# Geometry Optimization

### From NWChem

# Geometry Optimization with DRIVER

The DRIVER module is one of two drivers (also see documentation on STEPPER) to perform a geometry optimization function on the molecule defined by input using the GEOMETRY directive. Geometry optimization is either an energy minimization or a transition state optimization. The algorithm programmed in DRIVER is a quasi-newton optimization with line searches and approximate energy Hessian updates.

DRIVER is selected by default out of the two available modules to perform geometry optimization. In order to force use of DRIVER (e.g., because a previous optimization used STEPPER) provide a DRIVER input block (below) -- even an empty block will force use of DRIVER.

Optional input for this module is specified within the compound directive,

DRIVER (LOOSE || DEFAULT || TIGHT) GMAX <real value> GRMS <real value> XMAX <real value> XRMS <real value> EPREC <real eprec default 1e-7> TRUST <real trust default 0.3> SADSTP <real sadstp default 0.1> CLEAR REDOAUTOZ INHESS <integer inhess default 0> (MODDIR || VARDIR) <integer dir default 0> (FIRSTNEG || NOFIRSTNEG) MAXITER <integer maxiter default 20> BSCALE <real BSCALE default 1.0> ASCALE <real ASCALE default 0.25> TSCALE <real TSCALE default 0.1> HSCALE <real HSCALE default 1.0> PRINT ... XYZ <string xyz defaultfile_prefix>] NOXYZ END

On each optimization step a line search is performed. To speed up calculations (up to two times), it may be beneficial to turn off the line search using following directive:

set driver:linopt 0

## Convergence criteria

(LOOSE || DEFAULT || TIGHT) GMAX <real value> GRMS <real value> XMAX <real value> XRMS <real value>

The defaults may be used, or the directives LOOSE, DEFAULT, or TIGHT specified to use standard sets of values, or the individual criteria adjusted. All criteria are in atomic units. GMAX and GRMS control the maximum and root mean square gradient in the coordinates being used (Z-matrix, redundant internals, or Cartesian). XMAX and XRMS control the maximum and root mean square of the Cartesian step.

LOOSE DEFAULT TIGHT GMAX 0.00450 0.00045 0.000015 GRMS 0.00300 0.00030 0.00001 XMAX 0.01800 0.00180 0.00006 XRMS 0.01200 0.00120 0.00004

Note that GMAX and GRMS used for convergence of geometry may significantly vary in different coordinate systems such as Z-matrix, redundant internals, or Cartesian. The coordinate system is defined in the input file (default is Z-matrix). Therefore the choice of coordinate system may slightly affect converged energy. Although in most cases XMAX and XRMS are last to converge which are always done in Cartesian coordinates, which insures convergence to the same geometry in different coordinate systems.

The old criterion may be recovered with the input

gmax 0.0008; grms 1; xrms 1; xmax 1

## Available precision

EPREC <real eprec default 1e-7>

In performing a line search the optimizer must know the precision of the energy (this has nothing to do with convergence criteria). The default value of 1e-7 should be adjusted if less, or more, precision is available. Note that the default EPREC for DFT calculations is 5e-6 instead of 1e-7.

## Controlling the step length

TRUST <real trust default 0.3> SADSTP <real sadstp default 0.1>

A fixed trust radius (trust) is used to control the step during minimizations, and is also used for modes being minimized during saddle-point searches. It defaults to 0.3 for minimizations and 0.1 for saddle-point searches. The parameter sadstp is the trust radius used for the mode being maximized during a saddle-point search and defaults to 0.1.

## Maximum number of steps

MAXITER <integer maxiter default 20>

By default at most 20 geometry optimization steps will be taken, but this may be modified with this directive.

## Discard restart information

CLEAR

By default Driver reuses Hessian information from a previous optimization, and, to facilitate a restart also stores which mode is being followed for a saddle-point search. This option deletes all restart data.

## Regenerate internal coordinates

REDOAUTOZ

Deletes Hessian data and regenerates internal coordinates at the current geometry. Useful if there has been a large change in the geometry that has rendered the current set of coordinates invalid or non-optimal.

## Initial Hessian

INHESS <integer inhess default 0>

- 0 = Default ... use restart data if available, otherwise use diagonal guess.
- 1 = Use diagonal initial guess.
- 2 = Use restart data if available, otherwise transform Cartesian Hessian from previous frequency calculation.

