Rep:Mod:testKFL

TS BERNY CHAIR -HF

Animation of imaginary frequency

QST2 BOAT - HF

Animation of imaginary frequency

ETHENE + CIS BUTADIENE T.S

Animation of imaginary frequency

EXO TRANSITION STATE

Animation of imaginary frequency

ENDO TRANSITION STATE

Animation of imaginary frequency

ptimizing the "Chair" and "Boat" Transition Structures of Cope Rearrangement

In this section, the transition structures will be optimized to derive the most likely reaction pathway for Cope Rearrangement by first optimizing allyl fragments at the HF/3-21G level of theory. This is achieved using TS(Berny), freezing coordinates and Synchronous Transit-Guided Quasi-Newton (STQN) methods. The successes of these calculations are determined by the presence of an imaginary vibrational frequency and a small RMS gradient (signifying energy maxima in the PES surface).

All methods of calculating the transition structure require a starting geometry and initial estimate of the Hessian, which are then subsequently updated at each step of the reaction progress.[5]

Optimizing the Allyl Fragments

CH2CHCH2 fragment was built and optimised to a minimum using HF/3-21G level of theory. The results are tabulated here:

Structure Level of Calculations Total Energy (Hartrees) Log file source

Allyl Fragment HF/3-21G -115.82303995

The optimised allyl fragment will be used for building subsequent transition structures.

Using TS(Berny) For Chair Transition State Optimization

The assumption of using TS(Berny) calculation is that the guessed transition structure is accurate enough to compute the force constant matrix. To perform TS(Berny) calculations, the allyl fragment is duplicated and symmetrized. Unlike normal optimization process for other structures, the transition state structure requires precise geometry for the reactants as to ensure that there is a negative curvature in the reaction coordinate. Calculations must ensure that the starting structure has only one negative eigenvector so that the guessed structure can converge to the closest saddle point. As a result, the terminal ends of the allyl fragments are set at 2.2 Angs apart, with C1 symmetry for the entire structure. The results are tabulated here:

Structure Level of Calculations Total Energy (Hartrees) Imaginary Vibrational Frequency (cm-1) Log file source

Chair T.S of Cope Rearrangement HF/3-21G -231.61932227 -817.87

Vibrational frequency is expressed as:

And therefore, an imaginary frequency must arise from a negative force constant, f. Imaginary vibrational frequency results from negative eigenvalues of the Hessian, which is a force constant matrix. In most normal modes of frequency that are positive for ground state structures, due to the fact that the potential energy surface curves upwards at all directions. However, at the saddle point, the energy reaches a maximum in one direction while being a minimum in all other directions. As a result this difference, the force constant acts in the opposite direction for transition structures, leading to an imaginary vibrational frequency. An animation of the imaginary vibrational frequency shown on the transition state optimization is visualized below:

[IMAGINARY FREQUENCY FOR TS(BERNY) CHAIR]

Using Frozen Coordinate Method For Chair Transition State Optimization

Unlike the TS(Berny) method, the Frozen Coordinate Method first optimizes the rest of the structure that are not involved in the bond breaking and bond forming sites using HF/3-21G level of calculations. Subsequently, the coordinates of the bond breaking/forming sites are changed to 'derivatives', which would reflect the optimization of the inter-fragment distances leading to the transition state. In order to build the input structure, the initial inter-fragment distances are set to be 2.2 ‎Å.

Structure Level of Calculations Total Energy (Hartrees) Imaginary Vibrational Frequency (cm-1) Log file source

Chair T.S of Cope Rearrangement HF/3-21G -231.61932198 -818.06

Like the TS(Berny) calculations, the structure that is generated at the end of the optimization process is a transition state given the imaginary vibrational frequency.[IMAGINARY FREQUENCY FOR TS CHAIR FROZEN]

Summary of Optimizing Chair Transition State of Cope Rearrangement

Comparison between the two methods for optimizing the transition structure can be evaluated by the interatomic distance between the terminal carbons, which are the sites for the reaction.

Type of Calculations Interfragment Bond Length (‎Å)

TS(Berny) 2.02039

Frozen Coordinates 2.02172

As shown, the resultant geometries of the transition state using both methods are very similar, with bond lengths accurate to 10-2 Angs. Furthermore, the difference in the final total energy is also appreciably small, 0.00000029 Hartrees or ~0.000182 kcal/mol. From these results, it is reasonable to conclude that both methods produce the same structure at the same saddle point on the PES surface, which is the chair transition state. Close approximation to empirical results is possible for simple molecules due to simpler geometry involved in the process (the initial geometry is an important factor for the TS(Berny) calculations).

