Wavefunction symmetry

Nakatsuji H, Hirao K (1978) Cluster expansion of the wavefunction. symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell orbital theory. J Chem Phys 68 2053... 

H. Nakatsuji and K. Hirao, J. Chem. Phys., 68, 2053 (1978). Cluster Expansion of the Wavefunction. Symmetry-Adapted-Cluster Expansion, Its Variational Determination, and Extension of Open-Shell Orbital Theory. 

In addition, as far as electron-electron interaction is neglected, the tt electrons are subject to a potential with the full spatial symmetry of the CNT topology. The electronic wavefunctions can then be classified according to their transformation properties under the symmetry operations leaving the CNT invariant. As far as dipole approximation holds, both the linear and nonlinear optical response of a CNT are governed by matrix elements of the dipole operator between two electronic wavefunctions. Being the selection rules of dipole matrix elements essentially determined by the wavefunctions symmetry, it turns out that the optical properties of CNT are essentially rooted in their topology. 

Clary, D.C. (1994) Four-atom reaction dynamics,. 7. Phys. Chem. 98, 10678-10688. Pack, R.T. and Parker, G.A. (1987) Qtianttim reactive scattering in three dimensions tising hypersidierical (APH) coordinates. Theory, J. Chem. Phys. 87, 3888-3921. Truhlar, D.G., Mead, C.A. and Brandt, M.A. (1975) Time-Reversal Invariance, Representations for Scattering Wavefunctions, Symmetry of the Scattering Matrix, and Differential Cross-Sections, Adv. Che.m. Phys. 33, 295-344. 

Electrons, protons and neutrons and all other particles that have 5 = are known as fermions. Other particles are restricted to 5 = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fermions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection rules. It can be shown that the spin quantum number S associated with an even number of fermions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fermions, respectively, so the wavefunction symmetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number therefore behave like individual bosons and those with odd atomic number as fermions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

The two-electron wavefunctions, like the MOs they are made from, have symmetry properties described by the point group representations. The next section explains one significance of these overall wavefunction symmetries. 

We shall see that the correlation of wavefunction symmetries in such a manner as this is a powerful technique in understanding and predicting chemical behavior. 

We now focus on the maimer in which spin considerations affect wavefunction symmetry. The electrons are still identical particles, so our particle distribution must be... 

Most of our attention thus far has been with wavefunction symmetry and energy. However, understanding atomic spectroscopy or interatomic interactions (in reactions or scattering) requires close attention to angular momentum due to electronic orbital motion and spin. In this section we will see what possibilities exist for the total electronic angular momenta of atoms and how these various states are distinguished symbolically. 

If the nuclei of the atoms in the diatomic molecule have integer spins, they are bosons and the complete wavefunction (translational-nuclear-vibrational-electronic-rotational) must be symmetric with respect to exchange. If the nuclei of the atoms in the diatomic molecule have half-integer spins, then they are fermions and the complete wavefunction must be antisymmetric upon exchange. Given that P trans nd Pvii, are symmetric and eiect almost always symmetric for homonuclear diatomic molecules, the overall wavefunction symmetry behavior dictates what combinations of Prot and nuc are allowed. What we find ultimately is that the combinations of Prot and nujhave different degeneracies, so that the populations of molecules in various rotational states are skewed from normal expectations. 

FIGURE 18.2 This vibrational spectrum of acetylene, C2H2, shows intensity variations that are due to the effect of the nuclear wavefunctions symmetry on the degeneracies of the overall wavefunction of the molecule. This is one of the few direct consequences of nuclear wavefunctions in chemistry. Source L. W. Richards, J. Chem. Ed., 1996, 43 645. 

If one attempts to avoid the Scylla of spin contamination, which plagues calculations based on UHF wavefunctions, one encounters the Charybdis of symmetry breaking in ROHF wavefunctions. Symmetry breaking often makes ROHF geometry optimizations and energies even less reliable than their UHF counterparts, and RMP2 calculations will not solve the problems caused by symmetry-broken ROHF wavefunctions. 

The QRHF CC method has been applied by Stanton and co-workers (11,19) and the FSMRCC method has been applied by Kaldor (9,70) to consider the ground state of the NO3 radical system. In these approaches, the RHF orbitals for the nitrate anion are first solved, and a valence electron is then deleted to form the reference for the neutral. Although such a reference is not variationally optimal, it may be more suitable for the NO3 correlation problem than the standard UHF (unrestricted HF) reference. In UHF-based approaches, extensive orbital relaxation is involved in correcting for the symmetry-broken reference, while in the anion-based methods, the wavefunction symmetry conditions are satisfied from the start. 

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