ACES II User manual version 2.7.0 (PDF) - Quantum Theory Project

7.1.7 Internal coordinatesThe specification by internal coordinates is known as the Z matrix. Centers of the nucleiare expressed relative to previously defined centers by means of distances and angles. Thespecification includes a length, a bond angle, and a dihedral angle. The number associatedwith each atom is governed by its position in the Z matrix.The essentials of Z matrixconstruction can be illustrated by considering a Z matrix for a system of four atoms ABCD,not linked in any particular order.Arbitrary ABCD moleculeAB 1 ABC 1 AC 2 CABD 3 CD 2 DCB 1 TAUThe first line in the Z matrix contains the atomic symbol of one of the atoms, say A. Thesecond line specifies the position of a second atom, say B, relative to the first atom. Supposethat B is a distance AB from A. The second line then contains the atomic symbol B, followedby the number 1 (A is atom number 1), and a parameter label, AB, (“B 1 AB”). For thespecification of the third atom, a distance and an angle are needed. We may use the distancebetween atoms A and C and the angle CAB, or we may use the distance between atoms B andC and the angle CBA. In the first case the third line would have the form “C 1 AC 2 CAB”,while in the second case it would have the form “C 2 BC 1 CBA”. Finally, there is a linespecifying the position of D relative to the other atoms. This line must contain a distance,a bond angle, and a dihedral angle and could have the form “D 3 CD 2 DCB 1 TAU”, withTAU being the angle between the BCD and ABC planes.For a system with more than four atoms, the fifth and subsequent lines follow the samepattern as the fourth line of the example given above (i.e., they also contain a length,angle, and dihedral angle and the numbers of three previously specified centers). It shouldbe emphasized that this is a somewhat simplified description that would work for a tetraatomicmolecule such as hydrogen peroxide but would not be satisfactory for acetylene. Thelatter requires “dummy” atoms. These and several other tips for forming Z matrices arediscussed below.A more formal description of a line in the Z matrix input is as follows. Each line mayhave as many as seven entries. We consider the I th line. The contents are the I th elementof the ZSYM array, the 3 ∗ I th , (3 ∗ I − 1) th , and (3 ∗ I − 2) th elements of the NCON andPARNAM arrays. The ZSYM array is of length N, where N is the number of lines in theZ matrix (this includes those for any dummy and ghost atoms), and contains the chemicalsymbols of all the atoms in the Z matrix. The NCON and PARNAM arrays are of length3*N. NCON contains the numbers of atoms relative to which each atom is specified. ThePARNAM array contains the names of the lengths, angles, and dihedral angles contained in31