Symmetries and quantum numbers

The use of the term (ATj — 1) in the Coulomb interactions ensures that this Hamiltonian automatically contains the electron-nuclear and nuclear-nuclear interactions from the nuclear charges associated with the 7r-electrons. To see this, let us expand the Coulomb interaction  

The first term on the right-hand side is the electron-electron Coulomb interaction. The second term is the potential energy experienced by the electrons from 

The electron-electron interactions are usually treated using the semiempiri-cal Ohno or Mataga-Nishimoto potentials. These expressions are interpolations between a Coulomb potential, e /Aneorij, at large separations and U for the interaction between two electrons in the same orbital (rjj = 0). For bond-lengths in A and energies in eV the Ohno potential is 

Most linear conjugated molecules and polymers possess spatial symmetries, while cyclic polymers possess axial symmetry. Conjugated systems also possess an approximate particle-hole symmetry. These symmetries characterize the electronic 

Su-Schrieffer-Heeger (SSH) ometry Noninteracting electrons with dynamic nuclei 4 

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Fig. 49. Correlation between the energy levels of (1) free rotation of the symmetric top, and (2) torsion vibrations in the potential with symmetry Cj. Quantum numbers J and K enumerate rotational levels, n vibrational levels. Relative positions of A and E levels are shown on the right.

Figure 2. New proposed form of the periodic table based on symmetry of electronic configurations and quantum numbers.

For a linear molecule, the position of the symmetry axis (the molecule-fixed. z-axis) in space is specified by only two Euler angles, / and 7, which are respectively identical to the spherical polar coordinates 6 and (see Fig. 2.4). The third Euler angle, a, which specifies the orientation of the molecule-fixed x- and y-axes, is unaffected by molecular rotation but appears explicitly as an O- dependent phase factor in the rotational basis functions [Eq. (2.3.41)]. Cartesian coordinates in space- and molecule-fixed systems are related by the geometrical transformation represented by the 3x3 direction cosine matrix (Wilson et al., 1980, p. 286). The direction cosine matrix a given by Hougen (1970, p. 18) is obtained by setting a = 7t/2 (notation of Wilson et al, 1980 6 fi,4)=, x = oi 7t/2). The direction cosine matrix is expressed in terms of sines and cosines of 9 and 4>. Matrix elements (J M O la JMQ), evaluated in the JMQ) basis, of the direction cosines, are expressed in terms of the J, M, and quantum numbers. The direction cosine matrix elements of Hougen (1970, p. 31), Townes and Schawlow (1955, p. 96), and Table 2.1 assume the basis set definition derived from Eq. (2.3.40) and the phase choice a = 7t/2 ... 

Even though the quantum numbers I and lose their meaning in the case of the metal complex, symmetry properties based on group theory can be used to classify the states and to simplify the theory. Theoretical treatment of dd states may be slightly different depending on the symmetry and the number of d-elec-trons. In the following, we restrict ourselves to the symmetry of O. We will illustrate the theories for systems of one d-electron and of six d-electrons. 

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

In these equations, J and M are quantum numbers associated with the angular momentum operators and J, respectively. The number II = 0, 1 is a parity quantum number that specifies the symmetry or antisymmetry of the column vector with respect to the inversion of the nuclei through G. Note that the same parity quantum number II appears for and Also, the... 

In this assignment, we keep the symmetry species of the vibronic state in D3/, but indicate the vibrational quantum numbers for the Civ normal modes. The energy increases from left to right, and up to down,... 

For vei y small vibronic coupling, the quadratic terms in the power series expansion of the electronic Hamiltonian in normal coordinates (see Appendix E) may be considered to be negligible, and hence the potential energy surface has rotational symmetry but shows no separate minima at the bottom of the moat. In this case, the pair of vibronic levels Aj and A2 in < 3 become degenerate by accident, and the D3/, quantum numbers (vi,V2,/2) may be used to label the vibronic levels of the X3 molecule. When the coupling of the... 

Thus, we can use the approximate quantum number m to label such levels. Moreover, it may be shown [11] that (1) 3/m is one-half of an integer for the case with consideration of the GP effect, while it is an integer or zero for the case without consideration of the GP effect (2) the lowest level must have m = 0 and be a singlet with Ai symmetry in 53 when the GP effect is not taken into consideration, while the first excited level has m = 1 and corresponds to a doublet E conversely, with consideration of the GP effect, the lowest level must have m = j and be a doublet with E symmetry in S, while the first excited level corresponds to m = and is a singlet Ai. Note that such a reversal in the ordering of the levels was discovered previously by Hancock et al. [59]. Note further thatj = 3/m has a meaning similar to thej quantum numbers described after Eq. (59). The full set of quantum numbers would then be... 

Operators that eommute with the Hamiltonian and with one another form a partieularly important elass beeause eaeh sueh operator permits eaeh of the energy eigenstates of the system to be labelled with a eorresponding quantum number. These operators are ealled symmetry operators. As will be seen later, they inelude angular momenta (e.g., L2,Lz, S, Sz, for atoms) and point group symmetries (e.g., planes and rotations about axes). Every operator that qualifies as a symmetry operator provides a quantum number with whieh the energy levels of the system ean be labeled. 

In summary, the moleeular orbitals of a linear moleeule ean be labeled by their m quantum number, whieh plays the same role as the point group labels did for non-linear polyatomie moleeules, and whieh gives the eigenvalue of the angular momentum of the orbital about the moleeule s symmetry axis. Beeause the kinetie energy part of the... 

The orbitals of an atom are labeled by 1 and m quantum numbers the orbitals belonging to a given energy and 1 value are 21+1- fold degenerate. The many-eleetron Hamiltonian, H, of an atom and the antisymmetrizer operator A = (V l/N )Zp Sp P eommute with total =Zi (i), as in the linear-moleeule ease. The additional symmetry present in the spherieal atom refleets itself in the faet that Lx, and Ly now also eommute with H and A. However, sinee does not eommute with Lx or Ly, new quantum... 

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