Natural spin-orbitals

By expressing the mean-field interaction of an electron at r with the N- 1 other electrons in temis of a probability density pyy r ) that is independent of the fact that another electron resides at r, the mean-field models ignore spatial correlations among the electrons. In reality, as shown in figure B3.T5 the conditional probability density for finding one ofA - 1 electrons at r, given that one electron is at r depends on r. The absence of a spatial correlation is a direct consequence of the spin-orbital product nature of the mean-field wavefiinctions... 

One consequence of the spin-polarized nature of the effective potential in F is that the optimal Isa and IsP spin-orbitals, which are themselves solutions of F ( )i = 8i d >i, do not have identical orbital energies (i.e., 8isa lsP) and are not spatially identical to one another (i.e., (l)isa and (l)isp do not have identical LCAO-MO expansion coefficients). This resultant spin polarization of the orbitals in P gives rise to spin impurities in P. That is, the determinant Isa 1 s P 2sa is not a pure doublet spin eigenfunction although it is an eigenfunction with Ms = 1/2 it contains both S = 1/2 and S = 3/2 components. If the Isa and Is P spin-orbitals were spatially identical, then Isa Is P 2sa would be a pure spin eigenfunction with S = 1/2. 

As the actinides are a Second f series it is natural to expect similarities with the lanthanides in their magnetic and spectroscopic properties. However, while previous treatments of the lanthanides (p. 1242) provide a useful starting point in discussing the actinides, important differences are to be noted. Spin-orbit coupling is again strong (2000-4000 cm ) but, because of the greater exposure of the 5f... 

Before proving this theorem, we will make some general remarks about the nature of the one-electron functions ipk(x) or spin orbitals. For the two values of the spin coordinate f — 1, such a function y)k(r, f) has two space components... 

If the relation xp = jfl/t is substituted into Eq. III.8, we obtain an expansion of W into configurations of the natural spin orbitals... 

Calculations of the ground state of He (Shull and Lowdin 1958) have given the occupation numbers for the first natural spin orbitals as shown in Table V. These results show that the... 

TABLE V. Occupation Numbers for the First Natural Spin Orbitals in the Ground State of the He atom (Shull and Lowdin... 

Coulomb repulsion is breaking up the closed shell (Is)2, since it is energetically cheaper to have a small fraction (0.8137 %) of the electrons in the first natural spin orbital excited to the higher... 

The results reported in Table V were calculated by means of a basis consisting of four s functions, three p functions, two d functions, and one / function of the type of Eq. III.57 for rj — 2.2. The corresponding 20-term function had an energy of —2.901231 at.u.He, and, by going over to natural spin orbitals, one obtains a 10-term function with the same energy. 

Lowdin, P.-O., Phys. Rev. 97, 1474, 1490, 1509, Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals and convergence problems in the method of configuration interaction. II. Study of the ordinary Hartree-Fock approximation. III. Extension of the Har-tree-Fock scheme to include degenerate systems and correlation effects. ... 

Hirschfelder, J. O., and Lowdin, P.-O., "Long range interaction of two Is hydrogen atoms expressed in terms of natural spin-orbitals."... 

Again the left superscript indicates the spin-triplet nature of the arrangement. The letter A means that it is spatially (orbitally) one-fold degenerate and it is upper-case because we describe two-electron wavefunctions. The subscript is g because the product of d orbitals is even under the octahedral centre of inversion, and the right subscript 2 must remain a mystery for us once again. 

The basis for this formula is just the same as described above but, in this case, spin-orbit coupling admixes the higher-lying 2 2(g) term wavefunctions into the ground E(g). The coefficient 2 in Eq. (5.17) rather than the 4 in Eq. (5.16) arises from the different natures of the wavefunctions being mixed together. 

This is the most general expression obtained from a set of natural spin orbitals written in spinor form as... 

Levy, M., 1979, Universal Variational Functionals of Electron Densities, First Order Density Matrices, and Natural Spin Orbitals and Solution of the v-Representability Problem , Proc. Natl. Acad. Sci. USA, 16, 6062. 

Third, there are clear differences in the images for dissociation at the same wavelength while probing different spin-orbit states. Two effects contribute to these differences. One is the slight difference in energy in the atomic states, most easily seen in the data at 234 nm where the 0(3P2) data clearly has the strong contribution from vibrationally-excited O2 while the other spin-orbit states do not. In this case the threshold for the dynamical process that forms the vibrationally-excited products has been crossed by the 158.265 cm-1 of the spin-orbit excitation. The second effect is that due to the nature of the J-level. It is known that there is a v—J correlation from the angular fits as well as from the fact that when the polarization of the... 

As seen in the radiationless process, intercombinational radiative transitions can also be affected by spin-orbit interaction. As stated previously, spin-orbit coupling serves to mix singlet and triplet states. Although this mixing is of a highly complex nature, some insight can be gained by first-order perturbation theory. From first-order perturbation theory one can write a total wave function for the triplet state as... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

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