Three internal coordinates for each atom are required (except for atoms 1, 2, and 3, see below). These are arranged in standard z-matrix fashion.^[16] The three internal coordinates for the atoms are the bond length, bond angle, and dihedral angle. Each of these items is referenced to other atoms that have already been defined. In some cases, care must be exercised as to the order of atomic definition. AMPAC geometry input lines are “free-format”, that is the items do not need to be in specific columns to be read properly. The items must, however, be presented in the correct order. The standard form for a line of an internal coordinate z-matrix is listed below:

Symbol  Bond Length  Bond Angle  Dihedral Angle NA NB NC Spec

The symbol of an element can be defined by either letters (referencing specific elements on the periodic table) or atomic numbers. Differences between lower case and upper case characters are ignored.

When an element symbol is used by itself, the element’s standard atomic mass will be used in the calculation. This is what is desired in most cases but sometimes it is necessary to specify particular masses to be used in the calculation, e.g. kinetic isotope effect or vibrational frequency studies. To use the mass of a specific isotope, the number of the isotope is placed directly after the corresponding atomic symbol, with no spaces. Thus for carbon-13:

C13      1.54  1  109.50  1  180.00  1    3  2  1

If the specific isotope is not recognized by Ampac or a particular mass is needed, simply place the number after the atomic symbol (with no spaces). (To distinguish a user-specified mass from an isotope, it must contain a decimal point.) As an example, to give a carbon atom a mass of 13.005:

C13.005  1.54  1  109.50  1  180.00  1    3  2  1

Both modes of specifying the mass for an atom apply to Cartesian coordinates as well.

The optimization flags (BL, BA, and DA) are listed for the preceding geometric variable. If a flag is set to zero, the variable is not optimized in the course of the calculation. If the value is set to one, the parameter is allowed to vary. Definitions of reaction coordinates and grid search patterns can also be specified by way of the optimization flags (see Chapter 25, One-Dimensional Reaction Pathway, Chapter 26, Two-Dimensional Grid Search and the keyword pair STEP1=n.n, STEP2=m.m). N_A, N_B, and N_C are the numbers of previously defined atoms to which the geometric parameters of the present atom are referenced.

The bond length of an atom is with reference to one other atom, and is the distance from that atom in Angstroms. The value for a bond length must be greater than zero. In the case of the AMPAC input line, the bond length is referenced to atom N_A. While the term “bond length”^[17] is used, a better descriptor might be “interatomic distance”. For purposes of geometry definition, the atoms are not required to be chemically bonded either at the beginning or end of the calculation. The structure of a molecule is usually defined, however, by the bonding framework. This is easier to conceptualize and standard values for bond lengths are available in tabular form from a number of convenient sources.^[18] If atoms are defined via the bonding network, this also makes the interpretation of the output easier as these chemically important values are direct results of the calculation. Note that the first atom in a z-matrix does not have a bond length, as there is no preceding atom from which to reference the bond length. The second and all succeeding atoms have bond length values.

The bond angle^[19] of an atom is with reference to two other atoms and is simply the angle formed by the three atoms (See Figure 5.1, “Bond Lengths and Bond Angle for Water”). The values for bonds angles must be positive. In the protocol of AMPAC, this is the angle XN_AN_B (where X is the present atom). Note again that it is not required that the atoms used to define a bond angle be chemically connected to one another. It is not possible to define a bond angle until atom number three is listed. Thus atom 3 and all subsequent atoms have bond angles. AMPAC requires all bond angles to be positive.

The dihedral angle (also referred to as the torsional or tortional angle in some cases) is defined by reference to three previously defined atoms. The dihedral is defined as the angle of rotation about the N_AN_B axis (viewed in this manner) that is required to bring the atom under consideration (X) into alignment with atom N_C. (See Figure 5.2, “Newman Projection Showing Dihedral Angle Definition”) Dihedrals may either be positive or negative. The possible range is -360 to +360, although AMPAC will translate all dihedrals to -180 to +180. (Thus an input dihedral value of 240 will be output as -120.) It is required that the three atoms to which the dihedral of a fourth atom is referenced do not assume a straight line arrangement either as a part of the definition or as a result of the geometry optimization. The program checks for this condition at the beginning of the input and also at each step in the geometry optimization. This difficulty (e.g. encountered in the definition of alkynes) can be circumvented by the use of dummy atoms.

