几种常见的优化方法. 电子结构 几何机构. 稳定点 最小点. 函数. Taylor 展开： V(x) = V(x k ) + (x-x k )V’(x k ) +1/2 (x-x k ) 2 V’’(x k )+…. 当 x 是 3N 个变量的时候， V’(x k ) 成为 3Nx1 的向量，而 V’’(x k ) 成为 3Nx3N 的矩阵，矩阵元如：. Hessian. 一阶梯度法 a. Steepest descendent. 最速下降法. S k = -g k /|g k |. direction. gradient.
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V(x) = V(xk) + (x-xk)V’(xk) +1/2 (x-xk)2 V’’(xk)+…..
Sk = -gk/|gk|
robust, but slow
Usually more efficient than SD, also robust
Quasi-Newton方法： use an approximation of the inverse Hessian.
Form of approximation differs among methods
It was not until 1964 that MD was used to study a realistic molecular system, in which the atoms interacted via a Lennard-Jones potential.
After this point, MD techniques developed rapidly to
encompass diatomic species, water (which is still the
subject of current research today!), small rigid molecules,
flexible hydrocarbons and now even macromolecules
such as proteins and DNA.
These are all examples of continuous dynamical
simulations, and the way in which the atomic motion is
calculated is quite different from that in impulsive
simulations containing hard-core repulsions.
– Calculate equilibrium configurational properties in a similar
fashion to MC.
– Study transport properties (e.g. mean-squared displacement and
Extend the basic MD algorithm
– MD in the NVT, NpT and NpH ensembles
– The united atom approximation
– Constraint dynamics and SHAKE
– Rigid body dynamics
– Multiple time step algorithms
1.Dynamics of perfectly ‘hard’ particles can be solved exactly, but process becomes involved for many part (N-body problem). 2.Can use a numerical scheme that advances the system forward in time until a collision occurs.
3.Velocities of colliding particles (usually a pair!) then recalculated and system put into motion again.
4. Simulation proceeds by fits and starts, with a mean time between collisions related to the average kinetic energy of the particles.
5.Potentially very efficient algorithm, but collisions between particles of complex shape are not easy to solve, and cannot be generalised to continuous potentials.
1. By calculating the derivative of a macromolecular force
field, we can find the forces on each atom
as a function of its position.
2. Require a method of evolving the positions of the particles in space and time to produce a ‘true’ dynamical trajectory.
3. Standard technique is to solve Newton’s equations of
motion numerically, using some finite difference scheme,
which is known as integration.
4. This means that we advance the system by some small
time step Δt, recalculate the forces and velocities, and then
repeat the process iteratively.
5. Provided Δt is small enough, this produces an acceptable approximate solution to the continuous equations of motion.
1.The choice of time step is crucial: too short and phase space is sampled inefficiently, too long and the energy will fluctuate wildly and the simulation may become catastrophically unstable (“blow up”).
2. The instabilities are caused by the motion of atoms being extrapolated into regions where the potential energy is prohibitively high (e.g. atoms overlapping).
3. A good rule of thumb is that when simulating an atomic fluid, the time step should be comparable to the mean time between collisions (about 5 fs for Ar at 298K).
4. For flexible molecules, the time step should be an order of magnitude less than the period of the fastest motion (usually bond stretching: C—H around 10 fs so use 1 fs).
For first principles MD, as forces are evaluated from quantum mechanics, we are only concerned with the time-step.
Because the interactions are completely elastic and
pairwise acting, both energy and momentum are
conserved. Therefore, MD naturally samples from the
microcanonical or NVE ensemble.
As mentioned previously, the NVE ensemble is not very useful for studying real systems. We would like to be able to simulate systems at constant temperature or constant pressure.
(modified form of Newton II)
The is known as the Nosé-Hoover method (Melchionna
type) and the equations of motion are:
freeze the bond stretching motions of the hydrogens (or any other bond, in principle).
These multipliers can be determined by substituting the modified expressions for Newton II into our Verlet integration scheme and imposing the constraints that: