Contravariant basis

For each coordinate 2 in the full space, we may define a covariant basis vector 0R /02 and a contravariant basis vector 02 /0R, which obey orthogonality and completeness relations... 

A generalized set of reciprocal vectors for a constrained system is defined here to be any set off contravariant basis vectors b, ..., b- and K covariant basis... 

On the other hand, the reaction rate Jr r = 1/t) may be obtained by multiplying (4.43) with vectors of contravariant basis gr (see (A.89)). Inserting in such product from (4.33), from the relation between contra- and covariant bases in V (see (A.86)) and from (4.40), we obtain (by using of orthonormality of ) the relation between rates (reversal to (4.44))... 

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... 

If we use a contravariant basis e which provides the basis for, j/, a covariant tensor a can be written as... 

This shows that a covariant tensor is introduced on a contravariant basis e of the dual space, while a contravariant tensor a = ad e, (gi e j is given on a covariant basis ct which is defined by a(, tj) for tj V. Mixed tensors corresponding to (, u) and ( , are similarly introduced. 

Given a point on the interface r — S z, 9), there is a tangent plane passing through it. The contravariant basis vectors for the tangent plane are given by... 

Tensor notation may be applied to quantum chemical entities such as basis functions and matrix elements. For example, 1%, ) is a covariant tensor of rank one. Like before, superscripts, e.g., x ), denote contravariant tensors. Co- and contravariant basis functions are defined to be biorthogonal that is, they obey the conditions of... 

Co- and contravariant basis functions are nonorthogonal in general i.e., the following equation holds ... 

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). 

This may be confirmed by expanding an arbitary contravariant Cartesian vector (with a raised bead index) in a basis of a and m vectors and confirming that one recovers the original vector if such an expansion vector is left-multiplied by the RHS of Eq. (2.149). 

If the set 3N contravariant Cartesian vectors given by the / a vectors and K m vectors form a complete basis for 3N space of Cartesian vectors, which we will hereafter assume to be true, then they must also obey a completeness relation ... 

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. 

The gradient expression given above is not particularly useful since it appears in the MO basis. Following the discussion in Appendix C about covariant and contravariant representations, we may rewrite the gradient as... 

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). 

Even though we do not invoke the full machinery of tensor analysis (Butkov, 1968), it is useful to keep the distinction between contravariant and covariant components clear. To avoid conflicting notation we do not use upper and lower indices to denote contravariant and covariant indices. Instead, we will use the suffix ao (lower case letters) on tensors whose indices are all contravariant, and AO (capital letters) on tensors whose indices are all covariant. No special suffix is used in the MO basis. For example, using the two- and four-index trace operators the energy is... 

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. 

Consider a 3-D domain that can be adequately described by the generalized curvilinear coordinate system (u, v, w) and that its mappings are adequately smooth to allow consistent definitions. Then, any vector F can be decomposed into three components with respect to the contravariant a , a , a or the covariant a , a, a,a, linearly independent basis system as... 

To select the correct basis for (3.65), covariant (contravariant) components should be expressed as a function of their contravariant (covariant) counterparts, via the pertinent system metrics. Unfortunately, the differentiation of these metrics produces the demanding Cristoffel symbols that obstruct the solution of (3.65). This difficulty can be circumvented by classical... 

To determine the matrix, let us introduce the basis vectors and ey with respect to the plane throngh the vectors n,. and. The vector is perpendicnlar to the reference planes, whereas the vector Cy is parallel to them. Transforming these vectors into spherical basis vectors yields the contravariant spherical basis vectors [30] e"(nj ) and e (n J, which are rotated with respect to the vectors e (n,) and (n ) through the angles

[Pg.225]

In Seet. 4.2, we need veetor spaee with abasis whieh is formed by A linear independent vectors gp p =, ..., k) which are not generally perpendicular or of unit length [12, 18, 19]. Sueh nonorthogonal basis, we eall a contravariant one. Covariant components of the so called metric tensor are defined by... 

A metric tensor with matrix 9pq is obviously symmetrical and regular (this last assertion is necessary and sufficient for the linear independence of gp in the basis of k orthonormal vectors in this space, we obtain det g , as a product of two determinants first of them having the rows and second one having the columns formed from Cartesian components of gp and gq. Because of the linear independence of these k vectors, every determinant and therefore also det g , is nonzero and conversely). Contravariant components gP of the metric tensor are defined by inversion... 

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