In addition, the diagonal elements of the initial Hessian for internal coordinates may be scaled using separate factors for bonds, angles and torsions with the following

BSCALE <real bscale default 1.0> ASCALE <real ascale default 0.25> TSCALE <real tscale default 0.1>

These values typically give a two-fold speedup over unit values, based on about 100 test cases up to 15 atoms using 3-21g and 6-31g* SCF. However, if doing many optimizations on physically similar systems it may be worth fine tuning these parameters.

Finally, the entire Hessian from any source may be scaled by a factor using the directive

HSCALE <real hscale default 1.0>

It might be of utility, for instance, when computing an initial Hessian using SCF to start a large MP2 optimization. The SCF vibrational modes are expected to be stiffer than the MP2, so scaling the initial Hessian by a number less than one might be beneficial.

## Mode or variable to follow to saddle point

(MODDIR || VARDIR) <integer dir default 0> (FIRSTNEG || NOFIRSTNEG)

When searching for a transition state the program, by default, will take an initial step uphill and then do mode following using a fuzzy maximum overlap (the lowest eigen-mode with an overlap with the previous search direction of 0.7 times the maximum overlap is selected). Once a negative eigen-value is found, that mode is followed regardless of overlap.

The initial uphill step is appropriate if the gradient points roughly in the direction of the saddle point, such as might be the case if a constrained optimization was performed at the starting geometry. Alternatively, the initial search direction may be chosen to be along a specific internal variable (using the directive VARDIR) or along a specific eigen-mode (using MODDIR). Following a variable might be valuable if the initial gradient is either very small or very large. Note that the eigen-modes in the optimizer have next-to-nothing to do with the output from a frequency calculation. You can examine the eigen-modes used by the optimizer with

driver; print hvecs; end

The selection of the first negative mode is usually a good choice if the search is started in the vicinity of the transition state and the initial search direction is satisfactory. However, sometimes the first negative mode might not be the one of interest (e.g., transverse to the reaction direction). If NOFIRSTNEG is specified, the code will not take the first negative direction and will continue doing mode-following until that mode goes negative.

## Optimization history as XYZ files

XYZ [<string xyz default $fileprefix>] NOXYZ

The XYZ directive causes the geometry at each step (but not intermediate points of a line search) to be output into separate files in the permanent directory in XYZ format. The optional string will prefix the filename. The NOXYZ directive turns this off.

For example, the input

driver; xyz test; end

will cause files test-000.xyz, test-001.xyz, ... to be created in the permanent directory.

The script rasmolmovie in the NWChem contrib directory can be used to turn these into an animated GIF movie.

## Print options

The UNIX command "egrep '^@' < output" will extract a pretty table summarizing the optimization.

If you specify the NWChem input

scf; print none; end driver; print low; end task scf optimize

you'll obtain a pleasantly terse output.

For more control, these options for the standard print directive are recognized

- debug - prints a large amount of data. Don't use in parallel.
- high - print the search direction in internals
- default - prints geometry for each major step (not during the line search), gradient in internals (before and after application of constraints)
- low - prints convergence and energy information. At convergence prints final geometry, change in internals from initial geometry

and these specific print options

- finish (low) - print geometry data at end of calculation
- bonds (default) - print bonds at end of calculation
- angles (default) - print angles at end of calculation
- hvecs (never) - print eigen-values/vectors of the Hessian
- searchdir (high) - print the search direction in internals
- "internal gradient" (default) - print the gradient in internals
- sadmode (default) - print the mode being followed to the saddle point

# Geometry Optimization with STEPPER

The STEPPER module performs a search for critical points on the potential energy surface of the molecule defined by input using the GEOMETRY directive. Since STEPPER is not the primary geometry optimization module in NWChem the compound directive is required; the DRIVER module is the default. Input for this module is specified within the compound directive,

STEPPER ... END

The presence of the STEPPER compound directive automatically turns off the default geometry optimization tool DRIVER. Input specified for the STEPPER module must appear in the input file after the GEOMETRY directive, since it must know the number of atoms that are to be used in the geometry optimization. In the current version of NWChem, STEPPER can be used only with geometries that are defined in Cartesian coordinates. STEPPER removes translational and rotational components before determining the step direction (5 components for linear systems and 6 for others) using a standard Eckart algorithm. The default initial guess nuclear Hessian is the identity matrix.

