COORD

COORD=keyword   in $CONTRL

Diese verschiedenen Koordinatensysteme versteht man am besten anhand von Beispielen:

COORD=UNIQUE

$DATA-Input für Benzol

In der ersten Spalte der Geometriedaten steht das Elementsymbol gefolgt von der Ordnungszahl. Dann folgen die x-, y- und z-Koordinate (wie bei kartesischen Koordinaten). Die z-Achse ist die Hauptdrehachse; die z-Koordinaten für beide Atome sind 0.00000. Beide Atome liegen auf der y-Achse, ihr x-Wert ist 0.00000. Das C-Atom ist 1.39000 A vom Ursprung (dem Mittelpunkt des Benzolrings entfernt, das H-Atom ist 2.47000 A vom Ursprung entfernt.  Die Durchführung der Symmetrieoperation (D6h) führt zu den vollständigen Benzolkoordinaten (Man überzeuge sich im Output-File!). Da im regelmäßigen Sechseck der C-Abstand vom Zentrum gleich dem C-C-Abstand ist, entspricht der Wert von 1.39000 A dem C-C-Abstand. Der C-H-Abstand ergibt sich dann aus  2.47 A - 1.39 A = 1.08 A.  
Der Nachteil von UNIQUE ist, dass im Output auch nur cartesische Koordinaten ausgegeben werden. Man erhält also z.B. keine Bindungswinkel. Um diese zu erhalten, muss man die cartesischen Koordinaten in ein geeignetes Grafikprogramm eingeben, um sich die Winkel angeben zu lassen.

COORD=CART

$DATA-Inputfile für N₂O₄

Der Nachteil von CART ist, dass im Output auch nur cartesische Koordinaten ausgegeben werden. Man erhält also z.B. keine Bindungswinkel. Um diese zu erhalten, muss man die cartesischen Koordinaten in ein geeignetes Grafikprogramm eingeben, um sich die Winkel angeben zu lassen.

COORD=HINT (Hilderbrandt INTernal coordinates)

Es wird ein Koordinatensystem bestehend aus 4 Dummy-Atomen konstruiert. Das erste Atom sitzt im Koordinatenursprung, die anderen sitzen auf der x-, y- und z-Achse im Einheitsabstand. Die Konstruktion der Molekülgeometrie geschieht dadurch, dass die Atome zu den Dummy-Atomen in Beziehung gesetzt werden. Nur symmetrie-relevante Atome (unique atoms) werden angegeben. 

$DATA-Inputfile für H₂O

Nach dem Elementnamen (bis zu 10 Buchstaben; nur für output relevant) und der Ordnungszahl (Dezimalpunkt nicht vergessen!) folgt der Verbindungstyp (connection type): 
LC: Linear Connection. Dann der Bindungsabstand in A: 0.0. Dann der erste Bindungswinkel: 0.0. Dann der zweite Bindungswinkel: 0.0. Es folgt das Vorzeichen der Verbindung und bis zu vier Verbindungspunkte (connection points). O steht für origin, I: Einheitspunkt auf der x-Achse, J auf der y-Achse und K auf der z-Achse. Das Sauerstoffatom sitzt also im Ursprung des Koordinatensystems. 
Das H-Atom hat eine Bindungslänge zu O von 0.94 A, PCC: Planar Central Connection: der Bindungswinkel zum O-Atom beträgt 53° bezüglich dem Ursprung (O), der z- (K) und der x-Achse (I). 

COORD=ZMT

$DATA-Inputfile für N₂O₄

COORD=ZMTMPC

$DATA-Inputfile für N₂O₄

Ist ähnlich aufgebaut wie eine Z-Matrix. Die Konnektivität wird allerdings mit den letzten drei Zahlen einer Zeile definiert. Die Zahl hinter der Geometrieangabe bedeutet (für MOPAC-Programm): 1: Parameter optimieren; 0: nicht optimieren.

