ELSEVIER

THEO CHEM

Journal of Molecular Structure (Theochem) 39X-399 (1997) I-6

Computational methods for large molecules’

Walter Thiel

Organisch-chrnlisc.hrs Imtitut, Univrrsitiit Ziirich, Wir~terthurrrvtrussr 190, CH-8057 Ziirich, S,citzrrlnnd

Abstract

Recent methodological developments in our group include the extension of the semiempirical MNDO formalism to d orbitals, an approach to go beyond the MNDO model by incorporating additional one-electron terms (one- and two-center orthogonalization corrections, penetration integrals, effective core potentials), and the combination of quantum mechanical and molecular mechanical methods in QM/MM hybrid schemes. The theory and parametrization of these approaches, the statistical evaluation of the results, and selected applications are discussed, as well as an efficient implementation of analytic first and second derivatives in MNDO-type methods. 0 1997 Elsevier Science B.V.

Kqwords: Semiempirical methods; Hybrid methods; Analytic derivatives; MNDO; Orthogonalization corrections

1. Introduction

This article outlines a plenary lecture held at the WATOC 96 conference in Jerusalem. It summarizes

recent work from our group, without presenting

theoretical derivations or detailed numerical results which can be found in the original publications. A survey of the relevant literature and a more thorough methodological discussion are provided in a recent review [l].

At present, ab initio methods, density functional

methods, and semiempirical methods serve as the major computational tools of quantum chemistry. There is an obvious trade-off between accuracy and computational effort in these methods. The most accurate results are obtained from high-level corre- lated ab initio calculations which also require the

highest computational effort. On the other end of the spectrum, semiempirical MO calculations are very

’ Presented at WATOC ‘96, Jerusalem, Israel, 7- I2 July 1996.

fast, and it is therefore realistic to expect only a limited accuracy from such calculations. Actual appli- cations will usually have to balance the required accu- racy against the available computational resources. The combined use of several computational tools may well be the best approach to solve a given pro- blem, e.g. by initial explorations at the semiempirical level followed by more accurate density functional or ab initio calculations.

Comparisons of computation times with current quantum-chemical programs show [I] that semi- empirical calculations are faster than typical ab initio or density functional calculations by at least three orders of magnitude. This allows semiempirical investiga- tions on large molecules which are not practical with higher-level methods. Examples from fullerene chemistry illustrate this point: The structural prefer- ences and stabilities of fullerenes C, (20 5 II 5 960) can be understood on the basis of semiempirical MNDO calculations [2-41. The potential surfaces for annealing and fragmentation reactions of Chc, [5]

0166.1280/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO 166.1280(97)00039-O

2 W. Thiel/Jound of Molecular Structure (Theochew) 398-399 (1997) l-6

and for helium incorporation into ChO [6] can be In the MNDO/d approach, there are typically 13- explored by MNDO (with subsequent single-point 16 optimized parameters per element with an spd density functional calculations at MNDO optimized basis, with 4 parameters related solely to the d orbi- geometries) to elucidate the corresponding reaction tals. Hence, MNDO/d is less highly parametrized in mechanisms. Molecular dynamics runs on the the sp part than AM 1 or PM3 which employ typically MNDO potential surface for ChO + C& collisions ]7] 13-16 and 18 parameters, respectively, per element successfully simulate the elementary processes (deep (sp basis). In deliberately restricting the number of inelastic collisions, dimerization, and fragmentation) optimized parameters, MNDO/d resembles the origi- that are observed experimentally at different collision nal MNDO method which uses only 5-7 parameters energies [ 81. per element [9].

Considering the computational speed of semi- empirical methods, it is not surprising that they are generally less accurate than high-level quantum- chemical methods. There is a vast literature docu- menting the accuracy and the errors of the existing semiempirical methods (see references in [l]). Extensive statistical evaluations indicate that ground-state organic molecules are often treated

reasonably well whereas other systems are described less reliably. Moreover the deviations between theory and experiment are normally less systematic in semi- empirical than in ab initio calculations. In view of this situation, it is clearly desirable to improve the existing semiempirical methods (without sacrificing their computational efficiency), in order to enhance their overall accuracy and to extend their range of applicability.

