Invariance with respect to inversion - parity

There are orthogonal transformations which are not equivalent to any rotation, e.g., the matrix of inversion 

Now the reader will be taken by surprise. From what we have said, it follows that no molecule has a non-zero dipole moment. Indeed, the dipole moment is calculated as the mean value of the dipole moment operator, i.e. ix = on F- on) = This integral will be calculated very easily the integrand is antisymmetric with respect to inversion and therefore = 0. 

Is therefore the very meaning of the dipole moment, a quantity often used in chemistry and physics, a fairy tale If HCl has no dipole moment, then it is more understandable that H2 does not have either. All this seems absurd. What about this dipole moment  

The Hamiltonian is also invariant with respect to some other symmetry operations like changing the sign of the x coordinates of all particles, or similar operations which are products of inversion and rotation. If one changed the sign of all the x coordinates, it would correspond to a mirror reflection. Since rotational symmetry stems from space isotropy (which we will treat as trivial ), the mirror reflection may be identified with parity P. 

A consequence of inversion symmetry is that the wave functions have to be eigenfunctions of the inversion operator with eigenvalues II = 1, i.e. the wave function is symmetric, or II = —1, i.e. the wave function is antisymmetric. Any asymmetric wave function corresponding to a stationary state is therefore excluded ( illegal ). However, two optical isomers (enantiomers), corresponding to an object and its mirror image, do exist (Fig. 2.4).  

---

Invariance with Respect to Inversion-Parity Invariance with Respect to Charge Conjugation Invariance with Respect to the Symmetry of the Nuclear Framework Conservation of Total Spin Indices of Spectroscopic States... 

J, quantizes its component along the z axis, and II = 1 represents the parity with respect to the inversion. As to the invariance with respect to permutations of identical particles an acceptable wave function has to be antisymmetric with respect to the exchange of identical fermions, whereas it has to be s)mmetric when exchanging bosons. 

We conclude this chapter with a look at some more exotic properties, at least from the point of view of mainstream chemistry. In a 1949 article celebrating Einstein s 70th birthday, Dirac (1949) suggested that the laws of nature might not be invariant with respect to space inversion or time reversal. Special relativity only requires that physical laws be invariant with respect to the position and velocity of the observer, and any change in these can be effected though a series of (infinitesimal) transformations that do not involve reflections of time or space. Experimental evidence for processes that do not conserve parity under space inversion, P-odd processes, was eventually observed in nuclear p decay, contributing in turn to the development of the standard model for... 

---