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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.

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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$ .

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  • $\begingroup$ 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

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