From GAUSSIAN@cmchem.chem.cmu.edu Sat Jul 22 07:57:11 1995
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Date: Fri, 21 Jul 1995 13:41:42 -0400 (EDT)
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Subject: Clarification of Alternatives for Geometry Optimization in G94
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Gaussian 94 brings some significant changes in the default geometry
optimization procedures. We've received enough questions about these new
optimizations in redundant internal coordinates that we have decided to
prepare this note to clarify the discussion in the Gaussian 94 User's
Reference. Note these important points up front:

* Redundant internal coordinates are the default because in general they are
  much more efficient and reliable than any alternative.

* All of the Gaussian 92 procedures are still available. Use the
  Opt=Z-Matrix option to tell the program to use them.

* All optimization types--including partial optimizations--that were
  possible with Gaussian 92 can be performed in redundant internal
  coordinates. This note will detail the steps required for doing so.


TABLE OF CONTENTS

* Overview of Geometry Optimizations in Gaussian 94
* The Opt=AddRedundant Option
* Example: Adding Redundant Internal Coordinates
* Performing Partial Optimizations
* Displaying the Value of a Desired Coordinate
* Reading/Modifying a Structure from the Checkpoint File
* Modifying Optimized Structures (Why You Don't Need a Z-matrix)
* Performing Relaxed Potential Energy Surface Scans
* References


Overview of Geometry Optimizations in Gaussian 94

By default, Gaussian 94 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 [1]. Similarly, mixed internal and Cartesian coordinates were
shown to have some advantages for some cases [2] (among them, ease of use in
specifying certain types of molecules).

Pulay has demonstrated [3-5], 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 [6]. By default, Gaussian 94 performs optimizations
via the Berny algorithm in redundant internal coordinates; these new
procedures are the work of H. B. Schlegel and coworkers [7].  In Reference
7, it is demonstrated that these redundant internal coordinates are
also advantageous for optimizations of most types of molecules,
including transition states.

In addition to employing a new coordinate system, this optimization
procedure operates somewhat differently from those employed in earlier
versions of Gaussian:

* The choice of coordinate system for the starting molecular structure is,
  quite literally, irrelevant, and it has no effect on the way the
  optimization proceeds.  There is no requirement to choose Z-matrix
  coordinates carefully in order to achieve good performance. All of the
  efficiency factors in the various coordinate systems are of no
  consequence, since all structures are described by internally-generated
  redundant internal coordinates.

* All optimizations in redundant internal coordinates are full optimizations
  unless coordinates are explicitly frozen using the AddRedundant option.
  Including a separate constants variable section in the molecule
  specification or specifying parameters in the Z-matrix numerically instead
  of using variables DOES NOT result in any frozen variables. Furthermore,
  the optimization is not hindered if all of the variables in the Z-matrix
  are not linearly independent.

* The output for the optimized structure lists all of the redundant internal
  coordinates rather than merely those present in the original molecule
  specification. This will take some getting used to for experienced
  Gaussian users.

Optimizations in redundant internal coordinates DO make use of geometry
constraint information and numerical differentiation specifications, (using
a different input format--see below). Optimization in internal coordinates,
which was the default procedure in Gaussian 92, is still available, via the
Opt=Z-matrix option.


The Opt=AddRedundant Option

The Opt=AddRedundant options indicates that you will specify additional
redundant internal coordinate data (which appears in a separate input
section following the final molecule specification section). This is also
the mechanism for specifying a partial geometry optimization, for modifying
a given structure, and many other variations.

Additional redundant internal coordinates are specified using one of the
following forms:

FORMAT                                               USE
------------------------------------------------     --------------------
N1   N2   [N3   [N4]]  [value]                       Add/specify a coord.
N1   N2   [N3   [N4]]  [value]  F                    Specify a constraint.
N1   N2   [N3   [N4]]           R                    Remove a coordinate.
N1   N2   [N3   [N4]]  [value]  D                    Request numer. diff.
N1   N2   [N3   [N4]]  [value]  H diag-elem          Specify Hessian elem.
N1   N2   [N3   [N4]]  [value]  S #steps incr        Relaxed PES scan.

where N1 through N4 are numbers indicating atoms from the molecule
specification (numbering begins at 1 and any dummy atoms are not counted).
If only two atoms are specified, then the input is interpreted as specifying
a bond; three atoms specify a bond angle, and four atoms specify a dihedral
angle. The optional value sets the initial value for that coordinate.

Alternatively, a coordinate may be constrained to a specific value by
following the value with F code letter. Note that this constraint will be
met in the FINAL (optimized) structure, but not necessarily in the
initial, starting structure.

The R code may be specified to remove a redundant internal coordinate.
(Only the specified coordinate is removed; removing a bond does not
automatically remove any associated bond angles or dihedral angles.)

