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Optimization

This keyword requests that a geometry optimization be performed. The geometry will be adjusted until a stationary point on the potential surface is found. Analytic gradients will be used if available. For the Hartree-Fock, CIS, MP2, MP3, MP4(SDQ), CID, CISD, CCD, CCSD, QCISD, BD, CASSCF, and all DFT and semi-empirical methods, the default algorithm for both minimizations (optimizations to a local minimum) and optimizations to transition states and higher-order saddle points is the Berny algorithm using GEDIIS [Li06] in redundant internal coordinates [Pulay79, Fogarasi92, Pulay92, Baker93, Peng93, Peng96] (corresponding to the Redundant option). An brief overview of the Berny algorithm is provided in the final subsection of this discussion. The default algorithm for all methods lacking analytic gradients is the eigenvalue-following algorithm (Opt=EF).

Gaussian includes the STQN method for locating transition structures. This method, implemented by H. B. Schlegel and coworkers [Peng93, Peng96], uses a quadratic synchronous transit approach to get closer to the quadratic region of the transition state and then uses a quasi-Newton or eigenvector-following algorithm to complete the optimization. Like the default algorithm for minimizations, it performs optimizations by default in redundant internal coordinates. This method will converge efficiently when provided with an empirical estimate of the Hessian and suitable starting structures.

This method is requested with the QST2 and QST3 options. QST2 requires two molecule specifications, for the reactants and products, as its input, while QST3 requires three molecule specifications: the reactants, the products, and an initial structure for the transition state, in that order. The order of the atoms must be identical within all molecule specifications. See the examples for sample input for and output from this method.

Basic information as well as techniques and pitfalls related to geometry optimizations are discussed in detail in chapter 3 of Exploring Chemistry with Electronic Structure Methods [Foresman15]. For a review article on optimization and related subjects, see [Hratchian05a].

Gaussian 16 supports generalized internal coordinates (GIC), a facility which allows arbitrary redundant internal coordinates to be defined and used for optimization constraints and other purposes [Marenich17p]. There are several GIC-related options to Opt, and the GIC Info subsection describes using GICs as well as their limitations in the present implementation.

By default, optimizations search for a local minimum.

QST2

Search for a transition structure using the STQN method. This option requires the reactant and product structures as input, specified in two consecutive groups of title and molecule specification sections. Note that the atoms must be specified in the same order in the two structures. TS should not be specified with QST2.

QST3

Search for a transition structure using the STQN method. This option requires the reactant, product, and initial TS structures as input, specified in three consecutive groups of title and molecule specification sections. Note that the atoms must be specified in the same order within the three structures. TS should not be specified with QST3.

TS

Requests optimization to a transition state rather than a local minimum, using the Berny algorithm.

Saddle=N

Requests optimization to a saddle point of order N using the Berny algorithm.

Conical

Search for a conical intersection or avoided crossing using the state-averaged CASSCF method. Avoided is a synonym for Conical. Note that CASSCF=SlaterDet is needed in order to locate a conical intersection between a singlet state and a triplet state.

Options to Modify the Initial Geometry

ModRedundant

Except for any case when it is combined with the GIC option (see below), the ModRedundant option will add, delete, or modify redundant internal coordinate definitions (including scan and constraint information) before performing the calculation. This option requires a separate input section following the geometry specification; when used in conjunction with QST2 or QST3, a ModRedundant input section must follow each geometry specification. AddRedundant is synonymous with ModRedundant.

Lines in a ModRedundant input section use the following syntax:

[Type] N1 [N2 [N3 [N4]]] [A | F] [[min] max]
[Type] N1 [N2 [N3 [N4]]] B [[min] max]
[Type] N1 [N2 [N3 [N4]]] K | R [[min] max]
[Type] N1 [N2 [N3 [N4]]] D [[min] max]
[Type] N1 [N2 [N3 [N4]]] H diag-elem [[min] max]
[Type] N1 [N2 [N3 [N4]]] S nsteps stepsize [[min] max]

N1, N2, N3, and N4 are atom numbers or wildcards (discussed below). Atom numbering begins at 1, and any dummy atoms are not counted.

The atom numbers are followed by a one-character code letter indicating the coordinate modification to be performed; the action code is sometimes followed by additional required parameters as indicated above. If no action code is included, the default action is to add the specified coordinate. These are the available action codes:

A Activate the coordinate for optimization if it has been frozen.
F Freeze the coordinate in the optimization.
B Add the coordinate and build all related coordinates.
K Remove the coordinate and kill all related coordinates containing this coordinate.
R Remove the coordinate from the definition list (but not the related coordinates).
D Calculate numerical second derivatives for the row and column of the initial Hessian for this coordinate.
H Change the diagonal element for this coordinate in the initial Hessian to diag-elem.
S Perform a relaxed potential energy surface scan. Increment the coordinate by stepsize a total of nsteps times, performing an optimization from each resulting starting geometry.

An asterisk (*) in the place of an atom number indicates a wildcard. Min and Max only apply to coordinate specifications containing wildcards. The action then specified by the action code is taken only if the value of the coordinate is in the minmax range (or below maximum value if min is not given).

