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Geometry Optimization of MetalComplexes Using Natural InternalCoordinates: Representation of SkeletalDegrees of Freedom

´ATTILA BERCESThe Steacie Institute for Molecular Sciences, National Research Council, 100 Sussex Drive,Ottawa, Ontario K1A 0R6, Canada

Received 16 November 1995; accepted 19 March 1996

ABSTRACT

The geometry optimization using natural internal coordinates was applied fortransition metal complexes. The original definitions were extended here for theskeletal degrees of freedom which are related to the translational and rotationaldisplacements of the h n-bonded ligands. We suggest definitions for skeletalcoordinates of h n-bonded small unsaturated rings and chains. The performanceof geometry optimizations using the suggested coordinates were tested onvarious conformers of 14 complexes. Consideration was given to alternativerepresentations of the skeletal internal coordinates, and the performance ofoptimization is compared. Using the skeletal internal coordinates presentedhere, most transition metal complexes were optimized between 10 and 20geometry optimization cycles in spite of the usually poor starting geometry andcrude approximation for the Hessian. We also optimized the geometry of somecomplexes in Cartesian coordinates using the Hessian from a parametrizedredundant force field. We found that it took between two and three times asmany iterations to reach convergence in Cartesian coordinates than usingnatural internal coordinates. Q 1997 by John Wiley & Sons, Inc.

Introduction

ne of the most frequently calculated molecu-O lar properties performed by quantum me-chanical methods is geometry. The cost of these

E-mail: [email protected]

calculations greatly depends on the efficiency ofthe optimization algorithm and the choice of opti-mization variables. Recently, in this area of re-search the emphasis has been shifted toward find-ing the most suitable optimization variables.1, 2

Forgarasi et al. and Pulay and coworkers showedthat appropriately chosen internal coordinates canbe ideal optimization variables.1, 3 These natural

( )Journal of Computational Chemistry, Vol. 18, No. 1, 45]55 1997Q 1997 by John Wiley & Sons CCC 0192-8651 / 97 / 010045-11

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internal coordinates reduce both harmonic and an-harmonic coupling between coordinates to a mini-mum based purely on the topology of themolecules. The small harmonic coupling ensuresthat the method is successful even with a highlyapproximate Hessian. The small anharmonic cou-pling plays a significant role when the startingstructure is poor and large displacements arerequired to reach the energy minimum. Thesequalities make the natural internal coordinate opti-mization especially appropriate for the location ofequilibrium geometries for transition metal com-plexes. For these systems, neither the starting ge-ometry nor the Hessian can be estimated fairlyaccurately the way it could be done for organicmolecules based on molecular mechanics methods.However, for the application of natural internalcoordinates for metal complexes one has to con-sider the representation of certain characteristicinternal displacements of coordination com-pounds.

The internal coordinates of coordination com-pounds with h n-bonds fall into two categories:ligand coordinates and skeletal coordinates. Theskeletal movements represent the displacement ofan entire ligand with respect to the central atom.Some examples of these internal motions are lig-and]metal stretching, ligand tilt, and ligand inter-nal rotation, skeletal bending, or skeletal torsion ofthe molecule. The ligand coordinates, the regularbond stretches and deformations, are suitably rep-resented by the recommendation of Forgarasi etal.1 However, the natural internal coordinate rep-resentation of skeletal internal movements waspreviously not considered until very recently. Westudied the force fields of ferrocene, dibenzene-chromium, and benzene-chromium-tricarbonyl,and compared the force constants to that of thefree Cp and benzene rings.4 This comparison re-quired a physically meaningful definition of theskeletal internal movements. We studied a fewdifferent possibilities and found that the most suit-able representation involves internal coordinatesexpressed with the help of the centroid of theh n-bonded ligands. These coordinates providedsmall coupling force constants between the skele-tal and other coordinates. Usually the internal co-ordinate system that is appropriate for the normalcoordinate analysis, also performs well in geome-try optimization. Here we discuss the implementa-tion of these internal coordinates in geometry opti-mization and address the problem of optimizationof metal complexes in general. Figure 1 shows the

structures of metal complexes considered in thepresent study.

