Gaussian 03 Online Manual
This keyword requests that a geometry optimization be performed. The geometry will be adjusted until a stationary point on the potential surface is found. Gradients will be used if available. For the Hartree-Fock, CIS, MP2, MP3, MP4(SDQ), CID, CISD, CCD, CCSD, QCISD, 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 redundant internal coordinates [149,15] (specified by the Redundant option). The default algorithm for all methods lacking analytic gradients is the eigenvalue-following algorithm (Opt=EF). The Berny algorithm using internal coordinates (Opt=Z-matrix) is also available [136,148,529].
The remainder of this quite lengthy section discusses various aspects of geometry optimizations, and it includes these subsections:
Users should consult those subsection(s) that apply to their interests and needs.
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 . See also Appendix B if you are interested in details about setting up Z-matrices for various types of molecules.
GENERAL PROCEDURAL OPTIONS
If QST2 is specified, the title and molecule specification sections for both reactant and product structures are required as input as usual. The remaining M-2 points on the path are then generated by linear interpolation between the reactant and product input structures. The highest energy structure becomes the initial guess for the transition structure. At each step in the path relaxation, the highest point at each step is optimized toward the transition structure.
If QST3 is specified, a third set of title and molecule specification sections must be included in the input as a guess for the transition state as usual. The remaining M-3 points on the path are generated by two successive linear interpolations, first between the reactant and transition structure and then between the transition structure and product. By default, the central point is optimized to the transition structure, regardless of the ordering of the energies. In this case, M must be an odd number so that the points on the path may be distributed evenly between the two sides of the transition structure.
In the output for a simultaneous optimization calculation, the predicted geometry for the optimized transition structure is followed by a list of all M converged reaction path structures.
The treatment of the input reactant and product structures is controlled by other options: OptReactant, OptProduct, BiMolecular.
Note that the SCF wavefunction for structures in the reactant valley may be quite different from that of structures in the product valley. Guess=Always can be used to prevent the wavefunction of a reactant-like structure from being used as a guess for the wavefunction of a product-like structure.
Lines in a ModRedundant input section use the following syntax:
[Type] N1 [N2 [N3 [N4]]] [[+=]value] [A | F] [[min] max]]
[Type] N1 [N2 [N3 [N4]]] [[+=]value] B [[min] max]]
[Type] N1 [N2 [N3 [N4]]] K | R [[min] max]]
[Type] N1 [N2 [N3 [N4]]] [[+=]value] D [[min] max]]
[Type] N1 [N2 [N3 [N4]]] [[+=]value] H diag-elem [[min] max]]
[Type] N1 [N2 [N3 [N4]]] [[+=]value] 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. Value specifies a new value for the specified coordinate, and +=value increments the coordinate by value.
The atom numbers and coordinate value 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:
An asterisk (*) in the place of an atom number indicates a wildcard. Min and max then define a range (or maximum value if min is not given) for coordinate specifications containing wildcards. The action specified by the action code is taken only if the value of the coordinate is in the range.
Here are some examples of wildcard use:
When the action codes K and B are used with one or two atoms, the meaning of a wildcard is extended to include all applicable atoms, not just those involving defined coordinates.
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:
See the examples later in this section for illustrations of the use of this keyword.
COORDINATE SYSTEM SELECTION OPTIONS
When a Z-matrix without any variables is used for the molecule specification,and Opt=Z-matrix is specified, then the optimization will actually be performed in Cartesian coordinates.
Note that a variety of other coordinate systems, such as distance matrix coordinates, can be constructed using the ModRedundant option.
Energy (format D24.16)
The force constants are in lower triangular form: ((F(J,I),J=1,I),I=1,NAt3), where NAt3 is the number of Cartesian coordinates.
These options are available for the Berny algorithm only.
Mic120 says to use microiterations in L120 for ONIOM(MO:MM), even for mechanical embedding. This is the default for electronic embedding. Mic103 says to perform microiterations in L103 for ONIOM(MO:MM). It is the default for mechanical embedding, and it does not work for electronic embedding.
Specifies the Hessian update method. Keyword is one of: Powell,
BFGS, PDBFGS, ND2Corr, OD2Corr, D2CorrBFGS,
Bofill, D2CMix and None.
This method avoids the matrix diagonalizations. Consequently, the eigenvector following methods (Opt=TS) cannot be used in conjunction with it. QST2 and QST3 calculations are guided using an associated surface approximation, but this may not be as effective as the normal method involving eigenvector following.
OVERVIEW OF GEOMETRY OPTIMIZATIONS IN GAUSSIAN
By default, Gaussian performs the optimization in redundant internal coordinates. This is a change from previous versions of the program. There has been substantial controversy in recent years concerning the optimal coordinate system for optimizations. For example, Cartesian coordinates were shown to be preferable to internal coordinates (Z-matrices) for some cyclic molecules . Similarly, mixed internal and Cartesian coordinates were shown to have some advantages for some cases  (among them, ease of use in specifying certain types of molecules).