Here are some examples of wildcard use:

* All atoms specified by Cartesian coordinates.
* * All defined bonds.
3 * All defined bonds with atom 3.
* * * All defined valence angles.
* 4 * All defined valence angles around atom 4.
* * * * All defined dihedral angles.
* 3 4 * All defined dihedral angles around the bond connecting atoms 3 and 4.

By default, the coordinate type is determined from the number of atoms specified: Cartesian coordinates for 1 atom, bond stretch for 2 atoms, valence angle for 3 atoms, and dihedral angle for 4 atoms. Optionally, type can be used to designate these and additional coordinate types:

Analytic gradients are available for the HF, all DFT methods, CIS, MP2, MP3, MP4(SDQ), CID, CISD, CCD, CCSD, QCISD, CASSCF, and all semi-empirical methods.

The Tight, VeryTight, Expert, Eigentest and EstmFC options are available for the Berny algorithm only.

Optimizations of large molecules which have many very low frequency vibrational modes with DFT will often proceed more reliably when a larger DFT integration grid is requested (Int=UltraFine).

Output from Optimization Jobs. The string GradGradGrad… delimits the output from the Berny optimization procedures. On the first, initialization pass, the program prints a table giving the initial values of the variables to be optimized. For optimizations in redundant internal coordinates, all coordinates in use are displayed in the table (not merely those present in the molecule specification section):

 GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad
 Berny optimization.    The opt. algorithm is identified by the header format & this line.
 Initialization pass.
                   ----------------------------
                   !    Initial Parameters    !
                   ! (Angstroms and Degrees)  !
--------------------                          ----------------------
! Name  Definition              Value          Derivative Info.    !
--------------------------------------------------------------------
! R1    R(2,1)                  1.             estimate D2E/DX2   !
! R2    R(3,1)                  1.             estimate D2E/DX2   !
! A1    A(2,1,3)              104.5            estimate D2E/DX2   !
--------------------------------------------------------------------

The manner in which the initial second derivative are provided is indicated under the heading Derivative Info. In this case the second derivatives will be estimated.

Each subsequent step of the optimization is delimited by lines like these:

GradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGradGrad
Berny optimization.
Search for a local minimum.
Step number   4 out of a maximum of  20

Once the optimization completes, the final structure is displayed:

Optimization completed.
   -- Stationary point found.
                    ----------------------------
                    !   Optimized Parameters   !
                    ! (Angstroms and Degrees)  !
--------------------                            --------------------
! Name  Definition              Value          Derivative Info.    !
--------------------------------------------------------------------
! R1    R(2,1)                  0.9892         -DE/DX =    0.0002 !
! R2    R(3,1)                  0.9892         -DE/DX =    0.0002 !
! A1    A(2,1,3)              100.004          -DE/DX =    0.0001 !
--------------------------------------------------------------------

The redundant internal coordinate definitions are given in the second column of the table. The numbers in parentheses refer to the atoms within the molecule specification. For example, the variable R1, defined as R(2,1), specifies the bond length between atoms 1 and 2. The energy for the optimized structure will be found in the output from the final optimization step, which precedes this table in the output file.

Compound Jobs. Optimizations are commonly followed by frequency calculations at the optimized structure. To facilitate this procedure, the Opt keyword may be combined with Freq in the route section of an input file, and this combination will automatically generate a two-step job.

It is also common to follow an optimization with a single point energy calculation at a higher level of theory. The following route section automatically performs an HF/6-31G(d,p) optimization followed by an MP4/6-31G(d,p) single point energy calculation :

# MP4/6-31G(d,p)//HF/6-31G(d,p) Test

Note that the Opt keyword is not required in this case. However, it may be included if setting any of its options is desired.

Modifying Redundant Internal Coordinates. The following input file illustrates the method for modifying redundant internal coordinates within an input file:

X Cartesian coordinates.
B Bond length.
A Valence angle.
D Dihedral angle.
L Linear bend specified by three atoms (if N4 is -1) or by four atoms, where the fourth atom is used to determine the 2 orthogonal directions of the linear bend. In this case, min and max are each pairs of numbers, specifying the two orthogonal bending components.
# HF/6-31G(d) Opt=ModRedun Test  
 
Opt job  
 
0,1  
C1  0.000   0.000   0.000  
C2  0.000   0.000   1.505  
O3  1.047   0.000  -0.651  
H4 -1.000  -0.006  -0.484  
H5 -0.735   0.755   1.898  
H6 -0.295  -1.024   1.866  
O7  1.242   0.364   2.065  
H8  1.938  -0.001   1.499  
 
3  8 Adds hydrogen bond (but not angles or dihedrals).
2  1  3 Adds C-C-O angle.

This structure is acetaldehyde with an OH substituted for one of the hydrogens in the methyl group; the first input line for ModRedundant creates a hydrogen bond between that hydrogen atom and the oxygen atom in the carbonyl group. Note that this adds only the bond between these two atoms The associated angles and dihedral angles could be added as well using the B action code:

3  8  B

Displaying the Value of a Desired Coordinate. The second input line for ModRedundant specifies the C-C=O bond angle, ensuring that its value will be displayed in the summary structure table for each optimization step.