The sign convention for defining dihedral angles in AMPAC is as follows:

When looking down the N_A-N_B axis, the dihedral angle is positive if X can be rotated into congruence with N_C by moving clockwise less that 180. Conversely, the dihedral angle is negative if X can be rotated into congruence with N_C by moving counterclockwise less that 180.

Some examples of dihedral angle definitions are shown below. A dihedral angle cannot be specified until atom number four. All subsequent atoms have dihedrals.

The above examples illustrate some important principles for defining z-matrices and especially dihedral angles. First, there are several possible schemes that would properly define the geometry of either of these molecules. One advantage the scheme used possesses is that all components of the geometry specification correspond to actual internal coordinates of the molecule: bond lengths define actual bonds and so forth. While it is not always possible to implement this pattern, it is usually simplest.

Another advantage is the “modularity” of the hydrogen atom definitions. In both cases, some of the hydrogen atom dihedral angles are defined with respect to previously defined hydrogens. This allows easy rotation of both species about interatomic axes. For instance in the first example, the entire methyl group could be rotated by simply changing the dihedral of the first hydrogen. The other two hydrogens will “follow” as this one is rotated. Similarly, twist-ethylene could be produced in the second case by changing the dihedral angle of hydrogen 5 to 90 rather than 0. This type of organization is quite useful when a conformational search is being performed or a methyl rotation is interfering with location of a transition state. It is also generally useful to keep functional groups such as phenyls modularized. This allows relatively easy replacement for studies of substituent effects.

As noted above, it is not always possible to use “natural” bond patterns to define a molecule. A classic example is acetylene, C₂H₂. The first three atoms are easily defined:

C  0.00  0    0.00  0     0.00  0    0  0  0
C  1.20  1    0.00  0     0.00  0    1  0  0
H  1.10  1  180.00  1     0.00  0    1  2  0

There is some difficulty in defining the dihedral of the fourth atom. This atom may not be referenced to the previous three atoms because they fall on a straight line. This will cause the dihedral of the fourth hydrogen to become undefined and the program will generate an error message and halt computation.

The solution to this problem lies in adding “dummy atoms.” These atoms are place holders that have no chemical significance. They simply provide a point in space to which other atoms may be referred, and are commonly applied in the case of linear molecules as noted above. The AMPAC symbol for dummy atoms is “XX” or “Xx”, or they may be specified by the atomic number “99”. Continuing with the example of acetylene, the geometry is redefined below using two dummy atoms. (Note that the dummy atom geometric parameters are not optimized, as they have no chemical significance in this case.)

Another use of dummy atoms is for the definition of molecules with particular difficulties. A convenient example is a cyclic ring system such as benzene. (The AMPAC optimization routines handle the case of benzene with no difficulty, but it can be used to illustrate the point for other systems of similar types.) If the carbon skeleton of benzene or some other cyclic system is defined before the hydrogens, a C-C bond is left undefined (i.e. “dangling”):

Notice that the C₁-C₆ bond is not specifically defined in this z-matrix. In those cases where an optimization might fail, the problem usually arises when the first geometry optimization step causes a widening of the ring angles because there is no optimizable parameter “closing” the ring. Dummy atoms can be used to solve this case by “spinning” the atoms out from a central position like the spokes on a wheel. This eliminates all explicit bond specifications and the optimization will be more likely to proceed smoothly. Below is an example of the carbon skeleton of benzene defined using a central dummy atom:

Some care should be used in employing dummy atoms. In the examples above, the coordinates of the dummy atoms are not optimized. This insures that only the 3N-6 independent internal coordinates (N is the number of atoms) of the molecules are correctly described. If more than 3N-6 parameters are allowed to vary, dependencies will be present during the geometry optimizations which can produce spurious results. For this reason, optimization of dummy atom parameters should not be carried out except in special circumstances. As a general rule, dummy atoms can cause unexpected difficulties in the geometry optimization procedures and should be avoided when possible.

