From MOE  Manual

The Energy Minimize application in MOE calculates atomic coordinates that are local minima of a molecular energy function. Energy minimization is useful for molecule building, determining low energy conformations, conformational search and preparation for molecular dynamics simulations and vibrational analysis. The supported energy functions are:

• The current MOE forcefield (e.g. AMBER, CHARMM, MMFF94, etc.)
• The MOPAC semi-empirical energy functions (e.g. PM3, AM1, MNDO)

The MOPAC energy functions are fundamentally different from molecular mechanics forcefields in that they are capable of simulating chemical reactions. In forcefield minimization, the Energy Minimize application is capable of calculating constrained geometries through the use of MOE's chiral, distance, angle and dihedral restraints. In addition, atoms may be constrained to be frozen during the calculation.

Theory

Energy minimization consists of finding a set of atomic coordinates that correspond to a local minimum of the molecular energy function (such as the potential energy model). This is done by applying large scale non-linear optimization techniques to calculate a conformation (near to the starting geometry) for which the forces on the atoms are zero.

Forcefield Optimization. Non-linear optimization algorithms typically have the following structure [Gill 1981]. Let x_k denote the vector of atomic coordinates at step k of the procedure and let U be the energy function. Then,

1. Test for convergence. If the convergence criteria are satisfied (see below), then x_k is returned.
2. Compute the search direction. Compute a non-zero vector p_k called the search direction.
3. Compute the step size. Compute a non-zero scalar a_k, called the step size, for which U(x_k + a_k p_k) < U(x_k).
4. Advance. Set x_k+1 = x_k and k = k + 1 and go to Step 1.

The step size in Step 3 is computed by using a safeguarded bicubic interpolation search along the search direction. In Step 1, the optimization is terminated when any of the following three conditions are satisfied:

1. Root Mean Square gradient Test: |grad U(x_k)| < e_A sqrt(n), where e_A is a predefined constant and n is the number of unfixed atoms.
2. Iteration Limit Test: k > K, where K is a predefined upper limit on the maximum number of iterations.
3. Progress Tests: The following three conditions are simultaneously satisfied:

   U(x_k-1) - U(x_k) < T (1 + |U(x_k)|)
   |x_k-1 - x_k| < T ^1/2 (1 + |x_k|)
   |grad U(x_k)| <= T ^1/3 (1 + |U(x_k)|)

   In these conditions, T is a predefined constant indicating the number of significant figures in U that are required (the function test). For example, if T = 10^-6 then six figures of accuracy are required.

If the iteration limit is exceeded then it is typically due to a failure of the algorithm. Note that saddle points satisfy criterion 1; the conformation should be perturbed slightly and optimization restarted. If criterion 3 is satisfied then it is very likely that a local minimum has been achieved.

The choice of search direction p_k in Step 2 is what distinguishes the different nonlinear optimization methods. MOE uses a success of three methods to effect an energy minimization: Steepest Descent (SD), Conjugate Gradient (SG) and Truncated Newton (TN). In SD optimization p_k = - grad U(x_k) ; that is, the search proceeds along the direction of the forces. While intuitive, SD is extremely inefficient after a few iterations and it is only used when the gradient is extremely high. When the gradient is sufficiently small (but still quite high) the CG method is used. The Conjugate Gradients (CG) method improves upon SD by choosing the next search direction in a way so as to not undo the progress accomplished by the previous step. CG performs well in strained conditions; however, it exhibits poor convergence properties. Once the gradient is reasonable, the TN method is used.

The Truncated Newton (TN) method is the most efficient large-scale nonlinear optimization method known. It exhibits superlinear convergence even in highly nonlinear conditions. The Truncated Newton method attempts to use curvature information to improve convergence. The Newton direction p satisfies:

G_k(x_k) p = - grad U(x_k)

where G_k is the second derivative, or Hessian, of U. These equations are called the Newton equations. TN solves the Newton equations approximately using an iterative linear equation solver. The iterative linear equation solver (based on the Linear Conjugate Gradient method) is terminated after a small number of iterations; hence the name Truncated Newton.

Quantum Mechanical Optimization. The PM3, AM1 and MNDO Hamiltonians can be used as the energy model. These models are implemented in the external MOPAC program [Stewart; 1993]. MOPAC is a semi-empirical molecular orbital software program for the study of chemical structures, including reactions. MOPAC supports the MNDO, MINDO/3, AM1 and PM3 Hamiltonians (for the electron part of the calculation). For the most part, MOPAC has been parameterized for the organic elements (and some transition metals with full d shells). MOPAC can calculate geometries, transition states, vibrational spectra, thermodynamic quantities and force constants for molecules, radicals and ions. MOE will automatically run MOPAC to effect the energy minimization calculation for these Hamiltonians. MOPAC uses different optimization methodology from MOE and a complete description is beyond the scope of this document.

Chemical Computing Group Inc. (CCG) makes no copyright, confidentiality, or other intellectual property rights claim regarding the MOPAC 7.0 program which has been placed, by its authors, entirely into the public domain. CCG has limited its activities in relation to the MOPAC program to providing executables for various computer platforms and authoring interoperability software. CHEMICAL COMPUTING GROUP INC. AND ITS AFFILIATES DISCLAIM ALL WARRANTIES WITH REGARD TO THE MOPAC PROGRAM, INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS: THE MOPAC PROGRAM IS PROVIDED "AS-IS".

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