Rogers Computational Chemistry Using the PC

Generalizing the Newton–Raphson method of optimization (Chapter 1) to a surface in many dimensions, the function to be optimized is expanded about the many-dimensional position vector of a point x0

where g is the gradient vector corresponding to the first derivative of a onedimensional expansion and H is the Hessian matrix that corresponds to the second derivative. For a small gradient, this leads to (Jensen, 1999)

If the coordinate system has been transformed to the normal coordinate system by a unitary transformation U, the Hessian is diagonal and

where fi is the projection of the gradient along the Hessian eigenvector with the eigenvalue ei.

Having filled in some of the mathematical foundations of optimization procedures, we shall return to the practical calculation of quantities of everyday use to the chemist.

The Enthalpy of Formation

We shall modify the minimal ethylene file minimal.mm3 by placing an identifying name (Ethylene) starting in the first column of the first row of the input file and by placing a switch 1 in column 65 of the second line. This switch causes the enthalpy of formation option to be activated. An additional block of energy and enthalpy information, shown in File Segment 5-1, is generated by the MM3 calculation and added to the output file. The term ‘‘heat of formation,’’ more properly ‘‘enthalpy of formation,’’ means the enthalpy change f H298 brought about by the reaction forming a molecule from its elements in the standard state. Standard temperature is usually taken as 298.15 K. For example, f H298ðetheneÞ is the enthalpy of the reaction

2CðgrÞ þ 2 H2ðgÞ ¼ C2H4ðgÞ

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

File Segment 5-1 Heat of Formation and Strain Energy Output.

At the top of File Segment 5-1 is a heat of formation information block. Two sums are listed: One is a sum of normal bond enthalpies for ethylene, and the other is a sum selected from a parameter set of strainless bonds. Both sets of bond enthalpies have been empirically chosen. A group of molecules selected as ‘‘normal’’ generates one parameter set, and a group supposed to be ‘‘strainless’’ is selected to generate a second set of strainless bond enthalpies designated SBE in File Segment 5-1. The subject of parameterization has been treated in detail in Chapter 4. See Computer Projects 3-6 and 3-7 for the specific problem of bond enthalpies.

After initial estimates of BE and SBE have been made, application to a larger set of test molecules suggests adjustments to bring about the smallest discrepancy between the experimental and calculated values of f H298 (or other selected

146 COMPUTATIONAL CHEMISTRY USING THE PC

criteria) for the entire set. The success or failure of bond energy assignment is determined, of course, not so much by one’s ability to reproduce known f H298 values within the data set from which BEs are drawn but by the more important practical goal of correctly predicting unknown f H298 values from outside the test data set, leading to useful experiments and industrial applications.

Once the BEs and SBEs have been decided upon, the normal functioning of the MM program causes each bond to be multiplied by the number of times it appears in the computed molecule to find its contribution to the total bond enthalpy. In ethylene, 26:43 þ 4ð 4:59Þ ¼ 8:07 kcal mol 1. In File Segment 5-1, this sum is denoted BE. This whole procedure is essentially a conventional bond energy calculation.

The difference between an MM calculation of the enthalpy of formation and a bond energy scheme comes in the steric energy, which was shown in File 4-3. The sum of compression, bending, etc. energies is the steric energy, E ¼ 2:60 kcal mol 1 in File 4-3. This is added to BE, as is the partition function energy contribution (see below), PCE ¼ 2:40 kcal mol 1, to yield

8:07 þ 2:60 þ 2:40 ¼ 13:07 kcal mol 1

for the enthalpy of formation f H298 of ethylene. The same thing is done with the strainless bond energies except that there is no steric energy E to add. In ethylene, the strain energy is essentially zero, therefore normal f H298 is essentially the same as strainless f H298. This is not true in general.

The Partition Function Contribution. If we ignore vibrational motion for the moment, all translational and rotational modes of motion are fully excited, leading to an energy contribution of 12 RT per degree of freedom per mole or 3RT. Taking the ideal gas approximation for granted in calculating molar enthalpy (at this level, intermolecular interactions are assumed to be zero), H is RT higher than the energy. The total is 4RT. This accounts for the partition function contribution PFC ¼ 2:40 kcal mol 1.

The vibrational energy over all chemical bonds, i

would normally be added to this sum. This term is difficult to evaluate, and, depending on the way in which the bond enthalpy parameters are obtained, it may be an overcalculation anyway. If bond energies are taken from f H298, as they are for MM3, they already contain the thermal energy of vibration.

Finally, the system ‘‘knows’’ to print End of Ethylene at the end of the program because we included the name Ethylene in the first line of the input file.

Enthalpy of Reaction

If one knows the enthalpy of formation at 298 K of all the constituents of a chemical reaction, one knows the enthalpy of reaction, rH298. The calculation rests on the law that thermodynamic functions, of which enthalpy is one, sum to zero around any cyclical path, formally,

For example, in the hydrogenation of ethylene

CH2 CH2 þ H2 ! CH3 CH3

a diagram with enthalpy as the vertical axis can be drawn as in Fig. 5-4. Because there is no natural zero point for the enthalpy in classical thermo-

dynamics, we are free to define one, but only one, level in the enthalpy level diagram. The enthalpy levels in Fig. 5-4 are relative to a defined level of zero for the elements. The cyclical path includes f H298 values that are up for a positive H298 and down for a negative H298. The horizontal lines represent the standard states of ethylene, elements, and ethane. The rightmost arrow represents the transition from

with great care many years ago by Kistiakowsky (Kistiakowsky and Nickle, 1951), who found that rH298 ¼ 32:60 0:05 kcal mol 1. One should not expect this extraordinary precision and agreement between calculations and experimental measurements in general. Uncertainties of experiments and calculations are rarely better than 0:5 kcal mol 1; hence, the uncertainty in a calculated f H298 will not normally be better than the square root of the sum of squares of uncertainties inf H298, that is, not better than 0:7 kcal mol 1 for a simple A ! B reaction.

steric energies of the two butene isomers using the MM3 force field. Store the input files in the TINKER directory as .xyz files, for example, cbu2.xyz for cis-2-butene.