Computational Details

The reported calculations were carried out us-Ž .ing the Amsterdam density functional ADF pro-

gram system developed by Baerends and col-leagues5a and vectorized by Ravenek.5b Boerrigteret al.6 developed the numerical integration proce-dure. All optimized geometries in this study werecalculated based on the local density approxima-

Ž . 7ation LDA . Here, we are concerned with theperformance of geometry optimization, rather thanthe performance of the density functional calcula-tions in terms that reproduce experimental results.To obtain better agreement with experiment, gra-dient corrections to the exchange7b and correlation7c

potentials should have been included in theHamiltonian. The geometries were optimizedbased on the GDIIS technique8 using natural inter-nal coordinates. We interfaced the ADF programwith the GDIIS program4, 9 and implemented theskeletal coordinates discussed here.

Physical Nature of Skeletal Degreesof Freedom

From a topological point of view, there are twotypes of bonds between metals and organicmolecules. If the metal is connected to only onecarbon atom of a ligand, a monohapto bond isformed that is topologically very similar to thebonds of regular organic molecules. Accordingly,the definition of internal coordinates is also similarto that of organic molecules. On the other hand,the metal atom can bind to an organic ligandthrough several carbon atoms, forming a multicen-ter h n-bond. The most well-known examples ofthis bonding situation are the sandwich-type tran-sition metal complexes such as ferrocene anddibenzene-chromium. There are unique internalmotions associated with the h n-bonds that are dif-ferent from those of regular bonds. Consequently,to describe these internal motions, one has to useunusual internal coordinate representations. Inh n-bonded metal complexes the new types of in-ternal motions are the movement of a rigid frag-ment with respect to the central atom, which werefer to as skeletal degrees of freedom. For exam-ple, the C H ring of ferrocene can move away5 5from the metal, representing a metal]ligand

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GEOMETRY OPTIMIZATION OF TRANSITION METAL COMPLEXES

FIGURE 1. Structures of metal complexes.

stretching, or the ring can tilt or rotate around themolecular C axis.5

Any nonlinear molecule composed of two non-linear fragments possesses six more internal de-grees of freedom compared to the internal degrees

of freedom of its two fragments. The six newinternal degrees of freedom and the skeletal de-grees of freedom of metal complexes in particularare related to the translational and rotational dis-placements of the separate fragments. Accord-

JOURNAL OF COMPUTATIONAL CHEMISTRY 47

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ingly, a sensible approach to find the best repre-sentation of the skeletal degrees of freedom of acomplex is to regard them as an assembly of twofragments. As an example, we consider benzene-chromium-tricarbonyl as an assembly of benzeneand chromium-tricarbonyl. In Figure 2 we com-

Ž .pare the number of vibrational or internal , transi-tional, and rotational degrees of freedom betweenthe fragments and the complex.

All skeletal movements can be characterized aseither a translational or a rotational motion of theligand. Because translations and rotations of thefragments are relative to each other, we consider

Žthe movements of the benzene ring or the ligand.in general with respect to the rest of the complex.

For the skeletal internal motions resulting from thetranslational motion, the benzene ring can be re-garded as a point mass. To find the new degrees offreedom related to translation of the ligand, onecan consider a molecule in which the ligand ring isreplaced by an ordinary atom. Figure 3a, b showssuch situations; Figure 3a represents themetal]ligand stretch. Figure 3b shows one of thetwo degenerate ligand]metal]carbon bending co-ordinates.

The rest of the skeletal movements can be re-lated to the rotational motion of the free frag-ments. The rotation of the benzene ring around themolecular symmetry axis results in an internal

Ž .rotation Fig. 4a . The rotation of the benzene ringaround an axis in the plane of benzene results in aring; one of the two degenerate tilting coordinatesis shown in Fig. 4b.

