Concise Physical Chemistry

c15 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

238 EARLY QUANTUM THEORY: A SUMMARY

Atoms in their ground states areconservative systems. A conservative system is

one that is not running down. Our solar system is a conservative system but a clock

is not. This leads to the compact form

Hˆ=E

for thetime-independentSchrodinger equation for a conservative system, where ̈

=(x,y,z)is anamplitudefunction, andEis the total energy of the system.

We are interested in the time-independent form of the Schrodinger equation when ̈

we study the ground state structure and energy of atoms and molecules, which do

not change over small time intervals. This equation was generalized to cover many

electron atoms and molecules. As we shall see later, spectroscopictransitionswithin

atoms and molecules involve the time-dependent form of the equation.

Independently, Heisenberg was developing a parallel line of quantum reasoning

that led to his famousuncertainty principle. The Heisenberg and Schrodinger equa- ̈

tions can be shown to be mathematically equivalent. Both Heisenberg and Schrodinger ̈

received the Nobel Prize.

Almost immediately after publication of Schrodinger’s and Heisenberg’s initial ̈

papers on quantum theory, Born et al. (1926) proposed that the square of the wave

function|(r)|^2 dτ(strictly, its vector inner product〈|〉) describes undulations

in theprobabilityof finding a particle in an infinitesimal volume of spacedτ.For

most of our purposes, we shall regard the square of the wave function as governing

theprobable locationof an electron in a region of space containing many electrons.

15.3 MOLECULAR QUANTUM CHEMISTRY

As part of an emerging theory of molecular structure, the chemical bond was

associated with a region of high relative electron density between two nuclei. Heitler

and London (1927) took the lowest solution of the Schrodinger equation for one ̈

of theatomsin H 2 and combined it with an identical solution for the otheratomin

such a way as to obtain an approximate wave function for the chemical bond of the

hydrogenmolecule:

ψ=c 1  1 ±c 2  2

Thenewwavefunctionissaidtobealinear combinationof the two atomic wave

functions 1 and 2 , centered at their respective nuclei. By selecting appropriate

values ofc 1 andc 2 (which turn out to be equal), an energy minimum appears for the

combined wave function. In this notation, the atomic wave functionsare exact but

the H 2 moleular orbitalsψare approximate.

We are accustomed, from classical mechanics and thermodynamics, to thinking of

an energy minimum as representing a stable state. And so it is with the Heitler–London

energy minimum for the linear combination of atomicorbitalsin hydrogen. The

positive combinationψ=c 1  1 +c 2  2 leads to a stable state for two hydrogen

atoms at approximately the experimental bond distance (74 pm). This is the first