Pulay has demonstrated [540,541,542], however, that redundant internal coordinates are the best choice for optimizing polycyclic molecules, and Baker reached a similar conclusion when he compared redundant internal coordinates to Cartesian coordinates . By default, Gaussian performs optimizations via the Berny algorithm in redundant internal coordinates; these procedures are also the work of H. B. Schlegel and coworkers .
This optimization procedure operates somewhat differently from those traditionally employed in electronic structure programs (including Gaussian 94 and earlier versions):
Optimizations in redundant internal coordinates do make use of geometry constraint information and numerical differentiation specifications. See the examples subsection for details.
Optimizations in internal coordinates, which was the default procedure in Gaussian 92, is still available, via the Opt=Z-Matrix option.
WAYS OF GENERATING INITIAL FORCE CONSTANTS
Unless you specify otherwise, a Berny geometry optimization starts with an initial guess for the second derivative matrix-also known as the Hessian-which is determined using connectivity determined from atomic radii and a simple valence force field [149,544]. The approximate matrix is improved at each point using the computed first derivatives.
This scheme usually works fine, but for some cases, such as Z-matrices with unusual arrangements of dummy atoms, the initial guess may be so poor that the optimization fails to start off properly or spends many early steps improving the Hessian without nearing the optimized structure. In addition, for optimizations to transition states (see also below), some knowledge of the curvature around the saddle point is essential, and the default approximate Hessian must always be improved.
In these cases, there are several methods for providing improved force constants:
Redundant Internals Z-matrix 1 2 3 104.5 A 104.5 1 2 1.0 H 0.55 R 1.0 H 0.55
Redundant Internals Z-matrix 1 2 1.0 D R1 1.0 2 3 1.5 R2 1.5 1 2 3 104.5 D A1 104.5 D 2 3 4 110.0 A2 110.0
OPTIMIZING TO A TRANSITION STATE OR HIGHER-ORDER SADDLE POINT
Transition State Optimizations Using Synchronous Transit-Guided Quasi-Newton (STQN) Methods. Gaussian includes the STQN method for locating transition structures. This method, implemented by H. B. Schlegel and coworkers [149,150], 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.
Despite the superficial similarity, this method is very different from the Linear Synchronous Transit method for locating transition structures requested with the now-deprecated LST keyword. Opt=QST2 generates a guess for the transition structure that is midway between the reactants and products in terms of redundant internal coordinates, and it then goes on to optimize that starting structure to a first-order saddle point automatically. The Linear Synchronous Transit method merely locates a maximum along a path connecting two structures which may be used as a starting structure for a subsequent manually-initiated transition state optimization; LST does not locate a proper stationary point. In contrast, QST2 and QST3 do locate proper transition states.
Traditional Transition State Optimizations Using the Berny Algorithm. The Berny optimization program can also optimize to a saddle point using internal coordinates, if it is coaxed along properly. The options to request this procedure are Opt=TS for a transition state (saddle point of order 1) or Opt(Saddle=N) for a saddle point which is a maximum in N directions.
When searching for a local minimum, the Berny algorithm uses a combination of rational function optimization (RFO) and linear search steps to achieve speed and reliability (as described below). This linear search step cannot be applied when searching for a transition state. Consequently, transition state optimizations are much more sensitive to the curvature of the surface. A transition state optimization should always be started using one of the options described above for specifying curvature information. Without a full second derivative matrix the initial step is dependent on the choice of coordinate system, so it is best to try to make the reaction coordinate (direction of negative curvature) correspond to one or two redundant internal coordinates or Z-matrix variables (see the examples below).
In the extreme case in which the optimization begins in a region known to have the correct curvature (e.g., starting with Opt=CalcFC) and steps into a region of undesirable curvature, the Opt=CalcAll option may be useful. This is quite expensive, but the full optimization procedure with correct second derivatives at every point will usually reach a stationary point of correct curvature if started in the desired region. For suggestions on locating transition structures, refer to the literature .
