Supermatrix

So only the two-electron integrals with p > v, and X > a and [pv] > [Xa] need to be computed and stored. Dpv,Xa only appears in Gpv, and Gvp, whereas the original two-electron integrals contribute to other matrix elements as well. So it is much easier to form the Fock matrix by using the supermatrix D and modified density matrix P than the regular format of the two-electron integrals and standard density matrix. 

Now we are ready to formulate the basic idea of the correction algorithm in order to correct the four-indexed operator f1, it is enough to correct the two-indexed operators fc and f 1 in the supermatrix representation (7.100). The real advantage of this proposal is its compatibility with any definite way of f2 and P7 correction [61, 294], The matrix inversion demanded in (7.99) is divided into two stages. In the fi, v subspace it is possible to find the inverse matrix analytically with the help of the Frobenius formula that is well known in matrix algebra [295]. The... 

The supermatrix notation emphasizes the structure of the problem. Each diagonal operator drives a wavepacket, just as in the adiabatic case of Eq. (10), but here the motion of the wavepackets in different adiabatic states is mixed by the off-diagonal non-adiabatic operators. In practice, a single matrix is built for the operator, and a single vector for the wavepacket. The operator matrix elements in the basis set [Pg.384]

The symbol (Oaa denotes the energy difference between the two eigenstates, converted into angular frequency. The first term on the right-hand side (rhs) of Eq. (18) vanishes for the populations (oo a = 0) and describes the preces-sional motion for coherences. Rota pp is an element of the relaxation matrix (also called relaxation supermatrix) describing various decay and transfer processes in the spin system. Under certain conditions (secular approximation), one neglects the relaxation matrix elements unless the condition = pp is fulfilled. 

R is called the relaxation superoperator. Expanding the density operator in a suitable basis (e.g., product operators [7]), the a above acquires the meaning of a vector in a multidimensional space, and eq. (2.1) is thereby converted into a system of linear differential equations. R in this formulation is a matrix, sometimes called the relaxation supermatrix. The elements of R are given as linear combinations of the spectral density functions (a ), taken at frequencies corresponding to the energy level differences in the spin system. 

The supermatrix G, which contains the two-electron repulsion integrals, has elements defined by... 

At this point we have developed a method for handling symmetric one-electron operators a useful step, but far from adequate to cover all cases we are interested in, since we have said nothing yet about the much more demanding case of two-electron integrals. Let us consider first building the Fock matrix for a closed-shell SCF calculation. This is perhaps most easily expressed in terms of a P supermatrix with elements... 

Now, from our manipulations above we can easily see that F can be constructed from a skeleton matrix obtained using only a P2 list, since the Fock operator is totally symmetric. However, this far from ideal, since we would like to avoid the redundancies that arise unless we use the Pi list of two-electron integrals, or, here, supermatrix elements. We define first a matrix Y(IJKL) with the property that the IJ block is given by... 

The MCSCF optimization process is only the last step in the computational procedure that leads to the MCSCF wave function. Normally the calculation starts with the selection of an atomic orbital (AO) basis set, in which the molecular orbitals are expanded. The first computational step is then to calculate and save the one- and two-electron integrals. These integrals are commonly processed in different ways. Most MCSCF programs use a supermatrix (as defined in the closed shell HF operator) in order to simplify the evaluation of the energy and different matrix elements. The second step is then the construction of this super-matrix from the list of two-electron integrals. The MCSCF optimization procedure includes a step, where these AO integrals are transformed to MO basis. This transformation is most effectively performed with a symmetry blocked and ordered list of AO integrals. Step... 

On using eqs. (16.6.9) and (16.6.10) the matrices of the required representations found in (viii) give the elements Tk of the supermatrix as in step (vi) of Section 16.6, and these matrices, when multiplied by Tk(C t), are the space-group representations. 

A supermatrix is a matrix, each element of which is itself a matrix. The subscripts pq] on the LS of eq. (5) denote that the general term [pq] of the supermatrix A % B is the pqth term of the first matrix apq mutiplying the second matrix B. For example, the upper right-hand block of A B in eq. (4) is... 

Here the dot indicates contraction over Cartesian coordinates, tK is a component of the vector tjf(r) = (r — rK)/ATrsf r — rK 3, and M is the 3 x 3 block, corresponding to the Tfth and 7th atoms, of the so-called relay matrix, which gives an atomic representation of the molecular polarizability. The supermatrix M has dimension 3Na x 3Na, and is defined as ... 

If the supermatrix A becomes degenerate (at the point where the Hartree-Fock solution for which it is calculated loses its stability i.e. ceases to be a minimum of the energy functional) the inversion is not possible any more, but the Hartree-Fock picture of the electronic structure itself becomes invalid. In this case the above treatment obviously loses any sense. 

Here we introduce the notation ( . ..) for the scalar product of vectors whose components are numbered by the Cartesian shifts of the nuclei). Next, let h" be the supermatrix of the second derivatives of the matrix of the Fock operator with respect to the same shifts. As previously, we refer here to the supermatrix indexed by the pairs of nuclear shifts in order to stress that the elements of this matrix are themselves the 10 x 10 matrices of the corresponding second derivatives of the Fock operator with respect to the shifts. The contribution of the second order in the nuclear shifts can be given the form of the (super)matrix average over the vector of the nuclear shifts ... 

It is remarkable that the supermatrix A 1 is nothing [42] but the polarization propagator II for the CLS subsystem calculated for the symmetric molecule. With this we get ... 

As mentioned previously, the specifics of the central atoms in CCs are determined by the structure of the supermatrix II, which is in its turn predefined by the structure of the carrier space of the CLS group and by the number of electrons in it. Indeed, the supermatrix II of the polarization propagator is particularly simple in the basis of the eigenstates of the Fock operator Fq. Its matrix elements then are ... 

In the elements p and r, the Heaviside functions vpr and vrp are included to avoid bond backfolding since a bond cannot be backward when the previous bond was forward and vice versa. In the supermatrix G, cj, characterizes a lateral bond in layer i, p, a forward bond starting from layer i, and r,- a backward bond starting from layer i. q,-, p,-, and r,-depend on three kinds of parameters. The parameters ah, a, 0, and to arise from local chain stiffness and bond arrangements, Pi from the intermolecular interactions and pLi, pzi from the nearest-neighbor bond correlations. 

The supermatrix G differs from that proposed by DiMarzio and Rubin43 and employed by Scheutjens and Fleer12 and Cosgrove el al.16 since it accounts for the bond correlation and chain stiffness and excludes via the Heaviside functions the bondfolding. Compared to the supermatrix of Dill,44 it accounts, in addition, for the bond backtrack and bond correlation. 

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