Problem Set 9 Solutions

4
3. Continued
d). Estimate how the CPU time would scale relative to the minimal basis set calculation
for Hartree-Fock single point energy calculations on benzaldehyde using basis sets (b)
and (c).
The CPU time for a Hartree-Fock single point energy calculation scales approximately as the number of
basis functions to the power 3.5, K 3.5 .
Basis Set
No. Basis
Functions
Relative
CPU Time
minimal 46 1
6-31G(d) 124 (124/46) 3.5 = 32
6-311G(d,p) 180 (180/46) 3.5 = 119
4. Consider a linear combination of three gaussian type orbitals, called a contracted gaussian
function,
f (r) =
3
∑ c i G i (r) ,
i=1
where f (r) is an atomic orbital, and the c i s are linear coefficient.
gaussian function, given by € the expression
The function
G i (r) is a
€
€
G i (r) =
⎛⎛
⎜⎜
⎝⎝
2α i
π
3/ 4
⎞⎞
⎟⎟ e −α i r 2 .
⎠⎠
€
The reason that an atomic orbital like f (r) is expressed in terms of a linear combination
of gaussian functions is so € that it can better represent the cusp behavior of a true atomic
orbital in the vicinity of the nucleus.
€
An s-type atomic orbital is represented using the following parameters (in atomic units):
d 1 = 4.44635×10 –1 α 1 = 1.09818×10 –1
d 2 = 5.35328×10 –1 α 2 = 4.05771×10 –1
d 3 = 1.54329×10 –1 α 3 = 2.22766
a.) Plot the three orbitals, G 1 (r) , G 2 (r) , and G 3 (r), as functions of x.
relation that r 2 = x 2 + y 2 + z 2 , and plot a cut for x with y = z = 0.
To do this, use the
The three orbitals are € shown € plotted individually € on the next page. Notice that the functions range from
quite broad € to fairly narrow in extent.
€