5.7 Summary................................................................
Ab initio calculations rest on solving the Schr€odinger equation; the nature of the
necessary approximations determine the level of the calculation. In the simplest
approach, the Hartree–Fock method, the total molecular wavefunction C is
approximated as a Slater determinant composed of occupied spin orbitals (each
spin orbital is a product of a conventional spatial orbitalcand a spin function).
Writing the molecular energy as the expectation value of the wavefunction
(E ¼hC|H^|Ci), i.e. invoking the Schr€odinger equation, then differentiating
Ewith respect to the spin orbitals that compose the wavefunction (¼the Slater
determinant), we get the HF equations. To use these in practical calculations the
spatial orbitals are approximated as a linear combination (a weighted sum) of basis
functions. These are usually identified with atomic orbitals, but can really be any
mathematical functions that give a reasonable wavefunction, i.e. a wavefunction
which gives reasonable answers when we do the calculations. The main defect of
the HF method is that it does not treat electron correlation properly: each electron is
considered to move in an electrostatic field represented by the average positions of
the other electrons, whereas in fact electrons avoid each other better than this model
predicts, since any electronAreally sees any other electronBas a moving particle
and the two mutually adjust (correlate) their motions to minimize their interaction
energy. Electron correlation is treated better in post-HF methods, such as the
Møller–Plesset (MP), configuration interaction (CI), and coupled cluster (CC)
methods. These methods lower electron–electron interaction energy by allowing
the electrons to reside not just in conventionally occupied MOs (thenlowest MOs
for a 2n-electron species), but also in formally unoccupied MOs (virtual MOs).
The main uses of the ab initio method are calculating molecular geometries,
energies, vibrational frequencies, spectra (IR, UV, NMR), ionization energies and
electron affinities, and properties like dipole moments which are directly connected
with electron distribution. These calculations find theoretical and practical applica-
tions, since, for example, enzyme–substrate interactions depend on shapes and
charge distributions, reaction equilibria and rates depend on energy differences,
and spectroscopy plays an important role in identifying and understanding novel
molecules. The visualization of calculated phenomena, such as molecular vibra-
tions, charge distributions, and molecular orbitals, can be very important in inter-
preting the results of calculations.
References....................................................................
- General discussions of and references to ab initio calculations are found in: (a) Levine IN
(2000) Quantum chemistry, 5th edn. Prentice Hall, Engelwood Cliffs, NJ. (b) Lowe JP
(1993) Quantum chemistry, 2nd edn. Academic Press, New York. (c) Pilar FL (1990)
Elementary quantum chemistry, 2nd edn. McGraw-Hill, New York. (d) An advanced
book: Szabo A, Ostlund NS (1989) Modern quantum chemistry. McGraw-Hill, New York.
References 373