Spin functions general

In summary, for a homonuclear diatomic molecule there are generally (2/ + 1) (7+1) symmetric and (27+1)7 antisymmetric nuclear spin functions. For example, from Eqs. (50) and (51), the statistical weights of the symmetric and antisymmetric nuclear spin functions of Li2 will be and respectively. This is also true when one considers Li2 Li and Li2 Li. For the former, the statistical weights of the symmetric and antisymmetiic nuclear spin functions are and, respectively for the latter, they are and in the same order. 

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 ... 

A common feature of the Hartree-Fock scheme and the two generalizations discussed in Section III.F is that all physical results depend only on the two space density matrices p+ and p, which implies that the physical and mathematical simplicity of the model is essentially preserved. The differences lie in the treatment of the total spin in the conventional scheme, the basic determinant is a pure spin function as a consequence of condition 11.61, in the unrestricted scheme, the same determinant is a rather undetermined mixture of different spin states, and, in the extended scheme, one considers only the component of the determinant which has the pure spin desired. 

The density functional calculations were performed using the Vienna Ab Initio Simulation Package (VASP). ° The spin-polarized generalized gradient approximation, Perdue—Wang exchange correlation function, and ultrasoft pseudopotentials were used. ... 

So far in this chapter we have discussed the NOE theory in terms of general set of SOs 4>i x) or in terms of restricted SOs, which are constrained to have the same spatial function spin functions. In this section we are concerned only with closed-shell systems. Our molecules are thus allowed to have only an even number of electrons, with all electrons paired such that the spatial orbitals are doubly occupied. In this case of spin-compensated systems, the two nonzero blocks of the 1-RDM are the same ( D = D ) that is. 

The situation here is completely analogous to that obtained in the restricted open HE theory (ROHF). The states are not eigenfunctions of S, except when all the open-shell electrons have parallel spins (A p = 0 or = 0). This result is a consequence of the expansion (22) used to obtain Eq. (27). Actually, the spin decomposition, Eq. (22), for does not conserve in general the total spin S. However, we can form appropriate linear combinations of two-electron spin functions cri Sj, 8 81,52) that are simultaneously eigenfunctions of and S, and achieve a correct spin decomposition of the 2-RDM [85] ... 

When there are only two electrons the analysis is much simplified. Even quite elementary textbooks discuss two-electron systems. The simplicity is a consequence of the general nature of what is called the spin-degeneracy problem, which we describe in Chapters 4 and 5. For now we write the total solution for the ESE 4 (1, 2), where the labels 1 and 2 refer to the coordinates (space and spin) of the two electrons. Since the ESE has no reference at all to spin, 4 (1, 2) may be factored into separate spatial and spin functions. For two electrons one has the familiar result that the spin functions are of either the singlet or triplet type. 

We now have a significant difiference from the case of two electrons in a singlet state, namely, we have two spin functions to combine with spatial functions for a solution to the ESE rather than only one. For a doublet three-electron system our general solution must be... 

The -particle spatial (or spin) functions we work with are elements of a Hilbert space in which the permutations are operators. If H(l, 2,..., )andT(I, 2,..., n) are two such functions we generally understand that... 

We are now done with spin functions. They have done their job to select the correct irreducible representation to use for the spatial part of the wave function. Since we no longer need spin, it is safe to suppress the subscript in Eq. (5.110) and all of the succeeding work. We also note that the partition of the spatial function X is conjugate to the spin partition, i.e., 2"/ , 2. From now on, if we have occasion to refer to this partition in general by symbol, we will drop the tilde and represent it with a bare X. 

Eq.(21) requires that all occupied ground state orbitals be orthogonal to a linear combination of the excited state orbitals b j lvij ), which describes an excited electronic state. Eq.(22) requires the orthogonality of all occupied excited state orbital associated with a spin functions to the arbitrary vector Y7 IVoi ) from the subspace of the occupied ground state orbitals associated with a spin functions. In general, the coefficients 6° can be determined by minimizing the excited state Hartree-Eock energy. However, calculations show that the choice... 

In magnetic resonance we are often confronted with the problem of obtaining a solution to a Hamiltonian which has only spin operators. To find the allowed energies and eigenfunctions, we generally start out with a convenient set of spin functions < , which represent the spin system but are not eigenfunctions of the Hamiltonian,. The eigenfunction ifi can, however, be constructed from a linear sum of the s ... 

For dissimilar pairs, the parameter ys equals zero and we have Eq. 5.36. Like pairs of zero spin are bosons and all odd-numbered partial waves are ruled out by the requirement of even wavefunctions of the pair this calls for ys = 1. In general, for like pairs, the symmetry parameter ys will be between -1 and 1, depending on the monomer spins (fermions or bosons) and the various total spin functions of the pair. A simple example is considered below (p. 288ff.). If vibrational states are excited, the radial wavefunctions xp must be obtained from the vibrationally averaged potential, Fq(R). The functions gf(R) and gM(R) are similar to the pair distribution function, namely [294]... 

This equation is fully equivalent to the time-independent Schrodinger equation and its roots give the exact energies. Since the set (2.50) generally has an infinite number of functions, the secular determinant generally is of infinite order. In certain cases, however, one need only deal with a finite-order secular determinant. For example, in electron-spin (and nuclear-spin) problems, the number of basis spin functions is finite. Applications will be found in later chapters. 

As an example, the most general one-electron spin function c,a + c2/ has as its representative in the a,/ basis... 

We assume that the wave functions of a set of d orbitals are each of the general form specified by 9.2-1. We shall further assume that the spin function [jj% is entirely independent of the orbital functions and shall pay no further attention to it for the present. Since the radial function R(r) involves no directional variables, it is invariant to all operations in a point group and need concern us no further. The function 0(0) depends only upon the angle 0. Therefore, if all rotations are carried out about the axis from which 0 is measured (the z axis in Fig. 8.1), (0) will also be invariant. Thus, by always choosing the axes of rotation in this way (or, in other words, always quantizing the orbitals about the axis of rotation), only the function (< ) will be altered by rotations. The explicit form of the 4>(0) function, aside from a normalizing constant, is... 

The action of S on I na np> gives thus a term proportional to I na np> plus a sum over occupation number vectors, where the spin functions of an alpha orbital and a beta orbital have been flipped. Only permutations of the singly occupied orbitals are included. Inanp> is thus in general not an... 

The orthonormal basis generally prevalent in numerical simulations is composed of the product functions (henceforth, a), which are the products of the spin functions of the nuclei in all the possible variations. In this case, the fi index of the nucleus can be abandoned, because the order of the atomic functions already defines it. There are 2" product functions in the case of n spin-half nuclei. On this basis, the matrix of the Hamiltonian is composed of the following elements ... 

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