Meghan Handley CHEM 2411 TA: Brandon Wade TITLE: Separation of Mixtures by Fractional Distillation EXPERIMENT: September 9, 2010 LAB REPORT: September 16, 2010 PURPOSE: In this experiment, fractional
These densities are identical to one another and do not describe polarized orbital densities.
Therefore, the CI wavefunction which mixes the two configurations with like sign, when
analyzed in terms o
Chapter 9
Electronic Wavefuntions Must be Constructed to Have Permutational Antisymmetry
Because the N Electrons are Indistinguishable Fermions
I. Electronic Configurations
Atoms, linear molecules, an
C1 1s22s2  C2 [1s22px2 +1s22py2 +1s22pz2 ]
mixes its two configurations with opposite sign is of significance. As will be seen later in
Section 6, solution of the Schrdinger equation using t
where a = 3C2/C1 . The socalled polarized orbital pairs
(2s a 2px,y, or z) are formed by mixing into the 2s orbital an amount of the 2px,y, or z
orbital, with the mixing amplitude determined by the r
The most commonly employed tool for introducing such spatial correlations into
electronic wavefunctions is called configuration interaction (CI); this approach is described
briefly later in this Secti
the analogous quantity for the 2p orbital in the 1s22s2p configuration is:
< 2p h2/2me 2  4e2/r + V'SCF 2p> = 12.28 eV;
the corresponding value for the 1s orbital is (negative and) of even larger
with the N1 other electrons, they use a probability density N1(r') that is independent of
the fact that another electron resides at r. In fact, the socalled conditional probability
density for find
Interaction Energy (eV)
300
200
Fluctuation
SCF
100
0
100
2
1
0
Distance From Nucleus ()
1
2
As a function of the interelectron distance, the fluctuation potential decays to zero
more rapidly than
more than one state will arise; these states can differ in energy due to differences in how the
orbitals are occupied. In particular, if degenerate orbitals are partially occupied, many states
can ari
In particular, within the orbital model of electronic structure (which is developed
more systematically in Section 6), one can not construct trial wavefunctions which are
simple spinorbital products
and
S=1, M S = 0> = 2 1/2[ * + *].
The singlet spin state is:
S=0, M S = 0> = 2 1/2[ *  *].
To understand how the three triplet states have the same energy and why the singlet
state has a
two states with ML = 0 are called states. States with Lz eigenvalues of ML and  ML are
degenerate because the total energy is independent of which direction the electrons are
moving about the linear
can be formed by simply constructing combinations of the above three determinants with
Ms =1/2 which are orthogonal to the S = 3/2 combination given above and orthogonal to
each other. For example,

configurations with closedshell components, not all spin functions are possible because of
the antisymmetry of the wavefunction; in particular, any closedshell parts must involve
spin pairings for
Appendix (G). In the latter approach, one forms, consistent with the given electronic
configuration, the spin state having maximum Sz eigenvalue (which is easy to identify as
shown below and which cor
functions we need to form. When we include spinorbit coupling into H (this adds another
term to the potential that involves the spin and orbital angular momenta of the electrons),
L2, L z, S 2, S z n
Chapter 10
Electronic Wavefunctions Must Also Possess Proper Symmetry. These Include Angular
Momentum and Point Group Symmetries
I. Angular Momentum Symmetry and Strategies for Angular Momentum Coupli
= RL.
The singlet state can be reduced in like fashion:
21/2[ *  *] = 21/2 1/2[ LL  RR + RL LR  LL + RR  RL + LR ]
= 21/2 [ LL  RR ].
Notice that all three triplet st
Section 3 Electronic Configurations, Term Symbols, and
States
Introductory Remarks The Orbital, Configuration, and State Pictures of Electronic
Structure
One of the goals of quantum chemistry is to a
II. Even NElectron Configurations Are Not Mother Nature's True Energy States
Moreover, even singleconfiguration descriptions of atomic and molecular structure
(e.g., 1s22s22p4 for the Oxygen atom) d
3.
y
10
11
z
H
6
7
8
C
H
C
3
5
H
H
x
4
9
12
2px
2px
1
2
Using D2h symmetry and labeling the orbitals (f1f12) as shown above proceed by using
the orbitals to define a reducible representation.which ma
Pirrep =
irrep(R)R ,
R
may be used to find the SALCAOs for these irreducible representations.
PB2g =
B2g(R) R ,
R
PB2g f1 = (1)E f1 + (1)C2(z) f1 + (1)C2(y) f1 + (1)C2(x) f1 +
(1)i f1 + (1)(xy) f1
1 g
1 g
3u
2px 2py 2pz
2pz 2py 2px
3g
1 u
1 u
2s
2u
2s
2g
1u
1s
1s
1g
N2
N
N
The above diagram indicates how the SALCAOs are formed from the 1s,2s, and 2p N
atomic orbitals. It can be seen that there
1s probability density
2
1
0
0
1
2
3
4
r (bohr)
Plot of the orthogonalized 2s orbital probability density vs r; note there is one node.
2s probability density
0.3
0.2
0.1
0.0
0
2
4
6
8
10
r (bohr)
Plo
2b2
4a1
1b2
1u
1b1
2px
2py
1b1
2pz
1b2
2s
3a1
3a1
1b2
1a1
2a1
1g
2a1
1s
1a1
1a1
C
H2
C
H
H
z
y
x
This MO diagram is not an orbital correlation diagram but can be used to help generate one.
The energy
=
0.786798
(0.014892)(0.037268)(0.014892)
(0.037268)(5040.000000)(14999.893999)
=
23.286760
0.557496
<r> 33 =
r r2dr = <r> ij = 0.195391 3.908693
i
j
0
0.001118 0.786798 23.286760
Using these inte
C' =
1
S2
C=
1.010194
0.083258
0.006170
0.083258
1.014330
0.052991
0.006170
0.052991 1.004129
These new aos have been constructed to meet the orthonormalization requirement C'TSC' =
1 since:
1