Wave function stability

Rumer basis, spin functions, 199 Spin-Dipolar (SD) operator, 251 method, 322 Wave function stability, 76... 

The question of whether the energy is a minimum is closely related to the concept of wave function stability. If a lower energy RHF solution can be foimd, the wave function Is aidTfi insses OmgferTnvtaMttyr an RHE-typejwave function is a... 

Stabilizing resonances also occur in other systems. Some well-known ones are the allyl radical and square cyclobutadiene. It has been shown that in these cases, the ground-state wave function is constructed from the out-of-phase combination of the two components [24,30]. In Section HI, it is shown that this is also a necessary result of Pauli s principle and the permutational symmetry of the polyelectronic wave function When the number of electron pairs exchanged in a two-state system is even, the ground state is the out-of-phase combination [28]. Three electrons may be considered as two electron pairs, one of which is half-populated. When both electron pahs are fully populated, an antiaromatic system arises ("Section HI). 

As shown in Figure 27, an in-phase combination of type-V structures leads to another A] symmetry structures (type-VI), which is expected to be stabilized by allyl cation-type resonance. However, calculation shows that the two shuctures are isoenergetic. The electronic wave function preserves its phase when tr ansported through a complete loop around the degeneracy shown in Figure 25, so that no conical intersection (or an even number of conical intersections) should be enclosed in it. This is obviously in contrast with the Jahn-Teller theorem, that predicts splitting into A and states. 

Use a forced convergence method. Give the calculation an extra thousand iterations or more along with this. The wave function obtained by these methods should be tested to make sure it is a minimum and not just a stationary point. This is called a stability test. 

Each of the two first VB stmctures contributes 40% to the wave function, and each of the remaining three contributes 6%. The stability of benzene in the SCVB picture is due to resonance between these VB structures. It is furthermore straightforward to calculate the resonance energy by comparing the full SCVB energy with that ealeulated from a VB wave function omitting certain spin coupling functions. 

An obvious refinement of the simple theory for cobalt and nickel and their alloys can be made which leads to a significant increase in the calculated value of the Curie temperature. The foregoing calculation for nickel, for example, is based upon the assumption that the uncoupled valence electrons spend equal amounts of time on the nickel atoms with / = 1 and the nickel atoms with J = 0. However, the stabilizing interaction of the spins of the valence electrons and the parallel atomic moments would cause an increase in the wave function for the valence electrons in the neighborhood of the atoms with / = 1 and the parallel orientation. This effect also produces a change in the shape of the curve of saturation magnetization as a function of temperature. The details of this refined theory will be published later. 

The more incisive calculation of Springett, et al., (1968) allows the trapped electron wave function to penetrate into the liquid a little, which results in a somewhat modified criterion often quoted as 47r/)y/V02< 0.047 for the stability of the trapped electron. It should be noted that this criterion is also approximate. It predicts correctly the stability of quasi-free electrons in LRGs and the stability of trapped electrons in liquid 3He, 4He, H2, and D2, but not so correctly the stability of delocalized electrons in liquid hydrocarbons (Jortner, 1970). The computed cavity radii are 1.7 nm in 4He at 3 K, 1.1 nm in H2 at 19 K, and 0.75 nm in Ne at 25 K (Davis and Brown, 1975). The calculated cavity radius in liquid He agrees well with the experimental value obtained from mobility measurements using the Stokes equation p = eMriRr], with perfect slip condition, where TJ is liquid viscosity (see Jortner, 1970). Stokes equation is based on fluid dynamics. It predicts the constancy of the product Jit rj, which apparently holds for liquid He but is not expected to be true in general. 

The values of cion and c ,v determine the weights of the respective components, and reflect the relative stabilization of the VB states in solution e g. a polar solvent is expected to stabilize the ionic TBu+Cl-) relative to the covalent x Bu-Cl). By contrast, the HF wave function for BuCl is the (normalized) Slater determinant[22]... 

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