* * * Coordinate Choices * * *
Optimization in cartesian coordinates has a reputation of converging slowly. This is largely
due to the fact that translations and rotations are usually left in the problem. Numerical
problems caused by the small eigenvalues associated with these degrees of freedom are the
source of this poor convergence. The methods in GAMESS project the hessian matrix to
eliminate these degrees of freedom, which should not cause a problem. Nonetheless, Cartesian
coordinates are in general the most slowly convergent coordinate system.
The use of internal coordinates (see NZVAR in $CONTRL as well as $ZMAT) also eliminates
the six rotational and translational degrees of freedom. Also, when internal coordinates are
used, the GUESS hessian is able to use empirical information about bond stretches and bends. On
the other hand, there are many possible choices for the internal coordinates, and some of these
may lead to much poorer convergence of the geometry search than others. Particularly poorly
chosen coordinates may not even correspond to a quadratic surface, thereby ending all hope that
a quasi-Newton method will converge.
Internal coordinates are frequently strongly coupled. Because of this, Jerry Boatz has called
them "infernal coordinates"! A very common example to illustrate this might be a bond length
in a ring, and the angle on the opposite side of the ring. Clearly, changing one changes the other
simultaneously. A more mathematical definition of "coupled" is to say that there is a large off-diagonal
element in the hessian. In this case convergence may be unsatisfactory, especially with
a diagonal GUESS hessian, where a "good" set of internals is one with a diagonally dominant
hessian. Of course, if you provide an accurately computed hessian, it will have large off-diagonal
values where those are truly present. Even so, convergence may be poor if the
coordinates are coupled through large 3rd or higher derivatives. The best coordinates are
therefore those which are the most "quadratic".
One very popular set of internal coordinates is the usual "model builder" Z-matrix input,
where for N atoms, one uses N-1 bond lengths, N-2 bond angles, and N-3 bond torsions. The
popularity of this choice is based on its ease of use in specifying the initial molecular geometry.
Typically, however, it is the worst possible choice of internal coordinates, and in the case of
rings, is not even as good as Cartesian coordinates.However, GAMESS does not require this particular mix of the common types. GAMESS' only
requirement is that you use a total of 3N-6 coordinates, chosen from these 3 basic types, or
several more exotic possibilities. (Of course, we mean 3N-5 throughout for linear molecules).
These additional types of internal coordinates include linear bends for 3 collinear atoms, out of
plane bends, and so on. There is no reason at all why you should place yourself in a
straightjacket of N-1 bonds, N-2 angles, and N-3 torsions. If the molecule has symmetry, be
sure to use internals which are symmetrically related.
For example, the most effective choice of coordinates for the atoms in a four membered ring
is to define all four sides, any one of the internal angles, and a dihedral defining the ring pucker.
For a six membered ring, the best coordinates seem to be 6 sides, 3 angles, and 3 torsions. The
angles should be every other internal angle, so that the molecule can "breathe" freely. The
torsions should be arranged so that the central bond of each is placed on alternating bonds of the
ring, as if they were pi bonds in Kekule benzene. For a five membered ring, we suggest all 5
sides, 2 internal angles, again alternating every other one, and 2 dihedrals to fill in. The
internal angles of necessity skip two atoms where the ring closes. Larger rings should
generalize on the idea of using all sides but only alternating angles. If there are fused rings,
start with angles on the fused bond, and alternate angles as you go around from this position.
Rings and more especially fused rings can be tricky. For these systems, especially, we
suggest the Cadillac of internal coordinates, the "natural internal coordinates" of Peter Pulay.For a description of these, see
P.Pulay, G.Fogarosi, F.Pang, J.E.Boggs, J.Am.Chem.Soc. 101, 2550-2560 (1979 ).
G.Fogarasi, X.Zhou, P.W.Taylor, P.Pulay J.Am.Chem.Soc. 114, 8191-8201 (1992 ).