Ž .In the example of BzCr CO , the six new de-3grees of freedom, compared to the separate frag-ments, can be accounted for as one metal]ligandstretch, two ligand]metal]carbon deformations,one internal rotation, and two ligand tilts. Thisargument can be generalized to any complex, and

FIGURE 2. Internal and external degrees of freedom ofbenzene-chromium-tricarbonyl.

( ) ( )FIGURE 3. a Metal ]ligand stretching. b Ligand ]metal ]carbon bending.

it can be shown that the skeletal movements re-sulting from one ligand bonded in a multicentralfashion consists of three translational type andthree rotational type movements. This problem issimilar to finding the lattice vibrations of a singlecrystal, consisting of more than one molecule perunit cell. Further, similar to the vibrations in a

Žsingle crystal, the complete symmetry or local.symmetry analysis of all internal coordinates in-

cluding the skeletal ones can be made by themethod of factor group analysis developed byBhagavantam and Venkatarayudu.10

The representation of the translational-typeskeletal vibrations is fairly straightforward by in-troducing the centroid to represent the movementof the entire ligand and defining bond length andangles with the help of the centroid. Rotational-type movements can also be handled similarly,however, they require further considerations. Notethat rotation-type internal movements are notunique to transition metal complexes. The simplestexample is the internal rotation of ethane. How-ever, in the case of the internal rotation of the CH3group in organic molecules, the axis of rotationcoincides with the CC bond. Therefore, this rota-tion can be appropriately represented by dihedralangles around the CC bond. The rotational move-ments of a ligand in a metal complex can berepresented by bond angles and dihedral anglesthat involve the centroid, because the rotation isusually around the centroid-metal or centroid-carbon vectors.

( ) ( )FIGURE 4. a Internal rotation. b Ligand tilt.

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An example of the bending angles representingthe ring tilt is shown on Figure 5a. Six such bend-ing angles, a , a , . . . , a can be defined ford1 d2 d6

Ž .BzCr CO . Appropriately chosen linear combina-3tions of these six such bending angles can repre-sent the two orthogonal tilting motions of the ring.The internal rotation can be represented as a dihe-dral angle involving a ring carbon atom, the refer-ence point, the metal atom, and another carbonfrom the CO group, as shown on Figure 5b. Alto-gether 18 such dihedral angles can be defined for

Ž .BzCr CO , and the internal rotation can be ex-3pressed with an equally weighted linear combina-tion of all 18 of these dihedral angles.

In the Z-matrix optimization method it is alsopossible to introduce dummy atoms to facilitatethe definition of such bending and torsion anglesas shown in 4a and 4b. However, if one introducesa dummy atom in the scheme of the Z-matrixoptimization, it increases the number of degrees offreedom. Therefore, Z-matrix dummy atoms arenot appropriate for the present purpose. For thisreason, we avoided the use of the words dummyatom for the explanation of these internal coordi-nates. The introduction of such a reference pointthat does not increase the number of degrees offreedom was first addressed by Doman et al. inconnection with molecular mechanics type opti-mization of metallocenes.11 These authors sug-gested the elimination of the forces on the dummyatom by distributing them to real atoms. Kauppand Schleyer also used the centroid approach todetermine the energy profile of bent metallocenesas a function of the skeletal bending angle.12 How-ever, previously the definition of internal coordi-nates with the centroid was applied to only specialcases and not generalized for gradient based ge-ometry optimization.

For the transformation of forces between inter-nal and Cartesian coordinates, one needs thederivatives of the internal coordinates as a func-tion of Cartesian coordinates. Such differentiation

( )FIGURE 5. a Bending angle defined by a reference( )point. b Dihedral angle defined with a reference point.

involving dummy atoms or a reference point hasto be done by applying the chain rule,

