Contravariant quantities

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 ... 

Obviously, the contravariant AO-basis is not equivalent with the covariant basis because of the non-orthogonality of the AO basis functions. If we try to transform the contravariant quantity again to the MO basis with the AO-MO coefficients, we will get the following equation... 

The quantities /" are the contravariant components of a vector in the coordinate system X. They give an actual vector only when multiplied by the unit vector e = hvev. If the unit vectors along the coordinate lines have a scale inverse to the coordinate, e = ev/hv so that... 

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... 

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. 

The expression given by BCAH for elements of the constrained mobility within the internal subspace is based on inversion of the projection of the modified mobility within the internal subspace, rather than inversion of the projection (at of the mobility within the entire soft subspace. BCAH first define a tensor given by the projection of the modified friction tensor onto the internal subspace, which they denote by the symbol gat and refer to as a modified covariant metric tensor, which is equivalent to our CaT - They then define an inverse of this quantity within the subspace of internal coordinates, which they denote by g and refer to as a modified contravariant metric tensor, which is equivalent to our for afi = 1,..., / — 3. It is this last quantity that appears in their diffusion equation, given in Eq. (16.2-6) of Ref. 4, in place of our constrained mobility Within the space of internal coordinates, the two quantities are completely equivalent. 

In the AO-based implementation of the CCSD(F12) model we need the where the xy are the geminal indices (efectively they belong to the occupied space within this implementation, see Subsection 3.2) and yu belong to the covariant AO-basis. In the subsequent sections we will use the term covariant for the description of the quantities in the ordinary AO basis (denoted by upper-case AO left superscript) whereas the contravariant basis is used for the back-transformed quantities (vide infra) (denoted with lower-case ao left superscript). The final equation for the Al-noCABS V intermediate can be written in the following way... 

After the preliminaries presented above we can now precisely define vectors and tensors in Minkowski space by their transformation properties under Lorentz transformations. Each four-component quantity A, which features the same transformation property as the contravariant space-time vector as given by Eq. (3.12),... 

Any M-index quantity T with 4" components jg called a contravari-... 

Sets of quantities A, and BJ that behave according to (10.2.4a) or (10.2.4b) respectively under the given group of transformations are said to form the components of a rank-1 contravariant tensor in case (a) or a rank-1 covariant tensor in case (b). This anticipates the definition of rank-n tensors—which follow more general transformation rules but are very easily found. Thus for a 2-electron system we may construct an approximate (spatial) wavefunction using all product functions i(ri)i(r2), and if new orbitals are introduced according to (10.2.4a) then the new products will be... 

Note that we have already derived this equation by the help of tensor notation in the previous section. The overlap matrix S appears in a nonorthogonal basis and is important for correct contraction with co- and contravariant basis sets. Therefore, either PS or SP is a projector onto the occupied space depending on the tensor properties of the quantity to which it is applied. The same holds for the complementary projector onto the virtual space (1 — PS) or (1 — SP). 

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