Antisymmetry, electrons

The corresponding fiinctions i-, Xj etc. then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in tenns of these in precisely the fonn given by equation (AT 1.69). witli the caveat that each tenn refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fennions and the Pauli exclusion prmciple. Products of the nonnal coordinate fiinctions neveitlieless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron fiinctions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely usefiil conceptual model and a basis for more accurate calculations. 

More general situations have also been considered. For example. Mead [21] considers cases involving degeneracy between two Kramers doublets involving four electronic components a), a ), P), and P ). Equations (4) and (5), coupled with antisymmetry under lead to the following identities between the various matrix elements... 

X molecular spin orbitals must be different from one another in a way that satisfies the Exclusion Principle. Because the wave function IS written as a determinan t. in torch an gin g two rows of Ihe determinant corresponds to interchanging th e coordin ates of Ihe two electrons. The determinant changes sign according to the antisymmetry requirement. It also changes sign when tw O col-uni n s arc in tcrch an ged th is correspon ds to in Lerch an gin g two spin orbitals. 

A determinant is the most convenient way to write down the permitted functional forms of a polv electronic wavefunction that satisfies the antisymmetry principle. In general, if we have electrons in spin orbitals Xi,X2, , Xn (where each spin orbital is the product of a spatial function and a spin function) then an acceptable form of the wavefunction is ... 

Electronic Wavefuntions Must be Constructed to Have Permutational Antisymmetry Because the N Electrons are Indistinguishable Eermions... 

The Exclusion Principle is quantum mechanical in nature, and outside the realm of everyday, classical experience. Think of it as the inherent tendency of electrons to stay away from one another to be mutually excluded. Exclusion is due to the antisymmetry of the wave function and not to electrostatic coulomb repulsion between two electrons. Exclusion exists even in the absence of electrostatic repulsions. 

For an electronic wavefunction, antisymmetry is a physical requirement following from the fact that electrons are fermions. It is essentially a requirement that y agree with the results of experimental physics. More specifically, this requirement means that any valid wavefunction must satisfy the following condition ... 

In the Hartree-Fock model, where we take account of antisymmetry, it turns out that there is no correlation between the positions of electrons of opposite spin, yet,... 

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 fundamental laws which determine the behavior of an electronic system are the Schrodinger equation (Eq. II. 1) and the Pauli exclusion principle expressed in the form of the antisymmetry requirement (Eq. II.2). We note that even the latter auxiliary condition introduces a certain correlation between the movements of the electrons. 

This theorem follows from the antisymmetry requirement (Eq. II.2) and is thus an expression for Pauli s exclusion principle. In the naive formulation of this principle, each spin orbital could be either empty or fully occupied by one electron which then would exclude any other electron from entering the same orbital. This simple model has been mathematically formulated in the Hartree-Fock scheme based on Eq. 11.38, where the form of the first-order density matrix p(x v xx) indicates that each one of the Hartree-Fock functions rplt y)2,. . ., pN is fully occupied by one electron. 

Brickstock, A., and Pople, J. A., Phil. Mag. 44, 697, The spatial correlation of electrons in atoms and molecules. III. The influence of spin and antisymmetry on the correlation of electrons/ A discussion is made of the refinement needed, if the interelectronic repulsion is taken into account beyond the single determinant, b. 

The original VB wave function was introduced in the treatment of the hydrogen molecule by Heitler and London in 1932. This treatment considered only the one Is orbital on each hydrogen atom and assumed that the best wave function for a system of two electrons on two different atoms is a product of the two atomic Is orbitals i/ — XisXis- This wave function needs to be modified, however, to accommodate the antisymmetry of the wave function and to take into account the spin of the two electrons. 

The orbital phase continuity conditions stem from the intrinsic property of electrons. Electrons are fermions, and are described by wavefnnctions antisymmetric (change plus and minus signs) with respect to an interchange of the coordinates of an pair of particles. The antisymmetry principle is a more fnndamental principle than Pauli s exclusion principle. Slater determinants are antisymmetric, which is why the overlap integral between t(a c) given above has a negative... 

