Molecular System Database - Periodic Systems

Jump to Introduction to Symmetry and Periodicity, Differences from Simple Molecular Systems, or Descriptions of New and Augmented Tables

Introduction to Symmetry and Periodicity

The Discover program can perform simulations on periodic systems (crystals). Periodic systems are represented by a set of atoms in a unit cell. When a set of translation operators (space group operators) is applied to the atoms of the unit cell, space is filled, forming a periodic lattice. The unit cell may be two- or three-dimensional, and its shape is described by a set of cell parameters discussed below. In general, the origin of the unit cell is at (0,0,0), and at least one cell edge lies along the x, y, or z axis.

Systems with periodicity and symmetry include additional representational data in the System database tables. The following terms are used in describing these additional data columns for periodic systems:

Unit cell:
The local reference coordinate system for periodic systems. The cell axes are the intervals of periodicity for each dimension.
Asymmetric atoms:
The input-independent atom set known as the asymmetric unit. This atom set forms the original basis for generation of the periodic system.
Symmetry group:
The set of matrix transforms used to generate additional atoms in the system. These transforms form a mathematical group and are defined in terms of the Unit Cell local coordinate system.
Symmetry atoms:
The unique set of atoms generated by applying the symmetry group to the asymmetric unit of independent atoms. Any non-unique symmetry atoms are not included in the system Atom table.
P1 atoms:
The collection of all asymmetric and symmetry atoms in the system atom table--sometimes referred to as the unit volume. This atom set is the basis for infinite translation by the unit cell vectors to form the periodic system.
Lattice offset:
An ordered triple representing multiples of the cell vector lengths which are added to P1 Atom coordinates to perform translations. The lattice offsets form an infinite set and give rise to the infinite nature of periodic systems.
Ghost atoms:
The set of atoms resulting from lattice translation of P1 Atoms using lattice offsets. In the Discover program, Ghost Atoms are typically generated only on demand, as required to fill out the bonding of the system for a particular process, such as minimization.

Differences from Simple Molecular Systems

For a periodic system, two tables are used in addition to the basic set of tables: 

   Cell 

   P1Bond
In addition, two tables acquire some new columns: 

   Bond 

   Atom
The name P1Bond is not the actual name of the table in question. The real name is MainCell/P1Bond. However, P1Bond is an alias and can be used as if it were the actual name. This alias is used in the remainder of this discussion.

Descriptions of New and Augmented Tables

Cell Table

For periodic molecular systems, the Cell table has the following columns: 

   Periodicity                    (type short)
   AtomTable                      (type string)
   BondTable                      (type string)
   Axes                           (type short)
   CellParameter                  (type double(6))
   ParameterMatrixPosition        (type short(12))
   CellPermutation                (type short(3))
   Fractional2Cartesian           (type double(9))
   Cartesian2Fractional           (type double(9))
   Origin                         (type double(6))
   Orientation                    (type double(9))
   P1BondTable                    (type string)
A row of the Cell table corresponds to a unit cell.

The entry in the Periodicity column is 2 or 3, representing two-dimensional or three-dimensional periodicity, respectively.

The entry in the AtomTable column is currently MainCell/Atom in all cases. It indicates the name of the atom table associated with this cell definition.

The entry in the BondTable column is currently MainCell/Bond in all cases. It indicates the name of the bond table associated with this cell definition.

The entry in the Axes column specifies alignment of cell axes as follows: 

  0: zyx; i.e., c axis is aligned with the z axis,
                b axis is in the z-y plane 

  1: zxy; i.e., c axis is aligned with the z axis,
                a axis is in the z-x plane 

  2: xyz; i.e., a axis is aligned with the x axis,
                b axis is in the x-y plane 

  3: xzy; i.e., a axis is aligned with the x axis,
                c axis is in the x-z plane 

  4: yxz; i.e., b axis is aligned with the y axis,
                a axis is in the y-x plane 

  5: yzx; i.e., b axis is aligned with the y axis,
                c axis is in the y-z plane
The current default value is 2, corresponding to the xyz alignment in the Insight program.

The entry in the CellParameter column specifies cell vectors in terms of the Cartesian axes. Cell vectors are the three edges of the unit cell that are attached to the vertex at the origin. They are generally labeled a, b and c. The form of the entry depends upon the axis convention chosen as follows: 

  zyx: (a.x a.y b.y a.z b.z c.z) 

  zxy: (b.y b.x a.x b.z a.z c.z) 

  xyz: (c.z c.y b.y c.x b.x a.x) 

  xzy: (b.y b.z c.z b.x c.x a.x) 

  yxz: (c.z c.x a.x c.y a.y b.y) 

  yzx: (a.x a.z c.z a.y c.y b.y)
The entry in the ParameterMatrixPosition column specifies a cell parameter matrix permutation array used internally to handle different axis orientations.

The entry in the CellPermutation column specifies a coordinate permutation array used internally.

