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41 Molecular Orbital Theory (SCF Methods and Active Space SCF)

41 Molecular Orbital Theory (SCF Methods and Active Space SCF)

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Angular Overlap Model


Figure 4 Metal–ligand distances and e parameter values (in 10À3 cmÀ1) for Ni(II) amine complexes as

fitted from d–d spectra (stars, adopted from McClure26) and calculated by using overlap integrals (circles).

hindrance connected with the dialkylamino group in the order NH2, NHEt, NMe2, NEt2 ligands,

the NiÀN bond length gets longer and this is nicely reflected in decreasing values of the

parameter e. In Figure 4 we also plot values of e deduced by a scaling according to square of

the NiÀN overlap integral (calculated using a sp3 hybrid function for N) where we take

e ¼ 3583 cmÀ1 and R ¼ 2.15 A˚ (for trans-[Ni(NH3)4(NCS)2]) as reference. The data show that

for the NiÀN bond the values of e deduced from the spectra are consistent with Equation (2),

the behavior at larger distances being possibly influenced by additional longer-range crystal field

(ionic) contributions. The e parameters for the rhombic complexes also agree with those found

for the higher symmetry species: i.e., the notion of AOM parameter transferability seems to be

justified here.

Non-orthoaxial Complexes

The problem of parameter redundancy that might appear in high-symmetry complexes is

weakened when the coordination geometry deviates from orthoaxiality.8,29 Then, additional offdiagonal matrix elements appear that depend on electronic (AOM) and on geometrical parameters which are due to the angular geometry of the chromophore. Thus, insofar as the molecular

structure is known, AOM parameters can be derived from the analysis of d–d transitions. On the

other hand, the angular geometry of the chromophore can be deduced if sufficient spectroscopic

data are available (spectrum–structure correlations). An example for the latter case is given by the

spectroscopy on the [Cr(ox)3]3À impurity centers in CrIII-doped NaMg[Al(ox)3]Á8H2O.30 The

electronic spectrum in high-resolution including ground 4A2g and excited 2Eg zero-field splitting

could be rationalized only if a trigonal twist angle of 14 and, in addition, a trigonal compression

of the CrO6 polyhedron ( ¼ 4 ) were taken into account. Along the same lines, an interpretation

of the spectra of Mn5ỵ(d 2 ) doped into spodiosite and apatite oxidic host structures showed that

the geometry of the tetrahedral sites is significantly modified by Mn5ỵ when incorporated into

positions of P5ỵ with a smaller ionic radius.31 In a series of papers, Gerloch et al. have shown that

an accurate fit of AOM parameters is also possible by taking into account EPR and magnetic

susceptibility data.32–35 As has been argued earlier,28 electronic transition energies deduced from


Angular Overlap Model

spectral data allow one to deduce at most four parameters from four orbital energy differences in

ligand fields of low symmetry. It is noted, however, that the 5 Â 5 ligand field matrix depends on a

total of 14 different diagonal and off-diagonal matrix elements. All of these in low symmetry

affect the orbital and many-electronic wave functions. These matrix elements will also influence

the expectation values of the Zeeman and the orbital moment operators when applied on these

wavefunctions, and thus the g- and magnetic susceptibility tensors. The latter observables can be

used in turn to deduce a larger number of AOM parameters. The formalism behind this strategy

has been elaborated by Gerloch et al.32–35 A study on the optical spectra, the g- and magnetic

susceptibility tensor in tetrakis(diphenylmethylarsine oxide) nitrato cobalt(II) and nickel(II) serves

as a typical and well-documented example where as many as eight AOM parameters could be

derived from the spectra along with equations which, however, show parameter correlations (two

for Co and three for Ni).36 Further useful information can be obtained from EPR spectra, an

example for that is given in ref. 37.

