INDEX quantity

The formulation used here corresponds to the use of RI with the Coulomb norm [47], Although there are other formulations of the RI [48,49], we will not use them here. Because we need the matrix G-1 only at this level of the calculation, its absorption into three-index quantities can be exploited [50] as follows ... 

The density susceptibilities of the monomers can be expanded in terms of a single set of atomic orbitals, making of aA(r, r ) a two index quantity and thus greatly simplifying calculations of the dispersion term for large systems. See Section 7 for a more detailed discussion of this technique applied to the calculations of the dispersion energy. 

Second-order Moller-Plesset perturbation theory (MP2) is the computationally least expensive and most popular ab initio electron correlation method [4,15]. Except for transition metal compounds, MP2 equilibrium geometries are of comparable accuracy to DFT. However, MP2 captures long-range correlation effects (like dispersion) which are lacking in present-day density functionals. The computational cost of MP2 calculations is dominated by the integral transformation from the atomic orbital (AO) to the molecular orbital (MO) basis which scales as 0(N5) with the system size. This four-index transformation can be avoided by introduction of the RI integral approximation which requires just the transformation of three-index quantities and reduces the prefactor without significant loss in accuracy [36,37]. This makes RI-MP2 the most efficient alternative for small- to medium-sized molecular systems for which DFT fails. 

Now, these equations have been written with formal integration, but of course in the numerical implementation only the evaluation of the integrand at the grid points is required. Therefore, it is evident that the derivatives lyl, (Vpo)M and (Vpp)tyl can be evaluated on the grid independently of the indices pv, and so the four-index problem is decomposed into two independent two-index procedures, avoiding the potential computational bottleneck. By comparison, the resolution of the identity technique proposed by Komornicki and Fitzgerald [66] gives an approximate result in terms of a product of one two-index and two three-index quantities. 

For products of two matrices the trace operation is reminiscent of a scalar product it is a sum of indexed quantities just as the scaJar product is a sum of TO indexed quantities. The two-index quantities and Rsr may easily be numbered by the single index ... 

In this expansion we begin to try to make a distinction amongst the increasing profusion of sub- and super-scripts by using the fauniliar integer indexing labels i, j, k, ,... to label MOs (or MO-related quantities) and the less familiar indexing quantities r, s, f, ,... to label the basis functions (and quantities related to basis functions). ... 

In the same spirit it is possible to collapse four subscripts into two so that four-index quantities become analogues of matrices just as two-index quantities have been treated as vectors. Vectors and matrices of this type are often called supervectors and supermatrices, respectively. We shall use this notation later when molecular symmetry transformations are encountered. 

The strain components are converted to single-index quantities e , using (8.7) and, instead of (8.8), the rules... 

Prepare the three-index quantities Bq ap" (ccsdint routine)... 

The operator PJy used in Eqs. (95), (96), (97) is a permutation operator (called sometimes symmetrizer), the following equation illustrates its action on the arbitrary four-index quantity Ap ... 

For the correlation factor in the form of Eq. (60), the integrals are computed in the DF approximation by contracting the appropriate three-index quantities over the auxiliary index (step 2 Alg. 1)... 

The first commutator has been neglected in in both Eqs. (131) and (132), whereas the remaining two commutators were neglected only in the T2 equation. The removal of entire commutators assures the eize-extensivity of the CCSD(F12) energy [5, 41]. The Eqs. (130)-(132) are of a general form that is not yet suitable for the implementation. In the present work very often the expressions vector function and residual are used. They always refer to the many-index quantity, defined by the right hand sites of these equations. The working expressions of the coupled-cluster Ti, T2 and T2/ residuals are discussed in next subsections. 

Any M-index quantity T with 3" components Ti i2...i is called a tensor of the nth rank if its components transform like an n-fold product of vectors under Galilean transformations. 

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

For the case of a crystalline application, the fact that aU two-index quantities (Fock matrix in the PAO and in the WF representation, overlap matrix in the PAO representation) and aU four-index quantities (amplitudes, residuals, two-electron integrals) are translationally invariant was used. For instance. 

The sum (4.7.28) is still not the trace of a matrix. The situation is exactly the same as in the case treated in section 4.4.3. Therefore we proceed to rewrite the same sum (4.7.28) in an equivalent form which allows one to perform the summation using the rules of matrix multiplication. To do that we introduce the four index quantities defined by... 

In Equation 6-41, the indexed quantities q, bj, q, Vj and Wj are called column vectors of dimension n sometimes, they are vectors denoted by the bold-faced symbols a, b, c, v and w. The square matrix at the left reveals its obvious tridiagonal structure it is a special case of a diagonal banded matrix. We will not deal with matrix inversion in this book. Suffice it to say that the last (or the first) equation, which involves two unknowns only, is usually used to reduce the number of unknowns along each row, right up (or down) the matrix, thus resulting in a two or bidiagonal system. Repeating the process in the opposite direction yields the solution vector v. 

In Section 2.4 we found that the tensors that now appear in (12.2.14) were of importance in defining variational stability conditions . The 2-index quantities (VVi/)y and (Vi/V)y are elements of the matrices Q and M and if we collect the elements VQV into a matrix V then the basic equation for time evolution of the parameter values around p = po (i.e. d = 0) is... 

Consider the following examples for illustration A vector a in n-dimensional space is described completely by its n components a,. It may therefore be seen as a one-index quantity or a tensor of rank one. A matrix A has components A,/ (two indices) and is a rank two tensor. A tensor of rank three has n components, and its components have three indices, T, and so on. As a special case, scalars have only = 1 component and are tensors of rank zero. 

The AO-MP2 method introduced in 1993 by Haser applies screening criteria to the intermediate four-index quantities in order to reduce the computational scaling for larger molecules. Here, the Schwarz inequality introduced earlier in this review ... 

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