The entry in the Fractional2Cartesian column specifies the matrix which converts fractional coordinates (coordinates with respect to the cell vectors) to Cartesian coordinates in the format: (a.x a.y a.z b.x b.y b.z c.x c.y c.z). Thus F * F2C = C, where F represents fractional coordinates, C represents Cartesian coordinates and 

   F2C = (a.x a.y a.z)
         (b.x b.y b.z)
         (c.x c.y c.z)
Clearly the rows of F2C represent the cell vectors. For the xyz axis system the matrix reduces to 

   F2C = (a.x  0   0 )
         (b.x b.y  0 )
         (c.x c.y c.z)
where:

a.x = a
b.x = b cos
b.y = b sin
c.x = c cos
c.y = c (cos - cos cos ) / sin
c.z = sqrt [(c2 - (c.x)2 - (c.y)2)]
= angle between a and b
= angle between a and c
= angle between b and c

In a number of circumstances it is convenient to represent atom positions in fractional coordinates. Determination of atom position with respect to the cell is particularly straightforward with fractional coordinates. The fraction can be viewed as the fraction of the corresponding cell vector which the atom would traverse in moving from the origin to its current position, e.g., (0.5, 0.5, 0.5) is at the geometric center of the cell.

The entry in the Cartesian2Fractional column specifies the matrix which converts Cartesian coordinates to fractional coordinates in the format: 

(C2F11 C2F12 C2F13 C2F21 C2F22 C2F23 C2F31 C2F32 C2F33).
Thus C * C2F = F, where C represents Cartesian coordinates, F represents fractional coordinates and: 

   C2F = (C2F11 C2F12 C2F13)
         (C2F21 C2F22 C2F23)
         (C2F31 C2F32 C2F33)
The Origin and Orientation columns are not currently used. Cell origin is currently assumed to be (0,0,0), and cell orientation is currently assumed to be the identity matrix.

The entry in the P1BondTable column is currently MainCell/P1Bond in all cases. It indicates the name of the P1Bond table associated with the cell.

P1Bond Table

For periodic molecular systems, the P1Bond table has the following columns: 

   Atom-1                 (type rid)
   Atom-2                 (type rid)
   Order                  (type float)
   Bibond                 (type rid)
   XYZLatticeOffset       (type short(periodicity))
The first four columns are analogous to the columns of the same name in the Bond table in the basic non-periodic case. An entry in the XYZLatticeOffset column specifies an offset for Atom-2 such that Atom-1 and the offset version of Atom-2 are bonded. The offset can be trivial--(0 0 0)--which implies that Atom-1 and Atom-2 are bonded to one another. When the offset is non-trivial, no ``ghost atom'' corresponding to the offset is created initially, so the bond is initially implicit (i.e., not yet in the Atom table).

For periodic systems, the P1 Bond table is created in addition to the system Bond table described elsewhere. The P1 Bond table contains an additional column for a lattice offset which is applied to Atom-2 to give the bondmate for Atom-1. In this table, both atoms are P1 Atoms.

Similarly, the system Atom table is augmented to include two additional columns used to identify the P1 Atom parent and lattice offset for Ghost Atoms in the system. P1 Atoms all have NULL/(0,0,0) entries in these columns.

A P1 bond (i.e., an entry in the P1Bond table) represents an ``infinite bond'' consisting of the specified atom pair and all pairs constructed by applying an identical lattice offset to the atoms of the specified pair.

Bond Table Augmentation for Periodic Systems

For periodic molecular systems, the Bond table is created by making a copy of the first four columns of the P1Bond table and zeroing out any Atom-2 references corresponding to non-trivial lattice offsets. A P1Bond column (of type rid) is added to the Bond table so that each entry in Bond has a reference to its associated P1 bond. Thus, if you need an explicit bond, but find that Atom-2 is zero, you can use the related P1 bond to make a new ghost atom.

Given any P1 atom (i.e. an atom specified explicitly in a periodic system), you can find its bond mates by looking under Atom-1 in the Bond table. Its bond mates are either already there under Atom-2, or can be constructed using the P1Bond table. Now suppose you have a ghost atom formed by adding a lattice offset, L, to a parent P1 atom, P. How do you find its bond mates? We have established how to determine the bond mates of the P1 atom P. Applying L to each of these yields the bond mates of the original ghost atom. Thus, if Atom-2 + Loff is a bondmate of P obtained from P1Bond, then Atom-2 + Loff + L is bonded to the ghost atom.

Atom Table Augmentation for Periodic Systems

For periodic molecular systems, a P1Atom column (type rid) and an XYZLatticeOffset column (type short(3)) are added to the Atom table. Entries in these columns are null and (0 0 0), respectively, for the P1 atoms. For ghost atoms, a reference to the associated P1 atom is stored in the P1Atom column, and the associated offset is stored in the XYZLatticeOffset column.

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