Perturbation theory spin-projected

Figure 3.2. Equilibrium linear susceptibility in reduced units X = x Hi[/m) versus temperature for three different ellipsoidal systems with equation x ja +y lb + jc < I, resulting in a system of N dipoles arranged on a simple cubic lattice. The points shown are the projection of the spins to the xz plane. The probing field is applied along the anisotropy axes, which are parallel to the z axis. The thick lines indicate the equilibrium susceptibility of the corresponding noninteracting system (which does not depend on the shape of the system and is the same in the three panels) thin lines show the susceptibility including the corrections due to the dipolar interaction obtained by thermodynamic perturbation theory [Eq. (3.22)] the symbols represent the susceptibility obtained with a Monte Carlo method. The dipolar interaction strength is itj = d/ 2o = 0.02.

G, respectively. ROHF theory is more accurate than UHF theory in this case, presumably owing to moderate spin contamination in the latter. Projecting out the spin contamination at the PUHF level reduces the error by almost one half, while going to second-order perturbation theory (which introduces electron correlation and also probably reduces the spin contamination compared to UHF) provides an improvement of about the same order. 

The other area in which projection of an unrestricted result has received attention is projected Mpller-Plesset perturbation theory, the PUMPn methods [31], where n is the order of the perturbation theory. In cases in which the UHF approximation is a poor starting point (considerable spin contamination, for example), the convergence of the MP perturbation expansion can be slow and/or erratic. The PUMP methods apply projection operators to the perturbation expansion, although usually not full projection but simply annihilation of the leading contaminants. This approach has met with mixed success again, it represents a rather expensive modification to a technique that was originally chosen partly for its economy — seldom a recipe for success. 

Therefore, one would like to modify (symmetry adapt) the normal Raylei -Schrodinger perturbation theory such that the exchange effects are explicitly included in the lower order interaction energy expressions. This symmetry adaptation can be achieved by means of a projection operator, the antisymmetrizer A, that, operating on any N-electron space-spin fuiK tion (N = + N ), makes... 

Electronic structure calculations have been performed with the system of programs Gaussian94 (G94) [19]. Restricted Hartree-Fock theory (RHF) is used for closed shell systems, and Unrestricted Hartree-Fock theory (UHF) for open shell systems (radicals). The correlation energy corrections are introduced with Moller-Plesset perturbation theory up to second order, and results from spin projected calculations are used (PMP2 and PUMP2). 

The inner product N S depends on the relative projection of N and S, so all states with the same value of Mj = Ms + Mjv also have the same energy under the YsrN S interaction and can be coupled by it. It can be seen that for the rotational ground state, N = 0, Mn = 0 and thus the spin-rotation interaction cannot change Ms without changing Mj. Because Mj is a good quantum number, collisions between helium atoms and molecules in their rotational ground state cannot directly cause spin-depolarization (first-order perturbation theory predicts a Zeeman transition probability of zero). 

The spin projection methods discussed so far apply the projection after the wavefunction has been determined. A second approach is to construct the wavefunction so that there is no spin contamination from the beginning. This leads to methods such as ROHF and restricted open-shell M0ller-Plesset perturbation theory (ROMPn)," and to valence bond, MCSCF, and Cl and MRDCI methods that use spin-adapted configurations. 

---