Antisymmetry, of wave function

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. 

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. 

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 opposite of a creation operator is an annihilation operator a which removes orbital i from the wave function it is acting on. The a-a product of operators removes orbital j and creates orbital i, i.e. replaces the occupied orbital j with an unoccupied orbital i. The antisymmetry of the wave function is built into the operators as they obey the following anti-commutation relationships. 

It should be emphasized that not all normalizable hermitean matrices r(x x 2. . . x xlx2. . . xp) having the correct antisymmetry property are necessarily strict density matrices, i.e., are derivable from a wave function W. For instance, for p — N, it is a necessary and sufficient condition that the matrix JT is idempotent, so that r2 = r, Tr (JH) = 1. This means that the F-space goes conceptually outside the -space, which it fully contains. The relation IV. 5 has apparently a meaning within the entire jT-space, independent of whether T is connected with a wave function or not. The question is only which restrictions one has to impose on r in order to secure the validity of the inequality... 

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. 

Considering first the state with a total spin of 0, we note that since the spin wave function is antisymmetric with respect to interchanging the particle labels, the spatial part of the wave function should be symmetric in order to preserve the overall antisymmetry of the wave function. This leads to the following expression for the wave function ... 

Since in those forms of the UHF wave functions, one drops a constraint (either the need of a pure spin state in the first case or the Pauli antisymmetry rule in the second case), it is expected that the resulting wave function will give a lower energy than in the RHF case and thus introduce a part of the correlation energy. As shown in the table above, there is... 

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... 

First of all we note that the Fermi hole - which is due to the antisymmetry of the wave function - dominates by far the Coulomb hole. Second, another, very important property of the Fermi hole is that it, just like the total hole, integrates to -1... 

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... 

Let us explore first the nature of the integrand E cl for the limiting cases. At X = 0 we are dealing with an interaction free system, and the only component which is not included in the classical term is due to the antisymmetry of the fermion wave function. Thus, E j° is composed of exchange only, there is no correlation whatsoever.19 Hence, the X = 0 limit of the integral in equation (6-25) simply corresponds to the exchange contribution of a Slater determinant, as for example, expressed through equation (5-18). Remember, that E ° can... 

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... 

To understand the main idea behind DFT, consider the following. In the absence of magnetic fields, the many-electron Hamiltonian does not act on the electronic spin coordinates, and the antisymmetry and spin restrictions are directly imposed on the wave function (r j, v j,..., rvyv). Within the Bom-Oppenheimer approximation,... 

The Pauli antisymmetry principle tells us that the wave function (including spin degrees of freedom), and thus the basis functions, for a system of identical particles must transform like the totally antisymmetric irreducible representation in the case of fermions, or spin (for odd k) particles, and like the totally symmetric irreducible representation in the case of bosons, or spin k particles (where k may take on only integer values). 

An expression such as (19) is a linear combination of products of the one-electron wave function, and leads to a separation of (17) in a set of N independent equations for the N electrons. It embodies the antisymmetry required by the Pauli principle. 

The minimum requirements for a many-electron wave function, namely, antisymmetry with respect to interchange of electrons and indistinguishability of electrons, are satisfied by an antisymmetrized sum of products of one-electron wave functions (orbitals), ( 1),... 

Since H° is the sum of hydrogenlike Hamiltonians, the zeroth-order wave function is the product of hydrogenlike functions, one for each electron. We call any one-electron spatial wave function an orbital. To allow for electron spin, each spatial orbital is multiplied by a spin function (either a or 0) to give a spin-orbital. To introduce the required antisymmetry into the wave function, we take the zeroth-order wave function as a Slater determinant of spin-orbitals. For example, for the Li ground state, the normalized zeroth-order wave function is... 

Note that if the spin-orbital in column three of (1.259) were the same as that of either column one or column two, the determinant would vanish (since two columns would be identical). Hence antisymmetry of means that no two electrons can occupy the same spin-orbital in an approximate wave function in which electrons are assigned to orbitals. 

The phase factor r(n) is introduced in order to endow the antisymmetry of many-electron wave functions in the Fock space, as we soon will see. The definition that ai operating on an occupation number vector gives zero if spin... 

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