Concise Physical Chemistry

c17 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

A VARIATIONAL TREATMENT FOR THE HYDROGEN ATOM 273

# gen

hatom gen

02

h

10

S 1 1.0

0.280000 1.0

****

FILE 17.1 Gaussiangeninput for the hydrogen atom.

and algebra, but we choose to do it now by running the Gaussian©Ccomputer program,

using ageninput. Selecting a value ofα= 0 .28, one finds

Eψ=− 0. 4244 Eh

The computed value of the energy is less negative and is thereforehigherthan the

exact value of^12 Ehby about 15%.

17.3.2 A GAUSSIAN© HF Calculation ofEatom: Computer Files

Thegenkeyword in Gaussian permits you to input a value forαas the first entry in

the penultimate line of the input file, 0.280000 in File 17.1. We have done this for

α= 0. 18 , 0 .28, and 0.38. The energy results extracted from much larger Gaussian

output files are shown as File 17.2.

Two things are clear even from this tiny extract from the output. The value for

α= 0 .28 chosen from among the threeαvalues leads to the lowest energy, but

the optimizedα= 0 .28 isn’t much better than the other two choices. Evidently the

energy minimum is at the bottom of a rather shallow energy well because none of

the three energy values is obviously wrong. Changingαfrom .28 to .38 gives an

increase in the energy value of only 0.0107Eh. Changing the input value in the

other direction by 0.1 results in a slightly larger energy change. This small result

of 0.0107Ehis a little deceptive, however, because the hartree is such a large unit

relative to chemical problems. The conversion factor fromEhto kcal mol−^1 being

627.51 kcal mol−^1 , the calculated value in the hydrogen atom problem is a substantial

0. 0107 Eh( 627. 51 )= 6 .7 kcal mol−^1.

0.180000 HF=-0.4070275

0.280000 HF=-0.4244132

0.380000 HF=-0.4136982

FILE 17.2 Energies drawn from the Gaussiangenoutput file for the hydrogen atom.