MNDO/d parameters have been published [ 13- 151 for the following 12 elements: Na, Mg, Al, Si, P, S, Cl, Br, I, Zn, Cd, and Hg. According to extensive test calculations covering several properties and more than 600 molecules, MNDO/d provides significant improvements over the established semiempirical methods (MNDO, AMl, PM3), especially for hyper- valent compounds. The mean absolute error in

MNDO/d heats of formation amounts to 5.4 kcal mall’ for the complete validation set of 575 mole- cules, and is identical for the subsets of 508 normal- valent and 67 hypervalent molecules. By contrast, the established semiempirical methods with an sp basis yield heats of formation for hypervalent compounds that show much larger errors [ 13,151. In addition, MNDO/d consistently predicts the correct shape (point group) of each hypervalent molecule studied (e.g. Cz,. for ClF3) which is not true for the older methods with an sp basis [ 13,151.

2. MNDO with d orhitals

The established MNDO-type methods, i.e. MNDO

[9], AM1 [lo], and PM3 [l l] employ an sp basis without d orbitals in their original implementation. Therefore, they cannot be applied to most transition metal compounds, and difficulties are expected for hypervalent compounds of main-group elements where the importance of d orbitals for quantitative accuracy is well documented at the ab initio level.

To allow for an improved semiempirical descrip- tion of such compounds, the MNDO formalism has been extended to d orbitals [12]. The inclusion of d orbitals for second-row and heavier elements mainly requires a generalized semiempirical treatment of the two-electron interactions which has been specified in detail [12] and which forms the basis of our own MNDO/d parametrization [ 13- 151 and the indepen- dent PM3/tm parametrization [ 161.

Unpublished MNDO/d parameters [ 171 are avail- able for most of the heavier main-group elements and for the following transition metals: Ti, Zr, Hf, Fe, Ni, Pd, Cu, and Ag. Generally speaking, the para- metrization for the transition metals has proven to be more difficult than for the main-group elements. This is mostly due to the more complicated electronic structure of transition metal compounds, and partly also to the lack of reliable thermochemical reference data. During the parametrization, the proper reproduc- tion of the lowest electronic states of a given transition metal was enforced by a suitable choice of the one- center parameters, and some bond-specific a para- meters [ 151 were introduced for fine tuning. In spite of these measures, the results for elements in the mid- dle of a transition metal series (e.g. Fe) are still unsatisfactory, whereas those for elements at the left or right side of a series are more realistic.

For example, MNDO/d calculations for Zr [ 171

W. ThieVJoumal of Molecular Structure (Theochem) 39X-399 (1997) l-6 3

yield mean absolute errors for heats of formation, bond lengths, and vibrational frequencies which are comparable to those for main-group elements [ 151.

For further validation, the zirconocene catalysis of homogeneous olefin polymerization has been studied [18] for a model system, i.e. CpzZrCzH: + H2C=CH7 - CpzZrC4H$, in analogy to a recent density func- tional (DFT) investigation [ 191. The DFT geometries of reactants, products, and transition states are well reproduced by MNDO/d, whereas the intervening a

complexes are too unsymmetrical in MNDO/d. The relative energies of the various agostic isomers of the reactant (a, @ and the product (cw, p, y, 6) are reason- able in MNDO/d compared with DFT, and the MNDO/d reaction profile (including the transition state energies) lies between those calculated from ab initio SCF and DFT [ 181. In an overall perspective, the MNDO/d results for this model system seem realistic indicating that MNDO/d may become useful

for studying organometallic reactions.

3. Beyond MNDO: orthogonalization corrections

The established MNDO-type methods do not treat the Pauli exchange repulsions explicitly, but attempt to incorporate them through an effective atom-pair potential that is added to the core-core repulsion [ 11. When trying to improve the MNDO model, it would seem logical to include the Pauli exchange repulsions explicitly in the electronic calculation and to remove the effective atom-pair potential from the core-core repulsions. For the sake of consistency,

other one-electron terms of similar magnitude should then also be included explicitly, i.e. penetration inte- grals and core-valence interactions (effective core potentials).

These basic ideas have been implemented in two steps. First, the Pauli exchange repulsions have been introduced as valence-shell orthogonalization

corrections only in the one-center part of the core Hamiltonian [20]. In the second step, they have also been incorporated in the two-center part of the core Hamiltonian [21], i.e. in the resonance integrals. Both developments have been guided by analytical ab initio

formulas and numerical ab initio SCF results. Their implementations are actually quite similar: In both cases, a Gaussian minimal basis is used for technical

reasons, and most two-center interactions are

evaluated analytically followed by an appropriate Klopman-Ohno scaling. The valence-shell orthogo-

nalization corrections are represented in terms of the resonance integrals through a truncated and para- metrized series expansion. The resonance integrals contain a parametrized radial part while the angular

part is the same in the corresponding overlap integral. The first approach [20] has been parametrized for

the elements H, C, N, 0, and F. Compared with the established MNDO-type methods, it offers small con- sistent improvements for ground-state properties. Significant qualitative improvements are found in

several areas, particularly for excited states. The mean absolute error in vertical excitation energies is

0.28 eV, much lower than in AM 1 (1.20 eV) or PM3 (1.18 eV). These improvements are readily rationa- lized from the underlying theoretical model: The destabilization of antibonding molecular orbitals is greater than the stabilization of bonding molecular orbitals at the ab initio level. This effect is not taken

into account in the established MNDO-type methods whereas it is incorporated in the new approach through the orthogonalization corrections [Il. Hence, the excitation energies are raised in a natural manner by correcting for deficiencies inherent to the ZDO approximation [ 11.

