12 Simulated Annealing

One of the most difficult problems to solve in computational chemistry is the "multiple-minima" problem, or the location and characterization of geometric minima on a complex multidimensional potential energy surface. Since many chemical phenomena (especially in biological systems) depend on the structural arrangement of the molecule, determining the global minimum energy conformation is an important goal. For complex molecules, there can often be hundreds or thousands of reasonable candidate structures which are almost impossible to generate a priori and adequately explore. The simulated annealing strategy developed here is a heuristic algorithm to address this challenge of finding multiple minima.

Introduction

Modern strategies to minimize a function in several unconstrained variables were defined in the 1970s. These methods fall into the general category of "pseudo-Newton" approaches and make extensive use of the gradient of the function, as the only way to retain computational efficiency with large numbers of variables. The minimizers in AMPAC™ (BFGS, EF, DFP, TRUSTE, or NEWTON for energy and TS, TRUSTG, or LTRD for gradient) are all local minimizers in that they converge toward the minimum (or critical point) that is in the most favorable position when related to the starting point. This may not be the global minimum, but only the one that is lowest in energy in a specific region of the potential energy surface. When multiple minima do exist on the potential surface, it is often difficult to locate them directly. A tedious process of trial and error is usually followed by defining a variety of starting points around the area of viability for the optimizable parameters (a grid search). Thus, an open problem in theoretical optimization, the location of the absolute minimum of a function in many unconstrained variables, has direct application to one of the most persistent and annoying problems in computational chemistry, the "multiple-minima" dilemma. In 1983, Kirkpatrick et al. suggested a heuristic answer to this problem, making use of a method of simulation in statistical mechanics coupled with Boltzmann's law at decreasing temperatures.^[24] "Simulated annealing" was the name given the procedure.

A reasonable question for computational chemistry is not necessarily finding the minimum solution of the energy/geometry function defining the molecule, but collecting a variety of possible solutions, i.e. local geometric minima. If properly structured and implemented, the simulated annealing process can be used to collect an entire set of minima. The simulated annealing procedure is very slow in refining the exact location of a minimum, and specialized local methods (much more rapid) are used for exact location. A mixed strategy can be derived in which the approximate locations of the minima are generated by the annealing procedure and the final refinement is carried out from each starting place by one of the local optimization protocols mentioned above.

Simulated Annealing Theory

The theory and terminology behind simulated annealing comes from statistical mechanics (the behavior of systems with many degrees of freedom in thermal equilibrium at a finite temperature). The idea of annealing is to heat an object (e.g. a piece of metal) to a high temperature and then allow the object to slowly cool down. The high temperature allows the atoms to move from their original positions allowing the system to rearrange into a more stable state. As the system cools, it becomes trapped in its new configuration and the system will continue to evolve toward the lowest energy configuration in that local region. Slow cooling may also be followed by a rapid cooling ("quenching") which removes any residual heat, effectively locking the atoms in their new configuration.

Annealing can used to move a macroscopic system toward its most stable configuration (global minimum) and so has elicited interest in how it can be exploited to solve various types of problems.1 (In real systems, however, multiple heating/cooling cycles may be required and one is never certain that the true optimal configuration has been achieved.) In typical macroscopic systems (solid or liquid) the density of atoms is on the order of 1023 atoms per cubic centimeter. For such a system in thermal equilibrium, statistical mechanics shows that the behavior of the system is dominated by the most probable behavior and small fluctuations from this average behavior can be neglected. In this type of ensemble, each configuration, defined by a set of atomic positions {r_i}, of the system is weighted by its Boltzmann probability factor, exp(-E{r_i}/k_BT), where E{r_i} is the energy of the configuration, k_B is Boltzmann's constant, and T is temperature.

This Boltzmann factor was used to develop the Metropolis step,^[25] a simple algorithm that could be used to efficiently simulate a collection of atoms in equilibrium at a given temperature. At each step, a new geometry is generated as a random displacement from the current geometry. The energy at the new geometry is computed and the difference in energy between the current and the new geometries is given as ΔE. The probability that this new geometry is accepted, P(ΔE), is given by Metropolis' criterion:

( 12.1 )

Thus, if the new geometry is lower in energy than the current geometry, the step is always accepted. If the step leads to a higher energy, the step will only be accepted if a randomly generated number (between 0 and 1) is less than or equal to the Boltzmann probability factor. At high temperatures, the Boltzmann factor is close to one, so uphill steps (increasing energy) are frequently accepted allowing the system to climb up out of its local valley. At lower temperatures, the Boltzmann factor drops toward zero causing frequent rejection of uphill steps. Thus the system is prevented from escaping its local region and is progressively driven down in energy.

Strategies for Annealing Searches

The simple annealing procedure just defined is useful for the global minimum problem but is not well suited for finding multiple minima (or critical points). Also the final stage of annealing converges very slowly and so is inefficient for our purposes. Traditional energy (or gradient) minimizers are much more efficient for the purpose of converging on local extrema. This suggests that the best approach is a heuristic algorithm that effectively mixes the rapid global searching capability of annealing with the efficient local optimization of traditional minimizers. In AMPAC, this mixed annealing strategy is accomplished in three distinct steps.^[25]

Simulated Annealing

Each new geometry is generated as a random displacement from the current geometry. Displacements are generated randomly but must be scaled so that they fall within a boundary hypersphere of radius, R. To do this, a random number is generated between 0 and R and the length of the generated displacement is scaled to match this value. By default, a uniform random number is used for this step. Keyword GAUSSIAN forces the use of gaussian random number generator instead. (GAUSSIAN only affects the scaling of the displacement, not the generation of the displacement.) This generated displacement is then added to the existing geometry to produce a new displaced geometry. The energy is computed at this new geometry and is accepted as the new current geometry if it passes the Metropolis criterion (described above). (Note, that the Metropolis criterion can be generalized as will be discussed under the individual simulated annealing methods given later.) When a step is rejected, a new displacement from the current geometry is generated and tested until a successful step is found.

The first step of the simulated annealing phase is to "melt" the system by performing a Metropolis walk on the system at a high initial temperature, T_init. This walk is continued until the system reaches a steady state ("thermalization"). Subsequently, the temperature is lowered and again allowed to run until thermalization is achieved. Each new temperature, T_n, is determined by multiplying the previous temperature by factor, T_dec, where 0 < T_dec < 1. Thus the n^th temperature used is given by:

( 12.2 )

This process of alternately lowering the temperature and running until thermalization is achieved is continued until the system "freezes" and no further changes occur. (T_init and T_dec are set by keywords TEMP=n.n and TLAW=n.n respectively.)

The boundary radius, R, used in generating displacements is adjusted at each temperature, T_n. As a rule of thumb, the radius, R_n, is chosen to give a rejection ratio of approximately 0.5, where the rejection ratio is the number of points rejected by the Metropolis criterion divided by the total number of points. This adjustment to R_n can be handled automatically but is upper bounded by some value, R_max, to avoid possible excessive steps. A minimum boundary, R_min, can also be specified as f_min*R_max. If the radius R_n falls below R_min, the system is considered "frozen" and the annealing phase is ended. (R_max and f_min are set by STEP=n.n and STEPCV=n.n respectively.)

The Metropolis walk is continued at a given temperature until "thermalization" is achieved or the maximum number of steps, N_max, is exceeded. A fairly simple thermalization criterion that can be easily computed is used for this purpose.^[26] At every N_check points, E_σ (standard deviation of energy), ē (average energy), and e_min (minimum energy) is computed for that set of points and tested against our thermalization threshold, σ_therm. This thermalization criterion is given as:

( 12.3 )

(N_max, N_check, and σ_therm are set by NMAX=n, NCHECK=n, and STD=n.n, respectively.)

Metropolis walks are performed at successively lower temperatures until the system is deemed to be "frozen" (trapped in a local basin) and the simulated annealing phase is ended. The system is deemed frozen if (a) the boundary radius, R_n, is less than R_min; (b) the temperature falls below a certain threshold (usually T_init/100); or (c) the final geometry of two or three Markov chains is deemed to be the "same." By "same" here, we mean that the "distance" between the geometries falls below some threshold.

Filtering

This Metropolis walk on the PES at successive temperatures generates a series of successive geometries, which are known as Markov chains. These Markov chains can then be used to generate "candidates" for quenching (refinement). Since a typical Metropolis walk visits several minima in the course of its search at a given temperature, and since the walk concentrates on the lower regions of the PES with decreasing temperature, some minima may be visited many times at various temperatures. To avoid excessive minimization during the subsequent quenching phase, all of the candidates are analyzed on the basis of nearly equivalent conformations, accounting for translation, rotation, reflection and for permutations within a set of equivalent nuclei. (This process is also referred to as a "clustering sort".)

Since geometries close to each other on the Markov chain are generally associated with the same extrema and quenching is expensive, it is advisable not to quench every geometry. Instead, all of the Markov chains are divided into segments containing N_check geometries, with only the best conformation in each set being considered as a candidate for quenching. (N_check is set by NCHECK=n.)

A "distance" between conformations is calculated and candidates are clustered into closely related groups on this basis. This "distance" measures the similarity of different conformations taking into account translation, rotation, reflection, and for permutations within a set of equivalent nuclei.^[26] It has been found that only the smaller components of this "distance" criteria are important, so a band-pass filter is used to eliminate the larger component contributions. The band-pass filter is defined with central value BPF_ref and half-bandwidth of BPF_sig*BPF_ref. (These values are set by BPFREF=n.n and BPFSIG=n.n respectively.) Conformations related by a short "distance" are taken as being equivalent geometries. Candidate geometries from each set are then compared to identify redundant geometries. If the "distance" between candidate geometries is less than the value, Filt, the geometries are considered equivalent and one of the geometries is eliminated as redundant. (Filt is set by FILTER=n.n.)

Quenching

Each candidate geometry from Stage 2 then undergoes a local minimization ("quenching") for final refinement of the geometry. This minimization can occur in up to three distinct phases.

Phase 1

By default, TRUSTE (with ANNEAL) or TRUSTG (with TSANNEAL, GANNEAL, or MANNEAL) is used to optimize each geometry to the nearest minimum or critical point. This default can be overridden by explicitly including one of AMPAC's usual minimizers. For ANNE, an energy minimizer (BFGS, EF, or DFP) can be specified. For TSANNEAL, GANNEAL, or MANNEAL a gradient minimizer (TRUSTG) can be specified. (TS is not recommended since many geometries might be far from a critical point.)

