1 Computer Experiment* 15: Computational Coordination ...scienide2.uwaterloo.ca/~nooijen/Chem-440-computational/ORCA_labs... · oriented$modifications$are$known$collectively$as$“ligand$field$theory”$but$we$will$

1 Computer Experiment 15: Computational Coordination Chemistry

1.1 Background

1.1.1 Introduction The rich chemistry of transition metals gives rise to the large and important fields of

coordination chemistry, organometallic chemistry and branches of solid state

chemistry. In fact, transition metal complexes play an important role in biology and

chemical catalysis since they are able to catalyze a very large variety of reactions,

many of them of fundamental relevance to biological and industrial processes. The

high reactivity of transition metal complexes is intimately linked to their electronic

structure and the nature of the metal-‐ligand bonds. In fact, these bonds are typically

stronger than the weak interactions studied in Computer Experiment number 13

(section Error! Reference source not found., page Error! Bookmark not

defined.) but are weaker than the typical covalent bonds formed in organic

chemistry. Consequently, the metal-‐ligand bonds are easier to break and reform

which gives rise to low-‐energy pathways in chemical reactions. Furthermore, due to

the partially filled d-‐shells most of the transition metals are redox active and can

exist in different oxidation states (e.g. Mn can assume oxidation states from +VII to 0

or even -‐I). The potential at which the one-‐electron steps occur are a sensitive

function of the nature of the ligands and the coordination geometry and therefore be

easily tuned by suitable ligand design. Furthermore, transition metals typically

contain unpaired electrons in their partially filled d-‐shells which gives rise to

several accessible spin states. These spin states may show considerably different

reactivities and this leads to added complexity in studying their reactions.

Frequently, the systems even pass from one spin state to another surface during a

given catalytic cycle which opens many fascinating routes to unique reactions. Last

but not least, the partially filled d-‐shells and the variable metal-‐ligand bonds are

associated with a huge variety of fascinating spectroscopic phenomena. Thus, very

detailed insight into the geometric and electronic structure of the transition metal

sites can be obtained from spectroscopic studies and frequently these studies yield

the key to understanding the unique reactivities displayed by transition metal

complexes. In this Computer Experiment we will study some basic aspects of

theoretical coordination chemistry. We will stress the use of the model of ligand-‐

field theory (LFT) as a conceptual framework in which the results of actual MO

calculations can be readily and successfully interpreted.

1.1.2 Ligand Field Theory Beyond all advanced theoretical considerations, the major driving force for the

formation of coordination complexes is the electrostatic attraction of a positively

charged metal ion and a negatively charged ligand or a ligand with functional groups

that bear a partial negative charge – this is nicely seen from the computed

electrostatic potential of the typical ligand ehthylenediamine (Figure 1).

Figure 1: Principle of the formation of coordination compounds.

Traditionally, the properties of such coordination compounds are interepreted in

the framework of a theory that was worked out in the 1950s by physicists (mainly

Hans Bethe) and that is called “crystal field theory”. Slightly more chemistry

oriented modifications are known collectively as “ligand field theory” but we will

not distinguish between these terms. The essential idea is to focus the interest on

the open d-‐shells of the metals and study their interaction with their environment

which is assumed to somehow provide an electrostatic field in which the d-‐electrons

have to move. Thus, LFT takes no detailed account of chemical bonding but is

enormously successful in its domain of applicability. In fact, using LFT many

properties of large classes of complexes can be understood in a unified and simple

language. In particular, the concept of a “dN-‐metal” (N=number of d-‐electrons) that

is so central to inorganic chemistry would not be possible without ligand field

theory. The concepts are usually worked out on the example of octahedral

coordination complexes. While perfectly octahedral complexes may be rather rare,

approximately octahedral complexes are highly abundant and the conclusions

drawn from studying the octahedral case largely remain valid.

The shapes of the d-‐orbitals are well known and are shown again in Figure 2. The

ligands are assumed to be placed along the axes of the coordinate system such that

an octahedral field of negative point charges arises. This set of negative point

charges now starts to interaction with a “probe” electron that is placed into each of

the five d-‐orbitals in turn. Upon evaluating the energy of interaction of the probe

electron with the negative point charges1 it is found that the energy is different for

the different orbitals as shown in Figure 3.

Figure 2: The shapes of the five d-‐orbitals together with their common designation in terms of real spherical harmonics. The combination z2-‐r2 is usually only abbreviated as “z2”.

Thus, as a consequence from lowering the symmetry of the surrounding of the ion

from spherical (in the free ion) to octahedral (in the complex), the degeneracy of the

five d-‐orbitals is lifted. In fact, it is plausible from Figure 3 that an electron that

occupied the dx2-‐y2 or the dz2 orbital is – on average – more strongly repelled by the

negatively charged ligand than an electron that occupied the dxy, dxz or dyz orbitals.

A quantitative treatment (and in fact already the group theoretical treatment)

1 This will not be done here. The classic and rigorous mathematical text is due to J.S. Griffith The Theory of Transition-‐Metal Ions, Cambridge University Press, Cambridge, 1964. It may appear that this is an outdated book and an outdated theory – but its not! You will find many insights that rigorously carry over to the modern methods of quantum chemistry. A thorough understanding of ligand field theory teaches you an enormous amount about the N-‐electron problem in quantum chemistry. The intrinsic value of this understanding can hardly be overemphasized.

shows, that the set of five d-‐orbitals fall into a t2g subshell at lower energy and an eg

shell at higher energy.2

Figure 3: d-‐Orbital Splittings induced by an octahedral ligand field.

Following the rules of how to assign the many electron term symbol from the one-‐

electron MO scheme, it is evident that the configuration (t2g)1(eg)0 gives rise to a 2T2g

term3 while to higher energy we expect the configuration (t2g)0(eg)1 which gives rise

to a 2Eg term in Oh symmetry. Since the ‘T’ irreducible representation is three-‐

dimensional and ‘E’ is two-‐dimensional we say that 2T2g is threefold orbitally

degenerate and 2Eg is two-‐fold orbitally degenerate. The total degeneracy of the

state in the absence of spin-‐orbit coupling (SOC) and magnetic fields is (2S+1) times

the orbital degeneracy which amounts to six for 2T2g and four for 2Eg in the present

case. All of these ten states are observable and can be probed by spectroscopic

experiments. One of the simplest experiments is to induce a 2T2g→2Eg transition, by

application of a visible photon of suitable energy.4 The experiment provides a

quantitative measure of the d-‐orbital splitting Δ0 which is typically in the range 1-‐4

eV (8000-‐32000 cm-‐1). Already the example of [Ti(H2O)6]3+ in Figure 4 shows that this

2 Note carefully, that these energies are NOT drawn to scale. Above all, the repulsion leads to a strong increase of the energies of all five d-‐orbitals and the relatively small difference in the energy increase is the ligand-‐field splitting. Note also, that the final result does not depend on how we place the coordinate system. Thus, the result is said to be rotationally invariant. What does change if we rotate the coordinate system are only the designation of the orbitals. 3 Note: capital letters refer to many electron states and lowercase letters to one-‐electron orbitals. Furthermore, the term symbol carries as a left superscript the total multiplicity of the state which is 2S+1 where S is the total spin of this state. 4 This is only true on a superficial level. In fact, the optical spectroscopy of transition metal complexes becomes very complicated and very subtle rather quickly as one enters into more detail – try it, its worthwhile!

is a slight oversimplification since in reality we observe a broad and asymmetric

absorption band rather than a single line like in atomic spectroscopy.

Figure 4: Example of the measurement of the ligand field splitting in [Ti(H2O)6]

3+

However, we will not go into any further detail of the interpretation of the spectrum

here but merely pretend that it is possible extract reasonable values of the ligand-‐

field splitting Δ0 from the study of the absorption spectra of the complexes. Since the

magnitude of the splitting depends on the nature of the ligand, the ability of the

ligand to induce d-‐orbital splittings may be classified in the spectrochemical series

shown below.

Figure 5: The spectrochemical series.

On this basis CN-‐ is known to be a “strong-‐field ligand” since it gives rise to large

ligand field splittings and I-‐ is known to be a “weak field ligand” since it gives rise to

small ligand field splittings.5

A nice experimental confirmation that this way of thinking about the electronic

structure of transiton metal complexes makes sense is shown in Figure 6 where the

experimental enthalpies of formation of [M(H2O)6]2+ complexes is shown (M=first

row transition metal). 5 Please observe some irregularities here: hydroxide is a weaker field ligand than H2O despite the fact that it is actually negatively charged. This should not be the case if our reasoning about the origin of the ligand field splitting above would be correct. We will come back to this later.

Figure 6: Thermochemical confirmation of ligand field splittings: the ligand field stabilization energy (LFSE). Plotted are the enthalpies of formation of the hexaquo complexes for the divalent first-‐row transition metal ions. Open circles – raw experimental data. Full circles – after subtraction of the LSFE.

