Chapter 4

Complete Basis Set Models for Chemical Reactivity: from the Helium Atom to Enzyme Kinetics

George A. Petersson

Hall-Atwater Laboratories of Chemistry, Wesleyan University, Middletown, Connecticut

06459-0180, U.S.A.

1.INTRODUCTION

The qualitative idea of chemical reactivity is quantitatively expressed in the rate and extent of chemical reactions. These rate and equilibrium constants present a formidable challenge to theoretical predictions [1, 2]. The principal difficulty lies in the extreme sensitivity of the specific rate constant and the equilibrium constant

to small errors in the calculated barrier height		and enthalpy change
An error of only 1.4 kcal/mol in	these	energy	changes	leads
to an error of an order of magnitude in		or	at	room

temperature. Thus, one needs methods for calculating molecular energy changes with errors less than ca. 0.5 kcal/mol. This is a very demanding standard, requiring convergence of both the one-particle expansion (basis set) and the n-particle expansion (correlation energy) [2]. In the absence of near-degeneracies, the coupled cluster method of Bartlett [3] with Raghavachari’s perturbation treatment of triple excitations [4], CCSD(T), is a sufficiently accurate treatment of the n-particle problem [5]. The one-particle expansion is not so easily disposed of.

The slow convergence of the correlation energy with the oneelectron basis set expansion has provided the motivation for several attempts to extrapolate to the complete basis set limit [6-13]. Such extrapolations require a well defined sequence of basis sets and a model for the convergence of the resulting sequence of approximations to the

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J.Cioslowski (ed.), Quantum-Mechanical Prediction of Thermochemical Data, 99–130.

©2001 Kluwer Academic Publishers. Printed in the Netherlands.

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correlation energy. The various extrapolation schemes that have been proposed differ in both the method used to obtain a well defined sequence of one-electron basis sets and in the extrapolation model. The complete basis set (CBS) extrapolations described in this chapter employ the asymptotic convergence of pair natural orbital (PNO) expansions [8-10, 14].

Early work on atoms [15, 16] employed increasing sets of s, p, d,

... etc. basis functions, explicitly seeking convergence to the complete basis set limit. The power of such methods was greatly enhanced by the classic papers of Schwartz establishing the asymptotic convergence of the second-order Møller-Plessett (MP2) pair correlation energies [17] with the angular momentum expansions [6, 7],

where the exponents -3 and -5 apply to opposite spin		and equal spin
or	pairs, respectively. Extrapolations to the	complete basis set

limit using the asymptotic formulas of Schwartz were employed first by Bunge and later by Jankowski, Malinowski, and Polasik to establish a database of CBS-MP2 limits for closed-shell atoms [18, 19].

Several methods have been developed for establishing the MP2 limit for small molecules. We shall compare three of the most important methods, and a recently proposed combination of two of them that achieves a new level of efficiency in obtaining chemically accurate absolute MP2 energy limits. We conclude with a case study of the extension of these

approaches to enzyme kinetics,	namely the	isomerase-
catalyzed conversion of	17-dione to the	isomer.

2.PAIR NATURAL ORBITAL EXTRAPOLATIONS

Twenty years ago, we extended asymptotic extrapolations to polyatomic molecules by transformation of the Schwartz formulae to a symmetry-independent form based on the total number N of pair natural orbitals (PNOs) [8-10, 14],

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where the exponents -1 and -5/3 now apply to opposite spin and equal spin or pairs, respectively. The exclusion parameter can be determined as the solution of a quadratic equation [20]. Extrapolation of infinite-order (e.g. CCSD) pair energies:

requires attenuation by an interference factor	obtained from
the first-order wavefunction [20].

The algorithm employed for these PNO extrapolations selects the value of N giving the largest (i.e. most negative) value for the CBS pair energy with the constraint of Convergence to the exact CBS

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pair energy is ensured by systematically increasing as the basis set is expanded (Fig. 4.1). For example, we generally set equal to 5 for spd basis sets and equal to 10 for spdf basis sets. These nonlinear extrapolations are size-consistent only if the canonical SCF orbitals are localized prior to extrapolation of each of the individual pair energies to the CBS limit. We assume that N is large enough for the asymptotic form to be applicable and that the low-lying natural orbitals are accurately described with the basis set employed. Early implementations of these extrapolations served as polyatomic benchmarks for their time [21], but improvements in hardware and software now make more demanding standards possible.

3.CURRENT CBS MODELS

The order-by-order contributions to chemical energies, and thus the number of significant figures required, generally decrease with increasing order of perturbation theory. The general approach for our CBS-n models [20-25] is therefore to first determine the geometry and the zero-point energy (ZPE) at a low level of theory, and then perform a series of high-level single-point electronic energy calculations at this geometry, using large basis sets for the SCF calculation, medium basis sets for the MP2 calculation, and small basis sets for the higher-order calculations through order n. The components of each model have been selected to be balanced so that no single component dominates either the

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computer time or the error. The CBS-n models use the aforedescribed asymptotic extrapolation to reduce the error from truncation of the basis sets employed in calculation of the correlation energy. The compound model single-point energy is evaluated at a geometry determined at a lower level of theory (e.g. CBS-4M//UHF/3-21G) [24], which is again selected to achieve an accuracy consistent with the single-point energy (Table 4.1). Thus our fastest model, CBS-4M, employs HF/3-21G(*) geometries and frequencies, our intermediate model, CBS-QB3, uses B3LYP/6-311G(2d,d,p) geometries and frequencies, and our most accurate current model, CBS-QCI/APNO, incorporates atomic pair natural orbital (APNO) basis sets and employs QCISD/6-311G** geometries and frequencies [26]. These methods require small empirical corrections to achieve the desired accuracy for chemical energy differences (atomization energies, ionization potentials, and electron affinities) [27]. This sequence of models, namely CBS-4M, CBS-QB3, and CBS-QCI/APNO,