These are linear combinations of local coordinates, except in the case of rings. The examples
given in these two papers are very thorough.
An illustration of these types of coordinates is given in the example job EXAM25.INP,
distributed with GAMESS. This is a nonsense molecule, designed to show many kinds of functional
groups. It is defined using standard bond distances with a classical Z-matrix input, and the
angles in the ring are adjusted so that the starting value of the unclosed OO bond is also a
standard value.
Using Cartesian coordinates is easiest, but takes a very large number of steps to converge.
This however, is better than using the classical Z-matrix internals given in $DATA, which is
accomplished by setting NZVAR to the correct 3N-6 value. The geometry search changes the OO
bond length to a very short value on the 1st step, and the SCF fails to converge. (Note that if you
have used dummy atoms in the $DATA input, you cannot simply enter NZVAR to optimize in
internal coordinates, instead you must give a $ZMAT which involves only real atoms).
The third choice of internal coordinates is the best set which can be made from the simple
coordinates. It follows the advice given above for five membered rings, and because it includes
the OO bond, has no trouble with crashing this bond. It takes 20 steps to converge, so the
trouble of generating this $ZMAT is certainly worth it compared to the use of Cartesians.
Natural internal coordinates are defined in the final group of input. The coordinates are set
up first for the ring, including two linear combinations of all angles and all torsions within the
ring. After this the methyl is hooked to the ring as if it were a NH group, using the usual
terminal methyl hydrogen definitions. The H is hooked to this same ring carbon as if it were a
methine. The NH and the CH2 within the ring follow Pulay's rules exactly. The amount of input
is much greater than a normal Z-matrix. For example, 46 internal coordinates are given,
which are then placed in 3N-6=33 linear combinations. Note that natural internals tend to berich in bends, and short on torsions.
The energy results for the three coordinate systems which converge are as follows:
NSERCH Cart good Z-mat nat. int.
0 -48.6594935049 -48.6594935049 -48.6594935049
1 -48.6800538676 -48.6806631261 -48.6838361406
2 -48.6822702585 -48.6510215698 -48.6874045449
3 -48.6858299354 -48.6882945647 -48.6932811528
4 -48.6881499412 -48.6849667085 -48.6946836332
5 -48.6890226067 -48.6911899936 -48.6959800274
6 -48.6898261650 -48.6878047907 -48.6973821465
7 -48.6901936624 -48.6930608185 -48.6987652146
8 -48.6905304889 -48.6940607117 -48.6996366016
9 -48.6908626791 -48.6949137185 -48.7006656309
10 -48.6914279465 -48.6963767038 -48.7017273728
11 -48.6921521142 -48.6986608776 -48.7021504975
12 -48.6931136707 -48.7007305310 -48.7022405019
13 -48.6940437619 -48.7016095285 -48.7022548935
14 -48.6949546487 -48.7021531692 -48.7022569328
15 -48.6961698826 -48.7022080183 -48.7022570260
16 -48.6973813002 -48.7022454522 -48.7022570662
17 -48.6984850655 -48.7022492840
18 -48.6991553826 -48.7022503853
19 -48.6996239136 -48.7022507037
20 -48.7002269303 -48.7022508393
21 -48.7005379631
22 -48.7008387759
...
50 -48.7022519950
from which you can see that the natural internals are actually the best set. The $ZMAT exhibits
upward burps in the energy at step 2, 4, and 6, so that for the same number of steps, these
coordinates are always at a higher energy than the natural internals.
The initial hessian generated for these three columns contains 0, 33, and 46 force constants.
This assists the natural internals, but is not the major reason for its superior performance.
The computed hessian at the final geometry of this molecule, when transformed into the natural
internal coordinates is almost diagonal. This almost complete uncoupling of coordinates is what
makes the natural internals perform so well. The conclusion is of course that not all coordinate
systems are equal, and natural internals are the best. As another example, we have run the
ATCHCP molecule, which is a popular geometry optimization test, due to its two fused rings:
H.B.Schlegel, Int.J.Quantum Chem., Symp. 26, 253-264(1992 )
T.H.Fischer and J.Almlof, J.Phys.Chem. 96, 9768-9774(1992 )
J.Baker, J.Comput.Chem. 14, 1085-1100(1993 )
Here we have compared the same coordinate types, using a guess hessian, or a computed hessian.
