Wave symmetry breaking

Another aspect of wave function instability concerns symmetry breaking, i.e. the wave function has a lower symmetry than the nuclear framework. It occurs for example for the allyl radical with an ROHF type wave function. The nuclear geometry has C21, symmetry, but the Cay symmetric wave function corresponds to a (first-order) saddle point. The lowest energy ROHF solution has only Cj symmetry, and corresponds to a localized double bond and a localized electron (radical). Relaxing the double occupancy constraint, and allowing the wave function to become UHF, re-establish the correct Cay symmetry. Such symmetry breaking phenomena usually indicate that the type of wave function used is not flexible enough for even a qualitatively correct description. 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

One is purely formal, it concerns the departure from symmetry of an approximate solution of the Schrodinger equation for the electrons (ie within the Bom-Oppenheimer approximation). The most famous case is the symmetry-breaking of the solutions of the Hartree-Fock equations [1-4]. The other symmetry-breaking concerns the appearance of non symmetrical conformations of minimum potential energy. This phenomenon of deviation of the molecular structure from symmetry is so familiar, confirmed by a huge amount of physical evidences, of which chirality (i.e. the existence of optical isomers) was the oldest one, that it is well accepted. However, there are many problems where the Hartree-Fock symmetry breaking of the wave function for a symmetrical nuclear conformation and the deformation of the nuclear skeleton are internally related, obeying the same laws. And it is one purpose of the present review to stress on that internal link. 

For Sz=0 problems, where the ground state is a singlet state, the use of such a wave function appeared to give significantly lower energies than the orthodox symmetry-adapted solution in many problems, as illustrated below. Later on other types of symmetry breaking have been discovered and Fukutome [7] has given a systematics of the various HF instabilities in a fundamental paper. 

When the symmetry breaking of the wave function represents a biased procedure to decrease the weights of high energy VB stmctures which were fixed to umealistic values the tymmetry and single determinant constraints, one may expect that the valence CASSCF wave function will be symmetry-adapted, since this function optimizes the coefficients of all VB forms (the valence CASSCF is variational determination of the best valence space and of the best valence function, i.e. an optimal valence VB picture). In most problems the symmetry breaking should disappear when going to the appropriate MC SCF level. This is not always the case, as shown below. 

CAS F calculations are not a universal solution to symmetry-breaking of the wave functions, and for such weak resonance problems it is far more reliable to start from state average solutions which treat on an equal footing the two configurations which interact weakly. 

The symmetry breaking due to the vdW interaction, which allows the molecule to attach electron with additional partial waves... 

Symmetry breaking wave packet motion and absence of deuterium isotope efTect in ultra last excited state proton tranfer... 

Y -Hab(Hbb -E)- Hbd Haa -Hab(Hbb -E)"Hh If the Hamiltonian is a one-electron Hamiltonian, for example the Fock operator, the partitioning is done by basis functions, since the latter are usually centered on the atomic nuclei, which belong to donor (d), bridge (b) or acceptor (a). In the Hartree-Fock case, the total wave function is a Slater determinant. There may be problems with symmetry breaking in the symmetric case. Cl that includes the two localized solutions can solve this problem [29-31]. The problem is that the Hartree-Fock method gives energy advantage to a localized state, which holds true also in the unsymmetric case. 

Another way to computationally treat unpaired electrons is to employ restricted open-shell HF (ROHF) theory. Here, we encounter another pit-fall. It is an artifact called symmetry breaking (97). Whereas ROHF wave functions are pure spin states, the ROHF wave function may not retain the symmetry of the molecule. Suppose a molecule has C2V symmetry. The wave function should have the same symmetry, e.g., the orbital lobes on either side of the symmetry plane should be identical. However, with symmetry breaking, the two sides are not equal. The unsymmetrical ROHF wave function may even give lower energy than a physically correct (symmetrical)... 

A curious effect, prone to appear in near degeneracy situations, is the artifactual symmetry breaking of the electronic wave function [27]. This effect happens when the electronic wave function is unable to reflect the nuclear framework symmetry of the molecule. In principle, an approximate electronic wave function will break symmetry due to the lack of some kind of non-dynamical correlation. A typical example of this case is the allyl radical, which has C2v point group symmetry. If one removes the spatial and spin constraints of its ROHF wave function, a lower energy symmetry broken (Cs) solution is obtained. However, if one performs a simple CASSCF or a SCVB [28] calculation in the valence pi space, the symmetry breaking disappears. On the other hand, from the classical VB point of view, the bonding of the allyl radical is represented as a superposition of two resonant structures. 

The (n - 71 )1 3 excited states of the pyrazine molecule are a well-known case of wave function symmetry breaking [27,55]. An accidental degeneracy arises when one considers the valence electronic excitations within the equivalent nitrogen lone pairs. The pairs are in opposite and equivalent positions and, when there are two singly occupied orbitals in different symmetries, we will have a pair of accidentally degenerate states as shown below ... 

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