Ž .dq x , . . . , x , . . . , x x , . . . , x , . . . , xŽ .i k j R m n l

dx j

q q xi i R Ž .s q . 1 x x xj R j

Atoms j s k, . . . , l define the internal coordinate,while atoms j s m, . . . , n define the reference pointx . The position x is defined in the form of aR Rfunction of the coordinates of the defining atoms.The two most obvious choices for x are the centerRof mass and the geometrical central of the bondedatoms. The center of mass is expressed as

n1Ž . Ž .x x , . . . , x s m x 2ÝR m n i iM ism

where m and M are the atomic and fragmentimasses, respectively. The geometrical center is de-fined as

n1Ž . Ž .x x , . . . , x s x . 3ÝR m n in y m q 1 ism

We previously compared the consequences ofŽ . Ž .defining the reference point by eqs. 2 and 3 in

connection with the normal coordinate analysis offerrocene.4 The center of mass as a reference pointintroduces mass dependence of the internal coordi-nates. Therefore, the internal coordinate definitionsof different isotopomers of the same compoundare also different, resulting in a different set offorce constants for different isotopomers. Thesedifferences between force constants of isotopomersare the artifacts of the mass dependent referencepoint definitions.

Ž .When eq. 3 is used to define the referencepoint one has to specify which atoms of the frag-ments should be included in the definition. In our

Ž .example of BzCr CO , we introduced the geomet-3rical center of the carbon atoms of the benzenering. If we had included the hydrogen atoms aswell, the reference point would coincide with thegeometrical center of the carbon atoms for a planarring but it would lead to a different definition ofskeletal coordinates. The most physically meaning-ful reference point is the geometrical center of theatoms bonded to the metal atom. For example, if asix membered ring is bound in an h 4-bonded fash-ion, than only the four bonded carbons shoulddefine the reference point. Further examples aregiven in the next section.


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Skeletal Internal Coordinatesfor hn-Bonded Ligands

In this section we apply the principles describedabove to select the skeletal internal coordinates ofsome transition metal complexes with small aro-matic rings and unsaturated chains as ligands.Here we deal with only the skeletal coordinates;the rest of the coordinates can be defined by thestandard procedures.1 As discussed in the previoussection, one has to define four types of skeletalinternal coordinates: stretchings, bendings, inter-nal rotations, and ligand tiltings. The definition ofthe ligand tilting coordinates can be expressedwith a general formula; the rest of the skeletalcoordinate definitions depend on the actual struc-ture.

The tilting coordinates are represented by a pairof linear combinations of C]D]M angles a , whereD

Ž .C is a ring atom usually carbon , D is the refer-Ž .ence point, and M is the central metal see Fig. 5a .

The linear combination coefficients can be derived

from group theory, considering C local symme-nvtry for the n-membered h n-bonded ligands.

na � Ž . 4 Ž .q s cos k y 1 2prn a , 4Ý D k

ks1n

b � Ž . 4 Ž .q s sin k y 1 2prn a . 5Ý D kks1

EXAMPLES

Figure 1 shows several examples of transitionmetal complexes with ligands ranging from C F2 4to an eight membered ring. We included the inter-nal coordinates of selected complexes in Table I.

The simplest unsaturated ligand is the C F2 4represented in the Fe complex 1. In this particularsimple case, physically meaningful definition canalso be accomplished by regular stretching, bend-ing, and torsional coordinates or equivalently bythe principles outlined above.

The example of a three and a five memberedŽ .Ž .aromatic ring is represented by Ni C Cl C H3 3 5 5

TABLE I.Definition of Skeletal Internal Coordinates

Lig. Type Struct. No. Internal Coordinate Description

3h -3-ring 2a Ni ]D distance, where D is the centroid Skeletal stretchq = 2a y a y a , where a is shown in Fig. 4b Tilt aD2 D1 D3 Diq = a y a Tilt bD1 D3

jjq = Ý t , where t is a dihedral angle between C atoms Internal rotationi, j i D i D i( )of the ligand, the centroid D , the central metal atom

(and the bonded atoms of other ligands X i = 1, 2, 3, andj)j = 1, . . . , 5 in this case ; see Fig. 4a

A pair of perpendicular bending angles between D ]Ni ]D , Skeletal bend1 2where D and D are the centroids of the upper and1 2lower ring, respectively

