Proper orthogonal matrix

The group of all real orthogonal matrices of order 3 and determinant +1 will be denoted by 0(3). Such matrices correspond to pure rotation or proper rotation of the coordinate system. An orthogonal matrix with determinant —1 corresponds to the product of pure rotation and inversion. Such transformations are called improper rotations. The matrix corresponding to inversion is the negative of the unit matrix... 

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 ... 

If the selected orthogonal matrix does not permit one to estimate all the elTects, we can select a larger matrix or modify properly the requirements of the problem to be solved. 

This is a most useful result since we often need to calculate the inverse of a 3 x3 MR of a symmetry operator R. Equation (10) shows that when T(R) is real, I R)-1 is just the transpose of T(R). A matrix with this property is an orthogonal matrix. In configuration space the basis and the components of vectors are real, so that proper and improper rotations which leave all lengths and angles invariant are therefore represented by 3x3 real orthogonal matrices. Proper and improper rotations in configuration space may be distinguished by det T(R),... 

Thereby the matrix 11(F) denotes a K-dimensional permutation matrix and F (F) a 3 by 3 orthogonal matrix. The form of this representation follows from the fact that each isometric transformation maps the NC Xk, Zk, Mk onto a NC which by definition has the same set of distances, i.e. is isometric to NC Xk, Zk, Mk. Expressed alternatively, the nuclear configurations NC Xk( ), Zk, Mk and NC Xk(F 1 ( )), Zk, Mk are properly or improperly congruent up to permutations of nuclei with equal charge and mass for any F G ( ). The set of matrices Eq. (2.12) forms a representation of J d) by linear transformations and will furtheron be denoted by... 

As in section 3, the diabatic representations are obtained finding the orthogonal matrix T such that TT = P. Again, the diabatic representation is not unique because T is defined within an overall p-independent orthogonal transformation. In actual calculations, one has to manipulate the potential energy matrix V = Te(p)T, whose large dimensions are often the bottleneck in practice. Proper choice of T is therefore crucial. The practical implementation (see the final section) of hyperspherical harmonics as the proper diabatic set is of great perspective power. 

The pattern of such matrices is simple one has to put in some places sines, cosines, zeros and ones with the proper signs. This matrix is orthogonal, i.e. = U, which you may easily check. The product of two orthogonal matrices represents an orthogonal matrix, therefore any rotation corresponds to an orthogonal matrix. 

An interesting approach to the problem of calculating the local wave function of a fragment in interaction with its surrounding has been proposed by Kirtman and de Melo [88]. Their approach is based on the density matrix formulation of the HF problem. The zeroth order approximation to the density matrix of the total system is a simple direct sum of the density matrices of the fragments, provided that the AO basis is orthogonal by construction or it is properly orthogonalized. 

S is a 3 X 3 real, orthogonal matrix representation of one of the proper or improper rotations of the point group of the space group, v(S) is a vector associated with S which is smaller than any primitive translation vector and t(m) = zm. a., where a. are the primitive translation vectors and m. ... 

Due to the integral approximations used in the MNDO model, closed-shell Pauli exchange repulsions are not represented in the Hamiltonian, but are only included indirectly, e.g., through the effective atom-pair correction terms to the core-core repulsions [12], To account for Pauli repulsions more properly, the NDDO-based OM1 and OM2 methods [23-25] incorporate orthogonalization terms into the one-center or the one- and two-center one-electron matrix elements, respectively. Similar correction terms have also been used at the INDO level [27-31] and probably contribute to the success of methods such as MSINDO [29-31],... 

Thus the problem of deducing E and C is the determination of the proper non-orthogonal rotation matrix, R. 

All of the matrices we have just worked out, as well as all others which describe the transformations of a set of orthogonal coordinates by proper and improper rotations, are called orthogonal matrices. They have the convenient property that their inverses are obtained merely by transposing rows and columns. Thus, for example, the inverse of the matrix... 

In the octahedral CF the ground term 6Aig is not split by the spin-orbit interaction by means of the bilinear spin-spin interaction. Consequently all the MPs vanish gz - ge = gx - ge = D = E = /tip = 0. This is caused by the fact that the angular momentum components of the type (6A[g Lfj 4Ty) and (6Aig ifl 2Ty) vanish exactly due to the orthogonality of the spin functions of different spin multiplicities. Therefore, the simple SH formalism does not work properly, and we are left with the problem of a complete spin-orbit interaction matrix between the CFTs of different spin multiplicities. 

The coupling of the local microstates into the proper molecular states is provided by the diagonalisation of the spin Hamiltonian matrix. For the zero-field case the diagonalisation matrix represents an orthogonal transformation and its matrix elements relate to the combination of the Clebsch-Gordan coefficients. 

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