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On the calculation of principal axes coordinates in molecules with internal rotation
Journal of Molecular Structure, 91 (1983) 387-389 THEOCHEM Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
Short communication
ON TJY&CALCULATION OF PRINCIPAL AXES COORDINATES MOLECULES WITH INTERNAL ROTATION
IN
B. J. VAN DER VEKEN*, G. H. PIETERS and M. A. HERMAN University of Antwerp, RUCA, Labomtorium voor Anorganische Groenenborgerlaan 171, B 2020 Antwerpen (Belgium)
Scheikunde,
(Received 4 March 1982)
INTRODUCTION
A procedure is described which avoids the reversals in the directions of principal inertial axes which occur in the calculation of momentaneous principal axes systems for internally rotating molecules, thus allowing full automation of the calculations. In theoretical studies of internal rotation in molecules with one internal rotor, the kinetic aspects of internal rotation should also be considered [l] . It is therefore necessary to know the coordinates of the atoms of the molecule in the momentaneous principal axes system as a function of the internal rotation variable, the dihedral angle (J . If the coordinates are calculated in this manner for a sufficiently large number of dihedral angles, each coordinate can be expressed mathematically as a function of $ by means of a Fourier series analysis which elegantly allows the calculation of kinetic contributions
111. SINGLE ROTATION
METHOD
Starting from the atom coordinates in a randomly chosen orthonormal Cartesian axes system, as calculated for example by the method of Thompson [2], the coordinates in the principal axes system may be determined via a rotation of the original Cartesian system. This rotation is represented by the eigenvector matrix C, defined as IC=CA
where I is the inertial tensor I = $MX *Author for correspondence. 0166-1280/83/0000-0000/$03.00
o 1983 Elsevier Scientific Publishing Company
388
A is the diagonal matrix containing the principal moments of inertia, X is the matrix containing the coordinates of the atoms in the original Cartesian axes system and M is a diagonal matrix, containing the appropriate atomic masses. From matrix C the atom coordinate matrix X’ in the principal axes system may be calculated X’=XC The diagonalisation of the inertial tensor may be achieved by the Jacobi method (see e.g. ref. 3). However, during the calculation of principal axes coordinates as a function of 4, discontinuities for some molecules resulted. In other words, reversal of the direction of one of the principal axes occurred for some values of 4, the exact values depending on the molecule investigated. This is shown in Fig. 1 for the X coordinate of one of the terminal oxygen atoms in nitric acid: it is clear that for $J in the interval 150-210”) the X axis is calculated by the diagonalisation procedure to be in a direction opposite to that for the other values of $. In the case of the Jacobi method, this can be traced back to the choice of a positive square root in the calculation of sine and cosine values. It implies that for some values of 4, the diagonalisation method rotates the molecule in an opposite direction to reach its principal axis configuration. Other diagonalisation subroutines were tested, including the implicit QL method [4], but all showed the same failure, although not necessarily in the same $Jinterval. Such discontinuities prevent an automated extended treatment of the coordinates and although they are usually readily detected and corrected for manually, this rapidly becomes very time consuming when treating larger molecules.
X/A
t
.*.t*..
Fig. 1. X-coordinate in the momentaneous principal axes system for one of the terminal oxygen atoms in nitric acid. The variable $J denotes the dihedral angle between the H-O-N and NO, planes. +, without double rotation; o, with double rotation.
389
THE DOUBLE ROTATION
METHOD
In order to avoid the discontinuities mentioned above, a procedure has been evolved which uses double rotation. For a certain value of the dihedral angle, the new Cartesian coordinates are first calculated taking care to fix the origin and direction of the axes in a manner consistent with those for the previous values of the dihedral angle. The next step is the transformation of these coordinates using the principal axes transformation obtained for the previous value of the dihedral angle, i.e. the new molecule is rotated into the principal axes system of the molecule with the internal rotor in its previous position
x; = XjCi The subscript i of the rotation matrix C refers to the previous dihedral angle $i, the subscript j to the dihedral angle being treated. Obviously, this does not give the principal axis coordinates for the new molecule. However, by keeping the step size of the dihedral angle small, the inertial tensor of these new axes becomes quasi-diagonal. Consequent diagonalisation of this tensor implies rotation of the molecule through a very small angle for which no ambiguities in sign are expected
izj ’ X’MX’.
I’ =
J
IjLj = Lj Aj XI’ = Xi’Lj The procedure is then completed by taking the product of both rotation matrices. This results in the rotation matrix giving the transformation from the original Cartesian coordinates to the true principal axes coordinates
cj = CiLj It is evident that, for the first value of the dihedral angle, no pretransformation is required because at that moment the signs of the axes have not been previously fixed. From Fig. 1 it is clear that, indeed, no reversals occur with this procedure. Experience has also shown that for other molecules where reversals occurred, no problems arise from this treatment. REFERENCES 1 2 3 4
J. D. Lewis, T. B. MalIoy, T. T. Chao and J. Laane, J. Mol. Struct., 12 (1972) 427. H. B. Thompson, J. Chem. Phys., 47 (1967) 3407. T. R. Dickson, in The Computer and Chemistry, Freeman, San Francisco, 1968, p. 171. R. S. Martin and J. H. Wilkinson, Num. Math., 12 (1968) 377.
