Angaben zur Quelle [Bearbeiten]
Autor  Accelrys Inc. 
Titel  ForcefieldBased Simulations 
Ort  San Diego 
Datum  September 1998 
Anmerkung  http://www.esi.umontreal.ca/accelrys/life/insight2000.1/getdown.html 
URL  http://www.esi.umontreal.ca/accelrys/pdf/ffbs980.pdf 
Fragmente  11 
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II.1. Molecular dynamics simulations
Typical use of molecular dynamics (MD) include:  Searching the conformational space of alternative amino acid side chains in sitespecific mutation studies.  Identifying likely conformational states for highly flexible polymers or for flexible regions of macromolecules such as protein loops.  Producing sets of 3D structures consistent with distance and torsion constraints deduced from NMR experiments (simulated annealing).  Calculating free energies of binding, including solvation and entropy effects.  Probing the locations, conformations, and motions of molecules on catalyst surfaces.  Running diffusion calculations. In addition, simulation engines can be routinely used for:  Calculating normal modes of vibration and vibrational frequencies.  Analyzing intra molecular and inter molecular interactions in terms of residueresidue or moleculemolecule interactions, energy per residue, or interactions within a radius.  Calculating diffusion coefficients of small molecules in a polymer matrix.  Calculating thermal expansion coefficients of amorphous polymers.  Calculating the radial distribution of liquids and amorphous polymers.  Performing rigidbody comparisons between minimized conformations of the same or similar structures or between simulated and experimentally observed structures. At its simplest, molecular dynamics solves Newton’s equation of motion: (2.1) where F_{i} is the force, m_{i} is the mass and a_{i} is the acceleration of atom i.  Molecular dynamics
Typical uses of molecular dynamics include:
Other forcefieldbased calculations In addition, simulation engines can be routinely used for:
[page 4]
[page 192] At its simplest, molecular dynamics solves Newton's familiar equation of motion: Eq. 83 where F_{i} is the force, m_{i} is the mass, and a_{i} is the acceleration of atom i. 
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The force on atom i can be computed directly from the derivative of the potential energy V with respect to the coordinates r_{i}:
(2.2) The classical equations of motion are deterministic, which means that once the initial coordinates and velocities and other dynamic information at time t are known, the positions and velocities at time t + Δt are determined (calculated). The coordinates and velocities for a complete dynamics run are called the trajectory. The time step Δt depends on the integration method as well as the system itself. Time step should be small enough in order to avoid integration errors. Because of fast vibrational motions of the atoms, a time step of 1 fs is usually used. Although the initial coordinates are known, the initial velocities are randomly generated at the beginning of a dynamics run, according to the desired temperature. Analytical integration of equations such as (2.1) is not possible for large systems, so numerical integration should be performed. Molecular dynamics is usually applied to large systems. Energy evaluation is time consuming and the memory requirements are large. To generate the correct statistical ensembles, energy conservation is also important. Thus, the basic criteria for a good integrator for molecular simulations are as follows: • It should be fast, ideally requiring only one energy evaluation per time step. Variants of the Verlet [24] algorithm of integrating the equations of motion (Equation (2.1)) are perhaps the most widely used method in molecular dynamics. The advantages of Verlet integrators is that these methods require only one energy evaluation per step, require only modest memory, and also allow a relatively large time step to be used. 24. Verlet, L. Phys. Rev. 1967 , 159.  The force on atom i can be computed directly from the derivative of the potential energy V with respect to the coordinates r_{i}:
Eq. 84 Notice that classical equations of motion are deterministic. That is, once the initial coordinates and velocities are known, the coordi [page 193] nates and velocities at a later time can be determined. The coordinates and velocities for a complete dynamics run are called the trajectory. [...] Given the initial coordinates and velocities and other dynamic information at time t, the positions and velocities at time t + Δt are calculated. The timestep Δt depends on the integration method as well as the system itself. Although the initial coordinates are determined in the input file or from a previous operation such as minimization, the initial velocities are randomly generated at the beginning of a dynamics run, according to the desired temperature. [...] Criteria of Good Integrators in Molecular Dynamics Molecular dynamics is usually applied to a large system. Energy evaluation is time consuming and the memory requirement is large. To generate the correct statistical ensembles, energy conservation is also important. Thus, the basic criteria for a good integrator for molecular simulations are as follows:
[page 195] Verlet Leapfrog Integrator Variants of the Verlet (1967) algorithm of integrating the equations of motion (Eq. 84) are perhaps the most widely used method in molecular dynamics. The advantages of Verlet integrators is that these methods require only one energy evaluation per step, require only modest memory, and also allow a relatively large timestep to be used. Verlet, L. ``Computer experiments on classical fluids. I. Thermodynamical properties of LennardJones molecules", Phys. Rev., 159, 98103 (1967). 
