Concise Physical Chemistry

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

280 THE VARIATIONAL METHOD: ATOMS

hydrogen-likeatoms. A few STOs are

ψ 1 s=N 1 se−αr

ψ 2 s=N 2 sre−αr/^2

ψ 2 px=N 2 pxr^2 e−αr/^2

etc.

The normalization constantsNareN 1 s=(α^3 /π)^1 /^2 ,N 2 s=(α^3 / 96 π)^1 /^2 , and so on.

Clearly, the entire system depends uponα, which is an empirically fitted parameter.

The parameter is written as

α=Z−S

whereZis the atomic number (charge on the nucleus) andSis ashielding constant

which accounts for the diminution of nuclear charge experienced by an outer electron

owing to shielding by inner electrons.

This method can, with computational difficulty, be extended to atoms larger than

helium and to a few small molecules. In a molecular problem as simple as methane,

however, the dimension of the Slater determinant is 16×16. Clearly, molecular

problems are daunting to anyone attempting hand calculations. Practical applications

awaited widespread availability of powerful digital computers. Although we shall

soon consider more challenging problems of correlated wave functions, Slater orbitals

and the Hartree–Fock (HF) equations contain the essence of atomic and molecular

orbital theory. Much important structural and thermochemical information can be

gleaned from them (Hehre, 2006).

17.8 THE AUFBAU PRINCIPLE

Given the orbital structure of the elements in the first three rows of the periodic

table, one can predict, with reasonable certainty, their chemical properties. Hy-

drogen ionizes to form the H+ion (hydrated in aqueous media), and it can be

persuaded to take on one electron to form the H−hydride ion but helium does nei-

ther because its 1s orbital is “full” (Pauli). The electronic configuration of helium

is He 1s^2.

The first full row of the periodic table starts with lithium, which, like hydrogen,

can lose an electron, this time from the 2sorbital as its characteristic reaction. The 2s

probability density antinode is considerably more distant from the nucleus than the

1 sorbital of hydrogen, making this loss of an electron from Li even more facile than

the equivalent loss from H. Beryllium has two electrons in the 2sorbital. Electron

loss from Be takes place easily, but not as easily as Li because its nuclear charge is

one more than in Li. Be is a metal but is not as metallic as Li. Here the “metallic