Phase 2

If NEWTON (with ANNEAL) or LTRD (with TSANNEAL, GANNEAL, or MANNEAL) is specified, a second round of quenching is performed but this time using full Hessian methods. Prior to this phase, all of the candidates from phase 1 undergo filtering to remove any equivalent geometries and so reduce the number of geometries to minimize. This optimization step is expensive but can clean up if the initial minimization does poorly.

Phase 3

Typically, a user will freeze less important degrees of freedom throughout the simulated annealing search to focus the search on the more important degrees of freedom. (See Annealing Strategies below.) If keyword WHOLE is specified, all coordinates will be unfrozen and a final full optimization will be performed on each quenched geometry. To have this final optimization done in Cartesian coordinates, the keyword WHOLE=XYZ may be specified. This final phase is referred to as "relaxation."

To avoid the quenching stage all together, use NOQUENCH. When NOQUENCH is specified, the simulated annealing job stops just prior to the quenching stage without printing results but producing a restart file. The job can be finished by adding RESTART in the keyword line and rerunning the job. The final output of the procedure provides the main characteristics of the minima collected through the process. Additional output can be obtained by using the PRINT=n keyword.

Annealing Methods

In AMPAC, simulated annealing has been generalized to include four related strategies, each with their own strengths and weaknesses. The original Metropolis step gives the probability of accepting the step in terms of the change in energy during the step. To apply the Metropolis step and the simulated annealing strategy to more general problems, we can generalize from change in energy to a change in a generalized "cost function." For chemical systems, this opens the door to using the RMS gradient norm (gnorm) as the cost function in addition to the traditional energy (heat of formation). An additional generalization is that we can add penalty functions to the basic energy or gnorm. TSANNEAL, GANNEAL, and MANNEAL include an energy window penalty term that is directly added to the energy/gnorm used in the Metropolis step. This focuses the annealing search on energies in the window centered at F_ref and extends by S_ref in either direction. Configurations with an energy outside this window are penalized by a quadratic function with coefficient P_ref.

( 12.4 )

(See annealing keywords, FREF=n.n, SREF=n.n, and PREF=n.n.)

ANNEAL (ANNE)

The AMPAC keyword "ANNE" represents the standard simulated annealing strategy. The molecule's energy is used in the Metropolis step and quenching is performed using AMPAC's energy minimizers. ANNE is useful if one is only interested in minima. (Transition states and other critical points may occasionally still be found but this is uncommon.)

TSANNEAL (TSANN)

"TSANN" is a generalization of ANNE that uses a gradient minimizer instead of an energy minimizer in the final quenching step and thus is useful for locating transitions states rather than just minima. Like ANNE, the molecule's energy is used in the Metropolis step.

GANNEAL (GANN)

"GANN" is a second generalization of ANNE. Like TSANN, it uses a gradient minimizer for the final quenching step. However, it differs from TSANN and ANNE in that the RMS gradient norm (gnorm) is used in place of energy in the Metropolis step. Using gnorm focuses the simulated annealing search on all critical points rather than primarily on just minima.

MANNEAL (MANN)

"MANN" is the third generalization of ANNE. For the Metropolis step, MANN uses a combination of both the energy (like TSANN and ANNE) and gnorm (like GANN). The "cost function" for the Metropolis step is the minimum of the change in energy and Fctor3 times the change in gnorm. (Fctor3 determines the relative importance of the gnorm compared to energy in the Metropolis step and is 1.0 by default. This value can be set using the FCTOR3=n.n keyword.) As with TSANN and GANN, a gradient minimizer is used in the final quenching step and so is applicable to transition states and saddle points, not just minima. MANN can be thought of as a merger of both the TSANN and GANN annealing strategies.

Penalty Functions

Most annealing algorithms and other multiple-minima search procedures are applied in the context of molecular mechanics (MM). This is because a large number of independent energy calculations usually need to be carried out and MM is the only chemical modeling method efficient enough to do this. To minimize the number of such calculations required using the higher quality, but much slower semiempirical methods, the researcher can apply certain additional constraints to the preliminary search (prior to the use of semiempirical energy calculations), based on his chemical understanding of the problem. These constraints are treated as penalty functions to the genuine search criterion (energy (ANNE, TSANN), gradient norm (GANN), or both (MANN)). Enormous savings and enhanced efficiency can result. For example, in conformational problems involving aliphatic rings, up to 99% of the configurations generated correspond to geometries that are so poor that SCF convergence will be difficult. These conformations are discarded by the annealing procedure at a cost of only 1% of the computer time. The constraints presently implemented are

Explicit boundaries on changing coordinates (Cartesian or internal, with or without symmetry constraints). Such a constraint is actually seen as a periodic condition, as recommended in the original Metropolis algorithm (see annealing keywords LIMIT and AUTOLIMIT).
A severe penalty function is added if an interatomic distance drops below some threshold. This precludes atoms from collapsing onto one another (see annealing keyword PEN1).
A bounded molecular system can be defined in which it is forbidden for any molecular configuration to have a moment of inertia greater than some defined value (see annealing keyword PENA=n.n).
Indestructible chemical bonds can be defined by adding a penalty function if a bond leaves a specific range (see annealing keywords PEN2 and TOL=n).
MANN, GANN, and TSANN have an additional penalty function that defines a specific energy window that is to be the target of the simulated annealing search. The energy window is centered at F_ref and extends by S_ref in either direction. Configurations with an energy outside this window are penalized by a quadratic function with coefficient, P_ref. (See annealing keywords, FREF=n.n, SREF=n.n, and PREF=n.n.)

Annealing Strategies

While the annealing procedure in AMPAC is designed to somewhat automate the search for minima (and other critical points) on a potential energy surface, it is not a "black box." The judicious use of geometry input and the various penalty functions can make the search both complete in the quantum mechanical potential energy space and efficient computationally. It is neither chemically correct nor necessary to search ALL of the potential energy surface, when only a few geometric variables define all of the meaningful differences between minima and maxima. A few points to note:

Divide the potential energy search into primary and secondary degrees of freedom as the degrees of freedom are expressed in the optimizable geometric parameters. The secondary degrees of freedom can then be stringently constrained using the LIMIT keyword and wide variance can be allowed the primary degrees of freedom.
The TEMP=n.n keyword sets Tinit and can be used to allow the search to overcome larger energy barriers by increasing the value of "n.n". TEMP is expressed in kcal/mol and should be at least twice the value of the energy barrier to be overcome. The default value is 50 (200 in MANN or GANN). The annealing process will take substantially longer if TEMP is increased.
The TLAW=n.n keyword allows a wider search of the PES by slowing the rate of temperature decay, T_dec. A higher value for "n.n" results in a slower decay rate and correspondingly increases the computational cost.
The thoroughness with which the PES is searched at each temperature can be increased by decreasing the value for σ_therm (set by STD=n.n). This may cause the search to locate more remote minima on the PES, at each temperature. Note that a value of STD that is too small will cause a limited search to require an inordinate amount of time.
A smaller value of Filt (set by FILTER=n.n) can be used to retain configurations that are geometrically similar. Too small a value for Filt will result in a large number of virtually equivalent conformers being passed to the quenching routines. This approach should be used where conformers are separated by slight differences in geometry.
As the Markov chains are generated, they get sectioned into pieces by the annealing algorithm. Each piece hopefully refers to a specific minima on the surface. The length of these sections is governed by N_check (annealing keyword NCHECK=n). A smaller value of N_check increases the number of candidates for quenching, allowing a wider sampling of the Markov chain on the surface but at a higher computational cost. A large value for N_check reduces the computational cost but focuses the annealing search on a single global minima rather than multiple minima.

Simulated Annealing Dedicated Keywords

ANNEAL	Simulated annealing search for geometric minima.
AUTOLIMIT	Define default preliminary periodic boundaries.
BPFREF	Define central value of the band-pass filter.
BPFSIG	Define half-width of the band-pass filter.
CRUDE	Use crude rejection scheme.
FCTOR3	Determine balance between energy and gnorm (MANNEAL only).
FILTER	Determine equivalency of configurations during the clustering sort.
FREF	Define central value of the energy range.
GANNEAL	Simulated annealing search for extrema within an energy range.
GAUSSIAN	Use a Gaussian, rather than uniform, random number generator for geometry displacement.
LIMIT	Define periodic boundaries.
LTRD	Minimize gradient using full Hessian.
MANNEAL	Simulated annealing search for minima within an energy range.
MARK	All points of the Markov chains are written to channel 8.
NCHECK	Define interval for producing quenching candidates at each temperature.
NEWTON	Minimize energy using full Hessian.
NMAX	Define maximum value of criterion calls at a given temperature.
NOQUENCH	Skip quenching.
NRAND	Define random number seed value.
PENA	Activate penalty function on the molecule's moments of inertia.
PEN1	Activate close contact penalty function.
PEN2	Activate conformational penalty function.
PEN2GRP	Activate conformational penalty function within distinct groups.
PREF	Define the energy window penalty coefficient.
SREF	Specify half-width of the searched energy range.
STD	Define thermalization criterion.
STEP	Define maximum step size in the annealing search.
STEPCV	Define a lower bound for the step size (% of initial step).
TEMP	Starting "temperature" for the annealing procedure.
TEST	Print extra debugging output.
TLAW	Specify the decay constant in the temperature.
TOL	Permitted relative variation of a bond length from its initial value.
TSANNEAL	Simulated annealing search for extrema within an energy range.
WHOLE	End the quenching steps will full optimizations.

Example I ("Floppy" system)

The cyclohexane molecule has been chosen as an example of a molecule with relatively low rotational barriers. There are four conformers of the molecule that can be identified (chair, boat (flexible), twist or skew boat, and half-chair) and all of these are located by AMPAC's annealing protocol, using relaxed constraints and limited penalty functions.

