ADMA method

Matrix Assembler (ADMA) method, introduced for the generation of ab initio quality approximate density matrices for macromolecules [142-146], and for the computation of approximate macromolecular forces [146], among other molecular properties. 

The AFDF approach and the ADMA method have been reviewed in detail [142, 146]add here only a shortened version of the main features of these methods will be given. 

Whereas the first applications of the AFDF approach were based on a numerical combination of fuzzy fragment electron densities, each stored numerically as a set density values specified at a family of points in a three-dimensional grid, a more powerful approach is the generation of approximate macromolecular density matrices within the framework of the ADMA method [142-146]. A brief summary of the main steps in the ADMA method is given below. 

Within the MC-AFDF ADMA method, the management of multiple index assignments ofbasis orbitals and individual density matrix elements requires a series of index conversion relations. These relations are briefly reviewed below, using the notations of the original reference [143]. 

A more advanced AFDF approach, the ADMA method (Adjustable Density Matrix Assembler method) [39-41] is based on a density matrix database and the actual construction of a macromolecular density matrix. This technique, also reviewed in part in ref.[31], is suitable for the rapid computation of various additional molecular properties besides electron densities. 

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. 

The macromolecular density matrix P(K) is also a sparse matrix that simplifies its storage and subsequent computations. Using the macromolecular AO basis (that is stored as a list of appropriate indices referring to a standard list of AO basis sets), the macromolecular electron density is computed according to eq. (1). Using the ADMA method, approximate macromolecular... 

The Mulliken-Mezey AFDF scheme and the more general AFDF schemes - also serve as the basis for the adjustable density matrix assembler (ADMA) method. The ADMA method generates ab initio quality macromolecular density matrices, which can be used for the computation of a variety of ab initio quality properties for macromolecules. The ADMA method is also suitable for the calculation of ab initio quality electronic densities, however, additional molecular properties, such as forces and energies, can also be calculated. These options of the ADMA method are expected to be useful in macromolecular conformational analysis, geometry optimizations, and in computational studies of protein folding. 

Following the earlier, more detailed introductions to the ADMA method, a short description of the special feature of the ADMA technique is given subsequently. 

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-... 

Following the original description, the application of the ADMA method involves several steps ... 

This numerical problem of integration can be avoided using the ADMA technique. Within the ADMA method, the integration in Eq. (361) can be performed using the analytical expressions of macromolecular density matrices and AOs. As an option of the ADMA algorithm, the calculated ADMA Hellmann-Feynman forces can be used for macro-molecular geometry optimization and macromolecular conformational analysis. 

Fragmentation (AFDF) Principle, and the Adjustable Density Matrix Assembler (ADMA) Method. 172... 

Usually, the parent molecules M are confined to some limited size that allows rapid determination of the parent molecule density matrices within a conventional ab initio Hartree-Fock-Roothaan-Hall scheme, followed by the determination of the fragment density matrices and the assembly of the macro-molecular density matrix using the method described above. The entire iterative procedure depends linearly on the number of fragments, that is, on the size of the target macromolecule M. When compared to the conventional ab initio type methods of computer time requirements growing with the third or fourth power of the number of electrons, the linear scaling property of the ADMA method is advantageous. 

Specific aspects of the quantum chemical concept of local electron densities and functional groups of chemistry have been discussed, with emphasis on the Additive Fuzzy Density Fragmentation (AFDF) Principle, on the Adjustable Density Matrix Assembler (ADMA) Method of using a local density matrix formalism of fuzzy electron density fragments in macromolecular quantum chemistry, and on the fundamental roles of the holographic electron density theorem, local symmetry, and symmetry deficiency. 

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]. 

IV. An Approach to Low-Density Analysis Based on the ADMA Method.135... 

Note that the simplest, Mulliken-Mezey fragmentation scheme employed in the MEDLA method, and in the more advanced macromolecular density matrix approach of the ADMA method [34-37], corresponds to the special choice of v/y =... 

IV. AN APPROACH TO LOW-DENSITY ANALYSIS BASED ON THE ADMA METHOD... 

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. 

The same Mulliken-Mezey method, and the more elaborate alternative partitioning schemes serve as the basis of the ALDA (adjustable local density assembler) method and the ADMA (adjustable density matrix assembler) technique. The ALDA and ADMA methods generate geometry-adjustable electronic densities and macromolecular density matrices. The ADMA macromolecular density matrices can be computed without determining a macromolecular wave function, a feature advantageous in the macromolecular application of a variety of quantum-chemical... 

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,... 

Szekeres, Z., Exner, T, Mezey, P. G. (2005). Fuzzy fragment selection strategies, basis set dependence and HF-DFT comparisons in the applications of the ADMA method of macromolecular quantum chemistry. Int. J. Quantum Chem. 104, 847-860. 

Eckard S, Exner TE (2006) Generalized hybrid orbitals in the FA-ADMA method. Z Phys Chem 220 927-944... 

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