Short communication
ON TJY&CALCULATION OF PRINCIPAL AXES COORDINATES MOLECULES WITH INTERNAL ROTATION
IN
B. J. VAN DER VEKEN*, G. H. PIETERS and M. A. HERMAN University of Antwerp, RUCA, Labomtorium voor Anorganische Groenenborgerlaan 171, B 2020 Antwerpen (Belgium)
Scheikunde,
(Received 4 March 1982)
INTRODUCTION
A procedure is described which avoids the reversals in the directions of principal inertial axes which occur in the calculation of momentaneous principal axes systems for internally rotating molecules, thus allowing full automation of the calculations. In theoretical studies of internal rotation in molecules with one internal rotor, the kinetic aspects of internal rotation should also be considered [l] . It is therefore necessary to know the coordinates of the atoms of the molecule in the momentaneous principal axes system as a function of the internal rotation variable, the dihedral angle (J . If the coordinates are calculated in this manner for a sufficiently large number of dihedral angles, each coordinate can be expressed mathematically as a function of $ by means of a Fourier series analysis which elegantly allows the calculation of kinetic contributions
111. SINGLE ROTATION
METHOD
Starting from the atom coordinates in a randomly chosen orthonormal Cartesian axes system, as calculated for example by the method of Thompson [2], the coordinates in the principal axes system may be determined via a rotation of the original Cartesian system. This rotation is represented by the eigenvector matrix C, defined as IC=CA
where I is the inertial tensor I = $MX *Author for correspondence. 0166-1280/83/0000-0000/$03.00
o 1983 Elsevier Scientific Publishing Company
388
A is the diagonal matrix containing the principal moments of inertia, X is the matrix containing the coordinates of the atoms in the original Cartesian axes system and M is a diagonal matrix, containing the appropriate atomic masses. From matrix C the atom coordinate matrix X’ in the principal axes system may be calculated X’=XC The diagonalisation of the inertial tensor may be achieved by the Jacobi method (see e.g. ref. 3). However, during the calculation of principal axes coordinates as a function of 4, discontinuities for some molecules resulted. In other words, reversal of the direction of one of the principal axes occurred for some values of 4, the exact values depending on the molecule investigated. This is shown in Fig. 1 for the X coordinate of one of the terminal oxygen atoms in nitric acid: it is clear that for $J in the interval 150-210”) the X axis is calculated by the diagonalisation procedure to be in a direction opposite to that for the other values of $. In the case of the Jacobi method, this can be traced back to the choice of a positive square root in the calculation of sine and cosine values. It implies that for some values of 4, the diagonalisation method rotates the molecule in an opposite direction to reach its principal axis configuration. Other diagonalisation subroutines were tested, including the implicit QL method [4], but all showed the same failure, although not necessarily in the same $Jinterval. Such discontinuities prevent an automated extended treatment of the coordinates and although they are usually readily detected and corrected for manually, this rapidly becomes very time consuming when treating larger molecules.
X/A
t
.*.t*..
Fig. 1. X-coordinate in the momentaneous principal axes system for one of the terminal oxygen atoms in nitric acid. The variable $J denotes the dihedral angle between the H-O-N and NO, planes. +, without double rotation; o, with double rotation.
389
THE DOUBLE ROTATION
METHOD
In order to avoid the discontinuities mentioned above, a procedure has been evolved which uses double rotation. For a certain value of the dihedral angle, the new Cartesian coordinates are first calculated taking care to fix the origin and direction of the axes in a manner consistent with those for the previous values of the dihedral angle. The next step is the transformation of these coordinates using the principal axes transformation obtained for the previous value of the dihedral angle, i.e. the new molecule is rotated into the principal axes system of the molecule with the internal rotor in its previous position
x; = XjCi The subscript i of the rotation matrix C refers to the previous dihedral angle $i, the subscript j to the dihedral angle being treated. Obviously, this does not give the principal axis coordinates for the new molecule. However, by keeping the step size of the dihedral angle small, the inertial tensor of these new axes becomes quasi-diagonal. Consequent diagonalisation of this tensor implies rotation of the molecule through a very small angle for which no ambiguities in sign are expected
izj ’ X’MX’.
I’ =
J
IjLj = Lj Aj XI’ = Xi’Lj The procedure is then completed by taking the product of both rotation matrices. This results in the rotation matrix giving the transformation from the original Cartesian coordinates to the true principal axes coordinates
cj = CiLj It is evident that, for the first value of the dihedral angle, no pretransformation is required because at that moment the signs of the axes have not been previously fixed. From Fig. 1 it is clear that, indeed, no reversals occur with this procedure. Experience has also shown that for other molecules where reversals occurred, no problems arise from this treatment. REFERENCES 1 2 3 4
J. D. Lewis, T. B. MalIoy, T. T. Chao and J. Laane, J. Mol. Struct., 12 (1972) 427. H. B. Thompson, J. Chem. Phys., 47 (1967) 3407. T. R. Dickson, in The Computer and Chemistry, Freeman, San Francisco, 1968, p. 171. R. S. Martin and J. H. Wilkinson, Num. Math., 12 (1968) 377.