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52 AOMX: Angular Overlap Model Computation

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536



Molecular Orbital Theory (SCF Methods and Active Space SCF)



based methods, where the correlation energy is computed in a CI framework. The DFT methods,

which are based on the density alone, depend much less on the basis set.



2.41.8.2



Cu2+(NH3)3X



Next, we will look at complexes of the form Cu2ỵ(NH3)3X, where X is related to the thiolate

ligand in the blue copper proteins, e.g., SHÀ, OHÀ, SeHÀ, PH2À, and ClÀ. Such complexes have

been employed to explain why the blue copper proteins exhibit a trigonal structure, whereas most

Cu(II) complexes assume a tetragonal structure.40 For all these complexes, local minima representing both a tetragonal and a trigonal structure could be optimized. However, the relative

stability of the trigonal structure increases as we move down and to the left in the periodic table,

as can be seen in Table 3. It is also stabilized by negatively charged X ligands. The relative

energies were calculated by both the CASPT2 and B3LYP methods. The two methods give rather

similar results, with maximum and average differences of 18 kJ molÀ1 and 8 kJ molÀ1, and they

therefore give the same predictions of the most stable structure for all complexes, except for the

two complexes where the two geometries are almost degenerate, Cu(NH3)3(SH)(SH)ỵ and

Cu(NH3)3(PH2)ỵ.

Relativistic corrections (Darwin contact and massvelocity terms calculated at the CASSCF

level) are also given in the table. For most complexes, this correction is small and insignificant.

However, for three complexes (Cu(NH3)3(SH)ỵ, Cu(NH3)3(SeH)ỵ, and Cu(NH3)3(SH2)ỵ, the

corrections are large (14–16 kJ molÀ1) and positive (favoring the tetragonal state). The reason

for this is that relativistic corrections in general favor the structure with the lowest Cu 3d

population. For the three complexes exhibiting large relativistic effects, the Cu 3d population

for the tetragonal structure is 9.3–9.4, whereas it is 9.9 for the trigonal structure. For all the other

complexes the CASSCF Cu 3d are similar for the tetragonal and the trigonal structures, either

close to 9.3 or close to 9.9 (representing either d 9 or d10 states. This is not in accordance with the

B3LYP results, where the Cu 3d populations are always similar (within $0.1) for the two

geometries, but it varies continuously between 9.3 and 9.7 (in general, it is lower for complexes

with stable tetragonal states and higher for those with more stable trigonal states). This gives also

an explanation for the tetrahedral distortion of the complexes with stable trigonal structures (both

trigonal and tetrahedral): In the latter complexes much charge is donated from the large, soft, and

polarizable negatively charged X ligand, giving rise to an electronic structure close to Cuỵ, which

is closed-shell (d10) and therefore prefers a tetrahedral structure.



2.41.8.3



Electronic Spectra



Finally, we will discuss the electronic spectra of blue copper proteins. The absorption spectrum of

plastocyanin, the best studied blue copper protein, is dominated by a bright band at 16,700 cmÀ1

(600 nm), giving rise to its bright blue color. However, a more thorough investigation of the

experimental spectrum identifies at least six more absorption bands below 22,000 cmÀ1, as is

shown in Table 4.41 Several different methods have been used to interpret this spectrum, ranging

from the semi-empirical CNDO/S method, over various DFT methods (X and time-dependent

Table 3 Energy difference (kJ molÀ1) between the trigonal and tetragonal structures of

the various model complexes.

Model

Cu(NH3)42ỵ

Cu(NH3)3(OH2)2ỵ

Cu(NH3)3Clỵ

Cu(NH3)3(OH)ỵ

Cu(NH3)3(SH)ỵ

Cu(NH3)3(SeH)ỵ

Cu(NH3)3(PH2)ỵ

Cu(NH3)2(SH)(SH2)ỵ



B3LYP



CASPT2



CASPT2 ỵ Rel. Corr.



46.0

33.9

38.2

19.7

3.1

5.8

2.6

12.6



42.8

33.9

49.3

37.6

1.7

18.2

4.4

21.1



42.2

33.5

47.4

35.9

14.4

4.7

5.1

7.0



537



Molecular Orbital Theory (SCF Methods and Active Space SCF)



Table 4 The experimental spectrum of plastocyanin41 compared to spectra calculated with the X , CASPT2,

and time-dependent B3LYP methods.41,44,47 All excitation energies are given in cmÀ1. Significant oscillator

strengths are indicated in parentheses. The assignment is based on the CASPT2 calculations44 and the results

of the other methods are ordered so that excitations with the same character are found on the same row (even

if the authors of the X investigation give a different assignment of several bands in the experimental

spectrum41). The assignment invokes a coordinate system where the Cu ion is at the origin, the z-axis is

along the Cu–SMet bond, and the Cu–SCys bond is situated in the xy-plane. Two excitations studied with the

CASPT2 method could be studied only by severe approximations (see the text) and are therefore marked by

square brackets.

