Electron Pair Functions

Wavefunctions for two-electron systems have a number of special properties which result in drastic simplifications in their Cl treatment. It is one of the basic ideas of pair theories to take advantage of these features when dealing with n-electron systems since this leads to considerable technical advantages. 

First of all, two-electron functions factorize into a space part, f , and a spin part, a  

It has already been pointed out by Lowdin and ShulP that 0 can be expanded in terms of simple products of orbitals a,b  

The great advantage of Eqs. (19-21) becomes apparent if we consider linear transformations of orbitals from basis functions (a) to a  

The coefficient matrix specifying 0 according to Eq. (19) transforms as a tensor of rank 2. 

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Different electron-pair functions which belong to a selected final ionic state and their individual (incoherent) constants of proportionality ionization energies are given in eV, the constants of proportionality in kb(eV)-1 056. For the numerical values from the angle-integrated parts in numerical value has been given. 

Final ionic state Double-ionization energy Ef + State of electron-pair function Constant of proportionality [Pg.262]

The only permutations Pr which give non-zero contributions to (118) are those which permute the co-ordinates of an electron pair function, and transpositions Pwhich correspond to a term g from (117). The normalization integral is given simply by... 

An apparent next step is to describe the quantum mechanical state in terms of electron-pair functions, rather than one-electron functions [4], In fact the concept of electron pairs plays an important role in the theory of the chemical bond. 

The simplest form of the VB model follows the Heitler London method for the structure of the H2 molecule an approximate wavefunction is written as a (antisymmetrised) product of electron-pair functions. Each electron-pair function is a product of a (symmetric) spatial function and a (emtisymmetric) singlet spin function. The spatial function is chosen to model the chemist s intuition about the structure of the electron pair usually a simple symmetrised product of (hybrid) atomic orbitals. The resulting total approximate wavefunction will not be a determinant but, since it is a function which is antisymmetric with respect to exchange of electrons coordinates, it must be capable of being expanded as a linear combination of determinants. If polar structures are added to the VB model the same general result must obtain any antisymmetric function may be expressed as a linear combination of determinants of the space/spin functions ... 

The nitrogen in amines is sp hybridized, the nonbonding electron pair functioning as the equivalent of a substituent. This tetrahedral arrangement inverts rapidfy through a planar transition state. 

Stabilizing resonances also occur in other systems. Some well-known ones are the allyl radical and square cyclobutadiene. It has been shown that in these cases, the ground-state wave function is constructed from the out-of-phase combination of the two components [24,30]. In Section HI, it is shown that this is also a necessary result of Pauli s principle and the permutational symmetry of the polyelectronic wave function When the number of electron pairs exchanged in a two-state system is even, the ground state is the out-of-phase combination [28]. Three electrons may be considered as two electron pairs, one of which is half-populated. When both electron pahs are fully populated, an antiaromatic system arises ("Section HI). 

In the transition state region, the spin-pairing change mnst take place. At this nuclear configuration, the electronic wave function may be written as... 

By using the determinant fomi of the electronic wave functions, it is readily shown that a phase-inverting reaction is one in which an even number of election pairs are exchanged, while in a phase-preserving reaction, an odd number of electron pairs are exchanged. This holds for Htickel-type reactions, and is demonstrated in Appendix A. For a definition of Hilckel and Mbbius-type reactions, see Section III. 

The electronic wave functions of the different spin-paired systems are not necessarily linearly independent. Writing out the VB wave function shows that one of them may be expressed as a linear combination of the other two. Nevertheless, each of them is obviously a separate chemical entity, that can he clearly distinguished from the other two. [This is readily checked by considering a hypothetical system containing four isotopic H atoms (H, D, T, and U). The anchors will be HD - - TU, HT - - DU, and HU -I- DT],... 

A more general classification considers the phase of the total electronic wave function [13]. We have treated the case of cyclic polyenes in detail [28,48,49] and showed that for Hiickel systems the ground state may be considered as the combination of two Kekule structures. If the number of electron pairs in the system is odd, the ground state is the in-phase combination, and the system is aromatic. If the number of electron pairs is even (as in cyclobutadiene, pentalene, etc.), the ground state is the out-of-phase combination, and the system is antiaromatic. These ideas are in line with previous work on specific systems [40,50]. 