The latter set of runs is a test of the coordinates only, as the initial hessian information is
identical. The results show clearly the superiority of the natural internals, which like the
previous example, give an energy decrease on every step:
HESS=GUESS HESS=READ
Cartesians 65 41 stepsgood Z-matrix 32 23
natural internals 24 13
A final example is phosphinoazasilatrane, with three rings fused on a common SiN bond, in
which 112 steps in Cartesian space became 32 steps in natural internals. The moral is:
"A little brain time can save a lot of CPU time."
In late 1998, a new kind of internal coordinate method was included into GAMESS. This is
the delocalized internal coordinate (DLC) of
J.Baker, A. Kessi, B.Delley J.Chem.Phys. 105, 192-212(1996 )
although as is the usual case, the implementation is not exactly the same. Bonds are kept as
independent coordinates, while angles are placed in linear combination by the DLC process.
There are some interesting options for applying constraints, and other options to assist the
automatic DLC generation code by either adding or deleting coordinates. It is simple to use DLCs
in their most basic form:
$contrl nzvar=xx $end
$zmat dlc=.true. auto=.true. $end
Our initial experience is that the quality of DLCs is not as good as explicitly constructed natural
internals, which benefit from human chemical knowledge, but are almost always better than
carefully crafted $ZMATs using only the primitive internal coordinates (although we have seen
a few exceptions). Once we have more numerical experience with the use of DLC's, we will come
back and revise the above discussion of coordinate choices. In the meantime, they are quite
simple to choose, so give them a go.* * * The Role of Symmetry * * *
At the end of a successful geometry search, you will have a set of coordinates where the
gradient of the energy is zero. However your newly discovered stationary point is not
necessarily a minimum or saddle point!
This apparent mystery is due to the fact that the gradient vector transforms under the
totally symmetric representation of the molecular point group. As a direct consequence, a
geometry search is point group conserving. (For a proof of these statements, see J.W.McIver
and A.Komornicki, Chem.Phys.Lett., 10,303-306(1971 )). In simpler terms, the molecule
will remain in whatever point group you select in $DATA, even if the true minimum is in some
lower point group. Since a geometry search only explores totally symmetric degrees of
freedom, the only way to learn about the curvatures for all degrees of freedom is
RUNTYP=HESSIAN.
As an example, consider disilene, the silicon analog of ethene. It is natural to assume that
this molecule is planar like ethene, and an OPTIMIZE run in D2h symmetry will readily locate a
stationary point. However, as a calculation of the hessian will readily show, this structure is a
transition state (one imaginary frequency), and the molecule is really trans-bent (C2h ). A
careful worker will always characterize a stationary point as either a minimum, a transition
state, or some higher order stationary point (which is not of great interest!) by performing a
RUNTYP=HESSIAN.
The point group conserving properties of a geometry search can be annoying, as in the
preceding example, or advantageous. For example, assume you wish to locate the transition
state for rotation about the double bond in ethene. A little thought will soon reveal that ethene is
D2h , the 90 degrees twisted structure is D2d , and structures in between are D2 . Since the saddle
point is actually higher symmetry than the rest of the rotational surface, you can locate it byRUNTYP=OPTIMIZE within D2d symmetry. You can readily find this stationary point with the
diagonal guess hessian! In fact, if you attempt to do a RUNTYP=SADPOINT within D2d symmetry,
there will be no totally symmetric modes with negative curvatures, and it is unlikely that the
geometry search will be very well behaved.
Although a geometry search cannot lower the symmetry, the gain of symmetry is quite
possible. For example, if you initiate a water molecule optimization with a trial structure
which has unequal bond lengths, the geometry search will come to a structure that is indeed C2v
(to within OPTTOL, anyway). However, GAMESS leaves it up to you to realize that a gain of
symmetry has occurred.