Experimental

5,000

10,800

12,800

13,950

16,700

18,700

21,390

23,440

32,500

a



(.0031)

(.0114)

(.0043)

(.0496)

(.0048)

(.0035)



X



Assignmenta



CASPT2



B3LYP



4,119

10,974

13,117 (.0015)

13,493 (.0003)

17,571 (.1032)



4,206

9,441 (.0013)

12,827 (.0142)

13,673 (.0010)

18,364 (.0733)



4,527

8,691

11,942 (.046)

15,064

16,940 (.078)



*

dz2

dyz

dxz





20,599 (.0014)

[31,264]

[34,992]



20,267 (.0002)

20,806 (.0003)

21,327 (.0006)



25,313

15,895, 36,700

14,770, 52,894





Met

His



Singly occupied orbital in the excited state.



B3LYP calculations) to CASPT2.41–45 The results of the various calculations are also shown in

Table 4, together with calculated oscillator strengths and an assignment of the various excitations.

All methods agree that in the ground state, the singly-occupied orbital is comprised of the Cu

3dxy-orbital and a SCys 3p-orbital, forming an antibonding  interaction (some authors use a

coordinate system that is rotated 45  relative to ours).41,43 The bright blue line arises from the

excitation to the corresponding -bonding interaction, and its high intensity arises from the

strong overlap between these two orbitals. This interaction also explains the trigonal structure

of the plastocyanin site: By the -bond, SCys overlaps with two of the four lobes of the Cu 3dxyorbital. The two histidine ligands form normal  bonds to copper, overlapping with the remaining

two lobes of the singly occupied Cu 3dxy-orbital, whereas any additional ligand (methionine in

plastocyanin) can overlap only with doubly occupied orbitals, and therefore forms weak axial

interactions at long distances.

The normal Cu–SCys -antibonding interaction is found as the first excited state in plastocyanin, at an excitation energy of 5,000 cmÀ1. In some other proteins, e.g., nitrite reductase, this state

becomes the ground state, giving rise to a strongly tetrahedrally distorted (owing to the charge

transfer from SCys) tetragonal structure with  bonds to all four ligands.45,46 This state overlaps

strongly with the corresponding -bonding interaction, found at slightly higher energy than the 

bond (21,900 cmÀ1 in nitrite reductase), giving this enzyme a green colour.46 Other proteins exist

that have intermediate structures and spectra, e.g., cucumber basic protein and pseudoazurin.45.

Moreover, various mutant proteins have been constructed with other ligand sets (but still a

cysteine ligand) that have more tetragonal structures and even brighter excitations to the Cu–

SCys -orbital, giving them a yellow colour. In fact, the intensity ratio between the yellow and blue

bands of all blue copper proteins can be rationalized by the transition of the structure from

trigonal to tetragonal, e.g., as described by the angle between the planes defined by the N–Cu–N

and SCys–Cu–SMet atoms.45

Table 4 shows that the accuracy of the CASPT2 method is impressive for this complicated

system (a chromophore in a protein); the six lowest excitations are calculated with an error of less

than 1,000 cmÀ1. Owing to the size of the system, several approximations had to be invoked to

make the calculations possible. The chromophore was modeled by Cu(imidazole)2(SH)(SH2)ỵ, at

the crystal geometry and with a point-charge model of the surrounding protein. However, this

model is too small to give accurate results. Therefore, the excitation energies have been corrected

from

the

(by up to 2,600 cmÀ1) for truncation effects by using data

Cu(imidazole)2(SCH3)(S(CH3)2)ỵ model, optimized with the B3LYP method and Cs symmetry

(Figure 8). Moreover, the calculations had to be performed with quite small basis sets, e.g.,

without polarizing functions on the N, C, and H atoms. The two excitations with the highest

energy (charge-transfer excitations to the methionine and histidine residues, respectively) could be

studied only with the symmetric Cu(imidazole)2(SCH3)(S(CH3)2)ỵ model at the optimized

geometry. Therefore, these excitation energies are much less accurate, especially for the former