Using Wildcards in Redundant Internal Coordinates. A distance matrix coordinate system can be activated using the following input:

* * B Define all bonds between pairs of atoms
* * * K Remove all other redundant internal coordinates

The following input defines partial distance matrix coordinates to connect only the closest layers of atoms:

* * B 1.1 Define all bonds between atoms within 1.1 Å
* * * K Remove all other redundant internal coordinates

The following input sets up an optimization in redundant internal coordinates in which atoms N1 through Nn are frozen (such jobs may require the NoSymm keyword). Note that the lines containing the B action code will generate Cartesian coordinates for all of the coordinates involving the specified atom since only one atom number is specified:

N1 B Generate Cartesian coordinates involving atom N1
 
Nn B Generate Cartesian coordinates involving atom Nn
* F Freeze all Cartesian coordinates

The following input defines special “spherical” internal coordinates appropriate for molecules like C60 by removing all dihedral angles from the redundant internal coordinates:

* * * * R Remove all dihedral angles

Additional examples are found in the section on relaxed PES scans below.

Performing Partial Optimizations. The following job illustrates the method for freezing variables during an optimization:

# B3LYP/6-31G(d) Opt=ReadOpt   
 
Partial optimization of Fe2S2  
cluster with phenylthiolates.  
 
-2,1  
Fe 15.2630 -1.0091  7.0068  
S  14.8495  1.1490  7.0431  
Fe 17.0430  1.0091  7.0068  
S  17.4565 -1.1490  7.0431  
S  14.3762 -2.1581  8.7983  
C  12.5993 -2.1848  8.6878  
 
C  14.8285 -3.8823  3.3884  
H  14.3660 -3.3149  2.7071  
 
noatoms atoms=1-4 ReadOpt input.

The central cluster (the first four atoms) will be optimized while the phenylthiolates are frozen.

Restarting an Optimization. A failed optimization may be restarted from its checkpoint file by simply repeating the route section of the original job, adding the Restart option to the Opt keyword. For example, this route section restarts a B3LYP/6-31G(d) Berny optimization to a second-order saddle point:

%Chk=saddle2
# Opt=(TS,Restart,MaxCyc=50) Test

The model chemistry and starting geometry are retrieved from the checkpoint file. Options specifying the optimization type and procedure are required in the route section for the restart job (e.g., TS in the preceding example). Some parameter-setting options can be omitted to use the same values are for the original job, or they can be modified for the restarted job, such as MaxCycle in the example. Note that you must include CalcFC to compute the Hessian at the first point of the restarted job. Second derivatives are computed only when this option is present in the route section of the restarted job, regardless of whether it was specified for the original job.

Reading a Structure from the Checkpoint File. Redundant internal coordinate structures may be retrieved from the checkpoint file with Geom=Checkpoint as usual. The read-in structure may be altered by specifying Geom=ModRedundant as well; modifications have a form identical to the input for Opt=ModRedundant:

[Type] N1 [N2 [N3 [N4]]] [Action [Params]] [[Min] Max]]

Locating a Transition Structure with the STQN Method. The QST2 option initiates a search for a transition structure connecting specific reactants and products. The input for this option has this general structure (blank lines are omitted):

# HF/6-31G(d) Opt=QST2 # HF/6-31G(d) (Opt=QST2,ModRedun)
First title section First title section
Molecule specification for the reactants Molecule specification for the reactants
Second title section ModRedundant input for the reactants
Molecule specification for the products Second title section
  Molecule specification for the products
  ModRedundant input for the products (optional)

Note that each molecule specification is preceded by its own title section (and separating blank line). If the ModRedundant option is specified, then each molecule specification is followed by any desired modifications to the redundant internal coordinates.

Gaussian will automatically generate a starting structure for the transition structure midway between the reactant and product structures, and then perform an optimization to a first-order saddle point.

The QST3 option allows you to specify a better initial structure for the transition state. It requires the two title and molecule specification sections for the reactants and products as for QST2 and also additional, third title and molecule specification sections for the initial transition state geometry (along with the usual blank line separators), as well as three corresponding modifications to the redundant internal coordinates if the ModRedundant option is specified. The program will then locate the transition structure connecting the reactants and products closest to the specified initial geometry.

The optimized structure found by QST2 or QST3 appears in the output in a format similar to that for other types of geometry optimizations:

                    ----------------------------
                    !   Optimized Parameters   !
                    ! (Angstroms and Degrees)  !
---------------------                          ----------------------
! Name  Definition    Value    Reactant  Product  Derivative Info.  !
-------------------------------------------------------------------
! R1    R(2,1)        1.0836    1.083     1.084    -DE/DX =    0.  !
! R2    R(3,1)        1.4233    1.4047    1.4426   -DE/DX =   -0.   !
! R3    R(4,1)        1.4154    1.4347    1.3952   -DE/DX =   -0.   !
! R4    R(5,3)        1.3989    1.3989    1.3984   -DE/DX =    0.   !
! R5    R(6,3)        1.1009    1.0985    1.0995   -DE/DX =    0.   !
! …                                                              !
-------------------------------------------------------------------

In addition to listing the optimized values, the table includes those for the reactants and products.

