Modeling of Molecular Properties

Peter Comba

• Publisher: Wiley-VCH
• Year: 2011
• Print ISBN:
  9783527330218
• eBook ISBN:
  9783527636419
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physical & Theoretical Chemistry

Molecular modeling encompasses applied theoretical approaches and computational techniques to model structures and properties of molecular compounds and materials in order to predict and / or interpret their properties. The modeling covered in this book ranges from methods for small chemical to large biological molecules and materials. With its comprehensive coverage of important research fields in molecular and materials science, this is a must-have for all organic, inorganic and biochemists as well as materials scientists interested in applied theoretical and computational chemistry. The 28 chapters, written by an international group of experienced theoretically oriented chemists, are grouped into four parts: Theory and Concepts; Applications in Homogeneous Catalysis; Applications in Pharmaceutical and Biological Chemistry; and Applications in Main Group, Organic and Organometallic Chemistry. The various chapters include concept papers, tutorials, and research reports.

Part One

Theory and Concepts

Chapter 1

Accurate Dispersion-Corrected Density Functionals for General Chemistry Applications

Lars Goerigk and Stefan Grimme

1.1 Introduction

The aim of computational thermochemistry is to describe the energetic properties of chemical processes within an accuracy of 1 kcal mol⁻¹ or less (0.1–0.2 kcal mol⁻¹ for the relative energy of conformers). At the same time, the methods applied should not be too demanding in terms of necessary run times and hardware resources, which rules out highly accurate ab initio methods if larger, chemically relevant systems are to be considered. Whilst Kohn–Sham density functional theory [(KS-)DFT] offers an ideal solution to this dilemma [1, 2], the number of proposed exchange–correlation functionals is immense, and most of these suffer from severe problems. Very prominent examples are the self-interaction-error (SIE; also termed delocalization-error in many-electron systems) [3–6], and the lack of adequately describing long-range correlation effects, such as London-dispersion [7–10]. Moreover, the applicability of functionals to various problems is not broad but is rather specialized (see e.g., Ref. [11]) which, on occasion, makes their application very difficult for “non-experts.” In this chapter, two major contributions made by the authors' laboratories will be reviewed, both of which should help in identifying the goal of developing accurate, robust, and broadly applicable methods. These two techniques are: (i) double-hybrid density functionals (DHDFs) [12]; and (ii) an atom-pair wise London-dispersion correction scheme (DFT-D, DFT-D3) [13–15].

Both approaches have been implemented into many quantum chemistry codes, have attracted worldwide interest, and have often been applied very successfully. The theoretical background of both approaches will be reviewed in the following sections, with particular attention focused on the very recently developed PWPB95 functional [16] and the newest version of the dispersion correction (DFT-D3) [15]. Three examples are then described demonstrating the benefits of both approaches. First, a large benchmark study is discussed in Section 1.3.1, with attention focused on the PWPB95 functional and DFT-D3. A mechanistic study of B2PLYP and the DFT-D scheme is then detailed (see Section 1.3.2), to help understand the details of a recently reported reaction class. Finally, the description of excited states – and particularly of large chromophores – is shown to benefit from double-hybrid functionals (see Section 1.3.3).

1.2 Theoretical Background

1.2.1 Double-Hybrid Density Functionals

Double-hybrid density functionals are situated on the fifth rung in Perdew's scheme of “Jacob's ladder” [17], as they include virtual Kohn–Sham orbitals. Compared to hybrid-GGA functionals (fourth rung), where some part of the exchange functional is substituted by “exact” (HF) exchange, DHDFs additionally substitute some part of the correlation functional by mixing in a non-local perturbative correlation. This correlation part is basically obtained by a second-order M

ller–Plesset (MP2)-type treatment based on KS orbitals and eigenvalues. The first DHDF according to this idea is the B2PLYP functional by Grimme [12]. The first step in a double-hybrid calculation is the generation of Kohn–Sham orbitals from the hybrid-GGA portion of the DHDF. In the case of B2PLYP, this portion is denoted as B2LYP.

(1.1)

This hybrid-GGA part contains Becke 1988 (B88) [18] exchange

combined with non-local Fock-exchange

and Lee–Yang–Parr (LYP) [19, 20] correlation

. The

and

are mixing parameters for the “exact” Fock-exchange and perturbative correlation, respectively. A second-order perturbation treatment (PT2), based on the KS-orbitals and eigenvalues resulting from the B2LYP calculation, is carried out yielding the correlation energy

that is scaled by the mixing parameter,

. Thus, the final form of the B2PLYP exchange correlation energy is given by:

(1.2)

The two mixing parameters were fitted to the heats of formation (HOFs) of the G2/97 set; these parameters are

and

. Due to the perturbative contribution, B2PLYP formally scales with

, with

being the system size. However, if this step is evaluated using RI (density-fitting) schemes, the most time-consuming part is usually still the SCF and not the PT2 calculation.

Since B2PLYP, various other approaches have been reported, which are either modifications of B2PLYP [21–27] or are based on other (pure DFT-) exchange-correlation functionals [16, 28, 29]. These DHDFs usually differ in their amounts of Fock-exchange (between 50 and 82%). The impact of the Fock-exchange in a DHDF is depicted in Figure 1.1. Small amounts of

within common hybrid-functionals are good for main group thermochemical properties; however, these functionals suffer more from the SIE which, for example, influences the result for barrier heights. Too-large amounts, on the other hand, render density functionals (DFs) unstable when treating transition metal compounds. Smaller amounts of Fock-exchange effectively mimic the effect of treating static electron correlation, which makes the perturbative correction more stable (than e.g., MP2) in electronically complicated situations. Thus, DHDFs are also applicable to many open-shell problems for which a Hartree–Fock reference would strongly suffer from spin-contamination.

Figure 1.1 Effect of the amount of Fock-exchange in (double-)hybrid DFT calculations.

As a compromise to treat main group and transition metal chemistry equally well, a new DHDF was recently developed by the present authors which just contains 50% of Fock-exchange [16]. This is dubbed PWPB95, and is based on the Perdew–Wang (PW) GGA-exchange [30] and the Becke95 (B95) meta-GGA-correlation [31] functionals (inspired by Zhao's and Truhlar's PW6B95 hybrid-meta-GGA [32]). It is, thus, the first DHDF with meta-GGA ingredients:

(1.3)

In contrast to other DHDFs, for which inherent functional parameters (e.g.,

in B88) were not changed, PWPB95 is based on refitted PW and B95 parameters (three in the PW-exchange and two in the B95-correlation parts). Furthermore, PWPB95 includes a spin-opposite scaled second-order perturbative correlation contribution (OS-PT2) [33, 34]. Combined with an efficient Laplace transformation algorithm [35], this brings the formal scaling down from

to

with system size, which is the same as for conventional hybrid functionals.

The five inherent DFT parameters and the factor

were fitted on a fit set, covering various thermochemical energies (including noncovalent interactions). During the fitting procedure, the most recently developed empirical, atom-pairwise London-dispersion correction (DFT-D3)...

Table of contents

• Cover
• Related Titles
• Title Page
• Copyright
• Preface
• List of Contributors
• Part One: Theory and Concepts
• Part two: Applications in Homogeneous Catalysis
• Part Three: Applications in Pharmaceutical and Biological Chemistry
• Part Four: Applications in Main Group, Organic, and Organometallic Chemistry
• Index