The default in STEPPER is to minimize the energy as a function of the geometry with a maximum of 20 geometry optimization iterations. When this is the desired calculation, no input is required other than the STEPPER compound directive. However, the user also has the option of defining different tasks for the STEPPER module, and can vary the number of iterations and the convergence criteria from the default values. The input for these options is described in the following sections.

## MIN and TS -- Minimum or transition state search

The default is for STEPPER to minimize the energy with respect to the geometry of the system. This default behavior may be forced with the directive

MIN

STEPPER can also be used to find the transition state by following the lowest eigenvector of the nuclear Hessian. This is usually invoked by using the saddle keyword on the TASK directive (Section 5.10), but it may also be selected by specifying the directive

TS

in the STEPPER input.

## TRACK -- Mode selection

STEPPER has the ability to ``track* a specific mode during an optimization for a transition state search, the user can also have the module track the eigenvector corresponding to a specific mode. This is done by specifying the directive*

TRACK [nmode <integer nmode default 1>]

The keyword TRACK tells STEPPER to track the eigenvector corresponding to the integer value of <nmode> during a transition state walk. (Note: this input is invalid for a minimization walk since following a specific eigenvector will not necessarily give the desired local minimum.) The step is constructed to go up in energy along the nmode eigenvector and down in all other degrees of freedom.

## MAXITER -- Maximum number of steps

In most applications, 20 stepper iterations will be sufficient to obtain the energy minimization. However, the user has the option of specifying the maximum number of iterations allowed, using the input line,

MAXITER <integer maxiter default 20>

The value specified for the integer <maxiter> defines the maximum number of geometry optimization steps. The geometry optimization will restart automatically.

## TRUST -- Trust radius

The size of steps that can be taken in STEPPER is controlled by the trust radius which has a default value of 0.1. Steps are constrained to be no larger than the trust radius. The user has the option of overriding this default using the keyword TRUST, with the following input line,

TRUST <real radius default 0.1>

The larger the value specified for the variable radius, the larger the steps that can be taken by STEPPER. Experience has shown that for larger systems (i.e., those with 20 or more atoms), a value of 0.5, or greater, usually should be entered for <radius>.

## CONVGGM, CONVGG and CONVGE -- Convergence criteria

Three convergence criteria can be specified explicitly for the STEPPER calculations. The keyword CONVGGM allows the user to specify the convergence tolerance for the largest component of the gradient. This is the primary convergence criterion, as per the default settings, although all three criteria are in effect. this default setting is consistent with the other optimizer module DRIVER. The input line for CONVGGM has the following form,

CONVGGM <real convggm default 8.0d-04>

The keyword CONVGG allows the user to specify the convergence tolerance for the gradient norm for all degrees of freedom. The input line is of the following form,

CONVGG <real convgg default 1.0d-02>

The entry for the real variable <convgg> should be approximately equal to the square root of the energy convergence tolerance.

The energy convergence tolerance is the convergence criterion for the energy difference in the geometry optimization in STEPPER. It can be specified by input using a line of the following form,

CONVGE <real convge default 1.0d-04>

## Backstepping in STEPPER

If a step taken during the optimization is too large (e.g., the step causes the energy to go up for a minimization or down for a transition state search), the STEPPER optimizer will automatically ``backstep* and correct the step based on information prior to the faulty step. If you have an optimization that "backsteps" frequently then the initial trust radius should most likely be decreased.*

## Initial Nuclear Hessian Options

Stepper uses a modified Fletcher-Powell algorithm to find the transition state or energy minimum on the potential energy hypersurface. There are two files left in the user's permanent directory that are used to provide an initial hessian to the critical point search algorithm. If these files do not exist then the default is to use a unit matrix as the initial hessian. Once Stepper executes it generates a binary dump file by the name of name.stpr41 which will be used on all subsequent stepper runs and modified with the current updated hessian. The default file prefix is the "name" that is used (see START). It also stores the information for the last valid step in case the algorithm must take a "backstep". This file is the working data store for all stepper-based optimizations. This file is never deleted by default and is the first source of an initial hessian. The second source of an inital hessian is an ascii file that contains the lower triangular values of the initial hessian. This is stored in file name.hess, where "name" is again the default file prefix. This is the second source of an initial hessian and is the method used to incorporate an initial hessian from any other source (e.g., another ab initio code, a molecular mechanics code, etc.,). To get a decent starting hessian at a given point you can use the task specification task scf hessian, with a smaller basis set, which will by default generate the name.hess file. Then you may define your basis set of choice and proceed with the optimization you desire.