Parity

Certain quantum-mechanical operators have no classical analog. An example is the parity operator. Recall that the harmonic-oscillator wave functions are either even or odd. We shall show how this property is related to the parity operator. 

The parity operator ft is defined in terms of its effect on an arbitrary function/  

The parity operator replaces each Cartesian coordinate with its negative. For example, 

As with any quantum-mechanical operator, we are interested in the eigenvalues Cj and the eigenfunctions gj of the parity operator  

Since/is arbitrary, we conclude that fl equals the unit operator  

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The Hamiltonian considered above, which connmites with E, involves the electromagnetic forces between the nuclei and electrons. However, there is another force between particles, the weak interaction force, that is not invariant to inversion. The weak charged current mteraction force is responsible for the beta decay of nuclei, and the related weak neutral current interaction force has an effect in atomic and molecular systems. If we include this force between the nuclei and electrons in the molecular Hamiltonian (as we should because of electroweak unification) then the Hamiltonian will not conuuiite with , and states of opposite parity will be mixed. However, the effect of the weak neutral current interaction force is mcredibly small (and it is a very short range force), although its effect has been detected in extremely precise experiments on atoms (see, for... 

The conmron flash-lamp photolysis and often also laser-flash photolysis are based on photochemical processes that are initiated by the absorption of a photon, hv. The intensity of laser pulses can reach GW cm or even TW cm, where multiphoton processes become important. Figure B2.5.13 simnnarizes the different mechanisms of multiphoton excitation [75, 76, 112], The direct multiphoton absorption of mechanism (i) requires an odd number of photons to reach an excited atomic or molecular level in the case of strict electric dipole and parity selection rules [117],... 

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. 

At this point, it is important to note that as the potential energy surfaces are even in the vibrational coordinate (r), the same parity, that is, even even and odd odd transitions should be allowed both for nonreactive and reactive cases but due to the conical intersection, the diabatic calculations indicate that the allowed transition for the reactive case ate odd even and even odd whereas in the case of nomeactive transitions even even and odd odd remain allowed. 

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... 

Where the summation is over all 2n pemrutations P each with parity Cp. We use a short-hand notation ... 

The sign of the last term depends on the parity of the system. Note that in the first and last term (in fact, determinants), the spin-orbit functions alternate, while in all others there are two pairs of adjacent atoms with the same spin functions. We denote the determinants in which the spin functions alternate as the alternant spin functions (ASF), as they turn out to be important reference terms. 

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. 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of ineitia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes a,b,c). In order to detemiine the parity of the molecule through inversions in SF, we first rotate all the electrons and nuclei by 180° about the c axis (which is peipendicular to the molecular plane) and then reflect all the electrons in the molecular ab plane. The net effect is the inversion of all particles in SF. The first step has no effect on both the electronic and nuclear molecule-fixed coordinates, and has no effect on the electronic wave functions. The second step is a reflection of electronic spatial coordinates in the molecular plane. Note that such a plane is a symmetry plane and the eigenvalues of the corresponding operator then detemiine the parity of the electronic wave function. 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of inertia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes (a, b, c). In order to determine the parity of the molecule through inversions in SF, we first rotate all the displacement vectors... 

Consideration of stereochem-iitry. The parity or handedness - R/S or chjirans - of a stcreoccnter can be obtained by considering the sequence of the Morgan numbers of die atoms, similarly to CIP. Then the number of pairwise interclianges is counted until the numbers arc in ascending order (see Section 2,8,5). 

Figure 2 76. A typical 2D Molflle of (2R,3f,5P)-2-hydro)cy-3,5-heptadiene nitrile with stereochemical flags (parity values, etc.) in the gray columns. For further explanation, see the text.

Figure 2-77. Determination of parity value, a) First the structure is canonicalized. Only the Morgan numbers at the stereocenter are displayed here, b) The listing starts with the Morgan numbers of the atoms next to the stereocenter (1), according to certain rules. Then the parity value is determined by counting the number of permutations (odd = 1, even = 2).

Figure 3-22 shows a nucleophilic aliphatic substitution with cyanide ion as a nucleophile, i his reaction is assumed to proceed according to the S f2 mechanism with an inversion in the stereochemistry at the carbon atom of the reaction center. We have to assign a stereochemical mechanistic factor to this reaction, and, clearly, it is desirable to assign a mechanistic factor of (-i-1) to a reaction with retention of configuration and (-1) to a reaction with inversion of configuration. Thus, we want to calculate the parity of the product, of 3 reaction from the parity of the... 

Now we cany out the same reaction with thiolate as nucleophile, as shown in Figure 3-23. hlowever, we must now realize that the product has a parity of (+ 1) although we have the same mechanism as in Figure 3-22 with inversion of configuration. 

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