Carry out this computation using 1) Program TINKER (force field MM3) and 2) Program MM3 (force field MM3).

1) TINKER. Go to your DOS operating system, TINKER directory. Execute the command minimize. Respond to the choices offered:

Enter Cartesian Coordinate File Name: cbu2.xyz Enter Potential Parameter File Name: mm3

Enter RMS Gradient per Atom Criterion [0.01]: Enter (this is the default option)

Take the ‘‘Final Function Value’’ of TINKER as the steric energy for this calculation. Compare the results with each other and with a standard value from a good elementary organic chemistry text (e.g., Ege, 1994). Calculate isomH298 for the reaction cis ! trans and compare it with a standard text and with Kistiakowsky’s original value. Kistiakowsky’s original work was carried out at 355 K, but the temperature difference between 298 K and 355 K cancels for this isomerization.

The geometric output can be found as you would find any other file in DOS

D:\tinker>dir cbu*.*

Directory of D:\tinker

You can see the optimized geometry using the edit command.

D:\tinker>edit cbu2~1.xyz

2) The PCMODEL Interface. Using the draw option of the PCMODEL interface, construct the carbon atom skeleton of cis-2-butene. Use Add_B to add a bond in the 2-3 position, thereby making it a double bond. Click on H/AD and observe that the correct number of hydrogen atoms appears at plausible distances and angles to the carbon atoms. Hydrogen atoms at either end may not be visible because of the hydrogens in front of them. Later, we shall learn how to rotate the molecular image in three-dimensional space so as to make all atoms visible. This is not the optimized structure of cis-2-butene; it is the input or starting structure before minimization.

If you are running an updated version (V 8.0) of PC Model, click on force field!mm3. Omit this step for older versions. Click on Analyze (or compute depending on the version of PCMODEL) to obtain a menu of options. Select minimize. The geometry changes can be seen on the screen and a sequence of numbers appears in the right panel of the CRT screen, ending in Hf, the enthalpy of formation. This is the PCMODEL-MM3 calculated value of f H298 for cis-2- butene. Repeat the entire procedure for trans-2-butene and calculate the required

isomH298.

Rosenberg, 2000). Unfortunately, experimental values of heat capacities are not usually available over a wide temperature range and they present some computational problems as well [see Eq. (5-46)].

Population Energy Increments

There are two additive terms to the energy, POP and TORS, that have not been mentioned yet because they are zero in minimal ethylene. The POP term comes from higher-energy conformers. If the energy at the global minimum is not too far removed from one or more higher conformational minima, molecules will be distributed over the conformers according to the Boltzmann distribution

where Ni is the ratio of molecules in the ith high-energy conformational state to

N0

molecules in the ground state (see Computer Project 4-2). Ei E0 is the energy difference between the energetic conformer and the ground state. If there is degeneracy of any energy level, it is counted into the conformational mix. If there are several conformers not too far from the ground state, they will be simultaneously populated and all must be taken into account. The energy of the compound under investigation, as measured by experimental means, is that of the weighted average determined by the conformational mix at any selected temperature.

Exercise 5-2

A molecule has three nondegenerate conformers (Fig. 5-8). One is 450 10 23 J above the ground state, and the second high-energy conformer is 900 10 23 J above the ground state. What are the percentages of each of the three conformers relative to the total number of molecules in a sample of the normal conformational mixture at 300 K?

E2

E1

Solution 5-2

The exponent in the Boltzmann expression for the conformer at E1 is

where kB is Boltzmann’s constant. The exponent in the Boltzmann expression for the highest conformer is just twice this value

The ratio of conformers is

N1 ¼ eððE1 E0Þ=kB TÞ ¼ e 1:086 ¼ 0:3376

N0

between the middle conformer and the ground state and it is

N2 ¼ eððE2 E0Þ=kB TÞ ¼ e 2:172 ¼ 0:1139

N0

between the highest energy conformer and the ground state. The total number of molecules is

N ¼ N0 þ 0:3376N0 þ 0:1139N0 ¼ 1:452N0

so

Equal spacing between energy levels is not unusual. In the case of the harmonic oscillator, it is the rule.

Exercise 5-3

Suppose that the computed ground-state enthalpy of formation of a molecule is f H298 ¼130:0 kJ mol 1 but that it has two higher-energy conformers as described in Exercise 5-2. What is the expected experimental f H298 of the equilibrium conformational mixture?

Solution 5-3

The energies of the two higher-energy conformers described in Exercise 5-2 are

450 10 23Jð6:022 1023mol 1Þ ¼ 2:71 kJ mol 1

and

900 10 23Jð6:022 1023mol 1Þ ¼ 5:42 kJ mol 1