INTRODUCTION TO MOLECULAR MODELING
Lab assignment
Transition metal
compounds
Transition metals are
characterized by the gradual filling of the d-shells (and, in the lanthanides
and actinides, the f shells). In the first-row transition metals (Sc to Cu;
some people include Zn), the 3d orbitals fill up. The 4s and 3d orbital
compete. In K and Ca, the 4s is lower. However, from Sc on, the 3d is lower. In
spite of this, the 4s orbital are filled. This is a violation of the Aufbau (build-up) principle. It is caused by the
fact that the 3d orbitals are compact, much smaller in diameter than the 4s
orbitals, and therefore the electrons in the 3d orbitals repel each other
strongly. At one point, it becomes energetically more advantageous for the next
electron to go to the more diffuse 4s orbital that continue d=filling the d
shell. The electronic ground states of the atoms K-Zn are given below:
K (Ar) 4s
Ca (Ar) 4s2
Sc (Ar) 3d 4s2
Ti (Ar) 3d2 4s2
V (Ar) 3d3 4s2
Cr (Ar) 3d5 4s
Mn (Ar) 3d5 4s2
Fe (Ar) 3d6
4s2
Co (Ar) 3d7
4s2
Ni (Ar) 3d8
4s2
Cu (Ar) 3d104s
Zn (Ar) 3d10
4s2
Note the irregularity
at Cr and Cu, caused by the extra stability of the half-filled (3d5)
and filled (3d10) d-shells. (Ar) is an
argon-like closed shell, tightly bound.
Transition metal
complexes have a positive transition metal ion (most frequently +2 or +3),
surrounded by either negative ions (e.g. F-, Cl-,
CN-) or dipole molecules with a lone electron pair (F-, H2O,
NH3). The most common coordination is octahedral: the central atom
is surrounded by 6 ions or molecules, at the corners of an octahedron. The
negative end of dipole molecules faces the central ion.
The 5 d-type orbitals
in the atom are degenerate. However, in a complex they can have different
energies (the degeneracy is resolved).
In an octahedral complex, the 5 d orbitals split in two groups: the triply
degenerate t2g set (usually lower if the ligands,
as usual, are negative ions or lone pairs), and the doubly degenerate eg set. The magnitude of the splitting (the
energy difference eg – t2g)
depends mainly on the nature of the ligand. Negative
ions cause a larger split than dipolar molecules. Considering first-row
coordinating atoms, their bonding strength, and the eg
– t2g split
increases as F<O<N<C. Carbon has only a few compounds with lone pairs,
such as CO and CN- . However, these are strong ligands.
In transition metal compounds
with 4 to 7 d electrons, there is a conflict between the Aufbau
(build-up) principle and Hund’s rule. According to
the Aufbau principle, electrons will occupy the
lowest energy orbitals. For instance, if there are 4 d electrons,as in Mn(III), they should
all go to the lower t2g orbital. However, t2g has only 3 orbitals and thus can accept only 3 electrons with parallel
spin. If the fourth one occupies one of the t2g orbitals, it must have opposite
spin. However, according to Hund’s rule, parallel
spins are more stable. If we require that all 4 spins be parallel, the fourth
electron must go to the higher eg orbital.
The actual ground
state depends on the magnitude of the eg –
t2g splitting. If it is large, (strong ligand field), the Aufbau
(build-up) principle wins: the fourth (and fifth, sixth, ..)
electron goes to the lower t2g orbital. This is called the low spin case because the spin of the fourth (fifth, sixth,..) electron must be opposite to the first 3, and thus
cancels part of their spin. If the splitting is small (weak ligand field), Hund’s rule
wins, and the first 5 d electrons have all the same spin. This is the high-spin case. The figure above shows 6 electrons in the high-spin case.
Homework
1. Sketch the occupancy
of the t2g and eg orbitals for 1 to 10 d electrons, in both the weak field
and the strong field case.
Homework
2. Find an article in the
literature which uses and discusses molecular symmetry, and summarize it. The
best place to look is probably thre Spectrocopy section of the Journal of Chemical Physics. J.
Phys. Chem. A and JACS are also good places to look.
Lab exercise
Calculate the energy
and geometry of the iron(III) hexaquo
complex, Fe(H2O)3+, in both the high-spin and low-spin
form, and determine which is more stable, and by how much (in kcal/mol or
kJ/mol). Also, compare the geometries of the low-spin and high-spin complexes.
Which has the stronger bond?
Iron has 26 electrons,
18 of which are immobilized in the core, leaving 8 in the valence shell. Fe(III) has therefore 5 d electrons. In the high-spin form,
all are parallel, each occupying a different d orbital.
Constructing the
initial geometry for Fe(H2O)3+
is not trivial, as most modeling programs are geared toward organic molecules.
To help you, I am enclosing my hand-constructed initial geometry, assuming an Fe-O distance of 1.8 Å. The symmetry should come out as Th.
if everything is correct. You can control the spin state by giving, on the GEOM
line, MULT=6 (for the sextet), or MULT=2 (for the low-spin doublet).
Use the 3-21G basis,
at least initially. Later, use the m6-31G basis. (m
for modified). It is best to do the two basis set calculations separately. E.g.
for the high spin, 3-21G calculations the input is
%MEM=15
TITLE=Fe(H2O)6 3+, high spin
GEOM MULT=6 CHARGE=3
<Molecular
geometry, see below>
BASIS=3-21G
OPTI
SCF DFT=B3LYP
FORCE
JUMP
GEOM GEOP
If you want to
calculate the m6-31G basis results in the same job, change the basis. The 3-21G
geometry is not very good, so it is best to strat
with the initial geometry.
BASIS=m-6-31G
OPTI
SCF DFT=B3LYP LVSH=2
FORCE
JUMP
GEOM GEOP
Reaching SCF
convergence is more difficult for transition metals than for organic molecules.
The option LVSH=2 (level shift) artificially increases the HOMO-LUMO gap and
improves convergence for difficult systems. In calculations for simple organic
molecules, the Aufbau principle, i.e
the sequential fill-up of orbitals from below, is
usually automatically satisfied. For transition metals, the calculations may
converge to excited states which are higher that the lowest state of the same spin. Although no formal
proof exists, the Aufbau principle is almost always
obeyed within one spin, i.e. among the alpha spins and separately among the
beta spins. Check your calculation whether it satisfies the Aufbau
principle, both for the alpha and the beta spins. If not, one has to exchange
one occupied orbital with a virtual (unfilled) orbital. See the SWAP and SWAB
options in the GUESS command. GUESS is usually invoked automatically but it may
be necessary in difficult cases.
Lab report
Report the relative
energies for both basis sets. Also
compare the optimized geometries, in particular the Fe-O distances. Calculate
the complexation energy of Fe3+ with
water. For this, calculate the energy of the Fe3+ ion with the same
basis sets used for the complex. In this atomic calculation, turn off the
symmetry with SYMM=0 (the program is not designed for atoms). Also calculate
the (optimized) energy of water, again using the same basis sets. Which spin
state is more strongly bound, the high spin or the low
spin?
FE 0
0 0
O -1.8 0 0
H -2.4
-0.8 0
H -2.4 0.8 0
O 1.8 0 0
H 2.4
-0.8 0
H 2.4 0.8 0
O 0
-1.8 0
H 0
-2.4 0.8
H 0
-2.4 -0.8
O 0 1.8
0
H 0 2.4
0.8
H 0 2.4 -0.8
O 0 0 -1.8
H -0.8 0
-2.4
H 0.8 0
-2.4
O 0 0 1.8
H -0.8 0
2.4
H 0.8 0
2.4