The “Spec” item is used presently for entering partial charges for sparkles.

As well as internal coordinates, Cartesian coordinates (x, y, z) may be used for specification of geometries for AMPAC. This option allows easy interface with other modeling or graphics packages that provide or require the coordinates of atoms in Cartesian space. Each atom is defined by distances along three orthogonal axes and no connectivity list is provided. Following each coordinate is an optimization flag for the preceding coordinate. Note immediately that this type of geometry results in six more parameters to be optimized during the geometry search than an equivalent internal coordinate definition. Although generally of little consequence, this type of definition does require more calculation time than a z-matrix definition scheme. When reading a geometry from an input file, AMPAC will search for a connectivity list at the end of each line. If one is not located, AMPAC will assume that the geometry is in Cartesian coordinates and proceed with input on that basis.

There are several special rules that apply when input is provided in Cartesian coordinates. Unlike previous versions of AMPAC, users may now define optimization options for Cartesian coordinates. Also, the user should be aware that only internal coordinates are used for triatomic systems. (On input, AMPAC assumes a connectivity pattern for the first three atoms, and thus will not allow Cartesian coordinate to be utilized.) The keyword XYZ is not required to define a Cartesian geometry. Its use causes the entire calculation to be performed in Cartesian space, regardless of the initial geometry definition (internal or Cartesian space). (See the discussion of the XYZ keyword.) This is useful in eliminating possible problems with undefined dihedrals during optimizations or searches for transition states, such as might be the case with CHN. It is not necessary to use dummy atoms with Cartesian coordinate definitions.

Four extra “elements” are included in AMPAC. These items represent pure ionic charges, roughly equivalent to the following chemical entities:

For the purposes of discussion these entities are called “sparkles”. They all have an ionic radius of 0.7 Å so that any two sparkles of opposite sign will form an ion pair with an interatomic separation of 1.4 Å. They have a zero heat of atomization, no orbitals, and no ionization potential. They do not contribute to the orbital count, and cannot accept or donate electrons. Since they appear as uncharged species which immediately ionize, attention should be given to the charge on the whole system (see the CHARGE=n keyword). For example: the alkaline metal salt of formic acid would have the formula HCOO⁺, where + is the unipositive sparkle. The charge on the system would then be zero. A water molecule polarized by a positive sparkle would have the formula H₂O⁺, and the charge on the system would be +1. A sparkle is best conceptualized as a sphere of diameter 1.4 Å with the charge delocalized over its surface. Computationally, a sparkle is an integer charge at the center of a repulsion sphere of form e^-αr. The hardness of the sphere defined by this function is such that other atoms or sparkles can approach within about 2 Å quite easily, but only with great difficulty come closer than 1.4 Å.

Sparkles find utility in several situations. They can be used as counterions, e.g. for acid anions or for cations. They can also operate as polarization functions. A controlled, shaped electric field can be produced by combining one or more sparkles.

Sparkles may also now be specified with partial charges. This allows the user to construct electrostatic environments of any shape with exactly specified charges at precise locations. This approach may be useful in studying the interaction of a molecule with a simulated active site or receptor region. The partial charges for each sparkle are specified by using one of the four symbols given above and adding a field at the end of the line in the input file. (The field is labeled “Spec” above). The number in this field becomes the sparkle’s new charge. Note that charge accounting within AMPAC still relies on the actual symbols, not the assigned partial charge values. Sparkles may be assigned partial charge values from -20 to +20.

