Computer Animation of Molecular Vibrations: Ethane

E M Mortensen, R J Penick

February 1970

J Chemical Education

Ronald J. Penick DFSCS, U.S. Air Force Academy, Colorado Springs, and Earl M. Mortensen, University of Massachusetts, Amherst (current address, Department of Chemistry, The Cleveland State University, Cleveland, Ohio).

In teaching concepts of molecular motion the instructor is often at a disadvantage in conveying to the student the ideas and mental pictures he has gained through years of experience. In fact, with certain processes the instructor's mental images may even be somewhat nebulous. The difficulty in presenting the student with visual images is particularly true for more complex molecules as they translate, rotate, vibrate, interact with neighboring molecules, and possibly react. The exchange of energy between degrees of freedom in collisional processes can readily be discussed but much less easily illustrated.

For the purpose of aiding the instructor in teaching these concepts, a subcommittee [2] under the Teaching Aids Committee of the Advisory Council on College Chemistry have been exploring the feasibility of making computer animated films.

The subcommittee consisted of E. F. Bertaut, J. J. Lagowski, W.T. Lippincott, C. E. Rodriguez, and the authors.

Here, one phase of this subcommittee's work will be considered, viz the display and filming of the normal modes of molecular vibration. In particular, reference will be made to the vibrational motion of ethane, about which a film has been made.

Computer animation becomes almost a necessity when motion other than in the plane of the screen is considered and even for such planar or even linear motions the effort and cost associated with animating the motion by standard techniques is usually prohibitive. In addition, computer animation provides a precision which is difficult to attain using conventional methods. Once the computer programs have been written, it is usually relatively easy to produce animation for a number of systems so that the cost of producing these films should be relatively economical. For example, it would be very easy to animate the vibrations of a series of isotopic molecules if the programs used were not dependent upon the symmetry of the molecule. In fact, it is not inconceivable that films could be made to the special order of individuals to meet their particular course needs, especially if short 8 mm cartridge loops would be sufficient.

Theory

Since the purpose of determining the normal modes of vibration of a molecule was for visual display, the vibrational analysis was made using cartesian displacement coordinates rather than the more usual internal or symmetry coordinates. [1] The kinetic and potential energies may be written

T = ½ Σ_i m_i ξ_i²        (1)

V = ½ Σ_i Σ_j k_ij ξ_i ξ_j        (2)

where k_ij = ∂²V/∂ξ_i∂ξ_j)₀ (note that k_ij = k_ji) and ξ_i are the cartesian displacement coordinates. Unless otherwise indicated, the summations run from 1 to 3 N where N is the number of atoms. For later convenience a transformation to mass-weighted, cartesian displacement coordinates is made, i.e.

q_i = m_i ^½ ξ_i         (3)

so that

T = ½ Σ_iq^.i²

V = ½ Σ_i Σ_j f_ij q_i q_j

where f_ij / (m_i m_j)^½

It should be noted that f_ij = f_ji. The motion may be determined using Lagrange's eqautions, i.e.

d/dt ∂L/∂q^._i - ∂L/Vq_i = 0, i= 1,2,..., 3N

where L = T - V. Upon making the appropriate substitutions in the above equations, the set of equations

q^.._i + Σ_j f_ij Q_j = 0, i= 1,2,..., 3N        (4)

are obtained. A solution to this set of equations is given by

q_i = l_ik sin(λ_k^½t + ε_k)         (5)

where

λ_k = 4π²c²v_k>²         (6)

with the frequency v_k being expressed in terms of wave numbers. To demonstrate that eqn. (5) is a solution, it is substituted into eqn. (4), and the set of linear homogeneous equations

Σ_j(f_ij - δ_ij λ_k)l_jk = 0, i= 1,2,..., 3N         (7)

are obtained where δ_ij is the Kronecker delta. Except for the trivial solution l_jk=0, the above equation is only satisfied for those λ_k which satisfy the secular equation

|f_ij - δ_ij λ| = 0         (8)

