Matrix elements of the secular determinant for trial functions?

In summary, the conversation is about using the quotient rule to differentiate an equation for solving wavefunctions of multi-electron atoms with the variation method. The equation involves constants H and S, and an adjustable variable c. The person is looking for a website or resource to learn more about the quotient rule after getting a different answer than expected. The answer using the quotient rule is shown.
  • #1
MontavonM
7
0
This is the quantum part for solving wavefunctions of mulit-electron atoms that need to be approximated by the variation method.

Specifically we are supposed to differentiate this equation using the quotient rule :

E(c1,c2) = [(c1^2*H11 + 2c1c2H12 + c2^2*H22) / (c1^2*S11 + 2c1c2*S12 + c2^2S22)]

Our book only shows the way to do it using the product rule after you bring the bottom onto the initial left side of the equation...

H and S are constants, with c as the adjustable variable of the equation... I think.

I've finished the differentiation but I got a different answer than I should have (we were given the answer)... Could anyone refer me to a website or something that shows the quotient rule? Thanks in advance!
 
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  • #2
The answer using the quotient rule is as follows: E'(c1,c2) = [(2*c1*H11 + 2*c2*H12 - (c1^2*S11 + 2*c1*c2*S12 + c2^2*S22)*(2*c1*H11 + 2*c2*H12)) / (c1^2*S11 + 2*c1*c2*S12 + c2^2*S22)^2]
 

Related to Matrix elements of the secular determinant for trial functions?

1. What are matrix elements of the secular determinant?

Matrix elements of the secular determinant refer to the coefficients of the terms in the secular determinant equation. They are used to solve for the energy levels and wave functions of a system in quantum mechanics.

2. Why are matrix elements important in solving the secular determinant?

Matrix elements are important because they represent the physical properties of the system and can be used to calculate the energy levels and wave functions. Without the matrix elements, it would be impossible to solve the secular determinant and understand the behavior of the system.

3. How are matrix elements calculated for trial functions?

Matrix elements for trial functions are calculated using the overlap integral between the trial function and the Hamiltonian operator. This involves integrating the product of the two functions over the entire volume of the system.

4. What is the role of matrix elements in approximating the secular determinant?

Matrix elements play a crucial role in approximating the secular determinant by providing a numerical representation of the physical properties of the system. These elements are used to set up the secular determinant equation, which is then solved to approximate the energy levels and wave functions of the system.

5. Can matrix elements be negative?

Yes, matrix elements can be negative. The sign of the matrix element depends on the specific properties of the system and the functions being used. In some cases, negative matrix elements may indicate the presence of an anti-bonding state, while positive matrix elements may indicate a bonding state.

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