Semiempirical, subsystem-based methods for high-accuracy electronic structure calculations

I. Overview

Current electronic structure methods do not transfer information about different molecules between calculations, essentially starting each calculation from scratch. These methods work well for small molecules, but are too expensive to use on large molecules. Chemical insight tells us that the properties of molecular functional groups (e.g. methyl, phenyl, and other groups) are transferrable, such that these groups behave approximately the same in many different molecules. This suggests that we can do fast, high-accuracy calculations on large molecules by extracting this transferrable information from calculations on small molecules. We implement this idea by generating large data sets of relatively inexpensive, high-accuracy calculations on small molecules. We construct accurate models of the properties of molecular functional groups by using data-mining techniques to extract model parameters from these high-accuracy data sets. Finally, we model the properties of large molecules as the sum of the predicted properties of their functional groups. A schematic of this approach is shown in Figure 1. Our initial focus has been on developing accurate, functional-group-based treatments of electron correlation (see below).

II. Introduction to Computational Chemistry

Computational chemistry is the field of using computers to model, understand, and ultimately predict the behavior of molecules. The predictions of computational chemistry have been used to help design molecules that catalyze important chemical reactions, molecules that act as molecular switches, and drug molecules that bind to and inactivate infectious agents. (An introduction to the role of computational chemistry in drug design can be found in Chemical and Engineering News, 79(23), 69-74, June 4, 2001; http://pubs.acs.org/cen/coverstory/7923/7923drugdesign.html)

Molecules are made up of atoms, which bind to each other by sharing electrons. All of a molecule’s properties are a product of its structure. Thus, much of computational chemistry is focused on predicting the behavior of systems of many electrons. Predicting the behavior of a many-electron system is analogous to making predictions about a system of many classical particles, such as the planets in the solar system. In both cases, the equations that describe the system have been known for a long time (decades for the quantum-mechanical Schrodinger’s equations, centuries for Newton’s equations for classical particles). However, the equations typically cannot be solved analytically for more than two interacting particles, and calculating a solution on a computer becomes increasingly expensive for large systems. In both cases, making useful predictions requires judicious approximations.

IIa. Quantum Chemistry from First Principles

The standard methods for solving Schrodinger’s equation for a many-electron molecule are called “ab initio” methods, Latin for “from first principles”. These methods start every calculation from scratch, and do not transfer information between calculations. Ab initio calculations give highly accurate electronic structures for small molecules. Unfortunately, they are also very expensive. Many interesting molecules, such as proteins, are simply too large to be treated by ab initio methods.

IIb. Transferrability in Molecules

One way to reduce the expense of ab initio calculations is the “Tinkertoy approximation”. Chemistry is based on the idea that real molecules are like the Tinkertoy-style molecular modeling kits used in high-school chemistry. Atoms and their attendant electrons tend to form a few stable structural motifs whose shapes, and properties, are similar in many different molecules. Examples of such structural motifs include the functional groups of organic chemistry. These groups have many transferrable properties, the most dramatic of which may be their characteristic vibrational frequencies. Examples of some common organic functional groups, and ranges for their characteristic vibrational frequencies, can be found at http://chipo.chem.uic.edu/web1/ocol/spec/IRTable.htm

In classical physics, two particles connected by a spring will vibrate at a frequency that corresponds to the spring’s stiffness. Interatomic bonds can be thought of as “quantum springs” that can absorb light waves whose frequency (color) equals the bond’s vibrational frequency. (For example, a carbon-oxygen double bond vibrates at a frequency of around 5.25*10^14 cycles per second, and absorbs infra-red light at a characteristic frequency of ˜1750 cm^-1). The characteristic vibrational frequencies of a group are very similar in different molecules, such that infra-red absorbance spectrum of an organic molecule provides a good characterization of what functional groups it contains.

IIc. Semiempirical Quantum Chemistry: Saving Time with Transferrability

Electronic structure methods that incorporate this idea of similarity are referred to as “semiempirical”, Latin for “partly from experiment”. Semiempirical methods simplify the exact Schrodinger’s equation by using parameters that capture the transferable characteristics of different functional groups. These parameters are either fitted to reproduce experimental results, or obtained from accurate ab initio calculations on small molecules. An example is the popular ball-and-spring molecular mechanics approximation. This method predicts molecules’ structure by treating inter-atomic bonds as classical springs whose spring constants correspond to the characteristic vibrational frequencies mentioned above.

