Geometry Optimization¶
Code author: Rollin A. King and Alexander G. Heide
Section author: Rollin A. King, Alexander G. Heide, and Lori A. Burns
PSI4 carries out molecular optimizations using a Python module called optking. The optking program takes as input nuclear gradients and, optionally, nuclear second derivatives — both in Cartesian coordinates. The default minimization algorithm employs an empirical model Hessian, redundant internal coordinates, an RFO step with trust radius scaling, and the BFGS Hessian update.
The principal literature references include the introduction of redundant internal coordinates by Peng et al. [Peng:1996:49]. The general approach employed in this code is similar to the “model Hessian plus RF method” described and tested by Bakken and Helgaker [Bakken:2002:9160]. However, for separated fragments, we have chosen not to employ their “extraredundant” coordinates.
The internal coordinates are generated automatically based on an assumed bond connectivity. The connectivity is determined by testing if the interatomic distance is less than the sum of atomic radii times the value of COVALENT_CONNECT. If the user finds that some connectivity is lacking by default, then this value may be increased.
Warning
The selection of a Zmatrix input, and in particular the inclusion of dummy atoms, has no effect on the behavior of the optimizer, which begins from a Cartesian representation of the system.
Presently, by default, separate fragments are bonded by the nearest atoms, and the whole system is treated as if it were part of one molecule. However, with the option FRAG_MODE, fragments may instead be related by a minimal set of interfragment coordinates defined by reference points within each fragment. The reference points can be atomic positions (current default) or linear combinations of atomic positions (automatic use of principal axes is under development). These dimer coordinates can be directly specified through INTERFRAG_COORDS) See here <DimerSection_> for two examples of their use.
Basic Keywords¶
OPT_TYPE¶
Specifies minimum search, transitionstate search, or IRC following
Type: string
Possible Values: MIN, TS, IRC
Default: MIN
STEP_TYPE¶
Geometry optimization step type, either NewtonRaphson or Rational Function Optimization
Type: string
Possible Values: RFO, P_RFO, NR, SD, LINESEARCH
Default: RFO
GEOM_MAXITER¶
Maximum number of geometry optimization steps
Type: integer
Default: 50
G_CONVERGENCE¶
Set of optimization criteria. Specification of any MAX_*_G_CONVERGENCE or RMS_*_G_CONVERGENCE options will append to overwrite the criteria set here unless FLEXIBLE_G_CONVERGENCE is also on. See Table Geometry Convergence for details.
Type: string
Possible Values: QCHEM, MOLPRO, GAU, GAU_LOOSE, GAU_TIGHT, INTERFRAG_TIGHT, GAU_VERYTIGHT, TURBOMOLE, CFOUR, NWCHEM_LOOSE
Default: QCHEM
FULL_HESS_EVERY¶
Frequency with which to compute the full Hessian in the course of a geometry optimization. 0 means to compute the initial Hessian only, 1 means recompute every step, and N means recompute every N steps. The default (1) is to never compute the full Hessian.
Type: integer
Default: 1
Optimizing Minima¶
First, define the molecule and basis in the input.
molecule h2o {
O
H 1 1.0
H 1 1.0 2 105.0
}
set basis dz
Then the following are examples of various types of calculations that can be completed.
Optimize a geometry using default methods (RFO step):
optimize('scf')
Optimize using NewtonRaphson steps instead of RFO steps:
set step_type nr optimize('scf')
Optimize using finite differences of energies instead of gradients:
optimize('scf', dertype='energy')
Optimize while limiting the initial step size to 0.1 au:
set intrafrag_step_limit 0.1 optimize('scf')
Optimize while always limiting the step size to 0.1 au:
set {
intrafrag_step_limit 0.1
intrafrag_step_limit_min 0.1
intrafrag_step_limit_max 0.1
}
optimize('scf')
Optimize while calculating the Hessian at every step:
set full_hess_every 1
optimize('scf')
import optking
Hessian¶
If Cartesian second derivatives are available, optking can read them and transform them into internal coordinates to make an initial Hessian in internal coordinates. Otherwise, several empirical Hessians are available, including those of Schlegel [Schlegel:1984:333] and Fischer and Almlof [Fischer:1992:9770]. Either of these or a simple diagonal Hessian may be selected using the INTRAFRAG_HESS keyword.
All the common Hessian update schemes are available. For formulas, see Schlegel [Schlegel:1987:AIMQC] and Bofill [Bofill:1994:1].
The Hessian may be computed during an optimization using the FULL_HESS_EVERY keyword.
Transition States and Reaction Paths¶
Calculate a starting Hessian and optimize the “transition state” of linear water (note that without a reasonable starting geometry and Hessian, such a straightforward search often fails):
molecule h2o { O H 1 1.0 H 1 1.0 2 160.0 } set { basis dz full_hess_every 0 opt_type ts } optimize('scf')
At a transition state (planar HOOH), compute the second derivative, and then follow the intrinsic reaction path to the minimum:
molecule hooh { symmetry c1 H O 1 0.946347 O 2 1.397780 1 107.243777 H 3 0.946347 2 107.243777 1 0.0 } set { basis dzp opt_type irc geom_maxiter 50 } frequencies('scf') optimize('scf')
Constrained Optimizations¶
Optimize a geometry (HOOH) at a frozen dihedral angle of 90 degrees.
molecule { H O 1 0.90 O 2 1.40 1 100.0 H 3 0.90 2 100.0 1 90.0 } set optking { frozen_dihedral = (" 1 2 3 4 ") } optimize('scf')
To instead freeze the two OH bond distances
set optking { frozen_distance = (" 1 2 3 4 ") }
For bends, the corresponding keyword is “frozen_bend”.
To freeze the cartesian coordinates of atom 2
freeze_list = """
2 xyz
"""
set optking frozen_cartesian $freeze_list
To freeze only the y coordinates of atoms 2 and 3
freeze_list = """
2 y
3 y
"""
set optking frozen_cartesian $freeze_list
To optimize toward a value of 0.95 Angstroms for the distance between atoms 1 and 3, as well as that between 2 and 4
set optking {
ranged_distance = ("
1 3 0.949 0.95
2 4 0.949 0.95
")
}
Note
The effect of the frozen and ranged keywords is generally independent of how the geometry of the molecule was input (whether Zmatrix or Cartesian, etc.).. At this time; however, enforcing Cartesian constraints when using a zmatrix for molecular input is not supported. Freezing or constraining Cartesian coordinates requires Cartesian molecule input. If numerical errors results in symmetry breaking, while Cartesian constraints are active, symmetrization cannot occur and an error will be raised, prompting you to restart the job.
To scan the potential energy surface by optimizing at several fixed values of the dihedral angle of HOOH.
molecule hooh {
0 1
H 0.850718 0.772960 0.563468
O 0.120432 0.684669 0.035503
O 0.120432 0.684669 0.035503
H 0.850718 0.772960 0.563468
}
set {
basis ccpvdz
intrafrag_step_limit 0.1
}
lower_bound = [99.99, 109.99, 119.99, 129.99, 149.99]
upper_bound = [100, 110, 120, 130, 140, 150]
PES = []
for lower, upper in zip(lower_bound, upper_bound):
my_string = f"1 2 3 4 {lower} {upper}"
set optking ranged_dihedral = $my_string
E = optimize('scf')
PES.append((upper, E))
print("\n\tccpVDZ SCF energy as a function of phi\n")
for point in PES:
print("\t%5.1f%20.10f" % (point[0], point[1]))
To scan the potential energy surface without the RANGED_DIHEDRAL keyword, a zmatrix can be used.
Warning
Rotating dihedrals in large increments without allowing the molecule to relax in between increments can lead to unphysical geometries with overlapping functional groups in larger molecules, which may prevent successful constrained optimzations. Furthermore, such a relaxed scan of the PES does not always procude a result close to an IRC, or even a reaction path along which the energy changes in a continuous way.
molecule hooh {
0 1
H
O 1 0.95
O 2 1.39 1 103
H 3 0.95 2 103 1 D
D = 99
units ang
}
set {
basis ccpvdz
intrafrag_step_limit 0.1
frozen_dihedral (" 1 2 3 4 ")
}
dihedrals = [100, 110, 120, 130, 140, 150]
PES = []
for phi in dihedrals:
hooh.D = phi
E = optimize('scf')
PES.append((phi, E))
print("\n\tccpVDZ SCF energy as a function of phi\n")
for point in PES:
print("\t%5.1f%20.10f" % (point[0], point[1]))
MultiFragment Optimizations¶
In previous versions of optking, the metric for connecting atoms was increased until all atoms were connected. This is the current behavior for FRAG_MODE single. Setting FRAG_MODE to multi will now add a special set of intermolecular coordinates between fragments  internally referred to as DimerFrag coordinates (see here <DimerIntro_> for the brief description). For each pair of molecular fragments, a set of up to 3 reference points are chosen on each fragment. Each reference point will be either an atom or a linear combination of positions of atoms within that fragment. Stretches, bends, and dihedral angles between the two fragments will be created using these reference points. See Dimer coordinate table for how reference points are created. For a set of three dimers A, B, and C, sets of coordinates are created between each pair: AB, AC, and BC. Each DimerFrag may use different reference points. Creation of the intermolecular coordinates can be controlled through FRAG_REF_ATOMS and INTERFRAG_COORDS. FRAG_REF_ATOMS specifies which atoms (or linear combination of atoms) to use for the reference points and INTERFRAG_COORDS, which encompasses FRAG_REF_ATOMS, allows for constraints and labels to be added to the intermolecular coordinates.
Note
Manual specification of the interfragment coordinates is supported for power users, and provides complete control of fragments’ relative orientations. Setting INTERFRAG_MODE to multi should suffice in almost all cases. Dimer coordinate table. provides the name and ordering convention for the coordinates.
Basic multifragment optimization. Use automatically generated reference points.
memory 4GB
molecule mol {
0 1
O 0.5026452583 0.9681078610 0.4772692868
H 2.3292990446 1.1611084524 0.4772692868
H 0.8887241813 0.8340933116 0.4772692868

