Questions tagged [orbitals]

An orbital is the mathematical function describing the location and wave-like behaviour of an electron in either an atom ( for atomic orbitals) or in a molecule (molecular orbitals).

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Splitting of $sp$-System due to LS coupling

How do I explain the splitting of $sp$-system due to LS interaction? I think it is due to the magnetic field created due to the orbital motion of the electron, and that magnetic field splits the ...
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Are atoms still unstable in 4 spatial dimensions when the physical size of nuclei is accounted for?

Per this answer, depending on a dimensionless parameter hydrogen atoms in 4 spatial dimensions can be either unbound (i.e., nonexistent), stably bound dependent on boundary conditions, or unstable ...
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The uncertainty principle and the quantization of electron momentum in orbitals - Why are larger energy gaps considered more uncertain?

I am a chemist trying to understand how the uncertainty principle can be applied to the mechanics of atomic and molecular orbitals in order to have better intuitions about stability and reactivity. ...
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What is the standard explanation for tungstens abnormal high Pauling electronegativity?

The electronegativity of tungsten does not follow the common trend along the periodic table.
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Are the shapes of atomic orbitals direct consequence of the Schrödinger equation?

I am trying to understand whether the shapes of the orbitals are inevitable given the standard model. They would probably change if we change the fine tuning of the fundamental physical constants, ...
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Hydrogen atom quantum numbers distribution

The wave function of the hydrogen atom is given by $\psi_{n \ell m} \left( r, \theta, \varphi \right) = R_{n \ell} Y_{\ell}^{m} (\theta, \varphi)$ which depend on the quantum numbers $n, \ell, m$. I ...
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How many states are possible for two (indistinguishable) electrons with $n=2$ in an atom if we forget Pauli’s exclusion principle?

I have been told I can use $^8C_2 = 28$ to obtain the answer to this question, but I am doubtful of this result since I obtain $21$ by simply writing out the possible states as ($ml_{1}$, $ms_{1}$, $...
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Can the cases of multiple fermions in any spherically potential be approximated by the Schrodinger Equation for a single fermion?

In this video https://www.youtube.com/watch?v=tq_y1qOmUBE&t=783s it's mentioned that the structor of atoms of multiple particles can be approximated using the Schrodinger Equation of the Hydrogen ...
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How can an $n=25$ Rydberg state exist in a solid? What does the wave function look like?

Phys.org's April 15, 2022 Ancient Namibian stone could hold key to future quantum computers begins: A special form of light made using an ancient Namibian gemstone could be the key to new light-based ...
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Electronic bands and Aufbau principle

In atomic theory there is Aufbau principle stating (Wikipedia) The rule states that for a given electron configuration, the lowest energy term is the one with the greatest value of spin multiplicity. ...
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Formation of valence and conduction bands from atomic orbitals

In the wiki article on the electronic band structure illustrates on how the valence and conduction bands of group 4 semiconductors like diamond emerge from the 2s and 2p orbitals of the C atoms. ...
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Excitation of electrons from $3p$ to $3d$ in sulphur

In my book it is shown that sulphur in its excited state forms 6 covalent bonds with Fluorine to form SF6 but after the electron is excited does it not dexcite from $3d$ back to $3p$ and $3s$ by ...
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Does the $3s$ orbital contain the $2s$ orbital?

The following image is in my introductory chemistry textbook: It seems to imply that the $3s$ orbital “contains” the $2s$ and 1s orbital. That is, a $3s$ electron could find itself in a region of ...
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Hamiltonian of wavefunction of $2s$ and $2p$ orbitals

I have recently developed an interest in the analysis of a certain problem in atomic chemistry regarding the energies of $4s$ and $3d$ orbitals, and I observed that even though the wavefunction of $2s$...
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We know that nodes are regions where the probability of finding an electron is zero right? [duplicate]

According to the text I'm going through it says, that the probability density has always some value howsoever small it may be at finite distance from the nucleus. So this means that the probability of ...
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Quantum mechanically speaking: why is it that electrons get bound to a nucleus? ..and why doesn't the electron's wavefunction get infinitely small?