3h -3-chain 3 Same as above Skeletal stretch3Same as for h -3-ring Tilt a and b3Same as for h -3-ring, except X atoms are the three carbon Internal rotationj

atoms of the CO ligands, j = 1, 2, 3q = b y b where b is defined as the angle between Skeletal bendingD1 D3 D1

D ]Co ]C , j = 1, 2, 3 the carbon atoms of the CO ligandsjq = 2b y b y b Skeletal bendingD2 D1 D3

3h -5-ring 7a The centroid is placed in the geometrical center of the three Remarkbonded carbon atoms

Centroid, Al distance StretchSame as for three membered ring 2a or 3 Tilt a ]bThis involves three carbon atoms of the ring, i = 1, 2, 3, as Internal rotation

well as the two methyl C atoms, j = 1, 2The displacement of D, out of the plane defined by Al and Sk. out of plane

the two methyl carbons deformationq = g y b y b , where g is the C ]Al ]C angle, C are Sk. bend aD1 D 2 1 2

the methyl carbons

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complex in structure 2a. The corresponding coordi-nates for the three membered ring are included inTable I. Structure 3 shows an allyl complex,

Ž .Ž .Co C H CO , which is an example for a com-3 5 3plex with a three membered open chain. The tilt-ing coordinates can be defined the same way forthe open chain as for the ring.

The cyclobutadiene-iron-tricarbonyl complexshown in structure 4a represents an example for afour membered ring ligand. For the tilting coordi-nates of the two combinations obtained from eqs.Ž . Ž .4 and 5 , one obtains 1, 0, y1, 0 and 0, 1, 0, y1.These combinations, however, are not the mostideal for all conformations. For example, in a con-formation with the symmetry plane intersectingbonds, rather than atoms, the 1, 1, y1, y1 and1, y1, y1, 1 combinations are more appropriate.The complex in structure 5 represents the openchain analogue of structure 4a. The skeletal coordi-nates of structures 4 and 5 are almost identical.

The complexes in structures 6a, 7a, and 7b rep-resent the cyclopentadienyl complexes with h 5, h 3,and h 2 coordination modes. Although all of thesecomplexes involve the same ligand, the definitionof skeletal internal coordinates is different in eachcase. The skeletal internal coordinates are deter-mined by the coordination mode, rather than bythe ligand itself. For the h 5 coordinated ring, theC local symmetry was used to select the tilting5vand internal rotation coordinates; for the h 3 com-plex it is more appropriate to consider the three-fold local symmetry. Further, the h 2 coordinated

Ž .complex structure 7b is similar to the example ofthe C F complex in structure 1. For this complex2 4it is possible to define the internal coordinateswithout considering the centroid.

Examples of six membered rings are shown instructures 8a, 10a, 11, and 12a. Molecules in struc-tures 8 and 10 are analogous; the only difference isthat the benzene ring was changed to its inorganicanalogue borazine. Both complexes are h6 bonded,and the internal coordinate sets are identical. In

Ž . 6dibenzene-ruthenium structure 11 there is one hand one h 4 bonded benzene ring. The definitionsfor the h6 bonded ring are similar to that of theother benzene complexes. However, the skeletalmovements of the h 4 bonded benzene are definedanalogous to those of the butadiene complexes.The six membered ring of structure 12a is coordi-nated with two h 2 bonds. However, for the defini-tion of the skeletal coordinates it can be regardedas a complex with one h 4 coordination. Accord-ingly, the skeletal internal coordinates are analo-gous to those of the cyclobutadiene complex.

For the sake of completeness, we included acomplex with a seven membered h7 coordinatedring in structure 9. This example represents thelargest number of equivalently bound carbon

Ž .atoms in one ligand. The cycloocta-1,5-diene di-phosphino-methyliridium complex shown in struc-ture 13 represents the largest ligand and is also anexample for a single ligand occupying two differ-

Ž .ent an equatorial and an axial coordination sites.The Ir complex in structure 13 could be consideredas an h 4 coordinated or two h 2 coordinated lig-ands. Because this ligand occupies two differentcoordination sites, we prefer to regard the moleculeas two h 2 coordinated. Clearly the two metal lig-and bonds are not equivalent. Therefore, it is moreappropriate to represent them by two differentcentroid-metal bonds. We used trigonal bipirami-dal local symmetry around the central atom.