Input File (`annealing/anneal1.dat`)

  am1 rhf singlet t=auto anneal truste pen2 ncheck=10 noxyz limit
Cyclohexane
Annealing on cyclic system: PEN2, NCHECK, LIMIT
 C              0.000000  0    0.000000  0    0.000000  0    0    0    0
 C              1.520880  0    0.000000  0    0.000000  0    1    0    0
 C              1.520880  0  112.706930  1    0.000000  0    2    1    0
 C              1.520880  0  112.706930  1   -7.376720  1    3    2    1
 C              1.520880  0  112.706930  1  -46.310020  1    1    2    3
 C              1.520880  0  112.706930  1   50.819480  1    5    1    2
 H              1.121200  0  112.384140  1   75.006870  1    1    2    3
 H              1.121200  0  112.384140  1 -167.313050  1    1    2    3
 H              1.121200  0  112.384140  1  121.019560  1    2    3    1
 H              1.121200  0  112.384140  1 -121.144800  1    2    3    1
 H              1.121200  0  112.384140  1  113.455200  1    3    2    1
 H              1.121200  0  112.384140  1 -128.802830  1    3    2    1
 H              1.121200  0  112.384140  1  177.862400  1    4    3    2
 H              1.121200  0  112.384140  1  -63.807980  1    4    3    2
 H              1.121200  0  112.384140  1   57.505850  1    5    4    3
 H              1.121200  0  112.384140  1 -154.797280  1    5    4    3
 H              1.121200  0  112.384140  1  121.566360  1    6    5    4
 H              1.121200  0  112.384140  1 -122.040500  1    6    5    4
 0              0.000000  0    0.000000  0    0.000000  0    0    0    0
$$ limit - annealing boundaries   
  100 100 -180 100 -180 100 -180 100 -180 100 -180 100 -180 100    
    -180 100 -180 100 -180 100 -180 100 -180 100 -180 100 -180
    100 -180 100 -180
  130 130  180 130  180 130  180 130  180 130  180 130  180 130    
    180 130  180 130  180 130  180 130  180 130  180 130  180
    130  180 130  180
$$ end of extra data

	This is the extra input section marker for reaction path data. Note, that this marker can be shortened to "$$ limit". Details of these markers are found in “Extra Input Data”.
	These are the upper boundaries of the bond angles and dihedrals (in order as specified) as required for by the annealing keyword LIMIT.
	These are the upper boundaries of the bond angles and dihedrals (in order as specified) as required for by the annealing keyword LIMIT.

Archive File (`annealing/anneal1.arc`)

 Timestamp: 2004-02-12-14-26-42-0000028031-hpux

                     SUMMARY OF   AM1   CALCULATION
                                                       Feb-12-2004
                          AMPAC Version 8.13
                             Presented by:

                        Semichem, Inc.
                        PO Box 1649
                        Shawnee KS 66222
                        (913)268-3271
                        (913)268-3445 (fax)

 FORMULA: C6H12
 Cyclohexane
 Annealing on cyclic system: PEN2, NCHECK, LIMIT

     GEOMETRY NORMALLY RETURNED BY SIMULATED ANNEALING        
     SCF FIELD WAS ACHIEVED

          FINAL HEAT OF FORMATION =     -38.461440 kcal   
                                  =    -160.961127 kJ
          ELECTRONIC ENERGY       =   -4336.005512 eV
          CORE-CORE REPULSION     =    3401.111984 eV
          TOTAL ENERGY            =    -934.893527 eV
          GRADIENT NORM           =       0.154575
          RMS GRADIENT NORM       =       0.027763
          UNSTABLE MODE(S)        =       0 ( ESTIMATE  )
          DIPOLE                  =       0.000735 debyes
          NO. OF FILLED LEVELS    =      18 (OCC = 2)
          KOOPMAN IONISATION POTENTIAL =     10.92 eV
          MOLECULAR POINT GROUP   = C2H   0.100000
          FINAL GEOMETRY OBTAINED                                 CHARGE
 AM1 RHF SINGLET T=AUTO ANNEAL TRUSTE PEN2 NCHECK=10 NOXYZ LIMIT
 Cyclohexane
 Annealing on cyclic system: PEN2, NCHECK, LIMIT
  C     0.000000  0    0.000000  0    0.000000  0    0   0   0   -0.1551
  C     1.520880  0    0.000000  0    0.000000  0    1   0   0   -0.1551
  C     1.520880  0  111.165157  1    0.000000  0    2   1   0   -0.1552
  C     1.520880  0  111.161780  1   55.402521  1    3   2   1   -0.1547
  C     1.520880  0  111.164790  1  -55.406380  1    1   2   3   -0.1552
  C     1.520880  0  111.153391  1   55.431603  1    5   1   2   -0.1547
  H     1.121200  0  109.419507  1   65.581152  1    1   2   3    0.0782
  H     1.121200  0  109.528408  1 -176.560058  1    1   2   3    0.0768
  H     1.121200  0  109.547055  1  121.203418  1    2   3   1    0.0769
  H     1.121200  0  109.405925  1 -120.943019  1    2   3   1    0.0782
  H     1.121200  0  109.416876  1  -65.565386  1    3   2   1    0.0782
  H     1.121200  0  109.536594  1  176.593100  1    3   2   1    0.0768
  H     1.121200  0  109.446158  1 -176.926866  1    4   3   2    0.0768
  H     1.121200  0  109.309861  1   65.493109  1    4   3   2    0.0781
  H     1.121200  0   93.030707  1  109.421303  1    5   4   3    0.0782
  H     1.121200  0  143.754335  1 -124.577818  1    5   4   3    0.0769
  H     1.121200  0  109.444338  1 -121.316608  1    6   5   4    0.0767
  H     1.121200  0  109.324266  1  121.091565  1    6   5   4    0.0781
  0     0.000000  0    0.000000  0    0.000000  0    0   0   0
 Timestamp: 2004-02-12-14-26-42-0000028031-hpux

                     SUMMARY OF   AM1   CALCULATION
                                                       Feb-12-2004
                          AMPAC Version 8.13
                             Presented by:

                        Semichem, Inc.
                        PO Box 1649
                        Shawnee KS 66222
                        (913)268-3271
                        (913)268-3445 (fax)
  
 FORMULA: C6H12
 Cyclohexane
 Annealing on cyclic system: PEN2, NCHECK, LIMIT

     GEOMETRY NORMALLY RETURNED BY SIMULATED ANNEALING        
     SCF FIELD WAS ACHIEVED

          FINAL HEAT OF FORMATION =     -35.291019 kcal
                                  =    -147.692916 kJ
          ELECTRONIC ENERGY       =   -4341.342822 eV
          CORE-CORE REPULSION     =    3406.586775 eV
          TOTAL ENERGY            =    -934.756047 eV
          GRADIENT NORM           =       0.081059
          RMS GRADIENT NORM       =       0.014559
          UNSTABLE MODE(S)        =       0 ( ESTIMATE  )
          DIPOLE                  =       0.000643 debyes
          NO. OF FILLED LEVELS    =      18 (OCC = 2)
          KOOPMAN IONISATION POTENTIAL =     10.67 eV
          MOLECULAR POINT GROUP   = D2    0.100000
          FINAL GEOMETRY OBTAINED                                 CHARGE
 AM1 RHF SINGLET T=AUTO ANNEAL TRUSTE PEN2 NCHECK=10 NOXYZ LIMIT
 Cyclohexane
 Annealing on cyclic system: PEN2, NCHECK, LIMIT
  C     0.000000  0    0.000000  0    0.000000  0    0   0   0   -0.1535
  C     1.520880  0    0.000000  0    0.000000  0    1   0   0   -0.1540
  C     1.520880  0  112.213535  1    0.000000  0    2   1   0   -0.1535
  C     1.520880  0  111.556090  1  -29.933862  1    3   2   1   -0.1532
  C     1.520880  0  111.556646  1  -30.147321  1    1   2   3   -0.1535
  C     1.520880  0  111.569796  1   62.747080  1    5   1   2   -0.1537
  H     1.121200  0  109.546683  1   90.595080  1    1   2   3    0.0780
  H     1.121200  0  109.617463  1 -151.718093  1    1   2   3    0.0766
  H     1.121200  0  109.704583  1  121.283119  1    2   3   1    0.0759
  H     1.121200  0  108.979764  1 -121.705225  1    2   3   1    0.0759
  H     1.121200  0  109.545306  1   90.804138  1    3   2   1    0.0780
  H     1.121200  0  109.613895  1 -151.509686  1    3   2   1    0.0766
  H     1.121200  0  109.523089  1 -175.443835  1    4   3   2    0.0765
  H     1.121200  0  108.888534  1  -58.295260  1    4   3   2    0.0779
  H     1.121200  0  106.592919  1   51.245406  1    5   4   3    0.0780
  H     1.121200  0  137.683155  1 -166.341717  1    5   4   3    0.0766
  H     1.121200  0  109.638216  1  121.391228  1    6   5   4    0.0758
  H     1.121200  0  108.895026  1 -121.791528  1    6   5   4    0.0758
  0     0.000000  0    0.000000  0    0.000000  0    0   0   0
 Timestamp: 2004-02-12-14-26-42-0000028031-hpux

                     SUMMARY OF   AM1   CALCULATION
                                                       Feb-12-2004
                          AMPAC Version 8.13
                             Presented by:

                        Semichem, Inc.
                        PO Box 1649
                        Shawnee KS 66222
                        (913)268-3271
                        (913)268-3445 (fax)

 FORMULA: C6H12
 Cyclohexane
 Annealing on cyclic system: PEN2, NCHECK, LIMIT

     GEOMETRY NORMALLY RETURNED BY SIMULATED ANNEALING
     SCF FIELD WAS ACHIEVED                                   

          FINAL HEAT OF FORMATION =      68.531275 kcal
                                  =     286.803386 kJ
          ELECTRONIC ENERGY       =   -4203.164625 eV
          CORE-CORE REPULSION     =    3272.910650 eV
          TOTAL ENERGY            =    -930.253975 eV   
          GRADIENT NORM           =       0.181505
          RMS GRADIENT NORM       =       0.032599
          UNSTABLE MODE(S)        =       0 ( ESTIMATE  )
          DIPOLE                  =       0.602149 debyes
          NO. OF FILLED LEVELS    =      18 (OCC = 2)
          KOOPMAN IONISATION POTENTIAL =      7.47 eV
          MOLECULAR POINT GROUP   = C1    0.100000
          FINAL GEOMETRY OBTAINED                                 CHARGE
 AM1 RHF SINGLET T=AUTO ANNEAL TRUSTE PEN2 NCHECK=10 NOXYZ LIMIT
 Cyclohexane
 Annealing on cyclic system: PEN2, NCHECK, LIMIT
  C     0.000000  0    0.000000  0    0.000000  0    0   0   0   -0.1569
  C     1.520880  0    0.000000  0    0.000000  0    1   0   0   -0.1487
  C     1.520880  0  112.100498  1    0.000000  0    2   1   0   -0.1140
  C     1.520880  0  116.152940  1   91.481971  1    3   2   1   -0.3273
  C     1.520880  0  111.488101  1   49.899094  1    1   2   3   -0.1523
  C     1.520880  0  109.795911  1  -50.541673  1    5   1   2   -0.2934
  H     1.121200  0  109.479603  1  171.517575  1    1   2   3    0.0805
  H     1.121200  0  109.107766  1  -70.809525  1    1   2   3    0.0824
  H     1.121200  0  108.735853  1  120.973835  1    2   3   1    0.0821
  H     1.121200  0  109.308886  1 -121.517520  1    2   3   1    0.0942
  H     1.121200  0  103.628667  1  -24.175711  1    3   2   1    0.1761
  H     1.121200  0  115.236806  1 -131.929127  1    3   2   1    0.1170
  H     1.121200  0  119.544734  1  -23.967783  1    4   3   2    0.1186
  H     1.121200  0  119.264366  1  164.297526  1    4   3   2    0.1173
  H     1.121200  0  129.969972  1 -177.220511  1    5   4   3    0.0810
  H     1.121200  0  122.238784  1    0.185552  1    5   4   3    0.0767
  H     1.121200  0  113.876389  1  138.614763  1    6   5   4    0.0820
  H     1.121200  0  113.599372  1  -91.533744  1    6   5   4    0.0847
  0     0.000000  0    0.000000  0    0.000000  0    0   0   0
 Timestamp: 2004-02-12-14-26-42-0000028031-hpux