Stereochemistry and Reactivity

Structural predictions on the basis of LFSEs as calculated by using the AOM have been pioneered

by Burdett and others.38–40 An instructive implication for thermochemical substitution reactions

in square planar complexes was given by Lee.41 He calculated the activation energy which

accompanies the substitution of X by Y in the square pyramidal trans-[ML2TX] complex. Y

enters the coordination sphere at the axial position to form a square pyramid, which subsequently

reorients toward a trigonal bipyramid with X, Y, and T forming the trigonal plane. A simple

expression for the activation energy Ea in terms of -bond contributions and considering -back


Ea  DLFSE ẳ eT ỵ eX eL 1=2ịeY ỵ 4e T eT ị


has led to the following conclusions: trans-ligand T activities are in the reverse order of their

corresponding 10Dq values, i.e., IÀ > BrÀ > ClÀ > FÀ > OHÀ > NH3. Like 10Dq, eT values

decrease with increasing trans-activity. The trans-ligand T has the effect of reducing the values

of eX, and more effective T-ligands are those which also generate smaller eT values. Both effects

tend to decrease the activation energy Ea. The entering group Y contributes only À(1/2)eY and

tends also to decrease Ea as well. This is borne out by the fact that the nucleophilicity of amines

with respect to four-coordinate planar d 8 metals increases with increasing amine basicity.42

Finally, ligands T for which -back donation is operative tend to further reduce the activation

energy. In such cases e*T and eT are both negative, but |e*T| > |eT| leads to a negative

4(e*T À eT) contribution to Ea (Equation (5)). The latter result is also supported by more

sophisticated calculations,43 which show that if -bonding is possible in the transition state, it

dictates the reaction path.


The photochemical substitution in octahedrally coordinated complexes is usually governed by

Adamson’s rules:44 (i) the leaving ligand is located on the axis characterized by the weakest ligand

field; (ii) on the labilized axis, the leaving ligand is the one possessing the stronger ligand field. A

detailed AOM treatment that gives a rationale of both rules and goes further to cover exceptions

to these rules has been given by Vanquickenborne and Ceulemans.45,46 Their model approximates

the bonding energy of a given ligand in the ground state (I) and in the photoactive excited state

(I*) by summing up the orbital energies of occupied bonding and antibonding orbitals. Assuming

that bonding orbitals (mostly ligand centered) are stabilized to the same extent as the antibonding

ones (mostly 3d ) are destabilized, it follows that I and I* are just given by the number and

distribution of holes between the metal 3d-orbitals in a given electronic state. Instructive examples

are trans-[Cr(Leq)4(Lax)2] complexes. With a t2g3eg4 hole-configuration in the 4A2g ground state of

Cr(III), the bonding energies of Leq and Lax are calculated to be I(Lax) ẳ 2eax ỵ 2eax, and

I(Leq) ẳ 2eeq ỵ 2eeq, respectively. As expected, thermochemical reactions will lead to substitution

of the more weakly bound ligand. This can be easily judged based on reported AOM parameter

values for CrIII (see Figure 3). On the other hand, upon excitation the photoactive state has to be


Angular Overlap Model

considered, which usually is the lowest excited state. In tetragonal symmetry this depends on the

splitting of the 4T2g excited state. The state sequence can be derived from the ligand field

parameters involved:






Eð4 B2g Þ À Eð4 Eg ị ẳ 2eax

 e ị 3=2ịe e ị ẳ 1=2ị10Dq 10Dq ị


If one assumes 10Dqax < 10Dqeq, 4Eg becomes the photoactive state, otherwise 4B2g.

An inspection of the wave functions of 4Eg and 4B2g shows that a 4A2g ! 4Eg excitation corresponds to a dzx,yz1 ! (3/4) (dz21) ỵ (1/4) (dx2 y21) excitation, while 4A2g ! 4B2g is due to a

dxy1 ! dx2 À y21 excitation. Configurational mixing with higher 4Eg states increases (decreases) the

relative amount of dz2 (dx2 À y2) in 4Eg. Energy expressions for I*(Lax) and I*(Leq) in dependence on

the e and e parameters are listed in Table 1 for the excitations from dxy or dzx,yz into dz2 and

dx2 À y2. With 4B2g as the photoactive state (dxy1 ! dx2 À y21) the weakening of the CrÀLeq bond is

proportional to 10Dqeq, while the CrÀLax bond is not altered.