538



Molecular Orbital Theory (SCF Methods and Active Space SCF)



excitation, which is very sensitive to the geometry of the model and also to the size of the basis

sets. Finally, it should be noted that our assignment left one band unassigned, mainly on the basis

that this band is not present in the spectrum of the related protein nitrate reductase.45,46

The plastocyanin spectrum calculated with the time-dependent B3LYP method (using the

Cu(imidazole)2(SCH3)(S(CH3)2)ỵ model optimized with B3LYP without symmetry; no pointcharge model) is also included in Table 4.47 It can be seen that the result is quite similar to

both the CASPT2 and the experimental results for the six lowest excitations; the largest difference

to the CASPT2 is 1,500 cmÀ1 for the second excitation, and the largest difference to experiments is

1,700 cmÀ1 for the bright blue line. However, for the true charge-transfer excitations to the

methionine and histidine ligands, the difference is much larger. The B3LYP calculations show

one excitation to methionine at 21,327 cmÀ1, compared to the experimental line at 23,440 cmÀ1,

and the CASPT2 result at 31,264 cmÀ1. However, as was discussed above, for this excitation the

CASPT2 results are not reliable. Similarly, B3LYP gives four excitations to the two histidine

residues: two around 21,500 cmÀ1 and two close to 35,500 cmÀ1, all with a low calculated

intensity. In the experimental spectrum, there is only one line at 32,500 cmÀ1, and with

CASPT2, only one excitation could be studied, giving an energy of 34,992 cmÀ1. Thus, these

results indicate that B3LYP gives rise to spurious charge-transfer excitations. Similar results have

been obtained also for other (mostly organic) models (see for example the discussion of chargetransfer bands in polypeptides in ref. 48).

Finally, Table 4 also contains excitations for plastocyanin calculated with the density functional

X method.41 Once again, the result is similar to the other calculations and experiments for the

six lowest excitations, whereas the discrepancy is much larger for the charge-transfer excitations.

In general, X seems to give the least accurate results, except for the blue line (which may be

because the method was parameterized to reproduce the electronic spin resonance g-values for the

ground state). It should also be noted that the authors of this investigation made a different

assignment of several bands in the spectrum, based also on other spectroscopic experiments and

selection rules.41

In conclusion, we have seen that it is possible to study the spectrum of a chromophore in a

protein with theoretical methods. CASPT2 seems to give the most accurate results, provided that

a reasonable chemical model can be studied and proper active orbitals can be selected (five Cu 3dorbitals, five correlating Cu 3d 0 -orbitals, and all orbitals involved in charge-transfer excitations).

DFT, especially the time-dependent methods also gives reasonable results at a much lower cost

and with a smaller basis-set dependence. It should be noted, however, that the assignment of the

various excitations is much easier to perform at the CASPT2 level than with DFT (the orbitals are

more pure).



2.41.9



CONCLUSIONS



In this chapter we have tried to illustrate through some examples how a multiconfigurational model

describes the electronic structure in coordination compounds. The key concept has been the NOs,

which are the generalization of the HF orbitals to a situation where more than one electronic

configuration is needed to describe the electronic structure. The examples (with the exception of the

blue proteins) have been confined to the ground state. It is evident, however, that the multiconfigurational approach is even more essential in discussions of excited states and photochemistry.

Many theoretical studies of coordination compounds are today successfully performed using

DFT. This is all to the good, as long as one is aware of the pitfalls within this approach. The

problem of charge-transfer processes was mentioned above. The definition of spin and the

inclusion of spin-orbit coupling in heavier systems is another problem, which is of major concern

in studies of electronic spectroscopy and photochemical reactions. Strongly degenerate situations

can hardly be treated with DFT. One example is the Cr2 molecule, which was discussed above.

We have also tried to use DFT for the [Re2Cl8]2À ion, but failed to converge the calculations.

The virtue of a wave-function-based multiconfigurational approach is the complete generality,

meaning that any type of electronic structure may be studied with exactly defined spin and other

symmetry properties. The major problem with the approach is the size of the active space, which

limits the possibilities to compute the effects of dynamic electron correlation. Today the only

possible approach for large molecules with many electrons is CASPT2, which is limited with respect to

the active space and in some applications gives severe intruder-state problems. It is hoped that in the

near future we shall have access alternative methods where these limitations are removed.



Molecular Orbital Theory (SCF Methods and Active Space SCF)

2.41.10

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.



29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.



539



REFERENCES



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Andersson, K.; Malmqvist, P.-A˚.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. J. Phys. Chem. 1990, 94, 5483–5488.