The second approach [2 1 ] has presently been para- metrized for H, C, N, and 0. Its development has been motivated by a theoretical analysis [22,23] suggesting

that orthogonalization corrections to resonance inte- grals are essential for describing conformational prop- erties reliably. Our implementation of these

corrections [21] includes three-center contributions which reflect the stereochemical environment of each atom pair. As expected theoretically [22,23], the numerical results from our approach [21] show qualitative improvements for rotational barriers (e.g. ethane, hydrazine, hydrogen peroxide, formamide), for relative energies of isomers (e.g. n-pentane vs. neopentane), and for ring conformations (e.g. cyclo- pentane, cyclopentene). Moreover hydrogen bonds are reproduced much better than previously (e.g. water dimers, (HCOOH)*, strong H-bonds involving

cations). The barriers for typical pericyclic reactions are realistic (e.g. Diels-Alder, electrocyclic, Cope, and ene reactions). In the Diels-Alder reaction between butadiene and ethylene, the barrier is lower

4 W. Thiel/Jourml c~Molecular Structure (Theochem) 398-399 (1997) I-6

for the concerted pathway than for the biradicaloid one, and it decreases with increasing cyano substitu- tion (in agreement with experiment and ab initio

results). Judging from the presently available results

120,211, the explicit inclusion of Pauli exchange repulsions has led to qualitative improvements in several important areas which can be traced back to improvements in the underlying theoretical model. More work is needed to explore the limitations of these new approaches.

4. Hybrid methods

Hybrid methods are characterized by a combination of quantum-mechanical (QM) and molecular- mechanical (MM) potentials. They treat the

electronically important part of a large system by a quantum-chemical method (e.g. ab initio, DFT, or semiempirical) and the remainder by a classical force field. They can provide an appropriate descrip- tion for systems which are too large for a purely quantum-chemical approach (even at the semi- empirical level) and which contain regions that cannot be described classically (e.g. reactive centers with breaking and forming bonds, or chromophores where an electronic excitation takes place).

Many groups are currently involved with the devel-

opment of hybrid methods (see references in [l]). A typical example for the use of hybrid methods is the study of solvent effects [24] because in this case there is an obvious partitioning between the QM part (solute) and the MM part (solvent) of the system, with well-known QMIMM interactions. Another and more difficult application of hybrid methods concerns

enzymatic reactions where the boundary between the QM part (reactive region) and the MM part (protein environment) will generally cut through a covalent bond.

Our own research in this field has begun with a

combination of semiempirical QM methods and the MM3 force field [25,26] and has emphasized the development of a hierarchy of suitable QMNM coupling schemes [25]. The simplest such scheme (model A) employs a mechanical embedding of the QM subsystem and treats all QM/MM interactions as in the classical force field. Model B uses an electronic

embedding by evaluating the electrostatic QM/MM interactions from the QM electrostatic potential and the MM atomic charges, and by including QM polar- ization through calculating the electronic wave- function in the presence of the MM atomic charges.

Model C refines model B by adding MM polarization which is evaluated from the QM electric field and MM polarizabilities using a published dipole interaction model [27]. Model D corresponds to model C with an iterative self-consistent treatment of MM polariza-

tion. In our experience (MNDO/MM3), model D cannot be recommended because it is computationally much more demanding than model C and normally yields very similar results.

The coupling schemes outlined above can be imple- mented with any QM/MM combination. In the case of

MNDO/MM3, additional choices are required for models B-D since there is no unambiguous definition of electrostatic potentials in MNDO and since MM3 does not provide atomic charges. In our approach [26] these quantities are obtained from a special semi- empirical parametrization aimed at reproducing the corresponding ab initio RHF/6-3 lG* reference data. The electrostatic potentials are represented by a semi- empirical MNDO-type expression originally sug- gested by Ford and Wang 1281, and the atomic charges are obtained from a semiempirical formula- tion of a charge equilibration model [29] based on the electronegativity equalization principle. After suitable

calibration, this simple approach does indeed repro- duce the ab initio reference data fairly well [26] and therefore provides an efficient route to realistic electrostatic QM/MM interactions.