AddRedundant coordinate specifications may also optionally include
differentiation or constraint information. The D code indicates that the
coordinate is to be numerically differentiated when all or part of the
Hessian is being computed numerically. The H code is used for specifying
diagonal elements of the Hessian. The S code requests a relaxed potential
energy surface scan (discussed later in this note).

As the following examples will illustrate, the AddRedundant facility is very
general, and it can be used to make any of a wide variety of modifications
to a starting set of coordinates (it probably should have been named
ModRedundant).


Example: Adding Redundant Internal Coordinates

The following input file illustrates the method for including additional
redundant internal coordinates within a molecule specification:

# HF/6-31G(d) Opt=AddRedun 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

7  8 

This structure is acetaldehyde with an OH substituted for one of the
hydrogens in the methyl group; the AddRedundant input creates a hydrogen
bond between that hydrogen atom and the oxygen atom in the carbonyl group.
Note that the preceding input adds only the bond between these two atoms.
The associated angles and dihedral angles could be added as well if they
were desired.


Performing Partial Optimizations in Redundant Internal Coordinates

The following job illustrates the method for freezing variables during a
redundant internal coordinate optimization:

# HF/6-31G* Opt=AddRedundant 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 input for
the AddRedundant 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 and defaults to the value
from the Z-matrix for the second modified redundant internal coordinate.

(It is possible to request constraints which are inconsistent; in this case,
the optimized structure will be as close to the constrained values as possible,
in a least-squares sense.)

See pp. 114-115 of the Gaussian 94 User's Reference for additional
information about the AddRedundant option and its input.


Displaying the Value of a Desired Coordinate

If you want to determine the value of some particular coordinate in an
optimization, traditionally you had to define that quantity as one of the
Z-matrix variables to be optimized. Now, all that is necessary is to define
it as an additional redundant internal coordinate, again using
Opt=AddRedundant, and it value will be displayed at all points of the
optimization.


Reading/Modifying 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=Modify instead; modifications have a form
identical to the input for Opt=AddRedundant:

N1  N2  [N3  [N4]]  [new-value]  [A | F | D | S  #steps  increment]

where the first four items are atom numbers, new-value is an optional new
value to be assigned to the coordinate, and the final item is an optional
code letter: 

A=activate a coordinate
F=freeze a coordinate
D=numerically differentiate a coordinate
S=perform a relaxed PES scan. 

See pp. 78-79 of the Gaussian 94 User's Reference for more information.


Modifying Optimized Structures (Why You Don't Need a Z-matrix)

At present, redundant internal coordinate optimizations do not provide a
traditional Z-matrix as output. This will be disconcerting to experienced
users accustomed to getting the final structure in this form, especially
when they want to modify the final structure for a later run. The latter is
still complete feasible, and indeed is even easier than before. Simply use
the Cartesian coordinates version of the optimized structure as your
starting point (this is the same way you would obtain an editable copy of
the Z-matrix from an optimization in Z-matrix coordinates). It can be
generated by a route like this one:

# Guess=Only Geom=Check

(It can also be extracted from the checkpoint file using the NewZMat utility
or from an archive entry.) Once you have the structure in Cartesian
coordinates from the output of this job, you can use it in a variety of
ways:

* Add and/or remove atoms from it. Additional atoms may be specified in
  either Cartesian or internal coordinates.

* Modify it by substituting atoms or groups: For example, you could change a
  hydrogen to a methyl group by editing the structure, replacing the desired
  hydrogen with a carbon atoms, and then adding three additional hydrogen
  atoms bonded to that carbon. The latter could be given in internal
  coordinates:

  ...                      ...
  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 (we'll need to specify Opt=AddRedundant in the final job's
  route section) 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 status of one or more
variables, then you can use Geom=Modify as discussed previously.


Performing Relaxed Potential Energy Surface Scans

The same technique can be used to perform a relaxed potential energy surface
(PES) scan in redundant internal coordinates. 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 default operation of the Scan keyword in that a full
geometry optimization is performed at each point. Relaxed PES scans can be
requested with either the Scan or Opt keywords (Scan can also still be used
to request a traditional, energy-computation only PES scan).

Here is an example input file for a PES scan in which the geometry
optimizations are performed in redundant internal coordinates. It uses the
Opt keyword:

# HF/6-31G* Opt=AddRedundant Test

Relaxed PES Scan--opt at every point

0 1
O
H 1 1.0
H 1 1.0 2 120.

2 1 3 110.0 S 20 0.5

The scanning variables are set up via the AddRedundant option.  This input
says to vary the H-O-H angle from 110 degrees to 120 degrees in increments
of 0.5 degrees. See pp. 140-41 of the Gaussian 94 User's Reference for more
information about relaxed PES scans.

The Opt=Z-matrix keyword may also be used to perform a relaxed potential
energy surface (PES) scan. 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.  This may be more
convenient than a scan in redundant coordinates if symmetrically equivalent
variables are being scanned.

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
R1 1.1
A2 115.4 S 2 1.0
...