In general, at least three anchors are required as the basis for the loop, since the motion around a point requires two independent coordinates. However, symmetry sometimes requires a greater number of anchors. A well-known case is the Jahn-Teller degeneracy of perfect pentagons, heptagons, and so on, which will be covered in Section V. Another special case arises when the electronic wave function of one of the anchors is an out-of-phase combination of two spin-paired structures. One of the vibrational modes of the stable molecule in this anchor serves as the out-of-phase coordinate, and the loop is constructed of only two anchors (see Fig. 12). 

With 4) containing a normalization factor and all permutations over the atomic orbital wave functions i (1 = 1,2,... 2n). Likewise, if all electron pairs were exchanged in a cyclic manner, the product wave function, 4>b, has the fonn ... 

According to Eq. (A.4), if < 0, the ground state will be the in-phase combination, and the out-of-phase one, an excited state. On the other hand, if > 0, the ground state will be the out-of-phase combination, while the in-phase one is an excited state. This conclusion is far reaching, since it means that the electronic wave function of the ground state is nonsymmetric in this case, in contrast with common chemical intuition. We show that when an even number of electron pairs is exchanged, this is indeed the case, so that the transition state is the out-of-phase combination. 

Because of the quantum mechanical Uncertainty Principle, quantum m echanics methods treat electrons as indistinguishable particles, This leads to the Paiili Exclusion Pnn ciple, which states that the many-electron wave function—which depends on the coordinates of all the electrons—must change sign whenever two electrons interchange positions. That IS, the wave function must be antisymmetric with respect to pair-wise permutations of the electron coordinates. 

The reactivity of the individual O—P insecticides is determined by the magnitude of the electrophilic character of the phosphoms atom, the strength of the bond P—X, and the steric effects of the substituents. The electrophilic nature of the central P atom is determined by the relative positions of the shared electron pairs, between atoms bonded to phosphoms, and is a function of the relative electronegativities of the two atoms in each bond (P, 2.1 O, 3.5 S, 2.5 N, 3.0 and C, 2.5). Therefore, it is clear that in phosphate esters (P=0) the phosphoms is much more electrophilic and these are more reactive than phosphorothioate esters (P=S). The latter generally are so stable as to be relatively unreactive with AChE. They owe their biological activity to m vivo oxidation by a microsomal oxidase, a reaction that takes place in insect gut and fat body tissues and in the mammalian Hver. A typical example is the oxidation of parathion (61) to paraoxon [311-45-5] (110). 

Neutral compounds such as boron trifluoride and aluminum chloride form Lewis acid-base complexes by accepting an electron pair from the donor molecule. The same functional groups that act as lone-pair donors to metal cations can form complexes with boron trifluoride, aluminum chloride, and related compounds. 

When a Br nsted base functions catalytically by sharing an electron pair with a proton, it is acting as a general base catalyst, but when it shares the electron with an atom other than the proton it is (by definition) acting as a nucleophile. This other atom (electrophilic site) is usually carbon, but in organic chemistry it might also be, for example, phosphorus or silicon, whereas in inorganic chemistry it could be the central metal ion in a coordination complex. Here we consider nucleophilic reactions at unsaturated carbon, primarily at carbonyl carbon. Nucleophilic reactions of carboxylic acid derivatives have been well studied. These acyl transfer reactions can be represented by... 

Because the interelectronic cusp is difficult to describe well with one-electron basis functions, pair correlation energies converge much more slowly (as N" ) than SCF energies (which converge as f ). This fact makes the use of CBS extrapolations of the correlation energy very beneficial in terms of both accuracy and computational cost. 

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 relative importance of tlie different excitations may qualitatively be understood by noting tliat the doubles provide electron correlation for electron pairs, Quadruply excited determinants are important as they primarily correspond to products of double excitations. The singly excited determinants allow inclusion of multi-reference charactei in the wave function, i.e. they allow the orbitals to relax . Although the HF orbitals are optimum for the single determinant wave function, that is no longer the case when man) determinants are included. The triply excited determinants are doubly excited relative tc the singles, and can then be viewed as providing correlation for the multi-reference part of the Cl wave function. 

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