# 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:

what is the stm
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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
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What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
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Rafiq
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Rafiq
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
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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absolutely yes
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it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
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what is fullerene does it is used to make bukky balls
are you nano engineer ?
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fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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