Q-Chem 5.0 User’s Manual

9.1 Equilibrium Geometries and Transition-State Structures

9.1.1 Overview

Molecular potential energy surfaces rely on the Born-Oppenheimer separation of nuclear and electronic motion. Minima on such energy surfaces correspond to the classical picture of equilibrium geometries, and transition state structures correspond to first-order saddle points. Both equilibrium and transition-state structures are stationary points for which the energy gradient vanishes. Characterization of such critical points requires consideration of the eigenvalues of the Hessian (second derivative matrix): minimum-energy, equilibrium geometries possess Hessians whose eigenvalues are all positive, whereas transition-state structures are defined by a Hessian with precisely one negative eigenvalue. (The latter is therefore a local maximum along the reaction path between minimum-energy reactant and product structures, but a minimum in all directions perpendicular to this reaction path.

The quality of a geometry optimization algorithm is of major importance; even the fastest integral code in the world will be useless if combined with an inefficient optimization algorithm that requires excessive numbers of steps to converge. Q-Chem incorporates a geometry optimization package (Optimize—see Appendix A) developed by the late Jon Baker over more than ten years.

The key to optimizing a molecular geometry successfully is to proceed from the starting geometry to the final geometry in as few steps as possible. Four factors influence the path and number of steps:

Q-Chem controls the last three of these, but the starting geometry is solely determined by the user, and the closer it is to the converged geometry, the fewer optimization steps will be required. Decisions regarding the optimization algorithm and the coordinate system are generally made by the Optimize package (i.e., internally, within Q-Chem) to maximize the rate of convergence. Although users may override these choices in many cases, this is not generally recommended.

Level of Theory

Analytical

Maximum Angular

Analytical

Maximum Angular

(Algorithm)

Gradients

Momentum Type

Hessian

Momentum Type

DFT

✓

$h$

✓

$f$

HF

✓

$h$

✓

$f$

ROHF

✓

$h$

✗

  

MP2

✓

$h$

✗

  

(V)OD

✓

$h$

✗

  

(V)QCCD

✓

$h$

✗

  

CIS (except RO)

✓

$h$

✓

$f$

CFMM

✓

$h$

✗

  
Table 9.1: Gradients and Hessians currently available for geometry optimizations with maximum angular momentum types for analytical derivative calculations (for higher angular momentum, derivatives are computed numerically). Analytical Hessians are not yet available for meta-GGA functionals such as BMK and the M05 and M06 series.

Another consideration when trying to minimize the optimization time concerns the quality of the gradient and Hessian. A higher-quality Hessian (i.e., analytical versus approximate) will in many cases lead to faster convergence, in the sense of requiring fewer optimization steps. However, the construction of an analytical Hessian requires significant computational effort and may outweigh the advantage of fewer optimization cycles. Currently available analytical gradients and Hessians are summarized in Table 9.1.

Features of Q-Chem’s geometry and transition-state optimization capabilities include:

9.1.2 Job Control

Obviously a level of theory, basis set, and starting molecular geometry must be specified to begin a geometry optimization or transition-structure search. These aspects are described elsewhere in this manual, and this section describes job-control variables specific to optimizations.

JOBTYPE

Specifies the calculation.


TYPE:

STRING


DEFAULT:

Default is single-point, which should be changed to one of the following options.


OPTIONS:

OPT

Equilibrium structure optimization.

TS

Transition structure optimization.

RPATH

Intrinsic reaction path following.


RECOMMENDATION:

Application-dependent.


GEOM_OPT_HESSIAN

Determines the initial Hessian status.


TYPE:

STRING


DEFAULT:

DIAGONAL


OPTIONS:

DIAGONAL

Set up diagonal Hessian.

READ

Have exact or initial Hessian. Use as is if Cartesian, or transform

 

if internals.


RECOMMENDATION:

An accurate initial Hessian will improve the performance of the optimizer, but is expensive to compute.


GEOM_OPT_COORDS

Controls the type of optimization coordinates.


TYPE:

INTEGER


DEFAULT:

$-$1


OPTIONS:

0

Optimize in Cartesian coordinates.

1

Generate and optimize in internal coordinates, if this fails abort.

$-$1

Generate and optimize in internal coordinates, if this fails at any stage of the

 

optimization, switch to Cartesian and continue.

2

Optimize in $Z$-matrix coordinates, if this fails abort.

$-$2

Optimize in $Z$-matrix coordinates, if this fails during any stage of the

 

optimization switch to Cartesians and continue.


RECOMMENDATION:

Use the default; delocalized internals are more efficient.


GEOM_OPT_TOL_GRADIENT

Convergence on maximum gradient component.


TYPE:

INTEGER


DEFAULT:

300

$\equiv 300\times 10^{-6}$ tolerance on maximum gradient component.


OPTIONS:

$n$

Integer value (tolerance = $n \times 10^{-6}$).


RECOMMENDATION:

Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY must be satisfied.


GEOM_OPT_TOL_DISPLACEMENT

Convergence on maximum atomic displacement.


TYPE:

INTEGER


DEFAULT:

1200 $\equiv 1200 \times 10^{-6}$ tolerance on maximum atomic displacement.


