Computational Chemistry

Of^ ¼kf; O^¼operator (4.38)

is called aneigenvalue equation. The functionsfand constantsk that satisfy
Eq. (4.38) are eigenfunctions and eigenvalues, respectively, of the operatorO^.
The operatorH^is called the Hamiltonian operator, or simply the Hamiltonian.
The term is named after the mathematician Sir William Rowan Hamilton, who
formulated Newton’s equations of motion in a manner analogous to the quantum
mechanical equation4.36. Eigenvalue equations are very important in quantum
mechanics, and we shall again meet eigenfunctions and eigenvalues.
The eigenvalue formulation of the Schr€odinger equation is the starting point
for our derivation of the H€uckel method. We will apply Eq.4.36to molecules,
so in this contextHˆandcare the molecular Hamiltonian and wavefunction,
respectively.
From

H^c¼Ec

we get

cH^c¼Ec^2 (4.39)

Note that this is not the same asHˆc^2 ¼Ec^2 , just asx df(x)/dx, say, is not the
same asdxf(x)/dx. Integrating and rearranging we get

E¼

R

cH^cdv

R

c^2 dv

(4.40)

The integration variabledvindicates integration with respect to spatial coordi-
nates (x,y,zin a Cartesian coordinate system), and integration over all of space is
implied, since that is the domain of an electron in a molecule, and thus the
domain of the variables of the functionc. One might wonder why not simply
useE¼Hˆc/c; the problems with this function are that it goes to infinity asc
approaches zero, and it is not well-behaved with regard to finding a minimum by
differentiation.
Next we approximate the molecular wavefunctioncas a linear combination of
atomic orbitals (LCAO). The molecular orbital (MO) concept as a tool in interpret-
ing electronic spectra was formalized by Mulliken^23 starting in 1932 and building
on earlier (1926) work by Hund^24 [ 31 ] (recall that Mulliken coined the word

(^23) Robert Mulliken, born Newburyport, Massachusetts, 1896. Ph.D. University of Chicago. Pro-
fessor New York University, University of Chicago, Florida State University. Nobel Prize in
chemistry 1966, for the MO method. Died Arlington, Virginia, 1986.
(^24) Friedrich Hund, born Karlsruhe, Germany, 1896. Ph.D. Marburg, 1925, Professor Rostock,
Leipzig, Jena, Frankfurt, G€ottingen. Died G€ottingen, 1997.
4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 119