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Roos, B. O.; Andersson, K.; Fuălscher, M. P.; Serrano-Andre´s, L.; Pierloot, K.; Mercha´n, M.; Molina, V. J. Mol.

Struct. (Theochem.) 1996, 388, 257–276.

Chatt, J.; Duncanson, L. A. J. Chem Soc. 1953, 2939 .

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Comprehensive Coordination Chemistry II

ISBN (set): 0-08-0437486

Volume 2, (ISBN 0-08-0443249); pp 519539



2.42

Valence Bond Configuration

Interaction Model

F. TUCZEK

Christian-Albrechts-Universitaăt, Kiel, Germany

and

E. I. SOLOMON

Stanford University, CA, USA

2.42.1 INTRODUCTION

2.42.2 THE VBCI MODEL: COPPER(II) DIMERS (d 9– d 9, d 1– d 1)

2.42.3 APPLICATION TO COPPER PEROXO DIMERS:

DISTORTION OF THE Cu2O2 CORE

2.42.4 APPLICATION TO LINEAR AND BENT IRON(III)–IRON(III) OXO DIMERS (d5–d5)

2.42.5 EVALUATION OF U, Á, AND hdp FROM PHOTOELECTRON SPECTROSCOPY

2.42.6 SUMMARY AND CONCLUSION

2.42.7 REFERENCES



2.42.1



541

542

546

549

554

556

557



INTRODUCTION



Dimer formation from two mononuclear transition-metal units has important consequences for

spectroscopy.1 The most obvious example for this phenomenon is the appearance of highintensity, low-energy ‘‘dimer bands’’ in the UV–vis absorption spectra of binuclear copper2–4

and iron–oxo complexes.5–7 Several possible assignments for these features which are absent in

the spectra of corresponding monomeric complexes have been considered, including d ! d simultaneous pair excitations (SPEs) and low-energy charge-transfer (CT) transitions. With respect to

d–d spectra, intensity enhancement of ligand-field (LF) transitions of MnII,8–11 CrIII 12 and other

dimer combinations have also been well-known effects of dimer interactions. By analogy to the

ground state, energy splittings in the spin-flip LF excited states of these systems have been

analyzed in terms of a Heisenberg Hamiltonian. This formalism, however, does not account for

exchange interactions in LF excited states of orbital configurations different from that of the

ground state, which, e.g., applies to all LF states of CuII dimers. In this case the dimer splitting of

each excited state depends on additional interactions which probe individual superexchange

pathways between different combinations of d orbitals.13

CT spectra of transition-metal systems equally show evidence for binuclear interactions.14

Interest in this topic has been stimulated by the fact that metalloproteins often exhibit intense

bands at energies well below that of protein absorption, which are due to CT transitions from

ligands like oxide, sulfide, phenolate, thiolate, or peroxide to d-orbitals of the metal center(s) of

the active site. Apart from providing a characteristic spectroscopic fingerprint of the respective

metalloprotein in a specific physicochemical state, these transitions also sensitively probe

541



542



Valence Bond Configuration Interaction Model



metal-ligand bonding. Importantly, the ligand ! metal CT spectra of these ligands in mononuclear sites are clearly distinct from a situation where they act as a bridge between two or more

metal centers.15–17 In the past years, the correlation between binding geometry and CT spectrum

has been a major theme of bioinorganic electronic spectroscopy, an important aspect of these

studies having been to define how for a particular binding mode the CT spectrum of the monomer

(one metal center bound to the ligand) changes upon going to the dimer bridged by this ligand.

From group theory each monomer CT transition splits into two transitions in the dimer. The first

observation of this type of dimer splitting was made in a spectral comparison between an

end-on terminally azido coordinated CuII complex and the corresponding cis -1,3 azido

bridged dimer.18 If the dimer contains paramagnetic metal centers, each CT excited state in

addition exhibits a magnetic splitting due to coupling of a single electron in a ligand orbital

with these unpaired electrons in metal orbitals. In the case of CuII dimers, e.g., each excited

monomer state thus splits into four excited states, two singlets and two triplets. It has been found

that the CT singlet states apart from that splitting also exhibit a large shift to lower energy, which

has been interpreted as being due to a large antiferromagnetic interaction in the CT excited state.