The complex shown in structure 14 representsan example where the centroid of the h 5 bondedfragment is part of another five membered ring.

Performance of GeometryOptimization

The significance of the skeletal internal coordi-nate definitions can be shown by their applicationas optimization variables in the geometry opti-mization of metal complexes. We optimized thegeometry of the complexes described in the previ-ous two sections and the performance of this pro-cedure is discussed here. In addition, we com-pared the performance of the optimization withanother alternative definition of skeletal coordi-nates. Most skeletal internal coordinates could alsobe expressed without introducing the referencepoint. These alternatives are considered in thissection. Further, we compared the performancebetween optimization using Cartesian versus natu-ral internal coordinates.

In Table II we summarize the initial and finalgradients, the number of optimization steps, andthe largest internal coordinate changes in the laststep for each complex optimized in this study.Generally all optimizations converged in 10]20steps, depending on the symmetry, size, quality ofthe initial geometry, and the Hessian. As conver-gence criterion, we considered the geometry con-verged if the maximum gradient was below 0.001mdyn, the maximum stretching displacement was

˚less than 0.0001 A, and the maximum change inbending angles was below 0.001 rad. This criterionis sometimes too tight to be reached with density


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TABLE II.Performance of Geometry Optimization Using Skeletal Internal Coordinates

Step 1 Final Step

( ) ( )Gradient mdyn Gradient mdyn Displ.

˚( ) ( )Struct. No. Norm Max n Norm Max Max Stretches A Max Bendings rad

1 1.0616 0.4436 11 0.0008 0.0005 0.00012 0.000052a 1.1030 0.4958 13 0.0055 0.0030 0.00006 0.000242b 1.1026 0.4980 15 0.0064 0.0035 0.00005 0.000123 2.5071 2.2272 16 0.0018 0.0008 0.00014 0.000154a 1.7086 0.6337 11 0.0031 0.0017 0.00005 0.000024b 1.7084 0.6450 17 0.0090 0.0047 0.00018 0.000165 2.0526 0.9710 20 0.0095 0.0049 0.00008 0.000136a 0.3006 0.1256 12 0.0052 0.0024 0.00010 0.000216b 0.4086 0.3328 14 0.0033 0.0015 0.00013 0.000226c 0.9353 0.6899 15 0.0132 0.0050 0.00025 0.003006d 0.1921 0.1158 18 0.0200 0.0120 0.00031 0.000607a 0.4118 0.1770 13 0.0006 0.0003 0.00006 0.000067b 1.8839 1.7429 19 0.0037 0.0021 0.00024 0.001527c 0.3867 0.1890 16 0.0047 0.0037 0.00027 0.001377d 0.8725 0.7757 15 0.0039 0.0025 0.00033 0.003628a 0.8708 0.5411 8 0.0043 0.0013 0.00015 0.000028b 0.8509 0.5122 10 0.0087 0.0044 0.00022 0.000459 2.0927 0.8857 13 0.0026 0.0012 0.00014 0.0007510a 1.6311 0.5953 11 0.0023 0.0008 0.00008 0.0000410b 1.4159 0.6006 10 0.0036 0.0027 0.00005 0.0000512 1.2920 0.8469 10 0.0004 0.0003 0.00003 0.0000513 9.3092 4.8673 18 0.0045 0.0022 0.00040 0.0006214 0.1396 0.0505 10 0.0069 0.0026 0.00025 0.00124

functional calculations, where numerical evalua-tion of integrals cannot be avoided. When thegradient and the energy started oscillating for nu-merical reasons, we stopped the geometry opti-mization. This situation with structures 6c and 6dwas the most serious. However, the numericaluncertainty in the geometry is still an order ofmagnitude less that the absolute error of themethod compared to experimental structures.