The interpretation of the results is as follows: The enthalpy of formation is

increasing along the series. Thus, the later transiton metals hold the water ligands

more strongly than the early transition metals. This is first of all an effect of the

effective nuclear charge of the metal. Since the d-‐electrons shield the increasing

nuclear charge with increasing atomic number less effectively, the ligands “see” an

increased positive charge from the metal which leads to the slope of the line.

However, more interestingly is the “double-‐bowl” behavior of the experimental data.

This readily find an explanation from considering the energy of a given dN

configuration relative to the hypothetical case where all d-‐orbitals have the same

energy: thus occupation of a t2g level leads, on average, to an energy decrease of -‐

2/5Δ0 while occupation of an eg level leads to a 3/5Δ0 increase. Knowing the Δ0-‐

values from optical spectroscopy, the LFSE of a given dN configuration can be

subtracted from the experimental data to arrive at the full circles which now

perfectly fall on a straight line. Thus, in this way one has found a simple explanation

for a seemingly puzzling and irregular experimental result.

1.1.3 Electron-‐Electron Repulsion Effects A little contemplation shows that our discussion of the energies of given dN

configurations is somewhat oversimplified since up to now we have not taken

account of the basic fact that electrons are negatively charged and actually repel

each other.6 This leads to a number of interesting effects in coordination chemistry.

First of all, two rules-‐of-‐thumb should be mentioned: (a) electrons that occupied the

same orbital repel each other most strongly, (b) Electrons of opposite spin repel

each other more strongly than electrons of the same spin. Thus, the system will

want to assume the configuration where – as much as possible – the electrons have

the same spin and occupy different orbitals. This leads to interesting multiplet

splittings in the dN configurations where states of different multiplicity but the same

electron configuration have different energies.

Figure 7: Principles of electron-‐electron repulsion in LFT.

In LFT, one has the fortunate situation that electron-‐electron repulsion can be

described to a good approximation with a single parameter: the “Racah B-‐value”.

Thus, besides the ligand-‐field splitting parameter Δ0, the second important

parameter of LFT is the B-‐value. It is of the order of ~1000 cm-‐1 and state splittings

are typically multiples of B. The example in Figure 7 is the 1T1g-‐3T2g splitting (both

from the (t2g)2(eg)0 configuration!) is 4B and experimentally the splitting between

these states is 3000-‐4000 cm-‐1.

The electron-‐electron repulsion has a very interesting consequence: for several of

the dN configurations (actually d4-‐d7) it is not a priori clear which electron

configuration will be of lowest energy since there is a competition between ligand-‐

field splitting and electron-‐electron repulsion (Figure 8). For example, for the case of

6 As mentioned in the introduction, the electron-‐electron repulsion is the biggest challenge for theoretical chemistry. Its accurate calculation from first physical principles (the “electron correlation problem”) is very difficult and represents a major branch of research in quantum chemistry.

the d5 configuration, the high-‐spin state ((t2g)3(eg)2; 6A1g) minimizes the electron-‐

electron repulsion according to our rules of thumb but at the expense of promoting

two electrons into the higher-‐energy eg-‐orbitals. This can be avoided by forming the

alternative low-‐spin ((t2g)5(eg)0; 2T2g) configuration which maximizes the LSFE but at

the expense of a larger electron-‐electron repulsion. Thus, depending on the ligand

environment either one-‐ or the other alternative will be preferred: strong field

ligands lead to low-‐spin complexes and weak field ligands to high-‐spin complexes.

Experimentally, this is found to be true: [Fe(CN)6]3-‐ has a 2T2g ground state and

[FeCl6]3-‐ has a 6A1g ground state.

Figure 8: Competition between electron-‐electron repulsion and ligand field splitting to produce either high-‐spin or low-‐spin complexes.

Everything that has been said about the nature of the two dominating influences:

ligand field splitting and electron-‐electron repulsion can be nicely summarized and

systematized in the so-‐called Tanabe-‐Sugano diagrams (Figure 9). What is plotted

there on the X-‐axis is the ligand field strength in units of B while on the y-‐axis the

energy of a given state relative to the lowest state of the system is plotted.

Figure 9: General organization of a Tanabe-‐Sugano diagram.

For the d4-‐d7 electron configurations, the Tanabe-‐Sugano diagrams have two parts:

a weak field part, where the high-‐spin states is the ground state and a strong field

part, where the low-‐spin state is the ground state. At some critical ligand field

strength, a discontinous change from one ground state to the other occurs.

Experimentally, this is known to be possible: complexes with ligands that lead to

ligand field strengths close to the crossover point are know to be switchable from

one ground state to the other by external perturbation such as pressure. This leads

to the interesting field of “spin-‐crossover” chemistry which is a promising approach

to the design of molecular switches.

Secondly, the Tanabe-‐Sugano diagrams are excellent guides to the assignment of the

ligand-‐field absorption spectra of transiton metal complexes. Given, that the

selection rule ΔS=0 for an electric dipole allowed transition between two states is

fairly strong, one simply has to search in the Tanabe-‐Sugano diagram for states of

the same multiplicity as the ground state in order to find out how many ligand-‐field

bands are expected in the absorption spectrum.7 The only thing that is needed then

in order to determine Δ0 is a ruler and a reasonable estimate of B. Two examples of

such diagrams together with actual absorption spectra are shown in Figure 10.

7 Actually, even the spin-‐allowed ligand field bands are weak since they are parity forbidden in octahedral complexes. For example, in tetrahedral complexes there is no center of inversion and the observed ligand field bands are much stronger than in octahedral complexes. Spin-‐allowed ligand field bands may have ε’s of several hundred M-‐1 cm-‐1 while spin-‐forbidden ones typically have ε<1 M-‐1 cm-‐1

Figure 10: Example of a ligand field analysis for the d2 and d3configurations on the basis of Tanabe-‐Sugano diagrams.

A summary of the many electron terms together with the leading one-‐electron

configurations is shown in Table 1. Table 1: Ground-‐ and Ligand Field Excited States of Octahedral dN Complexes.

dN Ground Term Spin-‐Allowed Transitions Spin-‐Forbidden Transitions Oh Config. d9 2Eg t2g6eg3 2T2g(t2g5eg4) None

d8 3A2g t2g6eg2 3T2g(t2g5eg3) 3T1g(t2g5eg3) 3T1g(t2g4eg4) 1Eg(t2g6eg2) 1T2g(t2g5eg3)

d7 4T1g t2g5eg2 4T2g(t2g4eg3) 4T1g(t2g4eg3) 4A2g(t2g3eg4) 2Eg(t2g6eg1) 2A2g(t2g5eg2) 2T2g(t2g5eg2)

d6 5T2g t2g4eg2 5Eg(t2g3eg3) 3T2g(t2g5eg1)

d5 6A1g t2g3eg2 None 4T1g, 4T2g (t2g4eg1) 4A1g,4Eg,4T2g(t2g3eg2)

d4 5Eg t2g3eg1 5T2g(t2g2eg2) 3T1g(t2g2eg2)

d3 4A2g t2g3eg0 4T2g(t2g2eg1) 4T1g(t2g2eg1) 4T1g(t2g1eg2) 2Eg(t2g3) 2T1g(t2g3)

d2 3T1g t2g2eg0 3T2g(t2g1eg1) 3T1g(t2g1eg1) 1Eg(t2g2) 1T2g(t2g2) 1A1g(t2g2)

d1 2T2g t2g1eg0 2Eg(t2g0eg1) None

1.1.4 Other Coordination Geometries The entire chain of arguments developed so far is only valid for octahedral

complexes. However, in reality, coordination compounds exist in many different

effective geometries with coordination numbers ranging from 2-‐8. Each distinct

coordination geometry induces a different orbital splitting pattern and therefore

also a different set of molecular multiplets. For some of the more important

coordination geometries the orbital splitting patterns are shown in Figure 11.

Figure 11: Orbital splitting patterns in different coordination geometries.

While each of these geometries would require an elaborate analysis, it is clear that

the absorption spectra of coordination complexes react very sensitively to the

coordination geometry and consequently, the coordination geometry can be

inferred from a proper analysis of the absorption spectra (or other optical

techniques that probe the same transitions). We will not go into more detail here.

1.1.5 Inadequacy of Ligand Field Theory, Covalency, Nephelauxetic Effect

Below you will find a quote from one of the father’s of ligand field theory and the

dominating spectroscopist of his time:

What Jörgensen refers to so elaborately as “electrostatic romantics” is the crystal

field model that we have just discussed in the previous section. Indeed, there are

very good evidences from spectroscopic measurements that LFT is an

oversimplification. For example, consider the following EPR experiment:

Figure 12: The EPR spectrim of [Cu(imidazole)4]

2+ (left). On the right is the singly occupied orbital according to ligand field theory. On this basis, the sharp features in the middle of the spectrum can not be explained.

