Antisymmetry, principle

The postulate that electrons must be described by wavefunc-tions which are antisymmetric with respect to interchange of the coordinates (including spin) of a pair of electrons. As a consequence, two electrons occupying one orbital must differ in at least one quantum number. Synonymous with the Pauli exclusion principle. 

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A determinant is the most convenient way to write down the permitted functional forms of a polv electronic wavefunction that satisfies the antisymmetry principle. In general, if we have electrons in spin orbitals Xi,X2, , Xn (where each spin orbital is the product of a spatial function and a spin function) then an acceptable form of the wavefunction is ... 

The orbital phase continuity conditions stem from the intrinsic property of electrons. Electrons are fermions, and are described by wavefnnctions antisymmetric (change plus and minus signs) with respect to an interchange of the coordinates of an pair of particles. The antisymmetry principle is a more fnndamental principle than Pauli s exclusion principle. Slater determinants are antisymmetric, which is why the overlap integral between t(a c) given above has a negative... 

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,... 

The Pauli exclusion principle requires that no two electrons can occupy the same spin-orbital is a consequence of the more general Pauli antisymmetry principle ... 

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 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). 

This kind of wavefunction is called a Hartree Product, and it is not physically realistic. In the first place, it is an independent-electron model, and we know electrons repel each other. Secondly, it does not satisfy the antisymmetry principle due to Pauli which states that the sign of the wavefunction must be inverted under the operation of switching the coordinates of any two electrons, or... 

Another well-known property of determinants is that they vanish if they have two identical rows. This means that it is not possible to construct a non-vanishing antisymmetrised product in which two electrons in the same orbital have the same spin. Thus the rule that not more than two electrons must be assigned to any one space orbital follows as a direct consequence of the antisymmetry principle for product wave functions it had to be introduced as an extra postulate. 

Another important physical interpretation of the molecular-orbital determinant follows from an application of a similar argument to the columns. The elements of two columns become identical if two electrons have the same spin (a or [>) and are at the same point (, y, z). The determinant then vanishes and consequently the probability of such a configuration is zero. Such an argument does not apply to electrons of different spin, however. The antisymmetry principle operates, therefore, in such a way that electrons of the same spin are kept apart. We shall see in later sections that this is an important factor in determining stereochemical valence properties. 

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. 

Now suppose two electrons are placed one in each of these orbitals. The distribution of these electrons in their individual orbitals will simply be given by y/j2 and y/22. If we wish to examine the probability of various simultaneous positions of the two electrons, we have to consider the total wave function W, which will be an antisymmetric product with a form depending on the spins of the electrons. If we wish to investigate the effect of the antisymmetry principle on the spatial arrangement of the electrons, it is convenient to examine the case in which they both have the same spin a. Then the wave function will be of the form given in equations (6) and (8). If the factor a(I)a(2) is omitted. 

This sort of model can easily be generalised to deal with more than two electrons and other assignments of the spins. The case of most interest in molecular studies is that in which a set of molecular orbitals are all occupied by two electrons. Thus if there were two electrons, one of either spin, in both orbitals and y>2, the total wave function would be a 4 x 4 determinant. But most of the features of the two-electron model are retained. The system could be alternatively described as consisting of two electrons in each of the equivalent orbitals. The effect of the antisymmetry principle is then to keep electrons of the same spin apart, the motion of the two opposite spin-types being uncorrelated. 

These both have the same energy so that, in the absence of other determining factors, the electrons go one into each with the same spin (or an equivalent state). This means that they are kept apart by the antisymmetry principle and so the energy Is lowered by the reduction of Coulomb repulsion. In this rather exceptional case, therefore, the orbitals are not all doubly occupied and we cannot carry out any simple transformation into localised orbitals. 

The three quark color states are restricted to the color singlet, 1=1, which together with the fermion antisymmetry principle leads to requirement that the flavor-ordinary spin space must be totally symmetric i. e., I I I I This, in turn, leads to the following relationships between the flavor space and the ordinary spin space ... 

Electrons interchanged. When the many-electron function of a molecule is written in the form of a determinant, the fundamental antisymmetry principle (the Pauli exclusion principle) of quantum mechanics is satisfied. According to that principle an A-electron function must be antisymmetric, i.e. it must change sign whenever spatial and spin variables of any two electrons are interchanged ... 

If x = rs denotes the space-spin variable, we recall from first principles (Magnasco, 2007,2009a) that, for a normalized N-electronwavefunction satisfying the Pauli antisymmetry principle, the one-electron density function is defined as ... 

It is generally helpful to build into the expansion functions ( /) the symmetry properties of the system. According to the antisymmetry principle,... 

The origin of the nonvauishing Joule-Thomson effect is the effective repulsive (Fermions) and attractive (Bosons) potential exerted on the gas molecules, which arises from the different ways in which quantum states can be occupied in sy.stems obeying Fermi-Dirac and Boso-Einstein statistics, respectively [17]. In other words, the effective fields are a consequence of whether Pauli s antisymmetry principle, which is relativistic in nature [207], is applicable. Thus, a weakly degenerate Fermi gas will always heat up ((5 < 0), whereas a weakly degenerate Bose gas will cool down (5 > 0) during a Joule-Thomson expansion. These conclusions remain valid even if the ideal quantum gas is treated relativistically, which is required to understand... 

Dk is an antisymmetrized product of n spin orbitals which satisfies the indistinguishability and Pauli principles. The latter is actually a trivial consequence of a postulate of the greatest importance in quantum chemistry, i.e., the antisymmetry principle applying to total electronic wave... 

So it is seen that the joint utilization of the antisymmetry principle and of the formulas derived from the generalized Hellmann-Feynman theorem allows us to find a quantum-mechanical justification of the Lewis ideas (electron pairing and sharing) and to analyze the physical nature of the chemical bond. 

The most uniformly successful family of methods begins with the simplest possible n-electron wavefunction satisfying the Pauli antisymmetry principle - a Slater determinant [2] of one-electron functions % r.to) called spinorbitals. Each spinorbital is a product of a molecular orbital xpt(r) and a spinfunction a(to) or P(co). The V /.(r) are found by the self-consistent-field (SCF) procedure introduced [3] into quantum chemistry by Hartree. The Hartree-Fock (HF) [4] and Kohn-Sham density functional (KS) [5,6] theories are both of this type, as are their many simplified variants [7-16],... 

We now turn to the general case. What is an appropriate functional form of the wavefunction for a polyelectronic system (not necessarily an atom) with N electrons that satisfies the antisymmetry principle First, we note that the following functional form of the wavefunction is inappropriate ... 

This product of spin orbitals is xmacceptable because it does not satisfy the antisymmetry principle exchanging pairs of electrons does not give the negative of the wavefunction. This formulation of the wavefunction is known as a Hartree product. The energy of a system described by a Hartree product equals the sum of the one-electron spin orbitals. A key conclusion of the Hartree product description is that the probability of finding an electron at a particular point in space is independent of the probability of finding any... 

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