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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Why is it easier to throw electrons from a higher energy shell?

From what i understand, as we increase the radius of unlike charges, the potential energy of the system also increases, so electrons in the outer shell live in a higher potential system, i.e more ...
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Different versions for the Radial Solution of the Hydrogen Atom [closed]

Initially, I'm trying to prove if the following is true $$|\psi_{n00}(r=0)|^2 =\dfrac{1}{\pi n^3a_0^3}$$ I'm looking to solve some calculations made in Introductory Quantum Optics Christopher Gerry ...
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How did the Aufbau principle come to be? [duplicate]

How did the Aufbau principle come to be? Being that my education on chemistry doesn't go past highschool, i was never sure how the following diagram was conceived: I would like to understand how the ...
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Which element has the bigger atomic radius Cr or Mn?

In ncert Mn is bigger. In periodic table app, Cr is bigger.
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Tight-binding: aren't the electronic bands just the eigenvalues of the Hamiltonian?

In the tight-binding model we choose a set of atomic orbitals $\{\phi_1(\textbf{r}), ..., \phi_N(\textbf{r})\}$ and estimate a transfer matrix and hopping matrix $$ \begin{cases} S_{ij}(\textbf{k}) = \...
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„Orbital“ of an quark

Inspired by the idea of the electron orbitals ( probability of finding an electron in an atom) i was wondering what that would look like inside a proton or neutron for quarks. For simplicity consider ...
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Is it justified to take orbital as just a "dispersed region of charge" for applications in Organic chemistry?

When I read organic chemistry book in the past, it seems so a heuristic used to explain and understand reaction mechanism is that negative charge attracts positive charge (or indirectly as ...
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Why do you use $n_r = n -\ell -1$ as quantum number instead of $n$ for hydrogen atom?

I got two different quantum numbers for the same problem: Hydrogen atom without any interaction. Then, my energy is $$ E_n = -\frac{R_y}{n^2} $$ with the quantum number $n = 1, 2, 3, ....$ In another ...
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What is an orbital singlet?

For Helium atom we know there are spin singlet and spin triplet state corresponding to $S=0$ and $S=1$. But what if the electrons are more than two and what does singlet mean for orbital degree of ...
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Does laser trap atom cooling change the electron shell levels?

Do the electron orbits change during the cooling of the atom in a laser trap? When cooling down, do the electrons come closer to the nucleus of the atom or do they move away? Or do the electrons leave ...
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Why electrons in an atom don't radiate photons?

Consider the $s$ orbital of an helium atom. The electron can be anywhere around (or even in) the nucleus. Electrons, like all charges, create electric fields. When the atom interacts with other atoms ...
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Can the pilot wave theory explain why the circumference of an orbit has to be an integer multiple of the wavelength of the electron orbiting it?

Consider the atomic model proposed by Bohr. The velocity of an electron at any orbit is given by the following equation: $$v= n \dfrac{h}{2\pi mr}$$ Now, this equation stems from the fact that, quote, ...
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Why silver reflectivity decreases at around 350

This picture is from the site:https://www.zmescience.com/other/feature-post/gold-color-relativity/ I was wondering why the reflectivitiy of silver goes to 0 at around 350 nm. I understand why for gold ...
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How does an electron move in the $p$ orbital?

This is my first time learning about orbitals and I am very confused over how do electrons move around the nucleus in the $p$ orbital. Wouldn't it have to move out of the orbital where probability of ...
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Normalization-Formula for contracted Gaussian Orbitals

I am trying to write a python script that implements the Hartree-Fock-Routine. I am currently working on generating the basis-functions. While doing my research I found this website that explains how ...
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Do electrons with the same fundamental quantum number (ex: $n=1$), have the same average distance from nucleus?

For example, in the silicon atom, do the electrons located in $3s$ and $3p$, with $n=3$, have the same distance to the nucleus? I think in the diagram below, the arrangement of the electrons relative ...
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How to normalize cartesian gaussian type basis sets?

I am currently working on my graduation paper each high school student in Switzerland has to write. I chose quantum mechanics. To be more specific Hartree-Fock-Theory. I understand all the theory ...
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Do electrons in same orbital shell (ex: $3p$) have the same energy level?

Due to Pauli exclusion principle, each quantum state can be occupied by no more than one electron in a electron system. However, why there is more than one electron in the same energy level in the ...
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Existence of Subshell structures for molecules [closed]

For Hydrogen atom, we could explicitly solve Schrodinger's Equation, a partial differential equation. Using reparametrization, we arrive at series solution using Frobenius method. From this we see ...
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What is the shape of the path that electron makes in an atom?

If electrons make no transition to any of the other orbital or a shape, then what is the shape pf the path it makes while revolving around the nucleus? Does it have to do anything with the ...
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What causes the electric field in a Daniell Cell?

I would like to know what causes the e-field in a Daniell cell. From a chemistry point of view, I do understand that this is initiated by a redox reaction in which Gibbs free energy is decreased. The ...
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About why electrons in atom don't radiate(considering wave nature also) [duplicate]

It's said in textbooks that electrons won't radiate and fall into nucleus because matter wave of it's form a standing wave but could somebody explain why being a standing wave it doesn't radiate, even ...
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How does $s$ subshell not have a node in the center despite the nucleus being there?

In most images of $1s$ subshell I see that there's no node shown at the center, and even the formula $n-\ell-1$ gives 0 as the answer. But, isn't the nucleus experimentally proven to be at the center? ...
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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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1 answer
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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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