IðM À Leq Þ À I M Leq ị ẳ 1=4ị3eeq

 4e ị ¼ ð5=2ÞDq


With 4Eg as the photoactive state, the labilization of Leq and Lax is given by



IðM À Leq Þ À I M Leq ị ẳ 1=8ị3eeq

 4e Þ ¼ ð5=4ÞDq




IðM À Lax Þ À I Ã ðM Lax ị ẳ 1=4ị3eax

 4e ị ẳ 5=2ịDq


It follows that for a given value of 10Dq the CrÀLax bond absorbs twice as much of the

excitation energy as the CrÀLeq bond does. This difference increases upon configurational

mixing. If the quantity (I–I*) is the crucial parameter for the photo-substitution, Equations (8a)

and (8b) lead to Adamson’s rules: if 10Dqax > 10Dqeq, only the equatorial ligands are labilized

according to Equation (7). In the opposite case, i.e., 10Dqax < 10Dqeq, these equations show that

axial ligands are labilized at least twice as much as the equatorial ligands. This is the first of

Adamson’s rules. However, in either case, the labilization is proportional to the 10Dq value of the

ligand—this is equivalent to his second rule. Although deviations from this behavior might occur

in certain cases, correct predictions for the leaving ligand have been made for a series of trivalent

chromium and cobalt complexes.45

The leaving ligand treatment within the AOM has been extended46 to the following steps of the

photoisomerization process—the isomerization of the resulting five-coordinate species, followed

by the nucleophilic attack of the entering ligand. The electronic structure control of the latter two

steps was based on state correlation diagrams, but nicely translated into orbital pictures of the

Woodward–Hoffman rules of conservation of orbital symmetry.



s–d Mixing

When the symmetry is sufficiently low, some of the 3d-orbitals may become totally symmetric and

start to mix with 4s. This leads to a depression of the energies of those d-orbitals involved in such

Table 1 AOM expressions for the bonding energies of the Lax and Leq ligands in trans-[Cr(Leq)4(Lax)2]

complexes of CrIII in the ground state (I) and in the eg ! t2g singly excited configuration (I*). Ligand

labilization energy expressions (I–I*) are listed in parentheses.

Ligand ground state



2eax ỵ 2eax dxy!


2eeq ỵ 2eeq dxy!


! dz2

! dx2 y2

eax ỵ 2eax(eax)

eax ỵ 3eax(eaxeax)

7=4eeq ỵ 3eeq(1=4eeq eeq)

7=4eeq ỵ (5/2)eeq(1=4eeq (1/2)eeq)

2eax ỵ 2eax(0)

2eax ỵ 3eax(eax)

5=4eeq ỵ 3eeq(3=4eeq eeq)

5=4eeq ỵ (5/2)eeq(3=4eeq (1/2)eeq)


Angular Overlap Model

a mixing. A way of accounting for this effect is the extension of the nd basis to include the

(n ỵ 1)s-orbital. Then the AOM matrix element of Equation (1) must be supplemented with47,48



V sd

Slị Fj l ; ẩl ịẵesd lị1=2

ij ẳ Slị Fi l ; ẩl ịẵesd lị


where esd(l) represents the parameter for s–d mixing. It can be approximated by perturbation

theory as

esd lị <(n ỵ 1)sjVl jnd z2 >2 =j"ndz2 "(n ỵ 1)s j


The effect of sd mixing is very pronounced in square planar MA4 and in linear MA2 systems

and has been extensively studied for CuII and PtII complexes.49,50 For example, in the case of

quadratic [CuCl4]2À, having a single hole in the d subshell, AOM expressions for the transition

energies are readily obtained (see Table 2). Calculated values of e, e, and esd (5,350 cmÀ1,

1,200 cmÀ1, and 1,400 cmÀ1) are very close to those deduced from the spectrum of pseudotetrahedral [CuCl4]2À and imply a nice parameter transferability between the two geometries.51,52

The large effect of s–d mixing is reflected here by the relatively high energy of the 2A1g excited state.

