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2017 | OriginalPaper | Chapter

4. Remarks on Wave Function Theory and Methods

Authors : Dariusz Ke dziera, Anna Kaczmarek-Kedziera

Published in: Handbook of Computational Chemistry

Publisher: Springer International Publishing

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Abstract

Methods of computational chemistry seem to often be simply a melange of undecipherable acronyms. Frequently, the ability to characterize methods with respect to their quality and applied approximations or to ascribe the proper methodology to the physicochemical property of interest is sufficient to perform research. However, it is worth knowing the fundamental ideas underlying the computational techniques so that one may exploit the approximations intentionally and efficiently. This chapter is an introduction to quantum chemistry methods based on the wave function search in one-electron approximation.

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previous chapter The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics

next chapter Adiabatic, Born-Oppenheimer, and Non-adiabatic Approaches

Here, different subscripts appear on the both sides of the integral. The sum does not depend on the name of the summation index; thus, any subscript can be applied. However, one should not apply the same index on both sides of the integral, since it can cause the erroneous omission of the off-diagonal terms. Compare the overlap integral

$$ \begin{array}{rl}\left\langle \varPsi \Big|\varPsi \right\rangle &=\left\langle {\displaystyle \sum_i{c}_i{\chi}_i}\left|{\displaystyle \sum_j{c}_j{\chi}_j}\right.\right\rangle =\left\langle {c}_1{\chi}_1+{c}_2{\chi}_2+{c}_3{\chi}_3+\dots \left|{c}_1{\chi}_1+{c}_2{\chi}_2+{c}_3{\chi}_3+\dots \right.\right\rangle \\ {}&={c}_1^{*}{c}_1\left\langle {\chi}_1\left|{\chi}_1\right.\right\rangle +{c}_1^{*}{c}_2\left\langle {\chi}_1\left|{\chi}_2\right.\right\rangle +{c}_1^{*}{c}_3\left\langle {\chi}_1\left|{\chi}_3\right.\right\rangle +\dots \end{array} $$

The explicit writing of all terms shows that not only the integrals 〈χ _i|χ _i〉 with the same function on both sides are present, but also the contributions 〈χ _i|χ _j〉 with $ i\ne j $. Therefore, the diversification of the subscripts prevents mistakes.

In order to explain the antisymmetry requirement, we have to refer again to theory that is beyond the scope of the present chapter. Let us simply state here that wave functions must be antisymmetric without belaboring the point. This will mean that the exchange of the coordinates of the two electrons causes the wave function to change the sign: $ \Psi \left({\tau}_1,{\tau}_2\right)=-\Psi \left({\tau}_2,{\tau}_1\right). $

Since the value of the integral does not depend on the name of the variable, the obvious equality $ {\displaystyle {\int}_a^bf(x)dx}={\displaystyle {\int}_a^bf(y) dy} $ in the above case takes the form

$$ \left.\left\langle {\phi}_i(1){\phi}_j(2)\left|\frac{1}{r_{12}}\right|{\phi}_j(1){\phi}_i(2)\right\rangle \right|=\left\langle {\phi}_i(2){\phi}_j(1)\left|\frac{1}{r_{12}}\right|{\phi}_j(2){\phi}_i(1)\right\rangle . $$

Atkins, P., & Friedman, R. (2005). Molecular quantum mechanics. Oxford: Oxford University Press.

Cramer, C. J. (2004). Essentials of computational chemistry: Theories and models. Chichester: Wiley.

Griffiths, D. J. (2004). Introduction to quantum mechanics (2nd ed.). San Francisco: Benjamin Cummings.

Jensen, F. (2006). Introduction to computational chemistry. Chichester: Wiley.

Levine, I. N. (2008). Quantum chemistry (6th ed.). Englewood Cliffs: Prentice Hall.

Lowe, J. P., & Peterson, K. (2005). Quantum chemistry (3rd ed.). Boston: Academic.

McQuarrie, D. A., & Simon, J. D. (1997). Physical chemistry: A molecular approach. Sausalito: University Science.

Piela, L. (2007). Ideas of quantum chemistry. Amsterdam: Elsevier.

Ratner, M. A., & Schatz, G. C. (2000). Introduction to quantum mechanics in chemistry. Upper Saddle River: Prentice Hall.

Roos, B. O., & Widmark, P.-O. (Eds.). (2002). European summerschool in quantum chemistry. Lund: Lund University.

Szabo, A., & Ostlund, N. S. (1996). Modern quantum chemistry: Introduction to advanced electronic structure theory. Mineola: Dover.

Title: Remarks on Wave Function Theory and Methods
Authors: Dariusz Ke dziera
Anna Kaczmarek-Kedziera
Publisher: Springer International Publishing
Book: Handbook of Computational Chemistry
Print ISBN: 978-3-319-27281-8

Electronic ISBN: 978-3-319-27282-5

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-27282-5_3

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