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Let's say I want to calculate the wavelength of the photon emitted when an electron of an arbitrary element (let's say Carbon) drops from $n=4$ to $n=3$.

Correct me if I'm wrong, but I think I would have to find the change in the electron's energy value $\Delta E = E_i - E_f$, set that equal to $hc/\lambda$ and solve for the wavelength ($\lambda$).

How could I find $E_f$ and $E_i$? My textbook gives Bohr's Equation but it also says that it fails to generalize, it only works for singular electrons and is kind of inaccurate outside of hydrogen.

Does there exist a general equation to find these values?

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    $\begingroup$ There is a quite general equation, look for the Time independent Schrödinger Equation. But most of times it cannot be exactly solved. $\endgroup$ – user1420303 Apr 7 at 3:58
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Strictly speaking, for multi-electron atoms, the individual electrons don't actually have orbitals. However, many atoms are accurately described by the Hartree-Fock approximation, in which we can talk about orbitals for individual electrons, as described at What does an orbital mean in atoms with multiple electrons? What do the orbitals of Helium look like?. Your first sentence only makes sense within this approximation.

Moreover, for a multi-electron atom described within the Hartree-Fock approximation, the energy depends on both the quantum numbers $n$ and $l$, not just $n$ as with the hydrogen atom.

To answer your question: no, there is no general formula for the electron energy in terms of $n$ and $l$ for a general atom within the Hartree-Fock approximation. There are lots of different tricks that work more or less well for different atoms, but they all just give approximate answers.

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  • $\begingroup$ If I just want to find $E_f - E_i$ would Rydberg's Formula be sufficient? $\endgroup$ – Shrey Joshi Apr 7 at 5:24
  • $\begingroup$ @ShreyJoshi No, Rydberg's formula does not apply to multi-electron atoms. $\endgroup$ – tparker Apr 7 at 12:24
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The starting point for computing wave functions and energy levels of multi-electron atoms is an approximation to the time-independent Schrodinger equation called the Hartree-Fock method. (It can also be used on molecules.)

The basic idea is to approximate the exact $N$-electron wavefunction by an antisymmetrized product of $N$ single-electron wavefunctions called spin-orbitals. A variational method minimizing the energy is used to solve for the spin-orbitals. Instead of considering the interaction between each pair of electrons, each electron is treated as if it “feels” the average electrostatic potential of the nucleus plus the other electrons.

This is a numerical method carried out on computers, not an analytical method that produces a nice formula.

ADDENDUM

It took me awhile to write this. When I posted it, I saw that @tparker had already posted a similar answer.

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There are no closed form equations for the energy levels of multi-electron atoms. Computational atomic physics employs various methods to solve the time-independent Schroedinger equation for a given atom. While the calculated energy levels are not mathematically exact, they are excellent approximations (in terms of agreement with experimentally determined values).

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