Download Exam #2 Answers to Definitions (Problem #1)

Exam #2 Answers to Definitions
(Problem #1; 5 points for each subsection; 35 points total)
a) Born-Oppenheimer approximation assumes that the nuclei are stationary, and that electron
motion can be treated separately. Fixed bond distances and bond angles are assumed, and a
Hamiltonian operator is written for electronic motion only.
b) LCAO-MO method for constucting approximate molecular electronic wave functions.
LCAO-MO are molecular orbitals that are linear combinations of atomic orbitals. If
f1 , f 2 , f 3 ,... are a set of functions, then g is called a linear combination of these functions if it
equals a sum of these functions times constant coefficients:
g  c1 f 1  c 2 f 2  c3 f 3  ...
c) Variational Theorem as applied to finding an approximate ground-state energy and wave
function for a molecular (or an atomic) system
Variational Theorem: The expectation value of the energy calculated with any function 
obeying the same boundary conditions as the correct system wave functions is always greater
than the correct (exact) ground-state energy eigenvalue of the system.
* Hˆ dq

 E 
 E groundstate , where E ground state is the exact value of the ground-state energy
*


dq

The Variational Theorem is used to approximate ground-state energies and wave functions
by constructing trial wave functions,  . These trial wave functions contain variable parameters
( c j ). Then,  E  is minimized with respect to the variable parameters.
d) Pauli Principle as related to the pairwise permutation symmetry of a multi-electron wave
function.
For fermions (e.g. electrons, protons, neutrons, etc), the total wave function must change sign
under pairwise permutations of particle labels.
For bosons (e.g. photons, etc), the total wave function does NOT change sign under pairwise
permutations of particle labels.
When writing the total (spin and space) wave function for an electron, the Pauli Principle
requires that Pˆ i, j i, j   i, j 
Please note that a statement of the Pauli Exclusion Principle only earned partial credit.
e) Valence-bond representation of the ground electronic state wave function for molecular
hydrogen (H2); (Write out the wave function, including both the spin and spatial orbital part.
Be sure to define your notation).
The ground state of the hydrogen molecule has the following valence bond function:
VB  C1sA 11sB 2  1sB 11sA 212  12 where C is the normalization
constant; 1sA 11sB 2  1sB 11sA 2 is the space factor; and 12  12 is the
spin factor. Notation must be clearly defined for full credit.
f) Orbital overlap integrals as encountered in LCAO-MO calculations of molecular electronic
wave functions
S ab    a  b d
g) Inversion symmetry and angular momentum properties of
(1) a  g 2 s molecular orbital:
Inversion symmetry: the orbital does NOT change sign upon inversion, and has even
(gerade) parity.
Angular momentum: zero.
(2) a  u 2 p y molecular orbital
Inversion symmetry: the orbital changes sign upon inversion, and has odd (ungerade)
parity.
Angular momentum:  1 .