I have a pretty good intuitive understanding of quantum mechanics. But one thing that I don't really intuitively understand is why electrons end up in bound states. An electron might have some ...
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Old quantum theory and the energy levels of the hydrogen atom bounded states [closed]

I'm trying to obtain the energy levels of the hydrogen atom using the old quantum theory. In particular using the Bohr-Sommerfeld quantization rules \begin{equation} \displaystyle\int p_i dq_i = n_ih \...
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Partition Function of a Hydrogen Atom - How it is derived?

It is well-known that the Partition Function of Hydrogen Atom diverges if we calculate in naive manner. And I could find the partition function named Brillouin-Planck-Larkin partition function which ...
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Derivation of Bohr's formula for the atomic orbitals $2n^2$ by subsequent physicists

In his book VALENCE and The Structure of Atoms and Molecules Gilbert Newton Lewis wrote We may next consider a very recent idea advanced by Bohr (1921), which is not based so much upon deductions ...
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Excitation of mercury atom: Confused about statement in textbook

A physics textbook I am currently reading states that "if an electron with kinetic energy equal to 1.12 aJ collides with a single mercury atom in its ground state, the atom may be excited and ...
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Why is there destructive interference in an atom between protons and electrons?

"In energy wave theory, Planck’s relation describes the energy of a transverse wave, emitted or absorbed as an electron transitions energy levels in an atom. When an electron is contained within ...
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What do we mean when we say "an atom is at energy level 1"?

I know that when we say an atom is excited, we actually mean that one of its electrons either moves to another energy level or to another orbit. But when we say that an atom is in a certain energy ...
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Why is octaoxygen diamagnetic?

I tried considering that according to LCAO-MO theory $O_2$ is paramagnetic, which is confirmed by experimental evidence. Since octaoxygen has the crystal structure in figure, I thought there is a kind ...
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Cubed Root of an Expectation value $\left<\frac 1 {r^3}\right>$

I am afraid this question will reveal how little I know about expectation values but it cannot be helped. Say I have the expectation value: $$\left<\frac 1 {r^3}\right> = \frac {1}{(na_0)^3} \...
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Single-particle wavefunction in Slater determinant

The ground state of $N$ non-interacting fermions can be written using a Slater determinant as: $$ \Phi_{GS}(\textbf{r}_{1}, ..., \textbf{r}_{N}) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \phi_{\mu_{1}}(\...
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Algorithm to generate cartesian names for real spherical harmonics

Spherical Harmonics are used to describe atomic orbitals. For a physicist the complex spherical harmonics, indexed by $l, m$ with $-l \le m \le l$ are more natural because atomic states with spatial ...
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On non-existence of molecular helium (and or why that's equivalent to bond order zero)

Why does molecular orbital theory (MOT) imply that molecular helium does not exist? All answers I found in the web use following two standard arguments, but I not see why these are sufficient to ...
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Calculating the charge of an $sp_z$ hybridized orbital using charge density $\rho$

I have a homework problem that is asking to verify that the total charge for an electron is -1. The electron is in an $sp_z$ hybridized orbital with the shape: We are given the charge density as:$$\...
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Expectation value of $r^{-3}$ for $\ell=0$

I'm trying to calculate the hyperfine structure energy for Hydrogen. From Jackson's Electrodynamics book eq. 5.73 we have: $$H_{HFS}=\dfrac{\mu_{0}}{4\pi}\left[\dfrac{-8\pi}{3}\mu_{N}.\mu_{e}\delta(r)+...
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How an electron can have different energy levels if it is once declared that the energy level is fixed?

Let me be more clear about what I'm actually trying to ask. So, as we know for an isolated atom the energy level for an electron in $1^{st}$ shell is equal to $- 13.6$ $eV$ now if we shove that atom ...
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Giving an atom kinetic energy & momentum through impact with a larger body, does this affect the electronic state of the system?