Performing a Relaxed Potential Energy Surface Scan. The Opt=ModRedundant option may also be used to perform a relaxed potential energy surface (PES) scan. Like the facility provided by Scan, a relaxed PES scan steps over a rectangular grid on the PES involving selected internal coordinates. It differs from Scan in that a constrained geometry optimization is performed at each point.

Relaxed PES scans are available only for the Berny algorithm. If any scanning variable breaks symmetry during the calculation, then you must include NoSymm in the route section of the job, since it may fail with an error.

Redundant internal coordinates specified with the Opt=ModRedundant option may be scanned using the S code letter: N1 N2 [N3 [N4]] S steps step-size. For example, this input adds a bond between atoms 2 and 3, specifying three scan steps of 0.05 Å each:

2 3 S 3 0.05

Wildcards in the ModRedundant input may also be useful in setting up relaxed PES scans. For example, the following input is appropriate for a potential energy surface scan involving the N1-N2-N3-N4 dihedral angle:

N1 N2 N3 N4 S 20 2.0 Specify a relaxed PES scan of 20 steps in 2° increments

Cartesian coordinates can also include scan specifications:

X atom S x-steps x-size y-steps y-size z-steps z-size

For example, the following ModRedundant input performs a relaxed potential energy surface scan. Starting at the initial position of atom 1 and moving in five 0.2 Angstrom steps in the X direction and three 0.1 Angstrom steps in the Z direction:

X 1 S 5 0.2 0 0.0 3 0.1

The Berny geometry optimization algorithm in Gaussian is based on an earlier program written by H. B. Schlegel which implemented his published algorithm [Schlegel82]. The program has been considerably enhanced since this earlier version using techniques either taken from other algorithms or never published, and consequently it is appropriate to summarize the current status of the Berny algorithm here.

At each step of a Berny optimization the following actions are taken:

  • The Hessian is updated unless an analytic Hessian has been computed or it is the first step, in which case an estimate of the Hessian is made. Normally the update is done using an iterated BFGS for minima and an iterated Bofill for transition states in redundant internal coordinates, and using a modification of the original Schlegel update procedure for optimizations in internal coordinates. By default, this is derived from a valence force field [Schlegel84a], but upon request either a unit matrix or a diagonal Hessian can also be generated as estimates.
  • The trust radius (maximum allowed Newton-Raphson step) is updated if a minimum is sought, using the method of Fletcher [Fletcher80, Bofill94, Bofill95].
  • Any components of the gradient vector corresponding to frozen variables are set to zero or projected out, thereby eliminating their direct contribution to the next optimization step.
    If a minimum is sought, perform a linear search between the latest point and the best previous point (the previous point having lowest energy). If second derivatives are available at both points and a minimum is sought, a quintic polynomial fit is attempted first; if it does not have a minimum in the acceptable range (see below) or if second derivatives are not available, a constrained quartic fit is attempted. This fits a quartic polynomial to the energy and first derivative (along the connecting line) at the two points with the constraint that the second derivative of the polynomial just reach zero at its minimum, thereby ensuring that the polynomial itself has exactly one minimum. If this fit fails or if the resulting step is unacceptable, a simple cubic is fit is done.
    Any quintic or quartic step is considered acceptable if the latest point is the best so far but if the newest point is not the best, the linear search must return a point in between the most recent and the best step to be acceptable. Cubic steps are never accepted unless they are in between the two points or no larger than the previous step. Finally, if all fits fail and the most recent step is the best so far, no linear step is taken. If all fits fail and the most recent step is not the best, the linear step is taken to the midpoint of the line connecting the most recent and the best previous points.
  • If the latest point is the best so far or if a transition state is sought, a quadratic step is determined using the current (possibly approximate) second derivatives. If a linear search was done, the quadratic step is taken from the point extrapolated using the linear search and uses forces at that point estimated by interpolating between the forces at the two points used in the linear search. By default, this step uses the Rational Function Optimization (RFO) approach [Simons83, Banerjee85, Baker86, Baker87]. The RFO step behaves better than the Newton-Raphson method used in earlier versions of Gaussian when the curvature at the current point is not that desired. The old Newton-Raphson step is available as an option.
  • Any components of the step vector resulting from the quadratic step corresponding to frozen variables are set to zero or projected out.
  • If the quadratic step exceeds the trust radius and a minimum is sought, the step is reduced in length to the trust radius by searching for a minimum of the quadratic function on the sphere having the trust radius, as discussed by Jørgensen [Golab83]. If a transition state is sought or if NRScale was requested, the quadratic step is simply scaled down to the trust radius.
  • Finally, convergence is tested against criteria for the maximum force component, root-mean square force, maximum step component, and root-mean-square step. The step is the change between the most recent point and the next to be computed (the sum of the linear and quadratic steps).

Examples of Using GICs

Basic GIC input. Here is an example of using the generalized internal coordinates defined by the user from scratch for the geometry optimization of the water molecule.

# HF opt=readallgic

Title

 0 1
O    0.0000    0.0000    0.0000
H    0.0000    0.0000    1.3112
H    1.0354    0.0000   -0.6225

R(1,2)
R(1,3)
HOH=A(2,1,3)

The atomic indexes 1, 2, and 3 refer to the oxygen atom, the first and the second hydrogen atom, respectively. The first and the second expression define the O-H bonds, and the third one defines the H-O-H valence angle (with the user-provided label “HOH”). An excerpt of the output with a table containing the initial values of the GICs is shown below.

