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Energy Manipulations: Minimization and Dynamics

One can minimize the energy by adjusting the coordinates of all the atoms in order to reduce its value. Several minimization algorithms are provided. They include:

  • Steepest Descent (SD)
  • Conjugate Gradient (CONJ)
  • Adopted Basis Newton-Raphson (ABNR)
  • Newton-Raphson (NRAP)
  • Powell (POWE)
  • Truncated Newton Method (TNPACK)

Syntax for Energy Manipulation Commands

MINI     { SD     steepd-spec  } [ nonbond-spec ] [ hbond-spec ] -
         { CONJ   conj-spec    } [   INBFrq 0   ] [  IHBFrq 0  ] [NOUPdate]
         { ABNR   abnr-spec    }
         { NRAP   nrap-spec    }
         { POWEll powell_spec  }
         { TN     tnpack-spec  }

                [STEP real] [GRADient] [NUMErical]
                   [ frequency-spec ] [ tolerance-spec ] [ io-spec ] }

            [ CHEQ [CGMD int] [CGIN] [CGFC] [PBEQ] [QPOL [IPOL int] ] ]

hbond-spec::=     *Note Hbonds:(chmdoc/hbonds.doc).
nonbond-spec::=   *Note Nbonds:(chmdoc/nbonds.doc).

frequency-spec::= [NSTEP int] [IHBFrq int] [INBFrq int] [NPRInt int]

tolerance-spec::= [TOLENR real] [TOLGRD real] [TOLITR int] [TOLSTP real]

io-spec::= [DEBUG] [IUNCrd int [NSAVC int] ] [IUNXyz [NSAVX int] [MXYZ int] ]

conj-spec::= [NCGCyc int] [PCUT real] [PRTMin int]
                  [LATTice] [NOCOordinates]

powell-spec::=    [LATTice] [NOCOordinates]

steepd-spce::=  [NOENergy ] [LATTice] [NOCOordinates]

abnr-spec::= [EIGRng real] [MINDim int] [STPLim real] -
                [STRIct real] [ MASS ] [PSTRct real]
                  [LATTice] [NOCOordinates] [FMEM real]

nrap-spec::=  [TFREq real]

tnpack-spec::= [NCGCyc int]
                [PREC or NOPR] [USER or OURH]  [REST or QUAT]  [NOSC or SCHE]
                 [DEFS or SEAR] [LOWP or HIGP]  [IORD or NOOR] [PERM or NOPM]

Options common to minimization and dynamics

The following table describes the keywords which apply to all minimization methods.

Keyword Default Purpose
NSTEP 100 The number of steps to be taken. This is the number of cycles of minimization, not the number of energy evaluations.
INBFRQ 50 The frequency of regenerating the non-bonded list. The list is regenerated if the current step number modulo INBFRQ is zero and if INBFRQ is non-zero. Specifying zero prevents the non-bonded list from being regenerated at all.
IHBFRQ 50 The frequency of regenerating the hydrogen bond list. Analogous to INBFRQ
non-bond-spec   The specifications for generating the non-bonded list. See doc:nbonds.
hbond-spec   The specifications for generating the hydrogen bond list. See doc:hbonds
NPRINT 1 The frequency with which energies are printed during the course of dynamics or minimization.
GRADient   Minimize the magnitude of the gradient of the energy instead of the energy.
NUMErical   Forces will be determined by finite differences
IUNCrd -1 Unit to write out a “trajectory” file for the minimization
NSAVC 1 Frequency for writing out frames (only with IUNCrd)
DEBUg   Extra print for debug purposes

In the table which follows, keywords enclosed in square brackets means that one can choose one in the set. Such enclosed keywords do not expect a value after them. All other keywords are used for specifying values, see Syntax for Energy Manipulation Commands. The method column shows which method the keyword affects.

Keyword Default Method Purpose
[ CONJ ] CONJ   Do conjugate gradient minimization.
[ SD ] Do steepest descent minimization.
[ NRAP ] Do Newton-Raphson minimization.
[ ABNR ] Do Adopted Basis Newton-Raphson minimization,
[ TN ] Do Truncated-Newton minimization.
[MASS]     with mass weighted forces if specified.
STEP .02 ALL except TN Initial step size for the minimization algorithms. Reasonable values for the various methods are best determined by trial and error.
LATTice   ABNR With the CRYSTAL facility, also optimize the unit cell box size and/or shape.
NOCOords   ABNR With the CRYSTAL facility, only optimize the unit cell. This leaves coordinates unchanged, but allows the box size and/or shape to change.
PRTMIN 1 CONJ A flag indicating how much to print during minimization. If less than 2, the energy is printed only once each cycle. A setting of 2 shows the energy for each evaluation plus variables used in the method.
NCGCYC 100 CONJ The number of conjugate gradient cycles executed before the algorithm restarts.
PCUT .9999 CONJ If the cosine of the angle between the old and new P vector is greater than PCUT, the algorithm will be restarted. This prevents the algorithm from plodding down the same path repeatedly. If PRTMIN is less than 2, one effect of the restart is that the step size will go to its initial value. If this happens many times, you’ve converged.
EIGRNG .0005 ABNR The smallest eigenvalue (relative to the largest) that will be considered nonsingular.
MINDIM 5 ABNR The dimension of the basis set stored.
STPLIM 1.0 ABNR The maximum Newton Raphson step that will be allowed.
STRICT 0.1 ABNR The strictness of descent. The energy of a new step must be less than the previous best energy + STRICT for the new step to be accepted.
MASS false ABNR Use unweighted forces by default or mass weighted if specified. Mass weights converge more slowly but allow association with normal mode frequencies.
TFREQ 1.0 NRAP The smallest eigenvalue that is considered to be non-negative (i.e. do cubic fitting on all eigenvalues smaller than this).
TOLENR 0.0 ABNR A tolerance applied to the change in total energy change during a cycle of minimization (NCYCLE steps). If the energy change is less than or equal to TOLENR, the minimization routine will exit.
TOLGRD 0.0 ABNR A tolerance applied to the average gradient during a cycle of minimization. If the average gradient is less than or equal to TOLGRD, the routine will exit.
1.0 TN