The solution of eqn. (8) yields 3 N values of λ that will be designated by λ_k from which the frequencies of vibration ν_k can be determined using eqn. (6). For those motions associated with translation or rotation of the molecule, the corresponding λ_k, and therefore the ν_k, will vanish. For any λ_k obtained from the secular equation, the relative values of l_jk can be determined by solving the set of homogeneous, linear equations given by eqn. (7). It shall be assumed that the l_jk are normalized such that

Σ_j l_jk² = 1

Thus, it is shown that for λ_k obeying eqn. (8), eqn. (5) describes the displacement of atom i vibrating with mode k. Since eqn. (4) is homogeneous, linear combinations of the q_i given by eqn. (5) having values of λ_k satisfying eqn. (8) are also acceptable solutions of eqn. (4) indicating that any linear combination of vibrational modes is possible.

It should be noted that the secular determinant given by eqn. (8) is symmetrical which is the reason for the choice of a mass-weighted cartesian coordinate system. More efficient computational procedures are available for solving these symmetrical secular equations.

The force constants F_kl are more often expressed in terms of the internal coordinates R_k, rather than k_ij for cartesian displacement coordinates. The potential energy may be written in terms of the internal coordinates to a harmonic approximation by the equation

V = ½ Σ_k Σ_l F_kl R_k R_l        (9)

For small displacements the R_k can be expressed in terms of the ξ_i by an equation of the form

R_k = Σ_k A_ki ξ_i         (10)

where in general the matrix of elements A_ki is not square since a set of rotational and translational coordinates are not usually included with the internal set R_k. This is of advantage since one need not be concerned about the translation or rotation of the molecule which occurs when an atom is displaced along one of its cartesian coordinates. The matrix elements A_ki are determined from the geometry of the molecule and represent the most difficult part of the analysis. If eqn. (10) is substituted into eqn. (9) and the resulting expression compared with eq. (2), it can be shown that

K_ij = Σ_k Σ_l F_kl A_ki A_lj         (11)

In molecules such as ethane the potential energy is often expressed in terms of a set of internal coordinates which are redundant in order to preserve symmetry in the coordinates. Such redundancies present no problem here since the are independent, and eqn. (11) converts the force constants expressed in terms of a redundant set of internal coordinates to the cartesian set which does not contain redundancy.

The motion associated with the cartesian displacement coordinates ξ_i is given by

ξ_i= Σ_k N_k l_ik sin(2πcν_kl + ε_k)/m_i^½, i= 1,2,..., 3N         (12)

using eqns. (3), (5), and (6) where the summation is over those normal modes which are being considered and N_k is the normalizing factor which gives the correct vibrational amplitude for vibrational mode k. The expression for N_k is

N_k = (2ν_k + 1)^½N_k⁰

where ν is the vibrational quantum number and N_k⁰ is the normalizing factor for the zero-point vibrational amplitude associated with vibrational mode k and is given by

N_k⁰=1/2π (h/c_νk)^½

Calculations

The calculations required to animate the molecular vibrations were accomplished using two computer programs. The first program sets up and solves the secular determinant given by eqn. (8) using the Jacobi method. Input for this program consists of the masses of the atoms, the valence force constants F_kl, and the transformation matrix elements A_ki given in eq. (10). As output, the program provides the frequencies of vibration ν_k and the zero-point amplitudes N_k⁰ l_ik / m_i^½.

The second computer program calculates the motion of the atoms according to a harmonic oscillator approximation given by eqn. (12) and outputs the results on magnetic tape for later display on peripheral equipment. This program uses the output of the first program as input. Other input required are the cartesian coordinates of the atoms, angular coordinates to orient the molecule in space, and other information necessary to specify the mode or modes to be displayed. This program allows for any normal mode or any combination of normal modes to be plotted which have been calculated by the first program. In addition, the vibrational level of each normal mode and its phase may be selected as desired. Linear combinations of the degenerate modes may be used in place of the degenerate ones determined by the first program to allow more flexibility in displaying these modes. All amplitudes may be scaled by a constant factor to obtain the best visual display. For a scale factor of unity the amplitudes will be correct to a classical, harmonic oscillator approximation with the mode having energy (ν_k+½)hcν_k. All frequencies of vibration are scaled by the same factor (a factor of 10^-14 was used) so that motions displayed on film will have the correct relative frequency of vibration. The orientation of the molecule may be selected to give the best visual display for any given vibrational motion.