Sparkles can also be used in frequency calculations, but a mass must be provided in the form noted at the beginning of this section for isotopic mass specification (i.e. the symbol followed by the mass with no space between). If no mass is specified for a sparkle, a mass of 10,000 amu is used.

[Only active if CHN, CHECKCHN, or FULLCHN has been specified on the keyword line.] One or more additional z-matrices must be provided if the CHN algorithm is to be called using one of the above keywords. The first geometry in Section 3 of the input file is the initial (reactant) geometry for the reaction path. At least one more z-matrix must follow, each separated from the previous by a blank line or a line of zeros. The final z-matrix is taken to be the final (product) geometry for the reaction path. Any z-matrices between the initial and final geometries will be used to define additional points along the initial reaction path, prior to refinement by the CHN algorithm. See Chapter 8, CHN Methods for a more complete discussion of the CHN algorithm.

[Only active if CHAIN has been specified on the keyword line.] Z-matrices for left and right minima (respectively) must be provided if the CHAIN algorithm is to be used. These are in addition to the first geometry in Section 3 of the input file, which is the “guess” geometry for the transition state used to define the reaction path. The two minima must be separated by a blank line or a line of zeros.

[Only active if SYMMETRY is on the keyword line.] “Symmetry” as used in AMPAC is different than what most chemists think of as symmetry. In the case of geometry definitions for AMPAC, symmetry is used to set certain geometric parameters equal to one another during the course of a geometry optimization. This accomplishes two major objectives. First, the number of variables is decreased and the optimization proceeds more rapidly. Second, AMPAC’s symmetry option allows the user to impose specific constraints on the molecule in keeping with that molecule’s point group or some item of chemical interest. However, caution must be exercised in that imposition of symmetry is a chemical presupposition that may or may not be correct in the context of the problem. If a molecule should be symmetric (is a member of a particular point group), use of the proper symmetry will usually lead to a slightly lower energy than a full optimization where all independent coordinates are varied.

The symmetry utilities are invoked by using the keyword SYMMETRY and then providing a list of functions and relationships in the appropriate place in the input file. AMPAC presently accepts 20 different symmetry functions (see the SYMMETRY keyword description). These functions allow the internal (or Cartesian) coordinates of a molecule to be equated to one another, and in the case of dihedral angles can be defined as the negative of one another, or varied as sums and differences with respect to a reference angle. The method of symmetry definition is a series of lines at the end of the geometry specification section that is interpreted by the program if the SYMMETRY keyword is used. Each line of symmetry information has the following format:

Reference Atom  (separator)  Symmetry Function  (separator)  Dependent Atom(s)

In this arrangement, the Reference Atom is the atom whose geometric parameter is used as the beginning value and will be adjusted in the course of the calculation. Symmetry Function is the AMPAC defined symmetry function relating the reference and dependent atoms. The Dependent Atom(s) is one or more atoms to which the indicated symmetry function is applied. The separator can be either a blank space, a tab, or a comma.

The full list of available symmetry relations is as follows:

An example of a symmetry definition is the case of methane, where all four C-H bonds and all bond angles are treated as equivalent and the dihedral of atom 5 is defined as the negative of the dihedral of atom 4:

  am1 rhf singlet truste gradients bonds t=auto symmetry
Methane in Td symmetry
SYMMETRY
 C   0.000000  0    0.000000  0    0.000000  0    0    0    0
 H   1.100000  1    0.000000  0    0.000000  0    1    0    0
 H   1.100000  0  109.500000  1    0.000000  0    1    2    0
 H   1.100000  0  109.500000  0  120.000000  1    1    2    3
 H   1.100000  0  109.500000  0 -120.000000  0    1    2    3
 0   0.000000  0    0.000000  0    0.000000  0    0    0    0
$$ symmetry - constraints
  2,   1,     3, 4,  5
  3,   2,     4, 5
  4,  14,     5
$$ end of extra data

[Only active if the keyword WEIGHT is present along with PATH and T.V.] This section is used to provide weights to particular components of the oriented transition vector supplied for PATH. See the section called “Input File (irc/path_tv_weight.dat):” for an example of the use of WEIGHT.