What you see there is the EPR spectrum of [Cu(imidazole)4]2+ -‐ a typical

coordination complex. It is a typical square planar d9 complex like many others in

coordination chemistry. According to Figure 11 the unpaired electron resides in a

metal dx2-‐y2 orbital. The interesting sharp features in the EPR spectrum actually

represent hyperfine structure that arise from coupling of the unpaired electron to

four equivalent nitrogen nuclei of the imidazole ligand.8 Since we learned in

experiment number 9 (section Error! Reference source not found., page Error!

Bookmark not defined.) that the hyperfine structure is a measure for the

probability to meet the unpaired electron at the nucleus in question there is a

8 It is an interesting question to ask to the chemist how one can proof the origin of these signals? The answer is the following: nitrogen has two accessible isotopes 14N (the naturally most abundant) and 15N. 14N has a nuclear spin of I=1 while 15N has I=1/2. Furthermore the nuclear g-‐value of 14N is twice as large as that of 15N. Thus, synthesis of the imidazole ligand with 14N and 15N will lead to precisely predictable changes in the EPR spectrum and this is how the nature of the signals can be experimentally verified without any doubt.

mismatch between the predictions of LFT and the experimental observations:

According to LFT the SOMO is a pure metal orbital and there is no probability to find

the electron at the nitrogen nucleus – yet, the experiment proofs the contrary.

In fact, this experimental finding is readily explained on the basis of MO theory. The

shape of the SOMO is shown in Figure 13. It is evident that the SOMO indeed has dx2-‐y2

character on the central copper. However, the orbital is σ-‐antibonding with the

coordinating imidazole-‐nitrogens. Thus, part of the unpaired spin-‐density is

delocalized into these ligand atoms and this accounts for the experimental

observations. In fact, from the experimental data one is able to make a semi-‐

quantiative estimate of how much spin is required to be on the nitrogens in order to

explain the experimental data. In the present case, this amount to ~60-‐65% spin

population on the copper with some ~9-‐10% on each nitrogen.

Figure 13: Interpretation of the EPR spectrum of [Cu(imidazole)4]

2+ according to MO theory.

The “dilution” of the metal orbitals with ligand character has in fact also been noted

by the optical spectroscopists interested in LFT. They needed to account for

electron-‐electron repulsions that were smaller than expected from atomic

spectroscopy. They explained this finding by an increase of the size of the metal

orbitals and called the effect “nephelauxetic effect” (cloud expanding effect).

However, it finds a much more natural explanation in terms of MO theory. In fact,

the mixing of ligand character into the metal orbitals is referred to as metal-‐ligand

covalency. The more covalent the metal-‐ligand bond, the more ligand character is

mixed into the formally metal based MOs.

1.1.6 Molecular Orbitals of Octahedral Complexes The discussion above raises the question of how the MO theory of coordination

complexes works in general. This has been worked out in great detail by many

authors and it can be found in many textbooks. Therefore, we will be very brief in

this respect.

A typical MO scheme of a ML6 complex as it could be derived from extended Hückel

calculations9.

Thus, one considers that ionization potentials of transition metals are typically a few

eV lower than those of main group atoms. Consequently, one place the metal

orbitals nd (n+1)s and (n+1)p to 0th order at higher energy than the ligand p-‐

orbitals. Perturbation theory then tells us, that upon forming molecular orbitals

from fragment orbitals of correct symmetry, that the higher lying orbitals become

destabilized and antibonding and the lower lying orbitals become stabilized and

bonding. Furthermore, the smaller the energy gap between the mixing fragment

orbitals and the stronger their interaction10, the larger the admixture. Thus, the

metal d-‐orbitals of ligand field theory are the partially occupied antibonding orbitals

that can be found in MO calculations on coordination compounds. Some typical

bonding and antibonding components are shown in Figure 14. It is evident that the

metal-‐based t2g-‐orbitals correspond to π-‐antibonding MOs while the metal-‐based eg-‐

orbitals correspond to σ-‐antibonding MOs. Since π-‐interactions are typically weaker

than σ-‐interactions this accounts for the energetic ordering of the two-‐sets of

orbitals that was so luckily correctly predicted by LFT.

9 An interesting and subtle point is that this MO scheme could not be derived from Hartree-‐Fock calculations. In HF calculations, the metal orbitals typically fall energetically below the ligand orbitals but are still the first to be ionized. This arises because the electronic relaxation energy upon ionization is typically small for the ligand orbitals and very large for the metal orbitals. Consequently, the extended Hückel like arguments give a somewhat correct picture of the ionization events in complexes but the quantitative theory is more complicated. 10 In extended Hückel and other semiempirical theories, this interaction is taken to be proportional to the overlap between the metal and ligand fragment orbitals. (on also refers to these interaction parameters as “resonance integrals”)

Figure 14: MOs of octahedral ML6 complexes. The red box shows the five metal-‐based MOs that are the ones taken into account in ligand field theory.

It is also clear that since the M-‐L bonding orbitals are always fully occupied and the

metal based-‐d-‐orbitals are only partially occupied that there is a net transfer of

charge from the ligand to the metal. Thus, in MO theories one never finds partial

charges that come close the formal charges as measured by the oxidation state.

However, as long as the mixing between metal-‐ and ligand-‐orbitals is not extremely

excessive11 the concept of a dN configuration that is so central to LFT still remains

valid – one simply has to realize that the orbitals that one talks about are partially

delocalized onto the ligands. From this discussion it also becomes evident that it

does not make sense to try to compare the partial charges from a MO calculation of

whatever kind to the charge implied by the formal oxidation state.

1.1.7 Pi-‐Bonding and Pi-‐Backbonding There is one important class of ligands that do not perfectly correspond to the

discussion given above. These are ligands that have low-‐lying empty π*-‐orbitals.

Such orbitals can also interact with the metal-‐based t2g orbitals to form π-‐bonds.

11 For example exceeding 30-‐40% ligand character in the metal orbitals

However, in this case, the nature of the donor and acceptor orbitals is reversed

compared to the standard π-‐donor ligands discussed above. Thus, the interaction

stabilizes the metal t2g orbitals that now become bonding rather than antibonding.

Furthermore occupation of these orbitals leads to a backflow of charge from the

metal to the ligand – this type of bonding is therefore called backbonding.

Figure 15: orbitals in π -‐donor and π -‐acceptor complexes.

The effect is exemplified in Figure 15 – the standard π-‐donor complex [FeCl6]3-‐

(high-‐spin d5) features metal t2g based orbitals that are π-‐antibonding while those of

[Cr(CO)6] (low-‐spin d6) are π-‐bonding with CO-‐π* orbitals.

The effect of π-‐backbonding naturally has a very pronounced effect on the ligand-‐

field splitting. Thus, since the metal based t2g-‐orbitals are stabilized through

backbonding while the metal-‐eg based orbitals are destabilized due to σ-‐bonding, the

net effect is an increase in the ligand field splitting. The opposite effect is of course

true for standard π-‐donor complexes – the better the π-‐bond between the metal-‐

and the ligand the smaller the ligand field splitting. On this basis one comes now to a

much more appealing interpretation of the spectrochemical series (Figure 16): strong

field ligands are π-‐acceptor ligands while weak field ligands are good π-‐donor

ligands. In the middle of the spectrochemical series one simply finds ligands that

have no π-‐orbitals for either bonding or backbonding available (for example NH3 or

neutral aliphatic amines).

Figure 16: Interpretation of the spectrochemical series on the basis of MO theory.

1.2 Description of the Experiment The area of theoretical coordination chemistry is very large. Although this section is

somewhat lengthy, it is not possible to do more than to scratch the surface of it in

this course. Below you will find some problems in theoretical coordination

chemistry that are more or less representative. It will not be possible to work out all

of these problems. It is suggested to work out the first problem (1.2.1) and pick two

of the larger remaining ones. Problems 1.2.2 and 1.2.5 are small and require less

work. If these are chosen, at least one additional larger problem should be

investigated.

In this section we will use very small basis sets (SV for the metal and the oxygens

and STO-‐3G for the hydrogens) in order to keep the calculations fast. In practice it is

advisable to use at least polarized triple-‐ζ basis sets for the metals and at least

polarized double-‐ζ basis sets for the ligands in order to obtain reliable results.

As a generic “template” for our calculations, we will use approximately octahedral

fragments with water ligands. Appropriate metal-‐ligand bond distances are shown

in Table 2 for M(II) and M(III) ions of the first transition row. The complexes studied

will be of the form [M(H2O)5X]n+ where M is a first-‐row transition ion and X is an

additional ligand that will be varied in the course of the experiment. For X=H2O, the

highest possible symmetry that such complexes can assume is Th which is a genuine

cubic group.