Our semiempirical QM/MM hybrid method has been applied [25] to study proton affinities of alcohols and substituted pyridines, deprotonation enthalpies of alcohols and carbon acids, nucleophilic additions of lithium hydride to carbonyl compounds, hydrid trans- fer reactions in deprotonated hydroxyketones, and nucleophilic ring openings in oxiranes (using MNDO/MM3 and AMl/MM3). A comparison of the

results for models A-C allows to identify the appro- priate description of the QM/MM interactions in a given application. Not too surprisingly, the treatment of protonation and deprotonation requires an electro- nic embedding of the QM region: For example, the trends in the proton affinities and deprotonation enthalpies of alcohols are reproduced qualitatively

W. Thiel/Jourrd of Molecular Structure (Throchern) 398-399 (1997) lb6 5

only at levels B and C, respectively, indicating the general importance of electrostatic interactions in these processes as well as the special importance of

polarization for the anions. On the other hand, the hydrid transfer reactions in deprotonated hydroxyke- tones with rigid carbon skeletons can already be described at level A (mechanical embedding). In this case, the bicyclic or polycyclic backbones of the studied hydroxyketones impose characteristic transi- tion state geometries which determine the trends in the barriers for hydride transfer. Generally speaking, the discussed semiempirical QM/MM models (A-C) are not always accurate quantitatively, but they often reproduce qualitative trends reasonably [25].

In our opinion, hybrid methods are very promising tools for modeling large systems. There is consider- able flexibility in selecting a suitable QM component

(e.g. ab initio, DFT, or semiempirical) and a suitable MM component (e.g. a general-purpose or a specia- lized force field), in choosing the appropriate QM/ MM coupling scheme (e.g. models A-C), and in defining the QM/MM boundary (e.g. with regard to its position and the treatment of link atoms). This flexibility provides opportunities, but also calls for guidelines of how to achieve optimum performance with hybrid methods and how to avoid less favorable choices. These issues are currently under study.

5. Analytic derivatives

In semiempirical programs, open-shell molecules are usually described by the nonvariational half- electron (HE) RHF method [30]. Computation of the corresponding analytic gradient requires the solu- tion of the coupled-perturbed Hartree-Fock (CPHF) equations, which is accomplished in the published implementation [31] by an algorithm that scales as

NJ. Recent work in our group has shown [32] that the solution of the CPHF equations for semiempirical HE-RHF wavefunctions can be reformulated to scale as N3 when making use of the Z-vector method [33]. The implementation of this approach leads to dra-

matic reductions of the computation time for gradient evaluation in large molecules (e.g. from 100 139 s to

222 s in CggHzl with N =408, SCF time 605 s [32]). This development greatly facilitates semiempirical HE-RHF studies of open-shell molecules.

Harmonic force fields are commonly calculated from numerical finite differences of gradients in semi- empirical programs, and from analytic second deriva-

tives in ab initio programs. Both for semiempirical and ab initio SCF wavefunctions, the analytic second derivatives contain contributions from integral deri- vatives (direct terms) and from density matrix deriva- tives (CPHF terms). In the ab initio case, the

computational cost is usually dominated by the direct terms (which typically require a cpu time of about 10 Tscb) so that the analytic evaluation of the second derivatives will normally be more efficient than a numerical evaluation. By contrast, in the semi-

empirical case, the computational cost is dominated by the CPHF terms since the integral derivatives can be computed with almost negligible computational effort. A detailed analysis shows [34] that the analytic calculation of semiempirical harmonic force fields can only be competitive to the traditional finite-difference calculation if the CPHF problem is formulated in the A0 basis and is solved iteratively. In this case, the computational cost for a molecule with M atoms and N basic functions formally scales as MN’ both in the analytic and the numerical evaluation. A careful implementation [34] of analytic second derivatives

for MNDO-type semiempirical methods provides typical speedups of 4-8 compared with analogous numerical computations and exhibits a reliable con-

vergence over a wide range of molecules. The asymp- totic memory and disk storage requirements can be chosen to scale as low as N2 without significant degra- dation of performance. It is obvious that these advances facilitate force constant calculations for large molecules at the semiempirical SCF level.