Using the QST2 Method For Boat Transition State Optimization

The boat transition structure will be optimized using quadratic synchronous transit (QST2) method at HF/3-21G level of calculations. For the minimization process, QST2 method uses redundant internal coordinates as well as Newtonian laws to complete the optimization process. It operates by searching for a maximum along the quadratic region of the PES surface and minimum in all directions perpendicular to the curve that connect products with reactants.[6] As a result, the calculations are performed using the optimized reactant and product shown below: To be interpolated: optimized reactant (left) with optimized product (right)Calculations are performed using the Ci molecule optimized in the earlier section. Given the mechanisms of Cope rearrangement, corresponding carbon atoms in the reactants and products need to be labelled to reflect the rearrangement process. For example, the single bond between C3-C4 in the reactant will be broken to form a double bond between C2-C3 or between C4-C5. Similarly, another single bond needs to be formed between C1-C6, and hence a new carbon backbone is defined in the product. In addition to this, the structures of reactant and product need to be altered to resemble the transition state more closely. This involves changing the central C-C-C-C dihedral angle to 0o and then changing the C2-C3-C4 and C3-C4-C5 dihedral angles to 100o. The results calculated are tabulated below:

Structure Level of Calculations Total Energy (Hartrees) Imaginary Vibrational Frequency (cm-1) Log file source

Boat T.S of Cope Rearrangement HF/3-21G -231.60280201 -840.81

The presence of an imaginary vibrational frequency again validate that the transition state has been reached. Animating this particular vibrational frequency indicates that as the distance between C3-C4 lengthens, C1-C6 distance is compressed.

[ANIMATION FOR THE IMAGINARY VIBRATIONAL FREQUENCY]

The Intrinsic Reaction Coordinate for the Chair Transition State

Prediction of the final conformers is hard to visualize using the transition structure, and as a result, the intrinsic reaction coordinate allows the transition structure to be transformed to a structure corresponding to a local minimum on the PES surface. The IRC, which is defined as the path to a transition state point from a stable equilibrium point, is generated using the initial geometry corresponding to the transition state and initial force constants and therefore consider nuclei movement with infinitesimal velocities.[7] For this calculation, the reaction coordinate is computed in both directions and that the force constant is calculated at every step (this indicates that after every step, the force is re-evaluated to find the steepest gradient to a new position along the PES surface). A step of 100 IRC steps is performed, which yields the following reaction coordinate, shown below. Using the IRC calculations, it is shown that the final conformer is gauche2, with C2 symmetry point group.

Number of Steps IRC Path Final Conformer Final Conformer Total Energy (a.u.)

100

IRC path evaluated at 100 points for the Chair Transition State of 1,5-hexadiene during Cope Rearrangement

Gauche 1,5-hexadiene - gauche2 -231.69158000

125

IRC path evaluated at 125 points for the Chair Transition State of 1,5-hexadiene during Cope Rearrangement

Gauche 1,5-hexadiene - gauche2 -231.69158000

The IRC path shows that the calculations start from the transition state, the maxima point (labelled '0' on the Intrinsic Reaction Coordinate x axis) and possesses both directions to a constant minimum energy. In addition, the IRC path suggests that the reaction path is symmetrical with respect to either direction. Comparison between the 100 points and 125 points results indicated that the final conformer energy has remained constant, -231.69158000 a.u., and therefore 100 points for IRC calculations is sufficient.

Cope Rearrangement Activation Energies

HF/3-21G B3LYP/6-31G(d)

Electronic energy Sum of electronic and zero-point energies Sum of electronic and thermal energies Electronic energy Sum of electronic and zero-point energies Sum of electronic and thermal energies

          at 0K
      at 298.15K

          at 0K
      at 298.15K

Chair TS -231.619322 -231.466700 -231.461340 -234.55693 -234.414910 -234.408982

Boat TS -231.602802 -231.450915 -231.445288 -234.54308 -234.402356 -234.396012

Reactant (anti2) -231.692535 -231.539539 -231.532566 -234.611613 -234.469226 -234.461969

Table 8: Summary of optimization results using different levels of theory and structures.

The structures are first optimized at HF/3-21G level; the output of which are then re-optimized at the DFT level, at B3LYP/6-31G(d) level, in order to maximize the accuracy of the structures' energies. Table 5 illustrates that optimization + frequency calculations performed for the three structures, so that the thermochemistry results can be derived. By comparing either the zero-point energies or the 298 K - corrected temperatures between the ground state reactant with a transition state, the activation energy can be derived.

HF/3-21G HF/3-21G B3LYP/6-31G(d) B3LYP/6-31G(d) Experimental

Eact (Chair) 45.94 44.69 34.08 33.25 33.5 ± 0.5

Eact (Boat) 56.19 54.77 41.96 41.38 44.7 ± 2.0

Table 9: Comparison between the activation energies using different levels of theories and temperatures (all energy values in kcal/mol)

As shown on Table 6, calculations involving the DFT B3LYP/6-31G(d) level results in activation energies closer to experimental results, while HF/3-21G results in transition states that are not as extensively optimized, leading to an increased energy gap between the ground state reactant to both transition states (higher activation energies). In either the DFT theory or HF method, the chair transition structure provides a lower-energetic pathway to the product.

The conformations/geometries of either transition state converges upon re-optimization. This is shown below, where the corresponding bond lengths do not alter significantly from HF to DFT calculations.

HF/3-21G B3LYP/6-31G(d)

Boat T.S Allyl fragments separation (Angs) 2.13767 2.20639

Allyl C-C Bond (Angs) 1.38176 1.39327

Chair T.S Allyl fragments separation (Angs) 2.02055 1.96470

Allyl C-C Bond (Angs) 1.38922 1.40791

Table 7: Comparison between the resulting geometries from optimizing each transition structure using both DFT and HF methods.

Table 7 illustrates that using either method does not prevent the transition states from converging to the final, most stable geometry, given the small differences (~of 10-11 magnitude). As a result, a reasonable conclusion is that performing calculations at the expense of greater computational time does not necessarily impact geometries' convergence, but do affect energies more significantly, as shown by activation energies' discrepancies using either the chair or the boat transition state.