Table 2: Bond distances (Angström) for divalent and trivalent hexaquo ions of the first transiton row.

Divalent Trivalent Ti -‐ 2.028 V 2.128 1.992 Cr 2.052 1.959 Mn

2.192 1.991

Fe 2.114 1.995 Co 2.106 1.873 Ni 2.061 -‐ Cu 1.964 -‐ A Z-‐matrix for cubic symmetry is shown below in ORCA format (the example chosen

is that of [Ni(H2O)6]2+ with a S=1 ground state).

1.2.1 Some aspects of Ligand Field Theory with DFT Application of DFT to transition metal complexes is often successful but not without

its pitfalls. One problem in the present case, where we deal with highly symmetric

complexes is that DFT only applies to orbitally nondegenerate ground states. For

degenerate ground states, symmetry breakings occur which may lead to artifacts in

the calculations.12 Nevertheless, in this course, we will focus on dN configurations

with orbitally nondegenerate ground states.

Do a calculation on the complex [Cr(H2O)6]3+. The setup of the job can be done as

follows:

12 In practice this is usually not such a strong limitation since most systems of practical interest are not orbitally degenerate. Even those that are orbitally degenerate are Jahn-‐Teller active and become orbitally nondegenerate upon distortion.

%paras R_ML 2.061 # the metal-water-oxygen distance R_OH 0.9817 # the water O-H distance A_HOM 126.153 # the angle H-O-Metal R_MX 2.061 # the distance of the metal and the # external ligand. Here another water end * int 2 3 # The central metal: here nickel Ni 0 0 0 0.0 0.0 0.0 # the five coording water oxygens O 1 0 0 {R_ML} 0.0 0.0 O 1 2 0 {R_ML} 180.0 0.0 O 1 2 3 {R_ML} 90.0 0.0 O 1 2 4 {R_ML} 90.0 180.0 O 1 2 4 {R_ML} 90.0 270.0 # this is the coordinating atom of the external ligand X O 1 2 4 {R_MX} 90.0 90.0 # These are the hydrogens of the water ligands of M(H2O)5 H 2 1 3 {R_OH} {A_HOM} 90.0 H 2 1 3 {R_OH} {A_HOM} 270.0 H 3 1 2 {R_OH} {A_HOM} 90.0 H 3 1 2 {R_OH} {A_HOM} 270.0 H 4 1 2 {R_OH} {A_HOM} 0.0 H 4 1 2 {R_OH} {A_HOM} 180.0 H 5 1 2 {R_OH} {A_HOM} 180.0 H 5 1 2 {R_OH} {A_HOM} 0.0 H 6 1 2 {R_OH} {A_HOM} 90.0 H 6 1 2 {R_OH} {A_HOM} 270.0 # here come the remaining atoms of the external ligand X H 7 1 2 {R_OH} {A_HOM} 90.0 H 7 1 2 {R_OH} {A_HOM} 270.0 *

One common observation with calculations on transition metal complexes is that

the SCF calculations do not converge well and some precautions are necessary.13

Now look at the orbitals. In spin-‐unrestricted calculations we have separate spin-‐up

and spin-‐down MOs. In the present case we have a d3 system and expect that the

three spin-‐up HOMOs are mainly metal t2g in character and the two spin-‐up LUMOs

will have metal eg character – let us verify that this is so! Thus, we look at the spin-‐

up HOMO/LUMO gap: 39 1.0000 -0.759449 -20.6651 40 1.0000 -0.759446 -20.6651 41 1.0000 -0.759445 -20.6650 42 0.0000 -0.655032 -17.8239 43 0.0000 -0.655029 -17.8238

We observe that the three highest occupied and two lowest unoccupied orbitals are

degenerate (as expected). Now let us look at their composition: 36 37 38 39 40 41 -0.82938 -0.82938 -0.82938 -0.75945 -0.75945 -0.75945 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -------- -------- -------- -------- -------- -------- 0 Cr dxz 0.0 0.0 0.0 12.9 56.4 0.2 0 Cr dyz 0.0 0.0 0.0 0.2 0.1 69.2 0 Cr dxy 0.0 0.0 0.0 56.4 13.0 0.1 -0.65503 -0.65503 -0.49107 -0.29188 -0.29187 -0.29187 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -------- -------- -------- -------- -------- -------- 0 Cr s 0.0 0.0 61.6 0.0 0.0 0.0 0 Cr dz2 5.2 71.0 0.0 0.0 0.0 0.0 0 Cr dx2y2 71.0 5.2 0.0 0.0 0.0 0.0

13 Above we have chosen a recipe that is typical for many calculations on transition metal complexes with ORCA (The keyword “SlowConv” indicates that you expect slow convergence). In general obtaining convergence requires some experimentation. One commonly used trick is to use level shifting (like in the example above).

! UKS BP RI SlowConv NoFinalGrid SV(P) %scf shift shift 0.3 erroff 0 end end # geometric parameters %paras R_ML = 1.959 # the metal-water-oxygen distance R_OH = 0.9817 # the water O-H distance A_HOM = 126.153 # the angle H-O-Metal R_MX = 1.959 # the distance of the metal and the # external ligand. Here another water end

Indeed, the result is the one that we wanted as we see from the compositions of the

orbitals.14 Likewise, the five spin-‐down LUMOs are expected to represent the spin-‐

down counterparts of the metal t2g and metal eg orbitals – Let’s verify.

The first unoccupied spin-‐down orbitals: 39 0.0000 -0.652195 -17.7467 40 0.0000 -0.652191 -17.7466 41 0.0000 -0.652190 -17.7466 42 0.0000 -0.605269 -16.4698 43 0.0000 -0.605267 -16.4697

And their compositions: 36 37 38 39 40 41 -0.82555 -0.82555 -0.82554 -0.65219 -0.65219 -0.65219 1.00000 1.00000 1.00000 0.00000 0.00000 0.00000 -------- -------- -------- -------- -------- -------- 0 Cr dxz 0.0 0.0 0.0 16.2 62.3 11.6 0 Cr dyz 0.0 0.0 0.0 0.8 11.2 78.1 0 Cr dxy 0.0 0.0 0.0 73.1 16.6 0.4 42 43 44 45 46 47 -0.60527 -0.60527 -0.48831 -0.29023 -0.29022 -0.29022 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -------- -------- -------- -------- -------- -------- 0 Cr s 0.0 0.0 62.3 0.0 0.0 0.0 0 Cr dz2 6.9 73.8 0.0 0.0 0.0 0.0 0 Cr dx2y2 73.8 6.9 0.0 0.0 0.0 0.0

Interestingly, the orbital splittings in the spin-‐up and spin-‐down levels is quite

different. In general in DFT without admixture of Hartree-‐Fock exchange (as is the

case with the BP functional that we use here), the orbital energy difference between

occupied and virtual orbitals is a 0th order approximation to the excitation energy.

In the present case, we would predict a ligand field transiton around 2.8 eV

(~23000 cm-‐1). Let’s check by running a TD-‐DFT calculation. We calculate just six

excited states corresponding to the six ways of taking an electron out of a t2g orbital

and placing it into the eg orbital. In order to do this calculation include:

The result is: ------------------------------------------------------------------------------ ABSORPTION SPECTRUM ------------------------------------------------------------------------------ State Energy Wavelength fosc T2 TX TY TZ

14 However, they are not „pure“ dxy,dxz and dyz orbitals. This is because degenerate orbitals can arbitrarily mix among themselves without changing anything in the physics. These three orbitals simply span the three components of the t2g representation.

