# *OPTIMIZE¶

Geometry optimization directives.

This section controls the geometry optimization. The geometry optimization algorithm is based on the one of Dalton. The directives are therefore similar except for these few changes:

• No second-order algorithms are available since the molecular Hessian is not implemented.
• A few new keywords have been introduced (see below).

If **PROPERTIES or **ANALYZE is specified in the job input section together with *OPTIMIZE, then the property or analysis module is called in each optimization iteration and at the converged geometry.

Depending on convergence criteria for energies and gradients the values of the thresholds may be automatically adjusted.

## .NO SKIP¶

Don’t skip SSLL or SSSS gradient contributions when small. This keyword forces the LS and SS two-electron gradient to always be evaluated in all geometry iterations (depending on the integral flag).

The SL and SS two-electron integral contributions to the gradient are normally skipped if their contribution is estimated to be small. An “empirical” estimate of the norms of the LS and SS two-electron gradients based on the norm of the LL two-electron gradient. This trick is by default activated in a geometry optimization, since when the current geometry is far away from the equilibrium, e.g. the norm of the gradient is, say, 1.0, then there is no need to calculate the LS and/or SS two-electron gradient because they have a norm of, say, 0.001.

## .NUMGRA¶

Force the use of numerical gradient in geometry optimization.

## .TWOGRD¶

Include LL, SL, and SS integral contributions to the gradient (1 = on; 0 = off). The default is to turn all on:

.TWOGRD
1 1 1

## .BAKER¶

Baker’s convergence criteria [Baker1993] will be used. The minimum is then said to be found when the largest element of the gradient vector (in Cartesian or redundant internal coordinates) falls below 3.0D-4 and either the energy change from the last iteration is less than 1.0D-6 or the largest element of the predicted step vector is less 3.0D-4.

Convergence threshold for the gradient. The default is 1.0D-5.

## .ENERGY¶

Convergence threshold for the energy. The default is 1.0D-6.

## .STEP THRESHOLD¶

Convergence threshold for the step. The default is 1.0D-5.

## .PRINT¶

Set print level for this module. Read one more line containing print level. Default value is 0, any value higher than 12 gives debugging level output.

## .MAX IT¶

Read the maximum number of geometry iterations. The default value is 25.

## .TRUSTR¶

Set initial trust radius for calculation. This will also be the maximum step length for the first iteration. The trust radius is updated after each iteration depending on the ratio between predicted and actual energy change. The default trust radius is 0.5 a.u.

## .TR FAC¶

READ(LUCMD,*) TRSTIN, TRSTDE

Read two factors that will be used for increasing and decreasing the trust radius respectively. Default values are 1.2 and 0.7.

## .TR LIM¶

READ(LUCMD,*) RTENBD, RTENGD, RTRJMN, RTRJMX

Read four limits for the ratio between the actual and predicted energies. This ratio indicates how good the step is?that is, how accurately the quadratic model describes the true energy surface. If the ratio is below RTRJMN or above RTRJMX, the step is rejected. With a ratio between RTRJMN and RTENBD, the step is considered bad an the trust radius decreased to less than the step length. Ratios between RTENBD and RTENGD are considered satisfactory, the trust radius is set equal to the norm of the step. Finally ratios above RTENGD (but below RTRJMX) indicate a good step, and the trust radius is given a value larger than the step length. The amount the trust radius is increased or decreased can be adjusted with .TR FAC. The default values of RTENBD, RTENGD, RTRJMN and RTRJMX are 0.4, 0.8, -0.1 and 3.0 respectively.

## .MAX RE¶

READ(LUCMD,*) MAXREJ

Read maximum number of rejected steps in each iterations, default is 3.

## .NOTRUS¶

Turns off the trust radius, so that a full Newton step is taken in each iteration. This should be used with caution, as global convergence is no longer guaranteed. If long steps are desired, it is safer to adjust the initial trust radius and the limits for the actual/predicted energy ratio.

## .CONDIT¶

READ (LUCMD,*) ICONDI

Set the number of convergence criteria that should be fulfilled before convergence occurs. There are three different convergence thresholds, one for the energy, one for the gradient norm and one for the step norm. The possible values for this variable is therefore between 1 and 3. Default is 2. The three convergence thresholds can be adjusted with the keywords .ENERGY, .GRADIENT and .STEP THRESHOLD.

## .NOBREA¶

Disables breaking of symmetry. The geometry will be optimized within the given symmetry, even if a non-zero molecular Hessian index is found. The default is to let the symmetry be broken until a minimum is found with a molecular Hessian index of zero. This option only has effect when second-order methods are used.

## .SP BAS¶

READ(LUCMD,*) SPBSTX

Read a string containing the name of a basis set. When the geometry has converged, a single-point energy will be calculated using this basis set.

