Symmetrically correct wavefunctions

Eisfeld and Morokuma (S) went on to detail the non-trivial task of obtaining symmetrically correct wavefunctions for the NO3 ground state in CASSCF and MR-SDCL They found that a significantly large active space was necessary. Even the reasonable elimination of active orbitals that were almost always doubly occupied or almost always entirely empty resulted in symmetry breaking of the CASSCF wavefunction. 

Write, but do not evaluate, the integral in terms of all the relevant coordinates for first-order correction to the energy of the ground state lithium atom, f tf/lH iffodr. Do not normalize or symmetrize the wavefunction, do not include the spin terms, but do include the limits of integration. 

It is generally found that if one increases the flexibility of a single-determinantal wavefunction by allowing each space orbital to assume an independent form (rather than insisting on double occupation by an and a (1 electron for those orbitals which would otherwise be so occupied as dictated by the electronic configuration) that the asymptotic difficulties of the wavefunction are removed. Thus, the unrestricted Hartree-Fock method usually predicts the correct dissociation products of a molecular system.140 The symmetrical (C2 ,) insertion of Of3P) into Ha yields the 33i state of the HaO system. The electronic configuration of this state expressed in terms of the unrestricted set of orbitals is... 

How does the first-order correction alterthe wavefunction Recall that the perturbation raises the potential energy near the top of the box (near L) much more than near the bottom (near x = 0) therefore, we expect the probability of finding the particle near the bottom to be enhanced compared with that of finding it near the top. Because the zero-order ground-state wavefunction is positive throughout the interior of the box, we thus expect the wavefunction itself to be raised near the bottom of the box and lowered near the top. In fact, the correcticHi terms do just this. First, note that the basis wavefunctions with odd n are symmetric with respect to the center of the box therefore, they would have the same effect near the top of the box as near the bottom. The coefficients of these terms are zero they do not contribute to the correction. The even- basis functions all start positive near x = 0 and end negative near x = L therefore, such terms must be multiplied by positive coefficients (as the result provides) to enhance the wavefunction near the bottom and diminish it near the top. 

For the Diels-Alder reaction, the formation of cyclohexene from ethylene and butadiene, the LSDA and gradient-corrected DFT calculations agreed on a symmetric transition state structure, in accord with the wavefunction-based ab initio approaches. The cited experimental values ranged from 24.2 to 27.5 kcal/mol. Tliough the LSDA and gradient-corrected transition state structures were similar, the reaction barrier within the LSDA was severely underestimated, by about 20 kcal/mol, at 4.5 kcal/mol. The gradient-corrected DFT result was in much better agreement, at 18.7 kcal/mol. This result was better than the HF and MP2 values, which were 45.9 and 15.9 kcal/mol, respectively. 

Observe that the series diverges for certain values of the variables, producing nonsquare-integrable wavefunctions. Correct this by requiring that the series terminate. This requires that the truncated series be either symmetric or antisymmetric in the variable and also that p of Eq. (4-38) and (4-39) be equal to /(/ - -1) with / an integer. 

It is instructive to compare these results with the behavior of the Is state. In the first place, the effect of a uniform electric field on the Is level is zero, to first order, because (ls i/ ls) vanishes for reasons of symmetry. Only when first-order corrections are made to the 1 s wavefunction are energy effects seen, and these occur in the second-order energy terms. Therefore, the Is state gives a second-order Stark effect, but no first-order effect. The Is state gives no first-order effect because the spherically symmetric zeroth-order wavefunction has no electric dipole to interact with the field. But the proper zeroth-order wave functions for some of the = 2 states, given by Eqs. (12-81) and (12-82), do provide electric dipoles in opposite directions that interact with the field to produce first-order energies of opposite signs. 

For a system of either bosons or fermions, the wavefunction must have the correct properties of symmetry and antisymmetry. Particles with half-integral spin, such as electrons, are fermions and require antisymmetric wavefunctions. Particles with integral spin, such as photons, are bosons and require symmetric wavefunctions. The complete space-spin wavefunction of a system of two or more electrons must be antisymmetric to the permutation of any two electrons. Except in the simplest cases, the wavefunction for a system of n fermions is positive and negative in different regions of the 3 -dimensional space of the fermions. The regions are separated by one or more (3 - 1 )-dimensional hypersurfaces that cannot be specified except by solution of the Schrodinger equation. 

Like Heitler and London, Wang acknowledged that the correct ground state wavefunction must be symmetric with respect to the two electrons and the two nuclei [Wang, 1928, 582-583]. His wavefunction has the form ... 

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