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Verlet LeapFrog algorithm requires r(t), v(t Δt/2), and a(t), which are (respectively) the position, velocity, and acceleration at times t, t Δt/2, and t, and compute:
(2.3) (2.4) (2.5) where f(t + Δt) is evaluated from dV/dr at r(t + Δt).  The leapfrog algorithm
The Verlet leapfrog algorithm is as follows: Given r(t), v(t Δt/2), and a(t), which are (respectively) the position, velocity, and acceleration at times t, t Δt/2, and t, compute: where f(t + Δt) is evaluated from dV/dr at r(t + Δt). 
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AMBER Force Field
The AMBER energy expression contains a minimal number of terms. No cross terms are included. The functional forms of the energy terms used by AMBER are given in Equation (2.6).  Standard AMBER forcefield
The AMBER energy expression contains a minimal number of terms. No cross terms are included. The functional forms of the energy terms used by AMBER are given in Eq. 19. 
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A source is given, but it is not clear that the formulas and text are identical to the AccelrysDocumentation that was published three years later than the source given. 

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The analytic energy expressions for the ESFF are provided in Equation (2.7). Only diagonal terms are included.
where D and K are force constants, θ are bond angles, A and B are van der Waals parameters, χ is electronegativity, q are partial charges of atoms and r are distances between atoms. The bond energy is represented by a Morse functional form, where the bond dissociation energy D, the reference bond length r^{0}, and the anharmonicity parameters are needed. In constructing these parameters from atomic parameters, the force field utilizes not only the atom types and bond orders, but also considers whether the bond is endo or exo to 3, 4, or 5membered rings.  [page 37]
The analytic energy expressions for the ESFF forcefield are provided in Eq. 11. Only diagonal terms are included. The bond energy is represented by a Morse functional form, where the bond dissociation energy D, the reference bond length r^{0} , and the anharmonicity parameters are needed. In constructing these parameters from atomic parameters, the forcefield utilizes not only the atom types and bond orders, but also considers whether the bond is endo or exo to 3, 4, or 5membered rings. [...] [page 38] 
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The rules themselves depend on the electronegativity, hardness, and ionization of the atoms as well as atomic anharmonicities and the covalent radii and well depths. The latter quantities are fit parameters, and the former three are calculated.
The ESFF angle types are classified according to ring, symmetry, and πbonding information into five groups:
The rules that determine the parameters in the functional forms depend on the ionization potential and, for equatorial angles, the periodicity. In addition to these calculated quantities, the parameters are functions of the atomic radii and well depths of the central and end atoms of the angle, and, for planar angles, two overlap quantities and the 13 equilibrium distances. To avoid the discontinuities that occur in the commonly used cosine torsional potential when one of the valence angles approaches 180°, ESFF uses a functional form that includes the sine of the valence angles in the torsion. These terms ensure that the function goes smoothly to zero as either valence angle approaches 0° or 180°, as it should. The rules associated with this expression depend on the central bond order, ring size of the angles, hybridization of the atoms, and two atomic parameters for the central [atom, which is fit.]  [page 37]
The rules themselves depend on the electronegativity, hardness, and ionization of the atoms as well as atomic anharmonicities and the covalent radii and well depths. The latter quantities are fit parameters, and the former three are calculated. [page 38] The ESFF angle types are classified according to ring, symmetry, and πbonding information into five groups:
[page 39]
The rules that determine the parameters in the functional forms depend on the ionization potential and, for equatorial angles, the periodicity. In addition to these calculated quantities, the parameters are functions of the atomic radii and well depths of the central and end atoms of the angle, and, for planar angles, two overlap quantities and the 1–3 equilibrium distances. To avoid the discontinuities that occur in the commonly used cosine torsional potential when one of the valence angles approaches 180°, ESFF uses a functional form that includes the sine of the valence angles in the torsion. These terms ensure that the function goes smoothly to zero as either valence angle approaches 0° or 180°, as it should. The rules associated with this expression depend on the central bond order, ring size of the angles, hybridization of the atoms, and two atomic parameters for the central atom which is fit. 