                           ----------------------------
                           !    Initial Parameters    !
                           ! (Angstroms and Degrees)  !
 --------------------------                            --------------------------
 ! Name  Definition              Value          Derivative Info.                !
 --------------------------------------------------------------------------------
 ! R1    R(1,2)                  1.3112         estimate D2E/DX2                !
 ! R2    R(1,3)                  1.2081         estimate D2E/DX2                !
 ! HOH   A(2,1,3)              121.015          estimate D2E/DX2                !
 --------------------------------------------------------------------------------

Note that the labels “R1” and “R2” above were assigned by default. The coordinates R1=R(1,2) and R2=R(1,3) are parsed as pure distances and given here in Angstroms, and the HOH=A(2,1,3) is a pure valence angle in degrees.

# HF opt=readallgic

Title 

 0 1
O    0.0000    0.0000    0.0000
H    0.0000    0.0000    1.3112
H    1.0354    0.0000   -0.6225

OHSym1=(R(1,2)+R(1,3))/sqrt(2)
OHSym2=(R(1,2)-R(1,3))/sqrt(2)
HOH=A(2,1,3)

The first and the second expression in the example above define the symmetrized O-H bonds, and the third one is the H-O-H valence angle.

                           ----------------------------
                           !    Initial Parameters    !
                           ! (Angstroms and Degrees)  !
 --------------------------                            --------------------------
 ! Name   Definition             Value          Derivative Info.                !
 --------------------------------------------------------------------------------
 ! OHSym1 GIC-1                  3.3664         estimate D2E/DX2                !
 ! OHSym2 GIC-2                  0.1377         estimate D2E/DX2                !
 ! HOH    A(2,1,3)             121.015          estimate D2E/DX2                !
 --------------------------------------------------------------------------------
 NOTE: GIC-type coordinates are in arbitrary units.

The coordinates OHSym1 and OHSym2 are parsed as generic GICs and therefore given here in arbitrary units. The units are actually Bohrs in this case because the 2-1/2 factor is taken as dimensionless and the values of R(1,2) and R(1,3) are taken in Bohrs.

# HF opt=readallgic

Title

 0 1
O
H  1  1.3
H  1  1.2  2  120.

R12=SQRT[{X(2)-X(1)}^2+{Y(2)-Y(1)}^2+{Z(2)-Z(1)}^2]
R13=SQRT[{X(3)-X(1)}^2+{Y(3)-Y(1)}^2+{Z(3)-Z(1)}^2]
A0(Inactive)=DotDiff(2,1,3,1)/{R12*R13}
A213=ArcCos(A0)

The GIC input section above defines two bond distances and one valence angle expressed via Cartesian coordinates. The coordinate A0 is defined as the dot-product (DotDiff) of the vectors R12 and R13 divided by the product of their lengths, and it is selected as “inactive” (i.e., excluded from the geometry optimization). An excerpt of the output with a table containing the initial values of the GICs is shown below.

                           ----------------------------
                           !    Initial Parameters    !
                           ! (Angstroms and Degrees)  !
 --------------------------                            --------------------------
 ! Name  Definition              Value          Derivative Info.                !
 --------------------------------------------------------------------------------
 ! R12   GIC-1                   2.4566         estimate D2E/DX2                !
 ! R13   GIC-2                   2.2677         estimate D2E/DX2                !
 ! A213  GIC-3                   2.0944         estimate D2E/DX2                !
 --------------------------------------------------------------------------------
 NOTE: GIC-type coordinates are in arbitrary units.

The values of R12, R13, and the dot-product are calculated using the Cartesian coordinates given in Bohrs. The GIC arbitrary units are Bohrs (for R12 and R13) and radians (for A213).

The options that do not mention GIC and can be used with the Opt keyword should work as described—except for NoFreeze, which should not be combined with any GIC-related option. In the latter case, use the UnFreezeAll flag in the GIC input section, as described below.

This section discusses specifying generalized internal coordinates (GICs) in Gaussian input files. GICs have many potential uses: defining additional coordinates whose values are reported during geometry optimizations, freezing various structural parameters during the optimization of a molecular system, specifying parameters over which to perform a scan, defining constraints for geometry optimizations based on structural parameters or complex relationships between them, requesting calculation of parts of the Hessian, and other purposes.

The GIC input section is separated from the earlier input by a blank line. It has one or more lines containing coordinate definitions, expressions or standalone options. Here is a simple GIC input section for water illustrating some of the possible features:

R(1,2)                 Define a bond length coordinate for atoms 1 and 2
Bond2=R[1,3]           Define another bond length coordinate named Bond2
HOH(freeze)=A(2,1,3)   Define an optimization constraint: a bond angle coordinate named HOH (∠2-1-3)

For an optimization, these coordinates will result in the bond angle remaining fixed at its initial value and the two bond distances being optimized.

