Help with a derivation from a paper (diatomic molecular potential)

In summary, the conversation discusses the calculation of the expectation value for a new variable, X(r), in between two wavefunctions of a diatomic molecular potential described by an anharmonic oscillator potential. The screenshot refers to equations 2, 4, and 5, and claims that X(r) can be written as a Taylor series, leading to equation 6. A reference to the original paper is provided.
  • #1
Malamala
313
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Hello! I am confused about the derivation in the screenshot below. This is in the context of a diatomic molecular potential, but the question is quite general. Say that the potential describing the interaction between 2 masses, as a function of the radius between them is given by the anharmonic oscillator potential in eq 4., where ##r_e## is the equilibrium separation. What I need is to calculate the expectation value of a new variable, ##X(r)## in between 2 wavefunctions of such a potential, eq. 2 (please ignore eq. 3 and most of the comments in the paragraph after, as they are not related to my question). They Taylor expand ##X(r)## as in eq. 5 and then they claim that from there it follows that ##X_\nu## (eq. 2) is given by eq. 6. Can someone help me understand how to go from eq. 2, 4 and 5 to eq. 6? Thank you!
Screenshot 2023-06-06 at 1.35.40 PM.png
 
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  • #2
Can you explain what is ##X(r)## and give a reference to the source from which you took the screenshot?
 
  • #3
div_grad said:
Can you explain what is ##X(r)## and give a reference to the source from which you took the screenshot?
X(r) can be any function of r (well any function that can be written as a Taylor series around some value). The original paper is this: https://www.sciencedirect.com/science/article/abs/pii/0022285279900602
 
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FAQ: Help with a derivation from a paper (diatomic molecular potential)

What is the starting point for deriving a diatomic molecular potential?

The starting point for deriving a diatomic molecular potential typically involves the Schrödinger equation for a two-atom system. You need to consider both the electronic and nuclear components, often beginning with the Born-Oppenheimer approximation to separate the electronic motion from the nuclear motion.

How do I apply the Born-Oppenheimer approximation in this context?

The Born-Oppenheimer approximation assumes that the nuclei move much slower than the electrons due to their larger mass. This allows you to solve the electronic Schrödinger equation while treating the nuclear positions as fixed parameters. The resulting electronic energy serves as a potential for the nuclear motion.

What kind of potential functions are commonly used for diatomic molecules?

Common potential functions for diatomic molecules include the Morse potential, Lennard-Jones potential, and the Harmonic oscillator potential. Each of these has different forms and parameters that describe the interaction between two atoms at various distances.

How do I fit a potential function to experimental or computational data?

Fitting a potential function to data involves adjusting the parameters of the chosen potential model to minimize the difference between the predicted and observed values. This can be done using techniques like least squares fitting, and often requires computational tools to handle the complexity of the calculations.

What are some common pitfalls in deriving molecular potentials from a paper?

Common pitfalls include misinterpreting the approximations used, neglecting important interactions or corrections, and errors in the mathematical manipulations. It's crucial to carefully follow the derivation steps, verify each approximation's validity, and cross-check results with known data or additional sources.

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