A parameter which determines the desired accuracy of the computed solution. The following four convergence tests are checked: T1) f(x_{k-1})-f(x_k) < tolgrd (1+|f(x_k)|) T2) ||x_{k-1} - x_k|| < sqrt(tolgrd) (1+||x_k||) T3) ||g(x_k)|| < tolgrd^(1/3) (1+ ||f(x_k)||)/100 T4) ||g(x_k)|| < eg (1+ ||f(x_k)||)

If TOLGRD is equal to 0. in the input file, TOLGRD set to 10^(-8) in the calculation. If it is equal to 1., it is set to 10^(-12). eg is the square root of machine precision.

The routine will exit when either (T1,T2, and T3) are satisfied or (T4). (T4) is a useful test at the first Newton iteration or for comparison with other methods (see TNPACK paper).

TOLITR 100 ABNR CONJ The maximum number of energy evaluations allowed for a single step of minimization.
TOLSTP 0.0 ABNR A tolerance applied to the average step size during a cycle of minimization. If the average step size is less than or equal to TOLSTP, the routine will exit.
FMEM 0.0 ABNR

Memory factor. It is used to compute average gradient and step size according to the formula :

AVERAGE_VALUE = FMEM * AVERAGE_VALUE + (1-FMEM) * CURRENT_VALUE.

FMEM=0 means no memory (i.e current value is used) and FMEM=1 means infinitely long memory (i.e. initial value will be used all the time).

NOUP false ALL Turns off the list updates.
PREC or NOPR NOPR TN selects preconditioning (PREC) or no preconditioning (NOPR).
ANAL or FDIF ANAL TN selects option for Hd multiplication: ANAL for analytic version, FDIF for the finite-difference formula.
REST or QUAT REST TN specifies choice of PCG truncation test: residual (REST) or quadratic (QUAT).
SCOF or SCON SCOF TN specifies whether the scheduling subroutine is used (SCON for on, SCOF for off). The subroutine turns on preconditioning (if chosen) when the gradient is smaller than some tolerance, and uses steepest descent steps beforehand.
SRON or SROF SROF TN specifies whether the optimal search-vector subrou- tine is turned on (SRON) or off (SROF). This subrou- tine considers more than one possible descent directions at a Newton iteration and chooses the one that leads to greatest energy reduction. Additional energy + gradient evaluations are required.
IORD or NOOR NOOR TN specifies whether a reordering of M will be performed to minimize fill-in (IORD) or not (NOOR). This might be useful if M is very large and sparse. The reordering is done only once, but the savings are reflected in each-inner loop iteration where a linear system involving M is solved.
PERM or NOPM NOPM TN determines if the permutation array for reordering M is known when the current TNPACK call is made (PERM - known, NOPM - unknown).
NOEN FALSE SD only use the information of force to minimize a system. implemented for the case of minimizing a reaction path using the eudged elastic band method.
NSADD 0 NRAP sets the order of saddle point you want to find. NSADD=1 will search in the opposite direction of the most negative eigenvector (i.e. uphill) until a stationary point is located (i.e. transition state at NSADD=1).

Note that the following commands are equivalent:

  • ANAL = USER
  • FDIF = OURH
  • SCOF = NOSC
  • SCON = SCHE
  • SRON = DEFS
  • SROF = SEAR

Discussion of the various minimization methods

There are several different algorithms for minimizing the energy of the system. They all involve calculating the derivative of the potential energy, and possibly the second derivative, and using that information to adjust the coordinates in order to find a lower energy. In the descriptions of the algorithms below, a capitalized keyword at the beginning of each paragraph is the key word used to invoke that method. After the descriptions is a listing of all keywords associated with minimization.

The simplest minimization algorithm is steepest descent (SD). In each step of this iterative procedure, we adjust the coordinates in the negative direction of the gradient. It has one adjustable parameter, the step size, which determines how far to shift the coordinates at each step. The step size is adjusted depending on whether a step results in a lower energy. I.e., if the energy drops, we increase the step size by 20% to accelerate the convergence. If the energy rises, we overshot a minimum, so the step size is halved. Steepest descent does not converge in general, but it will rapidly improve a very poor conformation.