%tddft nroots 6 end

(cm-1) (nm) (D**2) (D) (D) (D) ------------------------------------------------------------------------------ 1 20591.4 485.6 0.000000000 0.00000 0.00000 -0.00002 0.00000 2 20591.5 485.6 0.000000000 0.00000 -0.00000 0.00000 0.00002 3 20591.8 485.6 0.000000000 0.00000 0.00000 -0.00003 0.00001 4 23600.1 423.7 0.000000000 0.00000 0.00001 -0.00000 0.00001 5 23600.7 423.7 0.000000000 0.00000 0.00001 -0.00001 -0.00001 6 23601.3 423.7 0.000000000 0.00000 0.00000 0.00002 0.00001

We see two excited states that are triply degenerate each and correspond to the

singly excited 4T1g and 4T2g terms expected from Table 1.15 Despite the much too

small basis set, the result does not even compare poorly with experiment, where

transitions are observed at 17,400 and 24,700 cm-‐1.16 Furthermore the average of

the two transition energies correspond well with the orbital energy difference. The

splitting between the two-‐states is one of the characterisitics of electron-‐electron

repulsion leading to different multiplets. The compositions of these transition shows

that they indeed correspond to the expected excitations: STATE 1: E= 0.093822 au 2.553 eV 20591.4 cm**-1 39a -> 43a : 0.040541 (c= -0.20134677) 40a -> 42a : 0.235055 (c= -0.48482442) 40a -> 43a : 0.091500 (c= 0.30248889) 41a -> 42a : 0.033109 (c= 0.18195871) 41a -> 43a : 0.591178 (c= 0.76888093) STATE 2: E= 0.093822 au 2.553 eV 20591.5 cm**-1 39a -> 43a : 0.102372 (c= 0.31995577) 40a -> 42a : 0.340978 (c= 0.58393339) 40a -> 43a : 0.178600 (c= -0.42261116) 41a -> 42a : 0.028003 (c= 0.16734110) 41a -> 43a : 0.340662 (c= 0.58366233) STATE 3: E= 0.093823 au 2.553 eV 20591.8 cm**-1 39a -> 42a : 0.835399 (c= -0.91400145) 40a -> 42a : 0.010739 (c= -0.10363150) 40a -> 43a : 0.138077 (c= -0.37158743) STATE 4: E= 0.107530 au 2.926 eV 23600.1 cm**-1 41a -> 42a : 0.921691 (c= -0.96004745) 41a -> 43a : 0.057615 (c= 0.24003050) STATE 5: E= 0.107533 au 2.926 eV 23600.7 cm**-1

15 Hint: you can find out which is which by using a group theoretical table and the output for the CD spectrum. The magnetic dipole components RX, RY, RZ transform as the molecular rotations. Thus, you have to find out whether a 4A2g→

4T1g or a 4A2g→

4T2g transition is magnetic dipole allowed! 16 However, this is not always the case for TD-‐DFT calculations. There is every reason to be critical about them and compare very careful to experiment! In many cases artifacts arise. Charge transfer states tend to appear MUCH too low in the spectrum, double excitations are entirely missing, neutral-‐to-‐ionic transitions are poorly predicted, spin-‐flip transitions can not be predicted etc. Some of this is discussed in Neese, F. (2006) A Critical Evaluation of DFT, including Time-‐Dependent DFT, Applied to Bioinorganic Chemistry. J. Biol. Inorg. Chem., 11, 702-‐711. Ab initio methods are covered in some detail in: Neese F.; Petrenko, T.; Ganyushin, D.; Olbrich, G. (2006) Advanced Aspects of ab initio Theoretical Spectroscopy of Open-‐Shell Transition Metal Ions. Coord. Chem. Rev., 205, 288-‐327

39a -> 42a : 0.122088 (c= 0.34941155) 39a -> 43a : 0.011888 (c= 0.10903325) 40a -> 42a : 0.341659 (c= -0.58451589) 40a -> 43a : 0.505530 (c= -0.71100658) STATE 6: E= 0.107536 au 2.926 eV 23601.3 cm**-1 39a -> 42a : 0.029369 (c= -0.17137471) 39a -> 43a : 0.828386 (c= 0.91015695) 40a -> 42a : 0.062645 (c= -0.25029038) 40a -> 43a : 0.070858 (c= 0.26619100)

In LFT, the energies of the two transitions are given by:

4A2g→4T2g : ΔE= Δ0

4A2g→4T1g : ΔE= Δ0+12B

Thus, from the calculations you can easily determined the values of Δ0 and B. In the

free Cr(III) ion, the B-‐value is known to be 950 cm-‐1. The ration of

B(complex)/B(free ion) is known as the nephelauxetic ratio. Is it larger or smaller

than unity? Why is it larger or smaller than unity?

JOB: • Run similar calculations for other d3 systems: [V(H2O)6]2+ and the

hypothetical [Ti(H2O)6]+ as well as the d8 system [Ni(H2O)]2+ as well as the

hypothetical [Cu(H2O)6]3+

• Look at the results of the population analysis. How does the partial charge on

the metal change. How does the spin-‐population at the metal change. Look at

the occupancies of the metal d-‐orbitals. What does that say about σ-‐ and π-‐

donation of the water ligands. How does it compare with the formal

occupations. Compare the spin-‐populations of the d3 and the d8 systems.

What do you observe?

• Calculate the first six ligand field excited states and determine Δ0 and B from

the computational results. Do the trends make sense in terms of your

chemical intuition?17 Calculate the nephelauxetic ratios – what do you include

17 The Racah-‐parameters of the free ions are: Ti2+=720 cm-‐1, Ti+=680 cm-‐1, V2+=765 cm-‐1, V3+=860 cm-‐1, Ni2+=1080 cm-‐1, Ni+= 1040 cm-‐1, Cu2+= 1240 cm-‐1, Cu+=1220 cm-‐1, Cu3+=1260 cm-‐1. In the d8 configuration, the first two ligand field excited states (3A2g→

3T2g,3T1g) are predicted to occur at Δ0 and Δ0+12B in a similar way as found for the the d3

configuration. For more extensive tabular material see Haberditzl, W. Quantenchemie 4. Komplexverbindungen, Hüthig, Heidelberg, 1979.

with respect to the ability of TD-‐DFT to model these values correctly.18 Will

the nephelauxetic ratios depend on the nature of the metal, its oxidation state

and the ligand? Please explain your answer.

• Study the dependence of Δ0 (and B) on the metal-‐ligand distance of

[Cr(H2O)6]3+: Does the result match your expectation? With which power of

the metal-‐ligand distance R do you expect Δ0 to vary based on conventional

ligand field arguments?

1.2.2 Inverted Bonding19 In many cases, you will observe that it is difficult to unambiguously identify the

metal based MOs. This is particularly so for later metals, higher oxidation states and

increasing numbers of unpaired electrons. The reason is that higher oxidation states

tend to stabilize the d-‐orbitals that consequently move closer to the ligand orbitals.

In addition, large number of singly occupied orbitals that are mainly centered on the

metal lead to a very large exchange stabilization which further suppresses the

energies of the occupied spin-‐up orbitals. As a consequence, these orbitals move

down in energy into the ligand based orbitals. This effect is more pronounuced the

more Hartree-‐Fock exchange you have in your density functional and is extremely

strong for the Hartree-‐Fock method itself. Yet, the ligand field picture is not invalid –

it is simply more difficult to recover it from the MO calculations.

• One useful way is to run the orbitals through a localization procedure and

look at the localized orbitals of mainly metal character.

• Another useful way to analyze the calculations is to look at the set of orbitals

that we call “quasi-‐restricted orbitals”. 20 The singly occupied ones will

correspond well with your ligand field expectations.

JOB:

18 The nephelauxetic ration are found to be 0.79 for [CrIII(H2O)6]

3+ and 0.77 for [NiII(H2O)6]2+.

19 A nice discussion is found in: I, J.; Noodleman, L.; Case, D.A. (2000) In: Solomon, E.I.; Lever, A.B.P (Eds.) Inorganic Electronic Structure and Spectroscopy, Vol. 1, John Wiley & sons, New York, pp 661ff 20 For their definition see: Neese, F. (2006) Importance of Direct Spin-‐Spin Coupling and Spin-‐Flip Excitations for the Zero-‐Field Splittings of Transition Metal Complexes: A Case Study, J. Am. Chem. Soc., 128, 10213-‐10222

• Do a calculation on [Fe(H2O)6]3+. Look at the canonical orbitals in both, the

spin-‐up and the spin-‐down sets. Where are the metal orbitals, what is their

composition, what are their energies, how do you interpret the result?

• Localize the orbitals using a %loc end block. Visualize the orbitals. Analyze

their composition by re-‐reading them into another calculation with the !

NoIter Keyword.

• Obtain a set of QRO’s by including the ! UNO keyword. Visualize them,

analyze them.

What does your result imply in terms of frontier molecular orbital theory? Where do

you think will the electron go upon reduction of the complex to the ferrous form?

1.2.3 Covalencies of Metal Ligand Bonds As mentioned in the introduction to this chapter, the coalvency of the metal ligand

bonds is an important factor in determining the physical and reactive properties of

transition metal complexes. We wish to understand covalency here as the ability of

the metal and the ligand to share electrons. A quantitative measure for this is how

much ligand character is mixed into the metal d-‐based orbitals and vice versa.21

Let us look at [M(H2O)6(X)]n+ complexes and vary X for fixed M and then M for fixed

X.

JOB:

• Choose X to be: CO, CN-‐, NH3, OH2, O2-‐, S2-‐, F-‐, Cl-‐, Br-‐ and I-‐.

