I would imagine if I have a system of $N$ atoms, each having atom specific parameters, given the position of each atom at a time $t$, I could calculate the potential energy of the system (only a single point though) P(t). However if I assume an initial configuration of the atoms at time $t$, can I write an ODE for the potential energy surface (curve) P(t)?

If I took a quantum perspective, lets say, where the position of these atoms could be inferred by the wave function, can I also create the random ODE which governs the potential energy curve P(t)? Given a particular wave function of course. Then I would expect I can look at E(P(t)) to have the expected energy surface?

I feel like these formulations have already been posed, but I can't find it in the literature. References would be greatly appreciated if these two problems have been posed before.

Thank you.


1 Answer 1


There is no simple ODE for $P(t)$. The discipline that deals with calculating your $P(t)$ is called molecular dynamics. A book on that topic will probably answer most of your questions. Here are the basics:

First some notation and convention: The potential energy is usually denoted $V$, so I will write $V(t)$ instead of $P(t)$. The expectation value is usually denoted $\langle \rangle$ so I will write $\langle V(t)\rangle$ instead of $E(P(t))$.

$V(t)$ is not the potential energy surface. The potential energy surface is defined as $V(R_1,R_2,...,R_{3N})$ where $R_i$ are the positions of the nuclei of the atoms (x,y and z; so there are $3N$). If you can calculate $V(R_1,R_2,...R_{3N})$ you can solve differential equations to get $R_i(t)$, which then give you $V(t)$.

In the classical case, the nuclei (mass $M_i$) follow Newton's equation $\frac{\partial^2R_i}{\partial t^2} = \frac{F_i}{M_i}$ with $F_i = -\frac{\partial}{\partial R_i}V(R_1,R_2,...,R_{3N})$. Solving this differential equation gives you all $R_i(t)$ which you can insert into $V(R_1,R_2,...R_{3N})$ to get $V(t)$.

In the quantum case, the differential equation you need to solve is the time-dependent Schrödinger equation $i\frac{\partial \Psi(R_1,R_2,...,R_{3N},t)}{\partial t} = \left( \sum_i-\frac{1}{2M_i}\frac{\partial^2}{\partial R_i^2} + V(R_1,R_2,...,R_{3N}) \right) \Psi(R_1,R_2,...,R_{3N},t)$. Solving this gives you $\Psi(R_1,R_2,...,R_{3N},t)$ and then you calculate the expectation value as $ \langle V(t) \rangle = \int \int \cdots \int \Psi^* V \Psi dR_1 dR_2 \cdots dR_N$ .

  • $\begingroup$ Hi Martin, first, thank you for the clean and concise answer. I understand everything you said perfectly. Thank you for taking the time to write this! $\endgroup$
    – Vogtster
    Jun 17, 2021 at 21:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.