Unfortunately, existing semiempirical methods are not as accurate as first-principles methods. Some phenomena, such as bond breaking, can only be accurately described using first-principles methods. Despite this, the approximation of transferrability should still apply to these phenomena. For example, bond-breaking phenomena should be transferrable, as the amount of work needed to break a chemical bond is closely related to that bond’s characteristic vibrational frequency.

III. A New Approach to Semiempirical Quantum Chemistry

Our work involves generating new semiempirical methods that take advantage of both transferability and the high-accuracy ab initio calculations that can be done on small molecules. The basic approach is as follows: To model a specific functional group, we first do high-accuracy ab initio calculations on that group in a set of small molecules, and in various environments. If the functional group’s behavior is truly transferable, these ab initio calculations should contain ALL the information needed to describe the functional group in any kind of molecule. We then use data mining methods to extract this transferable information, and parametrize a simple model to the results.

IV. Implementation of the Approach: A Semiempirical Model for Electron Correlation

The simplest way to treat a system of many interacting particles (electrons, planets, quarks, etc.) is to smear out the inter-particle interactions such that each particle only sees the average positions of its neighbors. The most important variable in this “mean-field” approximation is the particle density $^1D$: the average probability that a particle will be found at some position in space. The mean-field approximation effectively decouples the particles from each other, turning a complex system of N particles into N simple systems, each containing one particle and the particle densities of its (N-1) neighbors. (Mean-field calculations are solved by iteration: calculate the position of particle 1 from the densities (average positions) of particles 2 through N, then calculate the position of particle 2 using the densities of particles 3 through N and the new density of particle 1, and so on until the particle densities converge.)

The difference between a systems’s exact and mean-field solutions is referred to as correlation. Electron correlation is the most computationally expensive part of an electronic structure calculation. Unfortunately, it can also be a very important part: for example, dispersion (van der Waals) interactions are due entirely to electron correlation and cannot be modeled by a mean-field approximation. All electron correlation effects are contained in the electron pair density $^2D$, which contains the average positions of PAIRS of electrons. A schematic of the electron density and electron pair density of a simple molecule are shown in Figure 2.

We have developed a semiempirical model for electron correlation that assumes that correlation effects are transferrable. This model has two parts: a method for combining electron correlation effects from different functional groups, and a method for treating electron correlation effects in a functional group as a transferrable function of the group’s electron density.

IVa. Combining Correlation Effects from Different Functional Groups

We recently developed the “localized reduced density matrix”, or LRDM, approximation for treating electron correlation (J. Chem. Phys., 119, 1320-28 (2003)). This approximation allows us to decompose electron correlation effects into functional-group contributions, by assembling a large molecule’s electron pair density from the calculated pair densities of its functional groups.

IVb. Predicting Correlation Effects in a Single Functional Group as a Transferrable Function of Electron Density

Our functional-group-based decomposition of electron correlation effects allows us to build a model that takes advantage of the transferrability of electron correlation effects. We assume that the electron pair density of a functional group is a transferrable function of its electron density. (Density functional theory, which is the current standard for doing quantum chemical calculations on large molecules, makes a similar but more severe version of this assumption.) To model a functional group, we first build a database of high-accuracy calculations of the group’s electron density and electron pair density in a set of small molecules. Then, we use data mining methods to fit a function that predicts the group’s pair density $^2D$ as a function of its electron density $^1D$. Given [$^1D$]->[$^2D$] functions for all the functional groups in a molecule, we can model electron correlation effects in the molecule by (a) decomposing the molecule’s electron density into functional-group components, (b) predicting each functional group’s electron pair density using the prediction function, and (c) combining the predicted pair densities using LRDM.

We tested this method on two systems. The first system was a linear chain of ten hydrogen atoms, modeled as four identical, overlapping four-atom “functional groups”. The second system was the aldehyde (HOC-) functional group of a set of HOC-R molecules. In both cases, we were able to parametrize transferrable functions that gave very good predictions of the subsystem electron correlation effects. This work was recently accepted for publication in the Journal of Chemical Physics. Preprints may be obtained from the e-print archive at http://arXiv.org, indexed as physics/0311091. Figure 3 shows some representative results from that work.