• For the hypothetical complexes [CrIII(H2O)5(X)]3+/[CrIII(H2O)5(X-‐

)]2+/[CrIII(H2O)5(X2-‐)]2+ optimize the geometry of the complexes using RI-‐

BP86 and the small basis set defined above. Calculate the energies of the

system using the B3LYP functional in single point calculations and include

the COSMO(water) solvation model. Look at the metal-‐t2g and metal-‐eg

21 Please recall what we have said about the unitary invariance of the wavefunction (or Kohn-‐Sham determinant) with respect to rotations among occupied orbitals. Thus, there is no “absolute truth” in such numbers and they are of course not observables; yet, it is highly suggestive and chemically sensible to study the variations in metal-‐ligand bonding by looking at the metal-‐ligand bonding and antibonding orbitals.

derived orbitals. Localize the occupied orbitals and look at the LMO’s.22

Which conclusions can you draw about the nature of the bond? Discuss π-‐

bonding versus π-‐backbonding. How does it vary with the ligand? Which

changes do you observe in the populations of the metal d-‐orbitals?

• From the total energy calculations that you did you can compare the bond

energy of the ligand X relative to the hexaquo complex. To this end you have

to run a geometry optimization plus energy for the free ligands X. How does

the bond energy vary with the ligand and the oxidation state of the metal?23

Is it in accord with your chemical intuition?

• Repeat the calculations for fixed ligands with a lower-‐valent metal, for

example, the low-‐spin configuration (S=0) of Fe(II). How does the ability of

the metal for backbonding change?

1.2.4 Organometallics The field or organometallic chemistry is of growing importance due to the fact that

versatile and powerful catalysts can be designed on the basis of such complexes.

Here, we take a brief look at the “father” of all organometallic compounds – Fe(cp)2.

Below you find a set of reasonable coordinates for this complex:

JOB:

• Run a single point calculation using the BP86 functional and the small basis

set used throughout this computer experiment.

22 Recall that the localized orbitals never show a bond where there is none. Thus, if a bonding and an antibonding partner of a bond are both fully occupied, the localization procedure simply separates the two constituents into atomic or fragment orbitals. A bond only remains if the antibonding partner is in the virtual space. Thus, localization and orbital inspection is a good way to discuss the net bonding effects. You can visualize these localized orbitals with the orca_plot program and you can analyze their composition by running a calculation which have the myjob.loc orbitals as initial guess and ! noiter. 23 Alternatively, if you want to have an absolute value for the bond energy, optimize the geometry of [Cr(H2O)5]

3+/2+ and calculate its energy in the same way; run a geometry optimization of the free ligand and calculate its energy. Subtract the energies of reactants and products. However, in order to get more reliable values you also have to taken into account the basis set superposition error (BSSE) as explained in chapter Error! Reference source not found. on page 144). The basis sets we use here are so small that these undesired effects are likely to be large. Therefore, for the qualitative purposes of this section it is probably better to simply compare the bond energies using the hexaquo complex as a reference.

• Analyze the electronic structure of the complex using orbitals and

populations. What is the nature of the metal ligand bond?

• Do a calculation on the free cyclopentadienyl ligand and determine the

fragment orbitals that contribute most to the bonding. Draw a fragment

orbital correlation diagram.

• Use the BHLYP functional to calculate the first 15 electronically excited

states. What is their nature? Compare the transition energies with d-‐d

transition energies of the hexaquo complexes that you have studied above.

What do you observe? What is your interpretation? Determine the dominant

donor and acceptor orbital pairs. Work out the symmetry of the final

electronic states. Determine the selection rules for the electric dipole and

magnetic dipole transitions.

Structure of Fe(cp)2 used in this experiment (charge=0, multiplicity=1; closed shell)24

24 To the interested student: work out a strategy for computing the bond energy of the Fe(II)-‐cp-‐ bond and think about a suitable Werner type Fe(II) complex with which you can compare your results. A challenge is that you separate a neutral molecule into charged fragments; therefore you should use the charge compensation offered by the COSMO model and you may have to do it via a ligand exchange reaction.

Fe 0 0 0 0.000000 0.000 0.000 C 1 0 0 2.035741 0.000 0.000 C 2 1 0 1.399741 69.943 0.000 C 3 2 1 1.399822 108.010 59.725 C 4 3 2 1.399734 108.009 0.000 C 5 4 3 1.399880 107.981 359.931 H 1 2 3 2.949216 112.347 67.069 H 7 1 2 2.691289 62.854 207.853 H 8 7 1 2.691150 107.996 314.868 H 9 8 7 2.690913 108.001 0.000 H 10 9 8 2.690897 108.005 0.000 C 7 1 2 1.118895 28.137 142.474 C 8 7 1 1.118881 54.749 328.336 C 9 8 7 1.118864 54.753 13.469 C 10 9 8 1.118884 54.757 13.468 C 11 10 9 1.118850 54.762 13.464 H 2 1 3 1.100415 137.025 239.400 H 3 2 1 1.100553 125.205 225.905 H 4 3 2 1.100560 125.189 166.179 H 5 4 3 1.100431 125.231 166.185 H 6 5 4 1.119068 125.203 166.363

1.2.5 Conformation. A Famous Example The complex ion [CuCl4]2-‐ is known to exist in different crystals with varying

counterions. While Cu(II) normally prefers a square planar coordination, this ion is

found in various forms ranging from square-‐planar to almost tetrahedral. Here we

calculate the energy associated with this distortion. You can use the following input

file:

This input will lead the ORCA program to perform a “rigid scan” that will lead from a

square planar to a tetrahedral structure by defining a distortion angle θ (the final

angle should (180-‐109.4712)/2; you can also use a larger number of steps in order

to obtain a more smooth looking surface).

JOB • Do the calculation, plot the resulting potential energy surface, convert the

energies to relative energies in kcal/mol.

• Where do you find the minimum?

• Does it require a lot of energy to distort the molecule?

• What is the change in electronic structure? Examine two or three points along the surface. Look at the orbitals and populations; calculate the first four ligand field excited states. Do you think that spectroscopy will be a useful indicator of the coordination geometry?25

The latter calculation represents one of the cases where TD-‐DFT with normal GGA

functionals like BP86 fails badly and predicts LMCT states where there should be d-‐ 25 A very nice analysis of this problem can be found in: Solomon, E.I. Comments Inorg. Chem. (1984), 3, 227

! UKS BP RI SV DeMon/J SlowConv NoFinalGrid %scf shift shift 0.3 erroff 0 end end # geometric parameters %paras R_ML = 2.25 # the metal-water-oxygen distance Theta = 0,35,17 end * xyz -2 2 cu 0 0 0 cl { R_ML*cos(Theta) } 0 { R_ML*sin(Theta) } cl {-R_ML*cos(Theta) } 0 { R_ML*sin(Theta) } cl 0 { R_ML*cos(Theta) } { -R_ML*sin(Theta) } cl 0 {-R_ML*cos(Theta) } { -R_ML*sin(Theta) } *

d excitations (in the region below ~15000 cm-‐1). You should therefore do the latter

calculation with a functional that includes more HF exchange (BHLYP is

recommended which introduces as much as 50% HF exchange). You may also need

to calculate more than the desired four ligand field states in order to “catch” the

states of interest since they do not come in order of increasing orbital energy

difference. With 25 states you are on the safe side – please observe the assignment

of the transitions and see where the occupied metal orbitals are in the spin down set

(draw a MO diagram).

1.2.6 Reactivity: Powerful Oxidants. The interesting species [FeIVO(H2O)5]2+ has can be made26 and will be studied in this

computer experiment. It is a simple model for related FeIV-‐oxo species that are of

much relevance to chemistry and biochemistry since they are powerful oxidants.27

Here we will study some aspects of structure, bonding and energetics of such

species. First, of all – which spin state is the lowest one for this system – S = 1 or S =

2. Try to find out in a computer experiment and optimize the structure of the

species:

26 O. Pestovsky, S. Stoian, E.L. Bominaar, X.P. Shan, E. Münck, L. Que Jr., A. Bakac, Angew. Chem. Int. Ed. 44 (2005) 6871 27 The most intensely studied system is Cytochrome P450 which features a very complicated electronic structure. The active species (compound I) involves an iron-‐oxo group coupled to a porphyrin radical . For recent theoretical studies see: Shaik, S.; de Visser, S. P.; Ogliaro, F.; Schwarz, H.; Schröder, D. Curr. Opin. Chem. Biol. (2002), 6, 556; S. Shaik, D. Kumar, S.P. de Visser, A. Altun, W. Thiel, Chem. Rev. 105 (2005) 2279; Schöneboom, J.; Neese, F.; Thiel, W. (2005) J. Am. Chem. Soc., 127, 5840-‐5853;

• Repeat the optimization for the low-‐spin state. Analyze the electronic

structure of both species. What are the electron configurations in the two

states? How did the structure change? How did the changes in the structure

reflect the different electronic configurations?28 Is the oxo-‐group strongly

affected by the change in spin state?

• Recalculate the energies at the optimized geometries using the B3LYP

functional.29 What is the energy difference between the two states?30 Could

either be the ground state?