In general, Mother Nature usually chooses more symmetrical structures over less
symmetrical structures. Therefore you are probably better served to assume the higher
symmetry, perform the geometry search, and then check the stationary point's curvatures. The
alternative is to start with artificially lower symmetry and see if your system regains higher
symmetry. The problem with this approach is that you don't necessarily know which subgroup
is appropriate, and you lose the great speedups GAMESS can obtain from proper use of
symmetry. It is good to note here that "lower symmetry" does not mean simply changing the
name of the point group and entering more atoms in $DATA, instead the nuclear coordinates
themselves must actually be of lower symmetry.
* * * Practical Matters * * *
Geometry searches do not bring the gradient exactly to zero. Instead they stop when the
largest component of the gradient is below the value of OPTTOL, which defaults to a reasonable
0.0001. Analytic hessians usually have residual frequencies below 10 cm -1 with this degree of
optimization. The sloppiest value you probably ever want to try is 0.0005.
If a geometry search runs out of time, or exceeds NSTEP, it can be restarted. For
RUNTYP=OPTIMIZE, restart with the coordinates having the lowest total energy (do a string
search on "FINAL"). For RUNTYP=SADPOINT, restart with the coordinates having the smallest
gradient (do a string search on "RMS", which means root mean square). These are not
necessarily at the last geometry!
The "restart" should actually be a normal run, that is you should not try to use the restart
options in $CONTRL (which may not work anyway). A geometry search can be restarted by
extracting the desired coordinates for $DATA from the printout, and by extracting the
corresponding $GRAD group from the PUNCH file. If the $GRAD group is supplied, the program
is able to save the time it would ordinarily take to compute the wavefunction and gradient at the
initial point, which can be a substantial savings. There is no input to trigger reading of a $GRAD
group: if found, it is read and used. Be careful that your $GRAD group actually corresponds to
the coordinates in $DATA, as GAMESS has no check for this.
Sometimes when you are fairly close to the minimum, an OPTIMIZE run will take a first
step which raises the energy, with subsequent steps improving the energy and perhaps finding
the minimum. The erratic first step is caused by the GUESS hessian. It may help to limit the
size of this wrong first step, by reducing its radius, DXMAX. Conversely, if you are far from the
minimum, sometimes you can decrease the number of steps by increasing DXMAX.
When using internals, the program uses an iterative process to convert the internal
coordinate change into Cartesian space. In some cases, a small change in the internals will
produce a large change in Cartesians, and thus produce a warning message on the output. If these
warnings appear only in the beginning, there is probably no problem, but if they persist youcan probably devise a better set of coordinates. You may in fact have one of the two problems
described in the next paragraph. In some cases (hopefully very few) the iterations to find the
Cartesian displacement may not converge, producing a second kind of warning message. The fix
for this may very well be a new set of internal coordinates as well, or adjustment of ITBMAT in
$STATPT.
There are two examples of poorly behaved internal coordinates which can give serious
problems. The first of these is three angles around a central atom, when this atom becomes
planar (sum of the angles nears 360). The other is a dihedral where three of the atoms are
nearly linear, causing the dihedral to flip between 0 and 180. Avoid these two situations if you
want your geometry search to be convergent.
Sometimes it is handy to constrain the geometry search by freezing one or more coordinates,
via the IFREEZ array. For example, constrained optimizations may be useful while trying to
determine what area of a potential energy surface contains a saddle point. If you try to freeze
coordinates with an automatically generated $ZMAT, you need to know that the order of the
coordinates defined in $DATA is
y
y x r1
y x r2 x a3
y x r4 x a5 x w6
y x r7 x a8 x w9
and so on, where y and x are whatever atoms and molecular connectivity you happen to be using.