                     SUMMARY OF   AM1   CALCULATION
                                                       Feb-12-2004
                          AMPAC Version 8.13
                             Presented by:

                        Semichem, Inc.
                        PO Box 1649
                        Shawnee KS 66222
                        (913)268-3271
                        (913)268-3445 (fax)

 FORMULA: C6H12
 Cyclohexane
 Annealing on cyclic system: PEN2, NCHECK, LIMIT

     GEOMETRY NORMALLY RETURNED BY SIMULATED ANNEALING
     SCF FIELD WAS ACHIEVED

          FINAL HEAT OF FORMATION =      77.923830 kcal
                                  =     326.111227 kJ   
          ELECTRONIC ENERGY       =   -4311.602694 eV
          CORE-CORE REPULSION     =    3381.756010 eV
          TOTAL ENERGY            =    -929.846683 eV   
          GRADIENT NORM           =       0.170982
          RMS GRADIENT NORM       =       0.030709
          UNSTABLE MODE(S)        =       0 ( ESTIMATE  )
          DIPOLE                  =       1.341343 debyes
          NO. OF FILLED LEVELS    =      18 (OCC = 2)
          KOOPMAN IONISATION POTENTIAL =      8.73 eV
          MOLECULAR POINT GROUP   = C1    0.100000
          FINAL GEOMETRY OBTAINED                                 CHARGE
 AM1 RHF SINGLET T=AUTO ANNEAL TRUSTE PEN2 NCHECK=10 NOXYZ LIMIT
 Cyclohexane
 Annealing on cyclic system: PEN2, NCHECK, LIMIT
  C     0.000000  0    0.000000  0    0.000000  0    0   0   0   -0.1697
  C     1.520880  0    0.000000  0    0.000000  0    1   0   0   -0.1614
  C     1.520880  0  107.600269  1    0.000000  0    2   1   0   -0.1603
  C     1.520880  0  107.617051  1   19.233177  1    3   2   1   -0.1727
  C     1.520880  0  107.570855  1  -15.297778  1    1   2   3   -0.0993
  C     1.520880  0  118.200490  1 -131.340041  1    5   1   2   -0.4438
  H     1.121200  0  110.174369  1  104.276302  1    1   2   3    0.0895
  H     1.121200  0  110.722060  1 -138.188452  1    1   2   3    0.0880
  H     1.121200  0  110.890119  1  121.451761  1    2   3   1    0.0858
  H     1.121200  0  110.112298  1 -120.001738  1    2   3   1    0.0813
  H     1.121200  0  110.132638  1 -100.769198  1    3   2   1    0.0832
  H     1.121200  0  110.874004  1  140.640111  1    3   2   1    0.0848
  H     1.121200  0  111.301584  1 -134.563353  1    4   3   2    0.0914
  H     1.121200  0  110.988800  1  106.226878  1    4   3   2    0.0856
  H     1.121200  0   91.321534  1  101.005908  1    5   4   3    0.2108
  H     1.121200  0  101.575226  1 -102.115868  1    5   4   3    0.0620
  H     1.121200  0  116.872098  1 -170.709860  1    6   5   4    0.1226
  H     1.121200  0  116.849790  1   43.445186  1    6   5   4    0.1222
  0     0.000000  0    0.000000  0    0.000000  0    0   0   0

Each of the located and quenched minima is sequentially listed in the archive file.

Output File (`annealing/anneal1.out`)

 Timestamp: 2004-02-12-14-26-42-0000028031-hpux
 *******************************************************************************
                             AM1 CALCULATION RESULTS
 *******************************************************************************
 *                             AMPAC Version 8.13
 *                                Presented by:
 *
 *                           Semichem, Inc.
 *                           PO Box 1649
 *                           Shawnee KS 66222
 *                           (913)268-3271
 *                           (913)268-3445 (fax)
 *
 *  ANNE     - SIMULATED ANNEALING ON ENERGY ONLY
 *  TRUSTE   - MINIMISE ENERGY USING TRUST REGION
 *  T=AUTO   - AUTOMATIC DETERMINATION OF ALLOWED TIME
 *  NOXYZ    - CARTESIAN COORDINATES NOT TO BE PRINTED
 *  SINGLET  - IS THE REQUIRED SPIN MULTIPLICITY
 *  AM1      - THE AM1 HAMILTONIAN TO BE USED
 *******************************************************************************
 AM1 RHF SINGLET T=AUTO ANNEAL TRUSTE PEN2 NCHECK=10 NOXYZ LIMIT
 Cyclohexane
 Annealing on cyclic system: PEN2, NCHECK, LIMIT
    ATOM    CHEMICAL   BOND LENGTH    BOND ANGLE    TWIST ANGLE
   NUMBER   SYMBOL     (ANGSTROMS)     (DEGREES)     (DEGREES)
    (I)                   NA:I          NB:NA:I      NC:NB:NA:I   NA  NB  NC
      1     C
      2     C          1.52088                                     1
      3     C          1.52088        112.70693 *                  2   1
      4     C          1.52088        112.70693 *    -7.37672 *    3   2   1
      5     C          1.52088        112.70693 *   -46.31002 *    1   2   3
      6     C          1.52088        112.70693 *    50.81948 *    5   1   2
      7     H          1.12120        112.38414 *    75.00687 *    1   2   3
      8     H          1.12120        112.38414 *  -167.31305 *    1   2   3
      9     H          1.12120        112.38414 *   121.01956 *    2   3   1
     10     H          1.12120        112.38414 *  -121.14480 *    2   3   1
     11     H          1.12120        112.38414 *   113.45520 *    3   2   1
     12     H          1.12120        112.38414 *  -128.80283 *    3   2   1
     13     H          1.12120        112.38414 *   177.86240 *    4   3   2
     14     H          1.12120        112.38414 *   -63.80798 *    4   3   2
     15     H          1.12120        112.38414 *    57.50585 *    5   4   3
     16     H          1.12120        112.38414 *  -154.79728 *    5   4   3
     17     H          1.12120        112.38414 *   121.56636 *    6   5   4
     18     H          1.12120        112.38414 *  -122.04050 *    6   5   4

   MOLECULAR POINT GROUP            SYMMETRY CRITERIA
            C1                          0.10000000

          SINGLET STATE CALCULATION

        **  REFERENCES TO PARAMETERS  **

  H  (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)
  C  (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)

 *******************************************************************************
 *  KEYWORDS DEDICATED TO SIMULATED ANNEALING SECTION    
 *
 *  LIMIT    - EXPLICIT PERIODIC CONDITIONS GIVEN
 *  NCHECK=  - SET MARKOV CHAIN IN PIECES OF   10 LONG
 *  PEN2     - SWITCH ON MIN/MAX PENALTY ON INTERATOMIC DISTANCES
 *           - PENALTIES OPERATED BY SKEWED REBOUNDS
 *******************************************************************************
 STANDARD DEVIATION ON ENERGY   (KCAL)       0.00000055521
 STANDARD DEVIATION ON GRADIENT (KCAL/A,RD,RD)  0.00000000 0.00001427 0.00001282

 SIMULATED ANNEALING BY METROPOLIS SCHEME IN   31 VARIABLES      
     VERSION 2.0 (MAY 2000)
     THERMALIZATION THRESHOLD ON COST FNCT          0.080
     MAXIMUM STEP SIZE                            2.00000
     CONVERGENCE CRITERION ON STEP SIZE           0.06325

 MAXIMUM NUMBER OF STEPS AT EACH TEMPERATURE         1000
     FIRST TEMPERATURE (HOT) SET TO                200.00
     TEMPERATURE EXPONENTIAL DECAY                   0.80
     LAST TEMPERATURE (COLD) SET TO                  2.00
 RANDOM SEQUENCE INITIATOR                       -9876543
     MARKOV CHAIN STUDIED BY PIECES OF                 10
     No OF TEMPERATURES TO DETECT A FROZEN SYSTEM       2
 EQUIVALENCE THRESHOLD IN CLUSTERING ANALYSIS        1.00
     BAND-PASS FILTER NO 1 CENTERED AT               1.40 ANGSTROMS
     HALF BAND-WIDTH                                13.50 %
 PENALTIES ACTIVATED:                                            
     ON CHEMICAL BONDS, THRESHOLD:                   0.20
 PERIODIC BOUNDARY CONDITIONS (ANGSTROMS, DEGREES)
 LOWER BOUND  100.0000  100.0000 -180.0000  100.0000 -180.0000  100.0000   
             -180.0000  100.0000 -180.0000  100.0000 -180.0000  100.0000
             -180.0000  100.0000 -180.0000  100.0000 -180.0000  100.0000
             -180.0000  100.0000 -180.0000  100.0000 -180.0000  100.0000
             -180.0000  100.0000 -180.0000  100.0000 -180.0000  100.0000
             -180.0000
 TRIAL COORD  112.7069  112.7069   -7.3767  112.7069  -46.3100  112.7069
               50.8195  112.3841   75.0069  112.3841 -167.3131  112.3841
              121.0196  112.3841 -121.1448  112.3841  113.4552  112.3841
             -128.8028  112.3841  177.8624  112.3841  -63.8080  112.3841
               57.5059  112.3841 -154.7973  112.3841  121.5664  112.3841
             -122.0405
 UPPER BOUND  130.0000  130.0000  180.0000  130.0000  180.0000  130.0000
              180.0000  130.0000  180.0000  130.0000  180.0000  130.0000
              180.0000  130.0000  180.0000  130.0000  180.0000  130.0000
              180.0000  130.0000  180.0000  130.0000  180.0000  130.0000
              180.0000  130.0000  180.0000  130.0000  180.0000  130.0000
              180.0000

 COST FNCT IS ENERGY (KCAL/MOLE)
 PENALTY FUNCTION FOR LOCATION OF BONDS AND DETAILED MINIMUM DISTANCE CHECK   
 IS INITIALIZED WITH TOLERANCE 0.200
 LOCATED BONDS BETWEEN THE FOLLOWING ATOMS:
    123456789012345678
  1 .*..*.**..........
  2 *.*.....**........
  3 .*.*......**......
  4 ..*..*......**....
  5 *....*........**..
  6 ...**...........**
  7 *.................
  8 *.................
  9 .*................
  0 .*................
  1 ..*...............
  2 ..*...............
  3 ...*..............
  4 ...*..............
  5 ....*.............
  6 ....*.............
  7 .....*............
  8 .....*............

