**Quantum chemistry is the application of concepts developed in quantum mechanics to study the properties of molecules and their reactions. **

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Classical mechanics (CM), developed by Isaac Newton, describes the physics of large everyday objects. When the size of the objects becomes much smaller, to microscopic scales, the laws of Newtonian mechanics fail to explain the physics adequately. Quantum mechanics (QM) was formulated in the early 1900s to describe the behavior of matter at the atomic scale. This is critical in chemistry because the scale at which quantum mechanics becomes important is the scale at which electrons, atoms, and molecules exist.

## Differences Between Classical and Quantum Mechanics

Apart from the size of the object studied, there are several properties of QM which differ from CM that are important for understanding chemistry.

QM deals with objects that are very light. For example, fundamental particles of matter like electrons. CM governs the laws of matter that are heavier and macroscopic in nature.

CM is continuous. This means that physical quantities, like position, velocity, energy, or momentum of an object can change in very small amounts. By contrast, in QM, the behavior of objects is discrete or “quantized.” Discrete means there is a finite number of states that a particle can exist in. The energy of a quantum system cannot change its value continuously but can only occupy certain allowed values.

Newton’s equations govern CM. The solutions of Newton’s equations map the trajectory of the object studied. The solutions to the equations are deterministic and can be repeatedly predicted to be the same under the same experimental conditions.

The Schrodinger equation describes the evolution of a quantum system. The equation gives a wave function, which describes the particle as a wave. The wave is spread out over space and it doesn’t have well-defined values for its exact position or momentum. The uncertainties in the position and momentum are described by the Heisenberg Uncertainty Principle. Named after German physicist Werner Heisenberg, the Uncertainty Principle shows that the product of the uncertainty in position and the uncertainty in momentum is greater than or equal to a constant.

Therefore, the solutions to the Schrodinger equation are probabilistic. The wave function gives the probability for various possible outcomes, but it does not give exact outcomes like in a CM trajectory.

Schrödinger's wave equation can be solved exactly for simple quantum systems such as the hydrogen atom. However, for larger atoms and molecules that contain more than one electron, calculating the Schrödinger equation becomes difficult. Solutions are then based on established methods that give approximate results.

## Molecular Structure

The internal structure of a molecule is quite complex when compared to atoms. Molecules have internal degrees of freedom like rotational and vibrational states. The addition of extra degrees of freedom makes it difficult to compute structural models of the internal dynamics.

Molecules are linked together by sharing electrons in what is fundamentally known as covalent bonds. Mathematical simulation of covalent bonds takes two prominent approaches. The valence bond model and the molecular orbital model.

## Molecular Bond Theories

The valence bond model describes the creation of a bond orbital, where an electron from two atomic orbitals is shared. The valence bond model, while providing a simple quantum mechanical picture of chemical bonding, does not accurately describe molecules with more than two atoms.

In molecular orbital theory, the electron occupies a molecular orbital that is spread out over the entire molecule. This is different from the valence bond model in that it does not depend on electrons to be localized between two atoms that form a molecular bond. In the molecular orbital theory, summing up the total contributions of each atomic orbital forms the molecular orbital.

Molecular orbital theory calculations produce more accurate results when predicting molecular structure than valence bond theory. Consequently, quantum chemistry simulation software packages rely on molecular orbital methods to calculate molecular structure and dynamics.

## Outlook

Quantum chemical calculations have placed the known elements in the periodic table. Interaction of molecules and atoms with light and their spectroscopic behavior have also been studied in more depth through orbital theories. Further research efforts are undertaken to improve the valence bond and molecular orbital methods for calculating molecular properties.

Experimental data are commonly used to adjust and compare previous calculations to build the models. Computationally challenging quantum chemistry calculations are expected to be solved more efficiently when quantum computers become available.

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## References and Further Reading

Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik, Chemical Reviews 2019 *119* (19), 10856-10915 https://pubs.acs.org/doi/10.1021/acs.chemrev.8b00803

Yaoquan Tu, Aatto Laaksonen, Chapter 1 - Implementing Quantum Mechanics into Molecular Mechanics—Combined QM/MM Modeling Methods, Advances in Quantum Chemistry, Academic Press, Volume 59, 2010, Pages 1-15, https://linkinghub.elsevier.com/retrieve/pii/S0065327610590014

Whitaker, Andrew (1996). *Einstein, Bohr, and the Quantum Dilemma. *New York: Cambridge University Press.

Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan, Chemical Reviews 2020 *120* (22), 12685-12717 https://pubs.acs.org/doi/abs/10.1021/acs.chemrev.9b00829