Electronic spectra of linear CuCl253 and NiCl2,53,54 supported by later first principles calculations,55,56 show that the energy of the dz2 orbital (¼ e À 4esd) is very low, even lower than that of

the dzx,dyz orbitals (2e). This is again a manifestation of considerable s–d mixing. The resulting

dxy,x2Ày2 dz2 < dzx,dyz orbital sequence is consistent with a 2Å and with a 3S ground state for

CuX2 (X ¼ F, Cl) and NiCl2, respectively. Further examples of s–d mixing have been reported

with the unique paramagnetic NiO4 chromophore in Li2NiO257 and the NiO2N2 chromophore in

Ni[t-Bu2P(O¼NR)]2.58 In the latter case, alternative interpretations of the observed high-spin

ground state, which is very unusual for planar Ni(II) complexes, have been extensively discussed.59–61 Optical spectra of the linear chromophores NiO22À in K2[NiO2]62 and NiO23À in

K3[NiO2] and KNa2[NiO2]63 are other key examples illustrating the role of s–d mixing. In the

latter compounds a dz2> dzx,ddyz> dx2 À y2 order of d-orbital energies and 3Åg [NiII(d 8)] and 2Sg

[NiI(d 9)] ground states have been proposed based on optical and EPR spectra for NiI(d 9).62,63

For a review on this topic see ref. 64.

Orbital Phase Coupling (Orgel Effect)

Frontier orbitals in chelate ligands are frequently part of an extended -electron network. In such

cases perturbations from bonding and antibonding delocalized ligand -MOs, rather than separate

and independent ligand -functions, govern the respective LF potential. This effect was predicted

by Orgel early on65 and was first incorporated into the AOM by Ceulemans et al.66 Following

Atanasov et al.67 we consider a bidentate ligand L–L with in-phase ( ) and out-of-phase ()

combinations of the (L) orbitals (see Figure 5). In order to get appropriate metal-d functions of

the same symmetry (b1 or a2 in C2v), linear combinations according to Table 3 have to be taken,

which result in different parameters es and es0 for the respective three-center interactions (nonadditive model).

Within the new basis (d , d) the AOM matrix V s0 is diagonal, but the formalism requires a

representation in the initial basis. This can be easily obtained by a simple transformation, i.e.,

V s ¼ T À1 V s0 T (cf. Equation (7) in ref. 67). If we set es ¼ es0 , the degeneracy of the d-orbitals

restores the conventional (additive) model assuming holohedrized D4h symmetry.

Table 2 LF states and transition energies of square-planar [CuCl4]2À.

State energy

E(2B1g) ẳ 3e

E(2A1g) ẳ e ỵ 4esd

E(2Eg) ẳ 2e

E(2B2g) ẳ 4e

Transition energy

Exp. (cm1)51


2e ỵ 4esd

3e 2e

3e 4e





Angular Overlap Model

Figure 5 Parametrization for M(L–L) chelates with an extended -electron system (notation according to

local C2v symmetry) (adapted from Schoănherr,8 Figure 7).

Table 3 Symmetry adapted wave functions and interaction

parameters in a M(L–L) system for phase-coupled frontier

orbitals of -type.






M–L interaction

d ¼ (1/2) (dzx þ dyz)

d ¼ (1/2) (dzx À dyz)



For orthoaxial tris-chelate complexes the trigonal AOM matrix in the basis dxy, dyz, dzx is

readily derived by cyclic permutation of x, y, and z leading to


es ỵ es 0 ị


2 es

es 0 ị


2 es

es 0 ị






V trig

es ỵ es 0 ị

12 es es 0 ị 7

 ẳ 4 2 es es ị






















An inspection of Equation (11) shows that the phase-coupling effect leads to a splitting of the

t2g orbital subset with Dt2 (¼ "eg À "a1g) ¼ (3/2)(es À es0 ). Using the concept of phase-coupling,

the polarized electronic spectrum of [Cr(acac)3] (acac: acetylacetonate) could be rationalized,

which shows a distinct splitting of 800 cmÀ1 of the 4A2g ! 4T2g transition with an energy order


A1 < 4E.67,68 It should be noted that the geometry of the CrO6 chromophore is very close to an

octahedron in this complex, and the small deviations toward a trigonal elongation would lead to

much smaller splitting with opposite level ordering. However, since the acac ligand has a HOMO

and a LUMO of - and -type, respectively, Dt2 is positive, resulting in the correct level scheme.