6. Conclusions

The developments outlined in the previous sections promise to open new areas of application for semi- empirical methods and to improve their accuracy without comprising their computational efficiency. Semiempirical methods will therefore continue to be valuable tools for studying electronic effects in large molecules. Whenever technically feasible, such inves- tigations should be supplemented with appropriate higher-level calculations because the synergetic use of several computational tools is often expected to

6 W. Thiel/Journal of Moleculnr Structure (Theochem) 398-399 (1997) I-6

provide the best computational solution for a given

chemical problem.

Acknowledgements

The author wishes to thank his coworkers for their contributions, particularly D. Bakowies, M. Kolb, J.W.C. Lohrenz, S. Pachkovski, A. Voityuk, and W. Weber. Financial support of the Deutsche For- schungsgemeinschaft, the Alfried-Krupp-Stiftung and the Schweizerischer Nationalfonds is gratefully

acknowledged.

References

[I] W. Thiel, Adv. Chem. Phys. 93 (1996) 703.

[2] D. Bakowies, W. Thiel, J. Am. Chem. Sot. I I3 (1991) 3704.

[3] D. Bakowies. M. Btihl. W. Thiel, J. Am. Chem. Sot. I I7

(1995) 10113.

[4] D. Bakowiea, M. Btihl, W. Thiel, Chem. Phys. Lett. 247

(1995) 491.

[5] R.L. Murry, D.L. Snout, G.K. Odom, G.E. Scuseria, Nature

366 (I 993) 66.5.

[6] S. Patchkovskii. W. Thiel, J. Am. Chem. Sot. I I8 (1996)

7164.

[7] X. Long. R.L. Graham, C. Lee, S. Smithline, J. Chem. Phys.

100 (1994) 7223.

[S] E.E.B. Campbell, V. Schyja. R. Ehlich, I.V. Hertel, Phya. Rev.

Lett. 70 (1993) 263.

[9] M.J.S. Dewar, W. Thiel, J. Am. Chem. Sot. 99 (1977) 4899.

[IO] M.J.S. Dewar, E. Zoebisch, E.F. Healy, J.J.P. Stewart, J. Am.

Chem. Sot. 107 (1985) 3902.

[I I] J.J.P. Stewart, J. Comput. Chem. IO (1989) 209, 221.

[I21 W. Thiel. A.A. Voityuk, Theoret. Chim. Acta 81 (1992) 391.

[ 131 W. Thiel. A.A. Voityuk. Int. J. Quantum Chem. 44 (1992) 807.

[I41 W. Thiel, A.A. Voityuk. J. Mol. Struct. 313 (1994) 141.

[I51 W. Thiel, A.A. Voityuk. J. Phys. Chem. 100 (1996) 616.

[ 161 SPARTAN 4.0 (1995). Wavefunction. Irvine, CA, 1995.

[ 171 W. Thiel, A.A. Voityuk. to be published.

[IS] J.C.W. Lohrenz, W. Thiel, to be published.

[ 191 J.C.W. Lohrenz, T.K. Woo, T. Ziegler. J. Am. Chem. Sot. I I7

(1995) 12793.

[20] M. Kolb, W. Thiel, J. Comput. Chem. I4 (1993) 775.

1211 W. Weber, Ph.D. Thesis, Universitat Zurich, 1996.

1221 S. de Bruijn, Chem. Phya. Lett. 54 (1978) 399.

[23] S. de Bruijn. in: R. Carbo (Ed.). Current Aspects of Quantum

Chemistry, Elsevier, Amsterdam, 1982, pp. 25 l-272.

[24] J. Gao, in: K.B. Lipkowitz, D.B. Boyd (Eds.), Reviews m

Computational Chemistry, vol. 7, VCH. New York, 1995,

pp. 119-185.

[25] D. Bakowies. W. Thiel, J. Phys. Chem. 100 (1996) 10580.

[26] D. Bakowies, W. Thiel, J. Comput. Chem. I7 (1996) 87.

[27] B.T. Thole, Chem. Phys. 59 (I 98 I) 341.

[28] G.P. Ford, B. Wang, J. Comput. Chem. 14 (1993) 1101.

[29] A.K. Rappe, W.A. Goddard III, J. Phys. Chem. 95 (1991)

3358.

[30] M.J.S. Dewar, J.A. Hashmall, C.G. Venier. J. Am. Chem. Sot.

90 (1968) 1953.

[3l] M.J.S. Dewar, D.A. Liotard, J. Mol. Struct. (Theochem) 206

(1990) 123.

[32] S. Patchkovskii, W. Thiel. Theoret. Chim. Acta 93 (1996) 87.

[33] N.C. Handy, H.F. Schaefer. J. Chem. Phys. 81 (1984) 503 I.

[34] S. Patchkovskii. W. Thiel. J. Comput. Chem. 17 (1996) 1318.