Semi-empirical methods

In the vast and complex world of theoretical and computational chemistry, it is important to bridge the gap between quantum mechanics and the empirical inaccuracies observed in some experimental data. This is where semi-empirical methods play a key role. Their basis relies on the quantum mechanical treatment of atoms and molecules, but they simplify many aspects to make calculations feasible for large systems.

Introduction

Semi-empirical methods are a subset of quantum chemistry methods that simplify the calculations involved in predicting molecular structures, chemical reactions, and property estimations such as electronic configurations and energy levels. By using empirical data to support quantum mechanical equations, these methods offer a balance between comprehensive theories and computational efficiency.

What are semi empirical methods?

Semi-empirical methods, by their nature, take advantage of available experimental data to fill in gaps or inaccuracies that purely ab initio calculations might struggle with. Typically, these methods use reduced basis sets of atomic orbitals and make approximations to the electronic structure problem. The electronic Hamiltonian is simplified, often leading to parametrized expressions that fit against experimental data.

Basics of quantum chemical calculations

A brief overview of quantum chemical computations is necessary to understand semi-empirical methods. In their most basic form, quantum chemical methods aim to solve the Schrödinger equation:

        hΨ = eΨ
    

Where:

• H is the Hamiltonian of the system, which represents the total energy.
• Ψ is the wave function describing the quantum state of the system.
• E is the energy associated with that state.

Motivation and need

While ab initio methods provide an accurate depiction of many molecular systems without parameter dependence, they are computationally intensive. For larger molecules, these methods become impractical. Semi-empirical methods provide a more computationally efficient alternative by relying on some approximations and experimentally obtained parameters.

General semi-empirical methods

Extended Hückel theory (EHT)

One of the earliest forms of semi-empirical methods, the extended Hückel theory, focused on estimating molecular orbital energies without considering interactions between electrons beyond the nearest neighbors. Despite being simple, it laid the foundation for more sophisticated methods.

CNDO, INDO, and NDDO

These methods, derived from the self-consistent field (SCF) approach, make various approximations:

• CNDO: Complete neglect of differential overlap, assuming no differential overlap between atomic orbitals.
• INDO: Intermediate neglect of differential overlap provides an improvement by adding some overlap integrals.
• NDDO: Neglecting the diatomic differential overlap is a more sophisticated approach, which improves upon CNDO and INDO.

MNDO, AM1, and PM3

As the field evolved, methods such as MNDO (modified neglect of diatomic overlap), AM1 (Austin model 1), and PM3 (parameterized model number 3) were developed to provide better accuracy for a wider range of molecules.

Mathematical formulation and approximation

Hamiltonian approximation

In semi-empirical methods the Hamiltonian is often simplified to:

        H_ij≈S_ij*β
    

Where:

• H_ij is the Hamiltonian matrix element between orbitals i and j.
• S_ij is the overlap integral between orbitals.
• β is the empirical parameter adjusted to suit the experimental results.

Parameterization

Parameters that are important in semi-empirical methods are obtained by fitting calculations to experimental data such as ionization potentials, bond lengths and angles. This increases accuracy without extensive computational demands.

Visual Example

Consider the molecular orbital structure of ethylene calculated from semi-empirical methods. A simplified energy level diagram can be seen as follows:

        ,
        ,
        ,
        |x_* |
        ,
             homo lumo
    

This diagram shows the calculated energies of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), with semiempirical methods predicting the energy gap considering experimental data.

Application

Semi-empirical methods have been widely applied in various branches of chemistry:

• Organic Chemistry: Providing insight into reaction mechanisms and predicting product distributions.
• Materials Science: Helping to understand the electronic properties of conducting polymers.
• Pharmacology: Aiding drug design by predicting molecular interactions and conformational changes.

Boundaries

Despite their utility, semi-empirical methods face certain limitations:

• Parameter dependency: Since these methods rely heavily on parameterization, their accuracy can vary across different molecule types.
• Simplifying assumptions: Assumptions such as neglecting electron correlation can lead to inaccuracies for complex systems.

Conclusion

Semi-empirical methods stand at the intersection of empirical data and quantum mechanics, providing a practical approach to understanding the complexities of chemistry. While advances in computational capabilities continue to push the boundaries toward more accurate methods, semi-empirical models remain important, especially for large systems where computational efficiency is paramount. Through their hybrid nature, they bridge the gap between theoretical calculations and experimental observations, making substantial contributions to the toolkit of modern chemistry.

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