This in fact has been found to be the origin of the ‘‘dimer bands,’’ i.e., these are assigned as

singlet–singlet CT bands shifting into the visible part of the spectrum upon dimer formation.4

The valence bond configuration interaction (VBCI) model provides a coherent explanation for

all of these spectroscopic phenomena in terms of a valence-bond (VB) description which in a

minimum basis comprises the highest occupied orbitals (HOMOs) of the metal and the bridging

ligand. From the various configurations within this active space, ground (GS), charge–transfer

(CT) and metal–metal-CT (MMCT) states can be formed which interact by a number of electron

transfer integrals (i.e., covalent interactions). This configuration interaction (CI) scheme leads to

the above-mentioned energy shifts and splittings of charge transfer transitions.19 In addition, the

VBCI model allows quantitative predictions regarding the magnetic coupling in the electronic

ground state, which is expressed by the sign and magnitude of À2J. More specifically, it allows

the evaluation of CT excited-state contributions to the magnetic coupling constant of the

electronic ground state and thus correlates excited-state effects with ground-state properties. This

allows us to directly experimentally probe the various superexchange pathways involved in the

ground-state magnetic coupling À2J. In the present short review, the VBCI model is developed

first on a prototypical example and then applied to peroxo-bridged CuII and FeIII–oxo dimers.

A final paragraph deals with the evaluation of model parameters from photoelectron spectroscopy and the relationship between the VBCI and cluster-CI models.



2.42.2



THE VBCI MODEL: COPPER(II) DIMERS (d 9– d 9, d 1– d 1)



For the sake of simplicity, the VBCI model is exemplified on a copper(II) dimer bridged by a

ligand L (Figure 1). Each S = 1/2 CuII center contains nine electrons, the unpaired electron on

each half of the dimer being in the highest, singly occupied orbital (SOMO), which is an

antibonding combination of a metal d orbital (e.g., dx2Ày2) with orbitals of the surrounding

ligands, including the p orbital of the bridging ligand that mediates the coupling between the

two metals. It has proven useful to express the electronic states of this system in a localized

valence bond (VB) basis, i.e., use dx2Ày2 type orbitals dA and dB centered on Cua and Cub; the

connection to a delocalized (dimer MO) basis will be made below. In the VB description, the S = 0

and 1 singlet and triplet ground state wavefunctions of the CuII dimer are given as

1



(a)



ẫGS





1

ẳ p

2







ỵ ỵ 

dA p p dB 







Cub



Cua



(b)







 ỵ ỵ

dA p p dB 







Cua



1aị



Cub



L



Figure 1 Copper dimer bridged by a ligand L. (a) ground state. (b) CT excited state. Relevant ligand and

metal orbitals are included.



543



Valence Bond Configuration Interaction Model

3



1

ẫGS

MS ẳ 0ị ẳ p

2









 ỵ

 ỵ 

 ỵ ỵ

dA p p dB  ỵ dA À



p

p

d

B









ð1bÞ



These two states transform according to two different irreducible representations of the point

group of the dimer as the spatial part of the triplet wavefunction is antisymmetric with respect to

interchange of the two electrons (i.e., halves of the dimer), and that of the singlet is symmetric.

The energies of these states are evaluated from the Hamiltonian

H ẳ



4

X

iẳ1





hiị



4

X



e2 =rij



2ị



i < j;i ẳ 1



^ to Equation (1a) and (1b) gives an energy

containing one- and two-electron contributions. Applying H

difference between these two states represented by the exchange integral JAB. In the VBCI model,

however, energy splittings due to two-electron integrals are in general neglected (except for ‘‘on-site’’

interactions, i.e., both electrons i and j on one Cu center; see below) and thus singlet and triplet ground

states are considered as degenerate in zeroth order (Figure 2, ‘‘zeroth order singlets and triplets’’).

Charge–transfer (CT) transitions are possible by excitation of an electron from the bridging

ligand orbital into the SOMO’s dA and dB leading to two locally excited (i.e., CuIA,B) conCT

II

ground state (Figure 1b).

figurations ÈCT

A and ÈB , the other center remaining in the Cu

I

II

These two Cu Cu mixed-valence configurations are strictly degenerate at an energy Á above

the ground state, which corresponds to the energy difference between the metal dA,B and ligand

p-orbitals. Removal of the electron from the doubly occupied bridging ligand p orbital leaves an

unpaired electron in this orbital behind which couples with the other unpaired electron on a CuII

center to form the locally excited singlet and triplet charge transfer states

1



1



3



1

ẳ p

2



 



 ỵ





ỵ ỵ

dA dA ỵ





p dB  dA dA p dB 





3aị



1

ẩCT

B ẳ p

2



 





ỵ ỵ  ỵ ỵ

dB dB p dA  À dB dB p dA 



 





ð3bÞ



ÈCT

A



1

ÈCT

A M S ẳ 0ị ẳ p

2











ỵ ỵ 

ỵ þ

d A d A p d B  þ d A d A À



p

d

B











ð3cÞ



J (CT)



dA



configuration



dB



J (GS)

Zeroth order

singlets

Antiferromagnetism



Zeroth order

triplets



Figure 2 VBCI scheme of a bridged dimer. Left: GS, CT, MMCT, and DCT configurations. Second and

last column: 0th order singlets and triplets. Center: Levels and energies after GS–CT configuration and CI

with MMCT and DCT states.



