P, spin function

The notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... 

In the first of the two notations used in this equation, a singlet pair is associated with each combination of an a and a p spin function enclosed within a matching pair of left and right parentheses. The second notation indicates all singlet pairs explicitly. It is easiest to visualize the spin-coupling patterns (3) by means of the familiar five... 

Note that e disappearance of the exchange correlation derives from the orthogonality of the 0 and p spin functions, which causes the second integral in the second equality to be zero when integrated over either spin coordinate. 

The corresponding transformation of the spin-orbitals is obtained by multiplying (3 26) with an a or P spin function. When we make the transformation from one set of spin-orbitals to the other, the annihilation and creation operators will change. The following relations are easily established, by operating with the creation operator in the primed space on the vacuum state ... 

Bold quantities are operators, vectors, matrices or tensors. Plain symbols are scalars. a Polarizability a, P Spin functions a, p Dirac 4x4 spin matrices ap-jS Summation indices for basis functions F Fock operator or Fock matrix Fy, Eajd Fock matrix element in MO and AO basis Y Second hyperpolarizability yk Density matrix of order k gc Electronic g-factor... 

This problem can be avoided, however, if an appropriate open-shell perturbation theory is defined such that the zeroth-order Hamiltonian is diagonal in the truly spin-restricted molecular orbital basis. The Z-averaged perturbation theory (ZAPT) defined by Lee and Jayatilaka fulfills this requirement. ZAPT takes advantage of the symmetric spin orbital basis. For each doubly occupied spatial orbital and each unoccupied spatial orbital, the usual a and P spin functions are used, but for the singly occupied orbitals, new spin functions. 

The electronic wavefunction is a Slater determinant (or a combination of Slater determinants in the case of open-shell systems) based on the occupied m.o.s, each one taken twice (if doubly occupied) and associated either with a or P spin functions (molecular spin-orbitals)... 

The a and are 4x4 matrices, and the relativistic wave function consequently has four components. Traditionally, these are labelled the large and small components, each having an a and P spin function (note the difference between the a and P matrices and a and P spin functions). The large component describes the electronic part of the wave function, while the small component describes the positronic (electron antiparticle) part of the wave function, and the a and P matrices couple these components. In the limit of c oo, the Dirac equation reduces to the Schrodinger equation, and the two... 

Here A is the usual antisymmetrizer (eq. (3.21)) and a bar above a MO indicates that the electron has a P spin function, no bar indicates an a spin function. 

The metals in Group 1 have one unpaired electron in the valence shell s orbital. This AO may be combined with a or p spin functions to form two spin orbitals. It is therefore possible to write down two Slater determinant wavefunctions for these atoms the ground states are doubly degenerate. Atoms that contain two or more unpaired electrons are more difficult to describe. For these species one is forced to form wavefunctions by linear combination of two or more Slater determinants. In the next section we shall deal with the simplest atom of this kind, viz. a helium atom excited to the 1x 25 electron configuration. 

If we instead use the orbitals Is and 2p, there are more Slater determinants. Every orbital has either a or p spin function. The orbital 2p may have different quantum niunbers m. Altogether there are 12 different functions ... 

The two-component wave function is called a Pauli spinor. We will show in a later chapter that for this basis the Pauli matrices form a representation of the spin operators such that ha = 2s. With the conventional choice of basis, that is, the eigenfunctions of ff, the upper component represents the part of the wave function with spin projection nts = j, or a spin, and the lower component the part with rus = -, or p spin. The primitive a and p spin functions are represented by the vectors... 

The Pauli principle for the electrons of atoms and molecules says that no two electrons can occupy the same spin-orbital, and this is obviously important in building up the orbital picture of the elements. As discussed later, this principle goes hand in hand with the indistinguishability of the electrons and with their half-integer intrinsic spin. For now, it is important to say that each distinct spatial orbital (e.g.. Is, 2s, 2pj, 2pg, 2p j) can be occupied by at most two electrons. This is because a given spatial orbital can be combined with either an a or a p spin function to form two and only two different spin-orbitals. 

Table 2. Transformation of q, and d-orbitals and a and p spin functions in the O point group. See the text for the notations for symmetry operations ...

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