Antisymmetrized wavefunctions

Various types of antisymmetric wavefunction can be obtained by applying different functions of the T operators to fi o. and the unknown coefficients together with the energy can be determined from the projection equations... 

These restrictions, imposed above on electrons, apply equally to all pariiqles that are represented by antisymmetric wavefunctions, the so-called Fermions. The condition that no more than one particle can occupy a given quantum state leads immediately to the expression for the number of possible combinations. If C nhgi) is the number of combinations that can be made with g, particles taken tii at a time,... 

The simplest solution to this problem is to construct an antisymmetric wavefunction using a linear combination of one-electron wavefunctions. For two electrons, this takes the following form ... 

Considering that only two electrons are involved in a transition, one on D and one on A, the properly antisymmetrized wavefunctions for the initial excited state Pi (D excited but not A) and the final excited state vFf (A excited but not D) can be written as... 

In 1979, an elegant proof of the existence was provided by Levy [10]. He demonstrated that the universal variational functional for the electron-electron repulsion energy of an A -representable trial 1-RDM can be obtained by searching all antisymmetric wavefunctions that yield a fixed D. It was shown that the functional does not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus the v-representability is not required, only Al-representability. As a result, the 1-RDM functional theories of preceding works were unified. A year later, Valone [19] extended Levy s pure-state constrained search to include all ensemble representable 1-RDMs. He demonstrated that no new constraints are needed in the occupation-number variation of the energy functional. Diverse con-strained-search density functionals by Lieb [20, 21] also afforded insight into this issue. He proved independently that the constrained minimizations exist. 

A g-density, pg, is N-representable if and only if there exists some set of antisymmetric wavefunctions,, , and weighting coefficients, p,, for which Eq. (2) holds. 

Two orthogonal orbitals that interact, yAB 0, are not completely localized they correspond to Lowdin orthogonal orbitals. Two symmetrical wavefunctions mix ( lA2 + B2) and AB)) giving rise to S0 and S2 while antisymmetric wavefunction A2 -B2) remains an eigenstate ... 

In other words, Ttn] is that antisymmetric wavefunction that minimizes < T + V > and yields the density n(r). 

To assign values to the molecular orbital coefficients, c, many computational methods apply Hartree-Fock theory (which is based on the variational method).44 This uses the result that the calculated energy of a system with an approximate, normalized, antisymmetric wavefunction will be higher than the exact energy, so to obtain the optimal wavefunction (of the single determinant type), the coefficients c should be chosen such that they minimize the energy E, i.e., dEldc = 0. This leads to a set of equations to be solved for cMi known as the Roothaan-Hall equations. For the closed shell case, the equations are... 

For this purpose it would seem profitable to review briefly in the next section some of the, perhaps, less well-known properties of the exact non-relativistic wavefunction, but which are, nevertheless, important when discussing VB theory. Also in this section a short description is given of the construction and manipulation of antisymmetric wavefunctions of more general form than a simple Slater determinant. This is then followed by a brief survey of some of the more commonly used spin functions. 

For one- or two-electron wavefunctions the space and spin parts can be factored. Assume that one of the electrons is in electronic state m, the other in electronic state n. Then one can write antisymmetrized wavefunctions of the type... 

The fundamental result (18) may now be used to construct a totally antisymmetric wavefunction with spin included. This will be, for any choice of A,... 

It is easy to write down the form of an antisymmetric wavefunction for a polyelectronic atom if we recognize that Eq. (5.13) can be written in the... 

There is an obvious vicious circle in this approach if the spatial distribution of each electron is one of the unknowns, how can we speak of averaged distributions The answer is an iterative numerical calculation as demonstrated originally by the British physicist Douglas Hartree in 1928. In 1930, the method was improved by the Russian physicist Vladimir Fock who adapted the method to antisymmetric wavefunctions as required by the Pauli principle. The Hartree-Fock method is a numerical calculation that can be summarized in the following steps ... 

This latter interpretation follows a model developed by J.W. Linnett in 1964 (ref. 107) in which the orbital concept is largely ignored in favour of spin correlation which is a consequence of the antisymmetrization of the total wavefunction demanded by the Pauli principle. In such a model, what matters are the most likely relative positions of the electrons. It can be shown that, with an antisymmetric wavefunction, electrons having parallel spins tend to be as far apart as possible around the nucleus of an atom. Let us take the carbon atom as an example. For its excited valence configuration 2s, 2p, the four electrons have preferably parallel spins (extension of Hund s rule to excited configurations) and, among the infinity of spatial arrangements, the most likely ones are those in which the four electrons define the vertices of a tetrahedron centred at the nucleus. In particular, for... 

Closely related to the strictly variational VB method described so far, there have been a number of recent approaches which use perturbation theory. All of these are characterized by the use of non-orthogonal functions and fully antisymmetrized wavefunctions. In addition the full Hamiltonian is used without a multipole expansion. 

In Hartree-Fock theory, the simplest wavefunction theory involving an antisymmetric wavefunction, the electron repulsion energy of an N-electron system is given by... 

Wavefunctions must be either symmetric (delete the minus sign from Equation 1.12) or antisymmetric in order to be consistent with the Born interpretation electrons being indistinguishable, W2 must be invariant with respect to an interchange of any pair of electrons, because the probability of finding e, in a volume element around the coordinates qej and ey around qe. must be the same when the labels / and j are exchanged. Both symmetric and antisymmetric wavefunctions would satisfy this condition, but the Pauli principle allows only antisymmetric wavefunctions. 

This formulation is not just a mathematical trick to form an antisymmetric wavefunction. Quantum mechanics specifies that an electron s location is not deterministic but rather consists of a probability density in this sense, it can be anywhere. This determinant mixes all of the possible orbitals of all of the electrons in the molecular system to form the wavefunction. 

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