Spin-orbitals coordinate representation

For molecules with an even number of electrons, the spin function has only single-valued representations just as the spatial wave function. For these molecules, any degenerate spin-orbit state is unstable in the symmetric conformation since there is always a nontotally symmetric normal coordinate along which the potential energy depends linearly. For example, for an - state of a C3 molecule, the spin function has species da and E that upon... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

The relation to the density matrix elements in the spin-orbital occupation numbers representation recovers from noticing that the rows of indices of spin-orbitals ki,k2,..., / y = K (defining a row of creation operators. ..a a , forming a basis Slater determinant) can be in the same manner considered as a set of electronic coordinates in the spin-orbital representation as is the list xi, X2,. .., xjv, ... 

Table 10.19. Spin-orbital single configuration representations for the ground and lowest excited electronic states of CH. Only the spatial and spin coordinates of the three highest-energy electrons are specified...

For the nonrelativistic case with neglect of spin—orbit coupling we separate the space and spin parts of the coordinate—spin representation of the orbital... 

The limit with respect to rj is taken because of integration techniques required in a Fourier transform from the time-dependent representation. Indices r and s refer to general, orthonormal spin-orbitals, r x) and os(x), respectively, where x is a space-spin coordinate. Matrix elements of the corresponding field operators, al and as/ depend on the N-electron reference state, N), and final states with N 1 electrons, labeled by the indices m and n. The propagator matrix is energy-dependent poles occur when E equals a negative VDE, Eq(N) — En(N — 1), or a negative VAE, Em(N +1) — Eq(N). 

The (iV ) factor included in these expressions ensures that the determinants are normalized when the orbitals are normalized. Eq. (41) gives an explicit representation of the antisymmetrizer. This summation is over the N permutations of electron coordinates for a fixed orbital order, or equivalently, over the permutations of spin-orbital labels for a fixed order of electrons. The exponent Pp is the number of interchanges required to bring a particular permuted order of electron coordinates, or of spin-orbital labels, back to the original order. Different expansion terms are generated when different spin orbitals are employed in the determinant. For convenience, we will choose this spin-orbital basis to be the direct product of the set of n spatial orbitals and the set of spin factors a, / . A particular spin orbital of this form may be written as where r (= 1 to n) labels the spatial orbital and spin factor, or simply as (j), where the combined index r (= 1 to 2n) labels both the spatial and spin components. The notation used will be clear from the context. 

In practice, there is quite a difference between atoms and molecules as far as the representation of the one-particle spectrum is concerned. In atomic physics, one often utilizes a radial-angular representation of the one-electron orbitals in order to allow for the analytic integration over all spin-angular coordinates of the system [35,36]. This so-called angular reduction will be briefly discussed below in Subsection 4.4. However, in order to exploit the symmetry of free atoms the reference state must coincide with a closed-shell determinant o> = < >>, i.e. a reference state which should not depend on the magnetic quantum numbers of the one-electron functions. Unfortunately, the complexity of the perturbation expansions increases very rapidly if the number of electrons in the physical states of interest differ from (the number of electrons in) the reference state. 

We may construct the Bloch functions by recalling from Section 2.4 that the creation operator creates an electron with spin a in the 7r-orbital localized on the nth site. Thus, projecting the Bloch state, A ), onto the coordinate representation, r), we have the Bloch function. 

The Hamiltonian (O Eq. 2.23) maintains full symmetry and is invariant under electronic permutations and under rotation-reflections of the electronic coordinates. Trial functions are usually constructed from atomic orbitals and from their spin-orbitals. Permutational antisymmetry is achieved by forming Slater determinants from the spin-orbitals. Rotational symmetry is usually realized by vector coupling of orbitals that form bases for representations of the rotation group SO(3). Spin-eigenfunctions too are achieved by vector coupling. ... 

DENSITY MATRICES IN SPIN-ORBITAL AND COORDINATE REPRESENTATIONS... 

The density matrices we have discussed so far in this section have all been given in the spin-orbital representation. We shall now see how these matrices are represented in the coordinate... 

---