Symmetries atomic electron wavefunction

Section treats the spatial, angular momentum, and spin symmetries of the many-electron wavefunctions that are formed as antisymmetrized products of atomic or molecular orbitals. Proper coupling of angular momenta (orbital and spin) is covered here, and atomic and molecular term symbols are treated. The need to include Configuration Interaction to achieve qualitatively correct descriptions of certain species electronic structures is treated here. The role of the resultant Configuration Correlation Diagrams in the Woodward-Hoffmann theory of chemical reactivity is also developed. 

Open-shell Pseudohamiltonians.—The majority of atoms do not have valence structures which can be represented by the fully closed-shell wavefunction of equation (14), and consequently ab initio pseudopotentials cannot be derived directly from the theory outlined above. Acceptable wavefunctions for such atoms require either more than one determinant or the use of the symmetry-equivalenced or generalized Hartree-Fock method, and usually include partially filled shells. The total all-electron wavefunction may be symbolically expressed in terms of four subspaces,... 

Before leaving the discussion of this area, let us consider a specific chemical example. The water molecule has C2V symmetry, hence its normal vibrational modes have A, Ai, B, or B2 symmetry. The three normal modes of H2O are pictorially depicted in Fig. 6.3.1. From these illustrations, it can be readily seen that the atomic motions of the symmetric stretching mode, iq, are symmetric with respect to C2, bending mode, i>2, also has A symmetry. Finally, the atomic motions of the asymmetric stretching mode, V3, is antisymmetric with respect to C2 and This example demonstrates all vibrational modes of a molecule must have the symmetry of one of the irreducible representations of the point group to which this molecule belongs. As will be shown later, molecular electronic wavefunctions may be also classified in this manner. 

The OBS-GMCSC method offers a practical approach to the calculation of multiconfiguration electronic wavefunctions that employ non-orthogonal orbitals. Use of simultaneously-optimized Slater-type basis functions enables high accuracy with limited-size basis sets, and ensures strict compliance with the virial theorem. OBS-GMCSC wavefunctions can yield compact and accurate descriptions of the electronic structures of atoms and molecules, while neatly solving symmetry-breaking problems, as illustrated by a brief review of previous results for the boron anion and the dilithium molecule, and by newly obtained results for BH3. 

From a more practical point of view, electronic transitions follow two types of selection rules because of the orbital and spin nature of the electronic wavefunction. The first, called the Laporte rule, requires that A/ = + 1 for the orbitals involved in the transition. It predicts, for instance, that electronic transitions for transition metal ions in 7 d symmetry (involving orbitals with d-p character) should be more intense than Laporte-forbid-den d-d transitions in Oh symmetry involving orbitals of the same character thus leading to A/ 0. By contrast, charge-transfer transitions are essentially Laporte-allowed since they concern orbitals involving different atoms with different characters. In the case of centrosymmetric complexes, this rule implies a change of parity u u and g -> g transitions (as, for... 

Before we enter into a more detailed discussion on the determination of the molecular electric quadrupole moments and on additivity rules for atom susceptibilities, we will draw some general conclusions from the theoretical expressions for the g- and -values given in Eqs. (1.2) and (1.4), respectively. We first restate that the perturbation sums are necessarily zero if the total electronic wavefunction (for simplicity we may tliink of a Slater determinant) has cylindrical symmetry with respect to the rotational axis in consideration. To see this, we recall that in cylindrical coordinates with a as the sjmimetry axis ... 

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. 

Because we start with three atomic orbitals, we can construct three molecular orbitals. The general procedure for determining the values of the coefficients Ca, Cb, and Cq for these three molecular orbitals is mathematically involved in the present case, however, we can use what we know about the symmetry of the problem and the general behavior of electron wavefunctions to obtain a very good qualitative picture of the molecular orbitals. First, the ozone molecule is symmetric, with a mirror plane bisecting the O—O—0 bond angle. Reflection of the molecular orbital through this... 

Figure 2c shows the electronic structure of graphene described by a simple tight-binding Hamiltonian the electronic wavefunctions from different atoms overlap. However, such an overlap between the Pz(it) orbital and the Px and Py orbitals is zero by symmetry. Thus, the Pz electrons form the 71 band, and they can be treated independently from the other valence electrons. The two sub-lattices lead to the formation of two bands, n and Jt, which intersect at the corners of the Brillouin zone. This yields the conical energy spectrum (Dirac cone, inset in Fig. 2c) near the points K and K, which are called Dirac points. The bottom cone (equivalent to the HOMO molecular orbital) is fully occupied, while the top cone (equivalent to the LUMO molecular orbital) is empty. The Fermi level Ep is chosen as the zero-energy reference and lies at the Dirac point. Consequently, graphene is a special semimetal or zero-band-gap semicondutor, whose intrinsic Fermi surface is reduced to the six points at the corners of the two-dimensional Brillouin zone. 

For an atom the dipole moment operator has inversion symmetry across the nucleus therefore the absorption integral is zero if both the electron wavefunctions are of the same parity, but non-zero if they are of different parity. Thus an s s orbital transition is forbidden , but s - p orbital transition is allowed . This is the Laporte selection rule (also known as the parity selection rule), perhaps the most commonly observed consequence of which is the low intensities of d-d transitions in transition metal complexes. By itself, the parity selection rule would suggest that an i -> / transition is allowed. However, consideration of conservation of angular momentum, restricts changes to those transitions in which A1 = 1. (The possibilities of an increase or decrease in / arise because of the vector nature of momenta, which can oppose or reinforce one another). 

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