0 1
C 0.8853463281 5.2235996493 0.5504918473
C 1.8139169342 2.1992967152 3.8040686146
C 2.8624456357 4.1143863257 0.5409035710
C 0.6240195463 4.8153482424 2.1904642137
C 0.1646305764 3.3031992532 3.8141619690
C 3.3271056135 2.6064153737 2.1669340785
H 0.5244823836 6.4459192939 0.7478283184
H 4.0823309159 4.4449979205 0.7680411190
H 2.2074914566 5.7109913627 2.2110247636
H 1.3768100495 2.9846751653 5.1327625515
H 4.9209603634 1.7288723155 2.1638694922
H 2.1923374156 0.9964630692 5.1155773223
nocom
units au
}
set {
basis 631+G
frag_mode MULTI
}
optimize("mp2")
Warning
The molecule input for psi4 has no effect upon optking, expect to provide Cartesian coordinates. Specifying independent fragments with the – seperator, will not trigger optking to add specific interfragment coordinates. Use FRAG_MODE multi.
Specify the reference points to use for coordinates via FRAG_REF_ATOMS. Each list corresponds to a fragment. A list of indices denotes a linear combination of the atoms. In this case, the first reference point for the second fragment is the center of the benzene ring. Indexing starts at 1, so the second fragment in this example starts at index 4.
memory 4GB
molecule mol {
0 1
O 0.5026452583 0.9681078610 0.4772692868
H 2.3292990446 1.1611084524 0.4772692868
H 0.8887241813 0.8340933116 0.4772692868