From Table II it is apparent that the perfor-mance of this optimization technique is in linewith previous results of natural coordinate opti-mizations, especially if one takes the poor startinggeometries into consideration. Also we used a sim-ple diagonal Hessian, with force constants esti-mated from the force field of ferrocene,

Ž .dibenzene-chromium, and BzCr CO .3

An alternative representation of the skeletal de-grees of freedom is possible with introducingmetal]carbon bonds. The ligand stretching coordi-

Ž .nates of BzCr CO can be expressed as the equally3

weighted linear combination of all bonds betweenŽthe metal and the bonded carbon atoms s , i si

.1, . . . , 6 of the ligands: q s s q s q s q s q1 2 3 4s q s . Another linear combination, q s 2 s q5 6 1

s y s y 2 s y s q s , can represent one of the2 3 4 5 6

tilting coordinates. The corresponding orthogonaltilting can be defined as q s s q s y s y s .2 3 5 6

These internal coordinates were used before inempirical normal coordinate analysis of Bz Cr and2

Ž . 13BzCr CO . We compared the performance of3

optimization using this alternative representationŽ .for BzCr CO , and the results are summarized in3

Table III. These results were obtained by replacingthe skeletal stretching and the tilting coordinatesby the appropriate linear combinations ofmetal]carbon bond stretchings, otherwise usingnatural coordinates including the skeletal internalcoordinates defined above. In Table III we includethe values of the skeletal stretching coordinatesand that of a CC stretching coordinate along withthe corresponding gradients in the first few andthe last iterations. We note here that the replace-ment of the tilting coordinates should not make

Ž .significant difference in the case of BzCr CO ,3

since these coordinates are fixed by the molecular

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TABLE III.Performance of Geometry Optimization Using Centroid and Metal – Carbon Bond Representations

Cr } Benzene Stretch

Centroid Metal } C Bond

Step Coord. Gradient Coord. Gradient

0 1.5822 y0.5411 5.1938 y0.29601 1.7626 0.1400 5.2925 y0.10802 1.7406 0.0377 5.3103 y0.11843 1.7288 0.0275 5.3331 y0.08714 1.7254 0.0243 5.3546 y0.05665 1.7158 0.0029 5.3989 y0.02748 1.7139 y0.0003Opt. 1.7139 5.590

C1 } C2 Stretch


Coord. Gradient Coord. Gradient

0 1.4115 y0.0189 1.4115 0.06141 1.4142 0.0545 1.4027 y0.03332 1.4106 0.0380 1.4051 0.00313 1.4070 0.0094 1.4055 y0.00064 1.4064 0.0048 1.4053 y0.00515 1.4058 y0.0015 1.4060 0.00158 1.4057 y0.0003Opt. 1.4057 1.4057

Gradient Norm


0 0.8708 0.72791 0.2861 0.27482 0.1487 0.17843 0.0622 0.10784 0.0463 0.08335 0.0215 0.06408 0.0042

( )BzCr CO staggered conformation.3

symmetry. Therefore, the difference in perfor-mance of optimization is due to the different defi-nition of the skeletal stretching coordinate.

It is clear from Table III that the centroid typecoordinate remarkably improves the overall per-formance of optimization. The explanation for thedifferent performance is due to the different cou-pling of the skeletal stretching coordinate with theCC stretches. To find an explanation for the cou-pling, one has to consider the derivatives of theskeletal stretching coordinates with respect to thedisplacement of the C atom in the plane of thering. For the centroid-type coordinate this deriva-tive is zero, while for the metal]carbon stretchesthis derivative is nonzero. To appreciate the mag-

nitude of this coupling, one can look at the kineticenergy matrix elements corresponding to these co-ordinates, which are listed in Table IV, for

Ž .BzCr CO . While the coupling G matrix element3between CC stretch and skeletal coordinates arezero for the centroid representation, this is about20% of the diagonal elements for the metal]carbonbond representation. This coupling explains thedifference in the performance of the optimizationbetween the two sets of coordinates. We alsolooked at the force constants in both representa-tions; however, that did not show significant dif-ference in the two representations.