OPTIONS:

$n$

Integer value (tolerance = $n \times 10^{-6}$).


RECOMMENDATION:

Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY must be satisfied.


GEOM_OPT_TOL_ENERGY

Convergence on energy change of successive optimization cycles.


TYPE:

INTEGER


DEFAULT:

100 $\equiv 100 \times 10^{-8}$ tolerance on maximum (absolute) energy change.


OPTIONS:

$n$ Integer value (tolerance = value $n \times 10^{-8}$).


RECOMMENDATION:

Use the default. To converge GEOM_OPT_TOL_GRADIENT and one of GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY must be satisfied.


GEOM_OPT_MAX_CYCLES

Maximum number of optimization cycles.


TYPE:

INTEGER


DEFAULT:

50


OPTIONS:

$n$

User defined positive integer.


RECOMMENDATION:

The default should be sufficient for most cases. Increase if the initial guess geometry is poor, or for systems with shallow potential wells.


GEOM_OPT_PRINT

Controls the amount of Optimize print output.


TYPE:

INTEGER


DEFAULT:

3

Error messages, summary, warning, standard information and gradient print out.


OPTIONS:

0

Error messages only.

1

Level 0 plus summary and warning print out.

2

Level 1 plus standard information.

3

Level 2 plus gradient print out.

4

Level 3 plus Hessian print out.

5

Level 4 plus iterative print out.

6

Level 5 plus internal generation print out.

7

Debug print out.


RECOMMENDATION:

Use the default.


GEOM_OPT_SYMFLAG

Controls the use of symmetry in Optimize.


TYPE:

LOGICAL


DEFAULT:

TRUE


OPTIONS:

TRUE

Make use of point group symmetry.

FALSE

Do not make use of point group symmetry.


RECOMMENDATION:

Use the default.


GEOM_OPT_MODE

Determines Hessian mode followed during a transition state search.


TYPE:

INTEGER


DEFAULT:

0


OPTIONS:

0

Mode following off.

$n$

Maximize along mode $n$.


RECOMMENDATION:

Use the default, for geometry optimizations.


GEOM_OPT_MAX_DIIS

Controls maximum size of subspace for GDIIS.


TYPE:

INTEGER


DEFAULT:

0


OPTIONS:

0

Do not use GDIIS.

-1

Default size = min(NDEG, NATOMS, 4) NDEG = number of molecular

 

degrees of freedom.

$n$

Size specified by user.


RECOMMENDATION:

Use the default or do not set $n$ too large.


GEOM_OPT_DMAX

Maximum allowed step size. Value supplied is multiplied by 10$^{-3}$.


TYPE:

INTEGER


DEFAULT:

300

= 0.3


OPTIONS:

$n$

User-defined cutoff.


RECOMMENDATION:

Use the default.


GEOM_OPT_UPDATE

Controls the Hessian update algorithm.


TYPE:

INTEGER


DEFAULT:

-1


OPTIONS:

-1

Use the default update algorithm.

0

Do not update the Hessian (not recommended).

1

Murtagh-Sargent update.

2

Powell update.

3

Powell/Murtagh-Sargent update (TS default).

4

BFGS update (OPT default).

5

BFGS with safeguards to ensure retention of positive definiteness
 

(GDISS default).


RECOMMENDATION:

Use the default.


GEOM_OPT_LINEAR_ANGLE

Threshold for near linear bond angles (degrees).


TYPE:

INTEGER


DEFAULT:

165 degrees.


OPTIONS:

$n$

User-defined level.


RECOMMENDATION:

Use the default.


FDIFF_STEPSIZE

Displacement used for calculating derivatives by finite difference.


TYPE:

INTEGER


DEFAULT:

100

Corresponding to 0.001 . For calculating second derivatives.


OPTIONS:

$n$

Use a step size of $n\times 10^{-5}$.


RECOMMENDATION:

Use the default except in cases where the potential surface is very flat, in which case a larger value should be used. See FDIFF_STEPSIZE_QFF for third and fourth derivatives.


Example 9.208  As outlined, the rate of convergence of the iterative optimization process is dependent on a number of factors, one of which is the use of an initial analytic Hessian. This is easily achieved by instructing Q-Chem to calculate an analytic Hessian and proceed then to determine the required critical point

$molecule
   0  1
   O
   H  1  oh
   H  1  oh 2 hoh

   oh  = 1.1
   hoh = 104
$end

$rem
   JOBTYPE    freq   Calculate an analytic Hessian
   METHOD     hf
   BASIS      6-31g(d)
$end

$comment
Now proceed with the Optimization making sure to read in the analytic 
Hessian (use other available information too).
$end

@@@

$molecule
   read
$end

$rem
   JOBTYPE            opt
   METHOD             hf
   BASIS              6-31g(d)
   SCF_GUESS          read
   GEOM_OPT_HESSIAN   read   Have the initial Hessian
$end

9.1.3 Hessian-Free Characterization of Stationary Points

Q-Chem allows the user to characterize the stationary point found by a geometry optimization or transition state search without performing a full analytical Hessian calculation, which is sometimes unavailable or computationally unaffordable. This is achieved via a finite difference Davidson procedure developed by Sharada et al.[535] For a geometry optimization, it solves for the lowest eigenvalue of the Hessian ($\lambda _1$) and checks if $\lambda _1 > 0$ (a negative $\lambda _1$ indicates a saddle point); for a TS search, it solves for the lowest two eigenvalues, and $\lambda _1 < 0$ and $\lambda _2 > 0$ indicate a transition state. The lowest eigenvectors of the updated P-RFO (approximate) Hessian at convergence are used as the initial guess for the Davidson solver.

