Rotation invariance

Herschel investigated the optical properties of quartz slabs, which were cut perpendicular to the long crystal axis [17], He found that crystals cut from left-handed quartz would invariably rotate the plane of linearly polarized light in a clockwise fashion, while crystals cut from right-handed quartz rotated the plane of linearly polarized light in a counterclockwise direction. Elsewhere Biot was performing extensive studies on the optical rotatory properties of certain compounds dissolved in fluid solutions, and it was at this time that the connection between optical activity and crystallographic properties was made. 

Other mles are based on numerical evaluation of the eigenvalues it is assumed that in the case of perfect independence among variables, the PC will be the same as the original variables (PCA represents an invariant rotation of axes) and will account for unitary variance in case of autoscaled data, thus a PC with an eigenvalue less than 1 contains less information of one original variable and could be discarded (this rule sometimes is also taken as eigenvalues... 

The group could be a geometrical group such as the group of all translations, rotations, scalings, etc. The desired classification of pattern is invariant under the action of the group, i.e. 

In Fig. 2a, we compare the modulus of the normal component of the magnetic induction B (r) provided by the sensor and the one calculated by the model. Because of the excitation s shape, the magnetic induction B° (r) is rotation invariant. So, we only represent the field along a radii. It s obvious that the sensor does not give only the normal component B = but probably provides a combination, may be linear, of... 

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

Finally, following Mead and Truhlar [10], it may be seen that an interchange of A and B is equivalent to a sign reversal of <() followed by a rotation perpendicular to the AB bond, under the latter of which Aab) is invariant and Fab) changes sign. The net effect is therefore to induce the tiansitions... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

A range of physicochemical properties such as partial atomic charges [9] or measures of the polarizabihty [10] can be calculated, for example with the program package PETRA [11]. The topological autocorrelation vector is invariant with respect to translation, rotation, and the conformer of the molecule considered. An alignment of molecules is not necessary for the calculation of their autocorrelation vectors. 

The length or dimension of the RDF code is independent of the number of atoms and the size of a molecule, unambiguous regarding the three-dimensional arrangement of the atoms, and invariant against translation and rotation of the entire molecule. 

With the exception of integral 22, (pppp Ipppp ). all the integrals can he computed a priori without loss ol rotational invariance. That IS. no integral depends on the value of another integral, except for this Iasi one. It can. however, be shown that... 

Ihc complete neglect of differential overlap (CNDO) approach of Pople, Santry and Segal u as the first method to implement the zero-differential overlap approximation in a practical fashion [Pople et al. 1965]. To overcome the problems of rotational invariance, the two-clectron integrals (/c/c AA), where fi and A are on different atoms A and B, were set equal to. 1 parameter which depends only on the nature of the atoms A and B and the ii ilcniuclear distance, and not on the type of orbital. The parameter can be considered 1.0 be the average electrostatic repulsion between an electron on atom A and an electron on atom B. When both atomic orbitals are on the same atom the parameter is written , A tiiid represents the average electron-electron repulsion between two electrons on an aiom A. 

Symmetry operators leave the eleetronie Hamiltonian H invariant beeause the potential and kinetie energies are not ehanged if one applies sueh an operator R to the eoordinates and momenta of all the eleetrons in the system. Beeause symmetry operations involve refleetions through planes, rotations about axes, or inversions through points, the applieation of sueh an operation to a produet sueh as H / gives the produet of the operation applied to eaeh term in the original produet. Henee, one ean write ... 

The eleetrostatie potential is not invariant under rotations of the eleetron about the x or y axes (those perpendieular to the moleeular axis), so Lx and Ly do not eommute with the Hamiltonian. Therefore, only Lz provides a "good quantum number" in the sense that the operator Lz eommutes with the Hamiltonian. 

The fact that H commutes with Ez, Ex, and Ey and hence E2 is a result of the fact that the total coulombic potential energies among all the electrons and the nucleus is invariant to rotations of all electrons about the z, x, or y axes (H does not commute with L ) since if... 

The use of this formula for integral 22 gives rotational invariance. 

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