Computational Chemistry

mechanical (ab initio, semiempirical, density functional) calculation. In any
case, since vibrational calculations are meaningful only at stationary points,
the surface usually excludes ZPE and thermal contributions to energy, and is a
hypothetical 0 K energy surface, corresponding at least roughly to the electronic
energy plus internuclear repulsion, cf. Eq.5.94, but with the ZPE term excluded
and theEterm obtained from any quantum mechanical method or from a
molecular mechanics surrogate of one by virtue of parameterization. Although
we may not explicitly consider potential energy here, electronic energy is partly,
and internuclear repulsion wholly, this kind of energy.
Symbol: potential energy on a Born–Oppenheimer surface (i.e. in a PES dia-
gram) is denoted inChapter 2byE. Other common designations areV(origin
obscure) andPE, and sometimesU, but this latter is best reserved for internal
energy.Equation: potential energy is the integral over the relevant distance of
the force, itself usually a function of distance.
2.Kinetic energy(translational energy) is the energy of motion, and is taken into
account for the motion of a molecule as a whole by a term (3/2)RT, (1/2)RTfor
each degree of freedom of motion;Ris the ideal gas constant andTthe tempera-
ture. Part of the electronic energy of a molecule is its electronic kinetic energy.
Symbol: kinetic energy is denoted byKEorT(origin obscure), although this
could occasionally be confused with temperature. Equation: in classical physics
kinetic energy is (1/2)mv^2. The electronic kinetic energy of a molecule can be
calculated from the Schr€odinger equation as explained inSection 5.2.

3. Theinternal energyof a molecule is the energy due to its electronic kinetic and
   potential energy, its internuclear potential energy and nuclear ZPE, its rotational
   energy, and its translational motion. Changes in internal energy are usually
   largely changes in bond energies, arising from changes in electronic kinetic
   and potential energies.
   Symbol: internal energy is denoted byU(occasionallyE), because in the
   alphabetUlies close to other thermodynamic quantities:Q(heat),R(the gas
   constant),S(entropy),T(temperature),V(volume) andW(work), andUwas not
   yet taken (ca. 1860). Clausius introduces the symbol by simply saying “... U
   denotes an arbitrary function ofvandt”[ 129 ].Equation: for a molecule we can
   write for the internal energy atTK (cf. Eq.5.94):

UT¼Etotal0K¼EtotalþEvrþ

3

2

RT $ð 5 : 171 Þ

whereEtotalis the electronic energyþinternuclear repulsion, not necessarily at

the Hartree–Fock level,Evris the total vibrational and rotational energy, and

(3/2)RTis the translational energy, (1/2)RTfor each translational degree of

freedom. Internal rotations tend to be regarded as low-energy vibrations,

although more realistic treatments are possible [ 130 ]. Rotation of the molecule

as a whole, and population of upper vibrational levels, is taken into account in

calculating by statistical mechanics the thermal contribution to the energy at

temperatures above 0 K [ 130 ]. Upper electronic levels are usually scarcely

significantly populated at “chemically accessible” temperatures.R, the gas

5.5 Applications of the Ab initio Method 293