Electron probability density

It is fortunate that whenever the SCF method is employed for any element or ion having more than one electron, the resulting wave functions always tend to resemble those of the hydrogen atom. Hence, the probability electron densities for the other elements can be compared to the hydrogen/c orbital shapes. However, these wave functions are not identical to those of hydrogen, and the following differences should be noted ... 

Here pyy r ) represents the probability density for finding the 1 electrons at r, and e / mutual Coulomb repulsion between electron density at r and r. ... 

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. 

Total spin den sity reflects th e excess probability of fin din g a versus P electrons in an open-shell system. Tor a system m which the a electron density is equal to the P electron density (for example, a closed-shell system), the spin density is zero. 

Chemists are able to do research much more efficiently if they have a model for understanding chemistry. Population analysis is a mathematical way of partitioning a wave function or electron density into charges on the nuclei, bond orders, and other related information. These are probably the most widely used results that are not experimentally observable. 

More recent developments are based on the finding, that the d-orbitals of silicon, sulfur, phosphorus and certain transition metals may also stabilize a negative charge on a carbon atom. This is probably caused by a partial transfer of electron density from the carbanion into empty low-energy d-orbitals of the hetero atom ( backbonding ) or by the formation of ylides , in which a positively charged onium centre is adjacent to the carbanion and stabilization occurs by ylene formation. 

As mentioned, highly ortho polymers are somewhat faster than others. This is probably due to a slight elevation in electron density that is the general case for para positions [91,96]. 

When written in this way it is clear what is happening. The mechanisms of these reactions are probably similar, despite the different p values. The distinction is that in Reaction 10 the substituent X is on the substrate, its usual location but in Reaction 15 the substituent changes have been made on the reagent. Thus, electron-withdrawing substituents on the benzoyl chloride render the carbonyl carbon more positive and more susceptible to nucleophilic attack, whereas electron-donating substituents on the aniline increase the electron density on nitrogen, also facilitating nucleophilic attack. The mechanism may be an addition-elimination via a tetrahedral intermediate ... 

At a physical level. Equation 35 represents a mixing of all of the possible electronic states of the molecule, all of which have some probability of being attained according to the laws of quantum mechanics. Full Cl is the most complete non-relativistic treatment of the molecular system possible, within the limitations imposed by the chosen basis set. It represents the possible quantum states of the system while modelling the electron density in accordance with the definition (and constraints) of the basis set in use. For this reason, it appears in the rightmost column of the following methods chart ... 

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]

In our discussion of the electron density in Chapter 5, I mentioned the density functions pi(xi) and p2(xi,X2). I have used the composite space-spin variable X to include both the spatial variables r and the spin variable s. These density functions have a probabilistic interpretation pi(xi)dridii gives the chance of finding an electron in the element dri d i of space and spin, whilst P2(X], X2) dt] d i dt2 di2 gives the chance of finding simultaneously electron 1 in dri dii and electron 2 in dr2di2- The two-electron density function gives information as to how the motion of any pair of electrons is correlated. For independent particles, these probabilities are independent and so we would expect... 

Since the coiTelation between opposite spins has both intra- and inter-orbital contributions, it will be larger than the correlation between electrons having the same spin. The Pauli principle (or equivalently the antisymmetry of the wave function) has the consequence that there is no intraorbital conelation from electron pairs with the same spin. The opposite spin correlation is sometimes called the Coulomb correlation, while the same spin correlation is called the Fermi correlation, i.e. the Coulomb correlation is the largest contribution. Another way of looking at electron correlation is in terms of the electron density. In the immediate vicinity of an electron, here is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, the corresponding phenomenon for electrons of the same spin is the Fermi hole. 

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. 

It has already been pointed out that nitrosation is probably the first step in diazotization. Ingold (1952) describes the reaction as N-nitrosation and classifies it as an electrophilic substitution, together with related processes such as the formation of 4-nitrosophenol, an example of a C-nitrosation. It was probably Adamson and Kenner (1934) who first applied these ideas to diazotization and realized that in aniline itself the electron density at the nitrogen atom is greater than in the anilinium ion, so that the base is more reactive. On the other hand, the nitrosoacidium ion (3.1), the addition product of nitrous acid and a proton, is a more powerful electrophilic reagent than the HN02 molecule. They therefore represented the first step of diazotization as in Scheme 3-5. 

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. 

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