The cost of this Hessian-free characterization method depends on the rate of convergence of the Davidson solver. For example, to characterize an energy minimum, it requires $2\times N_\ensuremath{\mathrm{}}{iter}$ total energy + gradient calculations, where $N_\ensuremath{\mathrm{}}{iter}$ is the number of iterations that the Davidson algorithm needs to converge, and “2" is for forward and backward displacements on each iteration. According to Ref. Sharada:2014, this method can be much more efficient than exact Hessian calculation for substantially large systems.

Note: At the moment, this method does not support QM/MM or systems with fixed atoms.

GEOM_OPT_CHARAC

Use the finite difference Davidson method to characterize the resulting energy minimum/transition state.


TYPE:

BOOLEAN


DEFAULT:

FALSE


OPTIONS:

FALSE

do not characterize the resulting stationary point.

TRUE

perform a characterization of the stationary point.


RECOMMENDATION:

Set it to TRUE when the character of a stationary point needs to be verified, especially for a transition structure.


GEOM_OPT_CHARAC_CONV

Overide the built-in convergence criterion for the Davidson solver.


TYPE:

INTEGER


DEFAULT:

0 (use the built-in default value 10$^{-5}$)


OPTIONS:

$n$

Set the convergence criterion to 10$^{-n}$.


RECOMMENDATION:

Use the default. If it fails to converge, consider loosening the criterion with caution.


Example 9.209  Geometry optimization of a triflate anion that converges to an eclipsed conformation, which is a first order saddle point. This is verified via the finite difference Davidson method by setting GEOM_OPT_CHARAC to TRUE.

$molecule
   -1 1
   C  0.00000 -0.00078  0.98436
   F -1.09414 -0.63166  1.47859
   S  0.00000  0.00008 -0.94745
   O  1.25831 -0.72597 -1.28972
   O -1.25831 -0.72597 -1.28972
   O  0.00000  1.45286 -1.28958
   F  1.09414 -0.63166  1.47859
   F  0.00000  1.26313  1.47663
$end

$rem
   JOBTYPE                    opt 
   METHOD                     BP86
   GEOM_OPT_DMAX              50  
   BASIS                      6-311+G*
   SCF_CONVERGENCE            8   
   THRESH                     14  
   SYMMETRY                   FALSE
   SYM_IGNORE                 TRUE
   GEOM_OPT_TOL_DISPLACEMENT  10  
   GEOM_OPT_TOL_ENERGY        10  
   GEOM_OPT_TOL_GRADIENT      10  
   GEOM_OPT_CHARAC            TRUE
$end

Example 9.210  TS search for alanine dipeptide rearrangement reaction beginning with a guess structure converges correctly. The resulting TS structure is verified using the finite difference Davidson method.

$molecule
   0    1   
   C          3.21659       -1.41022       -0.26053
   C          2.16708       -0.35258       -0.59607
   N          1.21359       -0.16703        0.41640
   C          0.11616        0.82394        0.50964
   C         -1.19613        0.03585        0.74226
   N         -2.18193       -0.02502       -0.18081
   C         -3.43891       -0.74663        0.01614
   O          2.19596        0.25708       -1.63440
   C          0.11486        1.96253       -0.53088
   O         -1.29658       -0.59392        1.85462
   H          3.25195       -2.14283       -1.08721
   H          3.06369       -1.95423        0.67666
   H          4.20892       -0.93714       -0.22851
   H          1.24786       -0.78278        1.21013
   H          0.25990        1.31404        1.47973
   H         -2.02230        0.38818       -1.10143
   H         -3.60706       -1.48647       -0.76756
   H         -4.29549       -0.06423        0.04327
   H         -3.36801       -1.25875        0.98106
   H         -0.68664        2.66864       -0.27269
   H          0.01029        1.65112       -1.56461
   H          1.06461        2.50818       -0.45885
$end

$rem
   JOBTYPE         freq
   EXCHANGE        B3LYP
   BASIS           6-31G
   SCF_MAX_CYCLES  250 
   SYMMETRY        false
   SYM_IGNORE      true
$end

@@@

$rem
   JOBTYPE              ts
   SCF_GUESS            read
   GEOM_OPT_DMAX        100 
   GEOM_OPT_MAX_CYCLES  1500
   EXCHANGE             B3LYP
   BASIS                6-31G
   MAX_SCF_CYCLES       250 
   GEOM_OPT_HESSIAN     read
   SYMMETRY             false
   SYM_IGNORE           true
   GEOM_OPT_CHARAC      true
$end

$molecule
   read
$end