## .PREOPT¶

READ (LUCMD,*) NUMPRE
DO I = 1, NUMPRE
END DO

First we read the number of basis sets that should be used for preoptimization, then we read those basis set names as strings. These sets will be used for optimization in the order they appear in the input. One should therefore place the smaller basis at the top. After the preoptimization, optimization is performed with the basis specified in the molecule input file.

## .VISUAL¶

Specifies that the molecule should be visualized, writing a VRML file of the molecular geometry. No optimization will be performed when this keyword is given. See also related keywords .VR-BON, .VR-COR, .VR-EIG and .VR-VIB.

## .VRML¶

Specifies that the molecule should be visualized. VRML files describing both the initial and final geometry will be written (as initial.wrl and final.wrl). The file final.wrl is updated in each iteration, so that it always reflects the latest geometry. See also related keywords .VR-BON, .VR-COR, .VR-EIG and .VR-VIB.

## .SYMTHR¶

READ(LUCMD,*) THRSYM

Determines the gradient threshold (in a.u.) for breaking of the symmetry. That is, if the index of the molecular molecular Hessian is non-zero when the gradient norm drops below this value, the symmetry is broken to avoid unnecessary iterations within the wrong symmetry. This option only applies to second-order methods and when the keyword .NOBREA is not present. The default value of this threshold is 0.005.

## .TRSTRG¶

Specifies that the level-shifted trust region method should be used to control the step. This is the default, so the keyword is actually redundant at the moment. Alternative step control methods are .RF and .GDIIS.

## .VR-BON¶

Only has effect together with .VRML or .VISUAL. Specifies that the VRML files should include bonds between nearby atoms. The bonds are drawn as grey cylinders, making it easier to see the structure of the molecule. If .VR-BON is omitted, only the spheres representing the different atoms will be drawn.

## .VR-EIG¶

Only has effect together with .VRML or .VISUAL. Specifies that the eigenvectors of the molecule (that is the eigenvectors of the molecular Hessian, which differs from the normal modes as they are not mass-scaled) should be visualized. These are written to the files eigv_###.wrl.

## .INITHE¶

Specifies that the initial molecular Hessian should be calculated (analytical molecular Hessian), thus yielding a first step that is identical to that of second-order methods. This provides an excellent starting point for first-order methods, but should only be used when the molecular Hessian can be calculated within a reasonable amount of time. It has only effect for first-order methods and overrides the keywords .INITEV and .INIRED. It has no effect when .HESFIL has been specified.

## .INITEV¶

READ(LUCMD,*) EVLINI

The default initial molecular Hessian for first-order minimizations is the identity matrix when Cartesian coordinates are used, and a diagonal matrix when redundant internal coordinates are used. If .INITEV is used, all the diagonal elements (and therefore the eigenvalues) are set equal to the value EVLINI. This option only has effect when first-order methods are used and .INITHE and .HESFIL are non-present.

## .HESFIL¶

Specifies that the initial molecular Hessian should be read from the file DALTON.HES(?). This applies to first-order methods, and the Hessian in the file must have the correct dimensions. This option overrides other options for the initial Hessian. Each time a Hessian is calculated or updated, it?s written to this file (in Cartesian coordinates). If an optimization is interrupted, it can be restarted with the last geometry and the molecular Hessian in DALTON.HES, minimizing the loss of information. Another useful possibility is to transfer the molecular Hessian from a calculation on the same molecule with another (smaller) basis and/or a cheaper wave function. Finally, one can go in and edit the file directly to set up a specific force field.

## .REJINI¶

Specifies that the molecular Hessian should be reinitialized after every rejected step, as a rejected step indicates that the molecular Hessian models the true potential surface poorly. Only applies to first-order methods.

## .STEEPD¶

Specifies that the first-order steepest descent method should be used. No update is done on the molecular Hessian, so the optimization will be guided by the gradient alone. The ?pure? steepest descent method is obtained when the molecular Hessian is set equal to the identity matrix. Each step will then be the negative of the gradient vector, and the convergence towards the minimum will be extremely slow. However, this option can be combined with other initial molecular Hessians in Cartesian or redundant internal coordinates, giving a method where the main feature is the lack of molecular Hessian updates (static molecular Hessian).

## .RANKON¶

Specifies that a first-order method with the rank one update formula should be used for optimization. This updating is also referred to as symmetric rank one (SR1) or Murtagh-Sargent (MS).

## .PSB¶

Specifies that a first-order method with the Powell-Symmetric-Broyden (PSB) update formula should be used for optimization.

## .DFP¶

Specifies that a first-order method with the Davidon-Fletcher-Powell (DFP) update formula should be used for optimization. May be used for both minimizations and transition state optimizations.