Taking the value of Dt2 as deduced from the spin-allowed transitions, the magnitude and the sign

of the excited state 2Eg splitting of 220–290 cmÀ1, as reported from the spectra of the Cr3ỵ- doped

Ga and Al host lattices, could also be reproduced.68 Recent ab initio calculations lend full support

to this first manifestation of the Orgel effect.69 The system [Cr(bpy)3]3ỵ presents an example in

which the ordering of HOMO () and LUMO ( ) frontier orbitals are just opposite to acac.70

Further examples, e.g., [Co(C10H20N8)]2ỵ, [Co(salen)], and [Co(amben)], which display the effect

of different types of phase-coupling on the order of energies of -orbitals, and therefore anisotropy of the g-tensor, can be found in Ceulemans et al.71 A detailed overview on the treatment of

conjugated bidentate ligands within the parametrization of the AOM has been given by Schaăffer

and Yamatera.72


Angular Overlap Model

Misdirected Ligand Lone Pairs

Metal–ligand bonding is usually such that it allows a decomposition of the M–L interaction into

components of -and -type. This implies a pseudo-symmetry of the electron density around a given

MÀL bond that is not lower than C2v (cf. Figure 1). A lowering of this symmetry to Cs or C1 may be

due to rigidity of a chelate ligand or to nonbonding lone pairs on a ligator atom. The misalignment

of ligand orbitals in such cases (misdirected valency) precludes a full separation of  and  metal–

ligand bonding. Following an idea of Liehr,73 the concept of misdirected valency was introduced

into the AOM by Gerloch et al.74–77 who used this model to explain the band splitting in planar CoII

and trigonal–bipyramidal Ni(II) complexes. In Figure 6 we illustrate this using a ligand orbital that

is misplaced by the angle  with respect to the M–L axis () in the chelate plane, and further by the

angle ! out of this plane. With such low M–L symmetry (C1), the angle  causes an off-diagonal

AOM matrix element ec between dz2 and dzx, while the angle ! introduces es between dz2 and dyz.

Thus the initial local AOM matrix, separating contributions of - and -type, does not remain

diagonal. A case study for a symmetric tris-chelate complex is given elsewhere.68 An illustrative

example is provided by NiÀNCS bonding, which has been thoroughly discussed by Gerloch et al.75,76

Surprisingly, the NCS ligand has been suggested to be a -donor in trans-[Ni(NH3)4(NCS)2]

(e > 0),78 but a -acceptor in trans-[Ni(en)2(NCS)2] (e < 0).28 However, accounting for the large

deviations from the linear Ni–N–CS arrangement in the [en] complex, the effect of bent bonding due

to the lone pair on N should result in a non-zero value for the e(NCS) parameter if spz-hybrids of

the free linear N–CS ligand, rather than a sp2-rehybridization, are assumed. It was shown in Duer et al.77

that accounting for the misaligned bonds in the [en] complex by using one additional e(NCS)

parameter, a positive value for e is derived from the spectral fit. Thus the -donor effect of NCSÀ

in the [en] complex is restored. Another impressive manifestation of misaligned valence orbitals

was reported by Dubicki et al. who investigated H2O bonding by analyzing the zero field splitting

of [Cr(H2O)6]3ỵ in the CsCr(SeO4)2Á12H2O crystal.37,79 Misdirected M—L bonds can also be

expected, for example, in TM-doped lattices.8,80 Due to highly charged cations in the second

coordination sphere of oxidic structures, this effect was manifested in spectra and bonding

parameters of coordination polyhedra of NiII and CuII.80 This is the main topic of Chapter 1.36.

Higher Coordination Spheres

Ligands (or atoms) extending beyond the first coordination sphere of TM act as electrostatic perturbers,

covalent forces being negligibly small at such large distances. In an AOM description, crystal field

contributions to the parameters e and e are assumed to decrease according to an a (1/R3) ỵ b (1/R5)

law.14 Interesting examples are the fine structures observed in the intraconfigurational transitions of

K3[Cr(CN)6],14 of CrIII in the spectrum of ruby,81 and of the MnVI ion doped in Cs2SO4.82 All these

systems exhibit an electrostatic influence from the asymmetric cationic surroundings of the chromophores

due to the outer ions Kỵ, Al3ỵ, or Csỵ, respectively. Recently, the contribution of the second coordination

sphere in CrV-doped YVO4 and YPO4 solids has been manifested both spectroscopically and by DFT.83

Vibronic Coupling

Vibronic coupling in TM complexes, such as the Jahn–Teller effect in electronically degenerate

states in octahedral or tetrahedral symmetries, leads to geometrical distortions resulting in lower


Figure 6

Misdirected valency: Displacement of ligand frontier orbital from ML bonding axis (adapted

from Schoănherr,8 Figure 8).