28 More interestingly for the synthetically oriented chemists: How do have to choose your ligand in order to force the system to adopt either one or the other ground state? 29 In “real life” you would probably do it similarly but with much larger basis sets; for example, the geometry optimization with TZVP and the final energy calculation with TZVPP. We do not do that in order to keep the computation times short. In a serious study you would also want to evaluate the zero-‐point energy and thermal contributions to the energy differences just as done in chapter Error! Reference source not found. on page 51. 30 Compare for example: Neese, F. (2006) J. Inorg. Biochem. 100, 716-‐726. The entire issue deals with Fe(IV) and you will find many other interesting articles there dealing with the structure, chemistry, spectroscopy and theory of such compounds.

! UKS BP RI SlowConv NoFinalGrid TightSCF Opt %scf shift shift 0.3 erroff 0 end end %basis basis sv newgto h "STO-3G" end end %paras R_ML = 1.995 R_OH = 0.9817 A_HOM = 126.153 R_MX = 1.64 end * int 2 5 Fe 0 0 0 0.0 0.0 0.0 O 1 0 0 {R_ML} 0.0 0.0 O 1 2 0 {R_ML} 180.0 0.0 O 1 2 3 {R_ML} 90.0 0.0 O 1 2 4 {R_ML} 90.0 180.0 O 1 2 4 {R_ML} 90.0 270.0 O 1 2 4 {R_MX} 90.0 90.0 H 2 1 3 {R_OH} {A_HOM} 90.0 H 2 1 3 {R_OH} {A_HOM} 270.0 H 3 1 2 {R_OH} {A_HOM} 90.0 H 3 1 2 {R_OH} {A_HOM} 270.0 H 4 1 2 {R_OH} {A_HOM} 0.0 H 4 1 2 {R_OH} {A_HOM} 180.0 H 5 1 2 {R_OH} {A_HOM} 180.0 H 5 1 2 {R_OH} {A_HOM} 0.0 H 6 1 2 {R_OH} {A_HOM} 90.0 H 6 1 2 {R_OH} {A_HOM} 270.0 *

Now let us study the hydrogen atom abstraction reaction:

[FeO(H2O)5]2+ + CH4 → [Fe(OH)(H2O)5]2+ + CH3•

JOB:

• Optimize the structures of the reactants and products (assume S=5/231)

using the RI-‐BP86 method as above.

• Recalculate the energies using B3LYP single point calculations.

• Is the reaction predicted to be feasible? If not, would this molecule be able to

oxidize weaker C-‐H bonds such as that of toluene?

• Analyze the structure of the reaction product. What are the major geometric

changes? What are the electronic changes brought about by protonating the

oxo group and reducing the metal? What happens to the Fe-‐O bond, what to

its covalency?

• Compare the capability of this iron-‐oxo species to two other systems: OH•

radical and the well-‐known species [VO(H2O)5]2+. The latter calculation

requires another set of geometry optimizations.

1.2.7 Unusual Bonds and Coordinated Radicals – Iron-‐Nitrosyls Questions of structure and bonding can become quite complicated when the

oxidation state of the ligand becomes ambiguous. A famous example are transition 31 Start from the following coordinates: Fe 0 0 0 0.000000 0.000 0.000 O 1 0 0 2.064887 0.000 0.000

O 1 2 0 2.063420 175.071 0.000

O 1 2 3 2.105932 89.618 85.198

O 1 2 4 2.106399 89.557 195.176

O 1 2 5 2.100535 87.401 82.433

O 1 2 6 1.768057 92.691 179.818

H 2 1 7 1.009685 122.106 359.695

H 2 1 7 1.008473 127.717 179.862

H 3 1 7 1.009525 121.926 0.061

H 3 1 7 1.008441 127.868 180.210

H 4 1 2 1.008208 123.239 15.085

H 4 1 2 1.008169 123.475 169.580

H 5 1 2 1.008168 123.401 190.709

H 5 1 2 1.008201 123.457 345.633

H 6 1 2 1.007542 124.625 90.015

H 6 1 2 1.007546 124.702 269.828

H 7 1 2 1.005566 179.700 45.575

metal nitrosyls. Here the NO ligand is believed to be able to coordinate as NO-‐(S=1),

NO-‐(S=0), NO•(S=1/2) and NO+(S=0) depending on the metal, its formal oxidation

state and the remaining ligand set. Since the interpretation is often ambiguous,

Enemark and Feltham have invented the {MNO}x notation, where x is the number of

metal-‐d plus NO-‐π* electrons. Thus, the reaction of Fe(II) with NO• yields a {FeNO}7

complex. We will study here the iron pentaaquo-‐NO complex – a molecule which is

surely well known to every chemist32 but has a quite complicated electronic

structure.33

• Optimize the geometry of [Fe(H2O)5(NO)]3+/2+/1+ in the spin states S=5/2,3/2

and 1 (you can initially assume an NO bond distance of 1.76 Angström and

Fe-‐O bonds of 2.12; an N-‐O distance of 1.2 Angström and an O-‐N-‐Fe angle of

170°). The standard input ! UKS BP RI SlowConv NoFinalGrid Opt

TightSCF should work.

• How does the N-‐O distance change with oxidation state? How does the Fe-‐N

distance change? What does it tell you about the oxidation state of the ligand?

(Compare to free NO+, NO• and NO-‐) Do you observe trans effects?

• Analyze the bonding using orbitals, spin-‐ and electron-‐populations, bond

orders and all the tricks that you have learned. 34 What is your best

description of the electronic structure in terms of dN configurations? In fact,

is the dN designation sensible in this case?

• Are there alternative spin states? May they be lower in energy than the ones

that you have calculated?

32 It is the “brown ring” complex in analytic chemistry. 33 Wanat, T.; Schneppensieper, G.; Stocher, G.; van Eldik, R.; Bill, E.; Wieghardt, K. Inorg. Chem. (2002), 41, 4; a very interesting combined experimental and theoretical study on a related system is found in: Brown, C. A.; Pavlosky, M. A.; Westre, T. E.; Zhang, Y.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 1995, 117, 715. 34 In fact, there is an additional trick that you can learn here – unrestricted corresponding orbitals (UCOs). In order to get them use ! UCO in the ORCA input file. This will produce a file called MyJob.uco and additional output. The additional output will consist of spatial overlaps of spin-‐up and spin-‐down orbitals. Overlaps close to zero or at least significantly smaller than one indicate a spin coupled system. In this case the spin-‐expectation value <S2> will significantly deviate from S(S+1) [Look in the output for this number]. What is your interpretation if you find such a spin coupled system here? Look at the UCO’s and observe which orbitals are spin coupled (if any). (For a discussion see Neese, F. J. Phys. Chem. Solids, 2004, 65, 781-‐785)

1.2.8 High-‐Spin/Low-‐Spin and Spin-‐Crossover An important field of investigation deals with transition metal complexes that have

two thermally accessible spin states – such complexes are spin-‐crossover systems;

in the LFT language they feature ligand field strengths that are just on the boarder of

the strong-‐field and the weak field part of the Tanabe-‐Sugano diagram. This case is

particularly frequently met in Fe(II) complexes, where the 5T2g (t2g)4(eg)2 and 1A1g

(t2g)6(eg)0 states are part of the spin-‐crossover system. A very accurate calculation of

the energy difference between these two states is obviously very complicated as one

needs to reach an accuracy that is on the order of the thermal energy (~200 cm-‐1).

Moreover, the spin-‐crossover is frequently driven by cooperative and entropic

effects which are particularly difficult to model with quantum chemical methods.

Nevertheless, it has been found to be an especially difficult problem for DFT

methods to come at predictions that are at least within a reasonable range and

predictions of different functionals were found to differ by very large margins.35

Here we simply study a very simple model for one frequently studied spin-‐crossover

complex [Fe(phen)2(NCS)2]. We model this complex by [Fe(NH3)4(NCS)2] in order to

save computation time. The optimized structure for the low-‐spin and the high-‐spin

state are shown below.

JOB:

• First look at the two structures given below. What is the main difference

between the high-‐spin and low-‐spin structure? How do the geometric

structures reflect the electronic structures.

• Run single point calculations on both structures using the UHF method as

well as the BHLYP B3LYP, BLYP, BP96 and LSD functionals. Compare your

results using the standard small basis set of this experiment. To which extent

do you want to rely on DFT methods in order to predict spin state energetics?

Is the Hartree-‐Fock method an attractive alternative?