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[The rules associated with this expression depend on the central bond order, ring size of the angles, hybridization of the atoms, and two atomic parameters for the central] atom, which is fit. The functional form of the outofplane energy is the same as in CFF91, where the coordinate (Φ) is an average of the three possible angles associated with the outofplane center. The single parameter that is associated with the central atom is a fit quantity.
The charges are determined by minimizing the electrostatic energy with respect to the charges under the constraint that the sum of the charges is equal to the net charge on the molecule. This is equivalent to equalization of electronegativities. The derivation of the rule begins with the following equation for the electrostatic energy: (2.8) where χ is the electronegativity and η the hardness. The first term is just a Taylor series expansion of the energy of each atom as a function of charge, and the second is the Coulomb interaction law between charges. The Coulomb law term introduces a geometry dependence that ESFF for the time being ignores, by considering only topological neighbors at effectively idealized geometries. Minimizing the energy with respect to the charges leads to the following expression for the charge on atom i: (2.9) where λ is the Lagrange multiplier for the constraint on the total charge, which physically is the equalized electronegativity of all the atoms. The Δχ term contains the geometryindependent remnant of the full Coulomb summation. Equations (2.8) and (2.9) give a totally delocalized picture of the charges in a relatively severe approximation. To obtain reasonable charges as judged by, for example, crystal packing calculations, some modifications to the above picture have been made.  [page 39]
The rules associated with this expression depend on the central bond order, ring size of the angles, hybridization of the atoms, and two atomic parameters for the central atom which is fit. The functional form of the outofplane energy is the same as in CFF91, where the coordinate (Φ) is an average of the three possible angles associated with the outofplane center. The single parameter that is associated with the central atom is a fit quantity. The charges are determined by minimizing the electrostatic energy with respect to the charges under the constraint that the sum of the charges is equal to the net charge on the molecule. This is equivalent to equalization of electronegativities. The derivation of the rule begins with the following equation for the electrostatic energy: [page 40] Eq. 12 where χ is the electronegativity and η the hardness. The first term is just a Taylor series expansion of the energy of each atom as a function of charge, and the second is the Coulomb interaction law between charges. The Coulomb law term introduces a geometry dependence that ESFF for the time being ignores, by considering only topological neighbors at effectively idealized geometries. Minimizing the energy with respect to the charges leads to the following expression for the charge on atom i: Eq. 13 where λ is the Lagrange multiplier for the constraint on the total charge, which physically is the equalized electronegativity of all the atoms. The ∆χ term contains the geometryindependent remnant of the full Coulomb summation. Eqns. 12 and 13 give a totally delocalized picture of the charges in a relatively severe approximation. To obtain reasonable charges as judged by, for example, crystal packing calculations, some modifications to the above picture have been made. 
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Metals and their immediate ligands are treated with the above prescription, summing their formal charges to get a net fragment charge. Delocalized π−systems are treated in an analogous fashion. σ−systems are treated using a localized approach in which the charges of an atom depend simply on its neighbors. Note that this approach, unlike the straightforward implementations based on the equalization of electronegativity, does include some resonance effects in the π−system.