The basic form of a coordinate is the following:

label(options)=expression

All of the components are optional. In the preceding examples, all components were present only in the third line. The first line contained only a coordinate expression, while the second line also contained a label without options. Note that options may also be placed following the expression:

HOH=A(2,1,3) Freeze

Labels are user-assigned identifiers for the coordinate. They are not case sensitive. Labels many contain letters and number, but must begin with a letter. If no label is specified, a generic one will be assigned by the program (e.g., R1, R2, A1, etc.). A parenthesized, comma-separated list of options can be included following the label if desired. Note that square brackets or braces may be substituted for parentheses anywhere in a coordinate definition.

Structural Parameters

Coordinates are defined by expressions. The simplest expressions simply identify a specific structural parameter within the molecule, using the following constructs. Note that an asterisk may be used as a wildcard for any atom number (see the examples).

R(i,j)

Define a bond coordinate between atoms i and j. B, Bond and Stretch are synonyms for R.

A(i,j,k)

Define a non-linear angle coordinate involving atoms i, j and k where the angle vertex is at atom j. Angle and Bend are synonyms for A.

D(i,j,k,l)

Define a dihedral angle between the plane containing atoms i, j and k and the plane containing atoms j, k and l. Dihedral and Torsion are synonyms for D.

L(i,j,k,l,M)

Define the linear bend coordinate involving atoms i, j and k where the angle vertex is at atom j. Linear and LinearBend are synonyms for L.

A linear bend definition has two components, indicated by M values of -1 and -2 for the first and second components, respectively (no other values are permitted). A linear bend is specified by defining its two orthogonal directions. These can be indicated in two ways:

  • For a nonlinear molecule with more than 3 atoms, a fourth atom which does not form a linear angle with i, j and k in any combination can be used. In this case, l can be set to its atom number. For example, the following may be used to specify a linear bend involving atoms 1, 2 and 3 using atom 6 to determine the two orthogonal directions:
    L(1,2,3,6,-1)
    L(1,2,3,6,-2)

    If l is set to -4, then the fourth atom will be determined automatically based on the molecular geometry.

  • The other method is to project the linear bend onto one of the coordinate system’s axial planes: the values of -1, -2 and -3 for l specify the YZ, XZ and XY planes (respectively). The value 0 may also be used to request that the appropriate plane be determined automatically:
    L(1,2,3,0,-1)
    L(1,2,3,0,-2)

X(i)

Define the x Cartesian coordinate for atom i. Cartesian(i,-1) and Cartesian(i,X) are synonyms, and Cartesian may be abbreviated as Cart.

Y(i)

Define the y Cartesian coordinate for atom i. Cartesian(i,-2) and Cartesian(i,Y) are synonyms, and Cartesian may be abbreviated as Cart.

Z(i)

Define the z Cartesian coordinate for atom i. Cartesian(i,-3) and Cartesian(i,Z) are synonyms, and Cartesian may be abbreviated as Cart.

XCntr(atom-list)
YCntr(atom-list)
ZCntr(atom-list)

Define x, y or z Cartesian coordinate for the geometric center (centroid) of a molecular fragment that contains specified atoms. The atom list is a comma-separated list of atom numbers and/or ranges. For example, XCntr(1,12-15,27) defines the x coordinate of the fragment containing atoms 1, 12, 13, 14, 15 and 27. If the atom list is omitted, it defaults to the entire molecule.

DotDiff(i,j,k,l)

Define the dot product (a·b) of the two Cartesian coordinate difference vectors a and b for atoms i, j, k and l determined as a = (XiXj, YiYj, ZiZj) and b = (XkXl, YkYl, ZkZl).

Compound Expressions

Complex expressions may be constructed by combining multiple items using one or more mathematical operations. The argument(s) A and B can be the labels of a previously defined coordinate, a valid GIC expression or even constants (integer or floating-point). The operation names are not case sensitive. The following operations are available:

  • Square root: SQRT(A).
  • Power of e: EXP(A) for eA.
  • Trigonometric functions: SIN(A), COS(A), TAN(A).
  • Inverse cosine: ARCCOS(A).
  • Addition: A+B
  • Subtraction: A–B
  • Multiplication: A*B
  • Division: A/B
  • Exponentiation: A**n for An (n is an integer). The form A^n is also accepted.

Here are some simple examples which define symmetrized OH bonds in water:

R12(inactive)=B(1,2)
R13(inactive)=B(1,3)
RSym  = (R12 + R13)/SQRT(2)
RASym = [Bond(1,2) - Bond(1,3)]/SQRT(2)

The first two coordinates are set as inactive since they are intermediates not intended to be used in the optimization. Line 3 illustrates an expression using previously defined labels, while line 4 shows the use of literal expressions with operators. Note that the argument to the square root function is the constant 2.

Options

A comma separated list of options can follow the coordinate label, enclosed in parentheses. Alternatively, options may follow the expression, separated from it and from one another by spaces. All options are case insensitive.

For the purposes of geometry optimizations, a coordinate can be designated as:

  • Active: The coordinate is part of the list of internal coordinates used in the geometry optimzation. In contrast, Inactive coordinates are not included in the set used for the geometry optimization. By default, active coordinates are unfrozen: allowed to change value (see the next bullet).
  • Frozen: A coordinate whose value is held constant during the course of a geometry optimization. The values of active, unfrozen coordinates change during a geometry optimization. The frozen or unfrozen status of inactive coordinates is irrelevant during an optimization.

