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Explicit factorization of external coordinates in constrained Statistical Mechanics models
, 2008
"... If a macromolecule is described by curvilinear coordinates or rigid constraints are imposed, the equilibrium probability density that must be sampled in Monte Carlo simulations includes the determinants of different massmetric tensors. In this work, we explicitly write the determinant of the massm ..."
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If a macromolecule is described by curvilinear coordinates or rigid constraints are imposed, the equilibrium probability density that must be sampled in Monte Carlo simulations includes the determinants of different massmetric tensors. In this work, we explicitly write the determinant of the massmetric tensor G and of the reduced massmetric tensor g, for any molecule, general internal coordinates and arbitrary constraints, as a product of two functions; one depending only on the external coordinates that describe the overall translation and rotation of the system, and the other only on the internal coordinates. This work extends previous results in the literature, proving with full generality that one may integrate out the external coordinates and perform Monte Carlo simulations in the internal conformational space of macromolecules. In addition, we give a general mathematical argument showing that the factorization is a consequence of the symmetries of the metric tensors involved. Finally, the determinant of the massmetric tensor G is computed explicitly in a set of curvilinear coordinates specially wellsuited for general branched molecules. PACS: 05.20.y, 36.10.k, 87.14.g, 87.15.v, 87.15.Aa, 89.75.k 1
1An Exact Expression to Calculate the Derivatives of PositionDependent Observables in Molecular Simulations with Flexible Constraints
"... In this work. we introduce an algorithm to compute the derivatives of physical observables along the constrained subspace when flexible constraints are imposed on the system (i.e., constraints in which the constrained coordinates are fixed to configurationdependent values). The presented scheme is ..."
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In this work. we introduce an algorithm to compute the derivatives of physical observables along the constrained subspace when flexible constraints are imposed on the system (i.e., constraints in which the constrained coordinates are fixed to configurationdependent values). The presented scheme is exact, it does not contain any tunable parameter, and it only requires the calculation and inversion of a subblock of the Hessian matrix of second derivatives of the function through which the constraints are defined. We also present a practical application to the case in which the sought observables are the Euclidean coordinates of complex molecular systems, and the function whose minimization defines the flexible constraints is the potential energy. Finally, and in order to validate the method, which, as far as we are aware, is the first of its kind in the literature, we compare it to the natural and straightforward finitedifferences approach in a toy system and in three molecules of biological relevance: methanol, Nmethylacetamide and a triglycine peptide. Author Summary
Splitvalence Gaussian basis sets and the heterolevel approximation in RHF and MP2
, 2008
"... Efficient model chemistries for peptides. I. ..."
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Efficient model chemistries for peptides. II. Basis set convergence in the B3LYP method
, 2008
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A mathematical introduction to HartreeFock SCF methods in Quantum Chemistry
, 2008
"... It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the manybody problem. — Max Born, 1960 We present here an introduction to HartreeFock theory in Quantum Chemistry. From the molecular Hamiltonian, using and discuss ..."
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It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the manybody problem. — Max Born, 1960 We present here an introduction to HartreeFock theory in Quantum Chemistry. From the molecular Hamiltonian, using and discussing the BornOppenheimer approximation, we arrive to the Hartree and HartreeFock equations for the electronic problem. Special emphasis is placed in the most relevant mathematical aspects of the theoretical derivation of the final equations, as well as in the results regarding the existence and uniqueness of their solutions. Additionally, the discretization of the oneelectron orbitals space is reviewed and the RoothaanHall formalism introduced. Finally, the basic underlying concepts related to the construction and selection of Gaussian basis sets are discussed focusing in algorithmic efficiency issues. The whole work is intentionally introductory and rather selfcontained, so that it may be useful for non experts that aim to use quantum chemical methods in interdisciplinary applications. Moreover, much material that is found scattered in the literature has been put together here to facilitate comprehension and to serve as a handy reference.
Proteins
, 2013
"... Exact and efficient calculation of Lagrange multipliers in constrained biological polymers: ..."
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Exact and efficient calculation of Lagrange multipliers in constrained biological polymers: