The Electron Density as Molecular Descriptor

It is not the aim of this chapter to give a detailed account of quantum chemistry, but it is important to introduce some of its elements that will be needed to comprehend the rest of the chapter. One of the basic postulates of quantum mechanics applied to chemistry states that for every molecule or state a wave function exists that is all-determining. This means that, once the wave function is known, every so-called observable property for an N-electron molecule may be obtained by straightforward integration as in Eq. [1]  

Despite the fundamental role played by the wave function in quantum chemistry, it has no physical meaning in itself. The square of the wave function given in Eq. [2], in contrast, is interpreted as the probability of finding an electron in the volume dr around r. Eor ease of notation, consider a wave function. The aforementioned probability density becomes 

It is this concept that plays the most important role in molecular quantum similarity. Consider now the wave function shown in Eq. [1]. It is well known how to reduce the dimensionality of the density function p in Eq. [2] by integration. The density function p is generated from the wave function, which belongs to H C), a Hilbert space over the complex field. The density function belongs to H(R+), a Hilbert space over the positive real field only. In other words, generating the density function occurs through the following rule  

If a spinless -th order density function is desired for an N-electron molecule, integration should be performed over (N- ) spatial coordinate sets and all N spin variables. The first-order electron density in a point r is then  

Note that in Eq. [4], the integration is carried out over all coordinates (spatial and spin) of the electrons 2 to N, and over the spin only for the first electron. 

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The next section, entitled Basic Aspects of Molecular Similarity, gives a general overview of molecular similarity and the usual vocabulary used by chemists in this field. In the section entitled The Electron Density as Molecular Descriptor, some elements of quantum chemistry will be presented. These sections should not be expected by the reader to offer a rigorous discussion of all aspects of such broad fields, and therefore, only the most important definitions and concepts will be introduced. The following sections will then address extensively the subject of molecular quantum similarity in both theoretical and practical aspects. An ample references list will help the interested reader to look up the more specialized literature. 

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