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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 det(U)=-1 if its rows/columns do not make up a right-handed system. When this happens, we need to compromise between two goals: maximizing Tr( Y T X' ) and respecting the constraint that det(U)=+1 . Therefore, we need to settle for the second largest value of Tr( Y T X' ) . 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 det(C)>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 sign(det(C)) . 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 V S W T
  • Compute d sign(det(C))
  • 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:

Equation relating the imaginary elements i, j and k

Properties of quaternion arithmetic follow directly from these equalities.
These equalities give rise to some unusual properties, especially with respect to multiplication.

Multiplication table for the imaginary elements i, j and k

Note that multiplication of i, j, and k is anti-commutative .

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

Definition of a quaternion

A quaternion is a number of the above form, where a, b, c, and d are real-valued scalars and i, j, and k are imaginary numbers as defined above.
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:

Quaternions p and q

Definition of quaternions p and q for later use.
Addition of p and q is fairly intuitive:

Addition of quaternions p and q

Quaternion addition closely resembles vector addition. Corresponding coefficients are added to yield the sum quaternion. This operation is associative and commutative.
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:

Dot (inner) product of p and q

The dot product of quaternions is analogous to the dot product of vectors.

Magnitude of quaternion p

As with vectors, the square of the magnitude of p is the dot product of p with itself.
Multiplication, however, is not, due to the definitions of i, j, and k:

Multiplication of quaternions p and q

This result can be confirmed by carrying out long multiplication of p and q. There is no analog in vector arithmetic for quaternion multiplication.
Quaternion multiplication also has two equivalent matrix forms which will become relevant later in the derivation of the alignment method:

Multiplication of quaternions p and q, matrix forms

Note that quaternions can be represented as column vectors with the imaginary components omitted. This allows vector notation to be used for many quaternion operations, including multiplication. The quaternion a + bi + cj + dk, for example, may be represented by a column vector of the form [a, b, c, d].
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:

Some properties of quaternion multiplication

Some useful properties. q* is the quaternion conjugate, a-bi-cj-dk

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, Geometric methods in structural computational biology. OpenStax CNX. Jun 11, 2007 Download for free at http://cnx.org/content/col10344/1.6
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