Density matrix orbital space

For the more restrictive CSF expansion spaces, such as PPMC and RCI expansions, it occurs that entire transition density vectors will vanish for particular combinations of orbital indices. It is most convenient if a logical flag is set during the construction of the three vectors, one flag for each vector, to indicate that it contains non-zero elements. This avoids the eflbrt required to check each element individually for these zero vectors. The updates of the elements of the matrix C that result from a particular transition density vector involve two DO loops one over the CSF index, which determines the second subscript of the matrix C, and the other over an orbital index, which, combined with a density matrix orbital index, is used to determine the first subscript of C. Either choice of the ordering of these two loops results in an outer product matrix assembly method. 

The main drawback of the chister-m-chister methods is that the embedding operators are derived from a wavefunction that does not reflect the proper periodicity of the crystal a two-dimensionally infinite wavefiinction/density with a proper band structure would be preferable. Indeed, Rosch and co-workers pointed out recently a series of problems with such chister-m-chister embedding approaches. These include the lack of marked improvement of the results over finite clusters of the same size, problems with the orbital space partitioning such that charge conservation is violated, spurious mixing of virtual orbitals into the density matrix [170], the inlierent delocalized nature of metallic orbitals [171], etc. 

In modest sized systems, we can treat the nondynamic correlation in an active space. For systems with up to 14 orbitals, the complete-active-space self-consistent field (CASSCF) theory provides a very satisfactory description [2, 3]. More recently, the ab initio density matrix renormalization group (DMRG) theory has allowed us to obtain a balanced description of nondynamic correlation for up to 40 active orbitals and more [4-13]. CASSCF and DMRG potential energy... 

For simplicity, we shall commonly refer to the Q-electron distribution function as the 2-density and the 2-electron reduced density matrix as the 2-ntatrix. In position-space discussions, the diagonal elements of the 2-ntatrix are commonly referred to as the 2-density. In this chapter, we will also refer to the diagonal element of orbital-space representation of the Q-vaatnx as the 2-density. 

Zgid, D., Nooijen, M. The density matrix renormalization group self-consistent field method Orbital optimization with the density matrix renormalization group method in the active space. J. Chem. Phys. 2008, 128, 144116. 

It is then possible to write the density matrix R which refer to the space spanned by the doubly occupied orbitals ... 

If we transform the MO s such that condition (5 11) is fulfilled, the resulting transition density matrix will be obtained in a mixed basis, and can subsequently be transformed to any preferred basis The generators Epq of course have to be redefined in terms of the bi-orthonormal basis, but this is a technical detail which we do not have to worry about as long as we understand the relation between (5 9) and the Slater rules. How can a transformation to a bi-orthonormal basis be carried out We assume that the two sets of MO s are expanded in the same AO basis set. We also assume that the two CASSCF wave functions have been obtained with the same number of inactive and active orbitals, that is, the same configurational space is used. Let us call the two matrices that transform the original non-orthonormal MO s [Pg.242]

The two-electron reduced density matrix is a considerably simpler quantity than the N-electron wavefunction and again, if the A -representability problem could be solved in a simple and systematic manner the two-matrix would offer possibilities for accurate treatment of very large systems. The natural expansion may be compared in form to the expansion of the electron density in terms of Kohn-Sham spin orbitals and it raises the question of the connection between the spin orbital space and the -electron space when working with reduced quantities, such as density matrices and the electron density. 

We have written the operator Fl(x) as a function of the combined space-spin coordinates X, because while the spin summations can be carried out in Jl(x) before calculating matrix elements, Kl(x) may connect spin-orbitals that are off-diagonal in the spin wavefunctions however in the special case of the density matrix p (xi, Xa) arising from a wavefunction that is a spin singlet (5 = 0) one can show that must also be diagonal. This leads to a useful simplification here since we can usually assume this property for Wlo, and it means that Vl(x) reduces to a (non-local) function of the space variable r only we can therefore consistently parameterize the matrix elements for the whole potential, (/bI Vl(x) j) without having to decompose them into different spin combinations for the Coulomb and exchange potentials. 

Though we can compare electron densities directly, there is often a need for more condensed information. The missing link in the experimental sequence are the steps from the electron density to the one-particle density matrix f(1,1 ) to the wavefunction. Essentially the difficulty is that the wavefunction is a function of the 3n space coordinates of the electrons (and the n spin coordinates), while the electron density is only a three-dimensional function. Drastic assumptions must be introduced, such as the description of the molecular orbitals by a limited basis set, and the representation of the density by a single Slater-determinant, in which case the idempotency constraint reduces the number of unknowns... 

This is the one-electron density matrix in orbital space for the ground state 0). For the present purpose we call it the density matrix. It is manifestly Hermitian. Therefore it is possible to find a unitary transformation U of the orbitals fi) that diagonalises it. The new orbitals a) are the natural orbitals... 

In this equation and k denote the standard Coulomb and exchange operators involving core orbital (pi, D and D(ji v) stand for the normalisation integral for the SC wavefunction and the elements of the first-order density matrix in the space of the SC orbitals, and and k, are generalised Coulomb and exchange operators with matrix elements = XikMq )AXp K u Xq) = At least Voutofthe Af... 

A way of overcoming this problem, is to use the concept of natural orbitals. The natural orbitals are those which diagonalize the density matrix, and the eigenvalues are the occupation numbers. Orbitals with occupation numbers significantly different from 0 or 2 (for a closed-shell system) are usually those which are the most important to include in tlie active space. An RHF wave function will have occupation numbers of exactly 0 or 2, and some electron correlation must be included to obtain orbitals with non-integer occupation numbers. This may for... 

According to the assumption we have made the change in the density matrix, ARX, due to the coulombic interaction between fragments will be more or less localized. It is tempting to set ARX = XL. By doing that, however, one is forced [8] to split off the local space from the remainder of the system to satisfy the idempotency condition. This results in an ordinary cluster model which does not allow electron transfer to or from the surroundings and, as we will see in Sect. 5, is unsuitable for our purposes. In order to properly embed the cluster we take advantage of the fact that the sum of the occupied and unoccupied molecular orbital (MO) spaces is identical to the total AO space. So, instead of ARX = XL, we write... 

Many of the wavefunction expansion forms discussed in Section IV result in sparse density matrices D and d. This was discussed for the ERMC wavefunction and its subsets but it is also true for other direct product type expansion spaces. With the RCI expansion, for example, the matrix D consists of 2 X 2 blocks along the diagonal the orbitals of each block are those associated with an electron pair. The matrix d is also sparse because of the orbital subspace occupation restrictions and the non-zero elements consist of those with two orbital indices belonging to one electron pair and with the other two indices corresponding to another (or the same) electron pair. It would be beneficial if this density matrix sparseness could be exploited when it exists. However, this must be done in such a way as to avoid any restrictions on the types of wavefunction expansions allowed. 

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