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The alternating short and long bonds preserve fourfold, but not eightfold, rotations. Including the molecular plane and inversion symmetry, the planar structure belongs to $D_{4h}$, the D4h molecular point group.

Use the printed axes: the short bonds are $1$–$2$, $3$–$4$, $5$–$6$ and $7$–$8$, and the $x$ and $y$ axes pass through the midpoints of opposite short bonds. Interchanging short and long bonds is not a molecular symmetry operation of this geometry.

The angular factors of the nickel $3d$ atomic orbitals transform like quadratic Cartesian functions. In the D4h molecular point group, the assignments areAll five are even under inversion symmetry, hence the $g$ suffixes. The pair $(d_{xz},d_{yz})$ is mixed by a quarter-turn and spans a two-dimensional irreducible representation.

In the Td molecular point group, the corresponding result isThere is no inversion label because $T_d$ has no inversion centre. In both cases the dimensions add to five.

With $A=S_{x_{\mathrm{Si}}}(h,k,l)$ and $B=S_{x_{\mathrm{Mn}}}(h,k,l)$,For $l=0$, the transformation in part c maps $(h,k,0)$ onto its Friedel pair with only a phase factor. Cyclic permutations give the same result when either other index vanishes. Friedel's law therefore survives for every such reflection, including any systematic absences.

For the $(111)$ reflection set $t=2\pi x$. ThenSince the purely positional sums obey $S_x(-\mathbf{h})=S_x(\mathbf{h}{)}^{\ast }$, the scalar resonant Friedel-pair intensity difference isup to the same positive experimental scale factor. In particular,Different positional phases can therefore produce a nonzero resonant intensity difference at $(111)$, allowing the opposite structural handednesses to be distinguished. The claimed violation is generic, not an unconditional identity: the source supplies no numerical internal coordinates. For a concrete noncentrosymmetric counterexample, take $x_{\mathrm{Si}}=1/8$ and $x_{\mathrm{Mn}}=5/8$. Their orbit sums at $(111)$ are $\sqrt{2}(1-i)$ and $-\sqrt{2}(1-i)$, so the difference vanishes even if $f^{\prime \prime }\ne 0$; each generic chemical orbit is noncentrosymmetric, so this cancellation does not require inversion symmetry. Accidental cancellation, or $f^{\prime \prime }=0$, must be excluded before asserting a violation for a particular specimen.

Under spatial inversion the electric polarization, a polar vector, changes from $\mathbf{P}$ to $-\mathbf{P}$. Neumann's principle requires an equilibrium property of a phase retaining inversion symmetry to be unchanged. Thus $\mathbf{P}=-\mathbf{P}$, forcing $\mathbf{P}=0$: a centrosymmetric phase cannot itself be ferroelectric. A transition can, of course, remove that symmetry.

For an inversion-symmetric parent phase the Landau theory free energy satisfies $F(\mathbf{P})=F(-\mathbf{P})$. Every polynomial term of odd total degree changes sign and must have zero coefficient. Restricting to one fixed direction gives an even scalar expansion in $P$.

Use the indicated out-of-plane displacements and the equivalence of the four oxygen sites and of the two carbon sites in the isolated planar oxalate ion. In $V_1$, the two oxygen atoms on one side move oppositely to those on the other side, and the two carbon atoms also move oppositely. Their dipole changes cancel. The mode is even under molecular inversion symmetry, which independently forbids an electric-dipole transition.

In $V_2$, all four oxygen atoms move together out of the plane while both carbon atoms move in the opposite direction. The positive and negative charge distributions acquire a relative out-of-plane displacement, producing a nonzero derivative of the molecular electric dipole moment.

In $V_3$, two oxygen atoms move out and two move in, in a checkerboard arrangement, while the carbon atoms are stationary. The equivalent oxygen contributions cancel. This mode illustrates that being odd under inversion symmetry is necessary but not sufficient for vibrational infrared activity: its full symmetry does not transform like any component of an electric dipole moment.These conclusions concern the isolated-molecule displacements drawn in the source; a different crystal environment can alter selection rules through additional symmetry breaking.

For a second-order response, $P_i^{(2)}({\omega }_3)={ϵ}_0{\sum }_{jk}{\chi }_{ijk}^{(2)}({\omega }_3;{\omega }_1,{\omega }_2)E_j({\omega }_1)E_k({\omega }_2)$, with the appropriate frequency conservation and negative-frequency convention for difference-frequency processes. Interchanging the two input field-frequency pairs gives intrinsic permutation symmetry of nonlinear susceptibility:For second-harmonic generation the input frequencies agree, so the last two Cartesian indices are symmetric and can be contracted into six input-pair labels. This reduces the unrestricted 27 Cartesian entries to at most 18 before crystal symmetries are imposed.

Crystal point-group transformations must leave the material tensor invariant and can force entries to vanish or become related. Under spatial inversion both electric field and polarization reverse, whereas the field product is even. A centrosymmetric bulk therefore has zero electric-dipole ${\chi }^{(2)}$. Surfaces and interfaces can break inversion symmetry, and higher-multipole processes are outside that simple prohibition.

Kleinman symmetry is a stronger approximate permutation symmetry used sufficiently far from resonances when dispersion can be neglected over the participating frequencies. It is not justified merely because two input frequencies are equal; resonant or appreciably dispersive media need the full frequency-dependent relations. These distinctions determine which polarization and propagation geometries have nonzero effective coupling, in addition to whether they satisfy phase matching.^[ref]

The Rashba effect, named for Emmanuel Rashba, is a momentum-dependent spin-orbit interaction associated with broken spatial inversion symmetry. In an isotropic two-dimensional model, choose $\lambda >0$ and write the Hamiltonian $H={\hslash }^2{k}^{2}I/(2m)+\lambda (\hat{\mathbf{z}}\times \mathbf{k})\cdot \mathbf{\sigma }$, where $I$ is the identity matrix and $\mathbf{\sigma }$ is the vector of Pauli matrices. The eigenvalues of the spin term give $E_{\pm }(k)={\hslash }^2{k}^2/(2m)\pm \lambda k$. Define $k_R=m\lambda /{\hslash }^2$ and $E_R={\hslash }^2{k}_R^2/(2m)$. The lower branch is $E_{-}(k)={\hslash }^2(k-k_R{)}^2/(2m)-E_R$ and has a ring of minima at $k=k_R$. A negative $\lambda$ reverses the spin directions; ordering the two energies still uses $|\lambda |$. The alternative parameter $q_R=2k_R$ must not be mistaken for the minimum radius.