544



Valence Bond Configuration Interaction Model

3



ẩCT

B M S



1

ẳ 0ị ẳ p

2





 ỵ





d B d B ỵ

p d A  ỵ









ỵ ỵ

d B d B p d A 







3dị



ỵ/ combinations of these local CT configurations finally generate the following spin- and

symmetry-adapted CT states of the dimer

Á

1 1 CT

ẫCT

ẩA ỵ 1 ẩCT

ỵ ẳ p

B

2



4aị



1 1 CT 1 CT

ẫCT

ẩB ẩB

ẳ p

2



4bị





1

3 CT

ẫCT

ẳ p 3 ẩCT



A ỵ ẩB

2



4cị



1 3 CT 3 CT

ẫCT

ẩA ẩB

¼ pffiffiffi

2



ð4dÞ



1



1



3



3



Thus, as a result of spin coupling (two s = 1/2 = > S = 1, 0) and spatial degeneracy (ỵ/

combinations of local excitations), each CT state of a monomeric subunit (one CuII center)

corresponds to 2 Â 2 = 4 states in the CuII–CuII dimer. Out of the four CT states given in

Equation (4), two have the same spin and spatial symmetry of the singlet and triplet ground

states (Equation (1)), respectively, which leads to configuration interaction (CI)

D



1



D

pffiffiffi

pffiffiffi D   E

pffiffiffi

 1 GS E

 3 GS E

 ˆ

 ˆ

ÉCT

¼ 3 ÉCT

¼ 2 dA hˆ p ¼ 2 hdA p ẳ 2 hdB p ẳ hdp

ỵ H ẫỵ

H ẫ



5ị



^ which scales with

hdA p is a matrix element of the one-electron part hˆ of the dimer Hamiltonian H

the overlap between the metal dA or dB and the bridging ligand p orbital and thus is a measure of

covalency present in the dimer. Typically, this transfer matrix element is on the order of several

1 GS

3 CT

thousand wavenumbers.20 For reasons of symmetry, 1ÉCT

À does not interact with ẫỵ and ẫỵ

does not interact with 3ÉGS

;

in

addition,

singlets

do

not

interact

with

triplets.

Thus,

the

triplet

GS

À

and CT energies result from the zero-order energies 0 (GS) and Á (CT) and the 2 Â 2 secular

determinant

3 GS 3 CT









 E

hdp 





ẳ 0





 hdp E 



6ị



3 CT

2

3 CT

As a consequence, the triplet CT states 3 ẫCT

ỵ and ÉÀ split by an energy hdp /Á with ÉÀ

3 GS

being at higher energy than 3ẫCT

,

and



is

lowered

by

the

same

amount

(Figure

2,

second





column from right). The same applies to the singlet CT excited and ground states, respectively,

2

only with the energetic sequence of the CT states inverted and 1 ẫGS

ỵ lowered by hdp /. Thus, as a

1 GS

3 GS

result of the GSCT interaction, the two ground states ẫỵ and ÉÀ are still degenerate, but

lowered in energy by the metal-bridging ligand interaction (Figure 2, columns ‘‘CI GS–CT’’). This

energy is twice as large as for an isolated CuII monomer binding to the ligand which in the dimer

forms the bridge.20

The GS and CT states considered so far do not exhaust the possible configurations available

from the three-orbital (two d, one p) four-electron scheme. Starting from a CT configuration, the

unpaired electron localized at CuA or CuB may be transferred into the bridging ligand orbital p,

leading to two CuIIIL("#)Cu(I) states



Valence Bond Configuration Interaction Model

1



ẫMMCT





1

ẳ p

2





 

 ỵ



  ỵ ỵ 





dA dA ỵ

p p ặdB dB p p





545

7ị



The energy of these states is the so-called Mott–Hubbard U.21,22 Starting from the CuII–L("#)–CuII

ground state, this transition effectively corresponds to a metal ! metal charge transfer (MMCT).

