Concise Physical Chemistry

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

17 The Variational Method: Atoms

Exact orbital solutions for the hydrogen atom cannot be replicated for other atoms

or molecules, even small ones, but they are of inestimable value in the mathematical

treatment of larger systems. The energy of nuclear attraction is very much larger than

the binding energies that dominate chemistry, so one can regard the chemical bond

as a perturbation on nuclear attraction. Thus the orbitals developed for H are the

foundation stones for larger systems of ions, atoms, and the interdependent groups

of atoms that we call molecules.

17.1 MORE ON THE VARIATIONAL METHOD

A precise expression of the variational equation for approximate wavefunctions is

〈E 0 〉=

∫∞

−∞

φ 0 ∗(τ)Hˆφ 0 (τ)dτ

∫∞

−∞

φ∗ 0 (τ)φ 0 (τ)dτ

In this notation, the subscripted 0 indicates the ground state energy and wave func-

tions, the angle brackets around〈E 0 〉indicate an expectation value, the asterisk (∗)

designates the complex conjugate of the function, andτis a variable indicating that

the integrals are to be taken over all space.

Concise Physical Chemistry,by Donald W. Rogers

Copyright©C2011 John Wiley & Sons, Inc.

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