 VISIT OF THE PES BY SIMULATED ANNEALING
 ---------------------------------------
 CALL No 1001 ,   CURRENT COST FNCT =  567.6646   BEST COST FNCT =  567.6646
                  EXPECTATION VALUE =  581.5595 + -    1.96688

 * * * SYSTEM LOOKS FROZEN AT TEMPERATURE   52.4288 KCAL
 NORMAL END AFTER  1669 CALLS TO COST FNCT
 ELAPSED TIME IN ANNEALING     15.63 SECONDS

 CLUSTERING ANALYSIS OF CONFORMATIONS
 ------------------------------------
 KEEP  50 CONFORMATIONS FROM 143 SELECTED AT PES VISIT

 QUENCHING OF REMAINING CONFORMATIONS
 ------------------------------------
 PERFORMED WITHOUT PERIODIC BOUNDARIES.
 PERFORMED WITHOUT PENALTIES ON INTERATOMIC DISTANCES.
 MINIMIZE ENERGY BY TRUST ALGORITHM

 QUENCHING No  1 -- STAGE 1 OF 1  (   0.26 SECONDS)
 ..................................................
          HEAT OF FORMATION       =   -35.291019 KCAL
          RMS GRADIENT NORM       =     0.014553 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE  
 CURRENT COORDINATES (ANGSTROMS, DEGREES)
  112.2135  111.5561  -29.9339  111.5566  -30.1473  111.5698   62.7471  109.5467
   90.5951  109.6175 -151.7181  109.7046  121.2831  108.9798 -121.7052  109.5453
   90.8041  109.6139 -151.5097  109.5231  184.5562  108.8885  -58.2953  106.5929
   51.2454  137.6832 -166.3417  109.6382  121.3912  108.8950 -121.7915

 QUENCHING No  2 -- STAGE 1 OF 1  (   0.11 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No  3 -- STAGE 1 OF 1  (   0.22 SECONDS)
 ..................................................
          HEAT OF FORMATION       =   -38.461440 KCAL
          RMS GRADIENT NORM       =     0.027765 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE
 CURRENT COORDINATES (ANGSTROMS, DEGREES)
  111.1652  111.1618   55.4025  111.1648  -55.4064  111.1534   55.4316  109.4195
   65.5812  109.5284 -176.5601  109.5471  121.2034  109.4059 -120.9430  109.4169
  -65.5654  109.5366  176.5931  109.4462  183.0731  109.3099   65.4931   93.0307
  109.4213  143.7543 -124.5778  109.4443 -121.3166  109.3243  121.0916

 QUENCHING No  4 -- STAGE 1 OF 1  (   0.11 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No  5 -- STAGE 1 OF 1  (   0.10 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No  6 -- STAGE 1 OF 1  (   0.19 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No  7 -- STAGE 1 OF 1  (   0.19 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No  8 -- STAGE 1 OF 1  (   0.12 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No  9 -- STAGE 1 OF 1  (   0.17 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 10 -- STAGE 1 OF 1  (   0.17 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 11 -- STAGE 1 OF 1  (   0.26 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 12 -- STAGE 1 OF 1  (   0.10 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 13 -- STAGE 1 OF 1  (   0.19 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 14 -- STAGE 1 OF 1  (   0.20 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 15 -- STAGE 1 OF 1  (   0.10 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 16 -- STAGE 1 OF 1  (   0.26 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 17 -- STAGE 1 OF 1  (   0.10 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 18 -- STAGE 1 OF 1  (   0.49 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 19 -- STAGE 1 OF 1  (   0.44 SECONDS)
 ..................................................
          HEAT OF FORMATION       =   -35.290916 KCAL
          RMS GRADIENT NORM       =     0.022375 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE
 CURRENT COORDINATES (ANGSTROMS, DEGREES)
  112.2185  111.5268   30.1417  111.5546   29.9627  111.5701  -62.7260  109.6190
  151.5200  109.5526  -90.7795  108.9714  121.6817  109.7084 -121.3023  109.5568
  -90.5920  109.6125 -208.2792  109.5288  175.3664  108.8972   58.2197  222.2632
  -13.5652  106.5276  -51.2156  109.6361 -121.4127  108.8905  121.7647

 QUENCHING No 20 -- STAGE 1 OF 1  (   0.17 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 21 -- STAGE 1 OF 1  (   0.10 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 22 -- STAGE 1 OF 1  (   0.22 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 23 -- STAGE 1 OF 1  (   0.17 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 24 -- STAGE 1 OF 1  (   0.10 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 25 -- STAGE 1 OF 1  (   0.41 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 26 -- STAGE 1 OF 1  (   0.18 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 27 -- STAGE 1 OF 1  (   0.17 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 28 -- STAGE 1 OF 1  (   0.11 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 29 -- STAGE 1 OF 1  (   0.26 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 30 -- STAGE 1 OF 1  (   0.39 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 31 -- STAGE 1 OF 1  (   0.10 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 32 -- STAGE 1 OF 1  (   0.39 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 33 -- STAGE 1 OF 1  (   0.11 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 34 -- STAGE 1 OF 1  (   0.23 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 35 -- STAGE 1 OF 1  (   0.25 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 36 -- STAGE 1 OF 1  (   0.10 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 37 -- STAGE 1 OF 1  (   0.23 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No 38 -- STAGE 1 OF 1  (   0.55 SECONDS)
 ..................................................
          HEAT OF FORMATION       =    77.923830 KCAL
          RMS GRADIENT NORM       =     0.030712 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE  
 CURRENT COORDINATES (ANGSTROMS, DEGREES)
  107.6003  107.6171   19.2332  107.5709  -15.2978  118.2005 -131.3400  110.1744
  104.2763  110.7221 -138.1885  110.8901 -238.5482  110.1123  239.9983  110.1326
 -100.7692  110.8740 -219.3599  111.3016 -134.5634  110.9888  106.2269   91.3215
  101.0059  101.5752 -102.1159  116.8721 -170.7099  116.8498   43.4452

 QUENCHING No 39 -- STAGE 1 OF 1  (   0.10 SECONDS)
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 QUENCHING No 47 -- STAGE 1 OF 1  (   0.55 SECONDS)
 ..................................................
          HEAT OF FORMATION       =    68.531275 KCAL
          RMS GRADIENT NORM       =     0.032605 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE
 CURRENT COORDINATES (ANGSTROMS, DEGREES)
  112.1005  116.1529   91.4820  111.4881   49.8991  109.7959  -50.5417  109.4796
  171.5176  109.1078  -70.8095  108.7359  120.9738  109.3089 -121.5175  103.6287
  -24.1757  115.2368 -131.9291  119.5447  -23.9678  119.2644  164.2975  129.9700
  182.7795  122.2388    0.1856  113.8764 -221.3852  113.5994  -91.5337

 QUENCHING No 48 -- STAGE 1 OF 1  (   0.19 SECONDS)
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 QUENCHING No 50 -- STAGE 1 OF 1  (   0.19 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.
 TIME CONSUMPTION OF STAGE 1:     11.09 SECONDS
 ON   5 POINTS AT END OF STAGE 1,   4 FULFILL STATIONARITY & ENERGETIC REQUESTS.

 DETAILS ON NON EQUIVALENT CRITICAL POINTS FOUND (ENERGY SORTED)    
 ---------------------------------------------------------------

 CRITICAL POINT No  1
 ....................
          HEAT OF FORMATION       =   -38.461440 KCAL
          RMS GRADIENT NORM       =     0.027765 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE  
          MOLECULAR POINT GROUP   = C2H   0.100000
    ATOM    CHEMICAL   BOND LENGTH    BOND ANGLE    TWIST ANGLE
   NUMBER   SYMBOL     (ANGSTROMS)     (DEGREES)     (DEGREES)
    (I)                   NA:I          NB:NA:I      NC:NB:NA:I   NA  NB  NC
      1     C
      2     C          1.52088                                     1
      3     C          1.52088        111.16516 *                  2   1
      4     C          1.52088        111.16178 *    55.40252 *    3   2   1
      5     C          1.52088        111.16479 *   -55.40638 *    1   2   3
      6     C          1.52088        111.15339 *    55.43160 *    5   1   2
      7     H          1.12120        109.41951 *    65.58115 *    1   2   3
      8     H          1.12120        109.52841 *  -176.56006 *    1   2   3
      9     H          1.12120        109.54706 *   121.20342 *    2   3   1
     10     H          1.12120        109.40592 *  -120.94302 *    2   3   1
     11     H          1.12120        109.41688 *   -65.56539 *    3   2   1
     12     H          1.12120        109.53659 *   176.59310 *    3   2   1
     13     H          1.12120        109.44616 *   183.07313 *    4   3   2
     14     H          1.12120        109.30986 *    65.49311 *    4   3   2
     15     H          1.12120         93.03071 *   109.42130 *    5   4   3
     16     H          1.12120        143.75433 *  -124.57782 *    5   4   3
     17     H          1.12120        109.44434 *  -121.31661 *    6   5   4
     18     H          1.12120        109.32427 *   121.09156 *    6   5   4


