Parity odd

Since Q is negative, and //ab,cl for tbe ground state must be a negative sign, it follows that the ground state for the odd parity case is the in-phase combination, while for the even parity case, the out-of-phase wave function is the ground state. 

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

In the lowest optieally excited state of the molecule, we have one eleetron (ti ) and one hole (/i ), each with spin 1/2 which couple through the Coulomb interaetion and can either form a singlet 5 state (5 = 0), or a triplet T state (S = 1). Since the electric dipole matrix element for optical transitions — ep A)/(me) does not depend on spin, there is a strong spin seleetion rule (AS = 0) for optical electric dipole transitions. This strong spin seleetion rule arises from the very weak spin-orbit interaction for carbon. Thus, to turn on electric dipole transitions, appropriate odd-parity vibrational modes must be admixed with the initial and (or) final electronic states, so that the w eak absorption below 2.5 eV involves optical transitions between appropriate vibronic levels. These vibronic levels are energetically favored by virtue... 

We assume that standard Coulomb-correlated models for luminescent polymers [11] properly described the intrachain electronic structure of m-LPPP. In this case intrachain photoexcitation generate singlet excitons with odd parity wavefunctions (Bu), which are responsible for the spontaneous and stimulated emission. Since the pump energy in our experiments is about 0.5 eV larger than the optical ran... 

The theoretical reason is as follows. Although the placing of the ligands in a tetrahedral molecule does not define a centre of symmetry, the d orbitals are nevertheless centrosymmetric and the light operator is still of odd parity and so d-d transitions remain parity and orbitally Al = 1) forbidden. It is the nuclear coordinates that fail to define a centre of inversion, while we are considering a... 

Figure 2. Diagram depicting interprocessor communication in the case of wrapped distribution of K states with two per processor. The figure illustrates 7=15, odd parity. The eight processors are labeled i = 0,1..., 7 and the corresponding K states that they contain to the right of each i. Arrows indicate communication between processors for each of the K states.

For A = 1, the eigenfunctions of IT are even functions of q, while for A = —1, they are odd functions of q. An even function of q is said to be of even parity, while odd parity refers to an odd function of q. Thus, the eigenfunctions of n are any well-behaved functions that are either of even or odd parity in their cartesian variables. 

These eigenfunctions are also eigenfunctions of the parity operator, leading to the conclusion that c = 1. Consequently, some eigenfunctions will be of even parity while all the others will be of odd parity. 

This integral vanishes because the unperturbed ground state of the hydrogen atom, the Is state, has even parity and z has odd parity. 

The macrostates can have either even or odd parity, which refers to their behavior under time reversal or conjugation. Let e, = 1 denote the parity of the th microstate, so that = e r). (It is assumed that each state is... 

Consider two variables, x = A,B, where A has even parity and B has odd parity. Then using the terminal velocity it follows that... 

It is clear that the tme nonequilibrium probability distribution requires an additional factor of odd parity. Figure 1 sketches the origin of the extra term. 

All the other linear terms vanish because they have opposite parity to the flux, (x(r)x(r))0 = 0. (This last statement is only true if the vector has pure even or pure odd parity, x(T) = x(T j. The following results are restricted to this case.) The static average is the same as an equilibrium average to leading order. That is, it is supposed that the exponential may be linearized with respect to all the reservoir forces except the zeroth one, which is the temperature, X()r = 1 /T, and hence xofT) = Tffl j, the Hamiltonian. From the definition of the adiabatic change, the linear transport coefficient may be written... 

Thus the potential matrix for even parity is identical to that for odd parity for K y 0. The centrifugal potential with the (J — ji2)2 term in Eq. (1), which is not diagonal in K in the BF representation, has matrix elements... 

Excited states formed by light absorption are governed by (dipole) selection rules. Two selection rules derive from parity and spin considerations. Atoms and molecules with a center of symmetry must have wavefunctions that are either symmetric (g) or antisymmetric (u). Since the dipole moment operator is of odd parity, allowed transitions must relate states of different parity thus, u—g is allowed, but not u—u or g—g. Similarly, allowed transitions must connect states of the same multiplicity—that is, singlet—singlet, triplet-triplet, and so on. The parity selection rule is strictly obeyed for atoms and molecules of high symmetry. In molecules of low symmetry, it tends to break down gradually however,... 

Figure 11.5 Energy versus nuclear charge plots for odd parity (A) and even parity (B) nuclides. Experimentally observed values of binding energy differences are reported in parentheses for comparative purposes. Erom Nuclear and Radio chemistry, G. Eriedlander and J. W. Kennedy, Copyright 1956 by John Wiley and Sons. Reprinted by permission of John Wiley Sons, Etd. 

When a polaron can move between two sites of equal energy, the state of the system (electron+phonons) will split into two states of even and odd parity, separated by an energy AE fc27t2/mpa2, which, when mp is large, may be quite small. The polarizability of this system will lead to the formation of a moment in a field F of order... 

Suppose one first considers electric-dipole and magnetic-dipole transitions. As is now well recognized, these are the major contributors to rare-earth absorption and emission spectra. We know that the electric-dipole operator transforms as a polar vector, that is, just as the coordinates (23, 24). This means that it has odd parity under an inversion operation. On the other hand, the magnetic-dipole operator transforms as an axial vector or pseudovector and of course must have even parity (23, 24). 

As stated in an earlier paragraph, the sharp emission and absorption lines observed in the trivalent rare earths correspond to/->/transitions, that is, between free ion states of the same parity. Since the electric-dipole operator has odd parity,/->/matrix elements of it are identically zero in the free ion. On the other hand, however, because the magnetic-dipole operator has even parity, its matrix elements may connect states of the same parity. It is also easily shown that electric quadrupole, and other higher multipole transitions are possible. 

Figure 3. In (a) the potential curve is unsymmetric with respect to the equilibrium position 0 of the nucleus. The crystal field in this case causes mixing of the even and odd parity states. In (b) there is symmetry with respect to the nucleus when it is at 0, but vibration carries the ion to the unsymmetric point P [from Ref. (25)].

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