ÁREA DE INFLUENCIA DEL PROYECTO BARRIO SAN JUAN

By viewing the tautomerisation classically, as a movement of classical particles across a potential energy surface, the quantum nature of the protons are ignored. This may be

(a) (b) (c)

Figure 6.6: Transition states of Pc undergoing a tautomerisation from the cis confor- mation (a) to the trans (c) via the saddle point conformation (b)

-2

-3

-4

-1

0

1

2

1000/T12.0

12.5

13.0

11.5

11.0

10.5

ln

R

T(K)

Figure 6.7: Experimental Arrhenius plot showing the dependence of the rate of tau- tomerisation,R, on temperature T. The barrier to tautomerisation is calculated to be 168 meV. Figure taken from [30].

10.5 11.0 11.5 12.0 12.5 13.0 1/T ((mK)−1₎ −3 −2 −1 0 1 2 3 4 ln k HTST 95 90 85T (K) 80 75

Figure 6.8: Theoretical Arrhenius plot showing the dependence of the rate constant calculated under harmonic transition state theory, kHTST_{, on temperature T. Also}

shown is a line of best fit, from which the barrier to tautomerisation is calculated to be 209 meV.

problematic, specifically in the case of the protons, which, due to their small mass, have more of a quantum nature and might be expected to exhibit tunnelling phenomena. Also, for such light nuclei, zero point energy (ZPE) effects might be important even at the temperatures for which the rate constant was measured experimentally.

A more complete treatment of the quantum effects involved with the calculation of transition rates involves path integrals [179] but this is computationally expensive. A common set of approximations that are used are called transition state theory (TST) and this is discussed in Section 2.5.

To calculate the rate constant of tautomerisation the harmonic approximation of TST (HTST) was used. The vibrational energies of the system in the harmonic approx- imation were calculated in the cis configuration and in the saddle point configuration (Figure 6.6 (b)), with all atoms held static except for the N and H atoms in the cav- ity. These frequencies were then used in Equation 2.92 to calculate the rate constant,

kHTST_{, and this is plotted in an Arrhenius plot in Figure 6.8.}

The calculated activation energy from Figure 6.8 is 209 meV, reduced from a value of 340 meV for the barrier to tautomerisation calculated by the NEB method, and much closer to the experimental value of 170 meV. Harmonic TST as used here includes ZPE

75 80 85 90 95 T (K) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 κ -1 ×10−11

Figure 6.9: Tunnelling correction,κ₋1, for Pc (Cu(110)), over the temperature range used for the Arrhenius plot, Figure 6.8. The rate constant corrected for tunnelling is given byκkHTST_.

effects, so this result shows that ZPE effects play a substantial role in lowering the effective barrier to tautomerisation.

Harmonic TST includes ZPE effects, but no tunnelling effects. As a first approxima- tion, the role that tunnelling plays can be ascertained by assuming that the tunnelling in the reaction coordinate direction can be separated from the other degrees of freedom. With this approximation, tunnelling effects can be incorporated by multiplying the rate constant from HTST by a factorκ, an expression for which is given in Eq. (2.103). The probability of a H atom tunnelling through the barrier,PF, was calculated by applying

the WKB approximation to the energy barrier between the cis conformation and the saddle point.

In Figure 6.9,κ−1 (the amount by which the rate constant incorporating tunnelling differs from the calculated rate constant incorporating no tunnelling effects) is plotted as a function of temperature. The tunnelling correction is very small compared to the size of the rate constant at the temperatures for which the Arrhenius plot was plotted, showing that in this temperature region tunnelling plays a minor role. The difference in the experimental activation energy and the NEB barrier is therefore seen to be predominantly down to ZPE effects and not tunnelling.

then, as shown by Eq. (2.93), the rate constant is expected to show Arrhenius like behaviour, but with the activation energy corrected by the difference in zero point energies between the cis state and the saddle point configuration.

The calculated difference between the total ZPE of the cis and the saddle configura- tions is ∆EZPE(cis−saddle) = 0.14 eV. When the N atoms are held static and only the

vibrational energies of the two H atoms inside the cavity are calculated the difference in the ZPEs are changed by less than 0.1 meV, showing that the most substantial con- tribution to ∆EZPE(cis−saddle) comes from the ZPE of the H atoms. The difference

in ZPEs leads to a reduction in the estimated effective barrier height given by:

Ebarrier,eff =Ebarrier−∆EZPE(cis−saddle) = 0.20 eV. (6.1)

This value is in close agreement with the activation energy calculated by HTST. This shows that the difference in the ZPEs is dominated by the contribution from the H atoms.

Applying a similar procedure to the trans and saddle configurations leads to a ∆EZPE(trans−saddle) = 0.13 eV, and reduces the effective depth of the energy well

for the trans configuration to be:

Ebarrier,eff(trans) =Ebarrier(trans)−∆EZPE(trans−saddle) = 0.02 eV. (6.2)

This small barrier is consistent with the trans state not being too short-lived to be observed experimentally and having a very short lifetime at 5 K.

The changes in barrier height for a deuterated Pc (D-Pc) molecule due to the quantum nature of deuterium atoms are simply calculated. This calculation gives values of ∆EZPE(cis−saddle,D) = 0.10 eV and ∆EZPE(trans−saddle,D) = 0.09 eV

which means that the effective barrier increases to a value of Ebarrier,eff = 0.24 eV and

the height of the barrier to the trans conformation becomesEbarrier,eff(trans) = 0.06 eV.

The changes in the ∆EZPE values upon deuteration are largely explained by the

atoms. The angular frequency of a D atom in the harmonic approximation is:

ωD=

p

mH/mDωH, (6.3)

where ωH is the vibrational frequency of the same mode of a H atom and mH and

mD are the masses of hydrogen and deuterium respectively.

That the ∆EZPE,Dvalues are, to very good approximation, given by the ∆EZPE for

hydrogen, multiplied by the p

mH/mD factor, is due to the significant contribution of

the hydrogen nuclei to the ∆EZPE value.

Due to experimental limitations the thermal activation process of D-Pc could not be measured. The threshold voltage for inducing tautomerisation with the STM is similar for both Pc and D-Pc molecules, although the average number of electrons inducing a tautomerisation at low biases is lower for D-Pc than for H-Pc. This is consistent with a larger activation energy, but since the process of STM-induced tautomerisation is complex this is not conclusive.

In summary, the over-estimation of the activation energy of tautomerisation by the calculated potential energy barrier is shown, to a large extent, to be due to ignoring the quantum nature of the H nuclei. When this is taken into account, in an approximate manner through the calculation of ZPEs, the activation energy is brought much closer into line with experiment. It is shown that the tautomerisation process is a two step process, proceeding in a step-wise fashion via the trans state and also that the optB86b- vdW functional of Klimeˇs and coworkers [74, 75] accurately describes the behaviour of the hydrogen bonds within the cavity with respect to giving the correct adsorbed conformer as observed by STM and the correct rate constant of tautomerisation (after corrections for ZPE effects) compared to that observed experimentally.