The preexponential factor is tied to the probability of some favorable situation
like a propitious collision and involves entropy, while the exponential term
reflects the energy barrier for the reaction.AandEaare usually approximately
constant over the limited range of laboratory interest. The modified version of
Eyring, Polanyi and Evans (the “Eyring equation”) lends itself directly to the
theoretical calculation of rate constants:k¼kBT/hexp(#DG{/RT), wherekBis
Boltzmann’s constant,his Planck’s constant, andDG{is the free energy of
activation. High-level calculation of rate constants is best done with a
specialized program like, e.g., Polyrate [ 133 ] for unimolecular reactions, using
RRKM (Rice–Ramsperger–Kassel–Marcus) theory [ 134 ].
Symbol:Ea.Equation: the activation energy calculated as the transition state
enthalpy minus the reactant enthalpy,DH{orE{, is related to the Arrhenius
activation energy for a gas-phase unimolecular reaction by [ 135 ]Ea¼DHzþRT¼DHzþ 2 :48 kJ mol#^1 $ð 5 : 175 Þat room temperature (298.15 K).
A nice mnemonic for the relationships among various forms of energy is [ 136 ]:UAHG–TS+ PVU – TS = A
U + PV = H
H – TS = G
A + PV = Gie.5.5.2.1b Energies: Preliminaries
Along with geometries (Section 5.5.1), the molecular features most frequently
sought from ab initio calculations are probably energies. An ab initio calculation
gives an energy quantity that represents the energy of the molecule (or atom)
relative to its constituent electrons and nuclei at rest at infinite separation; this
separated state is taken as the zero of energy. The ab initio energy of a species is
thus the negative of the energy needed to dissociate it completely, to infinite
separation, into the electrons and nuclei, with no kinetic energy left over, or the
negative of the energy given out when the electrons and nuclei “fall together” from
rest at infinite separation to form the species. This was pointed out for Hartree–Fock
energies (Section 5.2.3.6.4, in connection with Eq.5.93), and the infinite-separation
reference point also holds for correlated ab initio energies. By ab initio energy,
then, we normally mean the purely electronic energy (the kinetic and potential
296 5 Ab initio Calculations