We will soon encounter the enormous consequences of this antisymmetry principle, which represents the quantum-mechanical generalization of Pauli s exclusion principle ( no two electrons can occupy the same state ). A logical consequence of the probability interpretation of the wave function is that the integral of equation (1-7) over the full range of all variables equals one. In other words, the probability of finding the N electrons anywhere in space must be exactly unity,... 

However, billiard balls are a pretty bad model for electrons. First of all, as discussed above, electrons are fermions and therefore have an antisymmetric wave function. Second, they are charged particles and interact through the Coulomb repulsion they try to stay away from each other as much as possible. Both of these properties heavily influence the pair density and we will now enter an in-depth discussion of these effects. Let us begin with an exposition of the consequences of the antisymmetry of the wave function. This is most easily done if we introduce the concept of the reduced density matrix for two electrons, which we call y2. This is a simple generalization of p2(x1 x2) given above according to... 

This can only be true if p2 (xj, Xj) = 0. In other words, this result tells us that the probability of finding two electrons with the same spin at the same point in space is exactly zero. Hence, electrons of like spin do not move independently from each other. It is important to realize that this kind of correlation is in no way connected to the charge of the electrons but is a direct consequence of the Pauli principle. It applies equally well to neutral fermions and - also this is very important to keep in mind - does not hold if the two electrons have different spin. This effect is known as exchange or Fermi correlation. As we will show below, this kind of correlation is included in the Hartree-Fock approach due to the antisymmetry of a Slater determinant and therefore has nothing to do with the correlation energy E discussed in the previous chapter. 

Let us introduce another early example by Slater, 1951, where the electron density is exploited as the central quantity. This approach was originally constructed not with density functional theory in mind, but as an approximation to the non-local and complicated exchange contribution of the Hartree-Fock scheme. We have seen in the previous chapter that the exchange contribution stemming from the antisymmetry of the wave function can be expressed as the interaction between the charge density of spin o and the Fermi hole of the same spin... 

The method of assuring the antisymmetry of a system of electrons, asjfpr example in a polyelectronic atom, is to construct what is often called the Slater determinant.1 If the N elections are numbered 1,2,3,... and each can occupy a state a, b, c,.... the determinant... 

The Pauli exclusion principle requires that no two electrons can occupy the same spin-orbital is a consequence of the more general Pauli antisymmetry principle ... 

The physical consequence of this is that two electrons of the same spin have zero probability of occupying the same position in space that is, two same-spin electrons exclude each other in space. Since Y is continuous, there is only a small probability of finding two electrons of the same spin close to each other in space that is, the Pauli antisymmetry requirement... 

The Pauli antisymmetry principle is a requirement a many-electron wavefunction must obey. A many-electron wavefunction must be antisymmetric (i.e. changes sign) to the interchange of the spatial and spin coordinates of any pair of electrons i and/, that is ... 

For the conduction electrons, it is reasonable to consider that the inner-shell electrons are all localized on individual nuclei, in wave functions very much like those they occupy in the free atoms. The potential V should then include the potential due to the positively charged ions, each consisting of a nucleus plus filled inner shells of electrons, and the self-consistent potential (coulomb plus exchange) of the conduction electrons. However, the potential of an ion core must include the effect of exchange or antisymmetry with the inner-shell or core electrons, which means that the conduction-band wave functions must be orthogonal to the core-electron wave functions. This is the basis of the orthogonalized-plane-wave method, which has been successfully used to calculate band structures for many metals.41... 

The requirement of overall exchange antisymmetry of the /V-clcct.ron wavefunction [Pg.36]

As was mentioned previously, simple orbital products (electron configurations) must be converted into antisymmetrized orbital products (Slater determinants) in order to satisfy the Pauli principle. Thus, proper many-electron wavefunctions satisfy constraints of exchange antisymmetry that have no counterpart in pre-quantum theories. 

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