Antisymmetrized wave function, electronic

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

Consider what happens to the many-electron wave function when two electrons have identical coordinates. Since the electrons have the same coordinates, they are indistinguishable the wave function should be the same if they trade positions. Yet the Exclusion Principle requires that the wave function change sign. Only a zero value for the wave function can satisfy these two conditions, identity of coordinates and an antisymmetric wave function. Eor the hydrogen molecule, the antisymmetric wave function is a(l)b(l)-... 

Every normalizable antisymmetric wave function can be expressed as the sum of a series of Slater determinants built up from a complete basic set of one-electron functions. 

As pointed out in Section 7.2, electrons, protons, and neutrons have spin f. Therefore, a system of N electrons, or N protons, or N neutrons possesses an antisymmetric wave function. A symmetric wave function is not allowed. Nuclei of " He and atoms of " He have spin 0, while photons and nuclei have spin 1. Accordingly, these particles possess symmetric wave functions, never antisymmetric wave functions. If a system is composed of several kinds of particles, then its wave function must be separately symmetric or antisymmetric with respect to each type of particle. For example, the wave function for... 

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

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... 

In order to connect this variational principle to density functional theory we perform the search defined in equation (4-13) in two separate steps first, we search over the subset of all the infinitely many antisymmetric wave functions Px that upon quadrature yield a particular density px (under the constraint that the density integrates to the correct number of electrons). The result of this search is the wave function vFxin that yields the lowest... 

The requirement that electrons have antisymmetrical wave functions is called the Pauli principle, which can be stated as follows ... 

At this point, it is necessary to say a few words about the v-representability of the electron density. An electron density is said to be v-representable if it is associated with the antisymmetric wave function of the ground state, corresponding to an external potential v(r), which may or may not be a Coulomb potential. Not all densities are v-representable. Furthermore, the necessary and sufficient conditions for the v-representability of an electron density are unknown. Fortunately, since the /V-representability (see Section 4.2) of the electron density is a weaker condition than v-representability, one needs to formulate DFT only in terms of /V-representable densities without unduly worrying about v-representability. 

It is evident that Lennard-Jones was following the track opened by Lewis, by concentrating on the pair of electrons. To get some insight into P2(xi, x2), it is natural to start with the simplest antisymmetric wave function, a Slater determinant constructed by real orbitals. In this case, one obtains... 

As the electrons are indistinguishable in the antisymmetrized wave function, the one-electron scattering can be obtained by integration over all coordinates but those of they th electron. Summation over all equivalent electrons then leads to... 

The atom-centered models do not account explicitly for the two-center density terms in Eq. (3.7). This is less of a limitation than might be expected, because the density in the bonds projects quite efficiently in the atomic functions, provided they are sufficiently diffuse. While the two-center density can readily be included in the calculation of a molecular scattering factor based on a theoretical density, simultaneous least-squares adjustment of one- and two-center population parameters leads to large correlations (Jones et al. 1972). It is, in principle, possible to reduce such correlations by introducing quantum-mechanical constraints, such as the requirement that the electron density corresponds to an antisymmetrized wave function (Massa and Clinton 1972, Frishberg and Massa 1981, Massa et al. 1985). No practical method for this purpose has been developed at this time. 

In order for to embody the Pauli exclusion principle, it must be an antisymmetrized wave function. Antisymmetrization requires that exchange of any two electrons between orbitals or exchange of the spins between electrons in the same orbital causes 4/ to change sign. 

In this case wiy (i = 1,2,3) are antisymmetrical wave functions formed from atomic one-electron orbitals. The complete wave function was written as a linear combination of the ipi... 

The wave function for a molecule with a nuclear configuration that has a plane of symmetry must be either symmetric or antisymmetric in the plane. In the simple molecular-orbital treatment an antisymmetric wave function for the molecule results from occupancy of antisymmetric orbitals by an odd number of electrons. 

The zeroth-order antisymmetric wave functions of closed-subshell states, states that have only one electron outside a closed-subshell configuration, and states that are one electron short of having a closed-subshell configuration can be expressed as a single Slater determinant [e.g. (1.259)]. However, for open-subshell states in general, one has to take an appropriate linear combination of a few Slater determinants to get a state that is an eigenfunction of L2 and S2. 

We say that the wave function is antisymmetric. As it turns out, nature demands many-electron wave functions to be antisymmetric. Such an antisymmetric wave function can be written as a determinant ... 

These simple determinantal functions for two electrons suggest that we can construct antisymmetric wave functions for any number of electrons in a similar manner. Thus if we have a set of orbitals />, y>n each... 

The antisymmetry principle is also of great importance in understanding the dualism between localised and delocalised descriptions of electronic structure. We shall see that these are just different ways of building up the same total determinantal wave functions.1 This can be developed mathematically from general properties of determinants, but a clearer picture can be formed if we make a detailed study of the antisymmetric wave function for some highly simplified model systems. 

For the next example, consider a system of three electrons of the same spin occupying atomic orbitals 2s, 2px, and 2py. Here the antisymmetric wave function is... 

Particles can be classified as fermions or bosons. Fermions obey the Pauli principle and have antisymmetric wave functions and half-integer spins. (Neutrons, protons, and electrons are fermions.) Bosons do not obey the Pauli principle and have symmetric wave functions and integer spins. (Photons are bosons.)... 

The efficient way of constructing the wave function of the states of equivalent electrons permitted by the Pauli exclusion principle is by utilization of the methods of the coefficients of fractional parentage (CFP). The antisymmetric wave function xp(lNolLSMlMs) of a shell nlN is constructed in a recurrent way starting with the antisymmetric wave function of N— 1 electrons xp(lN lociLiSiMLlMsl). Let us construct the following wave function of coupled momenta ... 

However, this is not the case for the dN shell. For d3 there are two of the same 2D terms. This problem with the dN shell and, partially, the fN shell was solved by Racah in his paper [23] introducing the seniority quantum number. In accordance with formula (9.7) we can build the antisymmetric wave function of shell lN with the help of the CFP with one detached electron. However, we could also use the CFP with two detached electrons... 

Thus, the normalized, completely antisymmetrized wave function of JV-electron atom is, similarly to (10.8),... 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

It is to be stressed that, although the two-electron submatrix elements in (14.63) and (14.65) are defined relative to non-antisymmetric wave functions, some constraints on the possible values of orbital and spin momenta of the two particles are imposed in an implicit form by second-quantization operators. Really, tensorial products (14.40) and (14.42), when the sum of ranks is odd, are zero. Thus, the appropriate terms in (14.63) and (14.65) then also vanish. 

Here the wave function under the sign of summation is defined according to (9.6), however, the coordinates are specified in the order as indicated in the lower row. Multiplying (15.17) by the conjugate, completely antisymmetric wave function for N electrons and integrating gives... 

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