Invariance rotational

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... 

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. 

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

G. W. Trucks and M. J. Frisch, Rotational Invariance Properties of Pruned Grids for Numerical Integration, in preparation (1996). 

And what if the basis functions are centred on different atoms The CNDO solution to the problem is to take all possible integrals such as those above to be equal, and to assume that they depend only on the atoms A and B on which the basis functions are centred. This satisfies the rotational invariance requirement. In CNDO theory, we write the two-electron integrals as pab and they are taken to have the same value irrespective of the basis functions on atom A and/or atom B. They are usually calculated exactly, but assuming that the orbital in question is a Is orbital (for hydrogen) or a 2s orbital (for a first row atom). 

It should be obvious that if A vanishes, the phase degree of freedom has to become redundant, as seen later. It would be worth mentioning that similar configuration has been studied in other contexts [21-23], Note that the configuration in (44) breaks rotational invariance as well as translational invariance, but the latter invariance is recovered by an isospin rotation [26]. 

The above function is a one-center correlated Gaussian with exponential coefficients forming the symmetric matrix A]. <1) are rotationally invariant functions as required by the symmetry of the problem—that is, invariant with respect to any orthogonal transformation. To show the invariance, let U be any 3x3 orthogonal matrix (any proper or improper rotation in 3-space) that is applied to rotate the r vector in the 3-D space. Prove the invariance ... 

The Kronecker product with the identity ensures rotational invariance (sphericalness) elliptical Gaussians could be obtained by using a full n x n A matrix. In the former formulation of the basis function, it is difficult to ensure the square integrability of the functions, but this becomes easy in the latter formulation. In this format, all that is required is that the matrix, A, be positive definite. This may be achieved by constructing the matrix from a Cholesky decomposition A), = Later in this work we will use the notation... 

We can derive a similar result from the rotational invariance if we define a path y t) by... 

R. M. Erdahl, C. Garrod, B. Golli, and M. Rosina, The application of group theory to generate new representability conditions for rotationally invariant density matrices. J. Math. Phys. 20, 1366-1374 (1979). 

Mak L, Grandison S, Morris RJ (2008) An extension of spherical harmonics to region-based rotationally invariant descriptors for molecular shape description and comparison. J Mol Graph Model 26 1035-1045... 

The elements of Tr wq restrictions of rotation-invariant functions to the ball of radius R. NIq will apply the Stone-Weierslrass Theorem (Theorem 3.2) and Proposition 3.7 to show that Tr 0 y spans Br), which will imply thatJ 0 y spans... 

T complex scalar product space of rotation-invariant functions in... 

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