0 1
C 0.8853463281 5.2235996493 0.5504918473
C 1.8139169342 2.1992967152 3.8040686146
C 2.8624456357 4.1143863257 0.5409035710
C 0.6240195463 4.8153482424 2.1904642137
C 0.1646305764 3.3031992532 3.8141619690
C 3.3271056135 2.6064153737 2.1669340785
H 0.5244823836 6.4459192939 0.7478283184
H 4.0823309159 4.4449979205 0.7680411190
H 2.2074914566 5.7109913627 2.2110247636
H 1.3768100495 2.9846751653 5.1327625515
H 4.9209603634 1.7288723155 2.1638694922
H 2.1923374156 0.9964630692 5.1155773223
nocom
units au
}
set {
basis 631+G
frag_mode MULTI
# The line below specifies the reference points that will be used to construct the
# interfragment coordinates between the two fragments (called A and B).
# The format is the following:
# [[A1], [A2], [A3]], [[B1], [B2], [B3]]
#
# In terms of atoms within each fragment, the line below chooses, for water:
# H3 of water for the first reference point, O1 of water for the second reference point, and
# H2 of water for the third reference point.
# For benzene: the mean of the positions of all the C atoms, C2, one of the Carbon atoms,
# and C6, another one of the carbon atoms.
frag_ref_atoms [
[[3], [1], [2]], [[4, 5, 6, 7, 8, 9], [5], [9]]
]
}
optimize("mp2")
For even greater control, a dictionary can be passed to INTERFRAG_COORDS
The coordinates that are created between two dimers depend upon the number of atoms present The fragments A and B have up to 3 reference atoms each as shown in Dimer coordinate table. The interfragment coordinates are named and can be frozen according to their names as show in example below. For specifying reference points, use 1 based indexing.
name 
type 
atomlabels 
present, if 

