Abstract
Quantum chemical calculations rely on a few fortunate circumstances, like usually small relativistic and negligible electrodynamic (QED) corrections, and large nuclei-to-electrons mass ratio. Unprecedented progress in computer technology has revolutionized quantum chemistry, making it a valuable tool for experimenters. It is important for computational chemistry to elaborate methods that look at molecules in a multiscale way, provide its global and synthetic description, and compare this description with those for other molecules. Only such a picture can free researchers from seeing molecules as a series of case-by-case studies. Chemistry is a science of analogies and similarities, and computational chemistry should provide the tools for seeing this.
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Notes
- 1.
Computational chemistry contributed significantly to applied mathematics, because new methods had to be invented in order to treat the algebraic problems of a previously unknown scale (like for M of the order of billions), see, e.g., Roos (1972).
- 2.
That is, derived from the first principles of (non-relativistic) quantum mechanics.
- 3.
It is difficult to define what computational chemistry is. Obviously, whatever involves calculations in chemistry might be treated as part of it. This, however, sounds like a pure banality. The same is true with the idea that computational chemistry means chemistry that uses computers. It is questionable whether this problem needs any solution at all. If yes, the author sticks to the opinion that computational chemistry means quantitative description of chemical phenomena at the molecular level.
- 4.
Perhaps the best known is GAUSSIAN, elaborated by a large team headed by John Pople.
- 5.
The speed as well as the capacity of computer, memory increased about 100 billion times over a period of 40 years. This means that what now takes an hour of computations, would require in 1960 about 10,000 years of computing.
- 6.
In addition, we assume the computer is so clever, that it automatically rejects those solutions, which are not square-integrable or do not satisfy the requirements of symmetry for fermions and bosons. Thus, all non-physical solutions are rejected.
- 7.
Bond patterns are almost the same for different conformers.
- 8.
For a dipeptide one has something like ten energy minima, counting only the backbone conformations (and not counting the side chain conformations for simplicity). For a very small protein of, say, a hundred amino acids, the number of conformations is therefore of the order of 10100, a very large number exceeding the estimated number of atoms in the Universe.
- 9.
The low-frequency vibrations may be used as indicators to look at possible instabilities of the molecule, such as dissociation channels, formation of new bonds, etc. Moving all atoms, first according to a low-frequency normal mode vibration and continuing the atomic displacements according to the maximum gradient decrease, we may find the saddle point, and then, sliding down, detect the products of a reaction channel.
- 10.
The integration of \(\vert \Psi {\vert }^{2}\) is over the coordinates (space and spin ones) of all the electrons except one (in our case the electron 1 with the coordinates \(\mathbf{r},{\sigma }_{1}\)) and in addition the summation over its spin coordinate \(({\sigma }_{1})\). As a result one obtains a function of the position of the electron 1 in space: \(\rho (\mathbf{r})\). The wave function \(\Psi \) is antisymmetric with respect to exchange of the coordinates of any two electrons, and, therefore, \(\vert \Psi {\vert }^{2}\) is symmetric with respect to such an exchange. Hence, the definition of \(\rho \) is independent of the label of the electron we do not integrate over. According to this definition, \(\rho \) represents nothing else but the density of the electron cloud carrying N electrons, and is proportional to the probability density of finding an electron at position r.
- 11.
Strictly speaking the nuclear attractors do not represent critical points, because of the cusp condition (Kato 1957).
- 12.
We may also analyze \(\rho \) using a “magnifying glass” represented by − Δρ.
- 13.
One has to be aware of a related mathematical trap. Applying even the smallest uniform electric field immediately transforms the problem into one with metastable energy (the global minimum corresponding to dissociation of the system, with the energy equal to \(-\infty \)), see, e.g., Piela (2007), p. 642.
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Acknowledgments
The author is very grateful to his friends, Professor Andrzej J. Sadlej and Professor Leszek Z. Stolarczyk, for the joy of being with them, discussing all exciting aspects of chemistry, science and beyond; a part of them is included in the present chapter.
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Piela, L. (2012). From Quantum Theory to Computational Chemistry. A Brief Account of Developments. In: Leszczynski, J. (eds) Handbook of Computational Chemistry. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0711-5_1
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