An eigenvalue-following (mode walking) optimization method [146,147] can be requested by Opt=EF. This was sometimes superior to the Berny method in Gaussian 88, but since the RFO step  has now been incorporated into the Berny algorithm, EF is seldom preferable unless its ability to follow a particular mode is needed, or gradients are not available (in which case Berny can't be used anyway). This algorithm has a dimensioning limit of 50 active variables. By default, the lowest mode is followed. This is correct when already in a region of correct curvature and when the softest mode is to be followed uphill. This default can be overridden in two ways:
Ang1 104.5 4
# Opt=(EF,TS) HCN --> HNC transition state search This job deliberately follows the wrong (second) mode! 0,1 N C,1,CN H,1,CH,2,HCN CN 1.3 CH 1.20 10 Requests the second mode. HCN 60.0
By default, the Berny optimization program checks the curvature (number of negative eigenvalues) of its approximate second derivative matrix at each step of a transition state optimization. If the number is not correct (1 for a transition state), the job is aborted. A search for a minimum will often succeed in spite of bad real or approximate curvature, because the steepest descent and RFO parts of the algorithm will keep the optimization moving downward, although it may also indicate that the optimization has moved away from the desired minimum and is headed through a transition state and on to a different minimum. On the other hand, a transition state optimization has less chance of success if the curvature is wrong at the current point. However, the test can be suppressed with the NoEigenTest option. If NoEigenTest is used, it is best to MaxCycle to a small value (e.g. 5) and check the structure after a few iterations.
THE BERNY OPTIMIZATION ALGORITHM
The Berny geometry optimization algorithm in Gaussian is based on an earlier program written by H. B. Schlegel which implemented his published algorithm . 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:
CHANGE IN TRADITIONAL CONVERGENCE CRITERIA BEGINNING WITH GAUSSIAN 98
Gaussian 98 introduced one small but significant change in the criteria for determining when a geometry has converged. When the forces are two orders of magnitude smaller than the cutoff value (i.e., 1/100th of the limiting value), then the geometry is considered converged even if the displacement is larger than the cutoff value. This test was introduced to facilitate optimizations of large molecules which may have a very flat potential energy surface around the minimum.
The generation of redundant internal coordinates for weakly bound complexes was also updated with Gaussian 98. We include Hydrogen bonds automatically. In addition, in connecting different fragments which are only weakly bound (hydrogen-bonded and otherwise), all pairs of atoms with one atom in each fragment having distance within a factor of 1.3 of the closest pair have their distances added to the internal coordinates. If at least 3 such pairs are found, then no angles or dihedrals involving both fragments are added. However, if only 1 or two pairs of atoms are close, then the related angles and dihedrals are added in order to ensure a complete coordinate system. As usual, the ModRedundant option can be used to add or remove any coordinates manually.
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.
The examples in the subsection will focus on normal optimization procedures in Gaussian 03. However, at the end of the subsection, examples illustrating traditional, Z-matrix-based optimizations using the Berny algorithm will also be given.
Basic Optimization Input. Traditionally, geometry optimizations required a Z-matrix specifying both the starting geometry and the variables to be optimized. For example, the input file in the left column below could be used for such an optimization on water:
# HF/6-31G(d) Opt Test # HF/6-31G(d) Opt Test Water opt Water opt 0 1 0 1 O1 O 0.00 0.00 0.00 H1 O1 R H 0.00 0.00 1.00 H2 O1 R H1 A H 0.97 0.00 -0.25 Variables: R=1.0 A=104.5
This Z-matrix specifies the starting configuration of the nuclei in the water molecule. It also specifies that the optimization should determine the values of R and A which minimize the energy. Since the OH bond distance is specified using the same variable for both hydrogen atoms, this Z-matrix also imposes (appropriate) symmetry constraints on the molecule.
The Cartesian coordinate input in the right column is equivalent to the Z-matrix in the left column. In early versions of Gaussian, such input would lead to an optimization performed in Cartesian coordinates; however, by Gaussian 92, Z-matrix input could be used for optimizations in either coordinate system.
By contrast, beginning with Gaussian 98 these two input files are exactly equivalent, and this holds for Gaussian 03 as well. They both will result in a Berny optimization in redundant internal coordinates, giving identical final output.
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.
When a Z-matrix was used for the initial molecule specification, this output will be followed by an expression of the optimized structure in that format, whenever possible.
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.
More detailed information about the out put from geometry optimizations is provided in Chap. 3 of Exploring Chemistry with Electronic Structure Methods.
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.
Specifying Redundant Internal Coordinates. The following input file illustrates the method for specifying redundant internal coordinates within an input file:
# 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 2 1 3
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 line adds only the bond between these two atoms. The associated angles and dihedral angles would need to be added as well if they were desired.
