Wave function, antisymmetric

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. 

Dilfusion and Green s function QMC calculations are often done using a fixed-node approximation. Within this scheme, the nodal surfaces used define the state that is obtained as well as ensuring an antisymmetric wave function. 

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)-... 

Systems containing symmetric wave function components ate called Bose-Einstein systems (129) those having antisymmetric wave functions are called Fermi-Ditac systems (130,131). Systems in which all components are at a single quantum state are called MaxweU-Boltzmaim systems (122). Further, a boson is a particle obeying Bose-Einstein statistics, a fermion is one obeying Eermi-Ditac statistics (132). 

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. 

Such a wave function is known as a Slater determinant. In general, when we deal with antisymmetrized wave functions, we use a compact notation for the Slater determinant ... 

In equation (8.32) the operator P is any one of the N operators, including the identity operator, that permute a given order of particles to another order. The summation is taken over all N permutation operators. The quantity dp is always - -1 for the symmetric wave function Ps, but for the antisymmetric wave function Wa, dpis-l-l(—l)if the permutation operator P involves the... 

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. 

Since F(Q) is symmetric (antisymmetric), it may be expanded in terms of a complete set of symmetric (antisymmetric) wave functions v(Q) (we omit the subscript S, A)... 

These four antisymmetric wave functions are normalized if the single-particle spatial wave functions singlet state occurs... 

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 ... 

These two transcendental equations define a pair of closely spaced energy levels, respectively associated with symmetric and antisymmetric wave functions as defined by the arbitrary choice of D = C. 

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... 

In order to leam more about the nature of the intermolecular forces we will start with partitioning of the total molecular energy, AE, into individual contri butions, which are as close as possible to those we defined in intermolecular perturbation theory. Attempts to split AE into suitable parts were undertaken independently by several groups 83-85>. The most detailed scheme of energy partitioning within the framework of MO theory was proposed by Morokuma 85> and his definitions are discussed here ). This analysis starts from antisymmetrized wave functions of the isolated molecules, a and 3, as well as from the complete Hamiltonian of the interacting complex AB. Four different approximative wave functions are used to describe the whole system ... 

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. 

It is a simple exercise to show tliat if cr and tt are orthonormal then a and b are too. Let us now consider the antisymmetric wave function... 

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. 

Particles having half-integral spin and requiring antisymmetric wave functions are called fermions particles having integral spin and requiring symmetric wave functions are called bosons. 

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. 

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