RAB 
distance 
A0B0 
always 
theta_A 
angle 
A1A0B0 
A has > 1 atom 
theta_B 
angle 
A0B0B1 
B has > 1 atom 
tau 
dihedral 
A1A0B0B1 
A and B have > 1 atom 
phi_A 
dihedral 
A2A1A0B0 
A has > 2 atoms. Is not linear 
phi_B 
dihedral 
A0B0B1B2 
B has > 2 atoms. Is not linear 
A constrained optimization is performed where the orientation of the two fragments is fixed but the distance between the fragments and all intrafragment coordinates are allowed to relax. In this example, the centers of the benzene and thiophene rings are selected for the first reference points. The methyl groups carbon and one hydrogen are selected for the other two reference points on the first fragments. For fragment two, two carbons of the benzene ring are chosen for the other reference points.
memory 4GB
molecule mol {
C 1.258686 0.546935 0.436840
H 0.683650 1.200389 1.102833
C 0.699036 0.349093 0.396608
C 2.693370 0.550414 0.355311
H 3.336987 1.206824 0.952052
C 3.159324 0.343127 0.536418
H 4.199699 0.558111 0.805894
S 1.883829 1.212288 1.301525
C 0.786082 0.656530 0.606057
H 1.387673 0.016033 0.048976
H 1.054892 0.465272 1.651226
H 0.978834 1.708370 0.365860