The electronegativity and hardness in the above equations must be determined. In earlier force fields they were often determined from experimental ionization potentials and electron affinities; however, these spectroscopic states do not correspond to the valence states involved in molecules. For this reason, ESFF is based on electronegativities and hardnesses, calculated using density functional theory. The orbitals are (fractionally) occupied in ratios appropriate for the desired hybridization state, and calculations are performed on the neutral atom as well as on positive and negative ions. ESFF uses the 69 potential for the van der Waals interactions. Since the van der Waals parameters must be consistent with the charges, they are derived using rules that are consistent with the charges. Starting with the London formula: (2.10) where α is the polarizability and IP the ionization potential of the atoms, the polarizability, in a simple harmonic approximation, is proportional to n/IP where n is the number of electrons. Across any one row of the periodic table, the core electrons remain unchanged, so that the following form is reasonable: (2.11)  [page 40]
Metals and their immediate ligands are treated with the above prescription, summing their formal charges to get a net fragment charge. Delocalized π systems are treated in an analogous fashion. And σ systems are treated using a localized approach in which the charges of an atom depend simply on its neighbors. Note that this approach, unlike the straightforward implementations based on the equalization of electronegativity, does include some resonance effects in the π system. Electronegativity and hardness obtained by DFT The electronegativity and hardness in the above equations must be determined. In earlier forcefields they were often determined from experimental ionization potentials and electron affinities; however, these spectroscopic states do not correspond to the valence states involved in molecules. For this reason, ESFF is based on electronegativities and hardnesses, calculated using density functional theory as implemented in DMol. The orbitals are (fraction [page 41] ally) occupied in ratios appropriate for the desired hybridization state, and calculations are performed on the neutral atom as well as on positive and negative ions. van der Waals interactions ESFF uses the 6–9 potential for the van der Waals interactions. Since the van der Waals parameters must be consistent with the charges, they are derived using rules that are consistent with the charges. Derivations Starting with the London formula: Eq. 14 where α is the polarizability and IP the ionization potential of the atoms, the polarizability, in a simple harmonic approximation, is proportional to n / IP where n is the number of electrons. Across any one row of the periodic table, the core electrons remain unchanged, so that the following form is reasonable: Eq. 15 
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where a´ and b´ are adjustable parameters that should depend on just the period, and n_{eff} is the effective number of (valence) electrons. Further assuming that α is proportional to R^{3} and that another equivalent expression to that in Equation (2.10) is:
(2.12) where ε is a well depth, the following forms are deduced for the rules for van der Waals parameters: (2.13) The van der Waals parameters are affected by the charge of the atom. In ESFF we found it sufficient to modify the ionization potential (IP) of metal atoms according to their formal charge and hardness: (2.14) and for nonmetals to account for the partial charges when calculating the effective number of electrons. ESFF atom types are determined by hybridization, formal charge, and symmetry rules. In addition, the rules may involve bond order, ring size, and whether bonds are endo or exo to rings. For metal ligands the cistrans and axialequatorial positionings are also considered. The additions of these latter types affect only certain parameters (for example, bond order influences only bond parameters) and thus are not as powerful as complete atom types. In one sense they provide a further refinement of typing beyond atom types. The ESFF has been parameterized to handle all elements in the periodic table up to radon. It is recommended for organometallic systems and other systems for which other [force fields do not have parameters.]  [page 41]
where a´ and b´ are adjustable parameters that should depend on just the period, and n_{eff} is the effective number of (valence) electrons. Further assuming that α is proportional to R^{3} and that another equivalent expression to that in Eq. 14 is:
where ε is a well depth, the following forms are deduced for the rules for van der Waals parameters: Eq. 17 The van der Waals parameters are affected by the charge of the atom. In ESFF we found it sufficient to modify the ionization potential (IP) of metal atoms according to their formal charge and hardness: Eq. 18 [page 42] and for nonmetals to account for the partial charges when calculating the effective number of electrons. ESFF atom types (Table 32) are determined by hybridization, formal charge, and symmetry rules (Atomtyping rules in ESFF). In addition, the rules may involve bond order, ring size, and whether bonds are endo or exo to rings. For metal ligands the cis–trans and axial–equatorial positionings are also considered. The addition of these latter types affects only certain parameters (for example, bond order influences only bond parameters) and thus are not as powerful as complete atom types. In one sense they provide a further refinement of typing beyond atom types. The ESFF forcefield has been parameterized to handle all elements in the periodic table up to radon. It is recommended for organometallic systems and other systems for which other forcefields do not have parameters. 
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[It is recommended for organometallic systems and other systems for which other] force fields do not have parameters. ESFF is designed primarily for predicting reasonable structures (both intra and intermolecular structures and crystals) and should give reasonable structures for organic, biological, organometallic and some ceramic and silicate models. It has been used with some success for studying interactions of molecules with metal surfaces. Predicted intermolecular binding energies should be considered approximate.  It is recommended for organometallic systems and other systems for which other forcefields do not have parameters. ESFF is designed primarily for predicting reasonable structures (both intra and intermolecular structures and crystals) and should give reasonable structures for organic, biological, organometallic and some ceramic and silicate models. It has been used with some success for studying interactions of molecules with metal surfaces. Predicted intermolecular binding energies should be considered approximate. 
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