In the descriptions that follow, coordinates that “already exist” refers to previously-defined coordinates with the same label or the same value expression. Such coordinates may have been defined earlier in the input stream or retrieved from the checkpoint file from a previous job.

Active

If the specified coordinate does not already exist, build a new coordinate defined by the given expression, and flag it as active and unfrozen. If the coordinate was previously defined, then flag it as active and unfrozen (whatever its previous status). It is the default. Activate, Add and Build are synonyms for Active. May be abbreviated to A when specified following the expression.

Frozen

Build a coordinate defined by the expression if it does not exist, and flag the coordinate as active for geometry optimizations and freeze it at the current value.
Freeze is a synonym for Frozen. May be abbreviated to F when specified following the expression.

Inactive

If the coordiante does not already exist, build a new coordinate defined by the expression and flag it inactive. If the coordinate with the given label or for the given expression has been already built and flagged as active (frozen or unfrozen), then remove it from the geometry optimization by flagging it as inactive. Remove is a synonym for Inactive. May be abbreviated to R when specified following the expression.

Kill

Remove the coordinate from the list of internal coordinates used in geometry optimization along with any dependent coordinates by flagging all of them as inactive. The dependent coordinates include any coordinate that depends on the same atoms as the given coordinate. For example, R(1,5) Kill will result in removing the coordinate R(1,5)—the internuclear distance between atoms 1 and 5—as well as the valence angles, dihedral angles and any other coordinate that depends on the Cartesian coordinates of atoms 1 and 5 in combination with other atoms in the molecule. RemoveAll is a synonym for Kill. May be abbreviated to K when specified following the expression.

PrintOnly

Include the initial value of the coordinate in the starting geometry in the Gaussian output file, and the flag it as inactive).

Modify

A label must be included in the coordinate specification for this option. It replaces the old coordinate with the specified label with the new expression, and flags the newly modified coordinate as active and unfrozen.

Diff

Calculate numerical second derivatives for the row and column of the initial Hessian corresponding to this coordinate. May be abbreviated to D when specified following the expression.

FC=x

Change the diagonal element for the given coordinate in the initial Hessian to x, a floating-point number in atomic units. ForceConstant is a synonym for FC.

Value=x

Set the initial value for the given internal coordinate to x, a floating point value. The units for the value are those of the Gaussian program, as defined by the Units keyword (Angstroms or degrees by default). The current Cartesian coordinates will be adjusted to match this value as closely as possible. This option should be used cautiously and sparingly. It is far easier and more reliable to set the initial molecular structure as desired in a graphical environment like GaussView.

StepSize=x,NSteps=n

These options are used to specify a relaxed potential energy surface scan in which the coordinate is incremented by x a total of n times, and a constrained optimization is perfromed from each resulting starting geometry. x should be a positive floating-point number in atomic units, N should be an integer >1. When these options follow the expression, the comma separating them should be replaced by a space.

Min=min,Max=max

This option is used in combination with Active, Freeze or Inactive. It adds, freezes or makes inactive the coordinate when its value satisfies the condition minvaluemax. min and max are floating-point numbers in the units defined by the Units (Angstroms or degrees by default). If Min or Max is omitted, the condition becomes valuemax or minmin respectively. When these options follow the expression, the comma should be replaced by a space.

action OnlyIf condition

action IfNot condition

These options provide conditional coordinate operations. They can only be placed following the expression defining the current coordinate. Action is one of Active, Freeze or Inactive. The condition is a label or expression for another coordinate. The specified action will be performed for the current coordinate if the coordinate referred to in condition is active for OnlyIf or inactive for IfNot. Note that the conditional test applies only to the action specified preceding the option and not to other options that may be present in the coordinate specification.

Standalone Options

The following options are independent of coordinate definitions and apply globally. They should be specified alone on their input line.

FreezeAll

Freeze all internal coordinate previously added as active.

UnFreezeAll

Unfreeze all internal coordinates previously added as active frozen.

RemoveAll

Remove/inactivate all internal coordinate previously added as active (frozen or unfrozen).

Atom i action

Apply the specified action to the Cartesian coordinates of atom i. If i is an asterisk, then the action applies to all atoms. Action is one of Active, Freeze, UnFreeze, Remove (make inactive), RemoveAll and XYZOnly. These options are as defined above; XYZOnly says to remove any internal coordinates that depend on atom i but to add/retain the coordinates of that atom. The default action is Active.

Examples

The following example manipulates some automatically-generated coordinates, defines some new ones, and then uses wildcards to remove coordinates related to specific atoms:

R(5,9) freeze                 		Freeze bond distance R(5,9). 
R(8,9)                        		Add a new active coordinate R(8,9) with a default label.
Ang189 = A(1,8,9)             		Add a new active coordinate A(1,8,9) labeled Ang189.
R10(remove)                             Remove a coordinate labeled R10.
Dih6123(remove) = D(6,1,2,3)  		If D(6,1,2,3) exists, then remove the coordinate.
Dis79(freeze) = R(7,9)        		Freeze the coordinate R(7,9): if it is new, then label it Dis79; if it already exists, retain the old label.
G1 = (R16+R19)*0.529177       		Add a new coordinate labeled G1.
Ang189a(modify)=cos(g2)*57.29577951   	Change the definition of coordinate Ang189a.
R(11,*) remove                		Remove distances between atom 11 and any other atom.
D(*,1,17,*) remove            		Remove any dihedral built around the 1-17 bond.