[Only active if the keyword T.V. (but not T.V.=n) has been specified with either PATH or IRC.] This section is used to list the components of the oriented transition vector for the PATH (in internal coordinates) or IRC (in Cartesian coordinates) method to follow. See the section called “Input File (irc/irc.dat):” and the section called “Input File (irc/path_tv_weight.dat):” for examples of its application.

[Only active if one of ANNEAL, MANNEAL, GANNEAL or TSANNEAL has been specified along with PEN2GRP.] PEN2GRP is an extension of PEN2 that only applies the penalty for atoms belonging to the same group. Information defining the atomic groupings is read in here in the extra input section. The data must contain a list of positive integers, one for each atom in the molecule, in free format. The n^th integer represents fragment group to which the n^th atom is to be assigned. The group number may range from 1 to the total number of atoms. This input can be further simplified by grouping consecutive identical indices using the “n*m” abbreviation. For example, a sequence of 10 atoms all belonging to group 2 can be abbreviated as “10*2”. See Chapter 13, Simulated Annealing for a detailed explanation on the use of simulated annealing keywords.

[Only active if one of ANNEAL, MANNEAL, GANNEAL or TSANNEAL has been specified along with LIMIT.] This section is used to provide upper and lower bounds for each optimizable geometric parameter in the annealing procedure. The lower bound for each parameter is given first (in free format, up to 80 characters per line, in order of appearance on the z-matrix) and next the upper bound for each parameter in the same order and fashion as the lower bound). See Chapter 13, Simulated Annealing for a detailed explanation on the use of simulated annealing keywords.

[Only active if C.I.(n,m) is specified and RECLAS is also used.] This section describes the perturbation of MOs for use in the CI procedure. See Chapter 11, Configuration Interaction for a discussion of the use of RECLAS.

[Only active for reaction path calculations. The calculation is a reaction path if one of the geometric parameters is marked with a “-1” for the optimization flag.] One of the optimizable geometric parameters may be designated as the “reaction coordinate.” This is typically done in order to locate the position of an approximate transition state along a proposed reaction pathway. The definition is accomplished by marking that item with a -1 as its optimization flag. The values supplied in this section of the input file are then substituted one after another (the first value used is that in the geometry itself) and the requested properties computed at each value. See Chapter 25, One-Dimensional Reaction Pathway for a more complete description of the use of the reaction coordinate.

[Read for all job types.] This section allows the user to supply additional property information about the molecular system. This information has no effect on the calcuation but is passed through to the .out, .arc, and .vis files. However, in the case of multi-geometry job types (i.e. everything but single point calcuations, optimizations, and frequency jobs), this data will simply be surpressed. This is because this extra data is connected to the input configureation and so would not have the same meaning or relevance to output of multi-geomtry jobs.

[Read for all job types.] User-specified information about the connectivity of the input structure can be specified here. Currently, this is not usued during the calculation but is passed through to the .out, .arc, and .vis files. However, in the case of multi-geometry job types (i.e. everything but single point calcuations, optimizations, and frequency jobs), this data will simply be surpressed. This is because this extra data is connected to the input configureation and so would not have the same meaning or relevance to output of multi-geomtry jobs.

AMPAC 10 still supports the old style of extra input data. AMPAC will look first for the extra input section markers and if found will read data from that point in the file. If the specific section marker is missing, then AMPAC will begin reading whatever it finds just after the geometries. Because of this, each extra input section must be in the correct order listed below. If a mixture of the old and new formats is used, then all unmarked sections (old format) should be listed first in the order given below followed by all of the marked sections (new format). The format of the data within each section is the same regardless of whether it is using the old or new format, except that two of them require explicit termination by a blank line or line of zeros.

The newest extra data sections--external property data and input connectivity--will only be accepted in the new format.