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 distillation was performed to separate mixtures into th
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 of orbital pairs, places the electrons into orbitals ' =
Chapter 9
Electronic Wavefuntions Must be Constructed to Have Permutational Antisymmetry
Because the N Electrons are Indistinguishable Fermions
I. Electronic Configurations
Atoms, linear molecules, and non-linear molecules have orbitals which can be
label
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 the CI method in which two
configurations (e.g., |1s22s2
where a = 3C2/C1 . The so-called 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 ratio of C2 to C1 . As will be detailed
in Section 6, th
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 Section and in considerable detail in Section 6.
Briefly, on
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 magnitude. The
SCF average coulomb interaction between
with the N-1 other electrons, they use a probability density N-1(r') that is independent of
the fact that another electron resides at r. In fact, the so-called conditional probability
density for finding one of N-1 electrons at r', given that an electron
Interaction Energy (eV)
300
200
Fluctuation
SCF
100
0
-100
-2
-1
0
Distance From Nucleus ()
1
2
As a function of the inter-electron distance, the fluctuation potential decays to zero
more rapidly than does the SCF potential. For this reason, approaches in
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 arise and have energies which differ substantially because
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 spin-orbital products (i.e., an orbital multiplied by an or spin function for
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 different energy, and an energy different than the MS
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 molecule's axis (just a +1 and -1 orbitals are degenera
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,
| 1/2, 1/2> = 1/2[ | 1s2s3s | - | 1s2s3s | + 0x | 1s2s3s
configurations with closed-shell components, not all spin functions are possible because of
the antisymmetry of the wavefunction; in particular, any closed-shell parts must involve
spin pairings for each of the doubly occupied orbitals, and, as such, con
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 corresponds to a state with S equal to this maximum Sz
eig
functions we need to form. When we include spin-orbit 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 no longer commute with H. However, Jz = S z + Lz and J2
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 Coupling
Because the total Hamiltonian of a many-electron ato
= |RL|.
The singlet state can be reduced in like fashion:
2-1/2[ |*| - |*|] = 2-1/2 1/2[ |LL| - |RR| + |RL| |LR| - |LL| + |RR| - |RL| + |LR| ]
= 2-1/2 [ |LL| - |RR| ].
Notice that all three triplet states involve atomic orbital occupancy in which one elec
1/2
(1/N!)
detcfw_ j (r i ) =
1/2
(1/N!)
1(1) 2(1) 3(1). k (1).N(1)
1(2) 2(2) 3(2). k (2).N(2)
.
.
.
.
1() 2() 3().k ().N()
The antisymmetry of many-electron spin-orbital products places constraints on any
acceptable model wavefunction, which give rise
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 allow practicing chemists to use
knowledge of the electr
II. Even N-Electron Configurations Are Not Mother Nature's True Energy States
Moreover, even single-configuration descriptions of atomic and molecular structure
(e.g., 1s22s22p4 for the Oxygen atom) do not provide fully correct or highly accurate
represen
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 (f1-f12) as shown above proceed by using
the orbitals to define a reducible representation.which may be subsequently reduced to its
irreducible components
Pirrep =
irrep(R)R ,
R
may be used to find the SALC-AOs 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)(xz) f1 + (-1)(yz) f1
= (1) f1 + (-1) -f1 + (1) -
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 SALC-AOs are formed from the 1s,2s, and 2p N
atomic orbitals. It can be seen that there are 3g, 3u, 1 ux, 1 uy, 1 gx, and 1 gy SALCAOs. The Ha
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)
Plot of the orthogonalized 3s orbital probability density
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 levels on each side (C and H2) can be "superimposed" to
=
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 integrals one then proceeds to evaluate the expectation val
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 T
-1
-1
-1
2 C S S 2 C = CTS 2 S S 2 C = CTC = 1 .
S
Bu