Gaussian overlap distributions

The properties of Gaussian overlap distributions are easily inferred from those of the constituent Gaussians. We note in particular the trivial relationships... 

If the ion exchange column is loaded with several ions of similar charge, B, C, etc., elution curves may be obtained for each ion by the use of appropriate eluants. If the elution curves are sufficiently far apart, as in Fig. 7.2, a quantitative separation is possible only an incomplete separation is obtained if the elution curves overlap. Ideally the curves should approach a Gaussian (normal) distribution (Section 4.9) and excessive departure from this distribution may indicate faulty technique and/or column operating conditions. 

Resolution, on the other hand, is a more technical term. It refers to the distance between adjacent bands relative to their bandwidths and acknowledges the fact that proteins are distributed in Gaussian profiles with overlapping distributions. The numerical expression for resolution is obtained by dividing the distance between the centers of adjacent bands by some measure of their average bandwidths. It expresses the distance between band centers in units of bandwidth and gives a measure of the overlap between two adjacent bands. For preparative applications, when maximal purity is desired, two proteins to be isolated should be separated by at least a bandwidth. In many applications it is sufficient to be able to simply discern that two bands are distinct. In this case bands can be less than a bandwidth apart. 

In Eq. (13), Wi and W2 are measured at the base of each peak. For Gaussian peak distributions, these widths equal 4a, and a column resolution of 1.0 means that there is 4.6% overlap of peak 1 in peak 2 and vice versa a resolution of 1.5 means there is 0.3% overlap between the two peaks. Generally, the resolution for a given separation can be increased by lengthening the column and, thus, increasing the number of theoretical plates. Here again, the trade-off for this increased resolution is a longer retention time. 

The electrostatic interactions in a molecule are determined by the structure of the density matrix D. In constructing D from an atomic orbital or atomic spinor basis, we incorporate a lot of redundant information in the Gaussian overlap charge distributions, since most of the electron density is concentrated near the nuclei. One should therefore try to transform D into a block diagonal form in which each dense block corresponds to a one-centre density so that the... 

Alternatively, the McMurchie-Davidson scheme [133] can be used for the computation of the spatial part of the matrix elements of Hpy (see [106]). The central concept of this method is to expand the product of two Gaussians (the so-called overlap distribution) in terms of Hermite functions according to... 

In the case of a flexible-coil molecule, even under quiescent conditions, there is no abrupt cut-off of overlap with decreasing concentration, owing to the Gaussian probability distribution of a segment occupying a volume at some distance from the origin. 

Assuming Gaussian density distributions, we obtain in the limit of a complete overlap, rc,i = Tc,2, an expression identical to the internal excluded volume interaction energy as given by Eq. (2.91), apart from a factor 1/2. Omitting the numerical prefactor of order unity we can write... 

For the LS method, N particles with mass m are placed with random positions in a square box with unit length (L = 1) in 2D (or cube with unit length in 3D) with no particle overlaps and a Gaussian velocity distribution at temperature T. The particle positions are updated using energy- and momentum-conserving... 

Cartesian Gaussians and products of two such Gaussians (i.e. the Cartesian overlap distributions) play an important role in the evaluation of molecular integrals. In the present section, we prepare ourselves for the study of integration techniques by examining the analytical properties of single Gaussians and their overlap distributions. 

Since the two Gaussians factorize into the three Cartesian directions, we may factorize the overlap distribution in the same way... 

According to the Gaussian product mie (9.2.10), the overlap distribution (9.2.20) may be written as a single Gaussian positioned at the centre of charge Px . 

Having discussed the Cartesian Gaussian functions and their overlap distributions, we are now ready to consider the evaluation of the simple one-electron integrals. By simple, we here mean the standard molecular integrals that do not involve the Coulomb interaction. In the present section, we thus discuss the evaluation of overlap integrals and multipole-moment integrals by the Obara-Saika scheme [5], based on the translational invariance of the integrals. We also... 

Since the Hermite Gaussians are to be used as one-centre basis functions for the expansion of Cartesian overlap distributions (over which the integration wiU eventually be carried out), it is important to determine the integrals over these Gaussians. The integral over the x component of the Hermite Gaussians is given by... 

Before we go on to consider the use of Hermite Gaussians as basis functions for overlap distributions, let us establish the relationship between Hermite Gaussians and Hermite polynomials. From Section 6.6.6, we recall that the Hermite polynomials may be generated from the Rodrigues... 

Let us consider the expansion of two-centre Cartesian overlap distributions in Hermite Gaussians. Since the overlap distribution is a polynomial of degree i -p j in xp (9.2.24), it may be expanded exactly in the Hermite polynomials of degree t < i + j. We therefore write... 

From these expressions, the fiill set of Hermite-to-Cartesian expansion coefficients may be generated and the overlap distribution may then be expanded in Hermite Gaussians according to (9.5.1). 

OVERLAP DISTRIBUTIONS FROM HERMITE GAUSSIANS BY RECURSION... 

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