In the VBCI scheme the MMCT transition, however, is not allowed in this direct manner since the

corresponding metal–metal transfer matrix element hAB is generally small and therefore neglected.

In contrast to the ground and CT configurations which generate singlet and triplet states, the

MMCT states (Equation (7)) are now inherently of singlet character (due to the Pauli principle,

there is no possibility of a parallel alignment of electrons, either on the metal or the ligand

orbital); therefore, these states only interact with singlet CT states and selectively stabilize these

states below the energy of their triplet counterparts (see below).

Finally, starting from a CT configuration, the second electron in the ligand orbital may also be

excited into a metal orbital, leading to the double CT (DCT) configuration CuI–L(&&)–CuI. This

now directly corresponds to a single electronic state which is a singlet and totally symmetric:

1







ỵ ỵ 

dA dA dB dB 

ẫDCT











8ị



The energy of this state is denoted by EDCT. Based on the singlet GS, CT, MMCT, and DCT

states and taking into account the respective configuration interaction matrix elements, the VBCI

determinants for the 1ẫỵ and 1ẫ states are given by

1



ẫGS







 E







 hdp







 0







 0



1



ẫCT





1



1 DCT

ẫMMCT

ẫỵ





hdp



0



E



hdp



hdp



U E



p

2hdp



0











p



2hdp 

ẳ 0





0







E

E

0



1



ẫCT







 E









 hdp



1



ẫMMCT





hdp 

ẳ 0





U E 



9ị



DCT



The determinant for the 3ẫ states is given by Equation (6); for 3ẫỵ only one state exists.

Solution of Equations (6) and (9) leads to the VBCI energy-level scheme represented by the two

central rows of Figure 2. Several important points should be observed from this energy diagram:

The two singlet CT states are greatly lowered with respect to the two triplet CT states, which

leads to a large stabilization of the singlet ! singlet CT transitions of bridging ligands in

dimers. As a measure of this antiferromagnetic interaction in the excited state (ESAF), it is

meaningful to take (as in the ground state) the energy difference between the 3ÉÀ and the 1ẫỵ

state, which is approximately given by

2 J CT  E



3









E 1 ẫCT



ẫCT







h2dp

U







h2dp

EDCT =2ị



10ị



The first contribution is due to CI between the 1ẫỵ CT state and the MMCT state (Equation (7)),

the second one to CI with the DCT state (Equation (8)).

The singlet ground state component is depressed in energy below the triplet state GS

component as it interacts with a singlet CT state that is admixed with a (singlet) MMCT

state and thus is at lower energy than the corresponding triplet CT state. In fourth-order

perturbation theory, JGS is given by



À 2 J GS  E



Â3



h4dp

h4dp

Ã

Â

Ã

À E 1 ÉGS

¼ 2 þ 2

ÉGS

þ

þ

Á U

Á ðE DCT =2Þ



ð11Þ



546



Valence Bond Configuration Interaction Model



Note that the GS coupling again contains AF contributions from the MMCT and DCT

states. This mechanism is referred to as superexchange. Equation (11), however, differs from

Anderson superexchange in that not only the first term (MMCT) contributes to À2JGS through

intermediacy of the CT state but also the DCT state.23 Further, with  being the amount of CT

character mixed into the ground state,

 ¼ À



hdp

Á



ð12Þ



and U À Á % U, the GS coupling (Equation (11)) is related to the CT excited-state coupling (10) by





2J GS ẳ 2 2J CT



13ị



With hdp being on the order of 103 cmÀ1 and Á on the order of 104 cmÀ1,  is on the order of

0.5–0.1, and therefore À2JGS is one to two orders of magnitude smaller than À2JCT. The p

orbital provides a superexchange pathway between dA and dB. Knowing À2JCT and  (or hdp)

for a specific CT state, the contribution of the corresponding superexchange pathway to the

coupling constant À2JGS of the electronic ground state can be determined. This is of particular

interest if there are several unpaired electrons per mononuclear site and therefore several

superexchange pathways (see Section 2.42.4).

Both singlet and triplet CT states are split, and this splitting can be determined experimentally by, e.g., optical absorption spectroscopy if both transitions from the respective GS spin

component to each of the split components of the CT state are electric-dipole (ED) allowed.

This is the case if the dimer does not have a center of symmetry. In that case the ỵ/ would

correspond to g/u (gerade/ungerade) combinations and only the g $ u transition would have

ED intensity.