 CRITICAL POINT No  2
 ....................
          HEAT OF FORMATION       =   -35.291019 KCAL
          RMS GRADIENT NORM       =     0.014553 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE
          MOLECULAR POINT GROUP   = D2    0.100000
    ATOM    CHEMICAL   BOND LENGTH    BOND ANGLE    TWIST ANGLE
   NUMBER   SYMBOL     (ANGSTROMS)     (DEGREES)     (DEGREES)
    (I)                   NA:I          NB:NA:I      NC:NB:NA:I   NA  NB  NC
      1     C 
      2     C          1.52088                                     1
      3     C          1.52088        112.21354 *                  2   1
      4     C          1.52088        111.55609 *   -29.93386 *    3   2   1
      5     C          1.52088        111.55665 *   -30.14732 *    1   2   3
      6     C          1.52088        111.56980 *    62.74708 *    5   1   2
      7     H          1.12120        109.54668 *    90.59508 *    1   2   3
      8     H          1.12120        109.61746 *  -151.71809 *    1   2   3
      9     H          1.12120        109.70458 *   121.28312 *    2   3   1
     10     H          1.12120        108.97976 *  -121.70523 *    2   3   1
     11     H          1.12120        109.54531 *    90.80414 *    3   2   1
     12     H          1.12120        109.61389 *  -151.50969 *    3   2   1
     13     H          1.12120        109.52309 *   184.55617 *    4   3   2
     14     H          1.12120        108.88853 *   -58.29526 *    4   3   2
     15     H          1.12120        106.59292 *    51.24541 *    5   4   3
     16     H          1.12120        137.68315 *  -166.34172 *    5   4   3
     17     H          1.12120        109.63822 *   121.39123 *    6   5   4
     18     H          1.12120        108.89503 *  -121.79153 *    6   5   4
 

 CRITICAL POINT No  3
 ....................
          HEAT OF FORMATION       =    68.531275 KCAL
          RMS GRADIENT NORM       =     0.032605 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE  
          MOLECULAR POINT GROUP   = C1    0.100000
    ATOM    CHEMICAL   BOND LENGTH    BOND ANGLE    TWIST ANGLE
   NUMBER   SYMBOL     (ANGSTROMS)     (DEGREES)     (DEGREES)
    (I)                   NA:I          NB:NA:I      NC:NB:NA:I   NA  NB  NC
      1     C
      2     C          1.52088                                     1
      3     C          1.52088        112.10050 *                  2   1
      4     C          1.52088        116.15294 *    91.48197 *    3   2   1
      5     C          1.52088        111.48810 *    49.89909 *    1   2   3
      6     C          1.52088        109.79591 *   -50.54167 *    5   1   2
      7     H          1.12120        109.47960 *   171.51758 *    1   2   3
      8     H          1.12120        109.10777 *   -70.80952 *    1   2   3
      9     H          1.12120        108.73585 *   120.97384 *    2   3   1
     10     H          1.12120        109.30889 *  -121.51752 *    2   3   1
     11     H          1.12120        103.62867 *   -24.17571 *    3   2   1
     12     H          1.12120        115.23681 *  -131.92913 *    3   2   1
     13     H          1.12120        119.54473 *   -23.96778 *    4   3   2
     14     H          1.12120        119.26437 *   164.29753 *    4   3   2
     15     H          1.12120        129.96997 *   182.77949 *    5   4   3
     16     H          1.12120        122.23878 *     0.18555 *    5   4   3
     17     H          1.12120        113.87639 *  -221.38524 *    6   5   4
     18     H          1.12120        113.59937 *   -91.53374 *    6   5   4


 CRITICAL POINT No  4
 ....................
          HEAT OF FORMATION       =    77.923830 KCAL
          RMS GRADIENT NORM       =     0.030712 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE
          MOLECULAR POINT GROUP   = C1    0.100000
    ATOM    CHEMICAL   BOND LENGTH    BOND ANGLE    TWIST ANGLE
   NUMBER   SYMBOL     (ANGSTROMS)     (DEGREES)     (DEGREES)
    (I)                   NA:I          NB:NA:I      NC:NB:NA:I   NA  NB  NC
      1     C 
      2     C          1.52088                                     1
      3     C          1.52088        107.60027 *                  2   1
      4     C          1.52088        107.61705 *    19.23318 *    3   2   1
      5     C          1.52088        107.57085 *   -15.29778 *    1   2   3
      6     C          1.52088        118.20049 *  -131.34004 *    5   1   2
      7     H          1.12120        110.17437 *   104.27630 *    1   2   3
      8     H          1.12120        110.72206 *  -138.18845 *    1   2   3
      9     H          1.12120        110.89012 *  -238.54824 *    2   3   1
     10     H          1.12120        110.11230 *   239.99826 *    2   3   1
     11     H          1.12120        110.13264 *  -100.76920 *    3   2   1
     12     H          1.12120        110.87400 *  -219.35989 *    3   2   1
     13     H          1.12120        111.30158 *  -134.56335 *    4   3   2
     14     H          1.12120        110.98880 *   106.22688 *    4   3   2
     15     H          1.12120         91.32153 *   101.00591 *    5   4   3
     16     H          1.12120        101.57523 *  -102.11587 *    5   4   3
     17     H          1.12120        116.87210 *  -170.70986 *    6   5   4
     18     H          1.12120        116.84979 *    43.44519 *    6   5   4
 
 ELAPSED TIME IN QUENCHING     11.16 SECONDS
 CALLS TO COST FNCT IN QUENCHING:  973

 ENLARGED CLUSTERING ANALYSIS OF CRITICAL POINTS
 -----------------------------------------------
 TABLE OF NEARLY IDENTICAL CONFIGURATIONS AT THRESHOLD =  1.10 A     
 ENERGY     ,CONFIGURATIONS    (A "*" MEANS IDENTICAL)
                 2 4
    -38.4614   1*
    -35.2910   2.*
     68.5313   3..*
     77.9238   4...*

 NUMBER OF CONFIGURATIONS KEPT:   4

     FULL COMPUTATION TIME :     26.80 SECONDS
 Process Info: 26.8u 0.4s 0:28 97%

	This section lists the dedicated annealing keywords found by the program.
	Information on the methods and criteria that annealing will use is listed in this section of the output.
	This section lists the active penalty functions defined by the annealing keywords.
	The lower bounds, beginning values, and upper bounds for the optimizable parameters are summarized in this table. The user should carefully examine this table to ensure that the proper limits have been defined.
	For PEN2 or PEN2GRP, a table of located bonded atoms are noted here. This information details which bonds will be enforced during the course of the simulated annealing.
	These are the coordinates at the end of the annealing configuration search.
	This table describes the presorted configurations and eliminates identical members from consideration. In this case, the four final candidates are revealed as non-identical and so none are filtered out. The threshold for filtering is controlled by FILTER (default=1.0) and occurs after each major phase of simulated annealing. A smaller value of FILTER can be used to minimize the chance of missing a minima, but this noticeably increases the amount of time required in the quenching process.

Example II ("Rigid" System)

The following molecule (methyl-N-methyl-amide) has been selected for illustration of the annealing procedure in the case of a relatively rigid system:

Figure 12.1. Example of a Relatively Rigid System

As mentioned in “Strategies for Annealing Searches”, the general strategy to use is to divide the conformational search problem into primary and secondary degrees of freedom. Then a connectivity may be defined which allows an explicit description of the primary degrees of freedom. In this case, three primary degrees of freedom may be identified as the rotation about bonds C₁-N₃^A, C₄-N₃^B, and C₅-C₁^C. The following is a viable (but not necessarily unique) connectivity pattern for this system that produces explicit descriptions of the primary degrees of freedom. (Note that the superscripts A, B, and C above identify below which of the dihedrals are used to specify the rotations.)

 C      0.000000  0    0.000000  0    0.000000  0    0    0    0
 O      1.243077  1    0.000000  0    0.000000  0    1    0    0
 N      1.366966  1  121.946787  1    0.000000  0    1    2    0
 C      1.520000  1  120.615770  1   -0.000067^A 1    3    1    2
 C      1.540000  1  123.093733  1 -179.998834  1    1    2    3
 H      0.986177  1  121.170189  1  179.991444  1    3    1    4
 H      1.004440  1  108.958790  1    0.000000^B 1    4    3    1
 H      1.004440  1  108.958790  1  120.000000  1    4    3    7
 H      1.000000  1  109.885165  1 -120.000000  1    4    3    7
 H      1.004440  1  108.958790  1  180.000000^C 1    5    1    2
 H      1.004440  1  108.958790  1  120.000000  1    5    1   10
 H      1.000000  1  109.885165  1 -120.000000  1    5    1   10

As mentioned in “Geometry Specification”, this modular designation of the hydrogen atoms of a methyl group allows easy manipulation of the orientation of the entire group by altering one dihedral angle. In this example, the master degrees of freedom are defined by the dihedrals of particular atoms: the dihedral on H₇ for the C₄ methyl (C₄-N₃ rotation), the dihedral on H₁₀ for the C₅ methyl (C₅-C₁ rotation), and the dihedral on C₅ for the C₁-N₃ rotation. The LIMIT keyword can now be used to define boundaries for the search of conformational space. The secondary variables are not important to the annealing procedure and should be limited to about ±10% of their original value. (Note that all annealing boundary conditions are ignored during the local quenching of the located minima.) The definition of the problem in these terms requires some chemical knowledge and intuition, but saves enormous computational effort. The final input and results files for the problem is listed below.

Input File (`annealing/anneal_limit.dat`)

  am1 rhf singlet t=auto anneal truste filter=1.2 noxyz limit         
methyl-n-methylamide
Annealing on rigid system: LIMIT
 C              0.000000  0    0.000000  0    0.000000  0    0    0    0
 O              1.243077  1    0.000000  0    0.000000  0    1    0    0
 N              1.366966  1  121.946787  1    0.000000  0    1    2    0
 C              1.520000  1  120.615770  1    0.000000  1    3    1    2    
 C              1.540000  1  123.093733  1 -179.990000  1    1    2    3
 H              0.986177  1  121.170189  1  179.990000  1    3    1    4
 H              1.004440  1  108.958790  1  -60.000000  1    4    3    1    
 H              1.004440  1  108.958790  1  120.000000  1    4    3    7
 H              1.000000  1  109.885165  1 -120.000000  1    4    3    7
 H              1.004440  1  108.958790  1  120.000000  1    5    1    2    
 H              1.004440  1  108.958790  1  120.000000  1    5    1   10
 H              1.000000  1  109.885165  1 -120.000000  1    5    1   10
 0              0.000000  0    0.000000  0    0.000000  0    0    0    0
$$ limit - annealing boundaries   
  1.2  1.3 110   1.5 110 -180   1.5 100 -190   0.9 100  160   
  0.9 100 -180   0.9 100  100   0.9 100 -140   0.9 100 -180
  0.9 100  100   0.9 100 -140
  1.3  1.4 130   1.6 130  180   1.6 140 -170   1.1 130  200   
  1.1 130  180   1.1 130  140   1.1 130 -100   1.1 130  180
  1.1 130  140   1.1 130 -100
$$ end of extra data

	The keyword ANNEAL activates energy-based annealing. LIMIT is used to restrict the range the over which the geometry can vary. The FILTER=`n.n` keyword loosens the criteria (default =1.0) for elimination of a candidate geometry as already located. For situations such as distinguishing between methyl rotations this value should probably be raised to 1.4.
	The dihedrals on these lines comprise the three primary degrees of freedom.
	This is the extra input section marker for reaction path data. Note, that this marker can be shortened to "$$ limit". Details of these markers are found in “Extra Input Data”.
	These are the lower limits each geometric parameter can assume. Note that most are very close to their initial values but items # 6, 15, and 24 have a lower limit of -180.00°.
	These are the upper limits each geometric parameter can assume. As before, most are very close to their initial values but items # 6, 15, and 24 now have an upper limit of 180.00°. The range -180° to +180° represent a full circle rotation about the required bonds.