Angular Overlap Model


symmetric polyhedra. Bacci has proposed an interesting application of the AOM to evaluate

vibronic coupling constants.84,85 Later on, the first and second order vibronic coupling constants

were used to formulate the epikernel principle of Jahn–Teller coupling.86 The use of vibronic

coupling constants within the AOM formalism in order to explain geometrical structures and

electronic spectra in Jahn–Teller distorted tetrahedra of Ni(II) and Cu(II) is reviewed elsewhere.87

Polynuclear TM Systems

An extension of the AOM to polynuclear TM complexes without direct M—M bonds has been

given.88,89 Atanasov and Schmidtke have described a generalization to systems involving TM—

TM bonds.90 An application to the parametrization of exchange coupling constants within the

Anderson model of superexchange led to a quantification of the Goodenough–Kanamori rules.91

AOM parameters deduced from the spectra involve not only covalent (overlap) but also ionic

(crystal field) terms. Complexes of TM in their oxidation states ỵ2 and ỵ3 with ligands such as

F and O2À can be considered to a high degree as ionic, in contrast to ligands such as CNÀ. Thus

the large values of the AOM parameters for the former ligands may reflect crystal field contributions to the ligand field in addition to the covalent (overlap) energy. While multiplet energies in

TM complexes are affected both by ionicity and metal–ligand overlap, magnetic exchange

coupling between magnetic TM ions mediated via bridging ligands mirrors solely TM–L covalency. The use of phenomena such as exchange coupling and hyperfine transferred magnetic fields

to deduce the covalent terms of the ligand field metal–ligand interactions utilizing an AOM

parametrization of extended structures has been illustrated for KAlCuF6.92 A recent critical

review of this topic can be found in Schaăffer.93




(A1) Meaningful parametrization: A chemically meaningful interpretation of bonding parameters

deduced from spectroscopy permits one to explore chemical trends. As an example, the twodimensional spectrochemical series serves as a means to classify ligands according to their  and 

bonding behavior.

(A2) Low symmetry: The local structure of the covalent part of the LF potential allows for an

additive scheme of contributions from various ligands. Therefore, the AOM is designed to treat

isolated complexes of low symmetry as well as dissolved complexes, doped materials, etc. This

contrasts with the global character of the CF parametrization.

(A3) Accuracy: In particular cases multiplet splitting can be rationalized in high precision, e.g.,

energy separations of some ten wave numbers or much less (zero field splittings in EPR) have

been correctly reproduced. Thereby, even slight angular distortions of the chromophore from

higher symmetric arrangements can be derived (spectrum–structure correlations). This is far

beyond the ability of any other quantum chemical method up to date.

(A4) Flexibility: Peculiarities in the electronic structures of the TM or the ligands require specific

extensions of the relatively simple framework of the conventional AOM which can be easily

derived within the underlying MO approach. Moreover, the potential to treat multiplet splitting

of mononuclear TM complexes as well as exchange coupling in bridged TM complexes (which

cannot be treated in CF theory) opens new perspectives for applications of the AOM.


(D1) Limited basis: The power of the AOM is to calculate energy splitting of d n-states, while, on

the other hand, absolute energies cannot be derived. Because the set of explicitly used wave

functions is limited to the five d-orbitals, there is no reliable access to charge transfer states, which

generally affect the energy of d n-states.94 The latter also influence greatly the spectral intensities

of d–d transitions, which, therefore, cannot be reasonably predicted within the framework of the

LFT, although some effort has been undertaken for special95 and more general cases.96


Angular Overlap Model

(D2) Over-parametrization: The more ligator atoms are involved in mixed-ligand complexes the

larger is the number of AOM parameters to be determined with the inherent uncertainty in their

determination from experiment. This may lead to an over-parametrized scheme, in particular

when experimental data are scarce.






























































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transmitted in any form or by any means electronic, electrostatic, magnetic tape,

