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minimization_of_functions [2020/06/15 15:32] – add min_intersect.inp example mayminimization_of_functions [2024/01/08 13:24] (current) – external edit 127.0.0.1
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 +====== Minimization of functions ======
  
 +The minimization of general functions of one or more variables can be carried out using the command:
 +
 +''MINIMIZE'', //func, x$_1$//[, //x$_2$, x$_3$, …//]
 +
 +where //func// represents a function of up to 50 variables //x$_1$, x$_2$, …//. Two different optimization methods can be selected as described below which do or do not use numerical derivative information.
 +
 +The optimization method, as well as finer control over //func//, can be chosen using the ''METHOD'' directive
 +
 +''METHOD'', //key// [, //key1=value, key2=value, …//] [minimize:method]
 +
 +where //key// defines the optimization method. Valid options for //key// are:
 +
 +  * **''BFGS''** Broyden-Fletcher-Goldfarb-Shanno conjugate gradient method, which uses numerical gradients (default)
 +  * **''SIMPLEX''** Downhill simplex method, which uses only function evaluations
 +
 +Options to these methods, //key1, key2, …//, are:
 +
 +  * **''VARSCALE''=//vscale//** Optimization in space of scaled variables.\\
 +//vscale=0// no scaling (not recommended)\\
 +//vscale=1// optimization in the space of ln($x$)\\
 +//vscale=2// optimization in space of initial value scaling, e.g., $x_1/x_{1i}$ (default)
 +  * **''THRESH''=//thresh//** Required accuracy of either the gradient (BFGS) or parameters (SIMPLEX). The default is $1 \cdot 10^{-4}$ for BFGS and $1 \cdot 10^{-2}$ for SIMPLEX. Note that previously this pertained to the function value in the Simplex case.
 +  * **''VSTEP''=//epsd//** Step size for numerical gradients (BFGS) or initial SIMPLEX vertices
 +  * **''PROC''=//procname//** Specifies the procedure to be executed in each optimization step. This defines a complete function evaluation (if needed, numerical gradients will be evaluated using this procedure as well)
 +  * **''STARTCMD''=//command//** Specifies a start command. In each optimization step all input beginning with //command// to the current ''MINIMIZE'' is processed.
 +
 +Miscellaneous directives (separated by semicolons or linebreaks)
 +
 +  * **''MAXIT'',//maxit//** maximum number of optimization cycles. The default is 30 for BFGS and 100 for SIMPLEX.
 +
 +===== Examples =====
 +
 +==== Geometry optimization ====
 +
 +<code - examples/min_optgeo.inp>
 +***, Simple geometry optimization
 +
 +basis=vdz
 +
 +geometry={
 +O
 +H 1 r
 +H 1 r 2 theta}
 +
 +r=1.8
 +theta=104
 +
 +hf
 +mp2
 +
 +{minimize,energy,r,theta}
 +---
 +</code>
 +
 +==== Basis function optimizations ====
 +
 +<code - examples/basisopt_simple.inp>
 +***, Optimization of 2 d functions
 +
 +geometry={Ne}
 +
 +dexp=[2.0,1.0]
 +
 +basis={
 +sp,Ne,vdz;c;
 +d,Ne,dexp(1),dexp(2)
 +}
 +
 +hf
 +mp2
 +eval=energy
 +
 +minimize,eval,dexp(1),dexp(2)
 +---
 +</code>
 +
 +<code - examples/basisopt_proc.inp>
 +***, Optimization of 2 d functions
 +
 +geometry={Ne}
 +
 +dexp=[2.0,1.0]
 +
 +{minimize,eval,dexp(1),dexp(2)
 +method,bfgs,varscale=1,thresh=1e-5,proc=optd}
 +
 +
 +proc optd
 +
 +basis={
 +sp,Ne,vdz;c;
 +d,Ne,dexp(1),dexp(2)
 +}
 +
 +hf
 +mp2
 +eval=energy
 +
 +endproc
 +</code>
 +
 +<code - examples/basisopt_cv.inp>
 +***, MP2 optimization of core-valence cc-pCVDZ functions
 +
 +geometry={Ne}
 +
 +sexp=20.
 +pexp=30.
 +
 +{minimize,ecv,sexp,pexp
 +method,bfgs,varscale=1,thresh=1e-5,proc=myopt}
 +
 +
 +proc myopt
 +
 +basis={
 +spd,Ne,vdz;c;
 +s,Ne,sexp
 +p,Ne,pexp
 +}
 +
 +hf
 +{mp2;core,1}
 +eval=energy
 +
 +{mp2;core,0}
 +eall=energy
 +
 +ecv=eall-eval
 +
 +endproc
 +</code>
 +
 +==== Intersection points ====
 +
 +<code - examples/min_intersect.inp>
 +***, Try to find intersection seam of singlet-triplet CH2 as function of angle
 +
 +basis=vdz
 +
 +geometry={
 +C
 +H 1 r
 +H 1 r 2 theta}
 +
 +r=1.6
 +angles=[70,75,80,85,90]
 +
 +do i=1,#angles
 +
 + theta=angles(i)
 +
 + {minimize,ediff,r
 + method,bfgs,thresh=1e-5,proc=findit}
 +
 + ropt(i)=r
 + eseam(i)=etriplet
 + converge(i)=ediff
 +
 +enddo
 +
 +table,angles,ropt,eseam,converge
 +digit,0,3,5,6
 +
 +
 +proc findit
 +
 + {hf;wf,8,2,2}
 + etriplet=energy
 +
 + {hf;wf,8,1,0}
 + esinglet=energy
 +
 + ediff=(etriplet-esinglet)**2
 +
 +endproc
 +---
 +</code>