# Specifying Generalized Internal Coordinates

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).
• 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 then 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:

```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 and include it in the optimization. Its value will be reported in Å:
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.
```

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: 23 April 2020. [G16 Rev. C.01]