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from the publishers

Comprehensive Coordination Chemistry II

ISBN (set): 0-08-0437486

Volume 2, (ISBN 0-08-0443249); pp 443–455


Molecular Mechanics


University of Warwick, Coventry, UK





2.37.5 STRUCTURE Macrocycle Hole Size Metal Hydrides and Alkyls Transition States





















At the start of the 1980s, the theoretical methods for computing molecular structures of

coordination compounds were severely limited. Ab initio methods based on the Hartree-Fock

approximation and its extensions were too expensive and/or inaccurate,1 molecular density

functional theory (DFT) was still in its infancy2 and extended Huăckel theory (EHT), although

very successful at generating the molecular orbital structure, ignored the interactions necessary for

computing sensible bond lengths. In contrast, a new method had become popular for ‘‘organic’’

molecules, which was both extremely computationally efficient and also capable of reliable

accuracy. The method ignored quantum effects and was instead based on the much simpler

classical equations for bond stretching, angle bending, etc. This molecular mechanics (MM)

method was fast enough to enable all the internal degrees of freedom to be optimized simultaneously.3 After some earlier forays into CoIII chemistry, by the mid-1980s, inorganic chemists

took up the challenge and set about implementing MM for transition metal (TM) species.



The essence of MM is the strain energy Utotal that, in its simplest form, consists of four energy


U total ẳ


Er ỵ


E ỵ


E ỵ







Er is the total bond deformation energy,

E is the total angle deformation energy,



E is the total torsion deformation energy, Enb is the total nonbonded (van der Waals and

electrostatic) interaction energy, and the summations run over all in relevant internal coordinates.



Molecular Mechanics

The individual energy terms are described using simple expressions. For example, the bond stretch

term Er for the bond between atoms i and j of length rij can be treated via a simple Hooke’s law


Er ¼ 12 kr ðrij À r0 Þ2


where kr is a force constant and r0 is the ‘‘ideal’’ distance between the relevant atomic centers.

Similarly, the angle bending potential energy can be described by a harmonic function:

E ¼ 12 k ðijk À 0 Þ2


where k is the force constant for the angle ijk made by atoms i, j, and k and 0 is its ‘‘ideal’’

value. The periodicity of torsion angles requires a more complex equation:

E ẳ 12 k ẵ1 ỵ cosfmijkl ỵ offset ÞgŠ


where k is the height of the rotation barrier about the torsion angle ijkl, m is the periodicity,

and offset is the offset of the minimum energy from a staggered conformation. Nonbonded

van der Waals interactions can be computed from, for example, a Lennard-Jones 612


Enb vdWị ẳ




d 12





where dij is the distance between atoms i and j, and A and B are derived from atom-based

parameters. Nonbonded electrostatic interactions, Enb(elec), can be calculated via Coulombs law:

Enb elecị ẳ

qi qj

" dij


where qi and qj are the partial charges on atoms i and j and " is the dielectric constant.

In addition to these basic energy terms, a number of additional/alternative terms can be

considered: for example, a Morse function for bond length distortions, nonbonding terms for

L–M–L angle bending,4 out-of-plane deformation terms to enforce planarity in aromatic or sp2hybridized systems, explicit terms for hydrogen bonding, cross terms linking bond lengths to

angle bends, electronic terms to represent the ligand field stabilization energy,5 and so on.

The collection of energy terms and their associated parameters is often referred to as the force

field (FF).



The advantage of a FF approach is the high computational efficiency, which allows large systems

and/or many smaller systems (e.g., conformational searches, virtual high-throughput screening) to

be treated in a reasonable time. The main disadvantage of the FF is that the results are only as

good as the parameters. In particular, FF parameters developed for one class of compound may

not be transferable to another. Likewise, parameters developed with respect to a particular strain

energy expression will not in general be applicable to any other.

These issues are discussed by Comba and Hambley6 who also provide a detailed account of

how to derive FF parameters as do Norrby and Brandt.7 Computer programs which have been

applied to modeling various coordination compounds include versions of MM2 and MM3,8

SHAPES,9 and modified versions of MacroModel,10 CHARMM,11 AMBER,12 MOMEC,6

and DOMMINO.13 Most FFs in coordination chemistry employ MM214–16 or AMBER,12

or something based on these, to treat the ‘‘organic’’ parts of the molecule which may be

modified to account for the effects of binding to a metal cation.17 All the above schemes rely

on an extensive table of FF parameters spanning all the various combinations of bond, angles

torsions, etc.

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