# 0.4 Molecular distance measures  (Page 4/6)

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Where $V$ and ${W}^{T}$ are the matrices of left and right eigenvectors, respectively, of the covariance matrix $C=X{Y}^{T}$ . This formula ensures that $U$ is orthonormal (the reader should carry out the high-level matrix multiplication and verify this fact).

The only remaining detail to take care of is to make sure that $U$ is a proper rotation, as discussed before. It could indeed happen that $\mathrm{det\left(U\right)=-1}$ if its rows/columns do not make up a right-handed system. When this happens, we need to compromise between two goals: maximizing $\mathrm{Tr\left(}$ ${Y}^{T}\mathrm{X\text{'}}$ $\right)$ and respecting the constraint that $\mathrm{det\left(U\right)=+1}$ . Therefore, we need to settle for the second largest value of $\mathrm{Tr\left(}$ ${Y}^{T}\mathrm{X\text{'}}$ $\right)$ . It is easy to see what the second largest value is; since:

then the second largest value occurs when ${T}_{11}={T}_{22}=+1$ and ${T}_{33}=-1$ . Now, we have that $T$ cannot be the identity matrix as before, but instead it has the lower-right corner set to -1. Now we finally have a unified way to represent the solution. If $\mathrm{det\left(C\right)>0}$ , $T$ is the identity; otherwise, it has a -1 as its last element. Finally, these facts can be expressed in a single formula for the optimal rotation $U$ by stating:

where $d=\mathrm{sign\left(det\left(C\right)\right)}$ . In the light of the preceding derivation, all the facts that have been presented as a proof can be succinctly put as an algorithm for computing the optimal rotation to align two data sets $x$ and $y$ :

## Optimal rotation

• Build the 3xN matrices $X$ and $Y$ containing, for the sets $x$ and $y$ respectively, the coordinates for each of the N atoms after centering the atoms by subtracting the centroids.
• Compute the covariance matrix $C=X{Y}^{T}$
• Compute the SVD (Singular Value Decomposition) of $C=VS{W}^{T}$
• Compute $d=\mathrm{sign\left(det\left(C\right)\right)}$
• Compute the optimal rotation $U$ as

## Optimal alignment for lrmsd using quaternions

Another way of solving the optimal rotation for the purposes of computing the lRMSD between two conformations is to use quaternions . These provide a very compact way of representing rotations (only 4 numbers as compared to 9 or 16 for a rotation matrix) and are extremely easy to normalize after performing operations on them. Next, a general introduction to quaternions is given, and then they will be used to compute the optimal rotation between two point sets.

## Introduction to quaternions

Quaternions are an extension of complex numbers. Recall that complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the canonical imaginary number, equal to the square root of -1. Quaternions add two more imaginary numbers, j and k. These numbers are related by the set of equalities in the following figure:

These equalities give rise to some unusual properties, especially with respect to multiplication.

Given this definition of i, j, and k, we can now define a quaternion.

Based on the definitions of i, j and k, we can also derive rules for addition and multiplication of quaternions. Assume we have two quaternions, p and q, defined as follows: Addition of p and q is fairly intuitive: The dot product and magnitude of a quaternion also closely resemble those operations for vectors. Note that a unit quaternion is a quaternion with magnitude 1 under this definition: Multiplication, however, is not, due to the definitions of i, j, and k: Quaternion multiplication also has two equivalent matrix forms which will become relevant later in the derivation of the alignment method: These useful properties of quaternion multiplication can be derived easily using the matrix form for multiplication, or they can be proved by carrying out the products:

#### Questions & Answers

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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