Carbo index

Cosine similarity coefficient Also known as the Carbo index c. EllWiB s... 

Many applications of QSM use not the MQSM values as such but similarity indices. There exist many such indices and for a review, the reader is referred to Willett et al. [54]. One example that was touched upon above is the Carbo index. The Carbo index is one of the so-called C-class descriptors giving a value between 0 and 1 where a higher value indicates higher similarity. Although originally defined for overlap MQSM, it can be generalized to different operators f so that the Carbo index becomes... 

Besides the Carbo index another index that has been used in the field of QSM is the Hodgkin-Richards index, which uses an arithmetic mean instead of the geometric... 

According to some authors, the similarity between shape functions might give more interesting information than using the electron density [63]. In this context, it should be mentioned that the Carbo index retains the same value. Moreover, it has... 

The value of SI(II) also varies in the range 0 to 1. When pA = npB (n is constant and 0) is substituted into Eq. (15), the Carbo index will be unity, which means that the Carbo index represents the similarity of the shapes of electron density distributions but not of the magnitudes. When pA = npB is substituted into Eq. (16), the result will be SI(H) = 2n/(l + n2), which means that the Hodgkin index characterizes the similarity in the shape as well as in the magnitude of the electron density functions [104],... 

For field-based measures and overlap of electron density functions then the Carbo index can be used (49), which is equivalent to the Cosine coefficient. 

The Carbo index is sensitive to the shape of the property distributions, but not to their magnitude. 

With respect to the Carbo index, it is less sensitive to the shape of the property distribution but more sensitive to its magnitude [Hodgkin and Wchards, 1987 Richards and Hodgkin, 1988]. It is defined as ... 

Fig. 4. Projection of the point-molecule representation of the nine molecules using the Carbo index obtained from a gravitational-like similarity measure. The studied property is the boiling point...

Several measures of the shape and electronic similarity between pairs of molecules have been devised. The Carbo index was developed to enable the electron density of two molecules in some relative orientation to be compared [Carbo et ah 1980]. It is essentially the Cosine... 

The Carbo index Rab was originally proposed as a method of comparing molecules in terms of their electron density q [1066]. In a more general version (eq. 212) this similarity index can be applied to compare any structural properties and Pb of two molecules A and B. 

An alternative to the Carbo index is the Hodgkin index H b (eq. 213) [1067]. While the Carbo index is sensitive to the shape of the property distribution, the Hodgkin index reflects more its magnitude. 

The current version of this approach uses N x N matrices (N being the number of molecules included in the data set), in which the 3D similarity indices of each molecule as compared to every other molecule of the whole set are the non-diagonal elements of the similarity matrix [1064, 1065]. In an application to the affinities of 31 steroids to corticosteroid- and testosterone-binding globulins (same data as used in CoMFA studies, e.g. [36]), Carbo index similarity matrices were calculated for the shape and the electrostatic potentials of all molecules. Using the similarity... 

Gaussian electrostatic evaluations of the Carbo index provided better results than grid-based calculations, while in the case of shape the grid-based evaluations seemed to be more suitable. Grid distances of 0.05 nm for shape and 0.2 nm for electrostatics were sufficient to give a satisfactory description of the biological data. 

This generalized cosine index is often called the Carbo index. Naturally the Carbo index is limited to the range (0,1), where Cab = 1 means perfect similarity. Still more (dis)similarity measures have been introduced and will be discussed further in this chapter. The range of the Carbo index naturally agrees with the Schwartz integral inequality " ... 

Maximizing the Carbo index has been described in several previous scientific reports. McMahon and King described the use of gradient methods in 1997, and in the same year, Parretti et described the use of Monte Carlo techniques. In many studies, including the two just mentioned, these maximizations do not refer to quantum similarity, but instead they refer to maximizing the similarity in molecular electrostatic potentials, which is different. 

In the Carbo index, the denominator is the geometric mean of the self-similarities. Naturally, we could also think about using the arithmetic mean, which gives rise to the Hodgkin-Richards index " " " originally developed for comparisons of electrostatic potentials ... 

This index, however, has no direct geometrical interpretation, as has the Carbo index. 

In all three of these approaches, the similarity indices should be maximized. Both the Carbo index and the Hodgkin-Richards index were used by Boon et al. % whereas in the applications of AIM, the Cioslowski similarity measure was For the atom-centered approach, we can use all... 

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