Say I took a slow moving atom and hit it with a baseball bat to give it some extra kinetic energy. The atom is now moving away from me and has more kinetic energy than before. As far as I understand ...
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Atom orbitals physics vs. chemistry

Why draw chemists always $|2p_x\rangle$, $|2p_y\rangle$ and $|2p_z\rangle$ orbitals instead of $|2,1,m\rangle$? Since $|2p_x\rangle$ or $|2p_y\rangle$ are a superposition of energy eigenstates they ...
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Selection rules intuitive

The selection rules for hydrogen are: $\Delta l=\pm1$ and $\Delta m=0,\pm1$. The first makes intuitive sense because of the conservation of angular momentum and the fact that a photon has spin 1. But ...
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Aufbau principle and Stern Gerlach

There are quite a few questions similar to the one i am about to ask. However, non of the answers satisfied me so far. Here i go : In the Stern Gerlach Experiment, which was performed with a silver ...
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Are H$^1$ (1s) and H$^1$ (2s) identical particles?

As per Wikipedia, Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (such as atomic nuclei), as well as atoms and ...
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The unanswered question: Where do electrons get their ever-lasting circulating energy? [closed]

This question has been asked by many others like me, and in so many forums as well as here, and answered by many as well, and I still do not find the answer. The question is "Where do electrons ...
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Why octet atom are stable? And what make atom unstable?

I have search a lot that want make octet atom stable but all I found is that it need to fill the orbit! why . Why it need to fill the orbit. And I also search that why atom which don't have octet are ...
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Is that electron which jumps from one stationary state to another?

In classical physics book of Kleppner, I read that An atom can "jump" from one stationary state a to a lower b by emitting radiation with $E_a-E_b$. The frequency of the emitted "...
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How fast do electrons go from one energy level to another? Do they teleport (instantaneous movement)? [duplicate]

In the book called "Physics of Radiotherapy" of M. Faiz Khan, Ph.D. it says: On the basis of classical physics, an accelerating or revolving electron must radiate energy. This would result ...
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To which orbital does an excited electron in a hydrogen atom go?

For atoms other than hydrogen, each subshell within an energy level has distinct energy. For hydrogen, however, all orbitals within an energy level are degenerate (have the same energy). So, when the ...
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Wavefunction(s) of an $N$ electron atom

Consider an isolated atom consisting of $N$ electrons. Does each electron have a wavefunction of its own (as is generally spoken about in fields like spectroscopy) or does our isolated atom have a ...
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How does moving an electron to a higher level actually work?

I understand the photoelectric effect and I assumed until an hour ago that to excite an electron, the photon should have energy equal to the binding energy of the initial level plus the binding energy ...
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Why was Rutherford wrong and Bohr was right (then)?

I read that Rutherford was ruled out because he predicted that electrons revolve around the nucleus and Bohr stated that accelerating electrons emit electromagnetic radiation and should collide with ...
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Electrons remaining in an orbit and the emission of photons?

Electrons are not stationary around a nucleus, even if it remains in the same excitation state / in the same 'orbit'. As moving electrons should stir the EM field, they should be creating EM waves/...
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How do we choose last electron & write the quantum numbers for that element?

I was solving questions about writing the quantum numbers of the last electron of an element. For Ions like Na+ , Na , O , Cl it was easy. But for elements like Scandium , Yttrium , Iron I am having ...
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What type of energy electrons store on different orbitals?

It's said that electrons, situated on different energy levels inside an atom, possess different amounts of some energy. I'm interested in what is that energy. Is it kinetic energy? Surely, electrons ...
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Why do we multiply in the total wave function but add in the LCAO method?

If to particles with wave functions $\psi_a$ and $\psi_b$ are „combined“ their total wave function is given by: $$\psi(r_1,r_2)= A[\psi_a(r_1)\psi_b(r_2) \pm \psi_b(r_1)\psi_a(r_2)]$$ (+ for bosons ...
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How Quickly Does an Atom Transfer From One Orbital to Another? [duplicate]

Here's the research that I've done: The Schrödinger equation: $$i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle=\hat{H}|\psi(t)\rangle$$ Ok, it evolves as a function of time. That tells us something....
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$L$-$S$ coupling with more than two electrons in the last shell

Is there a easy way to calculate all the terms for complicate configurations with more than two electrons in the last shell? If I try to find the terms in $L$-$S$ coupling $[Ar]3d^3$ or $[Ar]3d^5$ ...
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Explaining Characteristic X-rays using Bohr's theory

Using Bohr's theory, the wavelength of photon emitted (when electron moves to a lower principal quantum number) for hydrogen like atoms is inversely proportional to the square of atomic number Z. But ...
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