Properties of Spin Functions

There is great advantage in writing a spin orbital as a function rather than a row vector. We therefore introduce spin functions a and p  

The function value need not be specified, since a and p components should not be added as functions anyway. However, the spin functions have to be orthonormalized and assuming that the spatial function cp is already orthonormalized, we must have 

If these rules are obeyed, nothing bad will happen. 

The projection of spin along the z-axis, as is in fact the meaning of s, and the square of the angular momentum (s s) (we write this as s ) correspond to conserved observables. We thus have to construct spin functions that are eigenfunctions also to the square of angular momentum. 

According to our definition of the spin operators (Equation 1.62), we have 

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Table 7 Transformation Properties of Spin Functions in Frequently Occurring Molecular Double Groups ...

For a detailed discussion of the properties of spin functions see Chapter 3 in the Handbook of Molecular Physics and Quantum Chemistry, volume 2, Molecular Electronic Structure by Karadakov [98]. 

We are interested in electron 3 with electron 1 residing at nucleus a with space coordinates (0, 0,0) and with spin coordinate cti = and with electron 2 located at nucleus b with coordinates (i , 0, 0) and 02 = — whereas the electron 3 itself has spin coordinate 03 — i. The square of the absolute value of the function 0uhf calculated for these values depends on 3, ys, Z3 and represents the conditional probability density distribution for finding electron 3 (provided electrons 1 and 2 have the fixed coordinates given above and denoted by Iq, 2q). So, let us calculate individual elements of the determinant 0[/hf(1o, 2o, 3), taking into account the properties of spin functions a and p (cf. p. 27) ... 

The Hamiltonian (3.4) is a function of the usual spatial coordinates x, y, z or r, 0, (j)). Electrons possess the intrinsic property of spin, however, which is to be thought of as a property in an independent, or orthogonal, space (spin space). Spin is actually a consequence of the theory of relativity but we shall merely graft on the property in an ad hoc fashion. The spin, s, of an electron (don t confuse with s orbitals ) takes the value 1/2 only. The z component of spin, m, takes (25 + 1) values of ms, ranging 5, 5-l,...-s. Thus for the single electron, = +1/2 or -1/2, also labelled a or p, or indicated by t or i. 

The fact that there are only two kinds of spin function (a and (1), leads to the conclusion that two electrons at most may occupy a given molecular orbital. Were a third electron to occupy the orbital, two different rows in the determinant would be the same which, according to the properties of determinants, would cause it to vanish (the value of the determinant would be zero). Thus, the notion that electrons are paired is really an artifact of the Hartree-Fock approximation. 

We turn now to a brief discussion of the symmetry properties of spin-coupled orbitals . Because they almost always have the form of deformed atomic functions, the effect of a spatial symmetry operation upon the orbitals is to permute them amongst themselves. Thus each symmetry operation corresponds to a certain permutation P. From this it can be seen that the effect of upon complete spin-coupled wavefunctions is to transform them amongst themselves as follows ... 

The key property of this function is that the diagonal element, denoted for brevity by Qc KK ti,T2), integrated over all positions of points ri and T2, will give a numerical measure of the spin coupling ... 

These lectures present an introduction to density functionals for non-relativistic Coulomb systems. The reader is assumed to have a working knowledge of quantum mechanics at the level of one-particle wavefunctions (r) [1]. The many-electron wavefunction f (ri,r2,..., rjv) [2] is briefly introduced here, and then replaced as basic variable by the electron density n(r). Various terms of the total energy are defined as functionals of the electron density, and some formal properties of these functionals are discussed. The most widely-used density functionals - the local spin density and generalized gradient... 

The most obvious new feature of the Dirac equation as compared with the standard nonrelativistic Schrodinger equation is the explicit appearance of spin through the term a p. Any spin operator trivially commutes with a spin-free Hamiltonian, but the introduction of spin-dependent terms may change this property, as demonstrated in the case of ji/-coupling. A further scrutiny of spin symmetry is therefore a natural first step in discussing the symmetry of the Dirac Hamiltonian. This requires a basis of spin functions on which to carry out the various operations, and a convenient choice is the familiar eigenfunctions of the operator, i, i) and j, — ), also called the a and spin functions. 

One of the most important aspects of spin functions is their group theoretical properties. In particular, the set of spin functions, for a given N and S, forms a basis for an irreducible representation of the group of permutations of N objects. This group is of order Nl and is often referred to as the symmetric group . It is denoted by the symbol Sn- Thus if P" denotes a permutation of the N spin coordinates [Pg.2675]

The Serber spin functions are particularly useful in displaying the spatial symmetry properties of spin-coupled wave functions, when it is obvious that those spin functions which do not lead to the required overall symmetry of the total wave function have zero spin-coupling coefficients. This has turned out to be of great utility in cases where the introduction of electron correlation leads to unexpected additional symmetries of, for example, the a electrons in a planar n system. Examples of this will be presented in due course. 

Let us discuss further the pemrutational symmetry properties of the nuclei subsystem. Since the elechonic spatial wave function t / (r,s Ro) depends parameti ically on the nuclear coordinates, and the electronic spacial and spin coordinates are defined in the BF, it follows that one must take into account the effects of the nuclei under the permutations of the identical nuclei. Of course. 

In this section, we extend the above discussion to the isotopomers of X3 systems, where X stands for an alkali metal atom. For the lowest two electronic states, the permutational properties of the electronic wave functions are similar to those of Lij. Their potential energy surfaces show that the baniers for pseudorotation are very low [80], and we must regard the concerned particles as identical. The Na atom has a nuclear spin " K, and K have nuclear... 

As discussed in preceding sections, FI and have nuclear spin 5, which may have drastic consequences on the vibrational spectra of the corresponding trimeric species. In fact, the nuclear spin functions can only have A, (quartet state) and E (doublet) symmetries. Since the total wave function must be antisymmetric, Ai rovibronic states are therefore not allowed. Thus, for 7 = 0, only resonance states of A2 and E symmetries exist, with calculated states of Ai symmetry being purely mathematical states. Similarly, only -symmetric pseudobound states are allowed for 7 = 0. Indeed, even when vibronic coupling is taken into account, only A and E vibronic states have physical significance. Table XVII-XIX summarize the symmetry properties of the wave functions for H3 and its isotopomers. 

In this chapter, we discussed the permutational symmetry properties of the total molecular wave function and its various components under the exchange of identical particles. We started by noting that most nuclear dynamics treatments carried out so far neglect the interactions between the nuclear spin and the other nuclear and electronic degrees of freedom in the system Hamiltonian. Due to... 

Tensile Properties. Tensile properties of nylon-6 and nylon-6,6 yams shown in Table 1 are a function of polymer molecular weight, fiber spinning speed, quenching rate, and draw ratio. The degree of crystallinity and crystal and amorphous orientation obtained by modifying elements of the melt-spinning process have been related to the tenacity of nylon fiber (23,27). 

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