C 6.955593 0.119764 1.395442
C 6.977905 0.135060 1.376787
C 7.111625 1.067403 0.697024
C 6.810717 1.314577 0.707746
C 6.821873 1.322226 0.678369
C 7.122781 1.059754 0.689090
H 7.226173 2.012097 1.240759
H 6.687348 2.253224 1.259958
H 6.707325 2.266920 1.222105
H 7.246150 1.998400 1.241304
O 6.944245 0.111984 2.805375
H 7.058224 0.807436 3.049180
C 6.990227 0.143507 2.907714
H 8.018305 0.274985 3.264065
H 6.592753 0.807024 3.281508
H 6.368443 0.968607 3.273516
nocom
unit angstrom
}
# Create a python dictionary and convert to string for pass through to optking
MTdimer = """{
"Natoms per frag": [12, 16],
"A Frag": 1,
"A Ref Atoms": [[1, 3, 4, 6, 8], [8], [11]],
"A Label": "methylthiophene",
"B Frag": 2,
"B Ref Atoms": [[13, 14, 15, 16, 17, 18], [13], [15]],
"B Label": "tyrosine",
"Frozen": ["theta_A", "theta_B", "tau", "phi_A", "phi_B"],
}"""
set {
basis 631+G
frag_mode MULTI
interfrag_coords $MTdimer
}
optimize("mp2")
Dealing with problematic optimizations¶
Although optking is continuously improved with robustness in mind, some attempted optimizations will inevitably fail to converge to the desired minima. For difficult cases, the following suggestions are made.
As for any optimizer, computing the Hessian and limiting the step size will successfully converge a higher percentage of cases. The default settings have been chosen because they perform efficiently for common, representative test sets. More restrictive, cautious steps are sometimes necessary.
DYNAMIC_LEVEL allows optking to change the method of optimization toward algorithms that, while often less efficient, may help to converge difficult cases. If this is initially set to 1, then optking, as poor steps are detected, will increase the dynamic level through several forms of more robust and cautious algorithms. The changes will reduce the trust radius, allow backward steps (partial line searching), add cartesian coordinates, switch to cartesian coordinates, and take steepestdescent steps.
The developers have found the OPT_COORDINATES set to “BOTH” which includes both the redundant internal coordinate set, as well as cartesian coordinates, works well for systems with long ‘arms’ or floppy portions of a molecule poorly described by local internals.
Optking does support the specification of ghost atoms. Certain internal coordinates such as torsions become poorly defined when they contain nearlinear bends. An internal error AlgError may be raised in such cases. Optking will avoid such coordinates when choosing an initial coordinate system; however, they may arise in the course of an optimization. In such cases, try restarting from the most recent geometry. Alternatively, setting OPT_COORDINATES to cartesian will avoid any internal coordinate difficulties altogether. These coordinate changes can be automatically performed by turning DYNAMIC_LEVEL to 1.
Warning
In some cases, such as the coordinate issues described above, optking will reset to maintain a consistent history. If an error occurs in Psi4 due to GEOM_MAXITER being exceeded but the final step report indicates that optking has not taken GEOM_MAXITER steps, such a reset has occured. Inspection will show that the step counter was reset to 1 somewhere in the optimization.
Convergence Criteria¶
Optking monitors five quantities to evaluate the progress of a geometry optimization. These are (with their keywords) the change in energy (MAX_ENERGY_G_CONVERGENCE), the maximum element of the gradient (MAX_FORCE_G_CONVERGENCE), the rootmeansquare of the gradient (RMS_FORCE_G_CONVERGENCE), the maximum element of displacement (MAX_DISP_G_CONVERGENCE), and the rootmeansquare of displacement (RMS_DISP_G_CONVERGENCE), all in internal coordinates and atomic units. Usually, these options will not be set directly. Primary control for geometry convergence lies with the keyword G_CONVERGENCE which sets the aforementioned in accordance with Table Geometry Convergence.
Max Energy 
Max Force 
RMS Force 
Max Disp 
RMS Disp 


NWCHEM_LOOSE [4] 
\(4.5 \times 10^{3}\) 
\(3.0 \times 10^{3}\) 
\(5.4 \times 10^{3}\) 
\(3.6 \times 10^{3}\) 

GAU_LOOSE [6] 
\(2.5 \times 10^{3}\) 
\(1.7 \times 10^{3}\) 
\(1.0 \times 10^{2}\) 
\(6.7 \times 10^{3}\) 

TURBOMOLE [4] 
\(1.0 \times 10^{6}\) 
\(1.0 \times 10^{3}\) 
\(5.0 \times 10^{4}\) 
\(1.0 \times 10^{3}\) 
\(5.0 \times 10^{4}\) 
\(4.5 \times 10^{4}\) 
\(3.0 \times 10^{4}\) 
\(1.8 \times 10^{3}\) 
\(1.2 \times 10^{3}\) 

CFOUR [4] 
\(1.0 \times 10^{4}\) 

\(1.0 \times 10^{6}\) 
\(3.0 \times 10^{4}\) 
\(1.2 \times 10^{3}\) 

\(1.0 \times 10^{6}\) 
\(3.0 \times 10^{4}\) 
\(3.0 \times 10^{4}\) 

INTERFRAG_TIGHT [7] 
\(1.0 \times 10^{6}\) 
\(1.5 \times 10^{5}\) 
\(1.0 \times 10^{5}\) 
\(6.0 \times 10^{4}\) 
\(4.0 \times 10^{4}\) 
\(1.5 \times 10^{5}\) 
\(1.0 \times 10^{5}\) 
\(6.0 \times 10^{5}\) 
\(4.0 \times 10^{5}\) 