The VBCI splitting of the triplet CT states, which is approximately given by hdp2/Á (vide

supra), corresponds to the HOMO/LUMO splitting of the dimer at the same level of approximation. This follows from consideration of the sym/antisym combinations of the copper

SOMOs and their interaction with the ligand HOMO generating one non-bonding and one

antibonding copper MO.19 In addition, the triplet CT state energies predicted by MO and VB

theory are identical. This allows one to obtain ab initio VBCI parameters hdp and Á from triplet

MO ground state or CT transition-state calculations (see below). Due to additional MMCT

and DCT interactions (Figure 2), the VBCI splitting of singlet CT states becomes smaller than

the limiting value hdp2/Á = ÁE(HOMO–LUMO) or can even be reversed with respect to the

order predicted by the prototypical VBCI scheme of Figure 2 (see next Section).



2.42.3



APPLICATION TO COPPER PEROXO DIMERS:

DISTORTION OF THE Cu2O2 CORE



The VBCI model has been employed for the interpretation of the optical absorption and CD

spectra of of peroxo bridged CuII dimers like oxy-Hemocyanin and corresponding small-molecule

analogs (Figure 3).15–17,24–26 Hemocyanin (Hc), the oxygen-transport protein of mollusks and

arthropods, contains a binuclear Cu(I) active site that reversibly binds dioxygen as peroxide in a

side-on bridging (–2:2) geometry (Scheme 1(A)).27 The highest occupied molecular orbitals of

peroxide are a doubly degenerate * set which split in energy upon bonding to a metal center: a

-bonding orbital within the plane of the metal–peroxo bond, *, (cf. Scheme 1(A)) and a

-bonding orbital vertical to this plane, *v. As - is much stronger than -bonding, the *

orbital is much lower in energy than the *v orbital, and the * ! dx2Ày2 CT transition is

expected to be at much higher energy than the *v ! dx2Ày2 transition. From overlap considerations

with respect to dx2Ày2, the intensity of the * will also be considerably higher than that of the *v

transition. In agreement with this qualitative picture, the two intense bands in the optical absorption

spectrum of oxy–Hc at 17,200 cmÀ1 (" = 1,000 MÀ1 cmÀ1) and 29,000 cmÀ1 (" = 20,000 MÀ1cmÀ1)

have been assigned to the ED allowed transitions to the *v and * CT states, respectively, and

the positive feature in the CD spectrum near 480 nm (21,000 cmÀ1; Á" = 1 MÀ1 cmÀ1) as the MD

allowed transition to the g component of the *v CT state (Figure 3).



547



Valence Bond Configuration Interaction Model



Figure 3 Optical absorption and CD spectrum of oxy-Hemocyanin and the model complex

[Cu2(N3PY2)O2]2ỵ. While the spectrum of oxy-Hc is typical for -2:2 Cu peroxo systems with an almost

planar Cu2O2 unit, the model complex has a bent side-on peroxo bridged (butterfly) structure (see text;

adapted from ref. 31).



π∗v



α



α



α



O

Cu



Cu



Cu



O



O



πσ∗



α



Cu



O



(a)



(b)

Scheme 1



In order to reproduce these transition energies with the VBCI model, parameters derived from

DFT calculations and spectroscopy were used.28 Initially, the active site of oxy-Hc was modeled

using a simplified [(NH3)4Cu2(O2)]2ỵ planar core (symmetry D2h), and the relative energies of the

MO triplet states were determined with SCF-X -SW transition-state calculations.29 From these

values the zeroth-order energies for the * CT (Á) and the *v CT states (Áv) as well as

the magnitude of the in-plane transfer element (hd) were calculated. The zeroth-order energy for

the MMCT state was assumed to be equal to the Mott–Hubbard energy U, which for CuII dimers

was found by PES to be 6.5 eV (cf. Section 2.42.5).30 In planar D2h symmetry, however, the outof-plane *v orbital has no overlap with the Cu dx2Ày2 orbitals, and hence the transfer matrix

element (hd)v between *v and dx2Ày2 vanishes; consequently, no splitting of the *v states is

predicted in this symmetry. In order to reproduce the experimentally observed splitting of these

states, the actual dimer geometry has to be considered. Inclusion of the transaxial ligands lowers

the symmetry to C2h, and due to the pyramidal coordination the quantization axis of each copper

center is tilted with respect to the Cu2O2 plane ( in Scheme 1(A)). This leads to a nonzero value

of the matrix element (hd)v. An estimate of its magnitude was obtained from the value of (hd)

and the relative, experimentally determined oscillator strengths of the * and *v absorption bands.



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