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 coordinate appropriate for molecules like C60  by removing all dihedral angles from the redundant internal coordinates:
* * * * R Remove all dihedral angles
The following input rotates the group about the N2-N3 bond by 10 degrees:
* N2 N3 * +=10.0 Add 10.0 to the values to dihedrals involving N2-N3 bond
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 a redundant internal coordinate optimization:
# HF/6-31G* Opt=ModRedundant Test Partial optimization 1 1 C H 1 R1 H 1 R1 2 A1 O 1 R2 2 A2 3 120.0 H 4 R3 3 A3 2 180.0 A1=120.0 ... R3=1.1 4 5 1.3 F 5 4 3 2 F
The structure is specified as a traditional Z-matrix, with its variables defined in a separate section. The final input section gives the values for the ModRedundant option. This input fixes the O-H bond and the dihedral angle for the final hydrogen atom. Note that any value specified in this manner need not be the same as the one listed in the preceding Z-matrix (as is the case for the O-H bond length); the structure is adjusted to enforce this constraint. The constrained value is optional. For example, in this case the value of second modified redundant internal coordinate defaults to the value from the Z-matrix (180.0).
Modifying Optimized Structures (Why You Don't Need a Z-matrix). Use the Cartesian coordinates version of the optimized structure as your starting point. It can be generated by a route like this one:
# Guess=Only Geom=Check
(It can also be extracted from an archive entry.) Once you have the structure in Cartesian coordinates, you can use it in a variety of ways:
H6 1.2 2.3 1.1 H6 1.2 2.3 1.1 H7 1.2 0.0 -.9 C7 1.2 0.0 -.9 H8 0.0 -.9 0.0 H8 0.0 -.9 0.0 H9 C7 R H5 A C2 180.0 H10 C7 R H6 A C2 180.0 H11 C7 R H8 A C2 -180.0 R=1.0 A=120.0 7 2 1.5
The new structure on the right also uses an additional redundant internal coordinate (specifying Opt=ModRedundant on the final job) to alter the bond distance for the new carbon atom which is replacing the hydrogen (bonded to atom 2).
If all you want to do is change the value or activate/frozen status of one or more variables, then you can use Geom=ModRedundant rather than this approach.
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 Berny optimization to a second-order saddle point:
# RHF/6-31G(d) Opt=(Saddle=2,Restart,MaxCyc=50) Test
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]]] [[+=]Value] [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:
# 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=Z-matrix and Opt=ModRedundant keywords may also be used to perform a relaxed potential energy surface (PES) scan. Like the scan facility provided by previous versions of Gaussian, a relaxed PES scan steps over a rectangular grid on the PES involving selected internal coordinates. It differs from the operation of the Scan keyword 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, or it will 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]] [[+=]value] S steps step-size. For example, this input adds a bond between atoms 2 and 3, setting its initial value to 1.0 Å, and specifying three scan steps of 0.05 Å each:
2 3 1.0 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. Note that all other dihedrals around the bond should be removed:
* N2 N3 * R Remove all dihedrals involving the N2-N3 bond N1 N2 N3 N4 S 20 2.0 Specify a relaxed PES scan of 20 steps in 2º increments
Full vs. Partial Optimizations. When it is performed in internal (Z-matrix) coordinates, the Berny optimization algorithm makes a distinction between full and partial optimizations. Full optimizations optimize all specified variables in order to find the lowest energy structure, while partial optimizations optimize only a specified subset of the variables. Note that the FOpt keyword form is used to request that the optimization variables be tested for linear independence prior to beginning the optimization.
Those variables whose values should be held fixed are specified in a separate input section, separated by the usual variables section by a blank line or a line containing a space in the first column and the string Constants:. For example, the following input file will optimize only the bond distance R, but not the angle A, which will be held fixed at 105.4 degrees throughout the optimization:
# HF/6-31G(d) Opt=Z-matrix Test Partial optimization for water 0 1 O H1 O R H2 O R H1 A Variables: R 1.0 Constants: A 105.4
Breaking Symmetry During an Optimization in Internal Coordinates. Below are two geometry specifications for water. The one on the left has been constrained to C2v symmetry; since the same variable is used for both bond lengths, their values will always be the same:
O O H 1 R1 H 1 R1 H 1 R1 2 A H 2 R2 2 A R1=0.9 R1=0.9 A=105.4 R2=1.1 A=105.4
By contrast, the Z-matrix on the right is unconstrained since the two bond lengths are specified by different variables having different initial values. Note that an optimization in redundant internal coordinates which begins from a C2v structure will retain that symmetry throughout the optimization.
Relaxed PES Scans. For Opt=Z-matrix, a relaxed PES scan is requested simply by tagging the Z-matrix variables whose values are to be incremented with the S code letter and the number of steps and the increment size. For example, the following input file requests a relaxed PES scan for the given molecule:
# HF/6-31G(d) Opt=Z-matrix Test Relaxed PES scan 0 1 O H 1 R1 C 1 R2 2 A2 ... Variables: R1 0.9 S 5 0.05 R2 1.1 A2 115.4 S 2 1.0 ...
This causes the variable R1 to be incremented five times, by 0.05 Å each time, and the variable A2 to be incremented twice, by 1 degree each time, resulting in a total of 18 geometry optimizations (the initial values for each variable also constitute a point within the scan).