Antisymmetric wavefunction

In order to make the overall wavefunction antisymmetric, this spatial wavefunction has to be multiplied by the antisymmetric spin function, a( )P(2) - a 2)P(l). The wavefunction to be used in calculations is therefore ... 

For atoms with more than two electrons, the wavefunctions must become more elaborate to satisfy the symmetrization principle. However, John Slater developed a general method for reliably generating many-electron spin-spatial wavefunctions, antisymmetric with respect to P21 (exchange of the electron labels 1 and 2), for any number of electrons. We call these wavefunctions Slater determinants, because they are obtained by taking the determinant of a matrix of possible one-electron wavefunctions. For example, for ground state He, there are two possible one-electron spin-spatial wavefunctions for each electron Isa and lsj8. We set up a 2 X 2 matrix in which each row corresponds to a different electron and each column to a different wavefunction ... 

In the circles in the following wavefunction, fill in the correct signs (+ or —) that make the overall wavefunction antisymmetric with respect to exchange of the labels on electrons 1 and 2. 

Assuming, as before, that the singlet state (symmetric spatial wavefunction, antisymmetric spin wavefunction) is the ground state, the spatial probability density at large R is given by... 

If, instead, we choose the antisymmetric combination for the spatial part of the wavefunction in the excited state, there are three possible symmetric combinations of spin wavefunctions that make the overall wavefunction antisymmetric ... 

Show that the phase factor in eq. (11.17) makes the wavefunction antisymmetric in integralchange of two particles. 

For Iran sition metals th c splittin g of th c d orbitals in a ligand field is most readily done using HHT. In all other sem i-ctn pirical meth -ods, the orbital energies depend on the electron occupation. HyperCh em s m oiccii lar orbital calcii latiori s give orbital cri ergy spacings that differ from simple crystal field theory prediction s. The total molecular wavcfunction is an antisymmetrized product of the occupied molecular orbitals. The virtual set of orbitals arc the residue of SCT calculations, in that they are deemed least suitable to describe the molecular wavefunction, ... 

However, such a function is not antisymmetric, since interchanging two of the r, s —equivalent to swapping the orbitals of two electrons—does not result in a sign change. Hence, this Hartree product is an inadequate wavefunction. 

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

In other words, the energy of the exact wavefunction serves as a lower bound to the energies calculated by any other normalized antisymmetric function. Thus, the problem becomes one of finding the set of coefficients that minimize the energy of the resultant wavefunction. 

When Hartree-Fock theory fulfills the requirement that 4 be invarient with respect to the exchange of any two electrons by antisymmetrizing the wavefunction, it automatically includes the major correlation effects arising from pairs of electrons with the same spin. This correlation is termed exchange correlation. The motion of electrons of opposite spin remains uncorrelated under Hartree-Fock theory, however. 

This simple wavefunction is antisymmetric to the exchange of electron names, and treats both space and spin. 

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... 

Pauli s original version of the exclusion principle was found lacking precisely because it ascribes stationary states to individual electrons. According to the new quantum mechanics, only the atomic system as a whole possesses stationary states. The original version of the exclusion principle was replaced by the statement that the wavefunction for a system of fermions must be antisymmetrical with respect to the interchange of any two particles (Heisenberg [1925], Dirac [1928]). 

There are two possible cases for the wavefunction of a system of identical fundamental particles such as electrons and photons. These are the symmetric and the antisymmetric cases. Experimental evidence shows that for fermions such as electrons and other particles of half integer spin the wavefunction must be anti-symmetric with respect to the interchange of particle labels. This... 

The total wavefunction, , is an antisymmetrized product of the one-electron functions i/q (a Slater determinant). The i/tj are called one-electron functions since they depend on the coordinates of only one electron this approximation is embedded in all MO methods. The effects that are missing when this approximation is used go under the general name of electron correlation. 

Since two electrons with symmetric space wavefunctions and antisymmetric space wavefunctions represent singlet and triplet states respectively, then obviously the triplet state (E ) is of lower energy than the singlet state E+) by an amount Had an attractive force... 

Introduction of the half-integral spin of the electrons (values h/2 and —fe/2) alters the above discussion only in that a spin coordinate must now be added to the wavefunctions which would then have both space and spin components. This creates four vectors (three space and one spin component). Application of the Pauli exclusion principle, which states that all wavefunctions must be antisymmetric in space and spin coordinates for all pairs of electrons, again results in the T-state being of lower energy [equations (9) and (10)]. 

The main difficulty in the theoretical study of clusters of heavy atoms is that the number of electrons is large and grows rapidly with cluster size. Consequently, ab initio "brute force" calculations soon meet insuperable computational problems. To simplify the approach, conserving atomic concept as far as possible, it is useful to exploit the classical separation of the electrons into "core" and "valence" electrons and to treat explicitly only the wavefunction of the latter. A convenient way of doing so, without introducing empirical parameters, is provided by the use of generalyzed product function, in which the total electronic wave function is built up as antisymmetrized product of many group functions [2-6]. 

HF (HF) theory is based on the idea that one takes an antisymmetrized product wavefunction and uses the variational principle to obtain the best possible approximation to the A -particle wavefunction that cannot be represented by such a single determinant. Thus, one inserts the single determinant into the Rayleigh-Ritz functional and performs a constraint variation of the orbitals. The results of the variational process are the famous HF equations that are satisfied by each of the orbitals ... 

The properties of the Slater determinant demonstrate immediately the Pauli exclusion principle, as usually taught. It reads No two electrons can have all four quantum numbers equal, that is to say that they cannot occupy the same quantum state. It is the direct result of the more general argument that the wavefunction must be antisymmetric under the permutation of any pair of (identical and indistinguishable) electrons. 

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 conclusion is then that the wavefunction representing a system composed of indistinguishable particles must be either symmetric or antisymmetric under the permutation operation. On purely physical grounds, this result is apparent, as the probability density must be independent of the permutation of indistinguishable particles or 1 (1,2) 2 = (2,1) 2. 

According to the argument presented above, any molecule must be described by wavefunctions that are antisymmetric with respect to the exchange of any two identical particles. For a homonuclear diatomic molecule, for example, thepossibility of permutation of the two identical nuclei must be considered. Although both the translational and vibrational wavefunctions are symmetric under such a permutation, die parity of the rotational wavefunction depends on the value of 7, the rotational quantum number. It can be shown that the wave-function is symmetric if J is even and antisymmetric if J is odd The overall... 

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 ... 

The exact form of the wavefunction is also conditioned, however, by the observation that electrons possess spin quantum numbers of either +f of Consequently, physically correct solutions to the Schrodinger equation (2.1) must be antisymmetric. Mathematically, this condition can be written as ... 

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 ... 

Excited states formed by light absorption are governed by (dipole) selection rules. Two selection rules derive from parity and spin considerations. Atoms and molecules with a center of symmetry must have wavefunctions that are either symmetric (g) or antisymmetric (u). Since the dipole moment operator is of odd parity, allowed transitions must relate states of different parity thus, u—g is allowed, but not u—u or g—g. Similarly, allowed transitions must connect states of the same multiplicity—that is, singlet—singlet, triplet-triplet, and so on. The parity selection rule is strictly obeyed for atoms and molecules of high symmetry. In molecules of low symmetry, it tends to break down gradually however,... 

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

---