Another qualitative explanation can be given bylooking at the displacements that these internal


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TABLE IV.Comparison of G Matrix Elements in Centroid andMe } C Bond Representations.

( ) ( )BzCr CO BzCr CO3 3Centroid Metal } C bond

CC stretchDiagonal 0.1667 0.1667Skeletal stretch 0.0000 0.0248Skeletal tilt 0.0000 y0.0254

Skeletal stretch, diagonal 0.0331 0.1355Skeletal tilt, diagonal 0.0754 0.0850

˚ ˚Units are mdyn / A, mdyn A / rad, and mdyn / rad forstretches, stretch bend, and bends, respectively.

coordinates represent. All internal displacementshave to leave the rest of the internal coordinatesunchanged. For this reason, when the CC distanceincreases in the metal]carbon bond representation,the metal]ring distance has to decrease to keep themetal]carbon distance unchanged. This conditionintroduces large coupling between the skeletalstretch and the CC stretching coordinates.

To show the significance of natural internal co-ordinates for the optimization of transition metalcomplexes, we also compared it with optimizationin Cartesian coordinates. The performance ofCartesian optimization largely depends on thequality of the Hessian used in the optimization.Therefore, to make it a fair comparison, we used aHessian from a parametrized redundant force fieldin the Cartesian optimization. We used theparametrized Hessian supplied with release 2.01 ofthe ADF program. The performance of Cartesianoptimization is compared to the natural coordinateoptimization in Table V for five examples. Thestarting geometries and convergence criteria were

TABLE V.Comparison of Performance Between Cartesian andNatural Internal Coordinate Optimizations.

Cartesian CoordinatesFinal Step

Natural Int.Displ.Struct. Coord. Gradient

˚( ) ( )No. No. Cycles n Max mdyn Max A

2a 13 52 0.0021 0.000823 16 50 0.0006 0.001017a 13 47 0.0005 0.0001110a 11 18 0.0003 0.0001113 18 61 0.0032 0.00081

the same for the two methods compared. In gen-eral, the natural internal coordinate optimization isabout two to three times more efficient than theCartesian optimization. One can also see from thiscomparison, that as the molecular symmetry islowered the internal coordinate optimization ismore and more advantageous. For highly symmet-rical molecules many of the harmonic and anhar-monic coupling constants are zero; therefore, theCartesian optimization can also be fairly efficient.For metal complexes, besides the inherent deficien-cies of the Cartesian optimization, the highly ap-proximate parametrization of the Hessian and thepoor starting geometries also contribute to thepoor performance.

Conclusions

In this paper we adopted the natural internalcoordinate optimization of Forgarasi et al.1 forapplications to metal complexes with skeletalbonds. We suggested appropriate representationsfor the skeletal stretchings, bendings, tilts, andinternal rotations of unsaturated organic ligandscomplexed to metals. We demonstrated how toconstruct internal coordinates using referencepoints defined by the geometrical center of bondedatoms. The application of these internal coordi-nates to geometry optimization showed that geom-etry optimization is very efficient using these coor-dinates in spite of the poor starting geometries andcrude approximations of the Hessian. We alsoshowed that the centroid approach is superiorto the definition of skeletal motions withmetal]carbon bonds. We also found that geometryoptimization in natural internal coordinates isabout two to three times more efficient than inCartesian coordinates using a parametrized forcefield to approximate the Hessian.

Acknowledgments

Part of this work was done at the University ofCalgary. I wish to thank my former supervisorProfessor Tom Ziegler, the Academic ComputingServices, and the Canadian Pacific Modelling Lab-oratory for providing computational resources. Iwould also like to thank my former colleagues Dr.John Lohrenz, Dr. Heiko Jacobsen, and Mr. TomWoo for their encouragement to pursue this pro-ject and for continuing to apply this method in

VOL. 18, NO. 154


their research. Financial support from the NationalResearch Council is gratefully acknowledged. Thisarticle is issued as NRCC No. 39093.

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geometry optimization of metal complexes using natural internal coordinates: representation of...

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