Wednesday, December 30, 2009

Linear Algebra in Chemistry

Determinants and the Eigenvalue problem

In 2nd year quantum chemistry you will come across this object: \begin{vmatrix} \alpha - E & \beta & 0 & 0 \\ \beta & \alpha -E & \beta & 0 \\ 0 & \beta & \alpha -E & \beta \\ 0 & 0 & \beta & \alpha -E \\ \end{vmatrix} = 0
You divide by β and set (α − E) / β to equal x to get:
\begin{vmatrix} x & 1 & 0 & 0 \\ 1 & x & 1 & 0 \\ 0 & 1 & x & 1 \\ 0 & 0 & 1 & x \\ \end{vmatrix} = 0
Expand this out and factorise it into two quadratic equations to give:
(x2 + x − 1)(x2 − x − 1) = 0
which can be solved using x = -b \pm etc.

Simultaneous equations as linear algebra

The above determinant is a special case of simultaneous equations which occurs all the time in chemistry, physics and engineering which looks like this:
\begin{pmatrix} ( a_{11} - \lambda ) x_1 + a_{12} x_2 + a_{13} x_3 ..... + a_{1n} x_n = 0 \\ a_{21} x_2 + ( a_{22} - \lambda ) x_1 + a_{23} x_3 ..... + a_{2n} x_n = 0 \\ ..... \\ ..... \\ a_{1n} x_n + a_{2n} x_n + a_{n3} x_3 ..... + ( a_{11} - \lambda ) x_1 = 0\\ \end{pmatrix}
This equation in matrix form is ({\mathbf A} - \lambda {\mathbf 1}) {\mathbf x} = 0 and the solution is {\rm Det }({\mathbf A} - \lambda {\mathbf 1}) = 0.
This is a polynomial equation like the quartic above. As you know polynomial equations have as many solutions as the highest power of x i.e. in this case n. These solutions can be degenerate for example the π orbitals in benzene are a degenerate pair because of the factorization of the x6 polynomial from the 6 Carbon-pz orbitals. In the 2nd year you may do a lab exercise where you make the benzene determinant and see that the polynomial is
(x2 − 4)(x2 − 1)(x2 − 1) = 0
from which the 6 solutions and the orbital picture are immediately obvious.
The use of matrix equations to solve arbitrarily large problems leads to a field of mathematics called linear algebra.
Alexander Abi Chaker (20091978)
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6 comments:

  1. jack khodrDecember 31, 2009 at 1:39 AM

    its very intresting how a very complicated problem can be solved by using linear algerbra

    ReplyDelete
  2. UnknownJanuary 1, 2010 at 8:22 PM

    Alexander,

    Thanks first for writing your full name and ID number at the end of you post, this will make it easier for me.

    Second, your Post is well explained and developed. But if you do not mind, can you provide us with source of your information (website, article, book, etc)??

    Thanks again,

    Zeina

    ReplyDelete
  3. UnknownJanuary 4, 2010 at 1:13 AM

    Alex,

    I just found out that you have been disrespectful to another student! right?? We will have to apologize to him by writing a comment, AND talk to me after class. otherwise, some action will be taken!

    Thanks, Zeina

    ReplyDelete
  4. UnknownJanuary 7, 2010 at 7:02 PM

    Alex,

    Your POST does not relate matrices to a rela life situation! Can you please delete it and provide us with another example???

    Thanks, Zeina

    ReplyDelete
  5. AnonymousApril 27, 2011 at 11:04 PM

    HEYY ..
    I read all this and its really interesting!
    matrices are used a lot in statistical modeling as in linear models. Also, in integer programming and operations research, and linear programming, as in economic and business and scientific models. Also, for transformations in computer graphics and font manipulation. No doubt in many other places..


    Christelle saadeh
    (20101931)
    thank you ..

    ReplyDelete
  6. Vibration Data LoggerJune 27, 2012 at 8:45 AM

    Hi,
    Thank you very much Disertation writing, We appreciate your interest and suggestions.

    ReplyDelete

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