## .BFGS¶

Specifies the use of a first-order method with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update formula for optimization. This is the preferred first-order method for minimizations, as this update is able to maintain a positive definite Hessian. Note that this also makes it unsuitable for transitions state optimization (where one negative eigenvalue is sought).

## .NEWTON¶

Specifies that a second-order Newton method should be used for optimization that is, the analytical molecular Hessian will be calculated at every geometry. By default the level-shifted trust region method will be used, but it is possible to override this by using one of the two keywords .RF or .GDIIS. Not implemented in DIRAC.

Use the 2nd order quadratic steepest descent method (missing in Dalton manual).

## .SCHLEG¶

Specifies that a first-order method with Schlegel’s updating scheme [dalton ref 152] should be used. This makes use of all previous displacements and gradients, not just the last, to update the molecular Hessian.

## .HELLMA¶

Use gradients and Hessians calculated using the Hellmann-Feynman approximation. Currently not working properly.

## .M-BFGS¶

A list of old geometries and gradients are kept. At each new point, displacements and gradient difference for the last few steps are calculated, and all of these are then used to sequentially update the molecular Hessian, the most weight being given to the last displacement and gradient difference. Each update is done using the BFGS formula, and it?s thus only suitable for minimizations. Only applies to first-order methods.

## .CARTES¶

Indicates that Cartesian coordinates should be used in the optimization. This is the default for second-order methods.

## .REDINT¶

Specifies that redundant internal coordinates should be used in the optimization. This is the default for first-order methods.

## .INIRED¶

Use a simple model Hessian [dalton ref 18] diagonal in redundant internal coordinates as the initial Hessian. All diagonal elements are determined based on an extremely simplified molecular mechanics model, yet this model provides Hessians that are good starting points for most systems, thus avoiding any calculation of the exact Hessian. This is the default for first-order methods.

## .1STORD¶

Use default first-order method. This means that the BFGS update will be used, and that the optimization is carried out in redundant internal coordinates. Same effect as the combination of the two keywords .BFGS and .REDINT. Since the BFGS method ensures a positive definite Hessian, the .BOFILL optimization method is used by default in case of searches for transition states.

## .2NDORD¶

Use default second-order method. Molecular Hessians will be calculated at every geometry. The level-shifted Newton method and Cartesian coordinates are used. Identical to specifying the keywords .NEWTON and .CARTES. Not implemented in DIRAC.

## .GRDINI¶

Specifies that the molecular Hessian should be reinitialized every time the norm of the gradient is larger than norm of the gradient two iterations earlier. This keyword should only be used when it?s difficult to obtain a good approximation to the molecular Hessian during optimization. Only applies to first-order methods.

## .DISPLA¶

READ (LUCMD,*) DISPLA

Read one more line containing the norm of the displacement vector to be used during numerical evaluation of the molecular gradient, as is needed when doing geometry optimizations with CI or MP2 wave functions. Default is 0.001 a.u.

## .CONSTR¶

READ (LUCMD, *) NCON
DO I = 1, NCON
ICNSTR(ICON) = 1
END DO

Request a constrained geometry optimization. Only works when using redundant internal coordinates. The number of primitive coordinates that should be frozen has to be specified (NCON), then a list follows with the individual coordinate numbers. The coordinate numbers can be found by first running Dalton(DIRAC?) with the .FINDRE keyword. Any number of bonds, angles and dihedral angles may be frozen. NOTE: Symmetry takes precedence over constraints, if you e.g. want to freeze just one of several symmetric bonds, symmetry must be lowered or switched off.

## .MODHES¶

Determine a new model molecular Hessian (see .INIMOD) at every geometry without doing any updating. The model is thus used in much the same manner as an exact molecular Hessian, though it is obviously only a relatively crude approximation to the analytical molecular Hessian.

## .REMOVE¶

READ (LUCMD, *) NREM
DO I = 1, NREM
ICNSTR(IREM) = 2
END DO

Only has effect when using redundant internal coordinates. Specifies internal coordinates that should be removed. The input is identical to the one for .CONSTRAINT, that is one has to specify the number of coordinates that should be removed, then the number of each of those internal coordinates. The coordinate numbers can first be determined by running with .FINDRE set. Removing certain coordinates can sometimes be useful in speeding up constrained geometry optimization, as certain coordinates sometimes “struggle” against the constraints. See also .NODIHE.

## .INIMOD¶

Use a simple model Hessian [dalton ref 18] diagonal in redundant internal coordinates as the initial Hessian. All diagonal elements are determined based on an extremely simplified molecular mechanics model, yet this model provides Hessians that are good starting points for most systems, thus avoiding any calculation of the exact Hessian. This is the default for first-order methods.