Note that if a specified coordinate already exists, then an entry adding it will result in an error (e.g., lines 1-3 above).

The following example first defines the centroids of two fragments. Then, it defines the interfragment distance as an optimization coordinate, setting its initial value to 5Å:

Define the center of Fragment 1, but don't include it in the optimization.
XC1(Inactive)=XCntr(1-10)
YC1(Inactive)=YCntr(1-10)
ZC1(Inactive)=ZCntr(1-10)
Define the center of Fragment 2, but don't include it in the optimization.
XC2(Inactive)=XCntr(11-21)
YC2(Inactive)=YCntr(11-21)
ZC2(Inactive)=ZCntr(11-21)
Define the distance F1-F2 in Angstroms, and include it in the optimization.
F1F2=sqrt[(XC1-XC2)^2+(YC1-YC2)^2+(ZC1-ZC2)^2]*0.529177

The following example requests a relaxed PES scan over the same coordinate:

F1F2(NSteps=10,StepSize=0.2)

The following example removes an angle coordinate generated by default if ≥179.9°, substituting a linear bend:

A(1,2,3) Remove Min=179.9           Remove angle coordinate if too large.  
L(1,2,3,0,-1) Add IfNot A(1,2,3)    Add linear bend only if the angle coordinate not active.
L(1,2,3,0,-2) Add IfNot A(1,2,3)

The following example removes an angle coordinate if it is ≤ the specified value, setting the corresponding force constant is set to 0.2 au. The latter applies whenever it is needed: as the initial force constant and the force constant to use should be variable be reactivated. The second line specifies the force constant for a bond coordinate:

A(1,2,3) Remove Min=3.139847 ForceConstant=0.2            
R(1,2) FC=0.5 

The following example sets the force constants for various coordinates. It also inactivates bond angle coordinates ≥ 179.8°:

R(1,*) FC=0.8
D(*,4,5,*) FC=0.4
A(*,1,*) FC=0.5
A(*,*,*) R Min=179.8

Limitations of GICs in the Current Implementation

In the current implementation, GICs can be successfully used for many purposes including optimization constraints and PES scans. However, there are potential problems with active composite coordinates including multiple dihedral angles. In general, coordinates comprised of combinations of bond distances and bond angles should behave well. Simple dihedral angles are also welll supported. Complex expressions involving multiple dihedral angles are acceptable for frozen coordinates and for PES scans. However, they should be avoided as active optimization coordinates.

In a non-GIC optimization, or one using GICs with only regular dihedrals, then the program is careful about the periodicity of these coordinates. For example, in deciding whether a step in the geometry is too big and needs to be scaled back, it recognizes that a change in value from 1 degree to 359 degrees is really a change of -2 degrees rather than 358 degrees. Similarly, in numerically differentiating the forces in order to update the Hessian, displacements between geometries in internal coordinates are needed, and the periodicity is accounted for. A problem can arise when a GIC is a combination of parts for which such periodicity is important, typically, combinations of multiple dihedral angles. For example, consider these GICs:

D1 = D(1,2,3,4)
D2 = D(5,6,7,8)
V1 = D1 + 2*D2

D1 and D2 are dihedral angles, but they are intermediates and are not used as variables in the optimization. Their periodicity is not currently recognized in the composite coordinate V1. Suppose they have values of 1 and 2 degrees at one geometry and 1 and 359 degress at the next. The change in the optimization variable V1 should be 0 + 2*(-3) = -6 degrees, but it is actually 0 + 2*(357) = 714 degrees, which looks like an enormous change. This will result in the optimization algorithm performing very poorly. V1 isn’t a simple periodic function; it is necessary to apply periodicity to its component parts as it is computed, which is not done in the current GIC implementation.

GIC Units in Gaussian Output

The values of the GICs defined as pure distances and angles (including valence angles, linear bends and dihedral angles/torsions) are computed from the Cartesian coordinates in atomic units (Bohrs) and stored internally in Bohrs and radians. However, for the user’s convenience, they are expressed as usual in Angstroms and degrees in the Gaussian output. In the case of a generic GIC (i.e., when the GIC is not a pure Cartesian coordinate, bond distance or angle), the GIC value is computed as a function of Cartesian coordinates and bond distances in Bohrs and angles in radians, combined with optional constants in user-defined units. Such generic GIC values (labeled as GIC) are computed, stored and output in these same units: i.e., if the GIC is a combination of bonds or a combination of valence angles, then the arbitrary units become Bohrs for the bonds and radians for the angles.

Use of ModRedundant Format Input

Modifications to the GICs can be read in using the ModRedundant format from the current internal coordinate algorithm. However, the old format is only available with the GICs that include only pure bond distances, bond angles or torsion angles. In addition, the old format and the new GIC format described above cannot be mixed together in the same input section.

Last updated on: 05 January 2017. [G16 Rev. A.03]


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