Electron density MEDLA method

An Application of the MEDLA Method for the Direct Computation of Electron Densities of Functional Groups... 

The additive fuzzy electron density fragmentation scheme of Mezey is the basis of the Molecular Electron Density Lego Assembler (MEDLA) method [67,70-72], reviewed in section 4. of this report, where additional details and applications in local shape analysis are discussed. The MEDLA method was used for the generation of the first ab initio quality electron densities for macromolecules such as proteins [71,72] and other natural products such as taxol [66],... 

There is no longer any inherent difficulty in computing reasonably accurate electron densities for small molecules by either of the two main computational approaches wavefunction methods [85,86] and density functional methodologies [87-89]. With the introduction of the MEDLA technique [67,70], ab initio quality electron densities can be computed for virtually any macromolecule, including... 

The application of the additive fuzzy electron density fragments for the building of electron densities of large molecules is called the Molecular Electron Density Lego Assembler method, or MEDLA method [5,37,66,67,70-72],... 

Several numerical tests and detailed comparisons of MEDLA electron densities to electron densities computed by traditional ab initio SCF technique using 3-21G and 6-31G basis sets have shown [67,71] that the MEDLA results are invariably of better quality than the standard 3-21G ab initio results, and the MEDLA results are virtually indistinguishable from the standard ab initio 6-31G basis set results obtained with the traditional Hartree-Fock method. 

The MEDLA method does not impose any size limitation on the fragments only the feasibilty of traditional ab initio calculations limits the actual size of the fragments and the size of the "coordination shell" around them in the small molecule imitating the actual surroundings within the target molecule. Electron densities of satisfactory accuracy have been obtained in all the test calculations. 

Whereas density functional theory guaranties that for the ground electronic state of molecules the electron density determines the energy, the actual construction of such energy functions from first principles is a problem of considerable complexity. The electron densities computed by the MEDLA method suggest various approximations to the molecular energy of large systems. 

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. 

Matrix Assembler (ADMA) method [18-21] generates a macromolecular density matrix P((p(K)) that can be used for the computation of a variety of molecular properties besides ab initio quality macromolecular electron densities. In electron density computations the accuracy of the ADMA macromolecular density matrix P(cp(K)) corresponds to that of a MEDLA result of an infinite resolution numerical grid. 

It is now possible to analyze macromolecular electron densities at a resolution far exceeding the resolution of current x-ray diffraction and other experimental and macromolecular computational techniques. The MEDLA method presents a new perspective for the analysis of global and local shape, molecular similarity, and complementarity. 

FIGURE 1 The fuzzy body of the electron density of a bovine insulin molecule is represented by three molecular isodensity contour surfaces (MlDCOs), for the density thresholds of 0.1, 0.01, and 0.001 a.u. (atomic unit), respectively, as computed using the MEDLA method. Bovine insulin was among the proteins selected for the first ab initio quality electron density computations for macromolecules. ... 

Since the introduction of the MEDLA method, three more recent developments have extended the applications of the Mulliken-Mezey and the more general AFDF schemes. These developments, all utilizing various representations of additive fuzzy subsets of molecular electron densities, are the... 

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. 

Within the quantum chemical description of molecular electron density clouds, a natural criterion, the Density Domain criterion, provides a quantum chemical definition for functional groups [14-18]. Furthermore, techniques that generate fuzzy electron density contributions for local molecular moieties that are analogous to the fuzzy electron density clouds of complete molecules, determined by the analytic Additive Fuzzy Density Fragmentation (AFDF) method [19-21], or the earlier numerical-grid MEDLA method [22,23], are also... 

The MEDLA method is based on the following electron density fragment additivity principle. A natural scheme for the implementation of this principle has been d cribed in [12]. Consider an LCAO ab initio wavefunction of a small molecule of some fixed conformation K. If n is the number of atomic orbitals three-dimensional position vector variable, and P is the n X n density matrix, then the electronic density p(r) of the molecule is given by... 

Some recent developments concerning macromolecular quantum chemistry, especially the first linear-scaling method applied successfully for the ab initio quality quantum-chemistry computation of the electron density of proteins, have underlined the importance and the applicability of quantum chemistry-based approaches to molecular similarity. These methods, the linear-scaling numerical Molecular Electron Density Lego Approach (MEDLA) method [6 9] and the more advanced and more generally applicable linear-scaling macromolecular density matrix method called Adjustable Density Matrix Assembler or ADMA method [10,11], have been employed for the calculation of ab initio quality protein electron densities and other... 

Several additional developments had very positive impact on the advances and applications of the topological shape analysis and similarity methods. Among these are the already mentioned establishment of the holographic property of the electron density clouds of real, boundary-less molecules [4,5] and the extension of many aspects of small-molecule quantum chemistry to macromolecules, such as proteins, by the linear-scaling MEDLA and ADMA methods [6-11]. 

Walker PD, Mezey PG. A new computational microscope for molecules high resolution MEDLA images of taxol and HIV-1 protease, using additive electron density fragmentation principles and fuzzy set methods. J Math Chem 1995 17 203-234. 

The MEDLA (molecular electron density loge assembler, or molecular electron density Lego assembler) method is a numerical technique, based on the AFDF principle, on a numerical electron density fragment databank, and on the direct assembly of numerical electron density fragments into a macromolecular electron density. 

Besides the possibilities of the computation of all molecular properties expressible in terms of density matrices, the ADMA method has other advantages. For example, if electron density computations are compared, then the accuracy of the electron density obtained using the ADMA macromolecular density matrix P((p(A0) corresponds to the ideal MEDLA result that could be obtained using an infinite resolution numerical grid. The memory requirements of the ADMA method is also substantially lower than that of the numerical MEDLA method since it takes much less memory to store density matrices than three-dimensional numerical grids of electron densities, especially if reasonably detailed electron densities are required. 

Mulliken, R. S. Unpublished comments. Topic of the discussions held at the crater of the volcano Teide, Tenerife, Canary Islands, June 20, 1976, between Prof. Mulliken and the author, on the fundamentals of local components of molecular wavefunctions, Mulliken s population analysis, and various atomic charge models based on overlap integrals. These discussions had a motivating role in the later development of the AFDF methods MEDLA and AOMA, both based on the realization by the author that Mulliken s population analysis without integration provides a simple, additive fiizzy electron density fragmentation (AFDF) scheme. 

According to detailed tests, ° the MEDLA method generates ab initio quality electronic densities at a level better than standard SCF calculations using 3-2IG bases. The MEDLA electronic densities are virtually indistinguishable from electronic densities obtained by standard SCF calculations at the 6-3IG basis set level. ... 

The MEDLA method, based on numerical electronic density data base, the more advanced, geometry-adjustable ALDA method,based on a fragment density matrix data base, and the ADMA method, " generating macromolecular density... 

Although not yet obvious from the deceptively simple form of equation (37), this equation, the electrostatic Hellmann-Feynman theorem, allows one to use the electronic density and the simple internuclear Coulomb interactions to describe the forces acting on the nuclei of the molecule. A simple, classical interpretation of this theorem provides the key to the use of macromolecular electronic densities, such as those obtained within the MEDLA, ALDA, or ADMA methods, for the computation of forces within the macromolecule. 

If molecular electronic densities p(R) of satisfactory accuracy can be computed for large molecules, using the MEDLA, ALDA, or ADMA methods, then a 3D integration in the first term, and a trivial summation in the second term of equation (36) provides the force acting on nucleus a of the molecule. Quantum-chemical forces,... 

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