Transformation contravariant

The index on the annihilation operator will usually continue to be written in the conventional subscript position, except when added emphasis of the contravariant transformation property is deemed useful in either case the same operator is intended. 

A further reduction occurs when we use only real unimodular matrices we then obtain the real orthogonal group R(m), in which U = U, and the distinction between co- and contravariant transformations disappears. 

Contravariant vectors, those that transform like the coordinate... 

This result shows that, by its transformation properties, Aljkl is equivalent to a covariant vector of rank two. This process of summing over a pair of contravariant and covariant indices is called contraction. It always reduces the rank of a mixed tensor by two and thus, when applied to a mixed tensor of rank two, the result is a scalar ... 

The quantities bij, 6U, and 6 are respectively called the components of covariant, contravariant or mixed tensors of the second order, if they transform according to the formulae... 

Since a, af = S0, or in tensor notation, a, aj = S, the reciprocal axes are contravariant and are written as a . As the Miller indices are the coordinates in the reciprocal base system, they must be covariant and are written as ht. Thus, the Miller indices transform like the direct axes, both being covariant. 

Placement of indices as superscripts or subscripts follows the conventions of tensor analysis. Contravariant variables, which transform like coordinates, are indexed by superscripts, and coavariant quantities, which transform like derivatives, are indexed by subscripts. Cartesian and generalized velocities and 2 thus contravariant, while Cartesian and generalized forces, which transform like derivatives of a scalar potential energy, are covariant. 

It has been already noted that the rate of a steady-state reaction can be regarded as a vector in the P-dimensional space specified by its components, which are the rates along the basic routes. In terms of linear algebra, the above result means that when the basis of routes is transformed the reaction rate vector along these routes is transformed contravariantly. 

Let us now turn to the internal variables. We can consider that one of the internal variables is a tensor of arbitrary rank and transforms as the coordinates do, that is, contravariantly... 

When we rotate a contravariant nxl column vector (for position, velocity, momentum, electric field, etc.) we premultiply it by an n x n rotation tensor R. When, instead, we transform the coordinate system in which such vectors are defined, then the coordinate system and, for example, the V operator are covariant 1 x n row vectors, which are transformed by the tensor R 1 that is the reciprocal of R. A "dot product" or inner product a b must be the multiplication of a row vector a by a column vector b, to give a single number (scalar) as the result. This will be expanded further in the discussion of special relativity (Section 2.13) and of crystal symmetry (Section 7.10). 

Contravariant. Next, we define the transformation laws for contravariant quantities ... 

It turns out that all these contravariant quantities use the inverse transformation Q defined in Eq. (7.13.3) above ... 

The inverse transformation of contravariant quantities will use the matrix P ... 

This transformation applies to contravariant quantities such as zone axes. If, instead, one is transforming a unit cell Ua = (a0b0c0) into a new cell Un = (anb cn), it is really a covariant quantity, which should be represented as a 1 x 4 row vector it transforms using the matrix inverse to Q, namely P3 ... 

The first term is calculated in the AO basis, requiring the transformation of to the contravariant representation. The second term contains the contribution... 

The two-electron contribution to Eq. (245) may be combined with the first term in Eq. (244) before transforming the effective densities to the contravariant AO representation. In this way transformations of differentiated AO integrals are completely avoided (Rice and Amos, 1985). In addition, the second term in Eq. (244) contains contributions from differentiated overlap matrices -(S(1), iF<0)) (0), which are easily calculated in the AO basis using the techniques described in Appendix E. The last contribution to Eq. (244) is easily calculated having transformed the Fock matrix [Pg.223]

In these expressions differentiation and one-index transformations refer to the g integrals only of the Fock matrix [Eqs. (235) and (236)], treating the t elements as densities. The Fock matrix density elements in (Dj 1 and to the contravariant representation. If first derivative integrals in the MO basis is reduced to two occupied and two unoccupied indices (Handy et al., 1986). Note that 7] + T2 [Eqs. (257) and (258)] has the same structure as the <2) part of the MRCI Hessian (129). 

In the first expression the integrals are in the covariant AO representation (in which they are calculated), and the one-index transformed density elements are in the contravariant representation (obtained from the MO basis in usual one- and two-electron transformations). The second expression is useful whenever the transformation matrix is calculated directly in the covariant AO representation and requires the transformation of the Fock matrix to the contravariant representation. The last expression is convenient when the number of perturbations is large, since it avoids the transformation of the covariant AO Fock matrix to the MO or contravariant AO representations. 

Since both sides of Eq. (33) transform as contravariant vectors, the minimum energy path is invariant to any coordinate transformation. From this point of view, the minimum energy path together with the stationary points can be regarded as a fundamental characteristics of an elementary process. 

II est clair qu un foncteur contravariant P de Vf/k dans (Ens) est un faisceau pour la topologie plate si et seulement si P transforme une somme directs en produit direct et si la suite... 

Whence, the operator -L/ acting on the vector field of the annihilation operators satisfies the transformation law for contravariant vectors.33 Making the analogous observations as before now for the annihilation operators, the action of the transformation operator on the annihilation operators can formally be written in the usual form of the transformation law for contravariant vectors,33... 

Finally, it should be stressed that the position of an index in a sequence is significant, since all operators (and coefficients) will eventually be written in antisymmetrized form. We can shed some light on the sign change for the transformation operator for covariant and contravariant tensors by examining the following equations ... 

In words, the transformation operator transforms a covariant vector into a covariant vector [cf. Eq. (54)], but the transformation operator transforms a contravariant vector into a contravariant rank 1 tensor that is not a traditional vector. Since Lrs is antisymmetric, the rank 1 contravariant tensor in Eq. (55) can be converted into a vector by interchanging indices, which results in a minus sign. However, in cases in which there is no ambiguity, the covariant and contravariant indices will be collimated to make the notation more compact. 

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