Archive File (`annealing/anneal_limit.arc`)

 Timestamp: 2004-02-12-14-31-21-0000028426-hpux

                     SUMMARY OF   AM1   CALCULATION
                                                       Feb-12-2004
                          AMPAC Version 8.13
                             Presented by:

                        Semichem, Inc.
                        PO Box 1649
                        Shawnee KS 66222
                        (913)268-3271
                        (913)268-3445 (fax)

 FORMULA: C3H7N1O1
 methyl-n-methylamide
 Annealing on rigid system: LIMIT

     GEOMETRY NORMALLY RETURNED BY SIMULATED ANNEALING        
     SCF FIELD WAS ACHIEVED

          FINAL HEAT OF FORMATION =     -47.271731 kcal   
                                  =    -197.832193 kJ
          ELECTRONIC ENERGY       =   -3372.727068 eV
          CORE-CORE REPULSION     =    2364.418633 eV
          TOTAL ENERGY            =   -1008.308435 eV
          GRADIENT NORM           =       0.149190
          RMS GRADIENT NORM       =       0.027238
          UNSTABLE MODE(S)        =       0 ( ESTIMATE  )
          DIPOLE                  =       3.512179 debyes
          NO. OF FILLED LEVELS    =      15 (OCC = 2)
          KOOPMAN IONISATION POTENTIAL =      9.92 eV
          MOLECULAR POINT GROUP   = C1    0.100000
          FINAL GEOMETRY OBTAINED                                 CHARGE
 AM1 RHF SINGLET T=AUTO ANNEAL TRUSTE FILTER=1.2 NOXYZ LIMIT
 methyl-n-methylamide
 Annealing on rigid system: LIMIT
  C     0.000000  0    0.000000  0    0.000000  0    0   0   0    0.3003
  O     1.247321  1    0.000000  0    0.000000  0    1   0   0   -0.3706
  N     1.381274  1  121.217766  1    0.000000  0    1   2   0   -0.3918
  C     1.426641  1  123.102360  1    1.456008  1    3   1   2   -0.0753
  C     1.509603  1  121.507009  1 -179.463187  1    1   2   3   -0.2429
  H     0.989976  1  119.068723  1  177.080008  1    3   1   4    0.2203
  H     1.123676  1  109.980498  1 -125.009286  1    4   3   1    0.0662
  H     1.123158  1  110.134607  1  120.186817  1    4   3   7    0.1180
  H     1.124048  1  110.341502  1 -119.620798  1    4   3   7    0.0633
  H     1.117236  1  108.665993  1   -3.459860  1    5   1   2    0.1176
  H     1.116524  1  110.290598  1  119.360944  1    5   1  10    0.0990
  H     1.116291  1  110.808171  1 -119.673319  1    5   1  10    0.0959
  0     0.000000  0    0.000000  0    0.000000  0    0   0   0
 Timestamp: 2004-02-12-14-31-21-0000028426-hpux

                     SUMMARY OF   AM1   CALCULATION
                                                       Feb-12-2004
                          AMPAC Version 8.13
                             Presented by:
  
                        Semichem, Inc.
                        PO Box 1649
                        Shawnee KS 66222
                        (913)268-3271
                        (913)268-3445 (fax)
  
 FORMULA: C3H7N1O1
 methyl-n-methylamide
 Annealing on rigid system: LIMIT

     GEOMETRY NORMALLY RETURNED BY SIMULATED ANNEALING
     SCF FIELD WAS ACHIEVED

          FINAL HEAT OF FORMATION =     -47.128642 kcal   
                                  =    -197.233366 kJ
          ELECTRONIC ENERGY       =   -3381.308377 eV
          CORE-CORE REPULSION     =    2373.006147 eV
          TOTAL ENERGY            =   -1008.302230 eV
          GRADIENT NORM           =       0.055038
          RMS GRADIENT NORM       =       0.010048
          UNSTABLE MODE(S)        =       0 ( ESTIMATE  )
          DIPOLE                  =       3.814106 debyes
          NO. OF FILLED LEVELS    =      15 (OCC = 2)
          KOOPMAN IONISATION POTENTIAL =      9.97 eV
          MOLECULAR POINT GROUP   = C1    0.100000
          FINAL GEOMETRY OBTAINED                                 CHARGE
 AM1 RHF SINGLET T=AUTO ANNEAL TRUSTE FILTER=1.2 NOXYZ LIMIT
 methyl-n-methylamide
 Annealing on rigid system: LIMIT
  C     0.000000  0    0.000000  0    0.000000  0    0   0   0    0.2992
  O     1.247966  1    0.000000  0    0.000000  0    1   0   0   -0.3725
  N     1.382298  1  118.897215  1    0.000000  0    1   2   0   -0.3919
  C     1.423936  1  123.418899  1  175.124633  1    3   1   2   -0.0688
  C     1.505861  1  121.882577  1 -178.940876  1    1   2   3   -0.2366
  H     0.993545  1  117.680890  1 -171.295993  1    3   1   4    0.2335
  H     1.125283  1  111.348601  1  -71.819842  1    4   3   1    0.0620
  H     1.123797  1  110.520694  1  120.183130  1    4   3   7    0.0760
  H     1.122056  1  109.513669  1 -120.077821  1    4   3   7    0.0856
  H     1.116618  1  110.802688  1 -126.904319  1    5   1   2    0.0961
  H     1.117198  1  108.782930  1  120.067874  1    5   1  10    0.1159
  H     1.117099  1  110.013179  1 -120.603896  1    5   1  10    0.1015
  0     0.000000  0    0.000000  0    0.000000  0    0   0   0

This is the data relating to the first minimum located by annealing, and is a refined version of the input structure pictured in Figure 12.2. This structure happens to be the global minimum on the PES and is referred to as the "trans-minimum."

Figure 12.2. Trans-minimum

This is the data relating to the second minimum located by annealing. It is referred to as the "cis-minimum" (Figure 12.3).

Figure 12.3. Cis-minimum

Output File (`annealing/anneal_limit.out`)

 Timestamp: 2004-02-12-14-31-21-0000028426-hpux
 *******************************************************************************
                             AM1 CALCULATION RESULTS
 *******************************************************************************
 *                             AMPAC Version 8.13
 *                                Presented by:
 *
 *                           Semichem, Inc.
 *                           PO Box 1649
 *                           Shawnee KS 66222
 *                           (913)268-3271
 *                           (913)268-3445 (fax)
 *
 *  ANNE     - SIMULATED ANNEALING ON ENERGY ONLY
 *  TRUSTE   - MINIMISE ENERGY USING TRUST REGION
 *  T=AUTO   - AUTOMATIC DETERMINATION OF ALLOWED TIME
 *  NOXYZ    - CARTESIAN COORDINATES NOT TO BE PRINTED
 *  SINGLET  - IS THE REQUIRED SPIN MULTIPLICITY
 *  AM1      - THE AM1 HAMILTONIAN TO BE USED
 *******************************************************************************
 AM1 RHF SINGLET T=AUTO ANNEAL TRUSTE FILTER=1.2 NOXYZ LIMIT
 methyl-n-methylamide
 Annealing on rigid system: LIMIT
    ATOM    CHEMICAL   BOND LENGTH    BOND ANGLE    TWIST ANGLE
   NUMBER   SYMBOL     (ANGSTROMS)     (DEGREES)     (DEGREES)
    (I)                   NA:I          NB:NA:I      NC:NB:NA:I   NA  NB  NC
      1     C
      2     O          1.24308 *                                   1
      3     N          1.36697 *      121.94679 *                  1   2
      4     C          1.52000 *      120.61577 *     0.00000 *    3   1   2
      5     C          1.54000 *      123.09373 *  -179.99000 *    1   2   3
      6     H          0.98618 *      121.17019 *   179.99000 *    3   1   4
      7     H          1.00444 *      108.95879 *   -60.00000 *    4   3   1
      8     H          1.00444 *      108.95879 *   120.00000 *    4   3   7
      9     H          1.00000 *      109.88517 *  -120.00000 *    4   3   7
     10     H          1.00444 *      108.95879 *   120.00000 *    5   1   2
     11     H          1.00444 *      108.95879 *   120.00000 *    5   1  10
     12     H          1.00000 *      109.88517 *  -120.00000 *    5   1  10

   MOLECULAR POINT GROUP            SYMMETRY CRITERIA
            CS                          0.10000000

          SINGLET STATE CALCULATION

        **  REFERENCES TO PARAMETERS  **

  H  (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)
  C  (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)
  N  (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)
  O  (AM1): M.J.S. DEWAR ET AL, J. AM. CHEM. SOC. 107 3902-3909 (1985)

 *******************************************************************************
 *  KEYWORDS DEDICATED TO SIMULATED ANNEALING SECTION
 *
 *  LIMIT    - EXPLICIT PERIODIC CONDITIONS GIVEN
 *  FILTER=  - SET THRESHOLD OF DISTINGUISHED CONFIGURATION TO 1.200
 *******************************************************************************
 STANDARD DEVIATION ON ENERGY   (KCAL)       0.00000055521
 STANDARD DEVIATION ON GRADIENT (KCAL/A,RD,RD)  0.00000720 0.00001370 0.00000962

 SIMULATED ANNEALING BY METROPOLIS SCHEME IN   30 VARIABLES
     VERSION 2.0 (MAY 2000)
     THERMALIZATION THRESHOLD ON COST FNCT          0.080
     MAXIMUM STEP SIZE                            2.00000
     CONVERGENCE CRITERION ON STEP SIZE           0.06325