35 See Reiher, M.; Salomon, O.; Hess, B.A. Theor. Chem. Acc. (2001), 107, 48-‐55; Paulsen, H.; Duelund, L.; Winkler, H.; Toftlund, H.; Trautwein, A.X. Inorg. Chem., (2001), 40, 2201-‐2203; Zhang, Y.; Oldfield J. Phys. Chem., (2003), 107, 4147-‐4150; Fouqueau, A.; Mer, S.; Casida, M.E.; Daku, L.M.L.; Hauser, A.; Mieva, T.; Neese, F. J. Chem. Phys., (2004) 120, 9473-‐9486

The Low-‐Spin Structure:

The High-‐Spin Structure:36

36 SCF convergence may be somewhat challenging for this species since the (“t2g”)

4(eg)2 configuration is almost

orbitally degenerate and the orbital energy difference between the spin-‐down HOMO and the spin-‐down LUMO is only ~0.2 eV! The calculations with the hybrid functionals are more likely to converge better and might be taken as input for the calculations that do not involve HF exchange. Try ! SlowConv and perhaps NRSCF Consult the ORCA manual for more details.

Fe 0 0 0 0.000000 0.000 0.000 N 1 0 0 2.049885 0.000 0.000 N 1 2 0 2.052645 171.400 0.000 N 1 2 3 2.075900 92.686 228.011 N 1 2 4 1.891473 87.079 183.611 N 1 2 5 2.076027 92.843 83.846 N 1 2 5 1.890882 87.097 260.185 C 7 1 2 1.216067 163.994 272.872 C 5 1 2 1.215828 165.562 59.575 S 8 7 1 1.670676 179.710 208.661 S 9 5 1 1.669960 179.595 121.297 H 2 1 7 1.039864 102.694 252.981 H 2 1 7 1.039853 101.663 5.151 H 2 1 7 1.036898 122.329 128.624 H 3 1 7 1.039350 102.218 352.185 H 3 1 7 1.039385 101.030 104.131 H 3 1 7 1.036609 122.740 227.615 H 4 1 2 1.043434 99.578 85.432 H 4 1 2 1.036103 114.396 201.937 H 4 1 2 1.036150 114.727 328.610 H 6 1 2 1.036117 114.516 29.302 H 6 1 2 1.036059 114.770 156.207 H 6 1 2 1.043065 99.101 272.880

1.2.9 Exchange Coupling Between Transition Metal Ions A final important subject to be mentioned in this course is the “magnetic” coupling

between open-‐shell transition metal ions. There is a large class of multinuclear

transition metal systems with constituent ions that – based on their electron count –

must have unpaired electrons. These unpaired electron on different sites typically

couple weakly to produce a ladder of spin states. This ladder may

phenomenologically be described by a so-‐called “Heisenberg Hamiltonian” that has

the form -‐2J*SA*SB where SA and SB are the spin-‐operators for sites ‘A’ and ‘B’ and J is

the “exchange coupling constant”. If it is negative, low-‐spin states are preferred and

the system is said to be “antiferromagnetically” coupled Positive J refers to

“ferromagnetic coupling”. The ordering of spin-‐states has important implications for

many branches of chemistry and physics. We will not enter a detailed discussion of

this important subject here.37,38

37 See for example L. Noodleman J. Chem. Phys. 74 (1981) 5737; L. Noodleman, E.R. Davidson Chem. Phys. 109 (1986) 131; A. Bencini, D. Gatteschi J. Am. Chem. Soc. 108 (1980) 5736; K. Yamaguchi, Y. Takahara, T. Fueno, in: V.H. Smith (Ed.), Applied Quantum Chemistry, 155; O. Kahn, 'Molecular Magnetism', VCH Publishers, New York, 1993; Calzado, C. J.; Cabrero, J.; Malrieu, J. P.; Caballol, R. J. Chem. Phys. (2002), 116, 2728 38 It is important to understand that this “magnetic coupling” has nothing to do with genuine magnetic interactions. The origin of the exchange coupling is purely electrostatic in origin and is intimately associated with the antisymmetry

Fe 0 0 0 0.000000 0.000 0.000 N 1 0 0 2.247312 0.000 0.000 N 1 2 0 2.248019 169.744 0.000 N 1 2 3 2.307572 93.715 223.307 N 1 2 4 2.037538 86.562 188.204 N 1 2 5 2.312841 93.535 80.789 N 1 2 5 2.040200 87.326 252.683 C 7 1 2 1.217279 165.572 266.841 C 5 1 2 1.217076 166.485 63.735 S 8 7 1 1.664217 179.384 344.402 S 9 5 1 1.664331 179.291 29.396 H 2 1 5 1.036127 101.817 352.840 H 2 1 5 1.037918 97.395 104.836 H 2 1 5 1.033648 124.170 226.576 H 3 1 5 1.036474 101.009 243.118 H 3 1 5 1.037390 97.718 355.185 H 3 1 5 1.033489 124.285 117.374 H 4 1 2 1.037219 96.213 87.978 H 4 1 2 1.032760 114.165 204.515 H 4 1 2 1.033161 113.380 331.970 H 6 1 2 1.033002 113.409 27.309 H 6 1 2 1.032651 114.389 155.169 H 6 1 2 1.037071 95.577 271.582

The accurate calculation of exchange coupling constants is quite challenging. In

principle, multideterminantal methods are required to give a proper description of

the low-‐spin states. The values of J are on the order of a few dozen to a few hundred

wavenumbers – thus one has to calculate small differences between large numbers

to high accuracy which is quite difficult. The multideterminantal (but not

multiconfigurational) nature of the low-‐spin states would seem to invalidate DFT

methods, which are based on a single determinant. However, the situation is not so

bad, if some precaution is taken. Consider for example the case of two interacting

Cu(II) ions. Since both are d9 systems, a singlet and a triplet arise. The triplet is

straightforward to calculate with standard DFT methods, but the singlet is not. In

fact, the best way to approach the problem in DFT that is known up to date is to

generate a so-‐called “broken-‐symmetry” solution in which one-‐electron localizes

with spin-‐up on one site and another one with spin-‐down on the second site. This

does not represent a correct singlet wavefunction but merely a determinant with

MS=0. Nevertheless, the physics of weakly interacting electrons is basically correctly

described. Together with some approximate formalisms, it is possible to extract a

value for J from a high-‐spin and a broken symmetry calculation. In ORCA, it is

possible to specifically ask for such solutions.

We study the famous case of bis-‐µ-‐hydroxy-‐bridged Cu(II)-‐dimers which represents

a classical case of magnetostructural correlations. It has been shown experimentally,

that a switch between ferromagnetic and antiferromagnetic coupling occurs as a

function of bridging angle α. The critical angle is found somewhere around 97°. You

can do a calculation and see whether you can qualitatively reproduce this.

JOB: • Run the job shown below. Vary the angle Alpha and search in the output for

the predicted exchange coupling constant (choose J(3) for plotting). Plot the exchange coupling versus angle and observe whether you can detect an change from J<0 to J>0.

of the N-‐particle wavefunction. There is no “exchange interaction” in nature. What we describe as “exchange interaction” is merely the combined action of antisymmetry and electron-‐electron repulsion.

! UKS BP RI SV DeMon/J SlowConv TightSCF %basis newgto h "STO-3G" end end %scf BrokenSym 1,1 end %paras Alpha = 95.0 ATilt = 60.0 AHNCu = 112.0 ANCuN = 94.7 RCuO = 1.970 ROH = 1.034 RNH = 1.066 RCuN = 2.060 end * int 2 3 Cu 0 0 0 0 0 0 Cu 1 0 0 {2*RCuO*sin(0.5*Alpha)} 0 0 O 1 2 0 {RCuO} {90-0.5*Alpha} 0 O 1 2 3 {RCuO} {90-0.5*Alpha} 180 H 3 4 1 {ROH} {180-ATilt} 90 H 4 3 1 {ROH} {180-ATilt} 90 N 1 2 3 {RCuN} {180-ANCuN/2} 0 N 1 2 3 {RCuN} {180-ANCuN/2} 180 N 2 1 3 {RCuN} {180-ANCuN/2} 0 N 2 1 3 {RCuN} {180-ANCuN/2} 180 H 7 1 2 {RNH} {AHNCu} 0 H 7 1 2 {RNH} {AHNCu} 120 H 7 1 2 {RNH} {AHNCu} 240 H 8 1 2 {RNH} {AHNCu} 0 H 8 1 2 {RNH} {AHNCu} 120 H 8 1 2 {RNH} {AHNCu} 240 H 9 2 1 {RNH} {AHNCu} 0 H 9 2 1 {RNH} {AHNCu} 120 H 9 2 1 {RNH} {AHNCu} 240 H 10 2 1 {RNH} {AHNCu} 0 H 10 2 1 {RNH} {AHNCu} 120 H 10 2 1 {RNH} {AHNCu} 240 *

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1 Computer Experiment* 15: Computational Coordination ...scienide2.uwaterloo.ca/~nooijen/Chem-440-computational/ORCA_labs... · oriented$modifications$are$known$collectively$as$“ligand$field$theory”$but$we$will$