Contravariant representations

These are covariant and contravariant representations of the Cartesian identity tensor, and inverses of each other. 

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... 

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. 

Representations of these and other tensors in an arbitrary system of coordinates may be constructed as follows. For each contravariant rank 2 Cartesian tensor T " (such as H ) or covariant tensor S v (such as m v), we define corresponding Riemannian representations... 

It is very important to have conditions, easy to verify in practice, for a contravariant functor F to be representable Certainly a necessary... 

We collect a column vector representation of these three contravariant quantities ... 

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]

It is worth the trouble to relate the properties of the covariant measuring vectors to those of the contravariant measuring vectors. We shall consider only molecules, which are not subject to constraints. By using the coordinate representation of Va, it follows that... 

Lemme 8.10. Soient S un pr sch ma localement noetherien, et a un foncteur contravariant defini sur Sch/S, a valeurs dans la ca-tegorie des ensembles. Pour que cr soit representable par un S-presch ma etale et separ , (il faut et) il suffit que [Pg.470]

Take as given a point x, a specific representation and five arbitrary numbers X°, X, ... X. We now combine the totality of all contravariant projective vectors of fixed index whose components assume the values X°, X, ... X in the given presentation, into a single geometrical objecf. The vectors of a... 

T, and (4) Tf. Alternative (1) is said to be fully covariant, (2) is fully cowtravariant, and the other two are mixed representations. In principle, one is free to formulate physical laws and quantum chemical equations in any of these alternative representations, because the results are independent of the choice of representation. Furthermore, by applying the metric tensors, one may convert between all of these alternatives. It turns out, however, that it is convenient to use representations (3) or (4), which are sometimes called the natural representation. In this notation, every ket is considered to be a covariant tensor, and every bra is contravariant, which is advantageous as a result of the condition of biorthogonality in the natural representation, one obtains equations that are formally identical to those in an orthogonal basis, and operator equations may be translated directly into tensor equations in this natural representation. On the contrary, in fully co- or contravariant equations, one has to take the metric into account in many places, leading to formally more difficult equations. 

This natural tensor equation is formally similar to the operator equation. If we wish to cast this equation into another (nonorthogonal) representation, we can do so by applying the metric tensor as described above. Let us, for example, rewrite Eq. [94] using the fully contravariant form of the density matrix ... 

Atomic basis functions in quantum chemistry transform like covariant tensors. Matrices of molecular integrals are therefore fully covariant tensors e.g., the matrix elements of the Fock matrix are F v = (Xn F Xv)- In contrast, the density matrix is a fully contravariant tensor, P = (x IpIx )- This representation is called the covariant integral representation. The derivation of working equations in AO-based quantum chemistry can therefore be divided into two steps (1) formulation of the basic equations in natural tensor representation, and (2) conversion to covariant integral representation by applying the metric tensors. The first step yields equations that are similar to the underlying operator or orthonormal-basis equations and are therefore simple to derive. The second step automatically yields tensorially correct equations for nonorthogonal basis functions, whose derivation may become unwieldy without tensor notation because of the frequent occurrence of the overlap matrix and its inverse. 

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