 MAXIMUM NUMBER OF STEPS AT EACH TEMPERATURE         1000
     FIRST TEMPERATURE (HOT) SET TO                 50.00
     TEMPERATURE EXPONENTIAL DECAY                   0.80
     LAST TEMPERATURE (COLD) SET TO                  0.50
 RANDOM SEQUENCE INITIATOR                       -9876543
     MARKOV CHAIN STUDIED BY PIECES OF                 20
     No OF TEMPERATURES TO DETECT A FROZEN SYSTEM       2
 EQUIVALENCE THRESHOLD IN CLUSTERING ANALYSIS        1.20
     BAND-PASS FILTER NO 1 CENTERED AT               1.40 ANGSTROMS
     HALF BAND-WIDTH                                13.50 %
 PERIODIC BOUNDARY CONDITIONS (ANGSTROMS, DEGREES)
 LOWER BOUND    1.2000    1.3000  110.0000    1.5000  110.0000 -180.0000  
                1.5000  100.0000 -190.0000    0.9000  100.0000  160.0000
                0.9000  100.0000 -180.0000    0.9000  100.0000  100.0000
                0.9000  100.0000 -140.0000    0.9000  100.0000 -180.0000
                0.9000  100.0000  100.0000    0.9000  100.0000 -140.0000
 TRIAL COORD    1.2431    1.3670  121.9468    1.5200  120.6158    0.0000
                1.5400  123.0937 -179.9900    0.9862  121.1702  179.9900
                1.0044  108.9588  -60.0000    1.0044  108.9588  120.0000
                1.0000  109.8852 -120.0000    1.0044  108.9588  120.0000
                1.0044  108.9588  120.0000    1.0000  109.8852 -120.0000
 UPPER BOUND    1.3000    1.4000  130.0000    1.6000  130.0000  180.0000
                1.6000  140.0000 -170.0000    1.1000  130.0000  200.0000
                1.1000  130.0000  180.0000    1.1000  130.0000  140.0000
                1.1000  130.0000 -100.0000    1.1000  130.0000  180.0000
                1.1000  130.0000  140.0000    1.1000  130.0000 -100.0000

 COST FNCT IS ENERGY (KCAL/MOLE)

 VISIT OF THE PES BY SIMULATED ANNEALING
 ---------------------------------------

 * * * SYSTEM LOOKS FROZEN AT TEMPERATURE    8.3886 KCAL
 NORMAL END AFTER  1131 CALLS TO COST FNCT
 ELAPSED TIME IN ANNEALING      6.98 SECONDS

 CLUSTERING ANALYSIS OF CONFORMATIONS
 ------------------------------------
 KEEP   4 CONFORMATIONS FROM  26 SELECTED AT PES VISIT

 QUENCHING OF REMAINING CONFORMATIONS
 ------------------------------------
 PERFORMED WITHOUT PERIODIC BOUNDARIES.
 MINIMIZE ENERGY BY TRUST ALGORITHM

 QUENCHING No  1 -- STAGE 1 OF 1  (   0.14 SECONDS)
 ..................................................
          HEAT OF FORMATION       =   -47.271731 KCAL
          RMS GRADIENT NORM       =     0.027239 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE
 CURRENT COORDINATES (ANGSTROMS, DEGREES)
    1.2473    1.3813  121.2178    1.4266  123.1024    1.4560    1.5096  121.5070
 -179.4632    0.9900  119.0687  177.0800    1.1237  109.9805 -125.0093    1.1232
  110.1346  120.1868    1.1240  110.3415 -119.6208    1.1172  108.6660   -3.4599
    1.1165  110.2906  119.3609    1.1163  110.8082 -119.6733

 QUENCHING No  2 -- STAGE 1 OF 1  (   0.13 SECONDS)
 ..................................................
          HEAT OF FORMATION       =   -47.128642 KCAL
          RMS GRADIENT NORM       =     0.010052 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE
 CURRENT COORDINATES (ANGSTROMS, DEGREES)
    1.2480    1.3823  118.8972    1.4239  123.4189  175.1246    1.5059  121.8826
 -178.9409    0.9935  117.6809  188.7040    1.1253  111.3486  -71.8198    1.1238
  110.5207  120.1831    1.1221  109.5137 -120.0778    1.1166  110.8027 -126.9043
    1.1172  108.7829  120.0679    1.1171  110.0132 -120.6039

 QUENCHING No  3 -- STAGE 1 OF 1  (   0.08 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.

 QUENCHING No  4 -- STAGE 1 OF 1  (   0.09 SECONDS)
 ..................................................
          * * * QUENCHING INTERRUPTED: POINT PREVIOUSLY FOUND.
 TIME CONSUMPTION OF STAGE 1:      0.44 SECONDS
 ON   2 POINTS AT END OF STAGE 1,   2 FULFILL STATIONARITY & ENERGETIC REQUESTS.

 DETAILS ON NON EQUIVALENT CRITICAL POINTS FOUND (ENERGY SORTED)   
 ---------------------------------------------------------------

 CRITICAL POINT No  1
 ....................
          HEAT OF FORMATION       =   -47.271731 KCAL   
          RMS GRADIENT NORM       =     0.027239 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE
          MOLECULAR POINT GROUP   = C1    0.100000
    ATOM    CHEMICAL   BOND LENGTH    BOND ANGLE    TWIST ANGLE
   NUMBER   SYMBOL     (ANGSTROMS)     (DEGREES)     (DEGREES)
    (I)                   NA:I          NB:NA:I      NC:NB:NA:I   NA  NB  NC
      1     C
      2     O          1.24732 *                                   1
      3     N          1.38127 *      121.21777 *                  1   2
      4     C          1.42664 *      123.10236 *     1.45601 *    3   1   2
      5     C          1.50960 *      121.50701 *  -179.46319 *    1   2   3
      6     H          0.98998 *      119.06872 *   177.08001 *    3   1   4
      7     H          1.12368 *      109.98050 *  -125.00929 *    4   3   1
      8     H          1.12316 *      110.13461 *   120.18682 *    4   3   7
      9     H          1.12405 *      110.34150 *  -119.62080 *    4   3   7
     10     H          1.11724 *      108.66599 *    -3.45986 *    5   1   2
     11     H          1.11652 *      110.29060 *   119.36094 *    5   1  10
     12     H          1.11629 *      110.80817 *  -119.67332 *    5   1  10


 CRITICAL POINT No  2
 ....................
          HEAT OF FORMATION       =   -47.128642 KCAL  
          RMS GRADIENT NORM       =     0.010052 KCAL/ANGSTROMS
          CRITICAL POINT INDEX    =            0  ESTIMATE
          MOLECULAR POINT GROUP   = C1    0.100000
    ATOM    CHEMICAL   BOND LENGTH    BOND ANGLE    TWIST ANGLE
   NUMBER   SYMBOL     (ANGSTROMS)     (DEGREES)     (DEGREES)
    (I)                   NA:I          NB:NA:I      NC:NB:NA:I   NA  NB  NC
      1     C
      2     O          1.24797 *                                   1
      3     N          1.38230 *      118.89722 *                  1   2
      4     C          1.42394 *      123.41890 *   175.12463 *    3   1   2
      5     C          1.50586 *      121.88258 *  -178.94088 *    1   2   3
      6     H          0.99354 *      117.68089 *   188.70401 *    3   1   4
      7     H          1.12528 *      111.34860 *   -71.81984 *    4   3   1
      8     H          1.12380 *      110.52069 *   120.18313 *    4   3   7
      9     H          1.12206 *      109.51367 *  -120.07782 *    4   3   7
     10     H          1.11662 *      110.80269 *  -126.90432 *    5   1   2
     11     H          1.11720 *      108.78293 *   120.06787 *    5   1  10
     12     H          1.11710 *      110.01318 *  -120.60390 *    5   1  10

 ELAPSED TIME IN QUENCHING      0.47 SECONDS
 CALLS TO COST FNCT IN QUENCHING:   82

 ENLARGED CLUSTERING ANALYSIS OF CRITICAL POINTS   
 -----------------------------------------------
 TABLE OF NEARLY IDENTICAL CONFIGURATIONS AT THRESHOLD =  1.32 A
 ENERGY     ,CONFIGURATIONS    (A "*" MEANS IDENTICAL)
                 2
    -47.2717   1*
    -47.1286   2.*

 NUMBER OF CONFIGURATIONS KEPT:   2

     FULL COMPUTATION TIME :      7.46 SECONDS
 Process Info: 7.6u 0.4s 0:09 88%

	These are the upper and lower boundaries along with the initial values for each coordinate.
	These are the coordinates at the end of the annealing quenching phase.
	This is the trans-minimum geometry.
	This is the cis-minimum geometry.
	This table describes the presorted configurations and eliminates identical members from consideration. The threshold for filtering is controlled by FILTER (default=1.0) and occurs after each major phase of simulated annealing. A smaller value of FILTER can be used to minimize the chance of missing a minimum, but this noticeably increases the amount of time required in the quenching process.

^[24] S. Kirkpatrick, C.D. Gelatt Jr., M.P. Vecchi. Science 1983, 220, 671.

^[25] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller. J. Chem. Phys. 1953, 21, 287.

^[26] F. Bockisch, D. Liotard, J.-C. Rayez, B. Duguay. Int. J. Quantum Chem. 1992, 44, 619.


Polarizability Methods		COSMO Solvation Model
Copyright © 1994, 1997, 2004, Semichem, Inc. All rights reserved.

12

Simulated Annealing

Introduction

Simulated Annealing Theory

Strategies for Annealing Searches

Simulated Annealing

Filtering

Quenching

Phase 1

Phase 2

Phase 3

Annealing Methods

ANNEAL (ANNE)

TSANNEAL (TSANN)

GANNEAL (GANN)

MANNEAL (MANN)

Penalty Functions

Annealing Strategies

Simulated Annealing Dedicated Keywords

Example I ("Floppy" system)

Input File (annealing/anneal1.dat)

Archive File (annealing/anneal1.arc)

Output File (annealing/anneal1.out)

Example II ("Rigid" System)

Input File (annealing/anneal_limit.dat)

Archive File (annealing/anneal_limit.arc)

Output File (annealing/anneal_limit.out)

Input File (`annealing/anneal1.dat`)

Archive File (`annealing/anneal1.arc`)

Output File (`annealing/anneal1.out`)

Input File (`annealing/anneal_limit.dat`)

Archive File (`annealing/anneal_limit.arc`)

Output File (`annealing/anneal_limit.out`)