A second method is the conjugate gradient technique (CONJ) which has better convergence characteristics (R. Fletcher & C.M. Reeves, The Computer Journal 7:149 (1964)). The method is iterative and makes use of the previous history of minimization steps as well as the current gradient to determine the next step. It can be shown that the method converges to the minimum energy in N steps for a quadratic energy surface where N is the number of degrees of freedom in the energy. Since several terms in the potential are quadratic, it requires less evaluations of the energy and gradient to achieve the same reduction in energy in comparison to steepest descent. Its main drawback is that with very poor conformations, it is more likely to generate numerical overflows than steepest descent. The algorithm used in CHARMM has a slightly better interpolation scheme and automatic step size selection.

A third method is the conjugate gradient powell method minimizer (POWE) (A. Brunger). Its efficiency is much improved over the Fletcher and Reeves method. The use of the POWELL minimizer is encouraged whenever ABNR is not possible. The POWELL minimizer has no INBFRQ or IHBFRQ feature. The use of CHARMM loops to mimic this important feature is encouraged. The CHARMM loop facilities allow harmonic constrained minimizations with periodic updates. In case of bad contacts or unlikely conformations the SHAKE method might give an error when the displacements become to large. Using a harmonic constraint setup with periodic updates solves this problem.

A fourth method involves solving the Newton-Raphson minimization equations iteratively (NRAP). This procedure requires the computation of the derivative of the gradient which is a matrix of size O(n**2). The procedure here is to find a point where the gradient will be zero (hopefully a minimum in energy) assuming that the potential is quadratic. The Newton-Raphson equations can be solved by a number of means, but the method adopted for CHARMM involves diagonalizing the second derivative matrix and then finding the optimum step size along each eigenvector. When there are one or more negative eigenvalues, a blind application of the equations will find a saddle point in the potential. To overcome this problem, a single additional energy and gradient determination is performed along the eigenvector displacement for each small or negative eigenvalue. From this additional data, the energy function is approximated by a cubic potential and the step size that minimizes this function is adopted. The advantages of this method are that the geometry cannot remain at a saddle point, as sometimes occurs with the previous procedures, and that the procedure converges rapidly when the potential is nearly quadratic (or cubic). The major disadvantage is that this procedure can require excessive storage requirements, O(n**2), and computation time, O(n**3), for large molecules. Thus we are currently restricted to systems with about 200 atoms or less for this method. IMAGES and SHAKE are currently unavailable with this method.

The fifth method available is an adopted basis Newton-Raphson method (ABNR) (D. J. States). This routine performs energy minimization using a Newton-Raphson algorithm applied to a subspace of the coordinate vector spanned by the displacement coordinates of the last (mindim) positions. The second derivative matrix is constructed numerically from the change in the gradient vectors, and is inverted by an eigenvector analysis allowing the routine to recognize and avoid saddle points in the energy surface. At each step the residual gradient vector is calculated and used to add a steepest descent step onto the Newton-Raphson step, incorporating new direction into the basis set. This method is the best for most circumstances. SHAKE is currently unavailable with this method.

The sixth method is the truncated-Newton (TN) minimization package TNPACK, developed by T. Schlick and A. Fogelson. TNPACK is based on the preconditioned linear conjugate-gradient technique for solving the Newton equations. The structure of the problem — sparsity of the Hessian — is exploited for preconditioning. Thorough experience with the new version of TNPACK in CHARMM has been described in the paper: Journal of Computational Chemistry: 15, 532–552, 1994. Through comparisons among the minimization algorithms available in CHARMM, we find that TNPACK compares favorably to ABNR in terms of CPU time when curvature information is calculated by a finite-difference of gradients (the “numeric” option of TNPACK). With the analytic option, TNPACK can converge more rapidly than ABNR for small and medium systems (up to 400 atoms) as well as large molecules that have reasonably good starting conformations; for large systems that are poorly relaxed (i.e., the initial Brookhaven Protein Data Bank structures are poor approximations to the minimum), TNPACK performs similarly to ABNR. SHAKE is currently unavailable with this method.

Barriers and Minima

The GRADient option causes the minimizers to find a zero of the target function (grad(V))^2. The square of the gradient replaces the energy in the minimizers. Depending on the initial condition (initial set of coordinates), the search can either be terminated in a minimum or in a saddle point of the potential energy function (a barrier). If the second derivative of the initial condition is negative BARI will look for a saddle point; if it is positive it will stop at a minimum. The second derivative matrix is employed to calculate first derivatives of the target function. As a result it is much slower compared to ABNR and NRAP in reaching a minimum. For minimum energy calculations: DO NOT USE THE GRADient OPTION.

The NSADD keyword turns on special code that follows positive eigenvectors thus searching for a saddle point. Care must be taken when choosing the starting structure for this code (i.e. you should not start the search from a true minima as the code can get confused about which eigenvector to follow). The best suggestion is to slightly perturb your structure in the direction you believe that transition state (or higher order saddle point) of interest to be.