Matrices fragment density

The fundamental tool for the generation of an approximately transferable fuzzy electron density fragment is the additive fragment density matrix, denoted by Pf for an AFDF of serial index k. Within the framework of the usual SCF LCAO ab initio Hartree-Fock-Roothaan-Hall approach, this matrix P can be derived from a complete molecular density matrix P as follows. 

This follows from the definition (35) of fragment density matrices P that implies exact additivity of these fragment density matrices, i.e., they add up to the density matrix P of the complete molecule,... 

With reference to the individual AO basis sets

parent molecules Ms of nuclear configurations Kt, on the one hand, and the macromolecular AO basis set cp (K) of the macromolecular density matrix P (cp (K)) associated with the macromolecular nuclear configuration K, on the other hand, the following mutual compatibility conditions are assumed ... 

In order to fulfill compatibility condition (a), the local coordinate system of each parent molecule Mk can always be reoriented, resulting in a simple similarity transformation of the original fragment density matrix P (qS(Kk)) into a compatible fragment density matrix P (cp (K)),... 

The final macromolecular density matrix P(A") is rather sparse. The index relations described above help to identify the non-zero matrix elements of P(A"), and the actual computations can be restricted to those. Utilizing these restrictions and carrying out a finite number of steps only for the non-zero matrix elements of each fragment density matrix P (< Kk)), an iterative process is used for the assembly of the macromolecular density matrix P(AT) ... 

The macromolecular density matrix built from such displaced local fragment density matrices does not necessarily fulfill the idempotency condition that is one condition involved in charge conservation. It is possible, however, to ensure idempotency for a macromolecular density matrix subject to small deformations of the nuclear arrangements by a relatively simple algorithm, based on the Lowdin transform-inverse Lowdin transform technique. 

This fragment density matrix Pk has the same nxn dimensions as that of the density matrix P of the complete molecule. 

Based on the fragment density matrix Pk for the k-th fragment, the electron density of Mezey s fuzzy density fragment pk(r) is defined as... 

Consider a molecule and assume that its nuclei are distributed into m mutually exclusive families, where m can take any positive integer value. As follows from the defining equation (23) of the fragment density matrix, the sum of Mezey s fragment density matrices Pk is equal to the density matrix P of the molecule ... 

Electron density decreases exponentially with distance that suggests that an Additive Fuzzy Density Fragmentation (AFDF) approach can be used for both a fuzzy decomposition and construction of molecular electron densities. The simplest AFDF technique is the Mulliken-Mezey density matrix fragmentation [12,13], that is the basis of both the Molecular Electron Density Loge Assembler (MEDLA) [14-17] and the Adjustable Density Matrix Assembler (ADMA) [18-21] macromolecular quantum chemistry methods. 

The construction of the macromolecular density matrix is the simplest if the fragment density matrices P cpfK )) obtained from small parent molecules fulfill the following mutual compatibility requirements ... 

A simple similarity transformation of a fragment density matrix P tpfKjj))... 

In order to be able to determine the index x from the element index i and serial index k of fragment density matrix pk(cp(Kk)), three quantities are introduced for each index k and nuclear family f " for which ck"k 0 ... 

Using only the nonzero elements of each (usually rather sparse) fragment density matrix P ((p(Kk)), the macromolecular density matrix P(K) is... 

Each of these methods is based on the AFDF approach. Within the framework of the conventional Hartree-Fock-Roothaan-Hall self-consistent field linear combination of atomic orbitals (LCAO) ab initio representation of molecular wave functions built from molecular orbitals (MOs), the AFDF principle can be formulated using fragment density matrices. For a complete molecule M of some nuclear configuration K, using an atomic orbital (AO) basis of a set of n AOs density matrix P can be determined using the coefficients of AOs in the occupied MOs. The electronic density p(r) of the molecule M, a function of the three-dimensional position variable r, can be written as... 

The simplest, Mulliken-Mezey additive fragment density matrix, AFDM P of dimensions nXn,is defined as follows ... 

Note that both the density matrix P of the complete molecule M and the additive fragment density matrix AFDM P of the kth fragment involve the same, full set of AOs. Based on the fragment density matrix P, the electron density p (r) of Mezey s kth additive fuzzy density fragment is defined as... 

Since the mutually exclusive nuclear families /j,/2,...,/, ...,/ , contain all the nuclei of molecule M, the sum of the fragment density matrices P is equal to the density matrix P of complete molecule M ... 

Within the framework of the SALDA method,restoring idem-potency of density matrices at displaced nuclear geometries involves the parent molecules, where for small geometry variations a new, approximate, but exactly idempotent density matrix, as well as the associated, improved fragment density matrices can be computed using the above method. 

The macromolecular density matrix constructed from the fragment density matrices within the ADMA framework represents the same level of accuracy as the electron densities obtained with the MEDLA and ALDA methods. The effects of interactions between local fragment representations are determined to the same level of accuracy within the ADMA, the MEDLA, and the ALDA approaches. The ADMA direct density matrix technique allows small readjustments of nuclear geometries, in a manner similar to the ALDA technique however, within the ADMA framework, the geometry readjustment can be carried out directly on the macromolecule. 

Appropriately defined, mutually compatible, additive fragment density matrices (MC-AFDM) P obtained from small parent molecules are combined to form the ADMA macromolecular density matrix P, where mutual compatibility implies that... 

The ADMA method relies on a fragment density matrix database where the fragment density matriees must fulfill condition (b). Condition (a) can be satisfied using a suitable transformation of the fragment density matrices to physically equivalent fragment density matrices defined with respect to a properly oriented AO basis set. Since the final, macromolecu-... 

The /rth fragment density matrix P (ip) is obtained from an ab initio calculation for the parent molecule M. within a local coordinate system. Vector (p Kr) represents the set of AOs of the parent molecule M., with reference to the local coordinate system. In a local coordinate system with axes aligned with those of the macromolecular coordinate system, vector represents the same sequence of AOs at the same nuclear centers. These two representations are related by the orthogonal matrix transformation T ... 

This fragment density matrix P " is used to build the ADMA macromolecular density matrix P. 

A given parent molecule either contains the complete nuclear family /. as part of the surroundings for the actual nuclear set fj of the fragment density matrix P or does not contain any part of nuclear family /. . In addition, some peripheral H nuclei (or possibly other nuclei) may be needed to construct a viable parent molecule to tie off dangling bonds. These extra nuclei are at large distances from the actual nuclear set of the fragment density matrix P. ... 

Ah initio computations are carried out for each parent molecule Mj(, and the fragment density matrices P (fragment density matrix is already available in the ADMA databank. 

Using the appropriate transformation matrix the AO basis of each fragment density matrix P (coordinate axes parellel to those of the common, macromolecular coordinate system, resulting in the fragment density matrix P. ... 

If needed, permutations of the AO functions are carried out for each fragment density matrix P to generate a block structure for P where the following statements hold ... 

The transformed row and column indices of each fragment density matrix are determined and the corresponding matrix element is added to the appropriate matrix element of P, following a simple rule If the condition... 

This procedure is carried out for each nonzero element of each fragment density matrix P ", resulting in the corresponding ADMA density matrix P of the target macromolecule. 

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