous distance i.e., at the instant when a single electron is scattered by a particular molecule, r is the value that is evoked by the measurement in accordance with the probability density of the molecular state. Thus, when electrons are scattered by an ensemble of molecules in a given vibrational state v, characterized by the wave function r /v(r), the molecular intensities, Iv(s), are obtained by averaging the electron diffraction operator over the vibrational probability density. 

The probability density of an electron with amplitude (wave function) / is /2. The s-type (spherical) wave functions, / for the first few principal quantum numbers (n = 1,2,3. ..) are ... 

Since the probability density of finding an electron at r is p(r)/N, one expects the probability density E2(r, r ) that one electron is at r and another at r, would be given by multiplying the probability density p(r)/N that an electron is found at r and the probability density that another electron (from the N 1 left) is found at r. Normally one would calculate the latter by subtracting from density p(r ), the average density of one electron p(r )/N and dividing the resulting expression by (N— 1). Thus,... 

Fig. 1-2. Energy distribution of electrons near the Fermi level, cf> in metal crystals c = electron energy f(.i) s distribution function (probability density) ZXe) = electron state density, = occupied...

Fig. 6-68. Surface states created by oovsdently adsorbed particles on semiconductor electrodes BL = bonding level in adsorption = electron donor level D ABL = antibonding level in adsorption = electron acceptor level A W. = probability density of adsorption-induced surface state.

The state density of electrons ZXe) in the reductant and oxidant particles is given in Eqn. 8-2 by the product of the probability density W(e) and the particle concentration c as has been shown in Eqns. 2-48 and 2-49 ... 

The solution of the Schrodinger equation gives mathematical form to the wave functions which describe the locations of electrons in atoms. The wave function is represented by /, which is such that its square, i /2, is the probability density of finding an electron. 

This mathematical model for the probability densities of various electron orbits allowed physicists to develop visualization tools. For example, in 1931, long before computer visualizations were possible, an article in Physics Review [Wh] featured a mechanical device (see Figure 7.3) designed to create images of the shapes of the electron orbitals (see Figure 7.4). There are many pictures of electron orbitals available on the internet. See for example [Co]. 

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