## .FINDRE¶

Determines the redundant internal coordinate system then quits without doing an actual calculation. Useful for setting up constrained geometry optimizations, where the numbers of individual primitive internal coordinates are needed.

## .CMBMOD¶

Uses a combination of the BFGS update and the model molecular Hessian (diagonal in redundant internal coordinates). The two have equal weight in the first iteration of the geometry optimization, then for each subsequent iteration the weight of the model Hessian is halved. Only suitable for minimizations.

## .RF¶

Use the rational function method [dalton ref.12] instead of level-shifted Newton which is the default. The RF method is often slightly faster than the level-shifted Newton, but also slightly less robust. For saddle point optimizations there’s a special partitioned rational function method (used automatically when both .RF and .SADDLE are set). However, this method is both slower and less stable than the default trust-region level-shifted image method (which is the default).

## .GDIIS¶

Use the Geometrical DIIS[dalton ref 151] algorithm to control the step. Works in much the same way as DIIS for wave functions. However, the rational function and level-shifted Newton methods are generally more robust and more efficient. Can only be used for minimizations.

## .DELINT¶

Use delocalized internal coordinates. These are built up as non-redundant linear combinations of the redundant internal coordinates. Performance is more or less the same as for the redundant internals, but the transformation of displacements (step) is slightly less stable.

## .NODIHE¶

No dihedral angles will be used as coordinates (just bonds and angles).

## .VR-COR¶

Draws x-, y- and z-axis in the VRML scenes with geometries. Somewhat useful if one is struggling to build a reasonable geometry by adjusting coordinates manually.

## .VR-VIB¶

Similar to .VR-EIG, but more useful as it draws the actual normal mode vectors (the mass-weighted eigenvectors). These are written to the files norm_###.wrl. Keyword only has effect when a vibrational analysis has been requested.

## .VR-SYM¶

Draws in all symmetry elements of the molecule as vectors (rotational axes) and semi-transparent planes (mirror planes).

## .M-PSB¶

This identical to .M-BFGS, except the PSB formula is used for the updating. Only applies to first-order methods, but it can be used for both minimizations and saddle point optimizations.

## .LINE S¶

Turns on line searching, using a quartic polynomial. By default this is turned off, as there seems to be no gain in efficiency. Can only be used for minimizations.

Indicates that a saddle point optimization should be performed rather than a minimization. The default method is to calculate the molecular Hessian analytically at the initial geometry, then update it using Bofill’s update. The optimization is performed in redundant internal coordinates and using the trust-region level-shifted image method to control the step. That is by default all the keywords .INITHE, .FILL and .REDINT are already set, but this can of course be overridden by specifying other keywords. If locating the desired transition state is difficult, and provided analytical molecular Hessians are available, it may sometimes be necessary to use the .NEWTON keyword so that molecular Hessians are calculated at every geometry.

## .MODE¶

READ(LUCMD,*) NSPMOD

Only has effect when doing saddle point optimizations. Determines which molecular Hessian eigenmode should be maximized (inverted in the image method). By default this is the mode corresponding to the lowest eigenvalue, i.e. mode 1. If an optimization does not end up at the correct transition state, it may be worthwhile following other modes (only the lower ones are usually interesting).

## .BOFILL¶

Bofill’s update [dalton ref 20] is the default updating scheme for transition state optimizations. It’s a linear combination of the symmetric rank one and the PSB updating schemes, automatically giving more weight to PSB whenever the rank one potentially is numerically unstable.

## .LINDH¶

Use original Roland Lindh r_ref with .MODHES (missing in Dalton manual).

## .GRD IN¶

Specifies that the molecular Hessian should be reinitialized every time the norm of the gradient is larger than norm of the gradient two iterations earlier. This keyword should only be used when it?s difficult to obtain a good approximation to the molecular Hessian during optimization. Only applies to first-order methods.

## .GRD SCREEN¶

READ (LUCMD,*) SCRGRD

Read the screening threshold for gradient calculations (missing in Dalton manual).

## .NOAUX¶

Only has effect when using redundant internal coordinates. The default for minimizations is to add auxiliary bonds between atoms that are up to two and half times further apart then regular (chemical) bonds. This increases the redundancy of the coordinate system, but usually speeds up the geometry optimization slightly. .NOAUX turns this off. For saddle point optimizations and constrained geometry optimization this is off by default (cannot be switched on).

## .BFGSR1¶

Use a linear combination of the BFGS and the symmetric rank one updating schemes in the same fashion as Bofill’s update. Only suitable for minimizations.

## .IPRGRD¶

Print level for DIRAC molecular gradient evaluation (missing in Dalton manual).