GAU_VERYTIGHT [6] 
\(2.0 \times 10^{6}\) 
\(1.0 \times 10^{6}\) 
\(6.0 \times 10^{6}\) 
\(4.0 \times 10^{6}\) 
Footnotes
For ultimate control, specifying a value for any of the five monitored options activates that criterium and overwrites/appends it to the criteria set by G_CONVERGENCE. Note that this revokes the special convergence arrangements detailed in notes [5] and [6] and instead requires all active criteria to be fulfilled to achieve convergence. To avoid this revokation, turn on keyword FLEXIBLE_G_CONVERGENCE.
Interface to GeomeTRIC¶
The GeomeTRIC optimizer developed by Wang and Song [Wang:2016:214108] may be used in place of Psi4’s native Optking optimizer. GeomeTRIC uses a translationrotationinternal coordinate (TRIC) system that works well for optimizing geometries of systems containing noncovalent interactions.
Use of the GeomeTRIC optimizer is specified with the engine
argument to
optimize()
. The optimization will respect the keywords G_CONVERGENCE
and GEOM_MAXITER. Any other GeomeTRICspecific options (including constraints)
may be specified with the optimizer_keywords
argument to optimize()
.
Constraints may be placed on cartesian coordinates, bonds, angles, and dihedrals, and they can be
used to either freeze a coordinate or set it to a specific value. See the GeomeTRIC github
for more information on keywords and JSON specification of constraints.
Optimize the water molecule using GeomeTRIC:
molecule h2o { O H 1 1.0 H 1 1.0 2 160.0 } set { maxiter 100 g_convergence gau } optimize('hf/ccpvdz', engine='geometric')
Optimize the water molecule using GeomeTRIC, with one of the two OH bonds constrained to 2.0 au and the HOH angle constrained to 104.5 degrees:
molecule h2o { O H 1 1.0 H 1 1.0 2 160.0 } set { maxiter 100 g_convergence gau } geometric_keywords = { 'coordsys' : 'tric', 'constraints' : { 'set' : [{'type' : 'distance', 'indices' : [0, 1], 'value' : 2.0 }, {'type' : 'angle', 'indices' : [1, 0, 2], 'value' : 104.5 }] } } optimize('hf/ccpvdz', engine='geometric', optimizer_keywords=geometric_keywords)
Optimize the benzene/water dimer using GeomeTRIC, with the 6 carbon atoms of benzene frozen in place:
molecule h2o { C 0.833 1.221 0.504 H 1.482 2.086 0.518 C 1.379 0.055 0.486 H 2.453 0.184 0.483 C 0.546 1.167 0.474 H 0.971 2.162 0.466 C 0.833 1.001 0.475 H 1.482 1.867 0.468 C 1.379 0.275 0.490 H 2.453 0.404 0.491 C 0.546 1.386 0.506 H 0.971 2.381 0.524  O 0.000 0.147 3.265 H 0.000 0.505 2.581 H 0.000 0.965 2.790 no_com no_reorient } set { maxiter 100 g_convergence gau } geometric_keywords = { 'coordsys' : 'tric', 'constraints' : { 'freeze' : [{'type' : 'xyz', 'indices' : [0, 2, 4, 6, 8, 10]}] } } optimize('hf/ccpvdz', engine='geometric', optimizer_keywords=geometric_keywords)
Output¶
The progress of a geometry optimization can be monitored by grepping the output file for the
tilde character (~
). This produces a table like the one below that shows
for each iteration the value for each of the five quantities and whether the criterion
is active and fulfilled (*
), active and unfulfilled ( ), or inactive (o
).
 ~
Step Total Energy Delta E MAX Force RMS Force MAX Disp RMS Disp ~
 ~
Convergence Criteria 1.00e06 * 3.00e04 * o 1.20e03 * o ~
 ~
1 38.91591820 3.89e+01 6.91e02 5.72e02 o 1.42e01 1.19e01 o ~
2 38.92529543 9.38e03 6.21e03 3.91e03 o 2.00e02 1.18e02 o ~
3 38.92540669 1.11e04 4.04e03 2.46e03 o 3.63e02 2.12e02 o ~
4 38.92548668 8.00e05 2.30e04 * 1.92e04 o 1.99e03 1.17e03 o ~
5 38.92548698 2.98e07 * 3.95e05 * 3.35e05 o 1.37e04 * 1.05e04 o ~
The full list of keywords for optking is provided in Appendix OPTKING.
Information on the Psithon function that drives geometry optimizations is provided
at optimize()
.
Important User Changes from cppoptking¶
FIXED_COORD keywords have been generalized to RANGED_COORD e.g. RANGED_DISTANCE
Detailed optimization is now printed through the python logging system. If more information about the optimization is needed. Please see <output_name>.log