Visually, the atoms of the molecule are displayed by circles with hidden or partially hidden atoms properly displayed as shown in Figure 1 and Figure 2. The diameter of each atom is twice the covalent radius with the atoms placed to have proper bond lengths and angles with respect to neighboring atoms. Each atom is represented by a circle containing 98 points (i.e., if no part of the circle is hidden). It was found that at least this number of points were necessary in order to adequately display intersecting circles from two atoms. The z coordinate is used to determine whether one atom is in front of another, and the radii of the circles representing the atoms are scaled by the value of the z coordinate. It thus appears as though the viewer is looking along the -z axis.

For each frame or picture, the points which make up the atoms and therefore the molecule are output on magnetic tape in such a format that it may be used to produce a Calcomp plot of the molecule as in Figure 1 or be used as input to a Stomberg-Carlson 4020 for display on a CRT which is then photographed giving the visual representation shown in Figure 2. Figure 3 is a multiple exposure used here to illustrate the asymmetrical stretching motion (A_2u) in ethane.

The programs were written and debugged on a Burroughs 5500 and a CDC 3600 system. The produetion of the film was made on the former system located at the Air Force Academy.

The set of force constants used to calculate the normal modes of ethane were obtained from Hansen and Dennison. [1] The vibrational modes obtained from these calculations were arranged and displayed according to stretching, CH₃, deformation, rocking, and tortional motions. Asymmetrical modes are displayed following their corresponding symmetrical ones to better illustrate the symmetry of the vibrations. The molecule was oriented in space in such a way as to effectively display the particular vibrational mode, and the vibrational amplitudes were scaled to better display them.

Discussion

The use of computer animation for displaying molecular motion provides the possibility of animating relatively complicated processes with an accuracy and low cost not matched by more conventional techniques. Computer animation can provide a means of providing films which will be useful to the instructor in illustrating concepts of molecular motion, which in the past he has only been able to do verbally or with crude sketches. These techniques are particularly useful in showing molecular vibrations of molecules such as ethane. Not only can the various stretching, bending, and other motions be illustrated, but also the symmetry of the vibrational motions and the effect of isotopic substitution.

Walton and Risen [2] have also applied computer animation to molecular vibrations. Their procedure allows the individual freedom to request that the computer display the modes and view which is desired on a CRT. This method has great merit in its teaching possibilities, particularly using the time sharing capabilities of current, high-speed computers. There are, however, relatively few institutions which have the necessary facilities for the on-line interaction, whereas equipment for showing computer animated films is nearly always readily available.

In addition to displaying molecular vibrations, other programs are being developed for animating the general motion of molecules by integrating Hamilton's equations of motion. Such treatments require that a knowledge of the potential energy as a function of internuclear separation be known. For some processes this potential energy surface is known, at least approximately. In addition, the positions and momenta of the atoms must be known at some initial time. The program used to do the vibrational analysis in the ethane film can be used to supply information for the initial conditions associated with the vibrational portion of the motion. The implementation of these techniques will allow the study of collisional processes by showing the exchange of energy between degrees of freedom as well as the study of chemical reactions. In the study of chemical reactions the importance of the geometry of the collision complex may be readily shown as well as the relative importance of vibrational, rotational, and translational energy on the probability of reaction.

We would like to acknowledge the financial assistance received from the Advisory Council on College Chemistry.

In addition we are grateful for the support provided by the Frank J. Seiler Research Laboratory, Office of Aerospace Research, United States Air Force Academy and the Research Computing Center of the University of Massachusetts at Amherst in providing computer support for this project. A special acknowledgment is expressed to Mr. R. N. Davis of the Lincoln Laboratories, MIT for his invaluable support and advice.

REFERENCES

(1) HANSEN, G. E., AND DENNISON, D. M., J. Chem. Phys., 20, 313 (1952).

(2) WALTON, J. S., AND RISEN, W. M., JR, J. CHEM. EDU., 46, 334 (1969).